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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... |
| pm2.01i 189 | Inference associated with ... |
| pm2.01d 190 | Deduction based on reducti... |
| pm2.6 191 | Theorem *2.6 of [Whitehead... |
| pm2.61 192 | Theorem *2.61 of [Whitehea... |
| pm2.65 193 | Theorem *2.65 of [Whitehea... |
| pm2.65i 194 | Inference for proof by con... |
| pm2.21dd 195 | A contradiction implies an... |
| pm2.65d 196 | Deduction for proof by con... |
| mto 197 | The rule of modus tollens.... |
| mtod 198 | Modus tollens deduction. ... |
| mtoi 199 | Modus tollens inference. ... |
| mt2 200 | A rule similar to modus to... |
| mt3 201 | A rule similar to modus to... |
| peirce 202 | Peirce's axiom. A non-int... |
| looinv 203 | The Inversion Axiom of the... |
| bijust0 204 | A self-implication (see ~ ... |
| bijust 205 | Theorem used to justify th... |
| impbi 208 | Property of the biconditio... |
| impbii 209 | Infer an equivalence from ... |
| impbidd 210 | Deduce an equivalence from... |
| impbid21d 211 | Deduce an equivalence from... |
| impbid 212 | Deduce an equivalence from... |
| dfbi1 213 | Relate the biconditional c... |
| dfbi1ALT 214 | Alternate proof of ~ dfbi1... |
| biimp 215 | Property of the biconditio... |
| biimpi 216 | Infer an implication from ... |
| sylbi 217 | A mixed syllogism inferenc... |
| sylib 218 | A mixed syllogism inferenc... |
| sylbb 219 | A mixed syllogism inferenc... |
| biimpr 220 | Property of the biconditio... |
| bicom1 221 | Commutative law for the bi... |
| bicom 222 | Commutative law for the bi... |
| bicomd 223 | Commute two sides of a bic... |
| bicomi 224 | Inference from commutative... |
| impbid1 225 | Infer an equivalence from ... |
| impbid2 226 | Infer an equivalence from ... |
| impcon4bid 227 | A variation on ~ impbid wi... |
| biimpri 228 | Infer a converse implicati... |
| biimpd 229 | Deduce an implication from... |
| mpbi 230 | An inference from a bicond... |
| mpbir 231 | An inference from a bicond... |
| mpbid 232 | A deduction from a bicondi... |
| mpbii 233 | An inference from a nested... |
| sylibr 234 | A mixed syllogism inferenc... |
| sylbir 235 | A mixed syllogism inferenc... |
| sylbbr 236 | A mixed syllogism inferenc... |
| sylbb1 237 | A mixed syllogism inferenc... |
| sylbb2 238 | A mixed syllogism inferenc... |
| sylibd 239 | A syllogism deduction. (C... |
| sylbid 240 | A syllogism deduction. (C... |
| mpbidi 241 | A deduction from a bicondi... |
| biimtrid 242 | A mixed syllogism inferenc... |
| biimtrrid 243 | A mixed syllogism inferenc... |
| imbitrid 244 | A mixed syllogism inferenc... |
| syl5ibcom 245 | A mixed syllogism inferenc... |
| imbitrrid 246 | A mixed syllogism inferenc... |
| syl5ibrcom 247 | A mixed syllogism inferenc... |
| biimprd 248 | Deduce a converse implicat... |
| biimpcd 249 | Deduce a commuted implicat... |
| biimprcd 250 | Deduce a converse commuted... |
| imbitrdi 251 | A mixed syllogism inferenc... |
| imbitrrdi 252 | A mixed syllogism inferenc... |
| biimtrdi 253 | A mixed syllogism inferenc... |
| biimtrrdi 254 | A mixed syllogism inferenc... |
| syl7bi 255 | A mixed syllogism inferenc... |
| syl8ib 256 | A syllogism rule of infere... |
| mpbird 257 | A deduction from a bicondi... |
| mpbiri 258 | An inference from a nested... |
| sylibrd 259 | A syllogism deduction. (C... |
| sylbird 260 | A syllogism deduction. (C... |
| biid 261 | Principle of identity for ... |
| biidd 262 | Principle of identity with... |
| pm5.1im 263 | Two propositions are equiv... |
| 2th 264 | Two truths are equivalent.... |
| 2thd 265 | Two truths are equivalent.... |
| monothetic 266 | Two self-implications (see... |
| ibi 267 | Inference that converts a ... |
| ibir 268 | Inference that converts a ... |
| ibd 269 | Deduction that converts a ... |
| pm5.74 270 | Distribution of implicatio... |
| pm5.74i 271 | Distribution of implicatio... |
| pm5.74ri 272 | Distribution of implicatio... |
| pm5.74d 273 | Distribution of implicatio... |
| pm5.74rd 274 | Distribution of implicatio... |
| bitri 275 | An inference from transiti... |
| bitr2i 276 | An inference from transiti... |
| bitr3i 277 | An inference from transiti... |
| bitr4i 278 | An inference from transiti... |
| bitrd 279 | Deduction form of ~ bitri ... |
| bitr2d 280 | Deduction form of ~ bitr2i... |
| bitr3d 281 | Deduction form of ~ bitr3i... |
| bitr4d 282 | Deduction form of ~ bitr4i... |
| bitrid 283 | A syllogism inference from... |
| bitr2id 284 | A syllogism inference from... |
| bitr3id 285 | A syllogism inference from... |
| bitr3di 286 | A syllogism inference from... |
| bitrdi 287 | A syllogism inference from... |
| bitr2di 288 | A syllogism inference from... |
| bitr4di 289 | A syllogism inference from... |
| bitr4id 290 | A syllogism inference from... |
| 3imtr3i 291 | A mixed syllogism inferenc... |
| 3imtr4i 292 | A mixed syllogism inferenc... |
| 3imtr3d 293 | More general version of ~ ... |
| 3imtr4d 294 | More general version of ~ ... |
| 3imtr3g 295 | More general version of ~ ... |
| 3imtr4g 296 | More general version of ~ ... |
| 3bitri 297 | A chained inference from t... |
| 3bitrri 298 | A chained inference from t... |
| 3bitr2i 299 | A chained inference from t... |
| 3bitr2ri 300 | A chained inference from t... |
| 3bitr3i 301 | A chained inference from t... |
| 3bitr3ri 302 | A chained inference from t... |
| 3bitr4i 303 | A chained inference from t... |
| 3bitr4ri 304 | A chained inference from t... |
| 3bitrd 305 | Deduction from transitivit... |
| 3bitrrd 306 | Deduction from transitivit... |
| 3bitr2d 307 | Deduction from transitivit... |
| 3bitr2rd 308 | Deduction from transitivit... |
| 3bitr3d 309 | Deduction from transitivit... |
| 3bitr3rd 310 | Deduction from transitivit... |
| 3bitr4d 311 | Deduction from transitivit... |
| 3bitr4rd 312 | Deduction from transitivit... |
| 3bitr3g 313 | More general version of ~ ... |
| 3bitr4g 314 | More general version of ~ ... |
| notnotb 315 | Double negation. Theorem ... |
| con34b 316 | A biconditional form of co... |
| con4bid 317 | A contraposition deduction... |
| notbid 318 | Deduction negating both si... |
| notbi 319 | Contraposition. Theorem *... |
| notbii 320 | Negate both sides of a log... |
| con4bii 321 | A contraposition inference... |
| mtbi 322 | An inference from a bicond... |
| mtbir 323 | An inference from a bicond... |
| mtbid 324 | A deduction from a bicondi... |
| mtbird 325 | A deduction from a bicondi... |
| mtbii 326 | An inference from a bicond... |
| mtbiri 327 | An inference from a bicond... |
| sylnib 328 | A mixed syllogism inferenc... |
| sylnibr 329 | A mixed syllogism inferenc... |
| sylnbi 330 | A mixed syllogism inferenc... |
| sylnbir 331 | A mixed syllogism inferenc... |
| xchnxbi 332 | Replacement of a subexpres... |
| xchnxbir 333 | Replacement of a subexpres... |
| xchbinx 334 | Replacement of a subexpres... |
| xchbinxr 335 | Replacement of a subexpres... |
| imbi2i 336 | Introduce an antecedent to... |
| bibi2i 337 | Inference adding a bicondi... |
| bibi1i 338 | Inference adding a bicondi... |
| bibi12i 339 | The equivalence of two equ... |
| imbi2d 340 | Deduction adding an antece... |
| imbi1d 341 | Deduction adding a consequ... |
| bibi2d 342 | Deduction adding a bicondi... |
| bibi1d 343 | Deduction adding a bicondi... |
| imbi12d 344 | Deduction joining two equi... |
| bibi12d 345 | Deduction joining two equi... |
| imbi12 346 | Closed form of ~ imbi12i .... |
| imbi1 347 | Theorem *4.84 of [Whitehea... |
| imbi2 348 | Theorem *4.85 of [Whitehea... |
| imbi1i 349 | Introduce a consequent to ... |
| imbi12i 350 | Join two logical equivalen... |
| bibi1 351 | Theorem *4.86 of [Whitehea... |
| bitr3 352 | Closed nested implication ... |
| con2bi 353 | Contraposition. Theorem *... |
| con2bid 354 | A contraposition deduction... |
| con1bid 355 | A contraposition deduction... |
| con1bii 356 | A contraposition inference... |
| con2bii 357 | A contraposition inference... |
| con1b 358 | Contraposition. Bidirecti... |
| con2b 359 | Contraposition. Bidirecti... |
| biimt 360 | A wff is equivalent to its... |
| pm5.5 361 | Theorem *5.5 of [Whitehead... |
| a1bi 362 | Inference introducing a th... |
| mt2bi 363 | A false consequent falsifi... |
| mtt 364 | Modus-tollens-like theorem... |
| imnot 365 | If a proposition is false,... |
| pm5.501 366 | Theorem *5.501 of [Whitehe... |
| ibib 367 | Implication in terms of im... |
| ibibr 368 | Implication in terms of im... |
| tbt 369 | A wff is equivalent to its... |
| nbn2 370 | The negation of a wff is e... |
| bibif 371 | Transfer negation via an e... |
| nbn 372 | The negation of a wff is e... |
| nbn3 373 | Transfer falsehood via equ... |
| pm5.21im 374 | Two propositions are equiv... |
| 2false 375 | Two falsehoods are equival... |
| 2falsed 376 | Two falsehoods are equival... |
| pm5.21ni 377 | Two propositions implying ... |
| pm5.21nii 378 | Eliminate an antecedent im... |
| pm5.21ndd 379 | Eliminate an antecedent im... |
| bija 380 | Combine antecedents into a... |
| pm5.18 381 | Theorem *5.18 of [Whitehea... |
| xor3 382 | Two ways to express "exclu... |
| nbbn 383 | Move negation outside of b... |
| biass 384 | Associative law for the bi... |
| biluk 385 | Lukasiewicz's shortest axi... |
| pm5.19 386 | Theorem *5.19 of [Whitehea... |
| bi2.04 387 | Logical equivalence of com... |
| pm5.4 388 | Antecedent absorption impl... |
| imdi 389 | Distributive law for impli... |
| pm5.41 390 | Theorem *5.41 of [Whitehea... |
| imbibi 391 | The antecedent of one side... |
| pm4.8 392 | Theorem *4.8 of [Whitehead... |
| pm4.81 393 | A formula is equivalent to... |
| imim21b 394 | Simplify an implication be... |
| pm4.63 397 | Theorem *4.63 of [Whitehea... |
| pm4.67 398 | Theorem *4.67 of [Whitehea... |
| imnan 399 | Express an implication in ... |
| imnani 400 | Infer an implication from ... |
| iman 401 | Implication in terms of co... |
| pm3.24 402 | Law of noncontradiction. ... |
| annim 403 | Express a conjunction in t... |
| pm4.61 404 | Theorem *4.61 of [Whitehea... |
| pm4.65 405 | Theorem *4.65 of [Whitehea... |
| imp 406 | Importation inference. (C... |
| impcom 407 | Importation inference with... |
| con3dimp 408 | Variant of ~ con3d with im... |
| mpnanrd 409 | Eliminate the right side o... |
| impd 410 | Importation deduction. (C... |
| impcomd 411 | Importation deduction with... |
| ex 412 | Exportation inference. (T... |
| expcom 413 | Exportation inference with... |
| expdcom 414 | Commuted form of ~ expd . ... |
| expd 415 | Exportation deduction. (C... |
| expcomd 416 | Deduction form of ~ expcom... |
| imp31 417 | An importation inference. ... |
| imp32 418 | An importation inference. ... |
| exp31 419 | An exportation inference. ... |
| exp32 420 | An exportation inference. ... |
| imp4b 421 | An importation inference. ... |
| imp4a 422 | An importation inference. ... |
| imp4c 423 | An importation inference. ... |
| imp4d 424 | An importation inference. ... |
| imp41 425 | An importation inference. ... |
| imp42 426 | An importation inference. ... |
| imp43 427 | An importation inference. ... |
| imp44 428 | An importation inference. ... |
| imp45 429 | An importation inference. ... |
| exp4b 430 | An exportation inference. ... |
| exp4a 431 | An exportation inference. ... |
| exp4c 432 | An exportation inference. ... |
| exp4d 433 | An exportation inference. ... |
| exp41 434 | An exportation inference. ... |
| exp42 435 | An exportation inference. ... |
| exp43 436 | An exportation inference. ... |
| exp44 437 | An exportation inference. ... |
| exp45 438 | An exportation inference. ... |
| imp5d 439 | An importation inference. ... |
| imp5a 440 | An importation inference. ... |
| imp5g 441 | An importation inference. ... |
| imp55 442 | An importation inference. ... |
| imp511 443 | An importation inference. ... |
| exp5c 444 | An exportation inference. ... |
| exp5j 445 | An exportation inference. ... |
| exp5l 446 | An exportation inference. ... |
| exp53 447 | An exportation inference. ... |
| pm3.3 448 | Theorem *3.3 (Exp) of [Whi... |
| pm3.31 449 | Theorem *3.31 (Imp) of [Wh... |
| impexp 450 | Import-export theorem. Pa... |
| impancom 451 | Mixed importation/commutat... |
| expdimp 452 | A deduction version of exp... |
| expimpd 453 | Exportation followed by a ... |
| impr 454 | Import a wff into a right ... |
| impl 455 | Export a wff from a left c... |
| expr 456 | Export a wff from a right ... |
| expl 457 | Export a wff from a left c... |
| ancoms 458 | Inference commuting conjun... |
| pm3.22 459 | Theorem *3.22 of [Whitehea... |
| ancom 460 | Commutative law for conjun... |
| ancomd 461 | Commutation of conjuncts i... |
| biancomi 462 | Commuting conjunction in a... |
| biancomd 463 | Commuting conjunction in a... |
| ancomst 464 | Closed form of ~ ancoms . ... |
| ancomsd 465 | Deduction commuting conjun... |
| anasss 466 | Associative law for conjun... |
| anassrs 467 | Associative law for conjun... |
| anass 468 | Associative law for conjun... |
| pm3.2 469 | Join antecedents with conj... |
| pm3.2i 470 | Infer conjunction of premi... |
| pm3.21 471 | Join antecedents with conj... |
| pm3.43i 472 | Nested conjunction of ante... |
| pm3.43 473 | Theorem *3.43 (Comp) of [W... |
| dfbi2 474 | A theorem similar to the s... |
| dfbi 475 | Definition ~ df-bi rewritt... |
| biimpa 476 | Importation inference from... |
| biimpar 477 | Importation inference from... |
| biimpac 478 | Importation inference from... |
| biimparc 479 | Importation inference from... |
| adantr 480 | Inference adding a conjunc... |
| adantl 481 | Inference adding a conjunc... |
| simpl 482 | Elimination of a conjunct.... |
| simpli 483 | Inference eliminating a co... |
| simpr 484 | Elimination of a conjunct.... |
| simpri 485 | Inference eliminating a co... |
| intnan 486 | Introduction of conjunct i... |
| intnanr 487 | Introduction of conjunct i... |
| intnand 488 | Introduction of conjunct i... |
| intnanrd 489 | Introduction of conjunct i... |
| adantld 490 | Deduction adding a conjunc... |
| adantrd 491 | Deduction adding a conjunc... |
| pm3.41 492 | Theorem *3.41 of [Whitehea... |
| pm3.42 493 | Theorem *3.42 of [Whitehea... |
| simpld 494 | Deduction eliminating a co... |
| simprd 495 | Deduction eliminating a co... |
| simprbi 496 | Deduction eliminating a co... |
| simplbi 497 | Deduction eliminating a co... |
| simprbda 498 | Deduction eliminating a co... |
| simplbda 499 | Deduction eliminating a co... |
| simplbi2 500 | Deduction eliminating a co... |
| simplbi2comt 501 | Closed form of ~ simplbi2c... |
| simplbi2com 502 | A deduction eliminating a ... |
| simpl2im 503 | Implication from an elimin... |
| simplbiim 504 | Implication from an elimin... |
| impel 505 | An inference for implicati... |
| mpan9 506 | Modus ponens conjoining di... |
| sylan9 507 | Nested syllogism inference... |
| sylan9r 508 | Nested syllogism inference... |
| sylan9bb 509 | Nested syllogism inference... |
| sylan9bbr 510 | Nested syllogism inference... |
| jca 511 | Deduce conjunction of the ... |
| jcad 512 | Deduction conjoining the c... |
| jca2 513 | Inference conjoining the c... |
| jca31 514 | Join three consequents. (... |
| jca32 515 | Join three consequents. (... |
| jcai 516 | Deduction replacing implic... |
| jcab 517 | Distributive law for impli... |
| pm4.76 518 | Theorem *4.76 of [Whitehea... |
| jctil 519 | Inference conjoining a the... |
| jctir 520 | Inference conjoining a the... |
| jccir 521 | Inference conjoining a con... |
| jccil 522 | Inference conjoining a con... |
| jctl 523 | Inference conjoining a the... |
| jctr 524 | Inference conjoining a the... |
| jctild 525 | Deduction conjoining a the... |
| jctird 526 | Deduction conjoining a the... |
| iba 527 | Introduction of antecedent... |
| ibar 528 | Introduction of antecedent... |
| biantru 529 | A wff is equivalent to its... |
| biantrur 530 | A wff is equivalent to its... |
| biantrud 531 | A wff is equivalent to its... |
| biantrurd 532 | A wff is equivalent to its... |
| bianfi 533 | A wff conjoined with false... |
| bianfd 534 | A wff conjoined with false... |
| baib 535 | Move conjunction outside o... |
| baibr 536 | Move conjunction outside o... |
| rbaibr 537 | Move conjunction outside o... |
| rbaib 538 | Move conjunction outside o... |
| baibd 539 | Move conjunction outside o... |
| rbaibd 540 | Move conjunction outside o... |
| bianabs 541 | Absorb a hypothesis into t... |
| pm5.44 542 | Theorem *5.44 of [Whitehea... |
| pm5.42 543 | Theorem *5.42 of [Whitehea... |
| ancl 544 | Conjoin antecedent to left... |
| anclb 545 | Conjoin antecedent to left... |
| ancr 546 | Conjoin antecedent to righ... |
| ancrb 547 | Conjoin antecedent to righ... |
| ancli 548 | Deduction conjoining antec... |
| ancri 549 | Deduction conjoining antec... |
| ancld 550 | Deduction conjoining antec... |
| ancrd 551 | Deduction conjoining antec... |
| impac 552 | Importation with conjuncti... |
| anc2l 553 | Conjoin antecedent to left... |
| anc2r 554 | Conjoin antecedent to righ... |
| anc2li 555 | Deduction conjoining antec... |
| anc2ri 556 | Deduction conjoining antec... |
| pm4.71 557 | Implication in terms of bi... |
| pm4.71r 558 | Implication in terms of bi... |
| pm4.71i 559 | Inference converting an im... |
| pm4.71ri 560 | Inference converting an im... |
| pm4.71d 561 | Deduction converting an im... |
| pm4.71rd 562 | Deduction converting an im... |
| pm4.24 563 | Theorem *4.24 of [Whitehea... |
| anidm 564 | Idempotent law for conjunc... |
| anidmdbi 565 | Conjunction idempotence wi... |
| anidms 566 | Inference from idempotent ... |
| imdistan 567 | Distribution of implicatio... |
| imdistani 568 | Distribution of implicatio... |
| imdistanri 569 | Distribution of implicatio... |
| imdistand 570 | Distribution of implicatio... |
| imdistanda 571 | Distribution of implicatio... |
| pm5.3 572 | Theorem *5.3 of [Whitehead... |
| pm5.32 573 | Distribution of implicatio... |
| pm5.32i 574 | Distribution of implicatio... |
| pm5.32ri 575 | Distribution of implicatio... |
| bianim 576 | Exchanging conjunction in ... |
| pm5.32d 577 | Distribution of implicatio... |
| pm5.32rd 578 | Distribution of implicatio... |
| pm5.32da 579 | Distribution of implicatio... |
| sylan 580 | A syllogism inference. (C... |
| sylanb 581 | A syllogism inference. (C... |
| sylanbr 582 | A syllogism inference. (C... |
| sylanbrc 583 | Syllogism inference. (Con... |
| syl2anc 584 | Syllogism inference combin... |
| syl2anc2 585 | Double syllogism inference... |
| sylancl 586 | Syllogism inference combin... |
| sylancr 587 | Syllogism inference combin... |
| sylancom 588 | Syllogism inference with c... |
| sylanblc 589 | Syllogism inference combin... |
| sylanblrc 590 | Syllogism inference combin... |
| syldan 591 | A syllogism deduction with... |
| sylbida 592 | A syllogism deduction. (C... |
| sylan2 593 | A syllogism inference. (C... |
| sylan2b 594 | A syllogism inference. (C... |
| sylan2br 595 | A syllogism inference. (C... |
| syl2an 596 | A double syllogism inferen... |
| syl2anr 597 | A double syllogism inferen... |
| syl2anb 598 | A double syllogism inferen... |
| syl2anbr 599 | A double syllogism inferen... |
| sylancb 600 | A syllogism inference comb... |
| sylancbr 601 | A syllogism inference comb... |
| syldanl 602 | A syllogism deduction with... |
| syland 603 | A syllogism deduction. (C... |
| sylani 604 | A syllogism inference. (C... |
| sylan2d 605 | A syllogism deduction. (C... |
| sylan2i 606 | A syllogism inference. (C... |
| syl2ani 607 | A syllogism inference. (C... |
| syl2and 608 | A syllogism deduction. (C... |
| anim12d 609 | Conjoin antecedents and co... |
| anim12d1 610 | Variant of ~ anim12d where... |
| anim1d 611 | Add a conjunct to right of... |
| anim2d 612 | Add a conjunct to left of ... |
| anim12i 613 | Conjoin antecedents and co... |
| anim12ci 614 | Variant of ~ anim12i with ... |
| anim1i 615 | Introduce conjunct to both... |
| anim1ci 616 | Introduce conjunct to both... |
| anim2i 617 | Introduce conjunct to both... |
| anim12ii 618 | Conjoin antecedents and co... |
| anim12dan 619 | Conjoin antecedents and co... |
| im2anan9 620 | Deduction joining nested i... |
| im2anan9r 621 | Deduction joining nested i... |
| pm3.45 622 | Theorem *3.45 (Fact) of [W... |
| anbi2i 623 | Introduce a left conjunct ... |
| anbi1i 624 | Introduce a right conjunct... |
| anbi2ci 625 | Variant of ~ anbi2i with c... |
| anbi1ci 626 | Variant of ~ anbi1i with c... |
| bianbi 627 | Exchanging conjunction in ... |
| anbi12i 628 | Conjoin both sides of two ... |
| anbi12ci 629 | Variant of ~ anbi12i with ... |
| anbi2d 630 | Deduction adding a left co... |
| anbi1d 631 | Deduction adding a right c... |
| anbi12d 632 | Deduction joining two equi... |
| anbi1 633 | Introduce a right conjunct... |
| anbi2 634 | Introduce a left conjunct ... |
| anbi1cd 635 | Introduce a proposition as... |
| an2anr 636 | Double commutation in conj... |
| pm4.38 637 | Theorem *4.38 of [Whitehea... |
| bi2anan9 638 | Deduction joining two equi... |
| bi2anan9r 639 | Deduction joining two equi... |
| bi2bian9 640 | Deduction joining two bico... |
| anbiim 641 | Adding biconditional when ... |
| bianass 642 | An inference to merge two ... |
| bianassc 643 | An inference to merge two ... |
| an21 644 | Swap two conjuncts. (Cont... |
| an12 645 | Swap two conjuncts. Note ... |
| an32 646 | A rearrangement of conjunc... |
| an13 647 | A rearrangement of conjunc... |
| an31 648 | A rearrangement of conjunc... |
| an12s 649 | Swap two conjuncts in ante... |
| ancom2s 650 | Inference commuting a nest... |
| an13s 651 | Swap two conjuncts in ante... |
| an32s 652 | Swap two conjuncts in ante... |
| ancom1s 653 | Inference commuting a nest... |
| an31s 654 | Swap two conjuncts in ante... |
| anass1rs 655 | Commutative-associative la... |
| an4 656 | Rearrangement of 4 conjunc... |
| an42 657 | Rearrangement of 4 conjunc... |
| an43 658 | Rearrangement of 4 conjunc... |
| an3 659 | A rearrangement of conjunc... |
| an4s 660 | Inference rearranging 4 co... |
| an42s 661 | Inference rearranging 4 co... |
| anabs1 662 | Absorption into embedded c... |
| anabs5 663 | Absorption into embedded c... |
| anabs7 664 | Absorption into embedded c... |
| anabsan 665 | Absorption of antecedent w... |
| anabss1 666 | Absorption of antecedent i... |
| anabss4 667 | Absorption of antecedent i... |
| anabss5 668 | Absorption of antecedent i... |
| anabsi5 669 | Absorption of antecedent i... |
| anabsi6 670 | Absorption of antecedent i... |
| anabsi7 671 | Absorption of antecedent i... |
| anabsi8 672 | Absorption of antecedent i... |
| anabss7 673 | Absorption of antecedent i... |
| anabsan2 674 | Absorption of antecedent w... |
| anabss3 675 | Absorption of antecedent i... |
| anandi 676 | Distribution of conjunctio... |
| anandir 677 | Distribution of conjunctio... |
| anandis 678 | Inference that undistribut... |
| anandirs 679 | Inference that undistribut... |
| sylanl1 680 | A syllogism inference. (C... |
| sylanl2 681 | A syllogism inference. (C... |
| sylanr1 682 | A syllogism inference. (C... |
| sylanr2 683 | A syllogism inference. (C... |
| syl6an 684 | A syllogism deduction comb... |
| syl2an2r 685 | ~ syl2anr with antecedents... |
| syl2an2 686 | ~ syl2an with antecedents ... |
| mpdan 687 | An inference based on modu... |
| mpancom 688 | An inference based on modu... |
| mpidan 689 | A deduction which "stacks"... |
| mpan 690 | An inference based on modu... |
| mpan2 691 | An inference based on modu... |
| mp2an 692 | An inference based on modu... |
| mp4an 693 | An inference based on modu... |
| mpan2d 694 | A deduction based on modus... |
| mpand 695 | A deduction based on modus... |
| mpani 696 | An inference based on modu... |
| mpan2i 697 | An inference based on modu... |
| mp2ani 698 | An inference based on modu... |
| mp2and 699 | A deduction based on modus... |
| mpanl1 700 | An inference based on modu... |
| mpanl2 701 | An inference based on modu... |
| mpanl12 702 | An inference based on modu... |
| mpanr1 703 | An inference based on modu... |
| mpanr2 704 | An inference based on modu... |
| mpanr12 705 | An inference based on modu... |
| mpanlr1 706 | An inference based on modu... |
| mpbirand 707 | Detach truth from conjunct... |
| mpbiran2d 708 | Detach truth from conjunct... |
| mpbiran 709 | Detach truth from conjunct... |
| mpbiran2 710 | Detach truth from conjunct... |
| mpbir2an 711 | Detach a conjunction of tr... |
| mpbi2and 712 | Detach a conjunction of tr... |
| mpbir2and 713 | Detach a conjunction of tr... |
| adantll 714 | Deduction adding a conjunc... |
| adantlr 715 | Deduction adding a conjunc... |
| adantrl 716 | Deduction adding a conjunc... |
| adantrr 717 | Deduction adding a conjunc... |
| adantlll 718 | Deduction adding a conjunc... |
| adantllr 719 | Deduction adding a conjunc... |
| adantlrl 720 | Deduction adding a conjunc... |
| adantlrr 721 | Deduction adding a conjunc... |
| adantrll 722 | Deduction adding a conjunc... |
| adantrlr 723 | Deduction adding a conjunc... |
| adantrrl 724 | Deduction adding a conjunc... |
| adantrrr 725 | Deduction adding a conjunc... |
| ad2antrr 726 | Deduction adding two conju... |
| ad2antlr 727 | Deduction adding two conju... |
| ad2antrl 728 | Deduction adding two conju... |
| ad2antll 729 | Deduction adding conjuncts... |
| ad3antrrr 730 | Deduction adding three con... |
| ad3antlr 731 | Deduction adding three con... |
| ad4antr 732 | Deduction adding 4 conjunc... |
| ad4antlr 733 | Deduction adding 4 conjunc... |
| ad5antr 734 | Deduction adding 5 conjunc... |
| ad5antlr 735 | Deduction adding 5 conjunc... |
| ad6antr 736 | Deduction adding 6 conjunc... |
| ad6antlr 737 | Deduction adding 6 conjunc... |
| ad7antr 738 | Deduction adding 7 conjunc... |
| ad7antlr 739 | Deduction adding 7 conjunc... |
| ad8antr 740 | Deduction adding 8 conjunc... |
| ad8antlr 741 | Deduction adding 8 conjunc... |
| ad9antr 742 | Deduction adding 9 conjunc... |
| ad9antlr 743 | Deduction adding 9 conjunc... |
| ad10antr 744 | Deduction adding 10 conjun... |
| ad10antlr 745 | Deduction adding 10 conjun... |
| ad2ant2l 746 | Deduction adding two conju... |
| ad2ant2r 747 | Deduction adding two conju... |
| ad2ant2lr 748 | Deduction adding two conju... |
| ad2ant2rl 749 | Deduction adding two conju... |
| adantl3r 750 | Deduction adding 1 conjunc... |
| ad4ant13 751 | Deduction adding conjuncts... |
| ad4ant14 752 | Deduction adding conjuncts... |
| ad4ant23 753 | Deduction adding conjuncts... |
| ad4ant24 754 | Deduction adding conjuncts... |
| adantl4r 755 | Deduction adding 1 conjunc... |
| ad5ant13 756 | Deduction adding conjuncts... |
| ad5ant14 757 | Deduction adding conjuncts... |
| ad5ant15 758 | Deduction adding conjuncts... |
| ad5ant23 759 | Deduction adding conjuncts... |
| ad5ant24 760 | Deduction adding conjuncts... |
| ad5ant25 761 | Deduction adding conjuncts... |
| adantl5r 762 | Deduction adding 1 conjunc... |
| adantl6r 763 | Deduction adding 1 conjunc... |
| pm3.33 764 | Theorem *3.33 (Syll) of [W... |
| pm3.34 765 | Theorem *3.34 (Syll) of [W... |
| simpll 766 | Simplification of a conjun... |
| simplld 767 | Deduction form of ~ simpll... |
| simplr 768 | Simplification of a conjun... |
| simplrd 769 | Deduction eliminating a do... |
| simprl 770 | Simplification of a conjun... |
| simprld 771 | Deduction eliminating a do... |
| simprr 772 | Simplification of a conjun... |
| simprrd 773 | Deduction form of ~ simprr... |
| simplll 774 | Simplification of a conjun... |
| simpllr 775 | Simplification of a conjun... |
| simplrl 776 | Simplification of a conjun... |
| simplrr 777 | Simplification of a conjun... |
| simprll 778 | Simplification of a conjun... |
| simprlr 779 | Simplification of a conjun... |
| simprrl 780 | Simplification of a conjun... |
| simprrr 781 | Simplification of a conjun... |
| simp-4l 782 | Simplification of a conjun... |
| simp-4r 783 | Simplification of a conjun... |
| simp-5l 784 | Simplification of a conjun... |
| simp-5r 785 | Simplification of a conjun... |
| simp-6l 786 | Simplification of a conjun... |
| simp-6r 787 | Simplification of a conjun... |
| simp-7l 788 | Simplification of a conjun... |
| simp-7r 789 | Simplification of a conjun... |
| simp-8l 790 | Simplification of a conjun... |
| simp-8r 791 | Simplification of a conjun... |
| simp-9l 792 | Simplification of a conjun... |
| simp-9r 793 | Simplification of a conjun... |
| simp-10l 794 | Simplification of a conjun... |
| simp-10r 795 | Simplification of a conjun... |
| simp-11l 796 | Simplification of a conjun... |
| simp-11r 797 | Simplification of a conjun... |
| pm2.01da 798 | Deduction based on reducti... |
| pm2.18da 799 | Deduction based on reducti... |
| impbida 800 | Deduce an equivalence from... |
| pm5.21nd 801 | Eliminate an antecedent im... |
| pm3.35 802 | Conjunctive detachment. T... |
| pm5.74da 803 | Distribution of implicatio... |
| bitr 804 | Theorem *4.22 of [Whitehea... |
| biantr 805 | A transitive law of equiva... |
| pm4.14 806 | Theorem *4.14 of [Whitehea... |
| pm3.37 807 | Theorem *3.37 (Transp) of ... |
| anim12 808 | Conjoin antecedents and co... |
| pm3.4 809 | Conjunction implies implic... |
| exbiri 810 | Inference form of ~ exbir ... |
| pm2.61ian 811 | Elimination of an antecede... |
| pm2.61dan 812 | Elimination of an antecede... |
| pm2.61ddan 813 | Elimination of two anteced... |
| pm2.61dda 814 | Elimination of two anteced... |
| mtand 815 | A modus tollens deduction.... |
| pm2.65da 816 | Deduction for proof by con... |
| condan 817 | Proof by contradiction. (... |
| biadan 818 | An implication is equivale... |
| biadani 819 | Inference associated with ... |
| biadaniALT 820 | Alternate proof of ~ biada... |
| biadanii 821 | Inference associated with ... |
| biadanid 822 | Deduction associated with ... |
| pm5.1 823 | Two propositions are equiv... |
| pm5.21 824 | Two propositions are equiv... |
| pm5.35 825 | Theorem *5.35 of [Whitehea... |
| abai 826 | Introduce one conjunct as ... |
| pm4.45im 827 | Conjunction with implicati... |
| impimprbi 828 | An implication and its rev... |
| nan 829 | Theorem to move a conjunct... |
| pm5.31 830 | Theorem *5.31 of [Whitehea... |
| pm5.31r 831 | Variant of ~ pm5.31 . (Co... |
| pm4.15 832 | Theorem *4.15 of [Whitehea... |
| pm5.36 833 | Theorem *5.36 of [Whitehea... |
| annotanannot 834 | A conjunction with a negat... |
| pm5.33 835 | Theorem *5.33 of [Whitehea... |
| syl12anc 836 | Syllogism combined with co... |
| syl21anc 837 | Syllogism combined with co... |
| syl22anc 838 | Syllogism combined with co... |
| bibiad 839 | Eliminate an hypothesis ` ... |
| syl1111anc 840 | Four-hypothesis eliminatio... |
| syldbl2 841 | Stacked hypotheseis implie... |
| mpsyl4anc 842 | An elimination deduction. ... |
| pm4.87 843 | Theorem *4.87 of [Whitehea... |
| bimsc1 844 | Removal of conjunct from o... |
| a2and 845 | Deduction distributing a c... |
| animpimp2impd 846 | Deduction deriving nested ... |
| pm4.64 849 | Theorem *4.64 of [Whitehea... |
| pm4.66 850 | Theorem *4.66 of [Whitehea... |
| pm2.53 851 | Theorem *2.53 of [Whitehea... |
| pm2.54 852 | Theorem *2.54 of [Whitehea... |
| imor 853 | Implication in terms of di... |
| imori 854 | Infer disjunction from imp... |
| imorri 855 | Infer implication from dis... |
| pm4.62 856 | Theorem *4.62 of [Whitehea... |
| jaoi 857 | Inference disjoining the a... |
| jao1i 858 | Add a disjunct in the ante... |
| jaod 859 | Deduction disjoining the a... |
| mpjaod 860 | Eliminate a disjunction in... |
| ori 861 | Infer implication from dis... |
| orri 862 | Infer disjunction from imp... |
| orrd 863 | Deduce disjunction from im... |
| ord 864 | Deduce implication from di... |
| orci 865 | Deduction introducing a di... |
| olci 866 | Deduction introducing a di... |
| orc 867 | Introduction of a disjunct... |
| olc 868 | Introduction of a disjunct... |
| pm1.4 869 | Axiom *1.4 of [WhiteheadRu... |
| orcom 870 | Commutative law for disjun... |
| orcomd 871 | Commutation of disjuncts i... |
| orcoms 872 | Commutation of disjuncts i... |
| orcd 873 | Deduction introducing a di... |
| olcd 874 | Deduction introducing a di... |
| orcs 875 | Deduction eliminating disj... |
| olcs 876 | Deduction eliminating disj... |
| olcnd 877 | A lemma for Conjunctive No... |
| orcnd 878 | A lemma for Conjunctive No... |
| mtord 879 | A modus tollens deduction ... |
| pm3.2ni 880 | Infer negated disjunction ... |
| pm2.45 881 | Theorem *2.45 of [Whitehea... |
| pm2.46 882 | Theorem *2.46 of [Whitehea... |
| pm2.47 883 | Theorem *2.47 of [Whitehea... |
| pm2.48 884 | Theorem *2.48 of [Whitehea... |
| pm2.49 885 | Theorem *2.49 of [Whitehea... |
| norbi 886 | If neither of two proposit... |
| nbior 887 | If two propositions are no... |
| orel1 888 | Elimination of disjunction... |
| pm2.25 889 | Theorem *2.25 of [Whitehea... |
| orel2 890 | Elimination of disjunction... |
| pm2.67-2 891 | Slight generalization of T... |
| pm2.67 892 | Theorem *2.67 of [Whitehea... |
| curryax 893 | A non-intuitionistic posit... |
| exmid 894 | Law of excluded middle, al... |
| exmidd 895 | Law of excluded middle in ... |
| pm2.1 896 | Theorem *2.1 of [Whitehead... |
| pm2.13 897 | Theorem *2.13 of [Whitehea... |
| pm2.621 898 | Theorem *2.621 of [Whitehe... |
| pm2.62 899 | Theorem *2.62 of [Whitehea... |
| pm2.68 900 | Theorem *2.68 of [Whitehea... |
| dfor2 901 | Logical 'or' expressed in ... |
| pm2.07 902 | Theorem *2.07 of [Whitehea... |
| pm1.2 903 | Axiom *1.2 of [WhiteheadRu... |
| oridm 904 | Idempotent law for disjunc... |
| pm4.25 905 | Theorem *4.25 of [Whitehea... |
| pm2.4 906 | Theorem *2.4 of [Whitehead... |
| pm2.41 907 | Theorem *2.41 of [Whitehea... |
| orim12i 908 | Disjoin antecedents and co... |
| orim1i 909 | Introduce disjunct to both... |
| orim2i 910 | Introduce disjunct to both... |
| orim12dALT 911 | Alternate proof of ~ orim1... |
| orbi2i 912 | Inference adding a left di... |
| orbi1i 913 | Inference adding a right d... |
| orbi12i 914 | Infer the disjunction of t... |
| orbi2d 915 | Deduction adding a left di... |
| orbi1d 916 | Deduction adding a right d... |
| orbi1 917 | Theorem *4.37 of [Whitehea... |
| orbi12d 918 | Deduction joining two equi... |
| pm1.5 919 | Axiom *1.5 (Assoc) of [Whi... |
| or12 920 | Swap two disjuncts. (Cont... |
| orass 921 | Associative law for disjun... |
| pm2.31 922 | Theorem *2.31 of [Whitehea... |
| pm2.32 923 | Theorem *2.32 of [Whitehea... |
| pm2.3 924 | Theorem *2.3 of [Whitehead... |
| or32 925 | A rearrangement of disjunc... |
| or4 926 | Rearrangement of 4 disjunc... |
| or42 927 | Rearrangement of 4 disjunc... |
| orordi 928 | Distribution of disjunctio... |
| orordir 929 | Distribution of disjunctio... |
| orimdi 930 | Disjunction distributes ov... |
| pm2.76 931 | Theorem *2.76 of [Whitehea... |
| pm2.85 932 | Theorem *2.85 of [Whitehea... |
| pm2.75 933 | Theorem *2.75 of [Whitehea... |
| pm4.78 934 | Implication distributes ov... |
| biort 935 | A disjunction with a true ... |
| biorf 936 | A wff is equivalent to its... |
| biortn 937 | A wff is equivalent to its... |
| biorfi 938 | The dual of ~ biorf is not... |
| biorfri 939 | A wff is equivalent to its... |
| biorfriOLD 940 | Obsolete version of ~ bior... |
| pm2.26 941 | Theorem *2.26 of [Whitehea... |
| pm2.63 942 | Theorem *2.63 of [Whitehea... |
| pm2.64 943 | Theorem *2.64 of [Whitehea... |
| pm2.42 944 | Theorem *2.42 of [Whitehea... |
| pm5.11g 945 | A general instance of Theo... |
| pm5.11 946 | Theorem *5.11 of [Whitehea... |
| pm5.12 947 | Theorem *5.12 of [Whitehea... |
| pm5.14 948 | Theorem *5.14 of [Whitehea... |
| pm5.13 949 | Theorem *5.13 of [Whitehea... |
| pm5.55 950 | Theorem *5.55 of [Whitehea... |
| pm4.72 951 | Implication in terms of bi... |
| imimorb 952 | Simplify an implication be... |
| oibabs 953 | Absorption of disjunction ... |
| orbidi 954 | Disjunction distributes ov... |
| pm5.7 955 | Disjunction distributes ov... |
| jaao 956 | Inference conjoining and d... |
| jaoa 957 | Inference disjoining and c... |
| jaoian 958 | Inference disjoining the a... |
| jaodan 959 | Deduction disjoining the a... |
| mpjaodan 960 | Eliminate a disjunction in... |
| pm3.44 961 | Theorem *3.44 of [Whitehea... |
| jao 962 | Disjunction of antecedents... |
| jaob 963 | Disjunction of antecedents... |
| pm4.77 964 | Theorem *4.77 of [Whitehea... |
| pm3.48 965 | Theorem *3.48 of [Whitehea... |
| orim12d 966 | Disjoin antecedents and co... |
| orim1d 967 | Disjoin antecedents and co... |
| orim2d 968 | Disjoin antecedents and co... |
| orim2 969 | Axiom *1.6 (Sum) of [White... |
| pm2.38 970 | Theorem *2.38 of [Whitehea... |
| pm2.36 971 | Theorem *2.36 of [Whitehea... |
| pm2.37 972 | Theorem *2.37 of [Whitehea... |
| pm2.81 973 | Theorem *2.81 of [Whitehea... |
| pm2.8 974 | Theorem *2.8 of [Whitehead... |
| pm2.73 975 | Theorem *2.73 of [Whitehea... |
| pm2.74 976 | Theorem *2.74 of [Whitehea... |
| pm2.82 977 | Theorem *2.82 of [Whitehea... |
| pm4.39 978 | Theorem *4.39 of [Whitehea... |
| animorl 979 | Conjunction implies disjun... |
| animorr 980 | Conjunction implies disjun... |
| animorlr 981 | Conjunction implies disjun... |
| animorrl 982 | Conjunction implies disjun... |
| ianor 983 | Negated conjunction in ter... |
| anor 984 | Conjunction in terms of di... |
| ioran 985 | Negated disjunction in ter... |
| pm4.52 986 | Theorem *4.52 of [Whitehea... |
| pm4.53 987 | Theorem *4.53 of [Whitehea... |
| pm4.54 988 | Theorem *4.54 of [Whitehea... |
| pm4.55 989 | Theorem *4.55 of [Whitehea... |
| pm4.56 990 | Theorem *4.56 of [Whitehea... |
| oran 991 | Disjunction in terms of co... |
| pm4.57 992 | Theorem *4.57 of [Whitehea... |
| pm3.1 993 | Theorem *3.1 of [Whitehead... |
| pm3.11 994 | Theorem *3.11 of [Whitehea... |
| pm3.12 995 | Theorem *3.12 of [Whitehea... |
| pm3.13 996 | Theorem *3.13 of [Whitehea... |
| pm3.14 997 | Theorem *3.14 of [Whitehea... |
| pm4.44 998 | Theorem *4.44 of [Whitehea... |
| pm4.45 999 | Theorem *4.45 of [Whitehea... |
| orabs 1000 | Absorption of redundant in... |
| oranabs 1001 | Absorb a disjunct into a c... |
| pm5.61 1002 | Theorem *5.61 of [Whitehea... |
| pm5.6 1003 | Conjunction in antecedent ... |
| orcanai 1004 | Change disjunction in cons... |
| pm4.79 1005 | Theorem *4.79 of [Whitehea... |
| pm5.53 1006 | Theorem *5.53 of [Whitehea... |
| ordi 1007 | Distributive law for disju... |
| ordir 1008 | Distributive law for disju... |
| andi 1009 | Distributive law for conju... |
| andir 1010 | Distributive law for conju... |
| orddi 1011 | Double distributive law fo... |
| anddi 1012 | Double distributive law fo... |
| pm5.17 1013 | Theorem *5.17 of [Whitehea... |
| pm5.15 1014 | Theorem *5.15 of [Whitehea... |
| pm5.16 1015 | Theorem *5.16 of [Whitehea... |
| xor 1016 | Two ways to express exclus... |
| nbi2 1017 | Two ways to express "exclu... |
| xordi 1018 | Conjunction distributes ov... |
| pm5.54 1019 | Theorem *5.54 of [Whitehea... |
| pm5.62 1020 | Theorem *5.62 of [Whitehea... |
| pm5.63 1021 | Theorem *5.63 of [Whitehea... |
| niabn 1022 | Miscellaneous inference re... |
| ninba 1023 | Miscellaneous inference re... |
| pm4.43 1024 | Theorem *4.43 of [Whitehea... |
| pm4.82 1025 | Theorem *4.82 of [Whitehea... |
| pm4.83 1026 | Theorem *4.83 of [Whitehea... |
| pclem6 1027 | Negation inferred from emb... |
| bigolden 1028 | Dijkstra-Scholten's Golden... |
| pm5.71 1029 | Theorem *5.71 of [Whitehea... |
| pm5.75 1030 | Theorem *5.75 of [Whitehea... |
| ecase2d 1031 | Deduction for elimination ... |
| ecase3 1032 | Inference for elimination ... |
| ecase 1033 | Inference for elimination ... |
| ecase3d 1034 | Deduction for elimination ... |
| ecased 1035 | Deduction for elimination ... |
| ecase3ad 1036 | Deduction for elimination ... |
| ccase 1037 | Inference for combining ca... |
| ccased 1038 | Deduction for combining ca... |
| ccase2 1039 | Inference for combining ca... |
| 4cases 1040 | Inference eliminating two ... |
| 4casesdan 1041 | Deduction eliminating two ... |
| cases 1042 | Case disjunction according... |
| dedlem0a 1043 | Lemma for an alternate ver... |
| dedlem0b 1044 | Lemma for an alternate ver... |
| dedlema 1045 | Lemma for weak deduction t... |
| dedlemb 1046 | Lemma for weak deduction t... |
| cases2 1047 | Case disjunction according... |
| cases2ALT 1048 | Alternate proof of ~ cases... |
| dfbi3 1049 | An alternate definition of... |
| pm5.24 1050 | Theorem *5.24 of [Whitehea... |
| 4exmid 1051 | The disjunction of the fou... |
| consensus 1052 | The consensus theorem. Th... |
| pm4.42 1053 | Theorem *4.42 of [Whitehea... |
| prlem1 1054 | A specialized lemma for se... |
| prlem2 1055 | A specialized lemma for se... |
| oplem1 1056 | A specialized lemma for se... |
| dn1 1057 | A single axiom for Boolean... |
| bianir 1058 | A closed form of ~ mpbir ,... |
| jaoi2 1059 | Inference removing a negat... |
| jaoi3 1060 | Inference separating a dis... |
| ornld 1061 | Selecting one statement fr... |
| dfifp2 1064 | Alternate definition of th... |
| dfifp3 1065 | Alternate definition of th... |
| dfifp4 1066 | Alternate definition of th... |
| dfifp5 1067 | Alternate definition of th... |
| dfifp6 1068 | Alternate definition of th... |
| dfifp7 1069 | Alternate definition of th... |
| ifpdfbi 1070 | Define the biconditional a... |
| anifp 1071 | The conditional operator i... |
| ifpor 1072 | The conditional operator i... |
| ifpn 1073 | Conditional operator for t... |
| ifptru 1074 | Value of the conditional o... |
| ifpfal 1075 | Value of the conditional o... |
| ifpid 1076 | Value of the conditional o... |
| casesifp 1077 | Version of ~ cases express... |
| ifpbi123d 1078 | Equivalence deduction for ... |
| ifpbi23d 1079 | Equivalence deduction for ... |
| ifpimpda 1080 | Separation of the values o... |
| 1fpid3 1081 | The value of the condition... |
| elimh 1082 | Hypothesis builder for the... |
| dedt 1083 | The weak deduction theorem... |
| con3ALT 1084 | Proof of ~ con3 from its a... |
| 3orass 1089 | Associative law for triple... |
| 3orel1 1090 | Partial elimination of a t... |
| 3orrot 1091 | Rotation law for triple di... |
| 3orcoma 1092 | Commutation law for triple... |
| 3orcomb 1093 | Commutation law for triple... |
| 3anass 1094 | Associative law for triple... |
| 3anan12 1095 | Convert triple conjunction... |
| 3anan32 1096 | Convert triple conjunction... |
| 3ancoma 1097 | Commutation law for triple... |
| 3ancomb 1098 | Commutation law for triple... |
| 3anrot 1099 | Rotation law for triple co... |
| 3anrev 1100 | Reversal law for triple co... |
| anandi3 1101 | Distribution of triple con... |
| anandi3r 1102 | Distribution of triple con... |
| 3anidm 1103 | Idempotent law for conjunc... |
| 3an4anass 1104 | Associative law for four c... |
| 3ioran 1105 | Negated triple disjunction... |
| 3ianor 1106 | Negated triple conjunction... |
| 3anor 1107 | Triple conjunction express... |
| 3oran 1108 | Triple disjunction in term... |
| 3impa 1109 | Importation from double to... |
| 3imp 1110 | Importation inference. (C... |
| 3imp31 1111 | The importation inference ... |
| 3imp231 1112 | Importation inference. (C... |
| 3imp21 1113 | The importation inference ... |
| 3impb 1114 | Importation from double to... |
| bi23imp13 1115 | ~ 3imp with middle implica... |
| 3impib 1116 | Importation to triple conj... |
| 3impia 1117 | Importation to triple conj... |
| 3expa 1118 | Exportation from triple to... |
| 3exp 1119 | Exportation inference. (C... |
| 3expb 1120 | Exportation from triple to... |
| 3expia 1121 | Exportation from triple co... |
| 3expib 1122 | Exportation from triple co... |
| 3com12 1123 | Commutation in antecedent.... |
| 3com13 1124 | Commutation in antecedent.... |
| 3comr 1125 | Commutation in antecedent.... |
| 3com23 1126 | Commutation in antecedent.... |
| 3coml 1127 | Commutation in antecedent.... |
| 3jca 1128 | Join consequents with conj... |
| 3jcad 1129 | Deduction conjoining the c... |
| 3adant1 1130 | Deduction adding a conjunc... |
| 3adant2 1131 | Deduction adding a conjunc... |
| 3adant3 1132 | Deduction adding a conjunc... |
| 3ad2ant1 1133 | Deduction adding conjuncts... |
| 3ad2ant2 1134 | Deduction adding conjuncts... |
| 3ad2ant3 1135 | Deduction adding conjuncts... |
| simp1 1136 | Simplification of triple c... |
| simp2 1137 | Simplification of triple c... |
| simp3 1138 | Simplification of triple c... |
| simp1i 1139 | Infer a conjunct from a tr... |
| simp2i 1140 | Infer a conjunct from a tr... |
| simp3i 1141 | Infer a conjunct from a tr... |
| simp1d 1142 | Deduce a conjunct from a t... |
| simp2d 1143 | Deduce a conjunct from a t... |
| simp3d 1144 | Deduce a conjunct from a t... |
| simp1bi 1145 | Deduce a conjunct from a t... |
| simp2bi 1146 | Deduce a conjunct from a t... |
| simp3bi 1147 | Deduce a conjunct from a t... |
| 3simpa 1148 | Simplification of triple c... |
| 3simpb 1149 | Simplification of triple c... |
| 3simpc 1150 | Simplification of triple c... |
| 3anim123i 1151 | Join antecedents and conse... |
| 3anim1i 1152 | Add two conjuncts to antec... |
| 3anim2i 1153 | Add two conjuncts to antec... |
| 3anim3i 1154 | Add two conjuncts to antec... |
| 3anbi123i 1155 | Join 3 biconditionals with... |
| 3orbi123i 1156 | Join 3 biconditionals with... |
| 3anbi1i 1157 | Inference adding two conju... |
| 3anbi2i 1158 | Inference adding two conju... |
| 3anbi3i 1159 | Inference adding two conju... |
| syl3an 1160 | A triple syllogism inferen... |
| syl3anb 1161 | A triple syllogism inferen... |
| syl3anbr 1162 | A triple syllogism inferen... |
| syl3an1 1163 | A syllogism inference. (C... |
| syl3an2 1164 | A syllogism inference. (C... |
| syl3an3 1165 | A syllogism inference. (C... |
| syl3an132 1166 | ~ syl2an with antecedents ... |
| 3adantl1 1167 | Deduction adding a conjunc... |
| 3adantl2 1168 | Deduction adding a conjunc... |
| 3adantl3 1169 | Deduction adding a conjunc... |
| 3adantr1 1170 | Deduction adding a conjunc... |
| 3adantr2 1171 | Deduction adding a conjunc... |
| 3adantr3 1172 | Deduction adding a conjunc... |
| ad4ant123 1173 | Deduction adding conjuncts... |
| ad4ant124 1174 | Deduction adding conjuncts... |
| ad4ant134 1175 | Deduction adding conjuncts... |
| ad4ant234 1176 | Deduction adding conjuncts... |
| 3adant1l 1177 | Deduction adding a conjunc... |
| 3adant1r 1178 | Deduction adding a conjunc... |
| 3adant2l 1179 | Deduction adding a conjunc... |
| 3adant2r 1180 | Deduction adding a conjunc... |
| 3adant3l 1181 | Deduction adding a conjunc... |
| 3adant3r 1182 | Deduction adding a conjunc... |
| 3adant3r1 1183 | Deduction adding a conjunc... |
| 3adant3r2 1184 | Deduction adding a conjunc... |
| 3adant3r3 1185 | Deduction adding a conjunc... |
| 3ad2antl1 1186 | Deduction adding conjuncts... |
| 3ad2antl2 1187 | Deduction adding conjuncts... |
| 3ad2antl3 1188 | Deduction adding conjuncts... |
| 3ad2antr1 1189 | Deduction adding conjuncts... |
| 3ad2antr2 1190 | Deduction adding conjuncts... |
| 3ad2antr3 1191 | Deduction adding conjuncts... |
| simpl1 1192 | Simplification of conjunct... |
| simpl2 1193 | Simplification of conjunct... |
| simpl3 1194 | Simplification of conjunct... |
| simpr1 1195 | Simplification of conjunct... |
| simpr2 1196 | Simplification of conjunct... |
| simpr3 1197 | Simplification of conjunct... |
| simp1l 1198 | Simplification of triple c... |
| simp1r 1199 | Simplification of triple c... |
| simp2l 1200 | Simplification of triple c... |
| simp2r 1201 | Simplification of triple c... |
| simp3l 1202 | Simplification of triple c... |
| simp3r 1203 | Simplification of triple c... |
| simp11 1204 | Simplification of doubly t... |
| simp12 1205 | Simplification of doubly t... |
| simp13 1206 | Simplification of doubly t... |
| simp21 1207 | Simplification of doubly t... |
| simp22 1208 | Simplification of doubly t... |
| simp23 1209 | Simplification of doubly t... |
| simp31 1210 | Simplification of doubly t... |
| simp32 1211 | Simplification of doubly t... |
| simp33 1212 | Simplification of doubly t... |
| simpll1 1213 | Simplification of conjunct... |
| simpll2 1214 | Simplification of conjunct... |
| simpll3 1215 | Simplification of conjunct... |
| simplr1 1216 | Simplification of conjunct... |
| simplr2 1217 | Simplification of conjunct... |
| simplr3 1218 | Simplification of conjunct... |
| simprl1 1219 | Simplification of conjunct... |
| simprl2 1220 | Simplification of conjunct... |
| simprl3 1221 | Simplification of conjunct... |
| simprr1 1222 | Simplification of conjunct... |
| simprr2 1223 | Simplification of conjunct... |
| simprr3 1224 | Simplification of conjunct... |
| simpl1l 1225 | Simplification of conjunct... |
| simpl1r 1226 | Simplification of conjunct... |
| simpl2l 1227 | Simplification of conjunct... |
| simpl2r 1228 | Simplification of conjunct... |
| simpl3l 1229 | Simplification of conjunct... |
| simpl3r 1230 | Simplification of conjunct... |
| simpr1l 1231 | Simplification of conjunct... |
| simpr1r 1232 | Simplification of conjunct... |
| simpr2l 1233 | Simplification of conjunct... |
| simpr2r 1234 | Simplification of conjunct... |
| simpr3l 1235 | Simplification of conjunct... |
| simpr3r 1236 | Simplification of conjunct... |
| simp1ll 1237 | Simplification of conjunct... |
| simp1lr 1238 | Simplification of conjunct... |
| simp1rl 1239 | Simplification of conjunct... |
| simp1rr 1240 | Simplification of conjunct... |
| simp2ll 1241 | Simplification of conjunct... |
| simp2lr 1242 | Simplification of conjunct... |
| simp2rl 1243 | Simplification of conjunct... |
| simp2rr 1244 | Simplification of conjunct... |
| simp3ll 1245 | Simplification of conjunct... |
| simp3lr 1246 | Simplification of conjunct... |
| simp3rl 1247 | Simplification of conjunct... |
| simp3rr 1248 | Simplification of conjunct... |
| simpl11 1249 | Simplification of conjunct... |
| simpl12 1250 | Simplification of conjunct... |
| simpl13 1251 | Simplification of conjunct... |
| simpl21 1252 | Simplification of conjunct... |
| simpl22 1253 | Simplification of conjunct... |
| simpl23 1254 | Simplification of conjunct... |
| simpl31 1255 | Simplification of conjunct... |
| simpl32 1256 | Simplification of conjunct... |
| simpl33 1257 | Simplification of conjunct... |
| simpr11 1258 | Simplification of conjunct... |
| simpr12 1259 | Simplification of conjunct... |
| simpr13 1260 | Simplification of conjunct... |
| simpr21 1261 | Simplification of conjunct... |
| simpr22 1262 | Simplification of conjunct... |
| simpr23 1263 | Simplification of conjunct... |
| simpr31 1264 | Simplification of conjunct... |
| simpr32 1265 | Simplification of conjunct... |
| simpr33 1266 | Simplification of conjunct... |
| simp1l1 1267 | Simplification of conjunct... |
| simp1l2 1268 | Simplification of conjunct... |
| simp1l3 1269 | Simplification of conjunct... |
| simp1r1 1270 | Simplification of conjunct... |
| simp1r2 1271 | Simplification of conjunct... |
| simp1r3 1272 | Simplification of conjunct... |
| simp2l1 1273 | Simplification of conjunct... |
| simp2l2 1274 | Simplification of conjunct... |
| simp2l3 1275 | Simplification of conjunct... |
| simp2r1 1276 | Simplification of conjunct... |
| simp2r2 1277 | Simplification of conjunct... |
| simp2r3 1278 | Simplification of conjunct... |
| simp3l1 1279 | Simplification of conjunct... |
| simp3l2 1280 | Simplification of conjunct... |
| simp3l3 1281 | Simplification of conjunct... |
| simp3r1 1282 | Simplification of conjunct... |
| simp3r2 1283 | Simplification of conjunct... |
| simp3r3 1284 | Simplification of conjunct... |
| simp11l 1285 | Simplification of conjunct... |
| simp11r 1286 | Simplification of conjunct... |
| simp12l 1287 | Simplification of conjunct... |
| simp12r 1288 | Simplification of conjunct... |
| simp13l 1289 | Simplification of conjunct... |
| simp13r 1290 | Simplification of conjunct... |
| simp21l 1291 | Simplification of conjunct... |
| simp21r 1292 | Simplification of conjunct... |
| simp22l 1293 | Simplification of conjunct... |
| simp22r 1294 | Simplification of conjunct... |
| simp23l 1295 | Simplification of conjunct... |
| simp23r 1296 | Simplification of conjunct... |
| simp31l 1297 | Simplification of conjunct... |
| simp31r 1298 | Simplification of conjunct... |
| simp32l 1299 | Simplification of conjunct... |
| simp32r 1300 | Simplification of conjunct... |
| simp33l 1301 | Simplification of conjunct... |
| simp33r 1302 | Simplification of conjunct... |
| simp111 1303 | Simplification of conjunct... |
| simp112 1304 | Simplification of conjunct... |
| simp113 1305 | Simplification of conjunct... |
| simp121 1306 | Simplification of conjunct... |
| simp122 1307 | Simplification of conjunct... |
| simp123 1308 | Simplification of conjunct... |
| simp131 1309 | Simplification of conjunct... |
| simp132 1310 | Simplification of conjunct... |
| simp133 1311 | Simplification of conjunct... |
| simp211 1312 | Simplification of conjunct... |
| simp212 1313 | Simplification of conjunct... |
| simp213 1314 | Simplification of conjunct... |
| simp221 1315 | Simplification of conjunct... |
| simp222 1316 | Simplification of conjunct... |
| simp223 1317 | Simplification of conjunct... |
| simp231 1318 | Simplification of conjunct... |
| simp232 1319 | Simplification of conjunct... |
| simp233 1320 | Simplification of conjunct... |
| simp311 1321 | Simplification of conjunct... |
| simp312 1322 | Simplification of conjunct... |
| simp313 1323 | Simplification of conjunct... |
| simp321 1324 | Simplification of conjunct... |
| simp322 1325 | Simplification of conjunct... |
| simp323 1326 | Simplification of conjunct... |
| simp331 1327 | Simplification of conjunct... |
| simp332 1328 | Simplification of conjunct... |
| simp333 1329 | Simplification of conjunct... |
| 3anibar 1330 | Remove a hypothesis from t... |
| 3mix1 1331 | Introduction in triple dis... |
| 3mix2 1332 | Introduction in triple dis... |
| 3mix3 1333 | Introduction in triple dis... |
| 3mix1i 1334 | Introduction in triple dis... |
| 3mix2i 1335 | Introduction in triple dis... |
| 3mix3i 1336 | Introduction in triple dis... |
| 3mix1d 1337 | Deduction introducing trip... |
| 3mix2d 1338 | Deduction introducing trip... |
| 3mix3d 1339 | Deduction introducing trip... |
| 3pm3.2i 1340 | Infer conjunction of premi... |
| pm3.2an3 1341 | Version of ~ pm3.2 for a t... |
| mpbir3an 1342 | Detach a conjunction of tr... |
| mpbir3and 1343 | Detach a conjunction of tr... |
| syl3anbrc 1344 | Syllogism inference. (Con... |
| syl21anbrc 1345 | Syllogism inference. (Con... |
| 3imp3i2an 1346 | An elimination deduction. ... |
| ex3 1347 | Apply ~ ex to a hypothesis... |
| 3imp1 1348 | Importation to left triple... |
| 3impd 1349 | Importation deduction for ... |
| 3imp2 1350 | Importation to right tripl... |
| 3impdi 1351 | Importation inference (und... |
| 3impdir 1352 | Importation inference (und... |
| 3exp1 1353 | Exportation from left trip... |
| 3expd 1354 | Exportation deduction for ... |
| 3exp2 1355 | Exportation from right tri... |
| exp5o 1356 | A triple exportation infer... |
| exp516 1357 | A triple exportation infer... |
| exp520 1358 | A triple exportation infer... |
| 3impexp 1359 | Version of ~ impexp for a ... |
| 3an1rs 1360 | Swap conjuncts. (Contribu... |
| 3anassrs 1361 | Associative law for conjun... |
| 4anpull2 1362 | An equivalence of two four... |
| ad5ant245 1363 | Deduction adding conjuncts... |
| ad5ant234 1364 | Deduction adding conjuncts... |
| ad5ant235 1365 | Deduction adding conjuncts... |
| ad5ant123 1366 | Deduction adding conjuncts... |
| ad5ant124 1367 | Deduction adding conjuncts... |
| ad5ant125 1368 | Deduction adding conjuncts... |
| ad5ant134 1369 | Deduction adding conjuncts... |
| ad5ant135 1370 | Deduction adding conjuncts... |
| ad5ant145 1371 | Deduction adding conjuncts... |
| ad5ant2345 1372 | Deduction adding conjuncts... |
| syl3anc 1373 | Syllogism combined with co... |
| syl13anc 1374 | Syllogism combined with co... |
| syl31anc 1375 | Syllogism combined with co... |
| syl112anc 1376 | Syllogism combined with co... |
| syl121anc 1377 | Syllogism combined with co... |
| syl211anc 1378 | Syllogism combined with co... |
| syl23anc 1379 | Syllogism combined with co... |
| syl32anc 1380 | Syllogism combined with co... |
| syl122anc 1381 | Syllogism combined with co... |
| syl212anc 1382 | Syllogism combined with co... |
| syl221anc 1383 | Syllogism combined with co... |
| syl113anc 1384 | Syllogism combined with co... |
| syl131anc 1385 | Syllogism combined with co... |
| syl311anc 1386 | Syllogism combined with co... |
| syl33anc 1387 | Syllogism combined with co... |
| syl222anc 1388 | Syllogism combined with co... |
| syl123anc 1389 | Syllogism combined with co... |
| syl132anc 1390 | Syllogism combined with co... |
| syl213anc 1391 | Syllogism combined with co... |
| syl231anc 1392 | Syllogism combined with co... |
| syl312anc 1393 | Syllogism combined with co... |
| syl321anc 1394 | Syllogism combined with co... |
| syl133anc 1395 | Syllogism combined with co... |
| syl313anc 1396 | Syllogism combined with co... |
| syl331anc 1397 | Syllogism combined with co... |
| syl223anc 1398 | Syllogism combined with co... |
| syl232anc 1399 | Syllogism combined with co... |
| syl322anc 1400 | Syllogism combined with co... |
| syl233anc 1401 | Syllogism combined with co... |
| syl323anc 1402 | Syllogism combined with co... |
| syl332anc 1403 | Syllogism combined with co... |
| syl333anc 1404 | A syllogism inference comb... |
| syl3an1b 1405 | A syllogism inference. (C... |
| syl3an2b 1406 | A syllogism inference. (C... |
| syl3an3b 1407 | A syllogism inference. (C... |
| syl3an1br 1408 | A syllogism inference. (C... |
| syl3an2br 1409 | A syllogism inference. (C... |
| syl3an3br 1410 | A syllogism inference. (C... |
| syld3an3 1411 | A syllogism inference. (C... |
| syld3an1 1412 | A syllogism inference. (C... |
| syld3an2 1413 | A syllogism inference. (C... |
| syl3anl1 1414 | A syllogism inference. (C... |
| syl3anl2 1415 | A syllogism inference. (C... |
| syl3anl3 1416 | A syllogism inference. (C... |
| syl3anl 1417 | A triple syllogism inferen... |
| syl3anr1 1418 | A syllogism inference. (C... |
| syl3anr2 1419 | A syllogism inference. (C... |
| syl3anr3 1420 | A syllogism inference. (C... |
| 3anidm12 1421 | Inference from idempotent ... |
| 3anidm13 1422 | Inference from idempotent ... |
| 3anidm23 1423 | Inference from idempotent ... |
| syl2an3an 1424 | ~ syl3an with antecedents ... |
| syl2an23an 1425 | Deduction related to ~ syl... |
| 3ori 1426 | Infer implication from tri... |
| 3jao 1427 | Disjunction of three antec... |
| 3jaob 1428 | Disjunction of three antec... |
| 3jaobOLD 1429 | Obsolete version of ~ 3jao... |
| 3jaoi 1430 | Disjunction of three antec... |
| 3jaod 1431 | Disjunction of three antec... |
| 3jaoian 1432 | Disjunction of three antec... |
| 3jaodan 1433 | Disjunction of three antec... |
| mpjao3dan 1434 | Eliminate a three-way disj... |
| 3jaao 1435 | Inference conjoining and d... |
| syl3an9b 1436 | Nested syllogism inference... |
| 3orbi123d 1437 | Deduction joining 3 equiva... |
| 3anbi123d 1438 | Deduction joining 3 equiva... |
| 3anbi12d 1439 | Deduction conjoining and a... |
| 3anbi13d 1440 | Deduction conjoining and a... |
| 3anbi23d 1441 | Deduction conjoining and a... |
| 3anbi1d 1442 | Deduction adding conjuncts... |
| 3anbi2d 1443 | Deduction adding conjuncts... |
| 3anbi3d 1444 | Deduction adding conjuncts... |
| 3anim123d 1445 | Deduction joining 3 implic... |
| 3orim123d 1446 | Deduction joining 3 implic... |
| an6 1447 | Rearrangement of 6 conjunc... |
| 3an6 1448 | Analogue of ~ an4 for trip... |
| 3or6 1449 | Analogue of ~ or4 for trip... |
| mp3an1 1450 | An inference based on modu... |
| mp3an2 1451 | An inference based on modu... |
| mp3an3 1452 | An inference based on modu... |
| mp3an12 1453 | An inference based on modu... |
| mp3an13 1454 | An inference based on modu... |
| mp3an23 1455 | An inference based on modu... |
| mp3an1i 1456 | An inference based on modu... |
| mp3anl1 1457 | An inference based on modu... |
| mp3anl2 1458 | An inference based on modu... |
| mp3anl3 1459 | An inference based on modu... |
| mp3anr1 1460 | An inference based on modu... |
| mp3anr2 1461 | An inference based on modu... |
| mp3anr3 1462 | An inference based on modu... |
| mp3an 1463 | An inference based on modu... |
| mpd3an3 1464 | An inference based on modu... |
| mpd3an23 1465 | An inference based on modu... |
| mp3and 1466 | A deduction based on modus... |
| mp3an12i 1467 | ~ mp3an with antecedents i... |
| mp3an2i 1468 | ~ mp3an with antecedents i... |
| mp3an3an 1469 | ~ mp3an with antecedents i... |
| mp3an2ani 1470 | An elimination deduction. ... |
| biimp3a 1471 | Infer implication from a l... |
| biimp3ar 1472 | Infer implication from a l... |
| 3anandis 1473 | Inference that undistribut... |
| 3anandirs 1474 | Inference that undistribut... |
| ecase23d 1475 | Deduction for elimination ... |
| 3ecase 1476 | Inference for elimination ... |
| 3bior1fd 1477 | A disjunction is equivalen... |
| 3bior1fand 1478 | A disjunction is equivalen... |
| 3bior2fd 1479 | A wff is equivalent to its... |
| 3biant1d 1480 | A conjunction is equivalen... |
| intn3an1d 1481 | Introduction of a triple c... |
| intn3an2d 1482 | Introduction of a triple c... |
| intn3an3d 1483 | Introduction of a triple c... |
| an3andi 1484 | Distribution of conjunctio... |
| an33rean 1485 | Rearrange a 9-fold conjunc... |
| 3orel2 1486 | Partial elimination of a t... |
| 3orel2OLD 1487 | Obsolete version of ~ 3ore... |
| 3orel3 1488 | Partial elimination of a t... |
| 3orel13 1489 | Elimination of two disjunc... |
| 3pm3.2ni 1490 | Triple negated disjunction... |
| nanan 1493 | Conjunction in terms of al... |
| dfnan2 1494 | Alternative denial in term... |
| nanor 1495 | Alternative denial in term... |
| nancom 1496 | Alternative denial is comm... |
| nannan 1497 | Nested alternative denials... |
| nanim 1498 | Implication in terms of al... |
| nannot 1499 | Negation in terms of alter... |
| nanbi 1500 | Biconditional in terms of ... |
| nanbi1 1501 | Introduce a right anti-con... |
| nanbi2 1502 | Introduce a left anti-conj... |
| nanbi12 1503 | Join two logical equivalen... |
| nanbi1i 1504 | Introduce a right anti-con... |
| nanbi2i 1505 | Introduce a left anti-conj... |
| nanbi12i 1506 | Join two logical equivalen... |
| nanbi1d 1507 | Introduce a right anti-con... |
| nanbi2d 1508 | Introduce a left anti-conj... |
| nanbi12d 1509 | Join two logical equivalen... |
| nanass 1510 | A characterization of when... |
| xnor 1513 | Two ways to write XNOR (ex... |
| xorcom 1514 | The connector ` \/_ ` is c... |
| xorass 1515 | The connector ` \/_ ` is a... |
| excxor 1516 | This tautology shows that ... |
| xor2 1517 | Two ways to express "exclu... |
| xoror 1518 | Exclusive disjunction impl... |
| xornan 1519 | Exclusive disjunction impl... |
| xornan2 1520 | XOR implies NAND (written ... |
| xorneg2 1521 | The connector ` \/_ ` is n... |
| xorneg1 1522 | The connector ` \/_ ` is n... |
| xorneg 1523 | The connector ` \/_ ` is u... |
| xorbi12i 1524 | Equality property for excl... |
| xorbi12d 1525 | Equality property for excl... |
| anxordi 1526 | Conjunction distributes ov... |
| xorexmid 1527 | Exclusive-or variant of th... |
| norcom 1530 | The connector ` -\/ ` is c... |
| nornot 1531 | ` -. ` is expressible via ... |
| noran 1532 | ` /\ ` is expressible via ... |
| noror 1533 | ` \/ ` is expressible via ... |
| norasslem1 1534 | This lemma shows the equiv... |
| norasslem2 1535 | This lemma specializes ~ b... |
| norasslem3 1536 | This lemma specializes ~ b... |
| norass 1537 | A characterization of when... |
| trujust 1542 | Soundness justification th... |
| tru 1544 | The truth value ` T. ` is ... |
| dftru2 1545 | An alternate definition of... |
| trut 1546 | A proposition is equivalen... |
| mptru 1547 | Eliminate ` T. ` as an ant... |
| tbtru 1548 | A proposition is equivalen... |
| bitru 1549 | A theorem is equivalent to... |
| trud 1550 | Anything implies ` T. ` . ... |
| truan 1551 | True can be removed from a... |
| fal 1554 | The truth value ` F. ` is ... |
| nbfal 1555 | The negation of a proposit... |
| bifal 1556 | A contradiction is equival... |
| falim 1557 | The truth value ` F. ` imp... |
| falimd 1558 | The truth value ` F. ` imp... |
| dfnot 1559 | Given falsum ` F. ` , we c... |
| inegd 1560 | Negation introduction rule... |
| efald 1561 | Deduction based on reducti... |
| pm2.21fal 1562 | If a wff and its negation ... |
| truimtru 1563 | A ` -> ` identity. (Contr... |
| truimfal 1564 | A ` -> ` identity. (Contr... |
| falimtru 1565 | A ` -> ` identity. (Contr... |
| falimfal 1566 | A ` -> ` identity. (Contr... |
| nottru 1567 | A ` -. ` identity. (Contr... |
| notfal 1568 | A ` -. ` identity. (Contr... |
| trubitru 1569 | A ` <-> ` identity. (Cont... |
| falbitru 1570 | A ` <-> ` identity. (Cont... |
| trubifal 1571 | A ` <-> ` identity. (Cont... |
| falbifal 1572 | A ` <-> ` identity. (Cont... |
| truantru 1573 | A ` /\ ` identity. (Contr... |
| truanfal 1574 | A ` /\ ` identity. (Contr... |
| falantru 1575 | A ` /\ ` identity. (Contr... |
| falanfal 1576 | A ` /\ ` identity. (Contr... |
| truortru 1577 | A ` \/ ` identity. (Contr... |
| truorfal 1578 | A ` \/ ` identity. (Contr... |
| falortru 1579 | A ` \/ ` identity. (Contr... |
| falorfal 1580 | A ` \/ ` identity. (Contr... |
| trunantru 1581 | A ` -/\ ` identity. (Cont... |
| trunanfal 1582 | A ` -/\ ` identity. (Cont... |
| falnantru 1583 | A ` -/\ ` identity. (Cont... |
| falnanfal 1584 | A ` -/\ ` identity. (Cont... |
| truxortru 1585 | A ` \/_ ` identity. (Cont... |
| truxorfal 1586 | A ` \/_ ` identity. (Cont... |
| falxortru 1587 | A ` \/_ ` identity. (Cont... |
| falxorfal 1588 | A ` \/_ ` identity. (Cont... |
| trunortru 1589 | A ` -\/ ` identity. (Cont... |
| trunorfal 1590 | A ` -\/ ` identity. (Cont... |
| falnortru 1591 | A ` -\/ ` identity. (Cont... |
| falnorfal 1592 | A ` -\/ ` identity. (Cont... |
| hadbi123d 1595 | Equality theorem for the a... |
| hadbi123i 1596 | Equality theorem for the a... |
| hadass 1597 | Associative law for the ad... |
| hadbi 1598 | The adder sum is the same ... |
| hadcoma 1599 | Commutative law for the ad... |
| hadcomb 1600 | Commutative law for the ad... |
| hadrot 1601 | Rotation law for the adder... |
| hadnot 1602 | The adder sum distributes ... |
| had1 1603 | If the first input is true... |
| had0 1604 | If the first input is fals... |
| hadifp 1605 | The value of the adder sum... |
| cador 1608 | The adder carry in disjunc... |
| cadan 1609 | The adder carry in conjunc... |
| cadbi123d 1610 | Equality theorem for the a... |
| cadbi123i 1611 | Equality theorem for the a... |
| cadcoma 1612 | Commutative law for the ad... |
| cadcomb 1613 | Commutative law for the ad... |
| cadrot 1614 | Rotation law for the adder... |
| cadnot 1615 | The adder carry distribute... |
| cad11 1616 | If (at least) two inputs a... |
| cad1 1617 | If one input is true, then... |
| cad0 1618 | If one input is false, the... |
| cadifp 1619 | The value of the carry is,... |
| cadtru 1620 | The adder carry is true as... |
| minimp 1621 | A single axiom for minimal... |
| minimp-syllsimp 1622 | Derivation of Syll-Simp ( ... |
| minimp-ax1 1623 | Derivation of ~ ax-1 from ... |
| minimp-ax2c 1624 | Derivation of a commuted f... |
| minimp-ax2 1625 | Derivation of ~ ax-2 from ... |
| minimp-pm2.43 1626 | Derivation of ~ pm2.43 (al... |
| impsingle 1627 | The shortest single axiom ... |
| impsingle-step4 1628 | Derivation of impsingle-st... |
| impsingle-step8 1629 | Derivation of impsingle-st... |
| impsingle-ax1 1630 | Derivation of impsingle-ax... |
| impsingle-step15 1631 | Derivation of impsingle-st... |
| impsingle-step18 1632 | Derivation of impsingle-st... |
| impsingle-step19 1633 | Derivation of impsingle-st... |
| impsingle-step20 1634 | Derivation of impsingle-st... |
| impsingle-step21 1635 | Derivation of impsingle-st... |
| impsingle-step22 1636 | Derivation of impsingle-st... |
| impsingle-step25 1637 | Derivation of impsingle-st... |
| impsingle-imim1 1638 | Derivation of impsingle-im... |
| impsingle-peirce 1639 | Derivation of impsingle-pe... |
| tarski-bernays-ax2 1640 | Derivation of ~ ax-2 from ... |
| meredith 1641 | Carew Meredith's sole axio... |
| merlem1 1642 | Step 3 of Meredith's proof... |
| merlem2 1643 | Step 4 of Meredith's proof... |
| merlem3 1644 | Step 7 of Meredith's proof... |
| merlem4 1645 | Step 8 of Meredith's proof... |
| merlem5 1646 | Step 11 of Meredith's proo... |
| merlem6 1647 | Step 12 of Meredith's proo... |
| merlem7 1648 | Between steps 14 and 15 of... |
| merlem8 1649 | Step 15 of Meredith's proo... |
| merlem9 1650 | Step 18 of Meredith's proo... |
| merlem10 1651 | Step 19 of Meredith's proo... |
| merlem11 1652 | Step 20 of Meredith's proo... |
| merlem12 1653 | Step 28 of Meredith's proo... |
| merlem13 1654 | Step 35 of Meredith's proo... |
| luk-1 1655 | 1 of 3 axioms for proposit... |
| luk-2 1656 | 2 of 3 axioms for proposit... |
| luk-3 1657 | 3 of 3 axioms for proposit... |
| luklem1 1658 | Used to rederive standard ... |
| luklem2 1659 | Used to rederive standard ... |
| luklem3 1660 | Used to rederive standard ... |
| luklem4 1661 | Used to rederive standard ... |
| luklem5 1662 | Used to rederive standard ... |
| luklem6 1663 | Used to rederive standard ... |
| luklem7 1664 | Used to rederive standard ... |
| luklem8 1665 | Used to rederive standard ... |
| ax1 1666 | Standard propositional axi... |
| ax2 1667 | Standard propositional axi... |
| ax3 1668 | Standard propositional axi... |
| nic-dfim 1669 | This theorem "defines" imp... |
| nic-dfneg 1670 | This theorem "defines" neg... |
| nic-mp 1671 | Derive Nicod's rule of mod... |
| nic-mpALT 1672 | A direct proof of ~ nic-mp... |
| nic-ax 1673 | Nicod's axiom derived from... |
| nic-axALT 1674 | A direct proof of ~ nic-ax... |
| nic-imp 1675 | Inference for ~ nic-mp usi... |
| nic-idlem1 1676 | Lemma for ~ nic-id . (Con... |
| nic-idlem2 1677 | Lemma for ~ nic-id . Infe... |
| nic-id 1678 | Theorem ~ id expressed wit... |
| nic-swap 1679 | The connector ` -/\ ` is s... |
| nic-isw1 1680 | Inference version of ~ nic... |
| nic-isw2 1681 | Inference for swapping nes... |
| nic-iimp1 1682 | Inference version of ~ nic... |
| nic-iimp2 1683 | Inference version of ~ nic... |
| nic-idel 1684 | Inference to remove the tr... |
| nic-ich 1685 | Chained inference. (Contr... |
| nic-idbl 1686 | Double the terms. Since d... |
| nic-bijust 1687 | Biconditional justificatio... |
| nic-bi1 1688 | Inference to extract one s... |
| nic-bi2 1689 | Inference to extract the o... |
| nic-stdmp 1690 | Derive the standard modus ... |
| nic-luk1 1691 | Proof of ~ luk-1 from ~ ni... |
| nic-luk2 1692 | Proof of ~ luk-2 from ~ ni... |
| nic-luk3 1693 | Proof of ~ luk-3 from ~ ni... |
| lukshef-ax1 1694 | This alternative axiom for... |
| lukshefth1 1695 | Lemma for ~ renicax . (Co... |
| lukshefth2 1696 | Lemma for ~ renicax . (Co... |
| renicax 1697 | A rederivation of ~ nic-ax... |
| tbw-bijust 1698 | Justification for ~ tbw-ne... |
| tbw-negdf 1699 | The definition of negation... |
| tbw-ax1 1700 | The first of four axioms i... |
| tbw-ax2 1701 | The second of four axioms ... |
| tbw-ax3 1702 | The third of four axioms i... |
| tbw-ax4 1703 | The fourth of four axioms ... |
| tbwsyl 1704 | Used to rederive the Lukas... |
| tbwlem1 1705 | Used to rederive the Lukas... |
| tbwlem2 1706 | Used to rederive the Lukas... |
| tbwlem3 1707 | Used to rederive the Lukas... |
| tbwlem4 1708 | Used to rederive the Lukas... |
| tbwlem5 1709 | Used to rederive the Lukas... |
| re1luk1 1710 | ~ luk-1 derived from the T... |
| re1luk2 1711 | ~ luk-2 derived from the T... |
| re1luk3 1712 | ~ luk-3 derived from the T... |
| merco1 1713 | A single axiom for proposi... |
| merco1lem1 1714 | Used to rederive the Tarsk... |
| retbwax4 1715 | ~ tbw-ax4 rederived from ~... |
| retbwax2 1716 | ~ tbw-ax2 rederived from ~... |
| merco1lem2 1717 | Used to rederive the Tarsk... |
| merco1lem3 1718 | Used to rederive the Tarsk... |
| merco1lem4 1719 | Used to rederive the Tarsk... |
| merco1lem5 1720 | Used to rederive the Tarsk... |
| merco1lem6 1721 | Used to rederive the Tarsk... |
| merco1lem7 1722 | Used to rederive the Tarsk... |
| retbwax3 1723 | ~ tbw-ax3 rederived from ~... |
| merco1lem8 1724 | Used to rederive the Tarsk... |
| merco1lem9 1725 | Used to rederive the Tarsk... |
| merco1lem10 1726 | Used to rederive the Tarsk... |
| merco1lem11 1727 | Used to rederive the Tarsk... |
| merco1lem12 1728 | Used to rederive the Tarsk... |
| merco1lem13 1729 | Used to rederive the Tarsk... |
| merco1lem14 1730 | Used to rederive the Tarsk... |
| merco1lem15 1731 | Used to rederive the Tarsk... |
| merco1lem16 1732 | Used to rederive the Tarsk... |
| merco1lem17 1733 | Used to rederive the Tarsk... |
| merco1lem18 1734 | Used to rederive the Tarsk... |
| retbwax1 1735 | ~ tbw-ax1 rederived from ~... |
| merco2 1736 | A single axiom for proposi... |
| mercolem1 1737 | Used to rederive the Tarsk... |
| mercolem2 1738 | Used to rederive the Tarsk... |
| mercolem3 1739 | Used to rederive the Tarsk... |
| mercolem4 1740 | Used to rederive the Tarsk... |
| mercolem5 1741 | Used to rederive the Tarsk... |
| mercolem6 1742 | Used to rederive the Tarsk... |
| mercolem7 1743 | Used to rederive the Tarsk... |
| mercolem8 1744 | Used to rederive the Tarsk... |
| re1tbw1 1745 | ~ tbw-ax1 rederived from ~... |
| re1tbw2 1746 | ~ tbw-ax2 rederived from ~... |
| re1tbw3 1747 | ~ tbw-ax3 rederived from ~... |
| re1tbw4 1748 | ~ tbw-ax4 rederived from ~... |
| rb-bijust 1749 | Justification for ~ rb-imd... |
| rb-imdf 1750 | The definition of implicat... |
| anmp 1751 | Modus ponens for ` { \/ , ... |
| rb-ax1 1752 | The first of four axioms i... |
| rb-ax2 1753 | The second of four axioms ... |
| rb-ax3 1754 | The third of four axioms i... |
| rb-ax4 1755 | The fourth of four axioms ... |
| rbsyl 1756 | Used to rederive the Lukas... |
| rblem1 1757 | Used to rederive the Lukas... |
| rblem2 1758 | Used to rederive the Lukas... |
| rblem3 1759 | Used to rederive the Lukas... |
| rblem4 1760 | Used to rederive the Lukas... |
| rblem5 1761 | Used to rederive the Lukas... |
| rblem6 1762 | Used to rederive the Lukas... |
| rblem7 1763 | Used to rederive the Lukas... |
| re1axmp 1764 | ~ ax-mp derived from Russe... |
| re2luk1 1765 | ~ luk-1 derived from Russe... |
| re2luk2 1766 | ~ luk-2 derived from Russe... |
| re2luk3 1767 | ~ luk-3 derived from Russe... |
| mptnan 1768 | Modus ponendo tollens 1, o... |
| mptxor 1769 | Modus ponendo tollens 2, o... |
| mtpor 1770 | Modus tollendo ponens (inc... |
| mtpxor 1771 | Modus tollendo ponens (ori... |
| stoic1a 1772 | Stoic logic Thema 1 (part ... |
| stoic1b 1773 | Stoic logic Thema 1 (part ... |
| stoic2a 1774 | Stoic logic Thema 2 versio... |
| stoic2b 1775 | Stoic logic Thema 2 versio... |
| stoic3 1776 | Stoic logic Thema 3. Stat... |
| stoic4a 1777 | Stoic logic Thema 4 versio... |
| stoic4b 1778 | Stoic logic Thema 4 versio... |
| alnex 1781 | Universal quantification o... |
| eximal 1782 | An equivalence between an ... |
| nf2 1785 | Alternate definition of no... |
| nf3 1786 | Alternate definition of no... |
| nf4 1787 | Alternate definition of no... |
| nfi 1788 | Deduce that ` x ` is not f... |
| nfri 1789 | Consequence of the definit... |
| nfd 1790 | Deduce that ` x ` is not f... |
| nfrd 1791 | Consequence of the definit... |
| nftht 1792 | Closed form of ~ nfth . (... |
| nfntht 1793 | Closed form of ~ nfnth . ... |
| nfntht2 1794 | Closed form of ~ nfnth . ... |
| gen2 1796 | Generalization applied twi... |
| mpg 1797 | Modus ponens combined with... |
| mpgbi 1798 | Modus ponens on biconditio... |
| mpgbir 1799 | Modus ponens on biconditio... |
| nex 1800 | Generalization rule for ne... |
| nfth 1801 | No variable is (effectivel... |
| nfnth 1802 | No variable is (effectivel... |
| hbth 1803 | No variable is (effectivel... |
| nftru 1804 | The true constant has no f... |
| nffal 1805 | The false constant has no ... |
| sptruw 1806 | Version of ~ sp when ` ph ... |
| altru 1807 | For all sets, ` T. ` is tr... |
| alfal 1808 | For all sets, ` -. F. ` is... |
| alim 1810 | Restatement of Axiom ~ ax-... |
| alimi 1811 | Inference quantifying both... |
| 2alimi 1812 | Inference doubly quantifyi... |
| ala1 1813 | Add an antecedent in a uni... |
| al2im 1814 | Closed form of ~ al2imi . ... |
| al2imi 1815 | Inference quantifying ante... |
| alanimi 1816 | Variant of ~ al2imi with c... |
| alimdh 1817 | Deduction form of Theorem ... |
| albi 1818 | Theorem 19.15 of [Margaris... |
| albii 1819 | Inference adding universal... |
| 2albii 1820 | Inference adding two unive... |
| 3albii 1821 | Inference adding three uni... |
| sylgt 1822 | Closed form of ~ sylg . (... |
| sylg 1823 | A syllogism combined with ... |
| alrimih 1824 | Inference form of Theorem ... |
| hbxfrbi 1825 | A utility lemma to transfe... |
| alex 1826 | Universal quantifier in te... |
| exnal 1827 | Existential quantification... |
| 2nalexn 1828 | Part of theorem *11.5 in [... |
| 2exnaln 1829 | Theorem *11.22 in [Whitehe... |
| 2nexaln 1830 | Theorem *11.25 in [Whitehe... |
| alimex 1831 | An equivalence between an ... |
| aleximi 1832 | A variant of ~ al2imi : in... |
| alexbii 1833 | Biconditional form of ~ al... |
| exim 1834 | Theorem 19.22 of [Margaris... |
| eximi 1835 | Inference adding existenti... |
| 2eximi 1836 | Inference adding two exist... |
| eximii 1837 | Inference associated with ... |
| exa1 1838 | Add an antecedent in an ex... |
| 19.38 1839 | Theorem 19.38 of [Margaris... |
| 19.38a 1840 | Under a nonfreeness hypoth... |
| 19.38b 1841 | Under a nonfreeness hypoth... |
| imnang 1842 | Quantified implication in ... |
| alinexa 1843 | A transformation of quanti... |
| exnalimn 1844 | Existential quantification... |
| alexn 1845 | A relationship between two... |
| 2exnexn 1846 | Theorem *11.51 in [Whitehe... |
| exbi 1847 | Theorem 19.18 of [Margaris... |
| exbii 1848 | Inference adding existenti... |
| 2exbii 1849 | Inference adding two exist... |
| 3exbii 1850 | Inference adding three exi... |
| nfbiit 1851 | Equivalence theorem for th... |
| nfbii 1852 | Equality theorem for the n... |
| nfxfr 1853 | A utility lemma to transfe... |
| nfxfrd 1854 | A utility lemma to transfe... |
| nfnbi 1855 | A variable is nonfree in a... |
| nfnt 1856 | If a variable is nonfree i... |
| nfn 1857 | Inference associated with ... |
| nfnd 1858 | Deduction associated with ... |
| exanali 1859 | A transformation of quanti... |
| 2exanali 1860 | Theorem *11.521 in [Whiteh... |
| exancom 1861 | Commutation of conjunction... |
| exan 1862 | Place a conjunct in the sc... |
| alrimdh 1863 | Deduction form of Theorem ... |
| eximdh 1864 | Deduction from Theorem 19.... |
| nexdh 1865 | Deduction for generalizati... |
| albidh 1866 | Formula-building rule for ... |
| exbidh 1867 | Formula-building rule for ... |
| exsimpl 1868 | Simplification of an exist... |
| exsimpr 1869 | Simplification of an exist... |
| 19.26 1870 | Theorem 19.26 of [Margaris... |
| 19.26-2 1871 | Theorem ~ 19.26 with two q... |
| 19.26-3an 1872 | Theorem ~ 19.26 with tripl... |
| 19.29 1873 | Theorem 19.29 of [Margaris... |
| 19.29r 1874 | Variation of ~ 19.29 . (C... |
| 19.29r2 1875 | Variation of ~ 19.29r with... |
| 19.29x 1876 | Variation of ~ 19.29 with ... |
| 19.35 1877 | Theorem 19.35 of [Margaris... |
| 19.35i 1878 | Inference associated with ... |
| 19.35ri 1879 | Inference associated with ... |
| 19.25 1880 | Theorem 19.25 of [Margaris... |
| 19.30 1881 | Theorem 19.30 of [Margaris... |
| 19.43 1882 | Theorem 19.43 of [Margaris... |
| 19.43OLD 1883 | Obsolete proof of ~ 19.43 ... |
| 19.33 1884 | Theorem 19.33 of [Margaris... |
| 19.33b 1885 | The antecedent provides a ... |
| 19.40 1886 | Theorem 19.40 of [Margaris... |
| 19.40-2 1887 | Theorem *11.42 in [Whitehe... |
| 19.40b 1888 | The antecedent provides a ... |
| albiim 1889 | Split a biconditional and ... |
| 2albiim 1890 | Split a biconditional and ... |
| exintrbi 1891 | Add/remove a conjunct in t... |
| exintr 1892 | Introduce a conjunct in th... |
| alsyl 1893 | Universally quantified and... |
| nfimd 1894 | If in a context ` x ` is n... |
| nfimt 1895 | Closed form of ~ nfim and ... |
| nfim 1896 | If ` x ` is not free in ` ... |
| nfand 1897 | If in a context ` x ` is n... |
| nf3and 1898 | Deduction form of bound-va... |
| nfan 1899 | If ` x ` is not free in ` ... |
| nfnan 1900 | If ` x ` is not free in ` ... |
| nf3an 1901 | If ` x ` is not free in ` ... |
| nfbid 1902 | If in a context ` x ` is n... |
| nfbi 1903 | If ` x ` is not free in ` ... |
| nfor 1904 | If ` x ` is not free in ` ... |
| nf3or 1905 | If ` x ` is not free in ` ... |
| empty 1906 | Two characterizations of t... |
| emptyex 1907 | On the empty domain, any e... |
| emptyal 1908 | On the empty domain, any u... |
| emptynf 1909 | On the empty domain, any v... |
| ax5d 1911 | Version of ~ ax-5 with ant... |
| ax5e 1912 | A rephrasing of ~ ax-5 usi... |
| ax5ea 1913 | If a formula holds for som... |
| nfv 1914 | If ` x ` is not present in... |
| nfvd 1915 | ~ nfv with antecedent. Us... |
| alimdv 1916 | Deduction form of Theorem ... |
| eximdv 1917 | Deduction form of Theorem ... |
| 2alimdv 1918 | Deduction form of Theorem ... |
| 2eximdv 1919 | Deduction form of Theorem ... |
| albidv 1920 | Formula-building rule for ... |
| exbidv 1921 | Formula-building rule for ... |
| nfbidv 1922 | An equality theorem for no... |
| 2albidv 1923 | Formula-building rule for ... |
| 2exbidv 1924 | Formula-building rule for ... |
| 3exbidv 1925 | Formula-building rule for ... |
| 4exbidv 1926 | Formula-building rule for ... |
| alrimiv 1927 | Inference form of Theorem ... |
| alrimivv 1928 | Inference form of Theorem ... |
| alrimdv 1929 | Deduction form of Theorem ... |
| exlimiv 1930 | Inference form of Theorem ... |
| exlimiiv 1931 | Inference (Rule C) associa... |
| exlimivv 1932 | Inference form of Theorem ... |
| exlimdv 1933 | Deduction form of Theorem ... |
| exlimdvv 1934 | Deduction form of Theorem ... |
| exlimddv 1935 | Existential elimination ru... |
| nexdv 1936 | Deduction for generalizati... |
| 2ax5 1937 | Quantification of two vari... |
| stdpc5v 1938 | Version of ~ stdpc5 with a... |
| 19.21v 1939 | Version of ~ 19.21 with a ... |
| 19.32v 1940 | Version of ~ 19.32 with a ... |
| 19.31v 1941 | Version of ~ 19.31 with a ... |
| 19.23v 1942 | Version of ~ 19.23 with a ... |
| 19.23vv 1943 | Theorem ~ 19.23v extended ... |
| pm11.53v 1944 | Version of ~ pm11.53 with ... |
| 19.36imv 1945 | One direction of ~ 19.36v ... |
| 19.36iv 1946 | Inference associated with ... |
| 19.37imv 1947 | One direction of ~ 19.37v ... |
| 19.37iv 1948 | Inference associated with ... |
| 19.41v 1949 | Version of ~ 19.41 with a ... |
| 19.41vv 1950 | Version of ~ 19.41 with tw... |
| 19.41vvv 1951 | Version of ~ 19.41 with th... |
| 19.41vvvv 1952 | Version of ~ 19.41 with fo... |
| 19.42v 1953 | Version of ~ 19.42 with a ... |
| exdistr 1954 | Distribution of existentia... |
| exdistrv 1955 | Distribute a pair of exist... |
| 4exdistrv 1956 | Distribute two pairs of ex... |
| 19.42vv 1957 | Version of ~ 19.42 with tw... |
| exdistr2 1958 | Distribution of existentia... |
| 19.42vvv 1959 | Version of ~ 19.42 with th... |
| 3exdistr 1960 | Distribution of existentia... |
| 4exdistr 1961 | Distribution of existentia... |
| weq 1962 | Extend wff definition to i... |
| speimfw 1963 | Specialization, with addit... |
| speimfwALT 1964 | Alternate proof of ~ speim... |
| spimfw 1965 | Specialization, with addit... |
| ax12i 1966 | Inference that has ~ ax-12... |
| ax6v 1968 | Axiom B7 of [Tarski] p. 75... |
| ax6ev 1969 | At least one individual ex... |
| spimw 1970 | Specialization. Lemma 8 o... |
| spimew 1971 | Existential introduction, ... |
| speiv 1972 | Inference from existential... |
| speivw 1973 | Version of ~ spei with a d... |
| exgen 1974 | Rule of existential genera... |
| extru 1975 | There exists a variable su... |
| 19.2 1976 | Theorem 19.2 of [Margaris]... |
| 19.2d 1977 | Deduction associated with ... |
| 19.8w 1978 | Weak version of ~ 19.8a an... |
| spnfw 1979 | Weak version of ~ sp . Us... |
| spfalw 1980 | Version of ~ sp when ` ph ... |
| spvw 1981 | Version of ~ sp when ` x `... |
| 19.3v 1982 | Version of ~ 19.3 with a d... |
| 19.8v 1983 | Version of ~ 19.8a with a ... |
| 19.9v 1984 | Version of ~ 19.9 with a d... |
| spimevw 1985 | Existential introduction, ... |
| spimvw 1986 | A weak form of specializat... |
| spsv 1987 | Generalization of antecede... |
| spvv 1988 | Specialization, using impl... |
| chvarvv 1989 | Implicit substitution of `... |
| 19.39 1990 | Theorem 19.39 of [Margaris... |
| 19.24 1991 | Theorem 19.24 of [Margaris... |
| 19.34 1992 | Theorem 19.34 of [Margaris... |
| 19.36v 1993 | Version of ~ 19.36 with a ... |
| 19.12vvv 1994 | Version of ~ 19.12vv with ... |
| 19.27v 1995 | Version of ~ 19.27 with a ... |
| 19.28v 1996 | Version of ~ 19.28 with a ... |
| 19.37v 1997 | Version of ~ 19.37 with a ... |
| 19.44v 1998 | Version of ~ 19.44 with a ... |
| 19.45v 1999 | Version of ~ 19.45 with a ... |
| equs4v 2000 | Version of ~ equs4 with a ... |
| alequexv 2001 | Version of ~ equs4v with i... |
| exsbim 2002 | One direction of the equiv... |
| equsv 2003 | If a formula does not cont... |
| equsalvw 2004 | Version of ~ equsalv with ... |
| equsexvw 2005 | Version of ~ equsexv with ... |
| cbvaliw 2006 | Change bound variable. Us... |
| cbvalivw 2007 | Change bound variable. Us... |
| ax7v 2009 | Weakened version of ~ ax-7... |
| ax7v1 2010 | First of two weakened vers... |
| ax7v2 2011 | Second of two weakened ver... |
| equid 2012 | Identity law for equality.... |
| nfequid 2013 | Bound-variable hypothesis ... |
| equcomiv 2014 | Weaker form of ~ equcomi w... |
| ax6evr 2015 | A commuted form of ~ ax6ev... |
| ax7 2016 | Proof of ~ ax-7 from ~ ax7... |
| equcomi 2017 | Commutative law for equali... |
| equcom 2018 | Commutative law for equali... |
| equcomd 2019 | Deduction form of ~ equcom... |
| equcoms 2020 | An inference commuting equ... |
| equtr 2021 | A transitive law for equal... |
| equtrr 2022 | A transitive law for equal... |
| equeuclr 2023 | Commuted version of ~ eque... |
| equeucl 2024 | Equality is a left-Euclide... |
| equequ1 2025 | An equivalence law for equ... |
| equequ2 2026 | An equivalence law for equ... |
| equtr2 2027 | Equality is a left-Euclide... |
| stdpc6 2028 | One of the two equality ax... |
| equvinv 2029 | A variable introduction la... |
| equvinva 2030 | A modified version of the ... |
| equvelv 2031 | A biconditional form of ~ ... |
| ax13b 2032 | An equivalence between two... |
| spfw 2033 | Weak version of ~ sp . Us... |
| spw 2034 | Weak version of the specia... |
| cbvalw 2035 | Change bound variable. Us... |
| cbvalvw 2036 | Change bound variable. Us... |
| cbvexvw 2037 | Change bound variable. Us... |
| cbvaldvaw 2038 | Rule used to change the bo... |
| cbvexdvaw 2039 | Rule used to change the bo... |
| cbval2vw 2040 | Rule used to change bound ... |
| cbvex2vw 2041 | Rule used to change bound ... |
| cbvex4vw 2042 | Rule used to change bound ... |
| alcomimw 2043 | Weak version of ~ ax-11 . ... |
| excomimw 2044 | Weak version of ~ excomim ... |
| alcomw 2045 | Weak version of ~ alcom an... |
| hbn1fw 2046 | Weak version of ~ ax-10 fr... |
| hbn1w 2047 | Weak version of ~ hbn1 . ... |
| hba1w 2048 | Weak version of ~ hba1 . ... |
| hbe1w 2049 | Weak version of ~ hbe1 . ... |
| hbalw 2050 | Weak version of ~ hbal . ... |
| 19.8aw 2051 | If a formula is true, then... |
| exexw 2052 | Existential quantification... |
| spaev 2053 | A special instance of ~ sp... |
| cbvaev 2054 | Change bound variable in a... |
| aevlem0 2055 | Lemma for ~ aevlem . Inst... |
| aevlem 2056 | Lemma for ~ aev and ~ axc1... |
| aeveq 2057 | The antecedent ` A. x x = ... |
| aev 2058 | A "distinctor elimination"... |
| aev2 2059 | A version of ~ aev with tw... |
| hbaev 2060 | All variables are effectiv... |
| naev 2061 | If some set variables can ... |
| naev2 2062 | Generalization of ~ hbnaev... |
| hbnaev 2063 | Any variable is free in ` ... |
| sbjust 2064 | Justification theorem for ... |
| sbt 2067 | A substitution into a theo... |
| sbtru 2068 | The result of substituting... |
| stdpc4 2069 | The specialization axiom o... |
| sbtALT 2070 | Alternate proof of ~ sbt ,... |
| 2stdpc4 2071 | A double specialization us... |
| sbi1 2072 | Distribute substitution ov... |
| spsbim 2073 | Distribute substitution ov... |
| spsbbi 2074 | Biconditional property for... |
| sbimi 2075 | Distribute substitution ov... |
| sb2imi 2076 | Distribute substitution ov... |
| sbbii 2077 | Infer substitution into bo... |
| 2sbbii 2078 | Infer double substitution ... |
| sbimdv 2079 | Deduction substituting bot... |
| sbbidv 2080 | Deduction substituting bot... |
| sban 2081 | Conjunction inside and out... |
| sb3an 2082 | Threefold conjunction insi... |
| spsbe 2083 | Existential generalization... |
| sbequ 2084 | Equality property for subs... |
| sbequi 2085 | An equality theorem for su... |
| sb6 2086 | Alternate definition of su... |
| 2sb6 2087 | Equivalence for double sub... |
| sb1v 2088 | One direction of ~ sb5 , p... |
| sbv 2089 | Substitution for a variabl... |
| sbcom4 2090 | Commutativity law for subs... |
| pm11.07 2091 | Axiom *11.07 in [Whitehead... |
| sbrimvw 2092 | Substitution in an implica... |
| sbbiiev 2093 | An equivalence of substitu... |
| sbievw 2094 | Conversion of implicit sub... |
| sbievwOLD 2095 | Obsolete version of ~ sbie... |
| sbiedvw 2096 | Conversion of implicit sub... |
| 2sbievw 2097 | Conversion of double impli... |
| sbcom3vv 2098 | Substituting ` y ` for ` x... |
| sbievw2 2099 | ~ sbievw applied twice, av... |
| sbco2vv 2100 | A composition law for subs... |
| cbvsbv 2101 | Change the bound variable ... |
| sbco4lem 2102 | Lemma for ~ sbco4 . It re... |
| sbco4 2103 | Two ways of exchanging two... |
| equsb3 2104 | Substitution in an equalit... |
| equsb3r 2105 | Substitution applied to th... |
| equsb1v 2106 | Substitution applied to an... |
| nsb 2107 | Any substitution in an alw... |
| sbn1 2108 | One direction of ~ sbn , u... |
| wel 2110 | Extend wff definition to i... |
| ax8v 2112 | Weakened version of ~ ax-8... |
| ax8v1 2113 | First of two weakened vers... |
| ax8v2 2114 | Second of two weakened ver... |
| ax8 2115 | Proof of ~ ax-8 from ~ ax8... |
| elequ1 2116 | An identity law for the no... |
| elsb1 2117 | Substitution for the first... |
| cleljust 2118 | When the class variables i... |
| ax9v 2120 | Weakened version of ~ ax-9... |
| ax9v1 2121 | First of two weakened vers... |
| ax9v2 2122 | Second of two weakened ver... |
| ax9 2123 | Proof of ~ ax-9 from ~ ax9... |
| elequ2 2124 | An identity law for the no... |
| elequ2g 2125 | A form of ~ elequ2 with a ... |
| elsb2 2126 | Substitution for the secon... |
| elequ12 2127 | An identity law for the no... |
| ru0 2128 | The FOL statement used in ... |
| ax6dgen 2129 | Tarski's system uses the w... |
| ax10w 2130 | Weak version of ~ ax-10 fr... |
| ax11w 2131 | Weak version of ~ ax-11 fr... |
| ax11dgen 2132 | Degenerate instance of ~ a... |
| ax12wlem 2133 | Lemma for weak version of ... |
| ax12w 2134 | Weak version of ~ ax-12 fr... |
| ax12dgen 2135 | Degenerate instance of ~ a... |
| ax12wdemo 2136 | Example of an application ... |
| ax13w 2137 | Weak version (principal in... |
| ax13dgen1 2138 | Degenerate instance of ~ a... |
| ax13dgen2 2139 | Degenerate instance of ~ a... |
| ax13dgen3 2140 | Degenerate instance of ~ a... |
| ax13dgen4 2141 | Degenerate instance of ~ a... |
| hbn1 2143 | Alias for ~ ax-10 to be us... |
| hbe1 2144 | The setvar ` x ` is not fr... |
| hbe1a 2145 | Dual statement of ~ hbe1 .... |
| nf5-1 2146 | One direction of ~ nf5 can... |
| nf5i 2147 | Deduce that ` x ` is not f... |
| nf5dh 2148 | Deduce that ` x ` is not f... |
| nf5dv 2149 | Apply the definition of no... |
| nfnaew 2150 | All variables are effectiv... |
| nfe1 2151 | The setvar ` x ` is not fr... |
| nfa1 2152 | The setvar ` x ` is not fr... |
| nfna1 2153 | A convenience theorem part... |
| nfia1 2154 | Lemma 23 of [Monk2] p. 114... |
| nfnf1 2155 | The setvar ` x ` is not fr... |
| modal5 2156 | The analogue in our predic... |
| nfs1v 2157 | The setvar ` x ` is not fr... |
| alcoms 2159 | Swap quantifiers in an ant... |
| alcom 2160 | Theorem 19.5 of [Margaris]... |
| alrot3 2161 | Theorem *11.21 in [Whitehe... |
| alrot4 2162 | Rotate four universal quan... |
| excom 2163 | Theorem 19.11 of [Margaris... |
| excomim 2164 | One direction of Theorem 1... |
| excom13 2165 | Swap 1st and 3rd existenti... |
| exrot3 2166 | Rotate existential quantif... |
| exrot4 2167 | Rotate existential quantif... |
| hbal 2168 | If ` x ` is not free in ` ... |
| hbald 2169 | Deduction form of bound-va... |
| sbal 2170 | Move universal quantifier ... |
| sbalv 2171 | Quantify with new variable... |
| hbsbw 2172 | If ` z ` is not free in ` ... |
| hbsbwOLD 2173 | Obsolete version of ~ hbsb... |
| sbcom2 2174 | Commutativity law for subs... |
| sbco4lemOLD 2175 | Obsolete version of ~ sbco... |
| sbco4OLD 2176 | Obsolete version of ~ sbco... |
| nfa2 2177 | Lemma 24 of [Monk2] p. 114... |
| ax12v 2179 | This is essentially Axiom ... |
| ax12v2 2180 | It is possible to remove a... |
| ax12ev2 2181 | Version of ~ ax12v2 rewrit... |
| 19.8a 2182 | If a wff is true, it is tr... |
| 19.8ad 2183 | If a wff is true, it is tr... |
| sp 2184 | Specialization. A univers... |
| spi 2185 | Inference rule of universa... |
| sps 2186 | Generalization of antecede... |
| 2sp 2187 | A double specialization (s... |
| spsd 2188 | Deduction generalizing ant... |
| 19.2g 2189 | Theorem 19.2 of [Margaris]... |
| 19.21bi 2190 | Inference form of ~ 19.21 ... |
| 19.21bbi 2191 | Inference removing two uni... |
| 19.23bi 2192 | Inference form of Theorem ... |
| nexr 2193 | Inference associated with ... |
| qexmid 2194 | Quantified excluded middle... |
| nf5r 2195 | Consequence of the definit... |
| nf5ri 2196 | Consequence of the definit... |
| nf5rd 2197 | Consequence of the definit... |
| spimedv 2198 | Deduction version of ~ spi... |
| spimefv 2199 | Version of ~ spime with a ... |
| nfim1 2200 | A closed form of ~ nfim . ... |
| nfan1 2201 | A closed form of ~ nfan . ... |
| 19.3t 2202 | Closed form of ~ 19.3 and ... |
| 19.3 2203 | A wff may be quantified wi... |
| 19.9d 2204 | A deduction version of one... |
| 19.9t 2205 | Closed form of ~ 19.9 and ... |
| 19.9 2206 | A wff may be existentially... |
| 19.21t 2207 | Closed form of Theorem 19.... |
| 19.21 2208 | Theorem 19.21 of [Margaris... |
| stdpc5 2209 | An axiom scheme of standar... |
| 19.21-2 2210 | Version of ~ 19.21 with tw... |
| 19.23t 2211 | Closed form of Theorem 19.... |
| 19.23 2212 | Theorem 19.23 of [Margaris... |
| alimd 2213 | Deduction form of Theorem ... |
| alrimi 2214 | Inference form of Theorem ... |
| alrimdd 2215 | Deduction form of Theorem ... |
| alrimd 2216 | Deduction form of Theorem ... |
| eximd 2217 | Deduction form of Theorem ... |
| exlimi 2218 | Inference associated with ... |
| exlimd 2219 | Deduction form of Theorem ... |
| exlimimdd 2220 | Existential elimination ru... |
| exlimdd 2221 | Existential elimination ru... |
| nexd 2222 | Deduction for generalizati... |
| albid 2223 | Formula-building rule for ... |
| exbid 2224 | Formula-building rule for ... |
| nfbidf 2225 | An equality theorem for ef... |
| 19.16 2226 | Theorem 19.16 of [Margaris... |
| 19.17 2227 | Theorem 19.17 of [Margaris... |
| 19.27 2228 | Theorem 19.27 of [Margaris... |
| 19.28 2229 | Theorem 19.28 of [Margaris... |
| 19.19 2230 | Theorem 19.19 of [Margaris... |
| 19.36 2231 | Theorem 19.36 of [Margaris... |
| 19.36i 2232 | Inference associated with ... |
| 19.37 2233 | Theorem 19.37 of [Margaris... |
| 19.32 2234 | Theorem 19.32 of [Margaris... |
| 19.31 2235 | Theorem 19.31 of [Margaris... |
| 19.41 2236 | Theorem 19.41 of [Margaris... |
| 19.42 2237 | Theorem 19.42 of [Margaris... |
| 19.44 2238 | Theorem 19.44 of [Margaris... |
| 19.45 2239 | Theorem 19.45 of [Margaris... |
| spimfv 2240 | Specialization, using impl... |
| chvarfv 2241 | Implicit substitution of `... |
| cbv3v2 2242 | Version of ~ cbv3 with two... |
| sbalex 2243 | Equivalence of two ways to... |
| sbalexOLD 2244 | Obsolete version of ~ sbal... |
| sb4av 2245 | Version of ~ sb4a with a d... |
| sbimd 2246 | Deduction substituting bot... |
| sbbid 2247 | Deduction substituting bot... |
| 2sbbid 2248 | Deduction doubly substitut... |
| sbequ1 2249 | An equality theorem for su... |
| sbequ2 2250 | An equality theorem for su... |
| stdpc7 2251 | One of the two equality ax... |
| sbequ12 2252 | An equality theorem for su... |
| sbequ12r 2253 | An equality theorem for su... |
| sbelx 2254 | Elimination of substitutio... |
| sbequ12a 2255 | An equality theorem for su... |
| sbid 2256 | An identity theorem for su... |
| sbcov 2257 | A composition law for subs... |
| sbcovOLD 2258 | Obsolete version of ~ sbco... |
| sb6a 2259 | Equivalence for substituti... |
| sbid2vw 2260 | Reverting substitution yie... |
| axc16g 2261 | Generalization of ~ axc16 ... |
| axc16 2262 | Proof of older axiom ~ ax-... |
| axc16gb 2263 | Biconditional strengthenin... |
| axc16nf 2264 | If ~ dtru is false, then t... |
| axc11v 2265 | Version of ~ axc11 with a ... |
| axc11rv 2266 | Version of ~ axc11r with a... |
| drsb2 2267 | Formula-building lemma for... |
| equsalv 2268 | An equivalence related to ... |
| equsexv 2269 | An equivalence related to ... |
| sbft 2270 | Substitution has no effect... |
| sbf 2271 | Substitution for a variabl... |
| sbf2 2272 | Substitution has no effect... |
| sbh 2273 | Substitution for a variabl... |
| hbs1 2274 | The setvar ` x ` is not fr... |
| nfs1f 2275 | If ` x ` is not free in ` ... |
| sb5 2276 | Alternate definition of su... |
| equs5av 2277 | A property related to subs... |
| 2sb5 2278 | Equivalence for double sub... |
| dfsb7 2279 | An alternate definition of... |
| sbn 2280 | Negation inside and outsid... |
| sbex 2281 | Move existential quantifie... |
| nf5 2282 | Alternate definition of ~ ... |
| nf6 2283 | An alternate definition of... |
| nf5d 2284 | Deduce that ` x ` is not f... |
| nf5di 2285 | Since the converse holds b... |
| 19.9h 2286 | A wff may be existentially... |
| 19.21h 2287 | Theorem 19.21 of [Margaris... |
| 19.23h 2288 | Theorem 19.23 of [Margaris... |
| exlimih 2289 | Inference associated with ... |
| exlimdh 2290 | Deduction form of Theorem ... |
| equsalhw 2291 | Version of ~ equsalh with ... |
| equsexhv 2292 | An equivalence related to ... |
| hba1 2293 | The setvar ` x ` is not fr... |
| hbnt 2294 | Closed theorem version of ... |
| hbn 2295 | If ` x ` is not free in ` ... |
| hbnd 2296 | Deduction form of bound-va... |
| hbim1 2297 | A closed form of ~ hbim . ... |
| hbimd 2298 | Deduction form of bound-va... |
| hbim 2299 | If ` x ` is not free in ` ... |
| hban 2300 | If ` x ` is not free in ` ... |
| hb3an 2301 | If ` x ` is not free in ` ... |
| sbi2 2302 | Introduction of implicatio... |
| sbim 2303 | Implication inside and out... |
| sbrim 2304 | Substitution in an implica... |
| sblim 2305 | Substitution in an implica... |
| sbor 2306 | Disjunction inside and out... |
| sbbi 2307 | Equivalence inside and out... |
| sblbis 2308 | Introduce left bicondition... |
| sbrbis 2309 | Introduce right biconditio... |
| sbrbif 2310 | Introduce right biconditio... |
| sbnf 2311 | Move nonfree predicate in ... |
| sbnfOLD 2312 | Obsolete version of ~ sbnf... |
| sbiev 2313 | Conversion of implicit sub... |
| sbievOLD 2314 | Obsolete version of ~ sbie... |
| sbiedw 2315 | Conversion of implicit sub... |
| axc7 2316 | Show that the original axi... |
| axc7e 2317 | Abbreviated version of ~ a... |
| modal-b 2318 | The analogue in our predic... |
| 19.9ht 2319 | A closed version of ~ 19.9... |
| axc4 2320 | Show that the original axi... |
| axc4i 2321 | Inference version of ~ axc... |
| nfal 2322 | If ` x ` is not free in ` ... |
| nfex 2323 | If ` x ` is not free in ` ... |
| hbex 2324 | If ` x ` is not free in ` ... |
| nfnf 2325 | If ` x ` is not free in ` ... |
| 19.12 2326 | Theorem 19.12 of [Margaris... |
| nfald 2327 | Deduction form of ~ nfal .... |
| nfexd 2328 | If ` x ` is not free in ` ... |
| nfsbv 2329 | If ` z ` is not free in ` ... |
| sbco2v 2330 | A composition law for subs... |
| aaan 2331 | Distribute universal quant... |
| eeor 2332 | Distribute existential qua... |
| cbv3v 2333 | Rule used to change bound ... |
| cbv1v 2334 | Rule used to change bound ... |
| cbv2w 2335 | Rule used to change bound ... |
| cbvaldw 2336 | Deduction used to change b... |
| cbvexdw 2337 | Deduction used to change b... |
| cbv3hv 2338 | Rule used to change bound ... |
| cbvalv1 2339 | Rule used to change bound ... |
| cbvexv1 2340 | Rule used to change bound ... |
| cbval2v 2341 | Rule used to change bound ... |
| cbvex2v 2342 | Rule used to change bound ... |
| dvelimhw 2343 | Proof of ~ dvelimh without... |
| pm11.53 2344 | Theorem *11.53 in [Whitehe... |
| 19.12vv 2345 | Special case of ~ 19.12 wh... |
| eean 2346 | Distribute existential qua... |
| eeanv 2347 | Distribute a pair of exist... |
| eeeanv 2348 | Distribute three existenti... |
| ee4anv 2349 | Distribute two pairs of ex... |
| ee4anvOLD 2350 | Obsolete version of ~ ee4a... |
| sb8v 2351 | Substitution of variable i... |
| sb8f 2352 | Substitution of variable i... |
| sb8ef 2353 | Substitution of variable i... |
| 2sb8ef 2354 | An equivalent expression f... |
| sb6rfv 2355 | Reversed substitution. Ve... |
| sbnf2 2356 | Two ways of expressing " `... |
| exsb 2357 | An equivalent expression f... |
| 2exsb 2358 | An equivalent expression f... |
| sbbib 2359 | Reversal of substitution. ... |
| sbbibvv 2360 | Reversal of substitution. ... |
| cbvsbvf 2361 | Change the bound variable ... |
| cleljustALT 2362 | Alternate proof of ~ clelj... |
| cleljustALT2 2363 | Alternate proof of ~ clelj... |
| equs5aALT 2364 | Alternate proof of ~ equs5... |
| equs5eALT 2365 | Alternate proof of ~ equs5... |
| axc11r 2366 | Same as ~ axc11 but with r... |
| dral1v 2367 | Formula-building lemma for... |
| drex1v 2368 | Formula-building lemma for... |
| drnf1v 2369 | Formula-building lemma for... |
| ax13v 2371 | A weaker version of ~ ax-1... |
| ax13lem1 2372 | A version of ~ ax13v with ... |
| ax13 2373 | Derive ~ ax-13 from ~ ax13... |
| ax13lem2 2374 | Lemma for ~ nfeqf2 . This... |
| nfeqf2 2375 | An equation between setvar... |
| dveeq2 2376 | Quantifier introduction wh... |
| nfeqf1 2377 | An equation between setvar... |
| dveeq1 2378 | Quantifier introduction wh... |
| nfeqf 2379 | A variable is effectively ... |
| axc9 2380 | Derive set.mm's original ~... |
| ax6e 2381 | At least one individual ex... |
| ax6 2382 | Theorem showing that ~ ax-... |
| axc10 2383 | Show that the original axi... |
| spimt 2384 | Closed theorem form of ~ s... |
| spim 2385 | Specialization, using impl... |
| spimed 2386 | Deduction version of ~ spi... |
| spime 2387 | Existential introduction, ... |
| spimv 2388 | A version of ~ spim with a... |
| spimvALT 2389 | Alternate proof of ~ spimv... |
| spimev 2390 | Distinct-variable version ... |
| spv 2391 | Specialization, using impl... |
| spei 2392 | Inference from existential... |
| chvar 2393 | Implicit substitution of `... |
| chvarv 2394 | Implicit substitution of `... |
| cbv3 2395 | Rule used to change bound ... |
| cbval 2396 | Rule used to change bound ... |
| cbvex 2397 | Rule used to change bound ... |
| cbvalv 2398 | Rule used to change bound ... |
| cbvexv 2399 | Rule used to change bound ... |
| cbv1 2400 | Rule used to change bound ... |
| cbv2 2401 | Rule used to change bound ... |
| cbv3h 2402 | Rule used to change bound ... |
| cbv1h 2403 | Rule used to change bound ... |
| cbv2h 2404 | Rule used to change bound ... |
| cbvald 2405 | Deduction used to change b... |
| cbvexd 2406 | Deduction used to change b... |
| cbvaldva 2407 | Rule used to change the bo... |
| cbvexdva 2408 | Rule used to change the bo... |
| cbval2 2409 | Rule used to change bound ... |
| cbvex2 2410 | Rule used to change bound ... |
| cbval2vv 2411 | Rule used to change bound ... |
| cbvex2vv 2412 | Rule used to change bound ... |
| cbvex4v 2413 | Rule used to change bound ... |
| equs4 2414 | Lemma used in proofs of im... |
| equsal 2415 | An equivalence related to ... |
| equsex 2416 | An equivalence related to ... |
| equsexALT 2417 | Alternate proof of ~ equse... |
| equsalh 2418 | An equivalence related to ... |
| equsexh 2419 | An equivalence related to ... |
| axc15 2420 | Derivation of set.mm's ori... |
| ax12 2421 | Rederivation of Axiom ~ ax... |
| ax12b 2422 | A bidirectional version of... |
| ax13ALT 2423 | Alternate proof of ~ ax13 ... |
| axc11n 2424 | Derive set.mm's original ~... |
| aecom 2425 | Commutation law for identi... |
| aecoms 2426 | A commutation rule for ide... |
| naecoms 2427 | A commutation rule for dis... |
| axc11 2428 | Show that ~ ax-c11 can be ... |
| hbae 2429 | All variables are effectiv... |
| hbnae 2430 | All variables are effectiv... |
| nfae 2431 | All variables are effectiv... |
| nfnae 2432 | All variables are effectiv... |
| hbnaes 2433 | Rule that applies ~ hbnae ... |
| axc16i 2434 | Inference with ~ axc16 as ... |
| axc16nfALT 2435 | Alternate proof of ~ axc16... |
| dral2 2436 | Formula-building lemma for... |
| dral1 2437 | Formula-building lemma for... |
| dral1ALT 2438 | Alternate proof of ~ dral1... |
| drex1 2439 | Formula-building lemma for... |
| drex2 2440 | Formula-building lemma for... |
| drnf1 2441 | Formula-building lemma for... |
| drnf2 2442 | Formula-building lemma for... |
| nfald2 2443 | Variation on ~ nfald which... |
| nfexd2 2444 | Variation on ~ nfexd which... |
| exdistrf 2445 | Distribution of existentia... |
| dvelimf 2446 | Version of ~ dvelimv witho... |
| dvelimdf 2447 | Deduction form of ~ dvelim... |
| dvelimh 2448 | Version of ~ dvelim withou... |
| dvelim 2449 | This theorem can be used t... |
| dvelimv 2450 | Similar to ~ dvelim with f... |
| dvelimnf 2451 | Version of ~ dvelim using ... |
| dveeq2ALT 2452 | Alternate proof of ~ dveeq... |
| equvini 2453 | A variable introduction la... |
| equvel 2454 | A variable elimination law... |
| equs5a 2455 | A property related to subs... |
| equs5e 2456 | A property related to subs... |
| equs45f 2457 | Two ways of expressing sub... |
| equs5 2458 | Lemma used in proofs of su... |
| dveel1 2459 | Quantifier introduction wh... |
| dveel2 2460 | Quantifier introduction wh... |
| axc14 2461 | Axiom ~ ax-c14 is redundan... |
| sb6x 2462 | Equivalence involving subs... |
| sbequ5 2463 | Substitution does not chan... |
| sbequ6 2464 | Substitution does not chan... |
| sb5rf 2465 | Reversed substitution. Us... |
| sb6rf 2466 | Reversed substitution. Fo... |
| ax12vALT 2467 | Alternate proof of ~ ax12v... |
| 2ax6elem 2468 | We can always find values ... |
| 2ax6e 2469 | We can always find values ... |
| 2sb5rf 2470 | Reversed double substituti... |
| 2sb6rf 2471 | Reversed double substituti... |
| sbel2x 2472 | Elimination of double subs... |
| sb4b 2473 | Simplified definition of s... |
| sb3b 2474 | Simplified definition of s... |
| sb3 2475 | One direction of a simplif... |
| sb1 2476 | One direction of a simplif... |
| sb2 2477 | One direction of a simplif... |
| sb4a 2478 | A version of one implicati... |
| dfsb1 2479 | Alternate definition of su... |
| hbsb2 2480 | Bound-variable hypothesis ... |
| nfsb2 2481 | Bound-variable hypothesis ... |
| hbsb2a 2482 | Special case of a bound-va... |
| sb4e 2483 | One direction of a simplif... |
| hbsb2e 2484 | Special case of a bound-va... |
| hbsb3 2485 | If ` y ` is not free in ` ... |
| nfs1 2486 | If ` y ` is not free in ` ... |
| axc16ALT 2487 | Alternate proof of ~ axc16... |
| axc16gALT 2488 | Alternate proof of ~ axc16... |
| equsb1 2489 | Substitution applied to an... |
| equsb2 2490 | Substitution applied to an... |
| dfsb2 2491 | An alternate definition of... |
| dfsb3 2492 | An alternate definition of... |
| drsb1 2493 | Formula-building lemma for... |
| sb2ae 2494 | In the case of two success... |
| sb6f 2495 | Equivalence for substituti... |
| sb5f 2496 | Equivalence for substituti... |
| nfsb4t 2497 | A variable not free in a p... |
| nfsb4 2498 | A variable not free in a p... |
| sbequ8 2499 | Elimination of equality fr... |
| sbie 2500 | Conversion of implicit sub... |
| sbied 2501 | Conversion of implicit sub... |
| sbiedv 2502 | Conversion of implicit sub... |
| 2sbiev 2503 | Conversion of double impli... |
| sbcom3 2504 | Substituting ` y ` for ` x... |
| sbco 2505 | A composition law for subs... |
| sbid2 2506 | An identity law for substi... |
| sbid2v 2507 | An identity law for substi... |
| sbidm 2508 | An idempotent law for subs... |
| sbco2 2509 | A composition law for subs... |
| sbco2d 2510 | A composition law for subs... |
| sbco3 2511 | A composition law for subs... |
| sbcom 2512 | A commutativity law for su... |
| sbtrt 2513 | Partially closed form of ~... |
| sbtr 2514 | A partial converse to ~ sb... |
| sb8 2515 | Substitution of variable i... |
| sb8e 2516 | Substitution of variable i... |
| sb9 2517 | Commutation of quantificat... |
| sb9i 2518 | Commutation of quantificat... |
| sbhb 2519 | Two ways of expressing " `... |
| nfsbd 2520 | Deduction version of ~ nfs... |
| nfsb 2521 | If ` z ` is not free in ` ... |
| hbsb 2522 | If ` z ` is not free in ` ... |
| sb7f 2523 | This version of ~ dfsb7 do... |
| sb7h 2524 | This version of ~ dfsb7 do... |
| sb10f 2525 | Hao Wang's identity axiom ... |
| sbal1 2526 | Check out ~ sbal for a ver... |
| sbal2 2527 | Move quantifier in and out... |
| 2sb8e 2528 | An equivalent expression f... |
| dfmoeu 2529 | An elementary proof of ~ m... |
| dfeumo 2530 | An elementary proof showin... |
| mojust 2532 | Soundness justification th... |
| nexmo 2534 | Nonexistence implies uniqu... |
| exmo 2535 | Any proposition holds for ... |
| moabs 2536 | Absorption of existence co... |
| moim 2537 | The at-most-one quantifier... |
| moimi 2538 | The at-most-one quantifier... |
| moimdv 2539 | The at-most-one quantifier... |
| mobi 2540 | Equivalence theorem for th... |
| mobii 2541 | Formula-building rule for ... |
| mobidv 2542 | Formula-building rule for ... |
| mobid 2543 | Formula-building rule for ... |
| moa1 2544 | If an implication holds fo... |
| moan 2545 | "At most one" is still the... |
| moani 2546 | "At most one" is still tru... |
| moor 2547 | "At most one" is still the... |
| mooran1 2548 | "At most one" imports disj... |
| mooran2 2549 | "At most one" exports disj... |
| nfmo1 2550 | Bound-variable hypothesis ... |
| nfmod2 2551 | Bound-variable hypothesis ... |
| nfmodv 2552 | Bound-variable hypothesis ... |
| nfmov 2553 | Bound-variable hypothesis ... |
| nfmod 2554 | Bound-variable hypothesis ... |
| nfmo 2555 | Bound-variable hypothesis ... |
| mof 2556 | Version of ~ df-mo with di... |
| mo3 2557 | Alternate definition of th... |
| mo 2558 | Equivalent definitions of ... |
| mo4 2559 | At-most-one quantifier exp... |
| mo4f 2560 | At-most-one quantifier exp... |
| eu3v 2563 | An alternate way to expres... |
| eujust 2564 | Soundness justification th... |
| eujustALT 2565 | Alternate proof of ~ eujus... |
| eu6lem 2566 | Lemma of ~ eu6im . A diss... |
| eu6 2567 | Alternate definition of th... |
| eu6im 2568 | One direction of ~ eu6 nee... |
| euf 2569 | Version of ~ eu6 with disj... |
| euex 2570 | Existential uniqueness imp... |
| eumo 2571 | Existential uniqueness imp... |
| eumoi 2572 | Uniqueness inferred from e... |
| exmoeub 2573 | Existence implies that uni... |
| exmoeu 2574 | Existence is equivalent to... |
| moeuex 2575 | Uniqueness implies that ex... |
| moeu 2576 | Uniqueness is equivalent t... |
| eubi 2577 | Equivalence theorem for th... |
| eubii 2578 | Introduce unique existenti... |
| eubidv 2579 | Formula-building rule for ... |
| eubid 2580 | Formula-building rule for ... |
| nfeu1 2581 | Bound-variable hypothesis ... |
| nfeu1ALT 2582 | Alternate proof of ~ nfeu1... |
| nfeud2 2583 | Bound-variable hypothesis ... |
| nfeudw 2584 | Bound-variable hypothesis ... |
| nfeud 2585 | Bound-variable hypothesis ... |
| nfeuw 2586 | Bound-variable hypothesis ... |
| nfeu 2587 | Bound-variable hypothesis ... |
| dfeu 2588 | Rederive ~ df-eu from the ... |
| dfmo 2589 | Rederive ~ df-mo from the ... |
| euequ 2590 | There exists a unique set ... |
| sb8eulem 2591 | Lemma. Factor out the com... |
| sb8euv 2592 | Variable substitution in u... |
| sb8eu 2593 | Variable substitution in u... |
| sb8mo 2594 | Variable substitution for ... |
| cbvmovw 2595 | Change bound variable. Us... |
| cbvmow 2596 | Rule used to change bound ... |
| cbvmo 2597 | Rule used to change bound ... |
| cbveuvw 2598 | Change bound variable. Us... |
| cbveuw 2599 | Version of ~ cbveu with a ... |
| cbveu 2600 | Rule used to change bound ... |
| cbveuALT 2601 | Alternative proof of ~ cbv... |
| eu2 2602 | An alternate way of defini... |
| eu1 2603 | An alternate way to expres... |
| euor 2604 | Introduce a disjunct into ... |
| euorv 2605 | Introduce a disjunct into ... |
| euor2 2606 | Introduce or eliminate a d... |
| sbmo 2607 | Substitution into an at-mo... |
| eu4 2608 | Uniqueness using implicit ... |
| euimmo 2609 | Existential uniqueness imp... |
| euim 2610 | Add unique existential qua... |
| moanimlem 2611 | Factor out the common proo... |
| moanimv 2612 | Introduction of a conjunct... |
| moanim 2613 | Introduction of a conjunct... |
| euan 2614 | Introduction of a conjunct... |
| moanmo 2615 | Nested at-most-one quantif... |
| moaneu 2616 | Nested at-most-one and uni... |
| euanv 2617 | Introduction of a conjunct... |
| mopick 2618 | "At most one" picks a vari... |
| moexexlem 2619 | Factor out the proof skele... |
| 2moexv 2620 | Double quantification with... |
| moexexvw 2621 | "At most one" double quant... |
| 2moswapv 2622 | A condition allowing to sw... |
| 2euswapv 2623 | A condition allowing to sw... |
| 2euexv 2624 | Double quantification with... |
| 2exeuv 2625 | Double existential uniquen... |
| eupick 2626 | Existential uniqueness "pi... |
| eupicka 2627 | Version of ~ eupick with c... |
| eupickb 2628 | Existential uniqueness "pi... |
| eupickbi 2629 | Theorem *14.26 in [Whitehe... |
| mopick2 2630 | "At most one" can show the... |
| moexex 2631 | "At most one" double quant... |
| moexexv 2632 | "At most one" double quant... |
| 2moex 2633 | Double quantification with... |
| 2euex 2634 | Double quantification with... |
| 2eumo 2635 | Nested unique existential ... |
| 2eu2ex 2636 | Double existential uniquen... |
| 2moswap 2637 | A condition allowing to sw... |
| 2euswap 2638 | A condition allowing to sw... |
| 2exeu 2639 | Double existential uniquen... |
| 2mo2 2640 | Two ways of expressing "th... |
| 2mo 2641 | Two ways of expressing "th... |
| 2mos 2642 | Double "there exists at mo... |
| 2mosOLD 2643 | Obsolete version of ~ 2mos... |
| 2eu1 2644 | Double existential uniquen... |
| 2eu1v 2645 | Double existential uniquen... |
| 2eu2 2646 | Double existential uniquen... |
| 2eu3 2647 | Double existential uniquen... |
| 2eu4 2648 | This theorem provides us w... |
| 2eu5 2649 | An alternate definition of... |
| 2eu6 2650 | Two equivalent expressions... |
| 2eu7 2651 | Two equivalent expressions... |
| 2eu8 2652 | Two equivalent expressions... |
| euae 2653 | Two ways to express "exact... |
| exists1 2654 | Two ways to express "exact... |
| exists2 2655 | A condition implying that ... |
| barbara 2656 | "Barbara", one of the fund... |
| celarent 2657 | "Celarent", one of the syl... |
| darii 2658 | "Darii", one of the syllog... |
| dariiALT 2659 | Alternate proof of ~ darii... |
| ferio 2660 | "Ferio" ("Ferioque"), one ... |
| barbarilem 2661 | Lemma for ~ barbari and th... |
| barbari 2662 | "Barbari", one of the syll... |
| barbariALT 2663 | Alternate proof of ~ barba... |
| celaront 2664 | "Celaront", one of the syl... |
| cesare 2665 | "Cesare", one of the syllo... |
| camestres 2666 | "Camestres", one of the sy... |
| festino 2667 | "Festino", one of the syll... |
| festinoALT 2668 | Alternate proof of ~ festi... |
| baroco 2669 | "Baroco", one of the syllo... |
| barocoALT 2670 | Alternate proof of ~ festi... |
| cesaro 2671 | "Cesaro", one of the syllo... |
| camestros 2672 | "Camestros", one of the sy... |
| datisi 2673 | "Datisi", one of the syllo... |
| disamis 2674 | "Disamis", one of the syll... |
| ferison 2675 | "Ferison", one of the syll... |
| bocardo 2676 | "Bocardo", one of the syll... |
| darapti 2677 | "Darapti", one of the syll... |
| daraptiALT 2678 | Alternate proof of ~ darap... |
| felapton 2679 | "Felapton", one of the syl... |
| calemes 2680 | "Calemes", one of the syll... |
| dimatis 2681 | "Dimatis", one of the syll... |
| fresison 2682 | "Fresison", one of the syl... |
| calemos 2683 | "Calemos", one of the syll... |
| fesapo 2684 | "Fesapo", one of the syllo... |
| bamalip 2685 | "Bamalip", one of the syll... |
| axia1 2686 | Left 'and' elimination (in... |
| axia2 2687 | Right 'and' elimination (i... |
| axia3 2688 | 'And' introduction (intuit... |
| axin1 2689 | 'Not' introduction (intuit... |
| axin2 2690 | 'Not' elimination (intuiti... |
| axio 2691 | Definition of 'or' (intuit... |
| axi4 2692 | Specialization (intuitioni... |
| axi5r 2693 | Converse of ~ axc4 (intuit... |
| axial 2694 | The setvar ` x ` is not fr... |
| axie1 2695 | The setvar ` x ` is not fr... |
| axie2 2696 | A key property of existent... |
| axi9 2697 | Axiom of existence (intuit... |
| axi10 2698 | Axiom of Quantifier Substi... |
| axi12 2699 | Axiom of Quantifier Introd... |
| axbnd 2700 | Axiom of Bundling (intuiti... |
| axexte 2702 | The axiom of extensionalit... |
| axextg 2703 | A generalization of the ax... |
| axextb 2704 | A bidirectional version of... |
| axextmo 2705 | There exists at most one s... |
| nulmo 2706 | There exists at most one e... |
| eleq1ab 2709 | Extension (in the sense of... |
| cleljustab 2710 | Extension of ~ cleljust fr... |
| abid 2711 | Simplification of class ab... |
| vexwt 2712 | A standard theorem of pred... |
| vexw 2713 | If ` ph ` is a theorem, th... |
| vextru 2714 | Every setvar is a member o... |
| nfsab1 2715 | Bound-variable hypothesis ... |
| hbab1 2716 | Bound-variable hypothesis ... |
| hbab 2717 | Bound-variable hypothesis ... |
| hbabg 2718 | Bound-variable hypothesis ... |
| nfsab 2719 | Bound-variable hypothesis ... |
| nfsabg 2720 | Bound-variable hypothesis ... |
| dfcleq 2722 | The defining characterizat... |
| cvjust 2723 | Every set is a class. Pro... |
| ax9ALT 2724 | Proof of ~ ax-9 from Tarsk... |
| eleq2w2 2725 | A weaker version of ~ eleq... |
| eqriv 2726 | Infer equality of classes ... |
| eqrdv 2727 | Deduce equality of classes... |
| eqrdav 2728 | Deduce equality of classes... |
| eqid 2729 | Law of identity (reflexivi... |
| eqidd 2730 | Class identity law with an... |
| eqeq1d 2731 | Deduction from equality to... |
| eqeq1dALT 2732 | Alternate proof of ~ eqeq1... |
| eqeq1 2733 | Equality implies equivalen... |
| eqeq1i 2734 | Inference from equality to... |
| eqcomd 2735 | Deduction from commutative... |
| eqcom 2736 | Commutative law for class ... |
| eqcoms 2737 | Inference applying commuta... |
| eqcomi 2738 | Inference from commutative... |
| neqcomd 2739 | Commute an inequality. (C... |
| eqeq2d 2740 | Deduction from equality to... |
| eqeq2 2741 | Equality implies equivalen... |
| eqeq2i 2742 | Inference from equality to... |
| eqeqan12d 2743 | A useful inference for sub... |
| eqeqan12rd 2744 | A useful inference for sub... |
| eqeq12d 2745 | A useful inference for sub... |
| eqeq12 2746 | Equality relationship amon... |
| eqeq12i 2747 | A useful inference for sub... |
| eqeqan12dALT 2748 | Alternate proof of ~ eqeqa... |
| eqtr 2749 | Transitive law for class e... |
| eqtr2 2750 | A transitive law for class... |
| eqtr3 2751 | A transitive law for class... |
| eqtri 2752 | An equality transitivity i... |
| eqtr2i 2753 | An equality transitivity i... |
| eqtr3i 2754 | An equality transitivity i... |
| eqtr4i 2755 | An equality transitivity i... |
| 3eqtri 2756 | An inference from three ch... |
| 3eqtrri 2757 | An inference from three ch... |
| 3eqtr2i 2758 | An inference from three ch... |
| 3eqtr2ri 2759 | An inference from three ch... |
| 3eqtr3i 2760 | An inference from three ch... |
| 3eqtr3ri 2761 | An inference from three ch... |
| 3eqtr4i 2762 | An inference from three ch... |
| 3eqtr4ri 2763 | An inference from three ch... |
| eqtrd 2764 | An equality transitivity d... |
| eqtr2d 2765 | An equality transitivity d... |
| eqtr3d 2766 | An equality transitivity e... |
| eqtr4d 2767 | An equality transitivity e... |
| 3eqtrd 2768 | A deduction from three cha... |
| 3eqtrrd 2769 | A deduction from three cha... |
| 3eqtr2d 2770 | A deduction from three cha... |
| 3eqtr2rd 2771 | A deduction from three cha... |
| 3eqtr3d 2772 | A deduction from three cha... |
| 3eqtr3rd 2773 | A deduction from three cha... |
| 3eqtr4d 2774 | A deduction from three cha... |
| 3eqtr4rd 2775 | A deduction from three cha... |
| eqtrid 2776 | An equality transitivity d... |
| eqtr2id 2777 | An equality transitivity d... |
| eqtr3id 2778 | An equality transitivity d... |
| eqtr3di 2779 | An equality transitivity d... |
| eqtrdi 2780 | An equality transitivity d... |
| eqtr2di 2781 | An equality transitivity d... |
| eqtr4di 2782 | An equality transitivity d... |
| eqtr4id 2783 | An equality transitivity d... |
| sylan9eq 2784 | An equality transitivity d... |
| sylan9req 2785 | An equality transitivity d... |
| sylan9eqr 2786 | An equality transitivity d... |
| 3eqtr3g 2787 | A chained equality inferen... |
| 3eqtr3a 2788 | A chained equality inferen... |
| 3eqtr4g 2789 | A chained equality inferen... |
| 3eqtr4a 2790 | A chained equality inferen... |
| eq2tri 2791 | A compound transitive infe... |
| iseqsetvlem 2792 | Lemma for ~ iseqsetv-cleq ... |
| iseqsetv-cleq 2793 | Alternate proof of ~ iseqs... |
| abbi 2794 | Equivalent formulas yield ... |
| abbidv 2795 | Equivalent wff's yield equ... |
| abbii 2796 | Equivalent wff's yield equ... |
| abbid 2797 | Equivalent wff's yield equ... |
| abbib 2798 | Equal class abstractions r... |
| cbvabv 2799 | Rule used to change bound ... |
| cbvabw 2800 | Rule used to change bound ... |
| cbvab 2801 | Rule used to change bound ... |
| eqabbw 2802 | Version of ~ eqabb using i... |
| dfclel 2804 | Characterization of the el... |
| elex2 2805 | If a class contains anothe... |
| issettru 2806 | Weak version of ~ isset . ... |
| iseqsetv-clel 2807 | Alternate proof of ~ iseqs... |
| 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... |
| cleqh 2857 | Establish equality between... |
| hbxfreq 2858 | A utility lemma to transfe... |
| hblem 2859 | Change the free variable o... |
| hblemg 2860 | Change the free variable o... |
| eqabdv 2861 | Deduction from a wff to a ... |
| eqabcdv 2862 | Deduction from a wff to a ... |
| eqabi 2863 | Equality of a class variab... |
| abid1 2864 | Every class is equal to a ... |
| abid2 2865 | A simplification of class ... |
| eqab 2866 | One direction of ~ eqabb i... |
| eqabb 2867 | Equality of a class variab... |
| eqabbOLD 2868 | Obsolete version of ~ eqab... |
| eqabcb 2869 | Equality of a class variab... |
| eqabrd 2870 | Equality of a class variab... |
| eqabri 2871 | Equality of a class variab... |
| eqabcri 2872 | Equality of a class variab... |
| clelab 2873 | Membership of a class vari... |
| clabel 2874 | Membership of a class abst... |
| sbab 2875 | The right-hand side of the... |
| nfcjust 2877 | Justification theorem for ... |
| nfci 2879 | Deduce that a class ` A ` ... |
| nfcii 2880 | Deduce that a class ` A ` ... |
| nfcr 2881 | Consequence of the not-fre... |
| nfcrALT 2882 | Alternate version of ~ nfc... |
| nfcri 2883 | Consequence of the not-fre... |
| nfcd 2884 | Deduce that a class ` A ` ... |
| nfcrd 2885 | Consequence of the not-fre... |
| nfcrii 2886 | Consequence of the not-fre... |
| nfceqdf 2887 | An equality theorem for ef... |
| nfceqi 2888 | Equality theorem for class... |
| nfcxfr 2889 | A utility lemma to transfe... |
| nfcxfrd 2890 | A utility lemma to transfe... |
| nfcv 2891 | If ` x ` is disjoint from ... |
| nfcvd 2892 | If ` x ` is disjoint from ... |
| nfab1 2893 | Bound-variable hypothesis ... |
| nfnfc1 2894 | The setvar ` x ` is bound ... |
| clelsb1fw 2895 | Substitution for the first... |
| clelsb1f 2896 | Substitution for the first... |
| nfab 2897 | Bound-variable hypothesis ... |
| nfabg 2898 | Bound-variable hypothesis ... |
| nfaba1 2899 | Bound-variable hypothesis ... |
| nfaba1OLD 2900 | Obsolete version of ~ nfab... |
| nfaba1g 2901 | Bound-variable hypothesis ... |
| nfeqd 2902 | Hypothesis builder for equ... |
| nfeld 2903 | Hypothesis builder for ele... |
| nfnfc 2904 | Hypothesis builder for ` F... |
| nfeq 2905 | Hypothesis builder for equ... |
| nfel 2906 | Hypothesis builder for ele... |
| nfeq1 2907 | Hypothesis builder for equ... |
| nfel1 2908 | Hypothesis builder for ele... |
| nfeq2 2909 | Hypothesis builder for equ... |
| nfel2 2910 | Hypothesis builder for ele... |
| drnfc1 2911 | Formula-building lemma for... |
| drnfc2 2912 | Formula-building lemma for... |
| nfabdw 2913 | Bound-variable hypothesis ... |
| nfabd 2914 | Bound-variable hypothesis ... |
| nfabd2 2915 | Bound-variable hypothesis ... |
| dvelimdc 2916 | Deduction form of ~ dvelim... |
| dvelimc 2917 | Version of ~ dvelim for cl... |
| nfcvf 2918 | If ` x ` and ` y ` are dis... |
| nfcvf2 2919 | If ` x ` and ` y ` are dis... |
| cleqf 2920 | Establish equality between... |
| eqabf 2921 | Equality of a class variab... |
| abid2f 2922 | A simplification of class ... |
| abid2fOLD 2923 | Obsolete version of ~ abid... |
| sbabel 2924 | Theorem to move a substitu... |
| neii 2927 | Inference associated with ... |
| neir 2928 | Inference associated with ... |
| nne 2929 | Negation of inequality. (... |
| neneqd 2930 | Deduction eliminating ineq... |
| neneq 2931 | From inequality to non-equ... |
| neqned 2932 | If it is not the case that... |
| neqne 2933 | From non-equality to inequ... |
| neirr 2934 | No class is unequal to its... |
| exmidne 2935 | Excluded middle with equal... |
| eqneqall 2936 | A contradiction concerning... |
| nonconne 2937 | Law of noncontradiction wi... |
| necon3ad 2938 | Contrapositive law deducti... |
| necon3bd 2939 | Contrapositive law deducti... |
| necon2ad 2940 | Contrapositive inference f... |
| necon2bd 2941 | Contrapositive inference f... |
| necon1ad 2942 | Contrapositive deduction f... |
| necon1bd 2943 | Contrapositive deduction f... |
| necon4ad 2944 | Contrapositive inference f... |
| necon4bd 2945 | Contrapositive inference f... |
| necon3d 2946 | Contrapositive law deducti... |
| necon1d 2947 | Contrapositive law deducti... |
| necon2d 2948 | Contrapositive inference f... |
| necon4d 2949 | Contrapositive inference f... |
| necon3ai 2950 | Contrapositive inference f... |
| necon3bi 2951 | Contrapositive inference f... |
| necon1ai 2952 | Contrapositive inference f... |
| necon1bi 2953 | Contrapositive inference f... |
| necon2ai 2954 | Contrapositive inference f... |
| necon2bi 2955 | Contrapositive inference f... |
| necon4ai 2956 | Contrapositive inference f... |
| necon3i 2957 | Contrapositive inference f... |
| necon1i 2958 | Contrapositive inference f... |
| necon2i 2959 | Contrapositive inference f... |
| necon4i 2960 | Contrapositive inference f... |
| necon3abid 2961 | Deduction from equality to... |
| necon3bbid 2962 | Deduction from equality to... |
| necon1abid 2963 | Contrapositive deduction f... |
| necon1bbid 2964 | Contrapositive inference f... |
| necon4abid 2965 | Contrapositive law deducti... |
| necon4bbid 2966 | Contrapositive law deducti... |
| necon2abid 2967 | Contrapositive deduction f... |
| necon2bbid 2968 | Contrapositive deduction f... |
| necon3bid 2969 | Deduction from equality to... |
| necon4bid 2970 | Contrapositive law deducti... |
| necon3abii 2971 | Deduction from equality to... |
| necon3bbii 2972 | Deduction from equality to... |
| necon1abii 2973 | Contrapositive inference f... |
| necon1bbii 2974 | Contrapositive inference f... |
| necon2abii 2975 | Contrapositive inference f... |
| necon2bbii 2976 | Contrapositive inference f... |
| necon3bii 2977 | Inference from equality to... |
| necom 2978 | Commutation of inequality.... |
| necomi 2979 | Inference from commutative... |
| necomd 2980 | Deduction from commutative... |
| nesym 2981 | Characterization of inequa... |
| nesymi 2982 | Inference associated with ... |
| nesymir 2983 | Inference associated with ... |
| neeq1d 2984 | Deduction for inequality. ... |
| neeq2d 2985 | Deduction for inequality. ... |
| neeq12d 2986 | Deduction for inequality. ... |
| neeq1 2987 | Equality theorem for inequ... |
| neeq2 2988 | Equality theorem for inequ... |
| neeq1i 2989 | Inference for inequality. ... |
| neeq2i 2990 | Inference for inequality. ... |
| neeq12i 2991 | Inference for inequality. ... |
| eqnetrd 2992 | Substitution of equal clas... |
| eqnetrrd 2993 | Substitution of equal clas... |
| neeqtrd 2994 | Substitution of equal clas... |
| eqnetri 2995 | Substitution of equal clas... |
| eqnetrri 2996 | Substitution of equal clas... |
| neeqtri 2997 | Substitution of equal clas... |
| neeqtrri 2998 | Substitution of equal clas... |
| neeqtrrd 2999 | Substitution of equal clas... |
| eqnetrrid 3000 | A chained equality inferen... |
| 3netr3d 3001 | Substitution of equality i... |
| 3netr4d 3002 | Substitution of equality i... |
| 3netr3g 3003 | Substitution of equality i... |
| 3netr4g 3004 | Substitution of equality i... |
| nebi 3005 | Contraposition law for ine... |
| pm13.18 3006 | Theorem *13.18 in [Whitehe... |
| pm13.181 3007 | Theorem *13.181 in [Whiteh... |
| pm2.61ine 3008 | Inference eliminating an i... |
| pm2.21ddne 3009 | A contradiction implies an... |
| pm2.61ne 3010 | Deduction eliminating an i... |
| pm2.61dne 3011 | Deduction eliminating an i... |
| pm2.61dane 3012 | Deduction eliminating an i... |
| pm2.61da2ne 3013 | Deduction eliminating two ... |
| pm2.61da3ne 3014 | Deduction eliminating thre... |
| pm2.61iine 3015 | Equality version of ~ pm2.... |
| mteqand 3016 | A modus tollens deduction ... |
| neor 3017 | Logical OR with an equalit... |
| neanior 3018 | A De Morgan's law for ineq... |
| ne3anior 3019 | A De Morgan's law for ineq... |
| neorian 3020 | A De Morgan's law for ineq... |
| nemtbir 3021 | An inference from an inequ... |
| nelne1 3022 | Two classes are different ... |
| nelne2 3023 | Two classes are different ... |
| nelelne 3024 | Two classes are different ... |
| neneor 3025 | If two classes are differe... |
| nfne 3026 | Bound-variable hypothesis ... |
| nfned 3027 | Bound-variable hypothesis ... |
| nabbib 3028 | Not equivalent wff's corre... |
| neli 3031 | Inference associated with ... |
| nelir 3032 | Inference associated with ... |
| nelcon3d 3033 | Contrapositive law deducti... |
| neleq12d 3034 | Equality theorem for negat... |
| neleq1 3035 | Equality theorem for negat... |
| neleq2 3036 | Equality theorem for negat... |
| nfnel 3037 | Bound-variable hypothesis ... |
| nfneld 3038 | Bound-variable hypothesis ... |
| nnel 3039 | Negation of negated member... |
| elnelne1 3040 | Two classes are different ... |
| elnelne2 3041 | Two classes are different ... |
| pm2.24nel 3042 | A contradiction concerning... |
| pm2.61danel 3043 | Deduction eliminating an e... |
| rgen 3046 | Generalization rule for re... |
| ralel 3047 | All elements of a class ar... |
| rgenw 3048 | Generalization rule for re... |
| rgen2w 3049 | Generalization rule for re... |
| mprg 3050 | Modus ponens combined with... |
| mprgbir 3051 | Modus ponens on biconditio... |
| raln 3052 | Restricted universally qua... |
| ralnex 3055 | Relationship between restr... |
| dfrex2 3056 | Relationship between restr... |
| nrex 3057 | Inference adding restricte... |
| alral 3058 | Universal quantification i... |
| rexex 3059 | Restricted existence impli... |
| rextru 3060 | Two ways of expressing tha... |
| ralimi2 3061 | Inference quantifying both... |
| reximi2 3062 | Inference quantifying both... |
| ralimia 3063 | Inference quantifying both... |
| reximia 3064 | Inference quantifying both... |
| ralimiaa 3065 | Inference quantifying both... |
| ralimi 3066 | Inference quantifying both... |
| reximi 3067 | Inference quantifying both... |
| ral2imi 3068 | Inference quantifying ante... |
| ralim 3069 | Distribution of restricted... |
| rexim 3070 | Theorem 19.22 of [Margaris... |
| ralbii2 3071 | Inference adding different... |
| rexbii2 3072 | Inference adding different... |
| ralbiia 3073 | Inference adding restricte... |
| rexbiia 3074 | Inference adding restricte... |
| ralbii 3075 | Inference adding restricte... |
| rexbii 3076 | Inference adding restricte... |
| ralanid 3077 | Cancellation law for restr... |
| rexanid 3078 | Cancellation law for restr... |
| ralcom3 3079 | A commutation law for rest... |
| ralcom3OLD 3080 | Obsolete version of ~ ralc... |
| dfral2 3081 | Relationship between restr... |
| rexnal 3082 | Relationship between restr... |
| ralinexa 3083 | A transformation of restri... |
| rexanali 3084 | A transformation of restri... |
| ralbi 3085 | Distribute a restricted un... |
| rexbi 3086 | Distribute restricted quan... |
| ralrexbid 3087 | Formula-building rule for ... |
| r19.35 3088 | Restricted quantifier vers... |
| r19.35OLD 3089 | Obsolete version of ~ 19.3... |
| r19.26m 3090 | Version of ~ 19.26 and ~ r... |
| r19.26 3091 | Restricted quantifier vers... |
| r19.26-3 3092 | Version of ~ r19.26 with t... |
| ralbiim 3093 | Split a biconditional and ... |
| r19.29 3094 | Restricted quantifier vers... |
| r19.29OLD 3095 | Obsolete version of ~ r19.... |
| r19.29r 3096 | Restricted quantifier vers... |
| r19.29rOLD 3097 | Obsolete version of ~ r19.... |
| r19.29imd 3098 | Theorem 19.29 of [Margaris... |
| r19.40 3099 | Restricted quantifier vers... |
| r19.30 3100 | Restricted quantifier vers... |
| r19.43 3101 | Restricted quantifier vers... |
| 3r19.43 3102 | Restricted quantifier vers... |
| 2ralimi 3103 | Inference quantifying both... |
| 3ralimi 3104 | Inference quantifying both... |
| 4ralimi 3105 | Inference quantifying both... |
| 5ralimi 3106 | Inference quantifying both... |
| 6ralimi 3107 | Inference quantifying both... |
| 2ralbii 3108 | Inference adding two restr... |
| 2rexbii 3109 | Inference adding two restr... |
| 3ralbii 3110 | Inference adding three res... |
| 4ralbii 3111 | Inference adding four rest... |
| 2ralbiim 3112 | Split a biconditional and ... |
| ralnex2 3113 | Relationship between two r... |
| ralnex3 3114 | Relationship between three... |
| rexnal2 3115 | Relationship between two r... |
| rexnal3 3116 | Relationship between three... |
| nrexralim 3117 | Negation of a complex pred... |
| r19.26-2 3118 | Restricted quantifier vers... |
| 2r19.29 3119 | Theorem ~ r19.29 with two ... |
| r19.29d2r 3120 | Theorem 19.29 of [Margaris... |
| r2allem 3121 | Lemma factoring out common... |
| r2exlem 3122 | Lemma factoring out common... |
| hbralrimi 3123 | Inference from Theorem 19.... |
| ralrimiv 3124 | Inference from Theorem 19.... |
| ralrimiva 3125 | Inference from Theorem 19.... |
| rexlimiva 3126 | Inference from Theorem 19.... |
| rexlimiv 3127 | Inference from Theorem 19.... |
| nrexdv 3128 | Deduction adding restricte... |
| ralrimivw 3129 | Inference from Theorem 19.... |
| rexlimivw 3130 | Weaker version of ~ rexlim... |
| ralrimdv 3131 | Inference from Theorem 19.... |
| rexlimdv 3132 | Inference from Theorem 19.... |
| ralrimdva 3133 | Inference from Theorem 19.... |
| rexlimdva 3134 | Inference from Theorem 19.... |
| rexlimdvaa 3135 | Inference from Theorem 19.... |
| rexlimdva2 3136 | Inference from Theorem 19.... |
| r19.29an 3137 | A commonly used pattern in... |
| rexlimdv3a 3138 | Inference from Theorem 19.... |
| rexlimdvw 3139 | Inference from Theorem 19.... |
| rexlimddv 3140 | Restricted existential eli... |
| r19.29a 3141 | A commonly used pattern in... |
| ralimdv2 3142 | Inference quantifying both... |
| reximdv2 3143 | Deduction quantifying both... |
| reximdvai 3144 | Deduction quantifying both... |
| ralimdva 3145 | Deduction quantifying both... |
| reximdva 3146 | Deduction quantifying both... |
| ralimdv 3147 | Deduction quantifying both... |
| reximdv 3148 | Deduction from Theorem 19.... |
| reximddv 3149 | Deduction from Theorem 19.... |
| reximddv3 3150 | Deduction from Theorem 19.... |
| reximssdv 3151 | Derivation of a restricted... |
| ralbidv2 3152 | Formula-building rule for ... |
| rexbidv2 3153 | Formula-building rule for ... |
| ralbidva 3154 | Formula-building rule for ... |
| rexbidva 3155 | Formula-building rule for ... |
| ralbidv 3156 | Formula-building rule for ... |
| rexbidv 3157 | Formula-building rule for ... |
| r19.21v 3158 | Restricted quantifier vers... |
| r19.21vOLD 3159 | Obsolete version of ~ r19.... |
| r19.37v 3160 | Restricted quantifier vers... |
| r19.23v 3161 | Restricted quantifier vers... |
| r19.36v 3162 | Restricted quantifier vers... |
| rexlimivOLD 3163 | Obsolete version of ~ rexl... |
| rexlimivaOLD 3164 | Obsolete version of ~ rexl... |
| rexlimivwOLD 3165 | Obsolete version of ~ rexl... |
| r19.27v 3166 | Restricted quantitifer ver... |
| r19.41v 3167 | Restricted quantifier vers... |
| r19.28v 3168 | Restricted quantifier vers... |
| r19.42v 3169 | Restricted quantifier vers... |
| r19.32v 3170 | Restricted quantifier vers... |
| r19.45v 3171 | Restricted quantifier vers... |
| r19.44v 3172 | One direction of a restric... |
| r2al 3173 | Double restricted universa... |
| r2ex 3174 | Double restricted existent... |
| r3al 3175 | Triple restricted universa... |
| r3ex 3176 | Triple existential quantif... |
| rgen2 3177 | Generalization rule for re... |
| ralrimivv 3178 | Inference from Theorem 19.... |
| rexlimivv 3179 | Inference from Theorem 19.... |
| ralrimivva 3180 | Inference from Theorem 19.... |
| ralrimdvv 3181 | Inference from Theorem 19.... |
| rgen3 3182 | Generalization rule for re... |
| ralrimivvva 3183 | Inference from Theorem 19.... |
| ralimdvva 3184 | Deduction doubly quantifyi... |
| reximdvva 3185 | Deduction doubly quantifyi... |
| ralimdvv 3186 | Deduction doubly quantifyi... |
| ralimdvvOLD 3187 | Obsolete version of ~ rali... |
| ralimd4v 3188 | Deduction quadrupally quan... |
| ralimd4vOLD 3189 | Obsolete version of ~ rali... |
| ralimd6v 3190 | Deduction sextupally quant... |
| ralimd6vOLD 3191 | Obsolete version of ~ rali... |
| ralrimdvva 3192 | Inference from Theorem 19.... |
| rexlimdvv 3193 | Inference from Theorem 19.... |
| rexlimdvva 3194 | Inference from Theorem 19.... |
| rexlimdvvva 3195 | Inference from Theorem 19.... |
| reximddv2 3196 | Double deduction from Theo... |
| r19.29vva 3197 | A commonly used pattern ba... |
| 2rexbiia 3198 | Inference adding two restr... |
| 2ralbidva 3199 | Formula-building rule for ... |
| 2rexbidva 3200 | Formula-building rule for ... |
| 2ralbidv 3201 | Formula-building rule for ... |
| 2rexbidv 3202 | Formula-building rule for ... |
| rexralbidv 3203 | Formula-building rule for ... |
| 3ralbidv 3204 | Formula-building rule for ... |
| 4ralbidv 3205 | Formula-building rule for ... |
| 6ralbidv 3206 | Formula-building rule for ... |
| r19.41vv 3207 | Version of ~ r19.41v with ... |
| reeanlem 3208 | Lemma factoring out common... |
| reeanv 3209 | Rearrange restricted exist... |
| 3reeanv 3210 | Rearrange three restricted... |
| 2ralor 3211 | Distribute restricted univ... |
| risset 3212 | Two ways to say " ` A ` be... |
| nelb 3213 | A definition of ` -. A e. ... |
| rspw 3214 | Restricted specialization.... |
| cbvralvw 3215 | Change the bound variable ... |
| cbvrexvw 3216 | Change the bound variable ... |
| cbvraldva 3217 | Rule used to change the bo... |
| cbvrexdva 3218 | Rule used to change the bo... |
| cbvral2vw 3219 | Change bound variables of ... |
| cbvrex2vw 3220 | Change bound variables of ... |
| cbvral3vw 3221 | Change bound variables of ... |
| cbvral4vw 3222 | Change bound variables of ... |
| cbvral6vw 3223 | Change bound variables of ... |
| cbvral8vw 3224 | Change bound variables of ... |
| rsp 3225 | Restricted specialization.... |
| rspa 3226 | Restricted specialization.... |
| rspe 3227 | Restricted specialization.... |
| rspec 3228 | Specialization rule for re... |
| r19.21bi 3229 | Inference from Theorem 19.... |
| r19.21be 3230 | Inference from Theorem 19.... |
| r19.21t 3231 | Restricted quantifier vers... |
| r19.21 3232 | Restricted quantifier vers... |
| r19.23t 3233 | Closed theorem form of ~ r... |
| r19.23 3234 | Restricted quantifier vers... |
| ralrimi 3235 | Inference from Theorem 19.... |
| ralrimia 3236 | Inference from Theorem 19.... |
| rexlimi 3237 | Restricted quantifier vers... |
| ralimdaa 3238 | Deduction quantifying both... |
| reximdai 3239 | Deduction from Theorem 19.... |
| r19.37 3240 | Restricted quantifier vers... |
| r19.41 3241 | Restricted quantifier vers... |
| ralrimd 3242 | Inference from Theorem 19.... |
| rexlimd2 3243 | Version of ~ rexlimd with ... |
| rexlimd 3244 | Deduction form of ~ rexlim... |
| r19.29af2 3245 | A commonly used pattern ba... |
| r19.29af 3246 | A commonly used pattern ba... |
| reximd2a 3247 | Deduction quantifying both... |
| ralbida 3248 | Formula-building rule for ... |
| rexbida 3249 | Formula-building rule for ... |
| ralbid 3250 | Formula-building rule for ... |
| rexbid 3251 | Formula-building rule for ... |
| rexbidvALT 3252 | Alternate proof of ~ rexbi... |
| rexbidvaALT 3253 | Alternate proof of ~ rexbi... |
| rsp2 3254 | Restricted specialization,... |
| rsp2e 3255 | Restricted specialization.... |
| rspec2 3256 | Specialization rule for re... |
| rspec3 3257 | Specialization rule for re... |
| r2alf 3258 | Double restricted universa... |
| r2exf 3259 | Double restricted existent... |
| 2ralbida 3260 | Formula-building rule for ... |
| nfra1 3261 | The setvar ` x ` is not fr... |
| nfre1 3262 | The setvar ` x ` is not fr... |
| ralcom4 3263 | Commutation of restricted ... |
| rexcom4 3264 | Commutation of restricted ... |
| ralcom 3265 | Commutation of restricted ... |
| rexcom 3266 | Commutation of restricted ... |
| rexcom4a 3267 | Specialized existential co... |
| ralrot3 3268 | Rotate three restricted un... |
| ralcom13 3269 | Swap first and third restr... |
| ralcom13OLD 3270 | Obsolete version of ~ ralc... |
| rexcom13 3271 | Swap first and third restr... |
| rexrot4 3272 | Rotate four restricted exi... |
| 2ex2rexrot 3273 | Rotate two existential qua... |
| nfra2w 3274 | Similar to Lemma 24 of [Mo... |
| hbra1 3275 | The setvar ` x ` is not fr... |
| ralcomf 3276 | Commutation of restricted ... |
| rexcomf 3277 | Commutation of restricted ... |
| cbvralfw 3278 | Rule used to change bound ... |
| cbvrexfw 3279 | Rule used to change bound ... |
| cbvralw 3280 | Rule used to change bound ... |
| cbvrexw 3281 | Rule used to change bound ... |
| hbral 3282 | Bound-variable hypothesis ... |
| nfraldw 3283 | Deduction version of ~ nfr... |
| nfrexdw 3284 | Deduction version of ~ nfr... |
| nfralw 3285 | Bound-variable hypothesis ... |
| nfralwOLD 3286 | Obsolete version of ~ nfra... |
| nfrexw 3287 | Bound-variable hypothesis ... |
| r19.12 3288 | Restricted quantifier vers... |
| reean 3289 | Rearrange restricted exist... |
| cbvralsvw 3290 | Change bound variable by u... |
| cbvrexsvw 3291 | Change bound variable by u... |
| cbvralsvwOLD 3292 | Obsolete version of ~ cbvr... |
| cbvralsvwOLDOLD 3293 | Obsolete version of ~ cbvr... |
| cbvrexsvwOLD 3294 | Obsolete version of ~ cbvr... |
| rexeq 3295 | Equality theorem for restr... |
| raleq 3296 | Equality theorem for restr... |
| raleqi 3297 | Equality inference for res... |
| rexeqi 3298 | Equality inference for res... |
| raleqdv 3299 | Equality deduction for res... |
| rexeqdv 3300 | Equality deduction for res... |
| raleqtrdv 3301 | Substitution of equal clas... |
| rexeqtrdv 3302 | Substitution of equal clas... |
| raleqtrrdv 3303 | Substitution of equal clas... |
| rexeqtrrdv 3304 | Substitution of equal clas... |
| raleqbidva 3305 | Equality deduction for res... |
| rexeqbidva 3306 | Equality deduction for res... |
| raleqbidvv 3307 | Version of ~ raleqbidv wit... |
| raleqbidvvOLD 3308 | Obsolete version of ~ rale... |
| rexeqbidvv 3309 | Version of ~ rexeqbidv wit... |
| rexeqbidvvOLD 3310 | Obsolete version of ~ rexe... |
| raleqbi1dv 3311 | Equality deduction for res... |
| rexeqbi1dv 3312 | Equality deduction for res... |
| raleqOLD 3313 | Obsolete version of ~ rale... |
| rexeqOLD 3314 | Obsolete version of ~ rale... |
| raleleq 3315 | All elements of a class ar... |
| raleleqOLD 3316 | Obsolete version of ~ rale... |
| raleqbii 3317 | Equality deduction for res... |
| rexeqbii 3318 | Equality deduction for res... |
| raleqbidv 3319 | Equality deduction for res... |
| rexeqbidv 3320 | Equality deduction for res... |
| cbvraldva2 3321 | Rule used to change the bo... |
| cbvrexdva2 3322 | Rule used to change the bo... |
| cbvrexdva2OLD 3323 | Obsolete version of ~ cbvr... |
| cbvraldvaOLD 3324 | Obsolete version of ~ cbvr... |
| cbvrexdvaOLD 3325 | Obsolete version of ~ cbvr... |
| sbralie 3326 | Implicit to explicit subst... |
| sbralieALT 3327 | Alternative shorter proof ... |
| sbralieOLD 3328 | Obsolete version of ~ sbra... |
| raleqf 3329 | Equality theorem for restr... |
| rexeqf 3330 | Equality theorem for restr... |
| rexeqfOLD 3331 | Obsolete version of ~ rexe... |
| raleqbid 3332 | Equality deduction for res... |
| rexeqbid 3333 | Equality deduction for res... |
| cbvralf 3334 | Rule used to change bound ... |
| cbvrexf 3335 | Rule used to change bound ... |
| cbvral 3336 | Rule used to change bound ... |
| cbvrex 3337 | Rule used to change bound ... |
| cbvralv 3338 | Change the bound variable ... |
| cbvrexv 3339 | Change the bound variable ... |
| cbvralsv 3340 | Change bound variable by u... |
| cbvrexsv 3341 | Change bound variable by u... |
| cbvral2v 3342 | Change bound variables of ... |
| cbvrex2v 3343 | Change bound variables of ... |
| cbvral3v 3344 | Change bound variables of ... |
| rgen2a 3345 | Generalization rule for re... |
| nfrald 3346 | Deduction version of ~ nfr... |
| nfrexd 3347 | Deduction version of ~ nfr... |
| nfral 3348 | Bound-variable hypothesis ... |
| nfrex 3349 | Bound-variable hypothesis ... |
| nfra2 3350 | Similar to Lemma 24 of [Mo... |
| ralcom2 3351 | Commutation of restricted ... |
| reu5 3356 | Restricted uniqueness in t... |
| reurmo 3357 | Restricted existential uni... |
| reurex 3358 | Restricted unique existenc... |
| mormo 3359 | Unrestricted "at most one"... |
| rmobiia 3360 | Formula-building rule for ... |
| reubiia 3361 | Formula-building rule for ... |
| rmobii 3362 | Formula-building rule for ... |
| reubii 3363 | Formula-building rule for ... |
| rmoanid 3364 | Cancellation law for restr... |
| reuanid 3365 | Cancellation law for restr... |
| rmoanidOLD 3366 | Obsolete version of ~ rmoa... |
| reuanidOLD 3367 | Obsolete version of ~ reua... |
| 2reu2rex 3368 | Double restricted existent... |
| rmobidva 3369 | Formula-building rule for ... |
| reubidva 3370 | Formula-building rule for ... |
| rmobidv 3371 | Formula-building rule for ... |
| reubidv 3372 | Formula-building rule for ... |
| reueubd 3373 | Restricted existential uni... |
| rmo5 3374 | Restricted "at most one" i... |
| nrexrmo 3375 | Nonexistence implies restr... |
| moel 3376 | "At most one" element in a... |
| cbvrmovw 3377 | Change the bound variable ... |
| cbvreuvw 3378 | Change the bound variable ... |
| rmobida 3379 | Formula-building rule for ... |
| reubida 3380 | Formula-building rule for ... |
| cbvrmow 3381 | Change the bound variable ... |
| cbvreuw 3382 | Change the bound variable ... |
| nfrmo1 3383 | The setvar ` x ` is not fr... |
| nfreu1 3384 | The setvar ` x ` is not fr... |
| nfrmow 3385 | Bound-variable hypothesis ... |
| nfreuw 3386 | Bound-variable hypothesis ... |
| rmoeq1 3387 | Equality theorem for restr... |
| reueq1 3388 | Equality theorem for restr... |
| rmoeq1OLD 3389 | Obsolete version of ~ rmoe... |
| reueq1OLD 3390 | Obsolete version of ~ reue... |
| rmoeqd 3391 | Equality deduction for res... |
| reueqd 3392 | Equality deduction for res... |
| reueqdv 3393 | Formula-building rule for ... |
| reueqbidv 3394 | Formula-building rule for ... |
| rmoeq1f 3395 | Equality theorem for restr... |
| reueq1f 3396 | Equality theorem for restr... |
| cbvreu 3397 | Change the bound variable ... |
| cbvrmo 3398 | Change the bound variable ... |
| cbvrmov 3399 | Change the bound variable ... |
| cbvreuv 3400 | Change the bound variable ... |
| nfrmod 3401 | Deduction version of ~ nfr... |
| nfreud 3402 | Deduction version of ~ nfr... |
| nfrmo 3403 | Bound-variable hypothesis ... |
| nfreu 3404 | Bound-variable hypothesis ... |
| rabbidva2 3407 | Equivalent wff's yield equ... |
| rabbia2 3408 | Equivalent wff's yield equ... |
| rabbiia 3409 | Equivalent formulas yield ... |
| rabbiiaOLD 3410 | Obsolete version of ~ rabb... |
| rabbii 3411 | Equivalent wff's correspon... |
| rabbidva 3412 | Equivalent wff's yield equ... |
| rabbidv 3413 | Equivalent wff's yield equ... |
| rabbieq 3414 | Equivalent wff's correspon... |
| rabswap 3415 | Swap with a membership rel... |
| cbvrabv 3416 | Rule to change the bound v... |
| rabeqcda 3417 | When ` ps ` is always true... |
| rabeqc 3418 | A restricted class abstrac... |
| rabeqi 3419 | Equality theorem for restr... |
| rabeq 3420 | Equality theorem for restr... |
| rabeqdv 3421 | Equality of restricted cla... |
| rabeqbidva 3422 | Equality of restricted cla... |
| rabeqbidvaOLD 3423 | Obsolete version of ~ rabe... |
| rabeqbidv 3424 | Equality of restricted cla... |
| rabrabi 3425 | Abstract builder restricte... |
| nfrab1 3426 | The abstraction variable i... |
| rabid 3427 | An "identity" law of concr... |
| rabidim1 3428 | Membership in a restricted... |
| reqabi 3429 | Inference from equality of... |
| rabrab 3430 | Abstract builder restricte... |
| rabbida4 3431 | Version of ~ rabbidva2 wit... |
| rabbida 3432 | Equivalent wff's yield equ... |
| rabbid 3433 | Version of ~ rabbidv with ... |
| rabeqd 3434 | Deduction form of ~ rabeq ... |
| rabeqbida 3435 | Version of ~ rabeqbidva wi... |
| rabbi 3436 | Equivalent wff's correspon... |
| rabid2f 3437 | An "identity" law for rest... |
| rabid2im 3438 | One direction of ~ rabid2 ... |
| rabid2 3439 | An "identity" law for rest... |
| rabeqf 3440 | Equality theorem for restr... |
| cbvrabw 3441 | Rule to change the bound v... |
| cbvrabwOLD 3442 | Obsolete version of ~ cbvr... |
| nfrabw 3443 | A variable not free in a w... |
| rabbidaOLD 3444 | Obsolete version of ~ rabb... |
| nfrab 3445 | A variable not free in a w... |
| cbvrab 3446 | Rule to change the bound v... |
| vjust 3448 | Justification theorem for ... |
| dfv2 3450 | Alternate definition of th... |
| vex 3451 | All setvar variables are s... |
| elv 3452 | If a proposition is implie... |
| elvd 3453 | If a proposition is implie... |
| el2v 3454 | If a proposition is implie... |
| el3v 3455 | If a proposition is implie... |
| el3v3 3456 | If a proposition is implie... |
| eqv 3457 | The universe contains ever... |
| eqvf 3458 | The universe contains ever... |
| abv 3459 | The class of sets verifyin... |
| abvALT 3460 | Alternate proof of ~ abv ,... |
| isset 3461 | Two ways to express that "... |
| cbvexeqsetf 3462 | The expression ` E. x x = ... |
| issetft 3463 | Closed theorem form of ~ i... |
| issetf 3464 | A version of ~ isset that ... |
| isseti 3465 | A way to say " ` A ` is a ... |
| issetri 3466 | A way to say " ` A ` is a ... |
| eqvisset 3467 | A class equal to a variabl... |
| elex 3468 | If a class is a member of ... |
| elexOLD 3469 | Obsolete version of ~ elex... |
| elexi 3470 | If a class is a member of ... |
| elexd 3471 | If a class is a member of ... |
| elex22 3472 | If two classes each contai... |
| prcnel 3473 | A proper class doesn't bel... |
| ralv 3474 | A universal quantifier res... |
| rexv 3475 | An existential quantifier ... |
| reuv 3476 | A unique existential quant... |
| rmov 3477 | An at-most-one quantifier ... |
| rabab 3478 | A class abstraction restri... |
| rexcom4b 3479 | Specialized existential co... |
| ceqsal1t 3480 | One direction of ~ ceqsalt... |
| ceqsalt 3481 | Closed theorem version of ... |
| ceqsralt 3482 | Restricted quantifier vers... |
| ceqsalg 3483 | A representation of explic... |
| ceqsalgALT 3484 | Alternate proof of ~ ceqsa... |
| ceqsal 3485 | A representation of explic... |
| ceqsalALT 3486 | A representation of explic... |
| ceqsalv 3487 | A representation of explic... |
| ceqsralv 3488 | Restricted quantifier vers... |
| gencl 3489 | Implicit substitution for ... |
| 2gencl 3490 | Implicit substitution for ... |
| 3gencl 3491 | Implicit substitution for ... |
| cgsexg 3492 | Implicit substitution infe... |
| cgsex2g 3493 | Implicit substitution infe... |
| cgsex4g 3494 | An implicit substitution i... |
| cgsex4gOLD 3495 | Obsolete version of ~ cgse... |
| ceqsex 3496 | Elimination of an existent... |
| ceqsexOLD 3497 | Obsolete version of ~ ceqs... |
| ceqsexv 3498 | Elimination of an existent... |
| ceqsexv2d 3499 | Elimination of an existent... |
| ceqsexv2dOLD 3500 | Obsolete version of ~ ceqs... |
| ceqsex2 3501 | Elimination of two existen... |
| ceqsex2v 3502 | Elimination of two existen... |
| ceqsex3v 3503 | Elimination of three exist... |
| ceqsex4v 3504 | Elimination of four existe... |
| ceqsex6v 3505 | Elimination of six existen... |
| ceqsex8v 3506 | Elimination of eight exist... |
| gencbvex 3507 | Change of bound variable u... |
| gencbvex2 3508 | Restatement of ~ gencbvex ... |
| gencbval 3509 | Change of bound variable u... |
| sbhypf 3510 | Introduce an explicit subs... |
| sbhypfOLD 3511 | Obsolete version of ~ sbhy... |
| spcimgft 3512 | Closed theorem form of ~ s... |
| spcimgfi1 3513 | A closed version of ~ spci... |
| spcimgfi1OLD 3514 | Obsolete version of ~ spci... |
| spcgft 3515 | A closed version of ~ spcg... |
| spcimgf 3516 | Rule of specialization, us... |
| spcimegf 3517 | Existential specialization... |
| vtoclgft 3518 | Closed theorem form of ~ v... |
| vtocleg 3519 | Implicit substitution of a... |
| vtoclg 3520 | Implicit substitution of a... |
| vtocle 3521 | Implicit substitution of a... |
| vtocleOLD 3522 | Obsolete version of ~ vtoc... |
| vtoclbg 3523 | Implicit substitution of a... |
| vtocl 3524 | Implicit substitution of a... |
| vtoclOLD 3525 | Obsolete version of ~ vtoc... |
| vtocldf 3526 | Implicit substitution of a... |
| vtocld 3527 | Implicit substitution of a... |
| vtocl2d 3528 | Implicit substitution of t... |
| vtoclef 3529 | Implicit substitution of a... |
| vtoclf 3530 | Implicit substitution of a... |
| vtoclfOLD 3531 | Obsolete version of ~ vtoc... |
| vtocl2 3532 | Implicit substitution of c... |
| vtocl3 3533 | Implicit substitution of c... |
| vtoclb 3534 | Implicit substitution of a... |
| vtoclgf 3535 | Implicit substitution of a... |
| vtoclg1f 3536 | Version of ~ vtoclgf with ... |
| vtoclgOLD 3537 | Obsolete version of ~ vtoc... |
| vtocl2gf 3538 | Implicit substitution of a... |
| vtocl3gf 3539 | Implicit substitution of a... |
| vtocl2g 3540 | Implicit substitution of 2... |
| vtocl3g 3541 | Implicit substitution of a... |
| vtoclgaf 3542 | Implicit substitution of a... |
| vtoclga 3543 | Implicit substitution of a... |
| vtocl2ga 3544 | Implicit substitution of 2... |
| vtocl2gaf 3545 | Implicit substitution of 2... |
| vtocl2gafOLD 3546 | Obsolete version of ~ vtoc... |
| vtocl3gaf 3547 | Implicit substitution of 3... |
| vtocl3gafOLD 3548 | Obsolete version of ~ vtoc... |
| vtocl3ga 3549 | Implicit substitution of 3... |
| vtocl3gaOLD 3550 | Obsolete version of ~ vtoc... |
| vtocl4g 3551 | Implicit substitution of 4... |
| vtocl4ga 3552 | Implicit substitution of 4... |
| vtocl4gaOLD 3553 | Obsolete version of ~ vtoc... |
| vtoclegft 3554 | Implicit substitution of a... |
| vtoclegftOLD 3555 | Obsolete version of ~ vtoc... |
| vtoclri 3556 | Implicit substitution of a... |
| spcgf 3557 | Rule of specialization, us... |
| spcegf 3558 | Existential specialization... |
| spcimdv 3559 | Restricted specialization,... |
| spcdv 3560 | Rule of specialization, us... |
| spcimedv 3561 | Restricted existential spe... |
| spcgv 3562 | Rule of specialization, us... |
| spcegv 3563 | Existential specialization... |
| spcedv 3564 | Existential specialization... |
| spc2egv 3565 | Existential specialization... |
| spc2gv 3566 | Specialization with two qu... |
| spc2ed 3567 | Existential specialization... |
| spc2d 3568 | Specialization with 2 quan... |
| spc3egv 3569 | Existential specialization... |
| spc3gv 3570 | Specialization with three ... |
| spcv 3571 | Rule of specialization, us... |
| spcev 3572 | Existential specialization... |
| spc2ev 3573 | Existential specialization... |
| rspct 3574 | A closed version of ~ rspc... |
| rspcdf 3575 | Restricted specialization,... |
| rspc 3576 | Restricted specialization,... |
| rspce 3577 | Restricted existential spe... |
| rspcimdv 3578 | Restricted specialization,... |
| rspcimedv 3579 | Restricted existential spe... |
| rspcdv 3580 | Restricted specialization,... |
| rspcedv 3581 | Restricted existential spe... |
| rspcebdv 3582 | Restricted existential spe... |
| rspcdv2 3583 | Restricted specialization,... |
| rspcv 3584 | Restricted specialization,... |
| rspccv 3585 | Restricted specialization,... |
| rspcva 3586 | Restricted specialization,... |
| rspccva 3587 | Restricted specialization,... |
| rspcev 3588 | Restricted existential spe... |
| rspcdva 3589 | Restricted specialization,... |
| rspcedvd 3590 | Restricted existential spe... |
| rspcedvdw 3591 | Version of ~ rspcedvd wher... |
| rspceb2dv 3592 | Restricted existential spe... |
| rspcime 3593 | Prove a restricted existen... |
| rspceaimv 3594 | Restricted existential spe... |
| rspcedeq1vd 3595 | Restricted existential spe... |
| rspcedeq2vd 3596 | Restricted existential spe... |
| rspc2 3597 | Restricted specialization ... |
| rspc2gv 3598 | Restricted specialization ... |
| rspc2v 3599 | 2-variable restricted spec... |
| rspc2va 3600 | 2-variable restricted spec... |
| rspc2ev 3601 | 2-variable restricted exis... |
| 2rspcedvdw 3602 | Double application of ~ rs... |
| rspc2dv 3603 | 2-variable restricted spec... |
| rspc3v 3604 | 3-variable restricted spec... |
| rspc3ev 3605 | 3-variable restricted exis... |
| 3rspcedvdw 3606 | Triple application of ~ rs... |
| rspc3dv 3607 | 3-variable restricted spec... |
| rspc4v 3608 | 4-variable restricted spec... |
| rspc6v 3609 | 6-variable restricted spec... |
| rspc8v 3610 | 8-variable restricted spec... |
| rspceeqv 3611 | Restricted existential spe... |
| ralxpxfr2d 3612 | Transfer a universal quant... |
| rexraleqim 3613 | Statement following from e... |
| eqvincg 3614 | A variable introduction la... |
| eqvinc 3615 | A variable introduction la... |
| eqvincf 3616 | A variable introduction la... |
| alexeqg 3617 | Two ways to express substi... |
| ceqex 3618 | Equality implies equivalen... |
| ceqsexg 3619 | A representation of explic... |
| ceqsexgv 3620 | Elimination of an existent... |
| ceqsrexv 3621 | Elimination of a restricte... |
| ceqsrexbv 3622 | Elimination of a restricte... |
| ceqsralbv 3623 | Elimination of a restricte... |
| ceqsrex2v 3624 | Elimination of a restricte... |
| clel2g 3625 | Alternate definition of me... |
| clel2 3626 | Alternate definition of me... |
| clel3g 3627 | Alternate definition of me... |
| clel3 3628 | Alternate definition of me... |
| clel4g 3629 | Alternate definition of me... |
| clel4 3630 | Alternate definition of me... |
| clel5 3631 | Alternate definition of cl... |
| pm13.183 3632 | Compare theorem *13.183 in... |
| rr19.3v 3633 | Restricted quantifier vers... |
| rr19.28v 3634 | Restricted quantifier vers... |
| elab6g 3635 | Membership in a class abst... |
| elabd2 3636 | Membership in a class abst... |
| elabd3 3637 | Membership in a class abst... |
| elabgt 3638 | Membership in a class abst... |
| elabgtOLD 3639 | Obsolete version of ~ elab... |
| elabgtOLDOLD 3640 | Obsolete version of ~ elab... |
| elabgf 3641 | Membership in a class abst... |
| elabf 3642 | Membership in a class abst... |
| elabg 3643 | Membership in a class abst... |
| elabgw 3644 | Membership in a class abst... |
| elab2gw 3645 | Membership in a class abst... |
| elab 3646 | Membership in a class abst... |
| elab2g 3647 | Membership in a class abst... |
| elabd 3648 | Explicit demonstration the... |
| elab2 3649 | Membership in a class abst... |
| elab4g 3650 | Membership in a class abst... |
| elab3gf 3651 | Membership in a class abst... |
| elab3g 3652 | Membership in a class abst... |
| elab3 3653 | Membership in a class abst... |
| elrabi 3654 | Implication for the member... |
| elrabf 3655 | Membership in a restricted... |
| rabtru 3656 | Abstract builder using the... |
| rabeqcOLD 3657 | Obsolete version of ~ rabe... |
| elrab3t 3658 | Membership in a restricted... |
| elrab 3659 | Membership in a restricted... |
| elrab3 3660 | Membership in a restricted... |
| elrabd 3661 | Membership in a restricted... |
| elrab2 3662 | Membership in a restricted... |
| elrab2w 3663 | Membership in a restricted... |
| ralab 3664 | Universal quantification o... |
| ralrab 3665 | Universal quantification o... |
| rexab 3666 | Existential quantification... |
| rexrab 3667 | Existential quantification... |
| ralab2 3668 | Universal quantification o... |
| ralrab2 3669 | Universal quantification o... |
| rexab2 3670 | Existential quantification... |
| rexrab2 3671 | Existential quantification... |
| reurab 3672 | Restricted existential uni... |
| abidnf 3673 | Identity used to create cl... |
| dedhb 3674 | A deduction theorem for co... |
| class2seteq 3675 | Writing a set as a class a... |
| nelrdva 3676 | Deduce negative membership... |
| eqeu 3677 | A condition which implies ... |
| moeq 3678 | There exists at most one s... |
| eueq 3679 | A class is a set if and on... |
| eueqi 3680 | There exists a unique set ... |
| eueq2 3681 | Equality has existential u... |
| eueq3 3682 | Equality has existential u... |
| moeq3 3683 | "At most one" property of ... |
| mosub 3684 | "At most one" remains true... |
| mo2icl 3685 | Theorem for inferring "at ... |
| mob2 3686 | Consequence of "at most on... |
| moi2 3687 | Consequence of "at most on... |
| mob 3688 | Equality implied by "at mo... |
| moi 3689 | Equality implied by "at mo... |
| morex 3690 | Derive membership from uni... |
| euxfr2w 3691 | Transfer existential uniqu... |
| euxfrw 3692 | Transfer existential uniqu... |
| euxfr2 3693 | Transfer existential uniqu... |
| euxfr 3694 | Transfer existential uniqu... |
| euind 3695 | Existential uniqueness via... |
| reu2 3696 | A way to express restricte... |
| reu6 3697 | A way to express restricte... |
| reu3 3698 | A way to express restricte... |
| reu6i 3699 | A condition which implies ... |
| eqreu 3700 | A condition which implies ... |
| rmo4 3701 | Restricted "at most one" u... |
| reu4 3702 | Restricted uniqueness usin... |
| reu7 3703 | Restricted uniqueness usin... |
| reu8 3704 | Restricted uniqueness usin... |
| rmo3f 3705 | Restricted "at most one" u... |
| rmo4f 3706 | Restricted "at most one" u... |
| reu2eqd 3707 | Deduce equality from restr... |
| reueq 3708 | Equality has existential u... |
| rmoeq 3709 | Equality's restricted exis... |
| rmoan 3710 | Restricted "at most one" s... |
| rmoim 3711 | Restricted "at most one" i... |
| rmoimia 3712 | Restricted "at most one" i... |
| rmoimi 3713 | Restricted "at most one" i... |
| rmoimi2 3714 | Restricted "at most one" i... |
| 2reu5a 3715 | Double restricted existent... |
| reuimrmo 3716 | Restricted uniqueness impl... |
| 2reuswap 3717 | A condition allowing swap ... |
| 2reuswap2 3718 | A condition allowing swap ... |
| reuxfrd 3719 | Transfer existential uniqu... |
| reuxfr 3720 | Transfer existential uniqu... |
| reuxfr1d 3721 | Transfer existential uniqu... |
| reuxfr1ds 3722 | Transfer existential uniqu... |
| reuxfr1 3723 | Transfer existential uniqu... |
| reuind 3724 | Existential uniqueness via... |
| 2rmorex 3725 | Double restricted quantifi... |
| 2reu5lem1 3726 | Lemma for ~ 2reu5 . Note ... |
| 2reu5lem2 3727 | Lemma for ~ 2reu5 . (Cont... |
| 2reu5lem3 3728 | Lemma for ~ 2reu5 . This ... |
| 2reu5 3729 | Double restricted existent... |
| 2reurmo 3730 | Double restricted quantifi... |
| 2reurex 3731 | Double restricted quantifi... |
| 2rmoswap 3732 | A condition allowing to sw... |
| 2rexreu 3733 | Double restricted existent... |
| cdeqi 3736 | Deduce conditional equalit... |
| cdeqri 3737 | Property of conditional eq... |
| cdeqth 3738 | Deduce conditional equalit... |
| cdeqnot 3739 | Distribute conditional equ... |
| cdeqal 3740 | Distribute conditional equ... |
| cdeqab 3741 | Distribute conditional equ... |
| cdeqal1 3742 | Distribute conditional equ... |
| cdeqab1 3743 | Distribute conditional equ... |
| cdeqim 3744 | Distribute conditional equ... |
| cdeqcv 3745 | Conditional equality for s... |
| cdeqeq 3746 | Distribute conditional equ... |
| cdeqel 3747 | Distribute conditional equ... |
| nfcdeq 3748 | If we have a conditional e... |
| nfccdeq 3749 | Variation of ~ nfcdeq for ... |
| rru 3750 | Relative version of Russel... |
| ru 3751 | Russell's Paradox. Propos... |
| ruOLD 3752 | Obsolete version of ~ ru a... |
| dfsbcq 3755 | Proper substitution of a c... |
| dfsbcq2 3756 | This theorem, which is sim... |
| sbsbc 3757 | Show that ~ df-sb and ~ df... |
| sbceq1d 3758 | Equality theorem for class... |
| sbceq1dd 3759 | Equality theorem for class... |
| sbceqbid 3760 | Equality theorem for class... |
| sbc8g 3761 | This is the closest we can... |
| sbc2or 3762 | The disjunction of two equ... |
| sbcex 3763 | By our definition of prope... |
| sbceq1a 3764 | Equality theorem for class... |
| sbceq2a 3765 | Equality theorem for class... |
| spsbc 3766 | Specialization: if a formu... |
| spsbcd 3767 | Specialization: if a formu... |
| sbcth 3768 | A substitution into a theo... |
| sbcthdv 3769 | Deduction version of ~ sbc... |
| sbcid 3770 | An identity theorem for su... |
| nfsbc1d 3771 | Deduction version of ~ nfs... |
| nfsbc1 3772 | Bound-variable hypothesis ... |
| nfsbc1v 3773 | Bound-variable hypothesis ... |
| nfsbcdw 3774 | Deduction version of ~ nfs... |
| nfsbcw 3775 | Bound-variable hypothesis ... |
| sbccow 3776 | A composition law for clas... |
| nfsbcd 3777 | Deduction version of ~ nfs... |
| nfsbc 3778 | Bound-variable hypothesis ... |
| sbcco 3779 | A composition law for clas... |
| sbcco2 3780 | A composition law for clas... |
| sbc5 3781 | An equivalence for class s... |
| sbc5ALT 3782 | Alternate proof of ~ sbc5 ... |
| sbc6g 3783 | An equivalence for class s... |
| sbc6 3784 | An equivalence for class s... |
| sbc7 3785 | An equivalence for class s... |
| cbvsbcw 3786 | Change bound variables in ... |
| cbvsbcvw 3787 | Change the bound variable ... |
| cbvsbc 3788 | Change bound variables in ... |
| cbvsbcv 3789 | Change the bound variable ... |
| sbciegft 3790 | Conversion of implicit sub... |
| sbciegftOLD 3791 | Obsolete version of ~ sbci... |
| sbciegf 3792 | Conversion of implicit sub... |
| sbcieg 3793 | Conversion of implicit sub... |
| sbcie2g 3794 | Conversion of implicit sub... |
| sbcie 3795 | Conversion of implicit sub... |
| sbciedf 3796 | Conversion of implicit sub... |
| sbcied 3797 | Conversion of implicit sub... |
| sbcied2 3798 | Conversion of implicit sub... |
| elrabsf 3799 | Membership in a restricted... |
| eqsbc1 3800 | Substitution for the left-... |
| sbcng 3801 | Move negation in and out o... |
| sbcimg 3802 | Distribution of class subs... |
| sbcan 3803 | Distribution of class subs... |
| sbcor 3804 | Distribution of class subs... |
| sbcbig 3805 | Distribution of class subs... |
| sbcn1 3806 | Move negation in and out o... |
| sbcim1 3807 | Distribution of class subs... |
| sbcbid 3808 | Formula-building deduction... |
| sbcbidv 3809 | Formula-building deduction... |
| sbcbii 3810 | Formula-building inference... |
| sbcbi1 3811 | Distribution of class subs... |
| sbcbi2 3812 | Substituting into equivale... |
| sbcal 3813 | Move universal quantifier ... |
| sbcex2 3814 | Move existential quantifie... |
| sbceqal 3815 | Class version of one impli... |
| sbeqalb 3816 | Theorem *14.121 in [Whiteh... |
| eqsbc2 3817 | Substitution for the right... |
| sbc3an 3818 | Distribution of class subs... |
| sbcel1v 3819 | Class substitution into a ... |
| sbcel2gv 3820 | Class substitution into a ... |
| sbcel21v 3821 | Class substitution into a ... |
| sbcimdv 3822 | Substitution analogue of T... |
| sbctt 3823 | Substitution for a variabl... |
| sbcgf 3824 | Substitution for a variabl... |
| sbc19.21g 3825 | Substitution for a variabl... |
| sbcg 3826 | Substitution for a variabl... |
| sbcgfi 3827 | Substitution for a variabl... |
| sbc2iegf 3828 | Conversion of implicit sub... |
| sbc2ie 3829 | Conversion of implicit sub... |
| sbc2iedv 3830 | Conversion of implicit sub... |
| sbc3ie 3831 | Conversion of implicit sub... |
| sbccomlem 3832 | Lemma for ~ sbccom . (Con... |
| sbccomlemOLD 3833 | Obsolete version of ~ sbcc... |
| sbccom 3834 | Commutative law for double... |
| sbcralt 3835 | Interchange class substitu... |
| sbcrext 3836 | Interchange class substitu... |
| sbcralg 3837 | Interchange class substitu... |
| sbcrex 3838 | Interchange class substitu... |
| sbcreu 3839 | Interchange class substitu... |
| reu8nf 3840 | Restricted uniqueness usin... |
| sbcabel 3841 | Interchange class substitu... |
| rspsbc 3842 | Restricted quantifier vers... |
| rspsbca 3843 | Restricted quantifier vers... |
| rspesbca 3844 | Existence form of ~ rspsbc... |
| spesbc 3845 | Existence form of ~ spsbc ... |
| spesbcd 3846 | form of ~ spsbc . (Contri... |
| sbcth2 3847 | A substitution into a theo... |
| ra4v 3848 | Version of ~ ra4 with a di... |
| ra4 3849 | Restricted quantifier vers... |
| rmo2 3850 | Alternate definition of re... |
| rmo2i 3851 | Condition implying restric... |
| rmo3 3852 | Restricted "at most one" u... |
| rmob 3853 | Consequence of "at most on... |
| rmoi 3854 | Consequence of "at most on... |
| rmob2 3855 | Consequence of "restricted... |
| rmoi2 3856 | Consequence of "restricted... |
| rmoanim 3857 | Introduction of a conjunct... |
| rmoanimALT 3858 | Alternate proof of ~ rmoan... |
| reuan 3859 | Introduction of a conjunct... |
| 2reu1 3860 | Double restricted existent... |
| 2reu2 3861 | Double restricted existent... |
| csb2 3864 | Alternate expression for t... |
| csbeq1 3865 | Analogue of ~ dfsbcq for p... |
| csbeq1d 3866 | Equality deduction for pro... |
| csbeq2 3867 | Substituting into equivale... |
| csbeq2d 3868 | Formula-building deduction... |
| csbeq2dv 3869 | Formula-building deduction... |
| csbeq2i 3870 | Formula-building inference... |
| csbeq12dv 3871 | Formula-building inference... |
| cbvcsbw 3872 | Change bound variables in ... |
| cbvcsb 3873 | Change bound variables in ... |
| cbvcsbv 3874 | Change the bound variable ... |
| csbid 3875 | Analogue of ~ sbid for pro... |
| csbeq1a 3876 | Equality theorem for prope... |
| csbcow 3877 | Composition law for chaine... |
| csbco 3878 | Composition law for chaine... |
| csbtt 3879 | Substitution doesn't affec... |
| csbconstgf 3880 | Substitution doesn't affec... |
| csbconstg 3881 | Substitution doesn't affec... |
| csbgfi 3882 | Substitution for a variabl... |
| csbconstgi 3883 | The proper substitution of... |
| nfcsb1d 3884 | Bound-variable hypothesis ... |
| nfcsb1 3885 | Bound-variable hypothesis ... |
| nfcsb1v 3886 | Bound-variable hypothesis ... |
| nfcsbd 3887 | Deduction version of ~ nfc... |
| nfcsbw 3888 | Bound-variable hypothesis ... |
| nfcsb 3889 | Bound-variable hypothesis ... |
| csbhypf 3890 | Introduce an explicit subs... |
| csbiebt 3891 | Conversion of implicit sub... |
| csbiedf 3892 | Conversion of implicit sub... |
| csbieb 3893 | Bidirectional conversion b... |
| csbiebg 3894 | Bidirectional conversion b... |
| csbiegf 3895 | Conversion of implicit sub... |
| csbief 3896 | Conversion of implicit sub... |
| csbie 3897 | Conversion of implicit sub... |
| csbied 3898 | Conversion of implicit sub... |
| csbied2 3899 | Conversion of implicit sub... |
| csbie2t 3900 | Conversion of implicit sub... |
| csbie2 3901 | Conversion of implicit sub... |
| csbie2g 3902 | Conversion of implicit sub... |
| cbvrabcsfw 3903 | Version of ~ cbvrabcsf wit... |
| cbvralcsf 3904 | A more general version of ... |
| cbvrexcsf 3905 | A more general version of ... |
| cbvreucsf 3906 | A more general version of ... |
| cbvrabcsf 3907 | A more general version of ... |
| cbvralv2 3908 | Rule used to change the bo... |
| cbvrexv2 3909 | Rule used to change the bo... |
| rspc2vd 3910 | Deduction version of 2-var... |
| difjust 3916 | Soundness justification th... |
| unjust 3918 | Soundness justification th... |
| injust 3920 | Soundness justification th... |
| dfin5 3922 | Alternate definition for t... |
| dfdif2 3923 | Alternate definition of cl... |
| eldif 3924 | Expansion of membership in... |
| eldifd 3925 | If a class is in one class... |
| eldifad 3926 | If a class is in the diffe... |
| eldifbd 3927 | If a class is in the diffe... |
| elneeldif 3928 | The elements of a set diff... |
| velcomp 3929 | Characterization of setvar... |
| elin 3930 | Expansion of membership in... |
| dfss2 3932 | Alternate definition of th... |
| dfss 3933 | Variant of subclass defini... |
| dfss3 3935 | Alternate definition of su... |
| dfss6 3936 | Alternate definition of su... |
| dfssf 3937 | Equivalence for subclass r... |
| dfss3f 3938 | Equivalence for subclass r... |
| nfss 3939 | If ` x ` is not free in ` ... |
| ssel 3940 | Membership relationships f... |
| ssel2 3941 | Membership relationships f... |
| sseli 3942 | Membership implication fro... |
| sselii 3943 | Membership inference from ... |
| sselid 3944 | Membership inference from ... |
| sseld 3945 | Membership deduction from ... |
| sselda 3946 | Membership deduction from ... |
| sseldd 3947 | Membership inference from ... |
| ssneld 3948 | If a class is not in anoth... |
| ssneldd 3949 | If an element is not in a ... |
| ssriv 3950 | Inference based on subclas... |
| ssrd 3951 | Deduction based on subclas... |
| ssrdv 3952 | Deduction based on subclas... |
| sstr2 3953 | Transitivity of subclass r... |
| sstr2OLD 3954 | Obsolete version of ~ sstr... |
| sstr 3955 | Transitivity of subclass r... |
| sstri 3956 | Subclass transitivity infe... |
| sstrd 3957 | Subclass transitivity dedu... |
| sstrid 3958 | Subclass transitivity dedu... |
| sstrdi 3959 | Subclass transitivity dedu... |
| sylan9ss 3960 | A subclass transitivity de... |
| sylan9ssr 3961 | A subclass transitivity de... |
| eqss 3962 | The subclass relationship ... |
| eqssi 3963 | Infer equality from two su... |
| eqssd 3964 | Equality deduction from tw... |
| sssseq 3965 | If a class is a subclass o... |
| eqrd 3966 | Deduce equality of classes... |
| eqri 3967 | Infer equality of classes ... |
| eqelssd 3968 | Equality deduction from su... |
| ssid 3969 | Any class is a subclass of... |
| ssidd 3970 | Weakening of ~ ssid . (Co... |
| ssv 3971 | Any class is a subclass of... |
| sseq1 3972 | Equality theorem for subcl... |
| sseq2 3973 | Equality theorem for the s... |
| sseq12 3974 | Equality theorem for the s... |
| sseq1i 3975 | An equality inference for ... |
| sseq2i 3976 | An equality inference for ... |
| sseq12i 3977 | An equality inference for ... |
| sseq1d 3978 | An equality deduction for ... |
| sseq2d 3979 | An equality deduction for ... |
| sseq12d 3980 | An equality deduction for ... |
| eqsstrd 3981 | Substitution of equality i... |
| eqsstrrd 3982 | Substitution of equality i... |
| sseqtrd 3983 | Substitution of equality i... |
| sseqtrrd 3984 | Substitution of equality i... |
| eqsstrid 3985 | A chained subclass and equ... |
| eqsstrrid 3986 | A chained subclass and equ... |
| sseqtrdi 3987 | A chained subclass and equ... |
| sseqtrrdi 3988 | A chained subclass and equ... |
| sseqtrid 3989 | Subclass transitivity dedu... |
| sseqtrrid 3990 | Subclass transitivity dedu... |
| eqsstrdi 3991 | A chained subclass and equ... |
| eqsstrrdi 3992 | A chained subclass and equ... |
| eqsstri 3993 | Substitution of equality i... |
| eqsstrri 3994 | Substitution of equality i... |
| sseqtri 3995 | Substitution of equality i... |
| sseqtrri 3996 | Substitution of equality i... |
| 3sstr3i 3997 | Substitution of equality i... |
| 3sstr4i 3998 | Substitution of equality i... |
| 3sstr3g 3999 | Substitution of equality i... |
| 3sstr4g 4000 | Substitution of equality i... |
| 3sstr3d 4001 | Substitution of equality i... |
| 3sstr4d 4002 | Substitution of equality i... |
| eqimssd 4003 | Equality implies inclusion... |
| eqimsscd 4004 | Equality implies inclusion... |
| eqimss 4005 | Equality implies inclusion... |
| eqimss2 4006 | Equality implies inclusion... |
| eqimssi 4007 | Infer subclass relationshi... |
| eqimss2i 4008 | Infer subclass relationshi... |
| nssne1 4009 | Two classes are different ... |
| nssne2 4010 | Two classes are different ... |
| nss 4011 | Negation of subclass relat... |
| nelss 4012 | Demonstrate by witnesses t... |
| ssrexf 4013 | Restricted existential qua... |
| ssrmof 4014 | "At most one" existential ... |
| ssralv 4015 | Quantification restricted ... |
| ssrexv 4016 | Existential quantification... |
| ss2ralv 4017 | Two quantifications restri... |
| ss2rexv 4018 | Two existential quantifica... |
| ssralvOLD 4019 | Obsolete version of ~ ssra... |
| ssrexvOLD 4020 | Obsolete version of ~ ssre... |
| ralss 4021 | Restricted universal quant... |
| rexss 4022 | Restricted existential qua... |
| ralssOLD 4023 | Obsolete version of ~ rals... |
| rexssOLD 4024 | Obsolete version of ~ rexs... |
| ss2ab 4025 | Class abstractions in a su... |
| abss 4026 | Class abstraction in a sub... |
| ssab 4027 | Subclass of a class abstra... |
| ssabral 4028 | The relation for a subclas... |
| ss2abdv 4029 | Deduction of abstraction s... |
| ss2abi 4030 | Inference of abstraction s... |
| abssdv 4031 | Deduction of abstraction s... |
| abssdvOLD 4032 | Obsolete version of ~ abss... |
| abssi 4033 | Inference of abstraction s... |
| ss2rab 4034 | Restricted abstraction cla... |
| rabss 4035 | Restricted class abstracti... |
| ssrab 4036 | Subclass of a restricted c... |
| ssrabdv 4037 | Subclass of a restricted c... |
| rabssdv 4038 | Subclass of a restricted c... |
| ss2rabdv 4039 | Deduction of restricted ab... |
| ss2rabi 4040 | Inference of restricted ab... |
| rabss2 4041 | Subclass law for restricte... |
| ssab2 4042 | Subclass relation for the ... |
| ssrab2 4043 | Subclass relation for a re... |
| rabss3d 4044 | Subclass law for restricte... |
| ssrab3 4045 | Subclass relation for a re... |
| rabssrabd 4046 | Subclass of a restricted c... |
| ssrabeq 4047 | If the restricting class o... |
| rabssab 4048 | A restricted class is a su... |
| eqrrabd 4049 | Deduce equality with a res... |
| uniiunlem 4050 | A subset relationship usef... |
| dfpss2 4051 | Alternate definition of pr... |
| dfpss3 4052 | Alternate definition of pr... |
| psseq1 4053 | Equality theorem for prope... |
| psseq2 4054 | Equality theorem for prope... |
| psseq1i 4055 | An equality inference for ... |
| psseq2i 4056 | An equality inference for ... |
| psseq12i 4057 | An equality inference for ... |
| psseq1d 4058 | An equality deduction for ... |
| psseq2d 4059 | An equality deduction for ... |
| psseq12d 4060 | An equality deduction for ... |
| pssss 4061 | A proper subclass is a sub... |
| pssne 4062 | Two classes in a proper su... |
| pssssd 4063 | Deduce subclass from prope... |
| pssned 4064 | Proper subclasses are uneq... |
| sspss 4065 | Subclass in terms of prope... |
| pssirr 4066 | Proper subclass is irrefle... |
| pssn2lp 4067 | Proper subclass has no 2-c... |
| sspsstri 4068 | Two ways of stating tricho... |
| ssnpss 4069 | Partial trichotomy law for... |
| psstr 4070 | Transitive law for proper ... |
| sspsstr 4071 | Transitive law for subclas... |
| psssstr 4072 | Transitive law for subclas... |
| psstrd 4073 | Proper subclass inclusion ... |
| sspsstrd 4074 | Transitivity involving sub... |
| psssstrd 4075 | Transitivity involving sub... |
| npss 4076 | A class is not a proper su... |
| ssnelpss 4077 | A subclass missing a membe... |
| ssnelpssd 4078 | Subclass inclusion with on... |
| ssexnelpss 4079 | If there is an element of ... |
| dfdif3 4080 | Alternate definition of cl... |
| dfdif3OLD 4081 | Obsolete version of ~ dfdi... |
| difeq1 4082 | Equality theorem for class... |
| difeq2 4083 | Equality theorem for class... |
| difeq12 4084 | Equality theorem for class... |
| difeq1i 4085 | Inference adding differenc... |
| difeq2i 4086 | Inference adding differenc... |
| difeq12i 4087 | Equality inference for cla... |
| difeq1d 4088 | Deduction adding differenc... |
| difeq2d 4089 | Deduction adding differenc... |
| difeq12d 4090 | Equality deduction for cla... |
| difeqri 4091 | Inference from membership ... |
| nfdif 4092 | Bound-variable hypothesis ... |
| nfdifOLD 4093 | Obsolete version of ~ nfdi... |
| eldifi 4094 | Implication of membership ... |
| eldifn 4095 | Implication of membership ... |
| elndif 4096 | A set does not belong to a... |
| neldif 4097 | Implication of membership ... |
| difdif 4098 | Double class difference. ... |
| difss 4099 | Subclass relationship for ... |
| difssd 4100 | A difference of two classe... |
| difss2 4101 | If a class is contained in... |
| difss2d 4102 | If a class is contained in... |
| ssdifss 4103 | Preservation of a subclass... |
| ddif 4104 | Double complement under un... |
| ssconb 4105 | Contraposition law for sub... |
| sscon 4106 | Contraposition law for sub... |
| ssdif 4107 | Difference law for subsets... |
| ssdifd 4108 | If ` A ` is contained in `... |
| sscond 4109 | If ` A ` is contained in `... |
| ssdifssd 4110 | If ` A ` is contained in `... |
| ssdif2d 4111 | If ` A ` is contained in `... |
| raldifb 4112 | Restricted universal quant... |
| rexdifi 4113 | Restricted existential qua... |
| complss 4114 | Complementation reverses i... |
| compleq 4115 | Two classes are equal if a... |
| elun 4116 | Expansion of membership in... |
| elunnel1 4117 | A member of a union that i... |
| elunnel2 4118 | A member of a union that i... |
| uneqri 4119 | Inference from membership ... |
| unidm 4120 | Idempotent law for union o... |
| uncom 4121 | Commutative law for union ... |
| equncom 4122 | If a class equals the unio... |
| equncomi 4123 | Inference form of ~ equnco... |
| uneq1 4124 | Equality theorem for the u... |
| uneq2 4125 | Equality theorem for the u... |
| uneq12 4126 | Equality theorem for the u... |
| uneq1i 4127 | Inference adding union to ... |
| uneq2i 4128 | Inference adding union to ... |
| uneq12i 4129 | Equality inference for the... |
| uneq1d 4130 | Deduction adding union to ... |
| uneq2d 4131 | Deduction adding union to ... |
| uneq12d 4132 | Equality deduction for the... |
| nfun 4133 | Bound-variable hypothesis ... |
| nfunOLD 4134 | Obsolete version of ~ nfun... |
| unass 4135 | Associative law for union ... |
| un12 4136 | A rearrangement of union. ... |
| un23 4137 | A rearrangement of union. ... |
| un4 4138 | A rearrangement of the uni... |
| unundi 4139 | Union distributes over its... |
| unundir 4140 | Union distributes over its... |
| ssun1 4141 | Subclass relationship for ... |
| ssun2 4142 | Subclass relationship for ... |
| ssun3 4143 | Subclass law for union of ... |
| ssun4 4144 | Subclass law for union of ... |
| elun1 4145 | Membership law for union o... |
| elun2 4146 | Membership law for union o... |
| elunant 4147 | A statement is true for ev... |
| unss1 4148 | Subclass law for union of ... |
| ssequn1 4149 | A relationship between sub... |
| unss2 4150 | Subclass law for union of ... |
| unss12 4151 | Subclass law for union of ... |
| ssequn2 4152 | A relationship between sub... |
| unss 4153 | The union of two subclasse... |
| unssi 4154 | An inference showing the u... |
| unssd 4155 | A deduction showing the un... |
| unssad 4156 | If ` ( A u. B ) ` is conta... |
| unssbd 4157 | If ` ( A u. B ) ` is conta... |
| ssun 4158 | A condition that implies i... |
| rexun 4159 | Restricted existential qua... |
| ralunb 4160 | Restricted quantification ... |
| ralun 4161 | Restricted quantification ... |
| elini 4162 | Membership in an intersect... |
| elind 4163 | Deduce membership in an in... |
| elinel1 4164 | Membership in an intersect... |
| elinel2 4165 | Membership in an intersect... |
| elin2 4166 | Membership in a class defi... |
| elin1d 4167 | Elementhood in the first s... |
| elin2d 4168 | Elementhood in the first s... |
| elin3 4169 | Membership in a class defi... |
| nel1nelin 4170 | Membership in an intersect... |
| nel2nelin 4171 | Membership in an intersect... |
| incom 4172 | Commutative law for inters... |
| ineqcom 4173 | Two ways of expressing tha... |
| ineqcomi 4174 | Two ways of expressing tha... |
| ineqri 4175 | Inference from membership ... |
| ineq1 4176 | Equality theorem for inter... |
| ineq2 4177 | Equality theorem for inter... |
| ineq12 4178 | Equality theorem for inter... |
| ineq1i 4179 | Equality inference for int... |
| ineq2i 4180 | Equality inference for int... |
| ineq12i 4181 | Equality inference for int... |
| ineq1d 4182 | Equality deduction for int... |
| ineq2d 4183 | Equality deduction for int... |
| ineq12d 4184 | Equality deduction for int... |
| ineqan12d 4185 | Equality deduction for int... |
| sseqin2 4186 | A relationship between sub... |
| nfin 4187 | Bound-variable hypothesis ... |
| nfinOLD 4188 | Obsolete version of ~ nfin... |
| rabbi2dva 4189 | Deduction from a wff to a ... |
| inidm 4190 | Idempotent law for interse... |
| inass 4191 | Associative law for inters... |
| in12 4192 | A rearrangement of interse... |
| in32 4193 | A rearrangement of interse... |
| in13 4194 | A rearrangement of interse... |
| in31 4195 | A rearrangement of interse... |
| inrot 4196 | Rotate the intersection of... |
| in4 4197 | Rearrangement of intersect... |
| inindi 4198 | Intersection distributes o... |
| inindir 4199 | Intersection distributes o... |
| inss1 4200 | The intersection of two cl... |
| inss2 4201 | The intersection of two cl... |
| ssin 4202 | Subclass of intersection. ... |
| ssini 4203 | An inference showing that ... |
| ssind 4204 | A deduction showing that a... |
| ssrin 4205 | Add right intersection to ... |
| sslin 4206 | Add left intersection to s... |
| ssrind 4207 | Add right intersection to ... |
| ss2in 4208 | Intersection of subclasses... |
| ssinss1 4209 | Intersection preserves sub... |
| ssinss1d 4210 | Intersection preserves sub... |
| inss 4211 | Inclusion of an intersecti... |
| ralin 4212 | Restricted universal quant... |
| rexin 4213 | Restricted existential qua... |
| dfss7 4214 | Alternate definition of su... |
| symdifcom 4217 | Symmetric difference commu... |
| symdifeq1 4218 | Equality theorem for symme... |
| symdifeq2 4219 | Equality theorem for symme... |
| nfsymdif 4220 | Hypothesis builder for sym... |
| elsymdif 4221 | Membership in a symmetric ... |
| dfsymdif4 4222 | Alternate definition of th... |
| elsymdifxor 4223 | Membership in a symmetric ... |
| dfsymdif2 4224 | Alternate definition of th... |
| symdifass 4225 | Symmetric difference is as... |
| difsssymdif 4226 | The symmetric difference c... |
| difsymssdifssd 4227 | If the symmetric differenc... |
| unabs 4228 | Absorption law for union. ... |
| inabs 4229 | Absorption law for interse... |
| nssinpss 4230 | Negation of subclass expre... |
| nsspssun 4231 | Negation of subclass expre... |
| dfss4 4232 | Subclass defined in terms ... |
| dfun2 4233 | An alternate definition of... |
| dfin2 4234 | An alternate definition of... |
| difin 4235 | Difference with intersecti... |
| ssdifim 4236 | Implication of a class dif... |
| ssdifsym 4237 | Symmetric class difference... |
| dfss5 4238 | Alternate definition of su... |
| dfun3 4239 | Union defined in terms of ... |
| dfin3 4240 | Intersection defined in te... |
| dfin4 4241 | Alternate definition of th... |
| invdif 4242 | Intersection with universa... |
| indif 4243 | Intersection with class di... |
| indif2 4244 | Bring an intersection in a... |
| indif1 4245 | Bring an intersection in a... |
| indifcom 4246 | Commutation law for inters... |
| indi 4247 | Distributive law for inter... |
| undi 4248 | Distributive law for union... |
| indir 4249 | Distributive law for inter... |
| undir 4250 | Distributive law for union... |
| unineq 4251 | Infer equality from equali... |
| uneqin 4252 | Equality of union and inte... |
| difundi 4253 | Distributive law for class... |
| difundir 4254 | Distributive law for class... |
| difindi 4255 | Distributive law for class... |
| difindir 4256 | Distributive law for class... |
| indifdi 4257 | Distribute intersection ov... |
| indifdir 4258 | Distribute intersection ov... |
| difdif2 4259 | Class difference by a clas... |
| undm 4260 | De Morgan's law for union.... |
| indm 4261 | De Morgan's law for inters... |
| difun1 4262 | A relationship involving d... |
| undif3 4263 | An equality involving clas... |
| difin2 4264 | Represent a class differen... |
| dif32 4265 | Swap second and third argu... |
| difabs 4266 | Absorption-like law for cl... |
| sscon34b 4267 | Relative complementation r... |
| rcompleq 4268 | Two subclasses are equal i... |
| dfsymdif3 4269 | Alternate definition of th... |
| unabw 4270 | Union of two class abstrac... |
| unab 4271 | Union of two class abstrac... |
| inab 4272 | Intersection of two class ... |
| difab 4273 | Difference of two class ab... |
| abanssl 4274 | A class abstraction with a... |
| abanssr 4275 | A class abstraction with a... |
| notabw 4276 | A class abstraction define... |
| notab 4277 | A class abstraction define... |
| unrab 4278 | Union of two restricted cl... |
| inrab 4279 | Intersection of two restri... |
| inrab2 4280 | Intersection with a restri... |
| difrab 4281 | Difference of two restrict... |
| dfrab3 4282 | Alternate definition of re... |
| dfrab2 4283 | Alternate definition of re... |
| rabdif 4284 | Move difference in and out... |
| notrab 4285 | Complementation of restric... |
| dfrab3ss 4286 | Restricted class abstracti... |
| rabun2 4287 | Abstraction restricted to ... |
| reuun2 4288 | Transfer uniqueness to a s... |
| reuss2 4289 | Transfer uniqueness to a s... |
| reuss 4290 | Transfer uniqueness to a s... |
| reuun1 4291 | Transfer uniqueness to a s... |
| reupick 4292 | Restricted uniqueness "pic... |
| reupick3 4293 | Restricted uniqueness "pic... |
| reupick2 4294 | Restricted uniqueness "pic... |
| euelss 4295 | Transfer uniqueness of an ... |
| dfnul4 4298 | Alternate definition of th... |
| dfnul2 4299 | Alternate definition of th... |
| dfnul3 4300 | Alternate definition of th... |
| noel 4301 | The empty set has no eleme... |
| nel02 4302 | The empty set has no eleme... |
| n0i 4303 | If a class has elements, t... |
| ne0i 4304 | If a class has elements, t... |
| ne0d 4305 | Deduction form of ~ ne0i .... |
| n0ii 4306 | If a class has elements, t... |
| ne0ii 4307 | If a class has elements, t... |
| vn0 4308 | The universal class is not... |
| vn0ALT 4309 | Alternate proof of ~ vn0 .... |
| eq0f 4310 | A class is equal to the em... |
| neq0f 4311 | A class is not empty if an... |
| n0f 4312 | A class is nonempty if and... |
| eq0 4313 | A class is equal to the em... |
| eq0ALT 4314 | Alternate proof of ~ eq0 .... |
| neq0 4315 | A class is not empty if an... |
| n0 4316 | A class is nonempty if and... |
| nel0 4317 | From the general negation ... |
| reximdva0 4318 | Restricted existence deduc... |
| rspn0 4319 | Specialization for restric... |
| n0rex 4320 | There is an element in a n... |
| ssn0rex 4321 | There is an element in a c... |
| n0moeu 4322 | A case of equivalence of "... |
| rex0 4323 | Vacuous restricted existen... |
| reu0 4324 | Vacuous restricted uniquen... |
| rmo0 4325 | Vacuous restricted at-most... |
| 0el 4326 | Membership of the empty se... |
| n0el 4327 | Negated membership of the ... |
| eqeuel 4328 | A condition which implies ... |
| ssdif0 4329 | Subclass expressed in term... |
| difn0 4330 | If the difference of two s... |
| pssdifn0 4331 | A proper subclass has a no... |
| pssdif 4332 | A proper subclass has a no... |
| ndisj 4333 | Express that an intersecti... |
| inn0f 4334 | A nonempty intersection. ... |
| inn0 4335 | A nonempty intersection. ... |
| difin0ss 4336 | Difference, intersection, ... |
| inssdif0 4337 | Intersection, subclass, an... |
| inindif 4338 | The intersection and class... |
| difid 4339 | The difference between a c... |
| difidALT 4340 | Alternate proof of ~ difid... |
| dif0 4341 | The difference between a c... |
| ab0w 4342 | The class of sets verifyin... |
| ab0 4343 | The class of sets verifyin... |
| ab0ALT 4344 | Alternate proof of ~ ab0 ,... |
| dfnf5 4345 | Characterization of nonfre... |
| ab0orv 4346 | The class abstraction defi... |
| ab0orvALT 4347 | Alternate proof of ~ ab0or... |
| abn0 4348 | Nonempty class abstraction... |
| rab0 4349 | Any restricted class abstr... |
| rabeq0w 4350 | Condition for a restricted... |
| rabeq0 4351 | Condition for a restricted... |
| rabn0 4352 | Nonempty restricted class ... |
| rabxm 4353 | Law of excluded middle, in... |
| rabnc 4354 | Law of noncontradiction, i... |
| elneldisj 4355 | The set of elements ` s ` ... |
| elnelun 4356 | The union of the set of el... |
| un0 4357 | The union of a class with ... |
| in0 4358 | The intersection of a clas... |
| 0un 4359 | The union of the empty set... |
| 0in 4360 | The intersection of the em... |
| inv1 4361 | The intersection of a clas... |
| unv 4362 | The union of a class with ... |
| 0ss 4363 | The null set is a subset o... |
| ss0b 4364 | Any subset of the empty se... |
| ss0 4365 | Any subset of the empty se... |
| sseq0 4366 | A subclass of an empty cla... |
| ssn0 4367 | A class with a nonempty su... |
| 0dif 4368 | The difference between the... |
| abf 4369 | A class abstraction determ... |
| eq0rdv 4370 | Deduction for equality to ... |
| eq0rdvALT 4371 | Alternate proof of ~ eq0rd... |
| csbprc 4372 | The proper substitution of... |
| csb0 4373 | The proper substitution of... |
| sbcel12 4374 | Distribute proper substitu... |
| sbceqg 4375 | Distribute proper substitu... |
| sbceqi 4376 | Distribution of class subs... |
| sbcnel12g 4377 | Distribute proper substitu... |
| sbcne12 4378 | Distribute proper substitu... |
| sbcel1g 4379 | Move proper substitution i... |
| sbceq1g 4380 | Move proper substitution t... |
| sbcel2 4381 | Move proper substitution i... |
| sbceq2g 4382 | Move proper substitution t... |
| csbcom 4383 | Commutative law for double... |
| sbcnestgfw 4384 | Nest the composition of tw... |
| csbnestgfw 4385 | Nest the composition of tw... |
| sbcnestgw 4386 | Nest the composition of tw... |
| csbnestgw 4387 | Nest the composition of tw... |
| sbcco3gw 4388 | Composition of two substit... |
| sbcnestgf 4389 | Nest the composition of tw... |
| csbnestgf 4390 | Nest the composition of tw... |
| sbcnestg 4391 | Nest the composition of tw... |
| csbnestg 4392 | Nest the composition of tw... |
| sbcco3g 4393 | Composition of two substit... |
| csbco3g 4394 | Composition of two class s... |
| csbnest1g 4395 | Nest the composition of tw... |
| csbidm 4396 | Idempotent law for class s... |
| csbvarg 4397 | The proper substitution of... |
| csbvargi 4398 | The proper substitution of... |
| sbccsb 4399 | Substitution into a wff ex... |
| sbccsb2 4400 | Substitution into a wff ex... |
| rspcsbela 4401 | Special case related to ~ ... |
| sbnfc2 4402 | Two ways of expressing " `... |
| csbab 4403 | Move substitution into a c... |
| csbun 4404 | Distribution of class subs... |
| csbin 4405 | Distribute proper substitu... |
| csbie2df 4406 | Conversion of implicit sub... |
| 2nreu 4407 | If there are two different... |
| un00 4408 | Two classes are empty iff ... |
| vss 4409 | Only the universal class h... |
| 0pss 4410 | The null set is a proper s... |
| npss0 4411 | No set is a proper subset ... |
| pssv 4412 | Any non-universal class is... |
| disj 4413 | Two ways of saying that tw... |
| disjr 4414 | Two ways of saying that tw... |
| disj1 4415 | Two ways of saying that tw... |
| reldisj 4416 | Two ways of saying that tw... |
| disj3 4417 | Two ways of saying that tw... |
| disjne 4418 | Members of disjoint sets a... |
| disjeq0 4419 | Two disjoint sets are equa... |
| disjel 4420 | A set can't belong to both... |
| disj2 4421 | Two ways of saying that tw... |
| disj4 4422 | Two ways of saying that tw... |
| ssdisj 4423 | Intersection with a subcla... |
| disjpss 4424 | A class is a proper subset... |
| undisj1 4425 | The union of disjoint clas... |
| undisj2 4426 | The union of disjoint clas... |
| ssindif0 4427 | Subclass expressed in term... |
| inelcm 4428 | The intersection of classe... |
| minel 4429 | A minimum element of a cla... |
| undif4 4430 | Distribute union over diff... |
| disjssun 4431 | Subset relation for disjoi... |
| vdif0 4432 | Universal class equality i... |
| difrab0eq 4433 | If the difference between ... |
| pssnel 4434 | A proper subclass has a me... |
| disjdif 4435 | A class and its relative c... |
| disjdifr 4436 | A class and its relative c... |
| difin0 4437 | The difference of a class ... |
| unvdif 4438 | The union of a class and i... |
| undif1 4439 | Absorption of difference b... |
| undif2 4440 | Absorption of difference b... |
| undifabs 4441 | Absorption of difference b... |
| inundif 4442 | The intersection and class... |
| disjdif2 4443 | The difference of a class ... |
| difun2 4444 | Absorption of union by dif... |
| undif 4445 | Union of complementary par... |
| undifr 4446 | Union of complementary par... |
| undifrOLD 4447 | Obsolete version of ~ undi... |
| undif5 4448 | An equality involving clas... |
| ssdifin0 4449 | A subset of a difference d... |
| ssdifeq0 4450 | A class is a subclass of i... |
| ssundif 4451 | A condition equivalent to ... |
| difcom 4452 | Swap the arguments of a cl... |
| pssdifcom1 4453 | Two ways to express overla... |
| pssdifcom2 4454 | Two ways to express non-co... |
| difdifdir 4455 | Distributive law for class... |
| uneqdifeq 4456 | Two ways to say that ` A `... |
| raldifeq 4457 | Equality theorem for restr... |
| r19.2z 4458 | Theorem 19.2 of [Margaris]... |
| r19.2zb 4459 | A response to the notion t... |
| r19.3rz 4460 | Restricted quantification ... |
| r19.28z 4461 | Restricted quantifier vers... |
| r19.3rzv 4462 | Restricted quantification ... |
| r19.9rzv 4463 | Restricted quantification ... |
| r19.28zv 4464 | Restricted quantifier vers... |
| r19.37zv 4465 | Restricted quantifier vers... |
| r19.45zv 4466 | Restricted version of Theo... |
| r19.44zv 4467 | Restricted version of Theo... |
| r19.27z 4468 | Restricted quantifier vers... |
| r19.27zv 4469 | Restricted quantifier vers... |
| r19.36zv 4470 | Restricted quantifier vers... |
| ralidmw 4471 | Idempotent law for restric... |
| rzal 4472 | Vacuous quantification is ... |
| rzalALT 4473 | Alternate proof of ~ rzal ... |
| rexn0 4474 | Restricted existential qua... |
| ralidm 4475 | Idempotent law for restric... |
| ral0 4476 | Vacuous universal quantifi... |
| ralf0 4477 | The quantification of a fa... |
| ralnralall 4478 | A contradiction concerning... |
| falseral0 4479 | A false statement can only... |
| raaan 4480 | Rearrange restricted quant... |
| raaanv 4481 | Rearrange restricted quant... |
| sbss 4482 | Set substitution into the ... |
| sbcssg 4483 | Distribute proper substitu... |
| raaan2 4484 | Rearrange restricted quant... |
| 2reu4lem 4485 | Lemma for ~ 2reu4 . (Cont... |
| 2reu4 4486 | Definition of double restr... |
| csbdif 4487 | Distribution of class subs... |
| dfif2 4490 | An alternate definition of... |
| dfif6 4491 | An alternate definition of... |
| ifeq1 4492 | Equality theorem for condi... |
| ifeq2 4493 | Equality theorem for condi... |
| iftrue 4494 | Value of the conditional o... |
| iftruei 4495 | Inference associated with ... |
| iftrued 4496 | Value of the conditional o... |
| iffalse 4497 | Value of the conditional o... |
| iffalsei 4498 | Inference associated with ... |
| iffalsed 4499 | Value of the conditional o... |
| ifnefalse 4500 | When values are unequal, b... |
| iftrueb 4501 | When the branches are not ... |
| ifsb 4502 | Distribute a function over... |
| dfif3 4503 | Alternate definition of th... |
| dfif4 4504 | Alternate definition of th... |
| dfif5 4505 | Alternate definition of th... |
| ifssun 4506 | A conditional class is inc... |
| ifeq12 4507 | Equality theorem for condi... |
| ifeq1d 4508 | Equality deduction for con... |
| ifeq2d 4509 | Equality deduction for con... |
| ifeq12d 4510 | Equality deduction for con... |
| ifbi 4511 | Equivalence theorem for co... |
| ifbid 4512 | Equivalence deduction for ... |
| ifbieq1d 4513 | Equivalence/equality deduc... |
| ifbieq2i 4514 | Equivalence/equality infer... |
| ifbieq2d 4515 | Equivalence/equality deduc... |
| ifbieq12i 4516 | Equivalence deduction for ... |
| ifbieq12d 4517 | Equivalence deduction for ... |
| nfifd 4518 | Deduction form of ~ nfif .... |
| nfif 4519 | Bound-variable hypothesis ... |
| ifeq1da 4520 | Conditional equality. (Co... |
| ifeq2da 4521 | Conditional equality. (Co... |
| ifeq12da 4522 | Equivalence deduction for ... |
| ifbieq12d2 4523 | Equivalence deduction for ... |
| ifclda 4524 | Conditional closure. (Con... |
| ifeqda 4525 | Separation of the values o... |
| elimif 4526 | Elimination of a condition... |
| ifbothda 4527 | A wff ` th ` containing a ... |
| ifboth 4528 | A wff ` th ` containing a ... |
| ifid 4529 | Identical true and false a... |
| eqif 4530 | Expansion of an equality w... |
| ifval 4531 | Another expression of the ... |
| elif 4532 | Membership in a conditiona... |
| ifel 4533 | Membership of a conditiona... |
| ifcl 4534 | Membership (closure) of a ... |
| ifcld 4535 | Membership (closure) of a ... |
| ifcli 4536 | Inference associated with ... |
| ifexd 4537 | Existence of the condition... |
| ifexg 4538 | Existence of the condition... |
| ifex 4539 | Existence of the condition... |
| ifeqor 4540 | The possible values of a c... |
| ifnot 4541 | Negating the first argumen... |
| ifan 4542 | Rewrite a conjunction in a... |
| ifor 4543 | Rewrite a disjunction in a... |
| 2if2 4544 | Resolve two nested conditi... |
| ifcomnan 4545 | Commute the conditions in ... |
| csbif 4546 | Distribute proper substitu... |
| dedth 4547 | Weak deduction theorem tha... |
| dedth2h 4548 | Weak deduction theorem eli... |
| dedth3h 4549 | Weak deduction theorem eli... |
| dedth4h 4550 | Weak deduction theorem eli... |
| dedth2v 4551 | Weak deduction theorem for... |
| dedth3v 4552 | Weak deduction theorem for... |
| dedth4v 4553 | Weak deduction theorem for... |
| elimhyp 4554 | Eliminate a hypothesis con... |
| elimhyp2v 4555 | Eliminate a hypothesis con... |
| elimhyp3v 4556 | Eliminate a hypothesis con... |
| elimhyp4v 4557 | Eliminate a hypothesis con... |
| elimel 4558 | Eliminate a membership hyp... |
| elimdhyp 4559 | Version of ~ elimhyp where... |
| keephyp 4560 | Transform a hypothesis ` p... |
| keephyp2v 4561 | Keep a hypothesis containi... |
| keephyp3v 4562 | Keep a hypothesis containi... |
| pwjust 4564 | Soundness justification th... |
| elpwg 4566 | Membership in a power clas... |
| elpw 4567 | Membership in a power clas... |
| velpw 4568 | Setvar variable membership... |
| elpwd 4569 | Membership in a power clas... |
| elpwi 4570 | Subset relation implied by... |
| elpwb 4571 | Characterization of the el... |
| elpwid 4572 | An element of a power clas... |
| elelpwi 4573 | If ` A ` belongs to a part... |
| sspw 4574 | The powerclass preserves i... |
| sspwi 4575 | The powerclass preserves i... |
| sspwd 4576 | The powerclass preserves i... |
| pweq 4577 | Equality theorem for power... |
| pweqALT 4578 | Alternate proof of ~ pweq ... |
| pweqi 4579 | Equality inference for pow... |
| pweqd 4580 | Equality deduction for pow... |
| pwunss 4581 | The power class of the uni... |
| nfpw 4582 | Bound-variable hypothesis ... |
| pwidg 4583 | A set is an element of its... |
| pwidb 4584 | A class is an element of i... |
| pwid 4585 | A set is a member of its p... |
| pwss 4586 | Subclass relationship for ... |
| pwundif 4587 | Break up the power class o... |
| snjust 4588 | Soundness justification th... |
| sneq 4599 | Equality theorem for singl... |
| sneqi 4600 | Equality inference for sin... |
| sneqd 4601 | Equality deduction for sin... |
| dfsn2 4602 | Alternate definition of si... |
| elsng 4603 | There is exactly one eleme... |
| elsn 4604 | There is exactly one eleme... |
| velsn 4605 | There is only one element ... |
| elsni 4606 | There is at most one eleme... |
| elsnd 4607 | There is at most one eleme... |
| rabsneq 4608 | Equality of class abstract... |
| absn 4609 | Condition for a class abst... |
| dfpr2 4610 | Alternate definition of a ... |
| dfsn2ALT 4611 | Alternate definition of si... |
| elprg 4612 | A member of a pair of clas... |
| elpri 4613 | If a class is an element o... |
| elpr 4614 | A member of a pair of clas... |
| elpr2g 4615 | A member of a pair of sets... |
| elpr2 4616 | A member of a pair of sets... |
| nelpr2 4617 | If a class is not an eleme... |
| nelpr1 4618 | If a class is not an eleme... |
| nelpri 4619 | If an element doesn't matc... |
| prneli 4620 | If an element doesn't matc... |
| nelprd 4621 | If an element doesn't matc... |
| eldifpr 4622 | Membership in a set with t... |
| rexdifpr 4623 | Restricted existential qua... |
| snidg 4624 | A set is a member of its s... |
| snidb 4625 | A class is a set iff it is... |
| snid 4626 | A set is a member of its s... |
| vsnid 4627 | A setvar variable is a mem... |
| elsn2g 4628 | There is exactly one eleme... |
| elsn2 4629 | There is exactly one eleme... |
| nelsn 4630 | If a class is not equal to... |
| rabeqsn 4631 | Conditions for a restricte... |
| rabsssn 4632 | Conditions for a restricte... |
| rabeqsnd 4633 | Conditions for a restricte... |
| ralsnsg 4634 | Substitution expressed in ... |
| rexsns 4635 | Restricted existential qua... |
| rexsngf 4636 | Restricted existential qua... |
| ralsngf 4637 | Restricted universal quant... |
| reusngf 4638 | Restricted existential uni... |
| ralsng 4639 | Substitution expressed in ... |
| rexsng 4640 | Restricted existential qua... |
| reusng 4641 | Restricted existential uni... |
| 2ralsng 4642 | Substitution expressed in ... |
| rexreusng 4643 | Restricted existential uni... |
| exsnrex 4644 | There is a set being the e... |
| ralsn 4645 | Convert a universal quanti... |
| rexsn 4646 | Convert an existential qua... |
| elunsn 4647 | Elementhood in a union wit... |
| elpwunsn 4648 | Membership in an extension... |
| eqoreldif 4649 | An element of a set is eit... |
| eltpg 4650 | Members of an unordered tr... |
| eldiftp 4651 | Membership in a set with t... |
| eltpi 4652 | A member of an unordered t... |
| eltp 4653 | A member of an unordered t... |
| el7g 4654 | Members of a set with seve... |
| dftp2 4655 | Alternate definition of un... |
| nfpr 4656 | Bound-variable hypothesis ... |
| ifpr 4657 | Membership of a conditiona... |
| ralprgf 4658 | Convert a restricted unive... |
| rexprgf 4659 | Convert a restricted exist... |
| ralprg 4660 | Convert a restricted unive... |
| rexprg 4661 | Convert a restricted exist... |
| raltpg 4662 | Convert a restricted unive... |
| rextpg 4663 | Convert a restricted exist... |
| ralpr 4664 | Convert a restricted unive... |
| rexpr 4665 | Convert a restricted exist... |
| reuprg0 4666 | Convert a restricted exist... |
| reuprg 4667 | Convert a restricted exist... |
| reurexprg 4668 | Convert a restricted exist... |
| raltp 4669 | Convert a universal quanti... |
| rextp 4670 | Convert an existential qua... |
| nfsn 4671 | Bound-variable hypothesis ... |
| csbsng 4672 | Distribute proper substitu... |
| csbprg 4673 | Distribute proper substitu... |
| elinsn 4674 | If the intersection of two... |
| disjsn 4675 | Intersection with the sing... |
| disjsn2 4676 | Two distinct singletons ar... |
| disjpr2 4677 | Two completely distinct un... |
| disjprsn 4678 | The disjoint intersection ... |
| disjtpsn 4679 | The disjoint intersection ... |
| disjtp2 4680 | Two completely distinct un... |
| snprc 4681 | The singleton of a proper ... |
| snnzb 4682 | A singleton is nonempty if... |
| rmosn 4683 | A restricted at-most-one q... |
| r19.12sn 4684 | Special case of ~ r19.12 w... |
| rabsn 4685 | Condition where a restrict... |
| rabsnifsb 4686 | A restricted class abstrac... |
| rabsnif 4687 | A restricted class abstrac... |
| rabrsn 4688 | A restricted class abstrac... |
| euabsn2 4689 | Another way to express exi... |
| euabsn 4690 | Another way to express exi... |
| reusn 4691 | A way to express restricte... |
| absneu 4692 | Restricted existential uni... |
| rabsneu 4693 | Restricted existential uni... |
| eusn 4694 | Two ways to express " ` A ... |
| rabsnt 4695 | Truth implied by equality ... |
| prcom 4696 | Commutative law for unorde... |
| preq1 4697 | Equality theorem for unord... |
| preq2 4698 | Equality theorem for unord... |
| preq12 4699 | Equality theorem for unord... |
| preq1i 4700 | Equality inference for uno... |
| preq2i 4701 | Equality inference for uno... |
| preq12i 4702 | Equality inference for uno... |
| preq1d 4703 | Equality deduction for uno... |
| preq2d 4704 | Equality deduction for uno... |
| preq12d 4705 | Equality deduction for uno... |
| tpeq1 4706 | Equality theorem for unord... |
| tpeq2 4707 | Equality theorem for unord... |
| tpeq3 4708 | Equality theorem for unord... |
| tpeq1d 4709 | Equality theorem for unord... |
| tpeq2d 4710 | Equality theorem for unord... |
| tpeq3d 4711 | Equality theorem for unord... |
| tpeq123d 4712 | Equality theorem for unord... |
| tprot 4713 | Rotation of the elements o... |
| tpcoma 4714 | Swap 1st and 2nd members o... |
| tpcomb 4715 | Swap 2nd and 3rd members o... |
| tpass 4716 | Split off the first elemen... |
| qdass 4717 | Two ways to write an unord... |
| qdassr 4718 | Two ways to write an unord... |
| tpidm12 4719 | Unordered triple ` { A , A... |
| tpidm13 4720 | Unordered triple ` { A , B... |
| tpidm23 4721 | Unordered triple ` { A , B... |
| tpidm 4722 | Unordered triple ` { A , A... |
| tppreq3 4723 | An unordered triple is an ... |
| prid1g 4724 | An unordered pair contains... |
| prid2g 4725 | An unordered pair contains... |
| prid1 4726 | An unordered pair contains... |
| prid2 4727 | An unordered pair contains... |
| ifpprsnss 4728 | An unordered pair is a sin... |
| prprc1 4729 | A proper class vanishes in... |
| prprc2 4730 | A proper class vanishes in... |
| prprc 4731 | An unordered pair containi... |
| tpid1 4732 | One of the three elements ... |
| tpid1g 4733 | Closed theorem form of ~ t... |
| tpid2 4734 | One of the three elements ... |
| tpid2g 4735 | Closed theorem form of ~ t... |
| tpid3g 4736 | Closed theorem form of ~ t... |
| tpid3 4737 | One of the three elements ... |
| snnzg 4738 | The singleton of a set is ... |
| snn0d 4739 | The singleton of a set is ... |
| snnz 4740 | The singleton of a set is ... |
| prnz 4741 | A pair containing a set is... |
| prnzg 4742 | A pair containing a set is... |
| tpnz 4743 | An unordered triple contai... |
| tpnzd 4744 | An unordered triple contai... |
| raltpd 4745 | Convert a universal quanti... |
| snssb 4746 | Characterization of the in... |
| snssg 4747 | The singleton formed on a ... |
| snssgOLD 4748 | Obsolete version of ~ snss... |
| snss 4749 | The singleton of an elemen... |
| eldifsn 4750 | Membership in a set with a... |
| eldifsnd 4751 | Membership in a set with a... |
| ssdifsn 4752 | Subset of a set with an el... |
| elpwdifsn 4753 | A subset of a set is an el... |
| eldifsni 4754 | Membership in a set with a... |
| eldifsnneq 4755 | An element of a difference... |
| neldifsn 4756 | The class ` A ` is not in ... |
| neldifsnd 4757 | The class ` A ` is not in ... |
| rexdifsn 4758 | Restricted existential qua... |
| raldifsni 4759 | Rearrangement of a propert... |
| raldifsnb 4760 | Restricted universal quant... |
| eldifvsn 4761 | A set is an element of the... |
| difsn 4762 | An element not in a set ca... |
| difprsnss 4763 | Removal of a singleton fro... |
| difprsn1 4764 | Removal of a singleton fro... |
| difprsn2 4765 | Removal of a singleton fro... |
| diftpsn3 4766 | Removal of a singleton fro... |
| difpr 4767 | Removing two elements as p... |
| tpprceq3 4768 | An unordered triple is an ... |
| tppreqb 4769 | An unordered triple is an ... |
| difsnb 4770 | ` ( B \ { A } ) ` equals `... |
| difsnpss 4771 | ` ( B \ { A } ) ` is a pro... |
| snssi 4772 | The singleton of an elemen... |
| snssd 4773 | The singleton of an elemen... |
| difsnid 4774 | If we remove a single elem... |
| eldifeldifsn 4775 | An element of a difference... |
| pw0 4776 | Compute the power set of t... |
| pwpw0 4777 | Compute the power set of t... |
| snsspr1 4778 | A singleton is a subset of... |
| snsspr2 4779 | A singleton is a subset of... |
| snsstp1 4780 | A singleton is a subset of... |
| snsstp2 4781 | A singleton is a subset of... |
| snsstp3 4782 | A singleton is a subset of... |
| prssg 4783 | A pair of elements of a cl... |
| prss 4784 | A pair of elements of a cl... |
| prssi 4785 | A pair of elements of a cl... |
| prssd 4786 | Deduction version of ~ prs... |
| prsspwg 4787 | An unordered pair belongs ... |
| ssprss 4788 | A pair as subset of a pair... |
| ssprsseq 4789 | A proper pair is a subset ... |
| sssn 4790 | The subsets of a singleton... |
| ssunsn2 4791 | The property of being sand... |
| ssunsn 4792 | Possible values for a set ... |
| eqsn 4793 | Two ways to express that a... |
| eqsnd 4794 | Deduce that a set is a sin... |
| eqsndOLD 4795 | Obsolete version of ~ eqsn... |
| issn 4796 | A sufficient condition for... |
| n0snor2el 4797 | A nonempty set is either a... |
| ssunpr 4798 | Possible values for a set ... |
| sspr 4799 | The subsets of a pair. (C... |
| sstp 4800 | The subsets of an unordere... |
| tpss 4801 | An unordered triple of ele... |
| tpssi 4802 | An unordered triple of ele... |
| sneqrg 4803 | Closed form of ~ sneqr . ... |
| sneqr 4804 | If the singletons of two s... |
| snsssn 4805 | If a singleton is a subset... |
| mosneq 4806 | There exists at most one s... |
| sneqbg 4807 | Two singletons of sets are... |
| snsspw 4808 | The singleton of a class i... |
| prsspw 4809 | An unordered pair belongs ... |
| preq1b 4810 | Biconditional equality lem... |
| preq2b 4811 | Biconditional equality lem... |
| preqr1 4812 | Reverse equality lemma for... |
| preqr2 4813 | Reverse equality lemma for... |
| preq12b 4814 | Equality relationship for ... |
| opthpr 4815 | An unordered pair has the ... |
| preqr1g 4816 | Reverse equality lemma for... |
| preq12bg 4817 | Closed form of ~ preq12b .... |
| prneimg 4818 | Two pairs are not equal if... |
| prneimg2 4819 | Two pairs are not equal if... |
| prnebg 4820 | A (proper) pair is not equ... |
| pr1eqbg 4821 | A (proper) pair is equal t... |
| pr1nebg 4822 | A (proper) pair is not equ... |
| preqsnd 4823 | Equivalence for a pair equ... |
| prnesn 4824 | A proper unordered pair is... |
| prneprprc 4825 | A proper unordered pair is... |
| preqsn 4826 | Equivalence for a pair equ... |
| preq12nebg 4827 | Equality relationship for ... |
| prel12g 4828 | Equality of two unordered ... |
| opthprneg 4829 | An unordered pair has the ... |
| elpreqprlem 4830 | Lemma for ~ elpreqpr . (C... |
| elpreqpr 4831 | Equality and membership ru... |
| elpreqprb 4832 | A set is an element of an ... |
| elpr2elpr 4833 | For an element ` A ` of an... |
| dfopif 4834 | Rewrite ~ df-op using ` if... |
| dfopg 4835 | Value of the ordered pair ... |
| dfop 4836 | Value of an ordered pair w... |
| opeq1 4837 | Equality theorem for order... |
| opeq2 4838 | Equality theorem for order... |
| opeq12 4839 | Equality theorem for order... |
| opeq1i 4840 | Equality inference for ord... |
| opeq2i 4841 | Equality inference for ord... |
| opeq12i 4842 | Equality inference for ord... |
| opeq1d 4843 | Equality deduction for ord... |
| opeq2d 4844 | Equality deduction for ord... |
| opeq12d 4845 | Equality deduction for ord... |
| oteq1 4846 | Equality theorem for order... |
| oteq2 4847 | Equality theorem for order... |
| oteq3 4848 | Equality theorem for order... |
| oteq1d 4849 | Equality deduction for ord... |
| oteq2d 4850 | Equality deduction for ord... |
| oteq3d 4851 | Equality deduction for ord... |
| oteq123d 4852 | Equality deduction for ord... |
| nfop 4853 | Bound-variable hypothesis ... |
| nfopd 4854 | Deduction version of bound... |
| csbopg 4855 | Distribution of class subs... |
| opidg 4856 | The ordered pair ` <. A , ... |
| opid 4857 | The ordered pair ` <. A , ... |
| ralunsn 4858 | Restricted quantification ... |
| 2ralunsn 4859 | Double restricted quantifi... |
| opprc 4860 | Expansion of an ordered pa... |
| opprc1 4861 | Expansion of an ordered pa... |
| opprc2 4862 | Expansion of an ordered pa... |
| oprcl 4863 | If an ordered pair has an ... |
| pwsn 4864 | The power set of a singlet... |
| pwpr 4865 | The power set of an unorde... |
| pwtp 4866 | The power set of an unorde... |
| pwpwpw0 4867 | Compute the power set of t... |
| pwv 4868 | The power class of the uni... |
| prproe 4869 | For an element of a proper... |
| 3elpr2eq 4870 | If there are three element... |
| dfuni2 4873 | Alternate definition of cl... |
| eluni 4874 | Membership in class union.... |
| eluni2 4875 | Membership in class union.... |
| elunii 4876 | Membership in class union.... |
| nfunid 4877 | Deduction version of ~ nfu... |
| nfuni 4878 | Bound-variable hypothesis ... |
| uniss 4879 | Subclass relationship for ... |
| unissi 4880 | Subclass relationship for ... |
| unissd 4881 | Subclass relationship for ... |
| unieq 4882 | Equality theorem for class... |
| unieqi 4883 | Inference of equality of t... |
| unieqd 4884 | Deduction of equality of t... |
| eluniab 4885 | Membership in union of a c... |
| elunirab 4886 | Membership in union of a c... |
| uniprg 4887 | The union of a pair is the... |
| unipr 4888 | The union of a pair is the... |
| unisng 4889 | A set equals the union of ... |
| unisn 4890 | A set equals the union of ... |
| unisnv 4891 | A set equals the union of ... |
| unisn3 4892 | Union of a singleton in th... |
| dfnfc2 4893 | An alternative statement o... |
| uniun 4894 | The class union of the uni... |
| uniin 4895 | The class union of the int... |
| ssuni 4896 | Subclass relationship for ... |
| uni0b 4897 | The union of a set is empt... |
| uni0c 4898 | The union of a set is empt... |
| uni0 4899 | The union of the empty set... |
| csbuni 4900 | Distribute proper substitu... |
| elssuni 4901 | An element of a class is a... |
| unissel 4902 | Condition turning a subcla... |
| unissb 4903 | Relationship involving mem... |
| unissbOLD 4904 | Obsolete version of ~ unis... |
| uniss2 4905 | A subclass condition on th... |
| unidif 4906 | If the difference ` A \ B ... |
| ssunieq 4907 | Relationship implying unio... |
| unimax 4908 | Any member of a class is t... |
| pwuni 4909 | A class is a subclass of t... |
| dfint2 4912 | Alternate definition of cl... |
| inteq 4913 | Equality law for intersect... |
| inteqi 4914 | Equality inference for cla... |
| inteqd 4915 | Equality deduction for cla... |
| elint 4916 | Membership in class inters... |
| elint2 4917 | Membership in class inters... |
| elintg 4918 | Membership in class inters... |
| elinti 4919 | Membership in class inters... |
| nfint 4920 | Bound-variable hypothesis ... |
| elintabg 4921 | Two ways of saying a set i... |
| elintab 4922 | Membership in the intersec... |
| elintabOLD 4923 | Obsolete version of ~ elin... |
| elintrab 4924 | Membership in the intersec... |
| elintrabg 4925 | Membership in the intersec... |
| int0 4926 | The intersection of the em... |
| intss1 4927 | An element of a class incl... |
| ssint 4928 | Subclass of a class inters... |
| ssintab 4929 | Subclass of the intersecti... |
| ssintub 4930 | Subclass of the least uppe... |
| ssmin 4931 | Subclass of the minimum va... |
| intmin 4932 | Any member of a class is t... |
| intss 4933 | Intersection of subclasses... |
| intssuni 4934 | The intersection of a none... |
| ssintrab 4935 | Subclass of the intersecti... |
| unissint 4936 | If the union of a class is... |
| intssuni2 4937 | Subclass relationship for ... |
| intminss 4938 | Under subset ordering, the... |
| intmin2 4939 | Any set is the smallest of... |
| intmin3 4940 | Under subset ordering, the... |
| intmin4 4941 | Elimination of a conjunct ... |
| intab 4942 | The intersection of a spec... |
| int0el 4943 | The intersection of a clas... |
| intun 4944 | The class intersection of ... |
| intprg 4945 | The intersection of a pair... |
| intpr 4946 | The intersection of a pair... |
| intsng 4947 | Intersection of a singleto... |
| intsn 4948 | The intersection of a sing... |
| uniintsn 4949 | Two ways to express " ` A ... |
| uniintab 4950 | The union and the intersec... |
| intunsn 4951 | Theorem joining a singleto... |
| rint0 4952 | Relative intersection of a... |
| elrint 4953 | Membership in a restricted... |
| elrint2 4954 | Membership in a restricted... |
| eliun 4959 | Membership in indexed unio... |
| eliin 4960 | Membership in indexed inte... |
| eliuni 4961 | Membership in an indexed u... |
| eliund 4962 | Membership in indexed unio... |
| iuncom 4963 | Commutation of indexed uni... |
| iuncom4 4964 | Commutation of union with ... |
| iunconst 4965 | Indexed union of a constan... |
| iinconst 4966 | Indexed intersection of a ... |
| iuneqconst 4967 | Indexed union of identical... |
| iuniin 4968 | Law combining indexed unio... |
| iinssiun 4969 | An indexed intersection is... |
| iunss1 4970 | Subclass theorem for index... |
| iinss1 4971 | Subclass theorem for index... |
| iuneq1 4972 | Equality theorem for index... |
| iineq1 4973 | Equality theorem for index... |
| ss2iun 4974 | Subclass theorem for index... |
| iuneq2 4975 | Equality theorem for index... |
| iineq2 4976 | Equality theorem for index... |
| iuneq2i 4977 | Equality inference for ind... |
| iineq2i 4978 | Equality inference for ind... |
| iineq2d 4979 | Equality deduction for ind... |
| iuneq2dv 4980 | Equality deduction for ind... |
| iineq2dv 4981 | Equality deduction for ind... |
| iuneq12df 4982 | Equality deduction for ind... |
| iuneq1d 4983 | Equality theorem for index... |
| iuneq12dOLD 4984 | Obsolete version of ~ iune... |
| iuneq12d 4985 | Equality deduction for ind... |
| iuneq2d 4986 | Equality deduction for ind... |
| nfiun 4987 | Bound-variable hypothesis ... |
| nfiin 4988 | Bound-variable hypothesis ... |
| nfiung 4989 | Bound-variable hypothesis ... |
| nfiing 4990 | Bound-variable hypothesis ... |
| nfiu1 4991 | Bound-variable hypothesis ... |
| nfiu1OLD 4992 | Obsolete version of ~ nfiu... |
| nfii1 4993 | Bound-variable hypothesis ... |
| dfiun2g 4994 | Alternate definition of in... |
| dfiun2gOLD 4995 | Obsolete version of ~ dfiu... |
| dfiin2g 4996 | Alternate definition of in... |
| dfiun2 4997 | Alternate definition of in... |
| dfiin2 4998 | Alternate definition of in... |
| dfiunv2 4999 | Define double indexed unio... |
| cbviun 5000 | Rule used to change the bo... |
| cbviin 5001 | Change bound variables in ... |
| cbviung 5002 | Rule used to change the bo... |
| cbviing 5003 | Change bound variables in ... |
| cbviunv 5004 | Rule used to change the bo... |
| cbviinv 5005 | Change bound variables in ... |
| cbviunvg 5006 | Rule used to change the bo... |
| cbviinvg 5007 | Change bound variables in ... |
| iunssf 5008 | Subset theorem for an inde... |
| iunss 5009 | Subset theorem for an inde... |
| ssiun 5010 | Subset implication for an ... |
| ssiun2 5011 | Identity law for subset of... |
| ssiun2s 5012 | Subset relationship for an... |
| iunss2 5013 | A subclass condition on th... |
| iunssd 5014 | Subset theorem for an inde... |
| iunab 5015 | The indexed union of a cla... |
| iunrab 5016 | The indexed union of a res... |
| iunxdif2 5017 | Indexed union with a class... |
| ssiinf 5018 | Subset theorem for an inde... |
| ssiin 5019 | Subset theorem for an inde... |
| iinss 5020 | Subset implication for an ... |
| iinss2 5021 | An indexed intersection is... |
| uniiun 5022 | Class union in terms of in... |
| intiin 5023 | Class intersection in term... |
| iunid 5024 | An indexed union of single... |
| iunidOLD 5025 | Obsolete version of ~ iuni... |
| iun0 5026 | An indexed union of the em... |
| 0iun 5027 | An empty indexed union is ... |
| 0iin 5028 | An empty indexed intersect... |
| viin 5029 | Indexed intersection with ... |
| iunsn 5030 | Indexed union of a singlet... |
| iunn0 5031 | There is a nonempty class ... |
| iinab 5032 | Indexed intersection of a ... |
| iinrab 5033 | Indexed intersection of a ... |
| iinrab2 5034 | Indexed intersection of a ... |
| iunin2 5035 | Indexed union of intersect... |
| iunin1 5036 | Indexed union of intersect... |
| iinun2 5037 | Indexed intersection of un... |
| iundif2 5038 | Indexed union of class dif... |
| iindif1 5039 | Indexed intersection of cl... |
| 2iunin 5040 | Rearrange indexed unions o... |
| iindif2 5041 | Indexed intersection of cl... |
| iinin2 5042 | Indexed intersection of in... |
| iinin1 5043 | Indexed intersection of in... |
| iinvdif 5044 | The indexed intersection o... |
| elriin 5045 | Elementhood in a relative ... |
| riin0 5046 | Relative intersection of a... |
| riinn0 5047 | Relative intersection of a... |
| riinrab 5048 | Relative intersection of a... |
| symdif0 5049 | Symmetric difference with ... |
| symdifv 5050 | The symmetric difference w... |
| symdifid 5051 | The symmetric difference o... |
| iinxsng 5052 | A singleton index picks ou... |
| iinxprg 5053 | Indexed intersection with ... |
| iunxsng 5054 | A singleton index picks ou... |
| iunxsn 5055 | A singleton index picks ou... |
| iunxsngf 5056 | A singleton index picks ou... |
| iunun 5057 | Separate a union in an ind... |
| iunxun 5058 | Separate a union in the in... |
| iunxdif3 5059 | An indexed union where som... |
| iunxprg 5060 | A pair index picks out two... |
| iunxiun 5061 | Separate an indexed union ... |
| iinuni 5062 | A relationship involving u... |
| iununi 5063 | A relationship involving u... |
| sspwuni 5064 | Subclass relationship for ... |
| pwssb 5065 | Two ways to express a coll... |
| elpwpw 5066 | Characterization of the el... |
| pwpwab 5067 | The double power class wri... |
| pwpwssunieq 5068 | The class of sets whose un... |
| elpwuni 5069 | Relationship for power cla... |
| iinpw 5070 | The power class of an inte... |
| iunpwss 5071 | Inclusion of an indexed un... |
| intss2 5072 | A nonempty intersection of... |
| rintn0 5073 | Relative intersection of a... |
| dfdisj2 5076 | Alternate definition for d... |
| disjss2 5077 | If each element of a colle... |
| disjeq2 5078 | Equality theorem for disjo... |
| disjeq2dv 5079 | Equality deduction for dis... |
| disjss1 5080 | A subset of a disjoint col... |
| disjeq1 5081 | Equality theorem for disjo... |
| disjeq1d 5082 | Equality theorem for disjo... |
| disjeq12d 5083 | Equality theorem for disjo... |
| cbvdisj 5084 | Change bound variables in ... |
| cbvdisjv 5085 | Change bound variables in ... |
| nfdisjw 5086 | Bound-variable hypothesis ... |
| nfdisj 5087 | Bound-variable hypothesis ... |
| nfdisj1 5088 | Bound-variable hypothesis ... |
| disjor 5089 | Two ways to say that a col... |
| disjors 5090 | Two ways to say that a col... |
| disji2 5091 | Property of a disjoint col... |
| disji 5092 | Property of a disjoint col... |
| invdisj 5093 | If there is a function ` C... |
| invdisjrab 5094 | The restricted class abstr... |
| disjiun 5095 | A disjoint collection yiel... |
| disjord 5096 | Conditions for a collectio... |
| disjiunb 5097 | Two ways to say that a col... |
| disjiund 5098 | Conditions for a collectio... |
| sndisj 5099 | Any collection of singleto... |
| 0disj 5100 | Any collection of empty se... |
| disjxsn 5101 | A singleton collection is ... |
| disjx0 5102 | An empty collection is dis... |
| disjprg 5103 | A pair collection is disjo... |
| disjxiun 5104 | An indexed union of a disj... |
| disjxun 5105 | The union of two disjoint ... |
| disjss3 5106 | Expand a disjoint collecti... |
| breq 5109 | Equality theorem for binar... |
| breq1 5110 | Equality theorem for a bin... |
| breq2 5111 | Equality theorem for a bin... |
| breq12 5112 | Equality theorem for a bin... |
| breqi 5113 | Equality inference for bin... |
| breq1i 5114 | Equality inference for a b... |
| breq2i 5115 | Equality inference for a b... |
| breq12i 5116 | Equality inference for a b... |
| breq1d 5117 | Equality deduction for a b... |
| breqd 5118 | Equality deduction for a b... |
| breq2d 5119 | Equality deduction for a b... |
| breq12d 5120 | Equality deduction for a b... |
| breq123d 5121 | Equality deduction for a b... |
| breqdi 5122 | Equality deduction for a b... |
| breqan12d 5123 | Equality deduction for a b... |
| breqan12rd 5124 | Equality deduction for a b... |
| eqnbrtrd 5125 | Substitution of equal clas... |
| nbrne1 5126 | Two classes are different ... |
| nbrne2 5127 | Two classes are different ... |
| eqbrtri 5128 | Substitution of equal clas... |
| eqbrtrd 5129 | Substitution of equal clas... |
| eqbrtrri 5130 | Substitution of equal clas... |
| eqbrtrrd 5131 | Substitution of equal clas... |
| breqtri 5132 | Substitution of equal clas... |
| breqtrd 5133 | Substitution of equal clas... |
| breqtrri 5134 | Substitution of equal clas... |
| breqtrrd 5135 | Substitution of equal clas... |
| 3brtr3i 5136 | Substitution of equality i... |
| 3brtr4i 5137 | Substitution of equality i... |
| 3brtr3d 5138 | Substitution of equality i... |
| 3brtr4d 5139 | Substitution of equality i... |
| 3brtr3g 5140 | Substitution of equality i... |
| 3brtr4g 5141 | Substitution of equality i... |
| eqbrtrid 5142 | A chained equality inferen... |
| eqbrtrrid 5143 | A chained equality inferen... |
| breqtrid 5144 | A chained equality inferen... |
| breqtrrid 5145 | A chained equality inferen... |
| eqbrtrdi 5146 | A chained equality inferen... |
| eqbrtrrdi 5147 | A chained equality inferen... |
| breqtrdi 5148 | A chained equality inferen... |
| breqtrrdi 5149 | A chained equality inferen... |
| ssbrd 5150 | Deduction from a subclass ... |
| ssbr 5151 | Implication from a subclas... |
| ssbri 5152 | Inference from a subclass ... |
| nfbrd 5153 | Deduction version of bound... |
| nfbr 5154 | Bound-variable hypothesis ... |
| brab1 5155 | Relationship between a bin... |
| br0 5156 | The empty binary relation ... |
| brne0 5157 | If two sets are in a binar... |
| brun 5158 | The union of two binary re... |
| brin 5159 | The intersection of two re... |
| brdif 5160 | The difference of two bina... |
| sbcbr123 5161 | Move substitution in and o... |
| sbcbr 5162 | Move substitution in and o... |
| sbcbr12g 5163 | Move substitution in and o... |
| sbcbr1g 5164 | Move substitution in and o... |
| sbcbr2g 5165 | Move substitution in and o... |
| brsymdif 5166 | Characterization of the sy... |
| brralrspcev 5167 | Restricted existential spe... |
| brimralrspcev 5168 | Restricted existential spe... |
| opabss 5171 | The collection of ordered ... |
| opabbid 5172 | Equivalent wff's yield equ... |
| opabbidv 5173 | Equivalent wff's yield equ... |
| opabbii 5174 | Equivalent wff's yield equ... |
| nfopabd 5175 | Bound-variable hypothesis ... |
| nfopab 5176 | Bound-variable hypothesis ... |
| nfopab1 5177 | The first abstraction vari... |
| nfopab2 5178 | The second abstraction var... |
| cbvopab 5179 | Rule used to change bound ... |
| cbvopabv 5180 | Rule used to change bound ... |
| cbvopab1 5181 | Change first bound variabl... |
| cbvopab1g 5182 | Change first bound variabl... |
| cbvopab2 5183 | Change second bound variab... |
| cbvopab1s 5184 | Change first bound variabl... |
| cbvopab1v 5185 | Rule used to change the fi... |
| cbvopab2v 5186 | Rule used to change the se... |
| unopab 5187 | Union of two ordered pair ... |
| mpteq12da 5190 | An equality inference for ... |
| mpteq12df 5191 | An equality inference for ... |
| mpteq12f 5192 | An equality theorem for th... |
| mpteq12dva 5193 | An equality inference for ... |
| mpteq12dv 5194 | An equality inference for ... |
| mpteq12 5195 | An equality theorem for th... |
| mpteq1 5196 | An equality theorem for th... |
| mpteq1d 5197 | An equality theorem for th... |
| mpteq1i 5198 | An equality theorem for th... |
| mpteq2da 5199 | Slightly more general equa... |
| mpteq2dva 5200 | Slightly more general equa... |
| mpteq2dv 5201 | An equality inference for ... |
| mpteq2ia 5202 | An equality inference for ... |
| mpteq2i 5203 | An equality inference for ... |
| mpteq12i 5204 | An equality inference for ... |
| nfmpt 5205 | Bound-variable hypothesis ... |
| nfmpt1 5206 | Bound-variable hypothesis ... |
| cbvmptf 5207 | Rule to change the bound v... |
| cbvmptfg 5208 | Rule to change the bound v... |
| cbvmpt 5209 | Rule to change the bound v... |
| cbvmptg 5210 | Rule to change the bound v... |
| cbvmptv 5211 | Rule to change the bound v... |
| cbvmptvg 5212 | Rule to change the bound v... |
| mptv 5213 | Function with universal do... |
| dftr2 5216 | An alternate way of defini... |
| dftr2c 5217 | Variant of ~ dftr2 with co... |
| dftr5 5218 | An alternate way of defini... |
| dftr5OLD 5219 | Obsolete version of ~ dftr... |
| dftr3 5220 | An alternate way of defini... |
| dftr4 5221 | An alternate way of defini... |
| treq 5222 | Equality theorem for the t... |
| trel 5223 | In a transitive class, the... |
| trel3 5224 | In a transitive class, the... |
| trss 5225 | An element of a transitive... |
| trin 5226 | The intersection of transi... |
| tr0 5227 | The empty set is transitiv... |
| trv 5228 | The universe is transitive... |
| triun 5229 | An indexed union of a clas... |
| truni 5230 | The union of a class of tr... |
| triin 5231 | An indexed intersection of... |
| trint 5232 | The intersection of a clas... |
| trintss 5233 | Any nonempty transitive cl... |
| axrep1 5235 | The version of the Axiom o... |
| axreplem 5236 | Lemma for ~ axrep2 and ~ a... |
| axrep2 5237 | Axiom of Replacement expre... |
| axrep3 5238 | Axiom of Replacement sligh... |
| axrep4v 5239 | Version of ~ axrep4 with a... |
| axrep4 5240 | A more traditional version... |
| axrep4OLD 5241 | Obsolete version of ~ axre... |
| axrep5 5242 | Axiom of Replacement (simi... |
| axrep6 5243 | A condensed form of ~ ax-r... |
| axrep6OLD 5244 | Obsolete version of ~ axre... |
| axrep6g 5245 | ~ axrep6 in class notation... |
| zfrepclf 5246 | An inference based on the ... |
| zfrep3cl 5247 | An inference based on the ... |
| zfrep4 5248 | A version of Replacement u... |
| axsepgfromrep 5249 | A more general version ~ a... |
| axsep 5250 | Axiom scheme of separation... |
| axsepg 5252 | A more general version of ... |
| zfauscl 5253 | Separation Scheme (Aussond... |
| sepexlem 5254 | Lemma for ~ sepex . Use ~... |
| sepex 5255 | Convert implication to equ... |
| sepexi 5256 | Convert implication to equ... |
| bm1.3iiOLD 5257 | Obsolete version of ~ sepe... |
| ax6vsep 5258 | Derive ~ ax6v (a weakened ... |
| axnulALT 5259 | Alternate proof of ~ axnul... |
| axnul 5260 | The Null Set Axiom of ZF s... |
| 0ex 5262 | The Null Set Axiom of ZF s... |
| al0ssb 5263 | The empty set is the uniqu... |
| sseliALT 5264 | Alternate proof of ~ sseli... |
| csbexg 5265 | The existence of proper su... |
| csbex 5266 | The existence of proper su... |
| unisn2 5267 | A version of ~ unisn witho... |
| nalset 5268 | No set contains all sets. ... |
| vnex 5269 | The universal class does n... |
| vprc 5270 | The universal class is not... |
| nvel 5271 | The universal class does n... |
| inex1 5272 | Separation Scheme (Aussond... |
| inex2 5273 | Separation Scheme (Aussond... |
| inex1g 5274 | Closed-form, generalized S... |
| inex2g 5275 | Sufficient condition for a... |
| ssex 5276 | The subset of a set is als... |
| ssexi 5277 | The subset of a set is als... |
| ssexg 5278 | The subset of a set is als... |
| ssexd 5279 | A subclass of a set is a s... |
| abexd 5280 | Conditions for a class abs... |
| abex 5281 | Conditions for a class abs... |
| prcssprc 5282 | The superclass of a proper... |
| sselpwd 5283 | Elementhood to a power set... |
| difexg 5284 | Existence of a difference.... |
| difexi 5285 | Existence of a difference,... |
| difexd 5286 | Existence of a difference.... |
| zfausab 5287 | Separation Scheme (Aussond... |
| elpw2g 5288 | Membership in a power clas... |
| elpw2 5289 | Membership in a power clas... |
| elpwi2 5290 | Membership in a power clas... |
| rabelpw 5291 | A restricted class abstrac... |
| rabexg 5292 | Separation Scheme in terms... |
| rabexgOLD 5293 | Obsolete version of ~ rabe... |
| rabex 5294 | Separation Scheme in terms... |
| rabexd 5295 | Separation Scheme in terms... |
| rabex2 5296 | Separation Scheme in terms... |
| rab2ex 5297 | A class abstraction based ... |
| elssabg 5298 | Membership in a class abst... |
| intex 5299 | The intersection of a none... |
| intnex 5300 | If a class intersection is... |
| intexab 5301 | The intersection of a none... |
| intexrab 5302 | The intersection of a none... |
| iinexg 5303 | The existence of a class i... |
| intabs 5304 | Absorption of a redundant ... |
| inuni 5305 | The intersection of a unio... |
| axpweq 5306 | Two equivalent ways to exp... |
| pwnss 5307 | The power set of a set is ... |
| pwne 5308 | No set equals its power se... |
| difelpw 5309 | A difference is an element... |
| class2set 5310 | The class of elements of `... |
| 0elpw 5311 | Every power class contains... |
| pwne0 5312 | A power class is never emp... |
| 0nep0 5313 | The empty set and its powe... |
| 0inp0 5314 | Something cannot be equal ... |
| unidif0 5315 | The removal of the empty s... |
| eqsnuniex 5316 | If a class is equal to the... |
| iin0 5317 | An indexed intersection of... |
| notzfaus 5318 | In the Separation Scheme ~... |
| intv 5319 | The intersection of the un... |
| zfpow 5321 | Axiom of Power Sets expres... |
| axpow2 5322 | A variant of the Axiom of ... |
| axpow3 5323 | A variant of the Axiom of ... |
| elALT2 5324 | Alternate proof of ~ el us... |
| dtruALT2 5325 | Alternate proof of ~ dtru ... |
| dtrucor 5326 | Corollary of ~ dtru . Thi... |
| dtrucor2 5327 | The theorem form of the de... |
| dvdemo1 5328 | Demonstration of a theorem... |
| dvdemo2 5329 | Demonstration of a theorem... |
| nfnid 5330 | A setvar variable is not f... |
| nfcvb 5331 | The "distinctor" expressio... |
| vpwex 5332 | Power set axiom: the power... |
| pwexg 5333 | Power set axiom expressed ... |
| pwexd 5334 | Deduction version of the p... |
| pwex 5335 | Power set axiom expressed ... |
| pwel 5336 | Quantitative version of ~ ... |
| abssexg 5337 | Existence of a class of su... |
| snexALT 5338 | Alternate proof of ~ snex ... |
| p0ex 5339 | The power set of the empty... |
| p0exALT 5340 | Alternate proof of ~ p0ex ... |
| pp0ex 5341 | The power set of the power... |
| ord3ex 5342 | The ordinal number 3 is a ... |
| dtruALT 5343 | Alternate proof of ~ dtru ... |
| axc16b 5344 | This theorem shows that Ax... |
| eunex 5345 | Existential uniqueness imp... |
| eusv1 5346 | Two ways to express single... |
| eusvnf 5347 | Even if ` x ` is free in `... |
| eusvnfb 5348 | Two ways to say that ` A (... |
| eusv2i 5349 | Two ways to express single... |
| eusv2nf 5350 | Two ways to express single... |
| eusv2 5351 | Two ways to express single... |
| reusv1 5352 | Two ways to express single... |
| reusv2lem1 5353 | Lemma for ~ reusv2 . (Con... |
| reusv2lem2 5354 | Lemma for ~ reusv2 . (Con... |
| reusv2lem3 5355 | Lemma for ~ reusv2 . (Con... |
| reusv2lem4 5356 | Lemma for ~ reusv2 . (Con... |
| reusv2lem5 5357 | Lemma for ~ reusv2 . (Con... |
| reusv2 5358 | Two ways to express single... |
| reusv3i 5359 | Two ways of expressing exi... |
| reusv3 5360 | Two ways to express single... |
| eusv4 5361 | Two ways to express single... |
| alxfr 5362 | Transfer universal quantif... |
| ralxfrd 5363 | Transfer universal quantif... |
| rexxfrd 5364 | Transfer existential quant... |
| ralxfr2d 5365 | Transfer universal quantif... |
| rexxfr2d 5366 | Transfer existential quant... |
| ralxfrd2 5367 | Transfer universal quantif... |
| rexxfrd2 5368 | Transfer existence from a ... |
| ralxfr 5369 | Transfer universal quantif... |
| ralxfrALT 5370 | Alternate proof of ~ ralxf... |
| rexxfr 5371 | Transfer existence from a ... |
| rabxfrd 5372 | Membership in a restricted... |
| rabxfr 5373 | Membership in a restricted... |
| reuhypd 5374 | A theorem useful for elimi... |
| reuhyp 5375 | A theorem useful for elimi... |
| zfpair 5376 | The Axiom of Pairing of Ze... |
| axprALT 5377 | Alternate proof of ~ axpr ... |
| axprlem1 5378 | Lemma for ~ axpr . There ... |
| axprlem2 5379 | Lemma for ~ axpr . There ... |
| axprlem3 5380 | Lemma for ~ axpr . Elimin... |
| axprlem4 5381 | Lemma for ~ axpr . If an ... |
| axpr 5382 | Unabbreviated version of t... |
| axprlem3OLD 5383 | Obsolete version of ~ axpr... |
| axprlem4OLD 5384 | Obsolete version of ~ axpr... |
| axprlem5OLD 5385 | Obsolete version of ~ axpr... |
| axprOLD 5386 | Obsolete version of ~ axpr... |
| zfpair2 5388 | Derive the abbreviated ver... |
| vsnex 5389 | A singleton built on a set... |
| snexg 5390 | A singleton built on a set... |
| snex 5391 | A singleton is a set. The... |
| prex 5392 | The Axiom of Pairing using... |
| exel 5393 | There exist two sets, one ... |
| exexneq 5394 | There exist two different ... |
| exneq 5395 | Given any set (the " ` y `... |
| dtru 5396 | Given any set (the " ` y `... |
| el 5397 | Any set is an element of s... |
| sels 5398 | If a class is a set, then ... |
| selsALT 5399 | Alternate proof of ~ sels ... |
| elALT 5400 | Alternate proof of ~ el , ... |
| dtruOLD 5401 | Obsolete version of ~ dtru... |
| snelpwg 5402 | A singleton of a set is a ... |
| snelpwi 5403 | If a set is a member of a ... |
| snelpwiOLD 5404 | Obsolete version of ~ snel... |
| snelpw 5405 | A singleton of a set is a ... |
| prelpw 5406 | An unordered pair of two s... |
| prelpwi 5407 | If two sets are members of... |
| rext 5408 | A theorem similar to exten... |
| sspwb 5409 | The powerclass constructio... |
| unipw 5410 | A class equals the union o... |
| univ 5411 | The union of the universe ... |
| pwtr 5412 | A class is transitive iff ... |
| ssextss 5413 | An extensionality-like pri... |
| ssext 5414 | An extensionality-like pri... |
| nssss 5415 | Negation of subclass relat... |
| pweqb 5416 | Classes are equal if and o... |
| intidg 5417 | The intersection of all se... |
| intidOLD 5418 | Obsolete version of ~ inti... |
| moabex 5419 | "At most one" existence im... |
| rmorabex 5420 | Restricted "at most one" e... |
| euabex 5421 | The abstraction of a wff w... |
| nnullss 5422 | A nonempty class (even if ... |
| exss 5423 | Restricted existence in a ... |
| opex 5424 | An ordered pair of classes... |
| otex 5425 | An ordered triple of class... |
| elopg 5426 | Characterization of the el... |
| elop 5427 | Characterization of the el... |
| opi1 5428 | One of the two elements in... |
| opi2 5429 | One of the two elements of... |
| opeluu 5430 | Each member of an ordered ... |
| op1stb 5431 | Extract the first member o... |
| brv 5432 | Two classes are always in ... |
| opnz 5433 | An ordered pair is nonempt... |
| opnzi 5434 | An ordered pair is nonempt... |
| opth1 5435 | Equality of the first memb... |
| opth 5436 | The ordered pair theorem. ... |
| opthg 5437 | Ordered pair theorem. ` C ... |
| opth1g 5438 | Equality of the first memb... |
| opthg2 5439 | Ordered pair theorem. (Co... |
| opth2 5440 | Ordered pair theorem. (Co... |
| opthneg 5441 | Two ordered pairs are not ... |
| opthne 5442 | Two ordered pairs are not ... |
| otth2 5443 | Ordered triple theorem, wi... |
| otth 5444 | Ordered triple theorem. (... |
| otthg 5445 | Ordered triple theorem, cl... |
| otthne 5446 | Contrapositive of the orde... |
| eqvinop 5447 | A variable introduction la... |
| sbcop1 5448 | The proper substitution of... |
| sbcop 5449 | The proper substitution of... |
| copsexgw 5450 | Version of ~ copsexg with ... |
| copsexg 5451 | Substitution of class ` A ... |
| copsex2t 5452 | Closed theorem form of ~ c... |
| copsex2g 5453 | Implicit substitution infe... |
| copsex2dv 5454 | Implicit substitution dedu... |
| copsex4g 5455 | An implicit substitution i... |
| 0nelop 5456 | A property of ordered pair... |
| opwo0id 5457 | An ordered pair is equal t... |
| opeqex 5458 | Equivalence of existence i... |
| oteqex2 5459 | Equivalence of existence i... |
| oteqex 5460 | Equivalence of existence i... |
| opcom 5461 | An ordered pair commutes i... |
| moop2 5462 | "At most one" property of ... |
| opeqsng 5463 | Equivalence for an ordered... |
| opeqsn 5464 | Equivalence for an ordered... |
| opeqpr 5465 | Equivalence for an ordered... |
| snopeqop 5466 | Equivalence for an ordered... |
| propeqop 5467 | Equivalence for an ordered... |
| propssopi 5468 | If a pair of ordered pairs... |
| snopeqopsnid 5469 | Equivalence for an ordered... |
| mosubopt 5470 | "At most one" remains true... |
| mosubop 5471 | "At most one" remains true... |
| euop2 5472 | Transfer existential uniqu... |
| euotd 5473 | Prove existential uniquene... |
| opthwiener 5474 | Justification theorem for ... |
| uniop 5475 | The union of an ordered pa... |
| uniopel 5476 | Ordered pair membership is... |
| opthhausdorff 5477 | Justification theorem for ... |
| opthhausdorff0 5478 | Justification theorem for ... |
| otsndisj 5479 | The singletons consisting ... |
| otiunsndisj 5480 | The union of singletons co... |
| iunopeqop 5481 | Implication of an ordered ... |
| brsnop 5482 | Binary relation for an ord... |
| brtp 5483 | A necessary and sufficient... |
| opabidw 5484 | The law of concretion. Sp... |
| opabid 5485 | The law of concretion. Sp... |
| elopabw 5486 | Membership in a class abst... |
| elopab 5487 | Membership in a class abst... |
| rexopabb 5488 | Restricted existential qua... |
| vopelopabsb 5489 | The law of concretion in t... |
| opelopabsb 5490 | The law of concretion in t... |
| brabsb 5491 | The law of concretion in t... |
| opelopabt 5492 | Closed theorem form of ~ o... |
| opelopabga 5493 | The law of concretion. Th... |
| brabga 5494 | The law of concretion for ... |
| opelopab2a 5495 | Ordered pair membership in... |
| opelopaba 5496 | The law of concretion. Th... |
| braba 5497 | The law of concretion for ... |
| opelopabg 5498 | The law of concretion. Th... |
| brabg 5499 | The law of concretion for ... |
| opelopabgf 5500 | The law of concretion. Th... |
| opelopab2 5501 | Ordered pair membership in... |
| opelopab 5502 | The law of concretion. Th... |
| brab 5503 | The law of concretion for ... |
| opelopabaf 5504 | The law of concretion. Th... |
| opelopabf 5505 | The law of concretion. Th... |
| ssopab2 5506 | Equivalence of ordered pai... |
| ssopab2bw 5507 | Equivalence of ordered pai... |
| eqopab2bw 5508 | Equivalence of ordered pai... |
| ssopab2b 5509 | Equivalence of ordered pai... |
| ssopab2i 5510 | Inference of ordered pair ... |
| ssopab2dv 5511 | Inference of ordered pair ... |
| eqopab2b 5512 | Equivalence of ordered pai... |
| opabn0 5513 | Nonempty ordered pair clas... |
| opab0 5514 | Empty ordered pair class a... |
| csbopab 5515 | Move substitution into a c... |
| csbopabgALT 5516 | Move substitution into a c... |
| csbmpt12 5517 | Move substitution into a m... |
| csbmpt2 5518 | Move substitution into the... |
| iunopab 5519 | Move indexed union inside ... |
| iunopabOLD 5520 | Obsolete version of ~ iuno... |
| elopabr 5521 | Membership in an ordered-p... |
| elopabran 5522 | Membership in an ordered-p... |
| elopabrOLD 5523 | Obsolete version of ~ elop... |
| rbropapd 5524 | Properties of a pair in an... |
| rbropap 5525 | Properties of a pair in a ... |
| 2rbropap 5526 | Properties of a pair in a ... |
| 0nelopab 5527 | The empty set is never an ... |
| brabv 5528 | If two classes are in a re... |
| pwin 5529 | The power class of the int... |
| pwssun 5530 | The power class of the uni... |
| pwun 5531 | The power class of the uni... |
| dfid4 5534 | The identity function expr... |
| dfid2 5535 | Alternate definition of th... |
| dfid3 5536 | A stronger version of ~ df... |
| epelg 5539 | The membership relation an... |
| epeli 5540 | The membership relation an... |
| epel 5541 | The membership relation an... |
| 0sn0ep 5542 | An example for the members... |
| epn0 5543 | The membership relation is... |
| poss 5548 | Subset theorem for the par... |
| poeq1 5549 | Equality theorem for parti... |
| poeq2 5550 | Equality theorem for parti... |
| poeq12d 5551 | Equality deduction for par... |
| nfpo 5552 | Bound-variable hypothesis ... |
| nfso 5553 | Bound-variable hypothesis ... |
| pocl 5554 | Characteristic properties ... |
| ispod 5555 | Sufficient conditions for ... |
| swopolem 5556 | Perform the substitutions ... |
| swopo 5557 | A strict weak order is a p... |
| poirr 5558 | A partial order is irrefle... |
| potr 5559 | A partial order is a trans... |
| po2nr 5560 | A partial order has no 2-c... |
| po3nr 5561 | A partial order has no 3-c... |
| po2ne 5562 | Two sets related by a part... |
| po0 5563 | Any relation is a partial ... |
| pofun 5564 | The inverse image of a par... |
| sopo 5565 | A strict linear order is a... |
| soss 5566 | Subset theorem for the str... |
| soeq1 5567 | Equality theorem for the s... |
| soeq2 5568 | Equality theorem for the s... |
| soeq12d 5569 | Equality deduction for tot... |
| sonr 5570 | A strict order relation is... |
| sotr 5571 | A strict order relation is... |
| sotrd 5572 | Transitivity law for stric... |
| solin 5573 | A strict order relation is... |
| so2nr 5574 | A strict order relation ha... |
| so3nr 5575 | A strict order relation ha... |
| sotric 5576 | A strict order relation sa... |
| sotrieq 5577 | Trichotomy law for strict ... |
| sotrieq2 5578 | Trichotomy law for strict ... |
| soasym 5579 | Asymmetry law for strict o... |
| sotr2 5580 | A transitivity relation. ... |
| issod 5581 | An irreflexive, transitive... |
| issoi 5582 | An irreflexive, transitive... |
| isso2i 5583 | Deduce strict ordering fro... |
| so0 5584 | Any relation is a strict o... |
| somo 5585 | A totally ordered set has ... |
| sotrine 5586 | Trichotomy law for strict ... |
| sotr3 5587 | Transitivity law for stric... |
| dffr6 5594 | Alternate definition of ~ ... |
| frd 5595 | A nonempty subset of an ` ... |
| fri 5596 | A nonempty subset of an ` ... |
| seex 5597 | The ` R ` -preimage of an ... |
| exse 5598 | Any relation on a set is s... |
| dffr2 5599 | Alternate definition of we... |
| dffr2ALT 5600 | Alternate proof of ~ dffr2... |
| frc 5601 | Property of well-founded r... |
| frss 5602 | Subset theorem for the wel... |
| sess1 5603 | Subset theorem for the set... |
| sess2 5604 | Subset theorem for the set... |
| freq1 5605 | Equality theorem for the w... |
| freq2 5606 | Equality theorem for the w... |
| freq12d 5607 | Equality deduction for wel... |
| seeq1 5608 | Equality theorem for the s... |
| seeq2 5609 | Equality theorem for the s... |
| seeq12d 5610 | Equality deduction for the... |
| nffr 5611 | Bound-variable hypothesis ... |
| nfse 5612 | Bound-variable hypothesis ... |
| nfwe 5613 | Bound-variable hypothesis ... |
| frirr 5614 | A well-founded relation is... |
| fr2nr 5615 | A well-founded relation ha... |
| fr0 5616 | Any relation is well-found... |
| frminex 5617 | If an element of a well-fo... |
| efrirr 5618 | A well-founded class does ... |
| efrn2lp 5619 | A well-founded class conta... |
| epse 5620 | The membership relation is... |
| tz7.2 5621 | Similar to Theorem 7.2 of ... |
| dfepfr 5622 | An alternate way of saying... |
| epfrc 5623 | A subset of a well-founded... |
| wess 5624 | Subset theorem for the wel... |
| weeq1 5625 | Equality theorem for the w... |
| weeq2 5626 | Equality theorem for the w... |
| weeq12d 5627 | Equality deduction for wel... |
| wefr 5628 | A well-ordering is well-fo... |
| weso 5629 | A well-ordering is a stric... |
| wecmpep 5630 | The elements of a class we... |
| wetrep 5631 | On a class well-ordered by... |
| wefrc 5632 | A nonempty subclass of a c... |
| we0 5633 | Any relation is a well-ord... |
| wereu 5634 | A nonempty subset of an ` ... |
| wereu2 5635 | A nonempty subclass of an ... |
| xpeq1 5652 | Equality theorem for Carte... |
| xpss12 5653 | Subset theorem for Cartesi... |
| xpss 5654 | A Cartesian product is inc... |
| inxpssres 5655 | Intersection with a Cartes... |
| relxp 5656 | A Cartesian product is a r... |
| xpss1 5657 | Subset relation for Cartes... |
| xpss2 5658 | Subset relation for Cartes... |
| xpeq2 5659 | Equality theorem for Carte... |
| elxpi 5660 | Membership in a Cartesian ... |
| elxp 5661 | Membership in a Cartesian ... |
| elxp2 5662 | Membership in a Cartesian ... |
| xpeq12 5663 | Equality theorem for Carte... |
| xpeq1i 5664 | Equality inference for Car... |
| xpeq2i 5665 | Equality inference for Car... |
| xpeq12i 5666 | Equality inference for Car... |
| xpeq1d 5667 | Equality deduction for Car... |
| xpeq2d 5668 | Equality deduction for Car... |
| xpeq12d 5669 | Equality deduction for Car... |
| sqxpeqd 5670 | Equality deduction for a C... |
| nfxp 5671 | Bound-variable hypothesis ... |
| 0nelxp 5672 | The empty set is not a mem... |
| 0nelelxp 5673 | A member of a Cartesian pr... |
| opelxp 5674 | Ordered pair membership in... |
| opelxpi 5675 | Ordered pair membership in... |
| opelxpii 5676 | Ordered pair membership in... |
| opelxpd 5677 | Ordered pair membership in... |
| opelvv 5678 | Ordered pair membership in... |
| opelvvg 5679 | Ordered pair membership in... |
| opelxp1 5680 | The first member of an ord... |
| opelxp2 5681 | The second member of an or... |
| otelxp 5682 | Ordered triple membership ... |
| otelxp1 5683 | The first member of an ord... |
| otel3xp 5684 | An ordered triple is an el... |
| opabssxpd 5685 | An ordered-pair class abst... |
| rabxp 5686 | Class abstraction restrict... |
| brxp 5687 | Binary relation on a Carte... |
| pwvrel 5688 | A set is a binary relation... |
| pwvabrel 5689 | The powerclass of the cart... |
| brrelex12 5690 | Two classes related by a b... |
| brrelex1 5691 | If two classes are related... |
| brrelex2 5692 | If two classes are related... |
| brrelex12i 5693 | Two classes that are relat... |
| brrelex1i 5694 | The first argument of a bi... |
| brrelex2i 5695 | The second argument of a b... |
| nprrel12 5696 | Proper classes are not rel... |
| nprrel 5697 | No proper class is related... |
| 0nelrel0 5698 | A binary relation does not... |
| 0nelrel 5699 | A binary relation does not... |
| fconstmpt 5700 | Representation of a consta... |
| vtoclr 5701 | Variable to class conversi... |
| opthprc 5702 | Justification theorem for ... |
| brel 5703 | Two things in a binary rel... |
| elxp3 5704 | Membership in a Cartesian ... |
| opeliunxp 5705 | Membership in a union of C... |
| opeliun2xp 5706 | Membership of an ordered p... |
| xpundi 5707 | Distributive law for Carte... |
| xpundir 5708 | Distributive law for Carte... |
| xpiundi 5709 | Distributive law for Carte... |
| xpiundir 5710 | Distributive law for Carte... |
| iunxpconst 5711 | Membership in a union of C... |
| xpun 5712 | The Cartesian product of t... |
| elvv 5713 | Membership in universal cl... |
| elvvv 5714 | Membership in universal cl... |
| elvvuni 5715 | An ordered pair contains i... |
| brinxp2 5716 | Intersection of binary rel... |
| brinxp 5717 | Intersection of binary rel... |
| opelinxp 5718 | Ordered pair element in an... |
| poinxp 5719 | Intersection of partial or... |
| soinxp 5720 | Intersection of total orde... |
| frinxp 5721 | Intersection of well-found... |
| seinxp 5722 | Intersection of set-like r... |
| weinxp 5723 | Intersection of well-order... |
| posn 5724 | Partial ordering of a sing... |
| sosn 5725 | Strict ordering on a singl... |
| frsn 5726 | Founded relation on a sing... |
| wesn 5727 | Well-ordering of a singlet... |
| elopaelxp 5728 | Membership in an ordered-p... |
| elopaelxpOLD 5729 | Obsolete version of ~ elop... |
| bropaex12 5730 | Two classes related by an ... |
| opabssxp 5731 | An abstraction relation is... |
| brab2a 5732 | The law of concretion for ... |
| optocl 5733 | Implicit substitution of c... |
| 2optocl 5734 | Implicit substitution of c... |
| 3optocl 5735 | Implicit substitution of c... |
| opbrop 5736 | Ordered pair membership in... |
| 0xp 5737 | The Cartesian product with... |
| csbxp 5738 | Distribute proper substitu... |
| releq 5739 | Equality theorem for the r... |
| releqi 5740 | Equality inference for the... |
| releqd 5741 | Equality deduction for the... |
| nfrel 5742 | Bound-variable hypothesis ... |
| sbcrel 5743 | Distribute proper substitu... |
| relss 5744 | Subclass theorem for relat... |
| ssrel 5745 | A subclass relationship de... |
| ssrelOLD 5746 | Obsolete version of ~ ssre... |
| eqrel 5747 | Extensionality principle f... |
| ssrel2 5748 | A subclass relationship de... |
| ssrel3 5749 | Subclass relation in anoth... |
| relssi 5750 | Inference from subclass pr... |
| relssdv 5751 | Deduction from subclass pr... |
| eqrelriv 5752 | Inference from extensional... |
| eqrelriiv 5753 | Inference from extensional... |
| eqbrriv 5754 | Inference from extensional... |
| eqrelrdv 5755 | Deduce equality of relatio... |
| eqbrrdv 5756 | Deduction from extensional... |
| eqbrrdiv 5757 | Deduction from extensional... |
| eqrelrdv2 5758 | A version of ~ eqrelrdv . ... |
| ssrelrel 5759 | A subclass relationship de... |
| eqrelrel 5760 | Extensionality principle f... |
| elrel 5761 | A member of a relation is ... |
| rel0 5762 | The empty set is a relatio... |
| nrelv 5763 | The universal class is not... |
| relsng 5764 | A singleton is a relation ... |
| relsnb 5765 | An at-most-singleton is a ... |
| relsnopg 5766 | A singleton of an ordered ... |
| relsn 5767 | A singleton is a relation ... |
| relsnop 5768 | A singleton of an ordered ... |
| copsex2gb 5769 | Implicit substitution infe... |
| copsex2ga 5770 | Implicit substitution infe... |
| elopaba 5771 | Membership in an ordered-p... |
| xpsspw 5772 | A Cartesian product is inc... |
| unixpss 5773 | The double class union of ... |
| relun 5774 | The union of two relations... |
| relin1 5775 | The intersection with a re... |
| relin2 5776 | The intersection with a re... |
| relinxp 5777 | Intersection with a Cartes... |
| reldif 5778 | A difference cutting down ... |
| reliun 5779 | An indexed union is a rela... |
| reliin 5780 | An indexed intersection is... |
| reluni 5781 | The union of a class is a ... |
| relint 5782 | The intersection of a clas... |
| relopabiv 5783 | A class of ordered pairs i... |
| relopabv 5784 | A class of ordered pairs i... |
| relopabi 5785 | A class of ordered pairs i... |
| relopabiALT 5786 | Alternate proof of ~ relop... |
| relopab 5787 | A class of ordered pairs i... |
| mptrel 5788 | The maps-to notation alway... |
| reli 5789 | The identity relation is a... |
| rele 5790 | The membership relation is... |
| opabid2 5791 | A relation expressed as an... |
| inopab 5792 | Intersection of two ordere... |
| difopab 5793 | Difference of two ordered-... |
| difopabOLD 5794 | Obsolete version of ~ difo... |
| inxp 5795 | Intersection of two Cartes... |
| inxpOLD 5796 | Obsolete version of ~ inxp... |
| xpindi 5797 | Distributive law for Carte... |
| xpindir 5798 | Distributive law for Carte... |
| xpiindi 5799 | Distributive law for Carte... |
| xpriindi 5800 | Distributive law for Carte... |
| eliunxp 5801 | Membership in a union of C... |
| opeliunxp2 5802 | Membership in a union of C... |
| raliunxp 5803 | Write a double restricted ... |
| rexiunxp 5804 | Write a double restricted ... |
| ralxp 5805 | Universal quantification r... |
| rexxp 5806 | Existential quantification... |
| exopxfr 5807 | Transfer ordered-pair exis... |
| exopxfr2 5808 | Transfer ordered-pair exis... |
| djussxp 5809 | Disjoint union is a subset... |
| ralxpf 5810 | Version of ~ ralxp with bo... |
| rexxpf 5811 | Version of ~ rexxp with bo... |
| iunxpf 5812 | Indexed union on a Cartesi... |
| opabbi2dv 5813 | Deduce equality of a relat... |
| relop 5814 | A necessary and sufficient... |
| ideqg 5815 | For sets, the identity rel... |
| ideq 5816 | For sets, the identity rel... |
| ididg 5817 | A set is identical to itse... |
| issetid 5818 | Two ways of expressing set... |
| coss1 5819 | Subclass theorem for compo... |
| coss2 5820 | Subclass theorem for compo... |
| coeq1 5821 | Equality theorem for compo... |
| coeq2 5822 | Equality theorem for compo... |
| coeq1i 5823 | Equality inference for com... |
| coeq2i 5824 | Equality inference for com... |
| coeq1d 5825 | Equality deduction for com... |
| coeq2d 5826 | Equality deduction for com... |
| coeq12i 5827 | Equality inference for com... |
| coeq12d 5828 | Equality deduction for com... |
| nfco 5829 | Bound-variable hypothesis ... |
| brcog 5830 | Ordered pair membership in... |
| opelco2g 5831 | Ordered pair membership in... |
| brcogw 5832 | Ordered pair membership in... |
| eqbrrdva 5833 | Deduction from extensional... |
| brco 5834 | Binary relation on a compo... |
| opelco 5835 | Ordered pair membership in... |
| cnvss 5836 | Subset theorem for convers... |
| cnveq 5837 | Equality theorem for conve... |
| cnveqi 5838 | Equality inference for con... |
| cnveqd 5839 | Equality deduction for con... |
| elcnv 5840 | Membership in a converse r... |
| elcnv2 5841 | Membership in a converse r... |
| nfcnv 5842 | Bound-variable hypothesis ... |
| brcnvg 5843 | The converse of a binary r... |
| opelcnvg 5844 | Ordered-pair membership in... |
| opelcnv 5845 | Ordered-pair membership in... |
| brcnv 5846 | The converse of a binary r... |
| csbcnv 5847 | Move class substitution in... |
| csbcnvgALT 5848 | Move class substitution in... |
| cnvco 5849 | Distributive law of conver... |
| cnvuni 5850 | The converse of a class un... |
| dfdm3 5851 | Alternate definition of do... |
| dfrn2 5852 | Alternate definition of ra... |
| dfrn3 5853 | Alternate definition of ra... |
| elrn2g 5854 | Membership in a range. (C... |
| elrng 5855 | Membership in a range. (C... |
| elrn2 5856 | Membership in a range. (C... |
| elrn 5857 | Membership in a range. (C... |
| ssrelrn 5858 | If a relation is a subset ... |
| dfdm4 5859 | Alternate definition of do... |
| dfdmf 5860 | Definition of domain, usin... |
| csbdm 5861 | Distribute proper substitu... |
| eldmg 5862 | Domain membership. Theore... |
| eldm2g 5863 | Domain membership. Theore... |
| eldm 5864 | Membership in a domain. T... |
| eldm2 5865 | Membership in a domain. T... |
| dmss 5866 | Subset theorem for domain.... |
| dmeq 5867 | Equality theorem for domai... |
| dmeqi 5868 | Equality inference for dom... |
| dmeqd 5869 | Equality deduction for dom... |
| opeldmd 5870 | Membership of first of an ... |
| opeldm 5871 | Membership of first of an ... |
| breldm 5872 | Membership of first of a b... |
| breldmg 5873 | Membership of first of a b... |
| dmun 5874 | The domain of a union is t... |
| dmin 5875 | The domain of an intersect... |
| breldmd 5876 | Membership of first of a b... |
| dmiun 5877 | The domain of an indexed u... |
| dmuni 5878 | The domain of a union. Pa... |
| dmopab 5879 | The domain of a class of o... |
| dmopabelb 5880 | A set is an element of the... |
| dmopab2rex 5881 | The domain of an ordered p... |
| dmopabss 5882 | Upper bound for the domain... |
| dmopab3 5883 | The domain of a restricted... |
| dm0 5884 | The domain of the empty se... |
| dmi 5885 | The domain of the identity... |
| dmv 5886 | The domain of the universe... |
| dmep 5887 | The domain of the membersh... |
| dm0rn0 5888 | An empty domain is equival... |
| rn0 5889 | The range of the empty set... |
| rnep 5890 | The range of the membershi... |
| reldm0 5891 | A relation is empty iff it... |
| dmxp 5892 | The domain of a Cartesian ... |
| dmxpOLD 5893 | Obsolete version of ~ dmxp... |
| dmxpid 5894 | The domain of a Cartesian ... |
| dmxpin 5895 | The domain of the intersec... |
| xpid11 5896 | The Cartesian square is a ... |
| dmcnvcnv 5897 | The domain of the double c... |
| rncnvcnv 5898 | The range of the double co... |
| elreldm 5899 | The first member of an ord... |
| rneq 5900 | Equality theorem for range... |
| rneqi 5901 | Equality inference for ran... |
| rneqd 5902 | Equality deduction for ran... |
| rnss 5903 | Subset theorem for range. ... |
| rnssi 5904 | Subclass inference for ran... |
| brelrng 5905 | The second argument of a b... |
| brelrn 5906 | The second argument of a b... |
| opelrn 5907 | Membership of second membe... |
| releldm 5908 | The first argument of a bi... |
| relelrn 5909 | The second argument of a b... |
| releldmb 5910 | Membership in a domain. (... |
| relelrnb 5911 | Membership in a range. (C... |
| releldmi 5912 | The first argument of a bi... |
| relelrni 5913 | The second argument of a b... |
| dfrnf 5914 | Definition of range, using... |
| nfdm 5915 | Bound-variable hypothesis ... |
| nfrn 5916 | Bound-variable hypothesis ... |
| dmiin 5917 | Domain of an intersection.... |
| rnopab 5918 | The range of a class of or... |
| rnopabss 5919 | Upper bound for the range ... |
| rnopab3 5920 | The range of a restricted ... |
| rnmpt 5921 | The range of a function in... |
| elrnmpt 5922 | The range of a function in... |
| elrnmpt1s 5923 | Elementhood in an image se... |
| elrnmpt1 5924 | Elementhood in an image se... |
| elrnmptg 5925 | Membership in the range of... |
| elrnmpti 5926 | Membership in the range of... |
| elrnmptd 5927 | The range of a function in... |
| elrnmpt1d 5928 | Elementhood in an image se... |
| elrnmptdv 5929 | Elementhood in the range o... |
| elrnmpt2d 5930 | Elementhood in the range o... |
| dfiun3g 5931 | Alternate definition of in... |
| dfiin3g 5932 | Alternate definition of in... |
| dfiun3 5933 | Alternate definition of in... |
| dfiin3 5934 | Alternate definition of in... |
| riinint 5935 | Express a relative indexed... |
| relrn0 5936 | A relation is empty iff it... |
| dmrnssfld 5937 | The domain and range of a ... |
| dmcoss 5938 | Domain of a composition. ... |
| rncoss 5939 | Range of a composition. (... |
| dmcosseq 5940 | Domain of a composition. ... |
| dmcosseqOLD 5941 | Obsolete version of ~ dmco... |
| dmcoeq 5942 | Domain of a composition. ... |
| rncoeq 5943 | Range of a composition. (... |
| reseq1 5944 | Equality theorem for restr... |
| reseq2 5945 | Equality theorem for restr... |
| reseq1i 5946 | Equality inference for res... |
| reseq2i 5947 | Equality inference for res... |
| reseq12i 5948 | Equality inference for res... |
| reseq1d 5949 | Equality deduction for res... |
| reseq2d 5950 | Equality deduction for res... |
| reseq12d 5951 | Equality deduction for res... |
| nfres 5952 | Bound-variable hypothesis ... |
| csbres 5953 | Distribute proper substitu... |
| res0 5954 | A restriction to the empty... |
| dfres3 5955 | Alternate definition of re... |
| opelres 5956 | Ordered pair elementhood i... |
| brres 5957 | Binary relation on a restr... |
| opelresi 5958 | Ordered pair membership in... |
| brresi 5959 | Binary relation on a restr... |
| opres 5960 | Ordered pair membership in... |
| resieq 5961 | A restricted identity rela... |
| opelidres 5962 | ` <. A , A >. ` belongs to... |
| resres 5963 | The restriction of a restr... |
| resundi 5964 | Distributive law for restr... |
| resundir 5965 | Distributive law for restr... |
| resindi 5966 | Class restriction distribu... |
| resindir 5967 | Class restriction distribu... |
| inres 5968 | Move intersection into cla... |
| resdifcom 5969 | Commutative law for restri... |
| resiun1 5970 | Distribution of restrictio... |
| resiun2 5971 | Distribution of restrictio... |
| resss 5972 | A class includes its restr... |
| rescom 5973 | Commutative law for restri... |
| ssres 5974 | Subclass theorem for restr... |
| ssres2 5975 | Subclass theorem for restr... |
| relres 5976 | A restriction is a relatio... |
| resabs1 5977 | Absorption law for restric... |
| resabs1i 5978 | Absorption law for restric... |
| resabs1d 5979 | Absorption law for restric... |
| resabs2 5980 | Absorption law for restric... |
| residm 5981 | Idempotent law for restric... |
| dmresss 5982 | The domain of a restrictio... |
| dmres 5983 | The domain of a restrictio... |
| ssdmres 5984 | A domain restricted to a s... |
| dmresexg 5985 | The domain of a restrictio... |
| resima 5986 | A restriction to an image.... |
| resima2 5987 | Image under a restricted c... |
| rnresss 5988 | The range of a restriction... |
| xpssres 5989 | Restriction of a constant ... |
| elinxp 5990 | Membership in an intersect... |
| elres 5991 | Membership in a restrictio... |
| elsnres 5992 | Membership in restriction ... |
| relssres 5993 | Simplification law for res... |
| dmressnsn 5994 | The domain of a restrictio... |
| eldmressnsn 5995 | The element of the domain ... |
| eldmeldmressn 5996 | An element of the domain (... |
| resdm 5997 | A relation restricted to i... |
| resexg 5998 | The restriction of a set i... |
| resexd 5999 | The restriction of a set i... |
| resex 6000 | The restriction of a set i... |
| resindm 6001 | When restricting a relatio... |
| resdmdfsn 6002 | Restricting a relation to ... |
| reldisjun 6003 | Split a relation into two ... |
| relresdm1 6004 | Restriction of a disjoint ... |
| resopab 6005 | Restriction of a class abs... |
| iss 6006 | A subclass of the identity... |
| resopab2 6007 | Restriction of a class abs... |
| resmpt 6008 | Restriction of the mapping... |
| resmpt3 6009 | Unconditional restriction ... |
| resmptf 6010 | Restriction of the mapping... |
| resmptd 6011 | Restriction of the mapping... |
| dfres2 6012 | Alternate definition of th... |
| mptss 6013 | Sufficient condition for i... |
| elimampt 6014 | Membership in the image of... |
| elidinxp 6015 | Characterization of the el... |
| elidinxpid 6016 | Characterization of the el... |
| elrid 6017 | Characterization of the el... |
| idinxpres 6018 | The intersection of the id... |
| idinxpresid 6019 | The intersection of the id... |
| idssxp 6020 | A diagonal set as a subset... |
| opabresid 6021 | The restricted identity re... |
| mptresid 6022 | The restricted identity re... |
| dmresi 6023 | The domain of a restricted... |
| restidsing 6024 | Restriction of the identit... |
| iresn0n0 6025 | The identity function rest... |
| imaeq1 6026 | Equality theorem for image... |
| imaeq2 6027 | Equality theorem for image... |
| imaeq1i 6028 | Equality theorem for image... |
| imaeq2i 6029 | Equality theorem for image... |
| imaeq1d 6030 | Equality theorem for image... |
| imaeq2d 6031 | Equality theorem for image... |
| imaeq12d 6032 | Equality theorem for image... |
| dfima2 6033 | Alternate definition of im... |
| dfima3 6034 | Alternate definition of im... |
| elimag 6035 | Membership in an image. T... |
| elima 6036 | Membership in an image. T... |
| elima2 6037 | Membership in an image. T... |
| elima3 6038 | Membership in an image. T... |
| nfima 6039 | Bound-variable hypothesis ... |
| nfimad 6040 | Deduction version of bound... |
| imadmrn 6041 | The image of the domain of... |
| imassrn 6042 | The image of a class is a ... |
| mptima 6043 | Image of a function in map... |
| mptimass 6044 | Image of a function in map... |
| imai 6045 | Image under the identity r... |
| rnresi 6046 | The range of the restricte... |
| resiima 6047 | The image of a restriction... |
| ima0 6048 | Image of the empty set. T... |
| 0ima 6049 | Image under the empty rela... |
| csbima12 6050 | Move class substitution in... |
| imadisj 6051 | A class whose image under ... |
| imadisjlnd 6052 | Deduction form of one nega... |
| cnvimass 6053 | A preimage under any class... |
| cnvimarndm 6054 | The preimage of the range ... |
| imasng 6055 | The image of a singleton. ... |
| relimasn 6056 | The image of a singleton. ... |
| elrelimasn 6057 | Elementhood in the image o... |
| elimasng1 6058 | Membership in an image of ... |
| elimasn1 6059 | Membership in an image of ... |
| elimasng 6060 | Membership in an image of ... |
| elimasn 6061 | Membership in an image of ... |
| elimasni 6062 | Membership in an image of ... |
| args 6063 | Two ways to express the cl... |
| elinisegg 6064 | Membership in the inverse ... |
| eliniseg 6065 | Membership in the inverse ... |
| epin 6066 | Any set is equal to its pr... |
| epini 6067 | Any set is equal to its pr... |
| iniseg 6068 | An idiom that signifies an... |
| inisegn0 6069 | Nonemptiness of an initial... |
| dffr3 6070 | Alternate definition of we... |
| dfse2 6071 | Alternate definition of se... |
| imass1 6072 | Subset theorem for image. ... |
| imass2 6073 | Subset theorem for image. ... |
| ndmima 6074 | The image of a singleton o... |
| relcnv 6075 | A converse is a relation. ... |
| relbrcnvg 6076 | When ` R ` is a relation, ... |
| eliniseg2 6077 | Eliminate the class existe... |
| relbrcnv 6078 | When ` R ` is a relation, ... |
| relco 6079 | A composition is a relatio... |
| cotrg 6080 | Two ways of saying that th... |
| cotrgOLD 6081 | Obsolete version of ~ cotr... |
| cotrgOLDOLD 6082 | Obsolete version of ~ cotr... |
| cotr 6083 | Two ways of saying a relat... |
| idrefALT 6084 | Alternate proof of ~ idref... |
| cnvsym 6085 | Two ways of saying a relat... |
| cnvsymOLD 6086 | Obsolete version of ~ cnvs... |
| cnvsymOLDOLD 6087 | Obsolete version of ~ cnvs... |
| intasym 6088 | Two ways of saying a relat... |
| asymref 6089 | Two ways of saying a relat... |
| asymref2 6090 | Two ways of saying a relat... |
| intirr 6091 | Two ways of saying a relat... |
| brcodir 6092 | Two ways of saying that tw... |
| codir 6093 | Two ways of saying a relat... |
| qfto 6094 | A quantifier-free way of e... |
| xpidtr 6095 | A Cartesian square is a tr... |
| trin2 6096 | The intersection of two tr... |
| poirr2 6097 | A partial order is irrefle... |
| trinxp 6098 | The relation induced by a ... |
| soirri 6099 | A strict order relation is... |
| sotri 6100 | A strict order relation is... |
| son2lpi 6101 | A strict order relation ha... |
| sotri2 6102 | A transitivity relation. ... |
| sotri3 6103 | A transitivity relation. ... |
| poleloe 6104 | Express "less than or equa... |
| poltletr 6105 | Transitive law for general... |
| somin1 6106 | Property of a minimum in a... |
| somincom 6107 | Commutativity of minimum i... |
| somin2 6108 | Property of a minimum in a... |
| soltmin 6109 | Being less than a minimum,... |
| cnvopab 6110 | The converse of a class ab... |
| cnvopabOLD 6111 | Obsolete version of ~ cnvo... |
| mptcnv 6112 | The converse of a mapping ... |
| cnv0 6113 | The converse of the empty ... |
| cnvi 6114 | The converse of the identi... |
| cnvun 6115 | The converse of a union is... |
| cnvdif 6116 | Distributive law for conve... |
| cnvin 6117 | Distributive law for conve... |
| rnun 6118 | Distributive law for range... |
| rnin 6119 | The range of an intersecti... |
| rniun 6120 | The range of an indexed un... |
| rnuni 6121 | The range of a union. Par... |
| imaundi 6122 | Distributive law for image... |
| imaundir 6123 | The image of a union. (Co... |
| imadifssran 6124 | Condition for the range of... |
| cnvimassrndm 6125 | The preimage of a superset... |
| dminss 6126 | An upper bound for interse... |
| imainss 6127 | An upper bound for interse... |
| inimass 6128 | The image of an intersecti... |
| inimasn 6129 | The intersection of the im... |
| cnvxp 6130 | The converse of a Cartesia... |
| xp0 6131 | The Cartesian product with... |
| xpnz 6132 | The Cartesian product of n... |
| xpeq0 6133 | At least one member of an ... |
| xpdisj1 6134 | Cartesian products with di... |
| xpdisj2 6135 | Cartesian products with di... |
| xpsndisj 6136 | Cartesian products with tw... |
| difxp 6137 | Difference of Cartesian pr... |
| difxp1 6138 | Difference law for Cartesi... |
| difxp2 6139 | Difference law for Cartesi... |
| djudisj 6140 | Disjoint unions with disjo... |
| xpdifid 6141 | The set of distinct couple... |
| resdisj 6142 | A double restriction to di... |
| rnxp 6143 | The range of a Cartesian p... |
| dmxpss 6144 | The domain of a Cartesian ... |
| rnxpss 6145 | The range of a Cartesian p... |
| rnxpid 6146 | The range of a Cartesian s... |
| ssxpb 6147 | A Cartesian product subcla... |
| xp11 6148 | The Cartesian product of n... |
| xpcan 6149 | Cancellation law for Carte... |
| xpcan2 6150 | Cancellation law for Carte... |
| ssrnres 6151 | Two ways to express surjec... |
| rninxp 6152 | Two ways to express surjec... |
| dminxp 6153 | Two ways to express totali... |
| imainrect 6154 | Image by a restricted and ... |
| xpima 6155 | Direct image by a Cartesia... |
| xpima1 6156 | Direct image by a Cartesia... |
| xpima2 6157 | Direct image by a Cartesia... |
| xpimasn 6158 | Direct image of a singleto... |
| sossfld 6159 | The base set of a strict o... |
| sofld 6160 | The base set of a nonempty... |
| cnvcnv3 6161 | The set of all ordered pai... |
| dfrel2 6162 | Alternate definition of re... |
| dfrel4v 6163 | A relation can be expresse... |
| dfrel4 6164 | A relation can be expresse... |
| cnvcnv 6165 | The double converse of a c... |
| cnvcnv2 6166 | The double converse of a c... |
| cnvcnvss 6167 | The double converse of a c... |
| cnvrescnv 6168 | Two ways to express the co... |
| cnveqb 6169 | Equality theorem for conve... |
| cnveq0 6170 | A relation empty iff its c... |
| dfrel3 6171 | Alternate definition of re... |
| elid 6172 | Characterization of the el... |
| dmresv 6173 | The domain of a universal ... |
| rnresv 6174 | The range of a universal r... |
| dfrn4 6175 | Range defined in terms of ... |
| csbrn 6176 | Distribute proper substitu... |
| rescnvcnv 6177 | The restriction of the dou... |
| cnvcnvres 6178 | The double converse of the... |
| imacnvcnv 6179 | The image of the double co... |
| dmsnn0 6180 | The domain of a singleton ... |
| rnsnn0 6181 | The range of a singleton i... |
| dmsn0 6182 | The domain of the singleto... |
| cnvsn0 6183 | The converse of the single... |
| dmsn0el 6184 | The domain of a singleton ... |
| relsn2 6185 | A singleton is a relation ... |
| dmsnopg 6186 | The domain of a singleton ... |
| dmsnopss 6187 | The domain of a singleton ... |
| dmpropg 6188 | The domain of an unordered... |
| dmsnop 6189 | The domain of a singleton ... |
| dmprop 6190 | The domain of an unordered... |
| dmtpop 6191 | The domain of an unordered... |
| cnvcnvsn 6192 | Double converse of a singl... |
| dmsnsnsn 6193 | The domain of the singleto... |
| rnsnopg 6194 | The range of a singleton o... |
| rnpropg 6195 | The range of a pair of ord... |
| cnvsng 6196 | Converse of a singleton of... |
| rnsnop 6197 | The range of a singleton o... |
| op1sta 6198 | Extract the first member o... |
| cnvsn 6199 | Converse of a singleton of... |
| op2ndb 6200 | Extract the second member ... |
| op2nda 6201 | Extract the second member ... |
| opswap 6202 | Swap the members of an ord... |
| cnvresima 6203 | An image under the convers... |
| resdm2 6204 | A class restricted to its ... |
| resdmres 6205 | Restriction to the domain ... |
| resresdm 6206 | A restriction by an arbitr... |
| imadmres 6207 | The image of the domain of... |
| resdmss 6208 | Subset relationship for th... |
| resdifdi 6209 | Distributive law for restr... |
| resdifdir 6210 | Distributive law for restr... |
| mptpreima 6211 | The preimage of a function... |
| mptiniseg 6212 | Converse singleton image o... |
| dmmpt 6213 | The domain of the mapping ... |
| dmmptss 6214 | The domain of a mapping is... |
| dmmptg 6215 | The domain of the mapping ... |
| rnmpt0f 6216 | The range of a function in... |
| rnmptn0 6217 | The range of a function in... |
| dfco2 6218 | Alternate definition of a ... |
| dfco2a 6219 | Generalization of ~ dfco2 ... |
| coundi 6220 | Class composition distribu... |
| coundir 6221 | Class composition distribu... |
| cores 6222 | Restricted first member of... |
| resco 6223 | Associative law for the re... |
| imaco 6224 | Image of the composition o... |
| rnco 6225 | The range of the compositi... |
| rnco2 6226 | The range of the compositi... |
| dmco 6227 | The domain of a compositio... |
| coeq0 6228 | A composition of two relat... |
| coiun 6229 | Composition with an indexe... |
| cocnvcnv1 6230 | A composition is not affec... |
| cocnvcnv2 6231 | A composition is not affec... |
| cores2 6232 | Absorption of a reverse (p... |
| co02 6233 | Composition with the empty... |
| co01 6234 | Composition with the empty... |
| coi1 6235 | Composition with the ident... |
| coi2 6236 | Composition with the ident... |
| coires1 6237 | Composition with a restric... |
| coass 6238 | Associative law for class ... |
| relcnvtrg 6239 | General form of ~ relcnvtr... |
| relcnvtr 6240 | A relation is transitive i... |
| relssdmrn 6241 | A relation is included in ... |
| relssdmrnOLD 6242 | Obsolete version of ~ rels... |
| resssxp 6243 | If the ` R ` -image of a c... |
| cnvssrndm 6244 | The converse is a subset o... |
| cossxp 6245 | Composition as a subset of... |
| relrelss 6246 | Two ways to describe the s... |
| unielrel 6247 | The membership relation fo... |
| relfld 6248 | The double union of a rela... |
| relresfld 6249 | Restriction of a relation ... |
| relcoi2 6250 | Composition with the ident... |
| relcoi1 6251 | Composition with the ident... |
| unidmrn 6252 | The double union of the co... |
| relcnvfld 6253 | if ` R ` is a relation, it... |
| dfdm2 6254 | Alternate definition of do... |
| unixp 6255 | The double class union of ... |
| unixp0 6256 | A Cartesian product is emp... |
| unixpid 6257 | Field of a Cartesian squar... |
| ressn 6258 | Restriction of a class to ... |
| cnviin 6259 | The converse of an interse... |
| cnvpo 6260 | The converse of a partial ... |
| cnvso 6261 | The converse of a strict o... |
| xpco 6262 | Composition of two Cartesi... |
| xpcoid 6263 | Composition of two Cartesi... |
| elsnxp 6264 | Membership in a Cartesian ... |
| reu3op 6265 | There is a unique ordered ... |
| reuop 6266 | There is a unique ordered ... |
| opreu2reurex 6267 | There is a unique ordered ... |
| opreu2reu 6268 | If there is a unique order... |
| dfpo2 6269 | Quantifier-free definition... |
| csbcog 6270 | Distribute proper substitu... |
| snres0 6271 | Condition for restriction ... |
| imaindm 6272 | The image is unaffected by... |
| predeq123 6275 | Equality theorem for the p... |
| predeq1 6276 | Equality theorem for the p... |
| predeq2 6277 | Equality theorem for the p... |
| predeq3 6278 | Equality theorem for the p... |
| nfpred 6279 | Bound-variable hypothesis ... |
| csbpredg 6280 | Move class substitution in... |
| predpredss 6281 | If ` A ` is a subset of ` ... |
| predss 6282 | The predecessor class of `... |
| sspred 6283 | Another subset/predecessor... |
| dfpred2 6284 | An alternate definition of... |
| dfpred3 6285 | An alternate definition of... |
| dfpred3g 6286 | An alternate definition of... |
| elpredgg 6287 | Membership in a predecesso... |
| elpredg 6288 | Membership in a predecesso... |
| elpredimg 6289 | Membership in a predecesso... |
| elpredim 6290 | Membership in a predecesso... |
| elpred 6291 | Membership in a predecesso... |
| predexg 6292 | The predecessor class exis... |
| dffr4 6293 | Alternate definition of we... |
| predel 6294 | Membership in the predeces... |
| predtrss 6295 | If ` R ` is transitive ove... |
| predpo 6296 | Property of the predecesso... |
| predso 6297 | Property of the predecesso... |
| setlikespec 6298 | If ` R ` is set-like in ` ... |
| predidm 6299 | Idempotent law for the pre... |
| predin 6300 | Intersection law for prede... |
| predun 6301 | Union law for predecessor ... |
| preddif 6302 | Difference law for predece... |
| predep 6303 | The predecessor under the ... |
| trpred 6304 | The class of predecessors ... |
| preddowncl 6305 | A property of classes that... |
| predpoirr 6306 | Given a partial ordering, ... |
| predfrirr 6307 | Given a well-founded relat... |
| pred0 6308 | The predecessor class over... |
| dfse3 6309 | Alternate definition of se... |
| predrelss 6310 | Subset carries from relati... |
| predprc 6311 | The predecessor of a prope... |
| predres 6312 | Predecessor class is unaff... |
| frpomin 6313 | Every nonempty (possibly p... |
| frpomin2 6314 | Every nonempty (possibly p... |
| frpoind 6315 | The principle of well-foun... |
| frpoinsg 6316 | Well-Founded Induction Sch... |
| frpoins2fg 6317 | Well-Founded Induction sch... |
| frpoins2g 6318 | Well-Founded Induction sch... |
| frpoins3g 6319 | Well-Founded Induction sch... |
| tz6.26 6320 | All nonempty subclasses of... |
| tz6.26i 6321 | All nonempty subclasses of... |
| wfi 6322 | The Principle of Well-Orde... |
| wfii 6323 | The Principle of Well-Orde... |
| wfisg 6324 | Well-Ordered Induction Sch... |
| wfis 6325 | Well-Ordered Induction Sch... |
| wfis2fg 6326 | Well-Ordered Induction Sch... |
| wfis2f 6327 | Well-Ordered Induction sch... |
| wfis2g 6328 | Well-Ordered Induction Sch... |
| wfis2 6329 | Well-Ordered Induction sch... |
| wfis3 6330 | Well-Ordered Induction sch... |
| ordeq 6339 | Equality theorem for the o... |
| elong 6340 | An ordinal number is an or... |
| elon 6341 | An ordinal number is an or... |
| eloni 6342 | An ordinal number has the ... |
| elon2 6343 | An ordinal number is an or... |
| limeq 6344 | Equality theorem for the l... |
| ordwe 6345 | Membership well-orders eve... |
| ordtr 6346 | An ordinal class is transi... |
| ordfr 6347 | Membership is well-founded... |
| ordelss 6348 | An element of an ordinal c... |
| trssord 6349 | A transitive subclass of a... |
| ordirr 6350 | No ordinal class is a memb... |
| nordeq 6351 | A member of an ordinal cla... |
| ordn2lp 6352 | An ordinal class cannot be... |
| tz7.5 6353 | A nonempty subclass of an ... |
| ordelord 6354 | An element of an ordinal c... |
| tron 6355 | The class of all ordinal n... |
| ordelon 6356 | An element of an ordinal c... |
| onelon 6357 | An element of an ordinal n... |
| tz7.7 6358 | A transitive class belongs... |
| ordelssne 6359 | For ordinal classes, membe... |
| ordelpss 6360 | For ordinal classes, membe... |
| ordsseleq 6361 | For ordinal classes, inclu... |
| ordin 6362 | The intersection of two or... |
| onin 6363 | The intersection of two or... |
| ordtri3or 6364 | A trichotomy law for ordin... |
| ordtri1 6365 | A trichotomy law for ordin... |
| ontri1 6366 | A trichotomy law for ordin... |
| ordtri2 6367 | A trichotomy law for ordin... |
| ordtri3 6368 | A trichotomy law for ordin... |
| ordtri4 6369 | A trichotomy law for ordin... |
| orddisj 6370 | An ordinal class and its s... |
| onfr 6371 | The ordinal class is well-... |
| onelpss 6372 | Relationship between membe... |
| onsseleq 6373 | Relationship between subse... |
| onelss 6374 | An element of an ordinal n... |
| oneltri 6375 | The elementhood relation o... |
| ordtr1 6376 | Transitive law for ordinal... |
| ordtr2 6377 | Transitive law for ordinal... |
| ordtr3 6378 | Transitive law for ordinal... |
| ontr1 6379 | Transitive law for ordinal... |
| ontr2 6380 | Transitive law for ordinal... |
| onelssex 6381 | Ordinal less than is equiv... |
| ordunidif 6382 | The union of an ordinal st... |
| ordintdif 6383 | If ` B ` is smaller than `... |
| onintss 6384 | If a property is true for ... |
| oneqmini 6385 | A way to show that an ordi... |
| ord0 6386 | The empty set is an ordina... |
| 0elon 6387 | The empty set is an ordina... |
| ord0eln0 6388 | A nonempty ordinal contain... |
| on0eln0 6389 | An ordinal number contains... |
| dflim2 6390 | An alternate definition of... |
| inton 6391 | The intersection of the cl... |
| nlim0 6392 | The empty set is not a lim... |
| limord 6393 | A limit ordinal is ordinal... |
| limuni 6394 | A limit ordinal is its own... |
| limuni2 6395 | The union of a limit ordin... |
| 0ellim 6396 | A limit ordinal contains t... |
| limelon 6397 | A limit ordinal class that... |
| onn0 6398 | The class of all ordinal n... |
| suceqd 6399 | Deduction associated with ... |
| suceq 6400 | Equality of successors. (... |
| elsuci 6401 | Membership in a successor.... |
| elsucg 6402 | Membership in a successor.... |
| elsuc2g 6403 | Variant of membership in a... |
| elsuc 6404 | Membership in a successor.... |
| elsuc2 6405 | Membership in a successor.... |
| nfsuc 6406 | Bound-variable hypothesis ... |
| elelsuc 6407 | Membership in a successor.... |
| sucel 6408 | Membership of a successor ... |
| suc0 6409 | The successor of the empty... |
| sucprc 6410 | A proper class is its own ... |
| unisucs 6411 | The union of the successor... |
| unisucg 6412 | A transitive class is equa... |
| unisuc 6413 | A transitive class is equa... |
| sssucid 6414 | A class is included in its... |
| sucidg 6415 | Part of Proposition 7.23 o... |
| sucid 6416 | A set belongs to its succe... |
| nsuceq0 6417 | No successor is empty. (C... |
| eqelsuc 6418 | A set belongs to the succe... |
| iunsuc 6419 | Inductive definition for t... |
| suctr 6420 | The successor of a transit... |
| trsuc 6421 | A set whose successor belo... |
| trsucss 6422 | A member of the successor ... |
| ordsssuc 6423 | An ordinal is a subset of ... |
| onsssuc 6424 | A subset of an ordinal num... |
| ordsssuc2 6425 | An ordinal subset of an or... |
| onmindif 6426 | When its successor is subt... |
| ordnbtwn 6427 | There is no set between an... |
| onnbtwn 6428 | There is no set between an... |
| sucssel 6429 | A set whose successor is a... |
| orddif 6430 | Ordinal derived from its s... |
| orduniss 6431 | An ordinal class includes ... |
| ordtri2or 6432 | A trichotomy law for ordin... |
| ordtri2or2 6433 | A trichotomy law for ordin... |
| ordtri2or3 6434 | A consequence of total ord... |
| ordelinel 6435 | The intersection of two or... |
| ordssun 6436 | Property of a subclass of ... |
| ordequn 6437 | The maximum (i.e. union) o... |
| ordun 6438 | The maximum (i.e., union) ... |
| onunel 6439 | The union of two ordinals ... |
| ordunisssuc 6440 | A subclass relationship fo... |
| suc11 6441 | The successor operation be... |
| onun2 6442 | The union of two ordinals ... |
| ontr 6443 | An ordinal number is a tra... |
| onunisuc 6444 | An ordinal number is equal... |
| onordi 6445 | An ordinal number is an or... |
| ontrciOLD 6446 | Obsolete version of ~ ontr... |
| onirri 6447 | An ordinal number is not a... |
| oneli 6448 | A member of an ordinal num... |
| onelssi 6449 | A member of an ordinal num... |
| onssneli 6450 | An ordering law for ordina... |
| onssnel2i 6451 | An ordering law for ordina... |
| onelini 6452 | An element of an ordinal n... |
| oneluni 6453 | An ordinal number equals i... |
| onunisuci 6454 | An ordinal number is equal... |
| onsseli 6455 | Subset is equivalent to me... |
| onun2i 6456 | The union of two ordinal n... |
| unizlim 6457 | An ordinal equal to its ow... |
| on0eqel 6458 | An ordinal number either e... |
| snsn0non 6459 | The singleton of the singl... |
| onxpdisj 6460 | Ordinal numbers and ordere... |
| onnev 6461 | The class of ordinal numbe... |
| iotajust 6463 | Soundness justification th... |
| dfiota2 6465 | Alternate definition for d... |
| nfiota1 6466 | Bound-variable hypothesis ... |
| nfiotadw 6467 | Deduction version of ~ nfi... |
| nfiotaw 6468 | Bound-variable hypothesis ... |
| nfiotad 6469 | Deduction version of ~ nfi... |
| nfiota 6470 | Bound-variable hypothesis ... |
| cbviotaw 6471 | Change bound variables in ... |
| cbviotavw 6472 | Change bound variables in ... |
| cbviota 6473 | Change bound variables in ... |
| cbviotav 6474 | Change bound variables in ... |
| sb8iota 6475 | Variable substitution in d... |
| iotaeq 6476 | Equality theorem for descr... |
| iotabi 6477 | Equivalence theorem for de... |
| uniabio 6478 | Part of Theorem 8.17 in [Q... |
| iotaval2 6479 | Version of ~ iotaval using... |
| iotauni2 6480 | Version of ~ iotauni using... |
| iotanul2 6481 | Version of ~ iotanul using... |
| iotaval 6482 | Theorem 8.19 in [Quine] p.... |
| iotassuni 6483 | The ` iota ` class is a su... |
| iotaex 6484 | Theorem 8.23 in [Quine] p.... |
| iotavalOLD 6485 | Obsolete version of ~ iota... |
| iotauni 6486 | Equivalence between two di... |
| iotaint 6487 | Equivalence between two di... |
| iota1 6488 | Property of iota. (Contri... |
| iotanul 6489 | Theorem 8.22 in [Quine] p.... |
| iotassuniOLD 6490 | Obsolete version of ~ iota... |
| iotaexOLD 6491 | Obsolete version of ~ iota... |
| iota4 6492 | Theorem *14.22 in [Whitehe... |
| iota4an 6493 | Theorem *14.23 in [Whitehe... |
| iota5 6494 | A method for computing iot... |
| iotabidv 6495 | Formula-building deduction... |
| iotabii 6496 | Formula-building deduction... |
| iotacl 6497 | Membership law for descrip... |
| iota2df 6498 | A condition that allows to... |
| iota2d 6499 | A condition that allows to... |
| iota2 6500 | The unique element such th... |
| iotan0 6501 | Representation of "the uni... |
| sniota 6502 | A class abstraction with a... |
| dfiota4 6503 | The ` iota ` operation usi... |
| csbiota 6504 | Class substitution within ... |
| dffun2 6521 | Alternate definition of a ... |
| dffun2OLD 6522 | Obsolete version of ~ dffu... |
| dffun2OLDOLD 6523 | Obsolete version of ~ dffu... |
| dffun6 6524 | Alternate definition of a ... |
| dffun3 6525 | Alternate definition of fu... |
| dffun3OLD 6526 | Obsolete version of ~ dffu... |
| dffun4 6527 | Alternate definition of a ... |
| dffun5 6528 | Alternate definition of fu... |
| dffun6f 6529 | Definition of function, us... |
| dffun6OLD 6530 | Obsolete version of ~ dffu... |
| funmo 6531 | A function has at most one... |
| funmoOLD 6532 | Obsolete version of ~ funm... |
| funrel 6533 | A function is a relation. ... |
| 0nelfun 6534 | A function does not contai... |
| funss 6535 | Subclass theorem for funct... |
| funeq 6536 | Equality theorem for funct... |
| funeqi 6537 | Equality inference for the... |
| funeqd 6538 | Equality deduction for the... |
| nffun 6539 | Bound-variable hypothesis ... |
| sbcfung 6540 | Distribute proper substitu... |
| funeu 6541 | There is exactly one value... |
| funeu2 6542 | There is exactly one value... |
| dffun7 6543 | Alternate definition of a ... |
| dffun8 6544 | Alternate definition of a ... |
| dffun9 6545 | Alternate definition of a ... |
| funfn 6546 | A class is a function if a... |
| funfnd 6547 | A function is a function o... |
| funi 6548 | The identity relation is a... |
| nfunv 6549 | The universal class is not... |
| funopg 6550 | A Kuratowski ordered pair ... |
| funopab 6551 | A class of ordered pairs i... |
| funopabeq 6552 | A class of ordered pairs o... |
| funopab4 6553 | A class of ordered pairs o... |
| funmpt 6554 | A function in maps-to nota... |
| funmpt2 6555 | Functionality of a class g... |
| funco 6556 | The composition of two fun... |
| funresfunco 6557 | Composition of two functio... |
| funres 6558 | A restriction of a functio... |
| funresd 6559 | A restriction of a functio... |
| funssres 6560 | The restriction of a funct... |
| fun2ssres 6561 | Equality of restrictions o... |
| funun 6562 | The union of functions wit... |
| fununmo 6563 | If the union of classes is... |
| fununfun 6564 | If the union of classes is... |
| fundif 6565 | A function with removed el... |
| funcnvsn 6566 | The converse singleton of ... |
| funsng 6567 | A singleton of an ordered ... |
| fnsng 6568 | Functionality and domain o... |
| funsn 6569 | A singleton of an ordered ... |
| funprg 6570 | A set of two pairs is a fu... |
| funtpg 6571 | A set of three pairs is a ... |
| funpr 6572 | A function with a domain o... |
| funtp 6573 | A function with a domain o... |
| fnsn 6574 | Functionality and domain o... |
| fnprg 6575 | Function with a domain of ... |
| fntpg 6576 | Function with a domain of ... |
| fntp 6577 | A function with a domain o... |
| funcnvpr 6578 | The converse pair of order... |
| funcnvtp 6579 | The converse triple of ord... |
| funcnvqp 6580 | The converse quadruple of ... |
| fun0 6581 | The empty set is a functio... |
| funcnv0 6582 | The converse of the empty ... |
| funcnvcnv 6583 | The double converse of a f... |
| funcnv2 6584 | A simpler equivalence for ... |
| funcnv 6585 | The converse of a class is... |
| funcnv3 6586 | A condition showing a clas... |
| fun2cnv 6587 | The double converse of a c... |
| svrelfun 6588 | A single-valued relation i... |
| fncnv 6589 | Single-rootedness (see ~ f... |
| fun11 6590 | Two ways of stating that `... |
| fununi 6591 | The union of a chain (with... |
| funin 6592 | The intersection with a fu... |
| funres11 6593 | The restriction of a one-t... |
| funcnvres 6594 | The converse of a restrict... |
| cnvresid 6595 | Converse of a restricted i... |
| funcnvres2 6596 | The converse of a restrict... |
| funimacnv 6597 | The image of the preimage ... |
| funimass1 6598 | A kind of contraposition l... |
| funimass2 6599 | A kind of contraposition l... |
| imadif 6600 | The image of a difference ... |
| imain 6601 | The image of an intersecti... |
| f1imadifssran 6602 | Condition for the range of... |
| funimaexg 6603 | Axiom of Replacement using... |
| funimaexgOLD 6604 | Obsolete version of ~ funi... |
| funimaex 6605 | The image of a set under a... |
| isarep1 6606 | Part of a study of the Axi... |
| isarep1OLD 6607 | Obsolete version of ~ isar... |
| isarep2 6608 | Part of a study of the Axi... |
| fneq1 6609 | Equality theorem for funct... |
| fneq2 6610 | Equality theorem for funct... |
| fneq1d 6611 | Equality deduction for fun... |
| fneq2d 6612 | Equality deduction for fun... |
| fneq12d 6613 | Equality deduction for fun... |
| fneq12 6614 | Equality theorem for funct... |
| fneq1i 6615 | Equality inference for fun... |
| fneq2i 6616 | Equality inference for fun... |
| nffn 6617 | Bound-variable hypothesis ... |
| fnfun 6618 | A function with domain is ... |
| fnfund 6619 | A function with domain is ... |
| fnrel 6620 | A function with domain is ... |
| fndm 6621 | The domain of a function. ... |
| fndmi 6622 | The domain of a function. ... |
| fndmd 6623 | The domain of a function. ... |
| funfni 6624 | Inference to convert a fun... |
| fndmu 6625 | A function has a unique do... |
| fnbr 6626 | The first argument of bina... |
| fnop 6627 | The first argument of an o... |
| fneu 6628 | There is exactly one value... |
| fneu2 6629 | There is exactly one value... |
| fnunres1 6630 | Restriction of a disjoint ... |
| fnunres2 6631 | Restriction of a disjoint ... |
| fnun 6632 | The union of two functions... |
| fnund 6633 | The union of two functions... |
| fnunop 6634 | Extension of a function wi... |
| fncofn 6635 | Composition of a function ... |
| fnco 6636 | Composition of two functio... |
| fnresdm 6637 | A function does not change... |
| fnresdisj 6638 | A function restricted to a... |
| 2elresin 6639 | Membership in two function... |
| fnssresb 6640 | Restriction of a function ... |
| fnssres 6641 | Restriction of a function ... |
| fnssresd 6642 | Restriction of a function ... |
| fnresin1 6643 | Restriction of a function'... |
| fnresin2 6644 | Restriction of a function'... |
| fnres 6645 | An equivalence for functio... |
| idfn 6646 | The identity relation is a... |
| fnresi 6647 | The restricted identity re... |
| fnima 6648 | The image of a function's ... |
| fn0 6649 | A function with empty doma... |
| fnimadisj 6650 | A class that is disjoint w... |
| fnimaeq0 6651 | Images under a function ne... |
| dfmpt3 6652 | Alternate definition for t... |
| mptfnf 6653 | The maps-to notation defin... |
| fnmptf 6654 | The maps-to notation defin... |
| fnopabg 6655 | Functionality and domain o... |
| fnopab 6656 | Functionality and domain o... |
| mptfng 6657 | The maps-to notation defin... |
| fnmpt 6658 | The maps-to notation defin... |
| fnmptd 6659 | The maps-to notation defin... |
| mpt0 6660 | A mapping operation with e... |
| fnmpti 6661 | Functionality and domain o... |
| dmmpti 6662 | Domain of the mapping oper... |
| dmmptd 6663 | The domain of the mapping ... |
| mptun 6664 | Union of mappings which ar... |
| partfun 6665 | Rewrite a function defined... |
| feq1 6666 | Equality theorem for funct... |
| feq2 6667 | Equality theorem for funct... |
| feq3 6668 | Equality theorem for funct... |
| feq23 6669 | Equality theorem for funct... |
| feq1d 6670 | Equality deduction for fun... |
| feq1dd 6671 | Equality deduction for fun... |
| feq2d 6672 | Equality deduction for fun... |
| feq3d 6673 | Equality deduction for fun... |
| feq2dd 6674 | Equality deduction for fun... |
| feq3dd 6675 | Equality deduction for fun... |
| feq12d 6676 | Equality deduction for fun... |
| feq123d 6677 | Equality deduction for fun... |
| feq123 6678 | Equality theorem for funct... |
| feq1i 6679 | Equality inference for fun... |
| feq2i 6680 | Equality inference for fun... |
| feq12i 6681 | Equality inference for fun... |
| feq23i 6682 | Equality inference for fun... |
| feq23d 6683 | Equality deduction for fun... |
| nff 6684 | Bound-variable hypothesis ... |
| sbcfng 6685 | Distribute proper substitu... |
| sbcfg 6686 | Distribute proper substitu... |
| elimf 6687 | Eliminate a mapping hypoth... |
| ffn 6688 | A mapping is a function wi... |
| ffnd 6689 | A mapping is a function wi... |
| dffn2 6690 | Any function is a mapping ... |
| ffun 6691 | A mapping is a function. ... |
| ffund 6692 | A mapping is a function, d... |
| frel 6693 | A mapping is a relation. ... |
| freld 6694 | A mapping is a relation. ... |
| frn 6695 | The range of a mapping. (... |
| frnd 6696 | Deduction form of ~ frn . ... |
| fdm 6697 | The domain of a mapping. ... |
| fdmd 6698 | Deduction form of ~ fdm . ... |
| fdmi 6699 | Inference associated with ... |
| dffn3 6700 | A function maps to its ran... |
| ffrn 6701 | A function maps to its ran... |
| ffrnb 6702 | Characterization of a func... |
| ffrnbd 6703 | A function maps to its ran... |
| fss 6704 | Expanding the codomain of ... |
| fssd 6705 | Expanding the codomain of ... |
| fssdmd 6706 | Expressing that a class is... |
| fssdm 6707 | Expressing that a class is... |
| fimass 6708 | The image of a class under... |
| fimassd 6709 | The image of a class is a ... |
| fimacnv 6710 | The preimage of the codoma... |
| fcof 6711 | Composition of a function ... |
| fco 6712 | Composition of two functio... |
| fcod 6713 | Composition of two mapping... |
| fco2 6714 | Functionality of a composi... |
| fssxp 6715 | A mapping is a class of or... |
| funssxp 6716 | Two ways of specifying a p... |
| ffdm 6717 | A mapping is a partial fun... |
| ffdmd 6718 | The domain of a function. ... |
| fdmrn 6719 | A different way to write `... |
| funcofd 6720 | Composition of two functio... |
| opelf 6721 | The members of an ordered ... |
| fun 6722 | The union of two functions... |
| fun2 6723 | The union of two functions... |
| fun2d 6724 | The union of functions wit... |
| fnfco 6725 | Composition of two functio... |
| fssres 6726 | Restriction of a function ... |
| fssresd 6727 | Restriction of a function ... |
| fssres2 6728 | Restriction of a restricte... |
| fresin 6729 | An identity for the mappin... |
| resasplit 6730 | If two functions agree on ... |
| fresaun 6731 | The union of two functions... |
| fresaunres2 6732 | From the union of two func... |
| fresaunres1 6733 | From the union of two func... |
| fcoi1 6734 | Composition of a mapping a... |
| fcoi2 6735 | Composition of restricted ... |
| feu 6736 | There is exactly one value... |
| fcnvres 6737 | The converse of a restrict... |
| fimacnvdisj 6738 | The preimage of a class di... |
| fint 6739 | Function into an intersect... |
| fin 6740 | Mapping into an intersecti... |
| f0 6741 | The empty function. (Cont... |
| f00 6742 | A class is a function with... |
| f0bi 6743 | A function with empty doma... |
| f0dom0 6744 | A function is empty iff it... |
| f0rn0 6745 | If there is no element in ... |
| fconst 6746 | A Cartesian product with a... |
| fconstg 6747 | A Cartesian product with a... |
| fnconstg 6748 | A Cartesian product with a... |
| fconst6g 6749 | Constant function with loo... |
| fconst6 6750 | A constant function as a m... |
| f1eq1 6751 | Equality theorem for one-t... |
| f1eq2 6752 | Equality theorem for one-t... |
| f1eq3 6753 | Equality theorem for one-t... |
| nff1 6754 | Bound-variable hypothesis ... |
| dff12 6755 | Alternate definition of a ... |
| f1f 6756 | A one-to-one mapping is a ... |
| f1fn 6757 | A one-to-one mapping is a ... |
| f1fun 6758 | A one-to-one mapping is a ... |
| f1rel 6759 | A one-to-one onto mapping ... |
| f1dm 6760 | The domain of a one-to-one... |
| f1ss 6761 | A function that is one-to-... |
| f1ssr 6762 | A function that is one-to-... |
| f1ssres 6763 | A function that is one-to-... |
| f1resf1 6764 | The restriction of an inje... |
| f1cnvcnv 6765 | Two ways to express that a... |
| f1cof1 6766 | Composition of two one-to-... |
| f1co 6767 | Composition of one-to-one ... |
| foeq1 6768 | Equality theorem for onto ... |
| foeq2 6769 | Equality theorem for onto ... |
| foeq3 6770 | Equality theorem for onto ... |
| nffo 6771 | Bound-variable hypothesis ... |
| fof 6772 | An onto mapping is a mappi... |
| fofun 6773 | An onto mapping is a funct... |
| fofn 6774 | An onto mapping is a funct... |
| forn 6775 | The codomain of an onto fu... |
| dffo2 6776 | Alternate definition of an... |
| foima 6777 | The image of the domain of... |
| dffn4 6778 | A function maps onto its r... |
| funforn 6779 | A function maps its domain... |
| fodmrnu 6780 | An onto function has uniqu... |
| fimadmfo 6781 | A function is a function o... |
| fores 6782 | Restriction of an onto fun... |
| fimadmfoALT 6783 | Alternate proof of ~ fimad... |
| focnvimacdmdm 6784 | The preimage of the codoma... |
| focofo 6785 | Composition of onto functi... |
| foco 6786 | Composition of onto functi... |
| foconst 6787 | A nonzero constant functio... |
| f1oeq1 6788 | Equality theorem for one-t... |
| f1oeq2 6789 | Equality theorem for one-t... |
| f1oeq3 6790 | Equality theorem for one-t... |
| f1oeq23 6791 | Equality theorem for one-t... |
| f1eq123d 6792 | Equality deduction for one... |
| foeq123d 6793 | Equality deduction for ont... |
| f1oeq123d 6794 | Equality deduction for one... |
| f1oeq1d 6795 | Equality deduction for one... |
| f1oeq2d 6796 | Equality deduction for one... |
| f1oeq3d 6797 | Equality deduction for one... |
| nff1o 6798 | Bound-variable hypothesis ... |
| f1of1 6799 | A one-to-one onto mapping ... |
| f1of 6800 | A one-to-one onto mapping ... |
| f1ofn 6801 | A one-to-one onto mapping ... |
| f1ofun 6802 | A one-to-one onto mapping ... |
| f1orel 6803 | A one-to-one onto mapping ... |
| f1odm 6804 | The domain of a one-to-one... |
| dff1o2 6805 | Alternate definition of on... |
| dff1o3 6806 | Alternate definition of on... |
| f1ofo 6807 | A one-to-one onto function... |
| dff1o4 6808 | Alternate definition of on... |
| dff1o5 6809 | Alternate definition of on... |
| f1orn 6810 | A one-to-one function maps... |
| f1f1orn 6811 | A one-to-one function maps... |
| f1ocnv 6812 | The converse of a one-to-o... |
| f1ocnvb 6813 | A relation is a one-to-one... |
| f1ores 6814 | The restriction of a one-t... |
| f1orescnv 6815 | The converse of a one-to-o... |
| f1imacnv 6816 | Preimage of an image. (Co... |
| foimacnv 6817 | A reverse version of ~ f1i... |
| foun 6818 | The union of two onto func... |
| f1oun 6819 | The union of two one-to-on... |
| f1un 6820 | The union of two one-to-on... |
| resdif 6821 | The restriction of a one-t... |
| resin 6822 | The restriction of a one-t... |
| f1oco 6823 | Composition of one-to-one ... |
| f1cnv 6824 | The converse of an injecti... |
| funcocnv2 6825 | Composition with the conve... |
| fococnv2 6826 | The composition of an onto... |
| f1ococnv2 6827 | The composition of a one-t... |
| f1cocnv2 6828 | Composition of an injectiv... |
| f1ococnv1 6829 | The composition of a one-t... |
| f1cocnv1 6830 | Composition of an injectiv... |
| funcoeqres 6831 | Express a constraint on a ... |
| f1ssf1 6832 | A subset of an injective f... |
| f10 6833 | The empty set maps one-to-... |
| f10d 6834 | The empty set maps one-to-... |
| f1o00 6835 | One-to-one onto mapping of... |
| fo00 6836 | Onto mapping of the empty ... |
| f1o0 6837 | One-to-one onto mapping of... |
| f1oi 6838 | A restriction of the ident... |
| f1ovi 6839 | The identity relation is a... |
| f1osn 6840 | A singleton of an ordered ... |
| f1osng 6841 | A singleton of an ordered ... |
| f1sng 6842 | A singleton of an ordered ... |
| fsnd 6843 | A singleton of an ordered ... |
| f1oprswap 6844 | A two-element swap is a bi... |
| f1oprg 6845 | An unordered pair of order... |
| tz6.12-2 6846 | Function value when ` F ` ... |
| fveu 6847 | The value of a function at... |
| brprcneu 6848 | If ` A ` is a proper class... |
| brprcneuALT 6849 | Alternate proof of ~ brprc... |
| fvprc 6850 | A function's value at a pr... |
| fvprcALT 6851 | Alternate proof of ~ fvprc... |
| rnfvprc 6852 | The range of a function va... |
| fv2 6853 | Alternate definition of fu... |
| dffv3 6854 | A definition of function v... |
| dffv4 6855 | The previous definition of... |
| elfv 6856 | Membership in a function v... |
| fveq1 6857 | Equality theorem for funct... |
| fveq2 6858 | Equality theorem for funct... |
| fveq1i 6859 | Equality inference for fun... |
| fveq1d 6860 | Equality deduction for fun... |
| fveq2i 6861 | Equality inference for fun... |
| fveq2d 6862 | Equality deduction for fun... |
| 2fveq3 6863 | Equality theorem for neste... |
| fveq12i 6864 | Equality deduction for fun... |
| fveq12d 6865 | Equality deduction for fun... |
| fveqeq2d 6866 | Equality deduction for fun... |
| fveqeq2 6867 | Equality deduction for fun... |
| nffv 6868 | Bound-variable hypothesis ... |
| nffvmpt1 6869 | Bound-variable hypothesis ... |
| nffvd 6870 | Deduction version of bound... |
| fvex 6871 | The value of a class exist... |
| fvexi 6872 | The value of a class exist... |
| fvexd 6873 | The value of a class exist... |
| fvif 6874 | Move a conditional outside... |
| iffv 6875 | Move a conditional outside... |
| fv3 6876 | Alternate definition of th... |
| fvres 6877 | The value of a restricted ... |
| fvresd 6878 | The value of a restricted ... |
| funssfv 6879 | The value of a member of t... |
| tz6.12c 6880 | Corollary of Theorem 6.12(... |
| tz6.12-1 6881 | Function value. Theorem 6... |
| tz6.12-1OLD 6882 | Obsolete version of ~ tz6.... |
| tz6.12 6883 | Function value. Theorem 6... |
| tz6.12f 6884 | Function value, using boun... |
| tz6.12cOLD 6885 | Obsolete version of ~ tz6.... |
| tz6.12i 6886 | Corollary of Theorem 6.12(... |
| fvbr0 6887 | Two possibilities for the ... |
| fvrn0 6888 | A function value is a memb... |
| fvn0fvelrn 6889 | If the value of a function... |
| elfvunirn 6890 | A function value is a subs... |
| fvssunirn 6891 | The result of a function v... |
| fvssunirnOLD 6892 | Obsolete version of ~ fvss... |
| ndmfv 6893 | The value of a class outsi... |
| ndmfvrcl 6894 | Reverse closure law for fu... |
| elfvdm 6895 | If a function value has a ... |
| elfvex 6896 | If a function value has a ... |
| elfvexd 6897 | If a function value has a ... |
| eliman0 6898 | A nonempty function value ... |
| nfvres 6899 | The value of a non-member ... |
| nfunsn 6900 | If the restriction of a cl... |
| fvfundmfvn0 6901 | If the "value of a class" ... |
| 0fv 6902 | Function value of the empt... |
| fv2prc 6903 | A function value of a func... |
| elfv2ex 6904 | If a function value of a f... |
| fveqres 6905 | Equal values imply equal v... |
| csbfv12 6906 | Move class substitution in... |
| csbfv2g 6907 | Move class substitution in... |
| csbfv 6908 | Substitution for a functio... |
| funbrfv 6909 | The second argument of a b... |
| funopfv 6910 | The second element in an o... |
| fnbrfvb 6911 | Equivalence of function va... |
| fnopfvb 6912 | Equivalence of function va... |
| fvelima2 6913 | Function value in an image... |
| funbrfvb 6914 | Equivalence of function va... |
| funopfvb 6915 | Equivalence of function va... |
| fnbrfvb2 6916 | Version of ~ fnbrfvb for f... |
| fdmeu 6917 | There is exactly one codom... |
| funbrfv2b 6918 | Function value in terms of... |
| dffn5 6919 | Representation of a functi... |
| fnrnfv 6920 | The range of a function ex... |
| fvelrnb 6921 | A member of a function's r... |
| foelcdmi 6922 | A member of a surjective f... |
| dfimafn 6923 | Alternate definition of th... |
| dfimafn2 6924 | Alternate definition of th... |
| funimass4 6925 | Membership relation for th... |
| fvelima 6926 | Function value in an image... |
| funimassd 6927 | Sufficient condition for t... |
| fvelimad 6928 | Function value in an image... |
| feqmptd 6929 | Deduction form of ~ dffn5 ... |
| feqresmpt 6930 | Express a restricted funct... |
| feqmptdf 6931 | Deduction form of ~ dffn5f... |
| dffn5f 6932 | Representation of a functi... |
| fvelimab 6933 | Function value in an image... |
| fvelimabd 6934 | Deduction form of ~ fvelim... |
| fimarab 6935 | Expressing the image of a ... |
| unima 6936 | Image of a union. (Contri... |
| fvi 6937 | The value of the identity ... |
| fviss 6938 | The value of the identity ... |
| fniinfv 6939 | The indexed intersection o... |
| fnsnfv 6940 | Singleton of function valu... |
| opabiotafun 6941 | Define a function whose va... |
| opabiotadm 6942 | Define a function whose va... |
| opabiota 6943 | Define a function whose va... |
| fnimapr 6944 | The image of a pair under ... |
| fnimatpd 6945 | The image of an unordered ... |
| ssimaex 6946 | The existence of a subimag... |
| ssimaexg 6947 | The existence of a subimag... |
| funfv 6948 | A simplified expression fo... |
| funfv2 6949 | The value of a function. ... |
| funfv2f 6950 | The value of a function. ... |
| fvun 6951 | Value of the union of two ... |
| fvun1 6952 | The value of a union when ... |
| fvun2 6953 | The value of a union when ... |
| fvun1d 6954 | The value of a union when ... |
| fvun2d 6955 | The value of a union when ... |
| dffv2 6956 | Alternate definition of fu... |
| dmfco 6957 | Domains of a function comp... |
| fvco2 6958 | Value of a function compos... |
| fvco 6959 | Value of a function compos... |
| fvco3 6960 | Value of a function compos... |
| fvco3d 6961 | Value of a function compos... |
| fvco4i 6962 | Conditions for a compositi... |
| fvopab3g 6963 | Value of a function given ... |
| fvopab3ig 6964 | Value of a function given ... |
| brfvopabrbr 6965 | The binary relation of a f... |
| fvmptg 6966 | Value of a function given ... |
| fvmpti 6967 | Value of a function given ... |
| fvmpt 6968 | Value of a function given ... |
| fvmpt2f 6969 | Value of a function given ... |
| fvtresfn 6970 | Functionality of a tuple-r... |
| fvmpts 6971 | Value of a function given ... |
| fvmpt3 6972 | Value of a function given ... |
| fvmpt3i 6973 | Value of a function given ... |
| fvmptdf 6974 | Deduction version of ~ fvm... |
| fvmptd 6975 | Deduction version of ~ fvm... |
| fvmptd2 6976 | Deduction version of ~ fvm... |
| mptrcl 6977 | Reverse closure for a mapp... |
| fvmpt2i 6978 | Value of a function given ... |
| fvmpt2 6979 | Value of a function given ... |
| fvmptss 6980 | If all the values of the m... |
| fvmpt2d 6981 | Deduction version of ~ fvm... |
| fvmptex 6982 | Express a function ` F ` w... |
| fvmptd3f 6983 | Alternate deduction versio... |
| fvmptd2f 6984 | Alternate deduction versio... |
| fvmptdv 6985 | Alternate deduction versio... |
| fvmptdv2 6986 | Alternate deduction versio... |
| mpteqb 6987 | Bidirectional equality the... |
| fvmptt 6988 | Closed theorem form of ~ f... |
| fvmptf 6989 | Value of a function given ... |
| fvmptnf 6990 | The value of a function gi... |
| fvmptd3 6991 | Deduction version of ~ fvm... |
| fvmptd4 6992 | Deduction version of ~ fvm... |
| fvmptn 6993 | This somewhat non-intuitiv... |
| fvmptss2 6994 | A mapping always evaluates... |
| elfvmptrab1w 6995 | Implications for the value... |
| elfvmptrab1 6996 | Implications for the value... |
| elfvmptrab 6997 | Implications for the value... |
| fvopab4ndm 6998 | Value of a function given ... |
| fvmptndm 6999 | Value of a function given ... |
| fvmptrabfv 7000 | Value of a function mappin... |
| fvopab5 7001 | The value of a function th... |
| fvopab6 7002 | Value of a function given ... |
| eqfnfv 7003 | Equality of functions is d... |
| eqfnfv2 7004 | Equality of functions is d... |
| eqfnfv3 7005 | Derive equality of functio... |
| eqfnfvd 7006 | Deduction for equality of ... |
| eqfnfv2f 7007 | Equality of functions is d... |
| eqfunfv 7008 | Equality of functions is d... |
| eqfnun 7009 | Two functions on ` A u. B ... |
| fvreseq0 7010 | Equality of restricted fun... |
| fvreseq1 7011 | Equality of a function res... |
| fvreseq 7012 | Equality of restricted fun... |
| fnmptfvd 7013 | A function with a given do... |
| fndmdif 7014 | Two ways to express the lo... |
| fndmdifcom 7015 | The difference set between... |
| fndmdifeq0 7016 | The difference set of two ... |
| fndmin 7017 | Two ways to express the lo... |
| fneqeql 7018 | Two functions are equal if... |
| fneqeql2 7019 | Two functions are equal if... |
| fnreseql 7020 | Two functions are equal on... |
| chfnrn 7021 | The range of a choice func... |
| funfvop 7022 | Ordered pair with function... |
| funfvbrb 7023 | Two ways to say that ` A `... |
| fvimacnvi 7024 | A member of a preimage is ... |
| fvimacnv 7025 | The argument of a function... |
| funimass3 7026 | A kind of contraposition l... |
| funimass5 7027 | A subclass of a preimage i... |
| funconstss 7028 | Two ways of specifying tha... |
| fvimacnvALT 7029 | Alternate proof of ~ fvima... |
| elpreima 7030 | Membership in the preimage... |
| elpreimad 7031 | Membership in the preimage... |
| fniniseg 7032 | Membership in the preimage... |
| fncnvima2 7033 | Inverse images under funct... |
| fniniseg2 7034 | Inverse point images under... |
| unpreima 7035 | Preimage of a union. (Con... |
| inpreima 7036 | Preimage of an intersectio... |
| difpreima 7037 | Preimage of a difference. ... |
| respreima 7038 | The preimage of a restrict... |
| cnvimainrn 7039 | The preimage of the inters... |
| sspreima 7040 | The preimage of a subset i... |
| iinpreima 7041 | Preimage of an intersectio... |
| intpreima 7042 | Preimage of an intersectio... |
| fimacnvinrn 7043 | Taking the converse image ... |
| fimacnvinrn2 7044 | Taking the converse image ... |
| rescnvimafod 7045 | The restriction of a funct... |
| fvn0ssdmfun 7046 | If a class' function value... |
| fnopfv 7047 | Ordered pair with function... |
| fvelrn 7048 | A function's value belongs... |
| nelrnfvne 7049 | A function value cannot be... |
| fveqdmss 7050 | If the empty set is not co... |
| fveqressseq 7051 | If the empty set is not co... |
| fnfvelrn 7052 | A function's value belongs... |
| ffvelcdm 7053 | A function's value belongs... |
| fnfvelrnd 7054 | A function's value belongs... |
| ffvelcdmi 7055 | A function's value belongs... |
| ffvelcdmda 7056 | A function's value belongs... |
| ffvelcdmd 7057 | A function's value belongs... |
| feldmfvelcdm 7058 | A class is an element of t... |
| rexrn 7059 | Restricted existential qua... |
| ralrn 7060 | Restricted universal quant... |
| elrnrexdm 7061 | For any element in the ran... |
| elrnrexdmb 7062 | For any element in the ran... |
| eldmrexrn 7063 | For any element in the dom... |
| eldmrexrnb 7064 | For any element in the dom... |
| fvcofneq 7065 | The values of two function... |
| ralrnmptw 7066 | A restricted quantifier ov... |
| rexrnmptw 7067 | A restricted quantifier ov... |
| ralrnmpt 7068 | A restricted quantifier ov... |
| rexrnmpt 7069 | A restricted quantifier ov... |
| f0cli 7070 | Unconditional closure of a... |
| dff2 7071 | Alternate definition of a ... |
| dff3 7072 | Alternate definition of a ... |
| dff4 7073 | Alternate definition of a ... |
| dffo3 7074 | An onto mapping expressed ... |
| dffo4 7075 | Alternate definition of an... |
| dffo5 7076 | Alternate definition of an... |
| exfo 7077 | A relation equivalent to t... |
| dffo3f 7078 | An onto mapping expressed ... |
| foelrn 7079 | Property of a surjective f... |
| foelrnf 7080 | Property of a surjective f... |
| foco2 7081 | If a composition of two fu... |
| fmpt 7082 | Functionality of the mappi... |
| f1ompt 7083 | Express bijection for a ma... |
| fmpti 7084 | Functionality of the mappi... |
| fvmptelcdm 7085 | The value of a function at... |
| fmptd 7086 | Domain and codomain of the... |
| fmpttd 7087 | Version of ~ fmptd with in... |
| fmpt3d 7088 | Domain and codomain of the... |
| fmptdf 7089 | A version of ~ fmptd using... |
| fompt 7090 | Express being onto for a m... |
| ffnfv 7091 | A function maps to a class... |
| ffnfvf 7092 | A function maps to a class... |
| fnfvrnss 7093 | An upper bound for range d... |
| fcdmssb 7094 | A function is a function i... |
| rnmptss 7095 | The range of an operation ... |
| fmpt2d 7096 | Domain and codomain of the... |
| ffvresb 7097 | A necessary and sufficient... |
| fssrescdmd 7098 | Restriction of a function ... |
| f1oresrab 7099 | Build a bijection between ... |
| f1ossf1o 7100 | Restricting a bijection, w... |
| fmptco 7101 | Composition of two functio... |
| fmptcof 7102 | Version of ~ fmptco where ... |
| fmptcos 7103 | Composition of two functio... |
| cofmpt 7104 | Express composition of a m... |
| fcompt 7105 | Express composition of two... |
| fcoconst 7106 | Composition with a constan... |
| fsn 7107 | A function maps a singleto... |
| fsn2 7108 | A function that maps a sin... |
| fsng 7109 | A function maps a singleto... |
| fsn2g 7110 | A function that maps a sin... |
| xpsng 7111 | The Cartesian product of t... |
| xpprsng 7112 | The Cartesian product of a... |
| xpsn 7113 | The Cartesian product of t... |
| f1o2sn 7114 | A singleton consisting in ... |
| residpr 7115 | Restriction of the identit... |
| dfmpt 7116 | Alternate definition for t... |
| fnasrn 7117 | A function expressed as th... |
| idref 7118 | Two ways to state that a r... |
| funiun 7119 | A function is a union of s... |
| funopsn 7120 | If a function is an ordere... |
| funop 7121 | An ordered pair is a funct... |
| funopdmsn 7122 | The domain of a function w... |
| funsndifnop 7123 | A singleton of an ordered ... |
| funsneqopb 7124 | A singleton of an ordered ... |
| ressnop0 7125 | If ` A ` is not in ` C ` ,... |
| fpr 7126 | A function with a domain o... |
| fprg 7127 | A function with a domain o... |
| ftpg 7128 | A function with a domain o... |
| ftp 7129 | A function with a domain o... |
| fnressn 7130 | A function restricted to a... |
| funressn 7131 | A function restricted to a... |
| fressnfv 7132 | The value of a function re... |
| fvrnressn 7133 | If the value of a function... |
| fvressn 7134 | The value of a function re... |
| fvn0fvelrnOLD 7135 | Obsolete version of ~ fvn0... |
| fvconst 7136 | The value of a constant fu... |
| fnsnr 7137 | If a class belongs to a fu... |
| fnsnbg 7138 | A function's domain is a s... |
| fnsnb 7139 | A function whose domain is... |
| fnsnbOLD 7140 | Obsolete version of ~ fnsn... |
| fmptsn 7141 | Express a singleton functi... |
| fmptsng 7142 | Express a singleton functi... |
| fmptsnd 7143 | Express a singleton functi... |
| fmptap 7144 | Append an additional value... |
| fmptapd 7145 | Append an additional value... |
| fmptpr 7146 | Express a pair function in... |
| fvresi 7147 | The value of a restricted ... |
| fninfp 7148 | Express the class of fixed... |
| fnelfp 7149 | Property of a fixed point ... |
| fndifnfp 7150 | Express the class of non-f... |
| fnelnfp 7151 | Property of a non-fixed po... |
| fnnfpeq0 7152 | A function is the identity... |
| fvunsn 7153 | Remove an ordered pair not... |
| fvsng 7154 | The value of a singleton o... |
| fvsn 7155 | The value of a singleton o... |
| fvsnun1 7156 | The value of a function wi... |
| fvsnun2 7157 | The value of a function wi... |
| fnsnsplit 7158 | Split a function into a si... |
| fsnunf 7159 | Adjoining a point to a fun... |
| fsnunf2 7160 | Adjoining a point to a pun... |
| fsnunfv 7161 | Recover the added point fr... |
| fsnunres 7162 | Recover the original funct... |
| funresdfunsn 7163 | Restricting a function to ... |
| fvpr1g 7164 | The value of a function wi... |
| fvpr2g 7165 | The value of a function wi... |
| fvpr1 7166 | The value of a function wi... |
| fvpr2 7167 | The value of a function wi... |
| fprb 7168 | A condition for functionho... |
| fvtp1 7169 | The first value of a funct... |
| fvtp2 7170 | The second value of a func... |
| fvtp3 7171 | The third value of a funct... |
| fvtp1g 7172 | The value of a function wi... |
| fvtp2g 7173 | The value of a function wi... |
| fvtp3g 7174 | The value of a function wi... |
| tpres 7175 | An unordered triple of ord... |
| fvconst2g 7176 | The value of a constant fu... |
| fconst2g 7177 | A constant function expres... |
| fvconst2 7178 | The value of a constant fu... |
| fconst2 7179 | A constant function expres... |
| fconst5 7180 | Two ways to express that a... |
| rnmptc 7181 | Range of a constant functi... |
| fnprb 7182 | A function whose domain ha... |
| fntpb 7183 | A function whose domain ha... |
| fnpr2g 7184 | A function whose domain ha... |
| fpr2g 7185 | A function that maps a pai... |
| fconstfv 7186 | A constant function expres... |
| fconst3 7187 | Two ways to express a cons... |
| fconst4 7188 | Two ways to express a cons... |
| resfunexg 7189 | The restriction of a funct... |
| resiexd 7190 | The restriction of the ide... |
| fnex 7191 | If the domain of a functio... |
| fnexd 7192 | If the domain of a functio... |
| funex 7193 | If the domain of a functio... |
| opabex 7194 | Existence of a function ex... |
| mptexg 7195 | If the domain of a functio... |
| mptexgf 7196 | If the domain of a functio... |
| mptex 7197 | If the domain of a functio... |
| mptexd 7198 | If the domain of a functio... |
| mptrabex 7199 | If the domain of a functio... |
| fex 7200 | If the domain of a mapping... |
| fexd 7201 | If the domain of a mapping... |
| mptfvmpt 7202 | A function in maps-to nota... |
| eufnfv 7203 | A function is uniquely det... |
| funfvima 7204 | A function's value in a pr... |
| funfvima2 7205 | A function's value in an i... |
| funfvima2d 7206 | A function's value in a pr... |
| fnfvima 7207 | The function value of an o... |
| fnfvimad 7208 | A function's value belongs... |
| resfvresima 7209 | The value of the function ... |
| funfvima3 7210 | A class including a functi... |
| ralima 7211 | Universal quantification u... |
| rexima 7212 | Existential quantification... |
| reximaOLD 7213 | Obsolete version of ~ rexi... |
| ralimaOLD 7214 | Obsolete version of ~ rali... |
| fvclss 7215 | Upper bound for the class ... |
| elabrex 7216 | Elementhood in an image se... |
| elabrexg 7217 | Elementhood in an image se... |
| abrexco 7218 | Composition of two image m... |
| imaiun 7219 | The image of an indexed un... |
| imauni 7220 | The image of a union is th... |
| fniunfv 7221 | The indexed union of a fun... |
| funiunfv 7222 | The indexed union of a fun... |
| funiunfvf 7223 | The indexed union of a fun... |
| eluniima 7224 | Membership in the union of... |
| elunirn 7225 | Membership in the union of... |
| elunirnALT 7226 | Alternate proof of ~ eluni... |
| elunirn2OLD 7227 | Obsolete version of ~ elfv... |
| fnunirn 7228 | Membership in a union of s... |
| dff13 7229 | A one-to-one function in t... |
| dff13f 7230 | A one-to-one function in t... |
| f1veqaeq 7231 | If the values of a one-to-... |
| f1cofveqaeq 7232 | If the values of a composi... |
| f1cofveqaeqALT 7233 | Alternate proof of ~ f1cof... |
| dff14i 7234 | A one-to-one function maps... |
| 2f1fvneq 7235 | If two one-to-one function... |
| f1mpt 7236 | Express injection for a ma... |
| f1fveq 7237 | Equality of function value... |
| f1elima 7238 | Membership in the image of... |
| f1imass 7239 | Taking images under a one-... |
| f1imaeq 7240 | Taking images under a one-... |
| f1imapss 7241 | Taking images under a one-... |
| fpropnf1 7242 | A function, given by an un... |
| f1dom3fv3dif 7243 | The function values for a ... |
| f1dom3el3dif 7244 | The codomain of a 1-1 func... |
| dff14a 7245 | A one-to-one function in t... |
| dff14b 7246 | A one-to-one function in t... |
| f1ounsn 7247 | Extension of a bijection b... |
| f12dfv 7248 | A one-to-one function with... |
| f13dfv 7249 | A one-to-one function with... |
| dff1o6 7250 | A one-to-one onto function... |
| f1ocnvfv1 7251 | The converse value of the ... |
| f1ocnvfv2 7252 | The value of the converse ... |
| f1ocnvfv 7253 | Relationship between the v... |
| f1ocnvfvb 7254 | Relationship between the v... |
| nvof1o 7255 | An involution is a bijecti... |
| nvocnv 7256 | The converse of an involut... |
| f1cdmsn 7257 | If a one-to-one function w... |
| fsnex 7258 | Relate a function with a s... |
| f1prex 7259 | Relate a one-to-one functi... |
| f1ocnvdm 7260 | The value of the converse ... |
| f1ocnvfvrneq 7261 | If the values of a one-to-... |
| fcof1 7262 | An application is injectiv... |
| fcofo 7263 | An application is surjecti... |
| cbvfo 7264 | Change bound variable betw... |
| cbvexfo 7265 | Change bound variable betw... |
| cocan1 7266 | An injection is left-cance... |
| cocan2 7267 | A surjection is right-canc... |
| fcof1oinvd 7268 | Show that a function is th... |
| fcof1od 7269 | A function is bijective if... |
| 2fcoidinvd 7270 | Show that a function is th... |
| fcof1o 7271 | Show that two functions ar... |
| 2fvcoidd 7272 | Show that the composition ... |
| 2fvidf1od 7273 | A function is bijective if... |
| 2fvidinvd 7274 | Show that two functions ar... |
| foeqcnvco 7275 | Condition for function equ... |
| f1eqcocnv 7276 | Condition for function equ... |
| fveqf1o 7277 | Given a bijection ` F ` , ... |
| f1ocoima 7278 | The composition of two bij... |
| nf1const 7279 | A constant function from a... |
| nf1oconst 7280 | A constant function from a... |
| f1ofvswap 7281 | Swapping two values in a b... |
| fvf1pr 7282 | Values of a one-to-one fun... |
| fliftrel 7283 | ` F ` , a function lift, i... |
| fliftel 7284 | Elementhood in the relatio... |
| fliftel1 7285 | Elementhood in the relatio... |
| fliftcnv 7286 | Converse of the relation `... |
| fliftfun 7287 | The function ` F ` is the ... |
| fliftfund 7288 | The function ` F ` is the ... |
| fliftfuns 7289 | The function ` F ` is the ... |
| fliftf 7290 | The domain and range of th... |
| fliftval 7291 | The value of the function ... |
| isoeq1 7292 | Equality theorem for isomo... |
| isoeq2 7293 | Equality theorem for isomo... |
| isoeq3 7294 | Equality theorem for isomo... |
| isoeq4 7295 | Equality theorem for isomo... |
| isoeq5 7296 | Equality theorem for isomo... |
| nfiso 7297 | Bound-variable hypothesis ... |
| isof1o 7298 | An isomorphism is a one-to... |
| isof1oidb 7299 | A function is a bijection ... |
| isof1oopb 7300 | A function is a bijection ... |
| isorel 7301 | An isomorphism connects bi... |
| soisores 7302 | Express the condition of i... |
| soisoi 7303 | Infer isomorphism from one... |
| isoid 7304 | Identity law for isomorphi... |
| isocnv 7305 | Converse law for isomorphi... |
| isocnv2 7306 | Converse law for isomorphi... |
| isocnv3 7307 | Complementation law for is... |
| isores2 7308 | An isomorphism from one we... |
| isores1 7309 | An isomorphism from one we... |
| isores3 7310 | Induced isomorphism on a s... |
| isotr 7311 | Composition (transitive) l... |
| isomin 7312 | Isomorphisms preserve mini... |
| isoini 7313 | Isomorphisms preserve init... |
| isoini2 7314 | Isomorphisms are isomorphi... |
| isofrlem 7315 | Lemma for ~ isofr . (Cont... |
| isoselem 7316 | Lemma for ~ isose . (Cont... |
| isofr 7317 | An isomorphism preserves w... |
| isose 7318 | An isomorphism preserves s... |
| isofr2 7319 | A weak form of ~ isofr tha... |
| isopolem 7320 | Lemma for ~ isopo . (Cont... |
| isopo 7321 | An isomorphism preserves t... |
| isosolem 7322 | Lemma for ~ isoso . (Cont... |
| isoso 7323 | An isomorphism preserves t... |
| isowe 7324 | An isomorphism preserves t... |
| isowe2 7325 | A weak form of ~ isowe tha... |
| f1oiso 7326 | Any one-to-one onto functi... |
| f1oiso2 7327 | Any one-to-one onto functi... |
| f1owe 7328 | Well-ordering of isomorphi... |
| weniso 7329 | A set-like well-ordering h... |
| weisoeq 7330 | Thus, there is at most one... |
| weisoeq2 7331 | Thus, there is at most one... |
| knatar 7332 | The Knaster-Tarski theorem... |
| fvresval 7333 | The value of a restricted ... |
| funeldmb 7334 | If ` (/) ` is not part of ... |
| eqfunresadj 7335 | Law for adjoining an eleme... |
| eqfunressuc 7336 | Law for equality of restri... |
| fnssintima 7337 | Condition for subset of an... |
| imaeqsexvOLD 7338 | Obsolete version of ~ rexi... |
| imaeqsalvOLD 7339 | Obsolete version of ~ rali... |
| fnimasnd 7340 | The image of a function by... |
| canth 7341 | No set ` A ` is equinumero... |
| ncanth 7342 | Cantor's theorem fails for... |
| riotaeqdv 7345 | Formula-building deduction... |
| riotabidv 7346 | Formula-building deduction... |
| riotaeqbidv 7347 | Equality deduction for res... |
| riotaex 7348 | Restricted iota is a set. ... |
| riotav 7349 | An iota restricted to the ... |
| riotauni 7350 | Restricted iota in terms o... |
| nfriota1 7351 | The abstraction variable i... |
| nfriotadw 7352 | Deduction version of ~ nfr... |
| cbvriotaw 7353 | Change bound variable in a... |
| cbvriotavw 7354 | Change bound variable in a... |
| nfriotad 7355 | Deduction version of ~ nfr... |
| nfriota 7356 | A variable not free in a w... |
| cbvriota 7357 | Change bound variable in a... |
| cbvriotav 7358 | Change bound variable in a... |
| csbriota 7359 | Interchange class substitu... |
| riotacl2 7360 | Membership law for "the un... |
| riotacl 7361 | Closure of restricted iota... |
| riotasbc 7362 | Substitution law for descr... |
| riotabidva 7363 | Equivalent wff's yield equ... |
| riotabiia 7364 | Equivalent wff's yield equ... |
| riota1 7365 | Property of restricted iot... |
| riota1a 7366 | Property of iota. (Contri... |
| riota2df 7367 | A deduction version of ~ r... |
| riota2f 7368 | This theorem shows a condi... |
| riota2 7369 | This theorem shows a condi... |
| riotaeqimp 7370 | If two restricted iota des... |
| riotaprop 7371 | Properties of a restricted... |
| riota5f 7372 | A method for computing res... |
| riota5 7373 | A method for computing res... |
| riotass2 7374 | Restriction of a unique el... |
| riotass 7375 | Restriction of a unique el... |
| moriotass 7376 | Restriction of a unique el... |
| snriota 7377 | A restricted class abstrac... |
| riotaxfrd 7378 | Change the variable ` x ` ... |
| eusvobj2 7379 | Specify the same property ... |
| eusvobj1 7380 | Specify the same object in... |
| f1ofveu 7381 | There is one domain elemen... |
| f1ocnvfv3 7382 | Value of the converse of a... |
| riotaund 7383 | Restricted iota equals the... |
| riotassuni 7384 | The restricted iota class ... |
| riotaclb 7385 | Bidirectional closure of r... |
| riotarab 7386 | Restricted iota of a restr... |
| oveq 7393 | Equality theorem for opera... |
| oveq1 7394 | Equality theorem for opera... |
| oveq2 7395 | Equality theorem for opera... |
| oveq12 7396 | Equality theorem for opera... |
| oveq1i 7397 | Equality inference for ope... |
| oveq2i 7398 | Equality inference for ope... |
| oveq12i 7399 | Equality inference for ope... |
| oveqi 7400 | Equality inference for ope... |
| oveq123i 7401 | Equality inference for ope... |
| oveq1d 7402 | Equality deduction for ope... |
| oveq2d 7403 | Equality deduction for ope... |
| oveqd 7404 | Equality deduction for ope... |
| oveq12d 7405 | Equality deduction for ope... |
| oveqan12d 7406 | Equality deduction for ope... |
| oveqan12rd 7407 | Equality deduction for ope... |
| oveq123d 7408 | Equality deduction for ope... |
| fvoveq1d 7409 | Equality deduction for nes... |
| fvoveq1 7410 | Equality theorem for neste... |
| ovanraleqv 7411 | Equality theorem for a con... |
| imbrov2fvoveq 7412 | Equality theorem for neste... |
| ovrspc2v 7413 | If an operation value is a... |
| oveqrspc2v 7414 | Restricted specialization ... |
| oveqdr 7415 | Equality of two operations... |
| nfovd 7416 | Deduction version of bound... |
| nfov 7417 | Bound-variable hypothesis ... |
| oprabidw 7418 | The law of concretion. Sp... |
| oprabid 7419 | The law of concretion. Sp... |
| ovex 7420 | The result of an operation... |
| ovexi 7421 | The result of an operation... |
| ovexd 7422 | The result of an operation... |
| ovssunirn 7423 | The result of an operation... |
| 0ov 7424 | Operation value of the emp... |
| ovprc 7425 | The value of an operation ... |
| ovprc1 7426 | The value of an operation ... |
| ovprc2 7427 | The value of an operation ... |
| ovrcl 7428 | Reverse closure for an ope... |
| elfvov1 7429 | Utility theorem: reverse c... |
| elfvov2 7430 | Utility theorem: reverse c... |
| csbov123 7431 | Move class substitution in... |
| csbov 7432 | Move class substitution in... |
| csbov12g 7433 | Move class substitution in... |
| csbov1g 7434 | Move class substitution in... |
| csbov2g 7435 | Move class substitution in... |
| rspceov 7436 | A frequently used special ... |
| elovimad 7437 | Elementhood of the image s... |
| fnbrovb 7438 | Value of a binary operatio... |
| fnotovb 7439 | Equivalence of operation v... |
| opabbrex 7440 | A collection of ordered pa... |
| opabresex2 7441 | Restrictions of a collecti... |
| opabresex2d 7442 | Obsolete version of ~ opab... |
| fvmptopab 7443 | The function value of a ma... |
| fvmptopabOLD 7444 | Obsolete version of ~ fvmp... |
| f1opr 7445 | Condition for an operation... |
| brfvopab 7446 | The classes involved in a ... |
| dfoprab2 7447 | Class abstraction for oper... |
| reloprab 7448 | An operation class abstrac... |
| oprabv 7449 | If a pair and a class are ... |
| nfoprab1 7450 | The abstraction variables ... |
| nfoprab2 7451 | The abstraction variables ... |
| nfoprab3 7452 | The abstraction variables ... |
| nfoprab 7453 | Bound-variable hypothesis ... |
| oprabbid 7454 | Equivalent wff's yield equ... |
| oprabbidv 7455 | Equivalent wff's yield equ... |
| oprabbii 7456 | Equivalent wff's yield equ... |
| ssoprab2 7457 | Equivalence of ordered pai... |
| ssoprab2b 7458 | Equivalence of ordered pai... |
| eqoprab2bw 7459 | Equivalence of ordered pai... |
| eqoprab2b 7460 | Equivalence of ordered pai... |
| mpoeq123 7461 | An equality theorem for th... |
| mpoeq12 7462 | An equality theorem for th... |
| mpoeq123dva 7463 | An equality deduction for ... |
| mpoeq123dv 7464 | An equality deduction for ... |
| mpoeq123i 7465 | An equality inference for ... |
| mpoeq3dva 7466 | Slightly more general equa... |
| mpoeq3ia 7467 | An equality inference for ... |
| mpoeq3dv 7468 | An equality deduction for ... |
| nfmpo1 7469 | Bound-variable hypothesis ... |
| nfmpo2 7470 | Bound-variable hypothesis ... |
| nfmpo 7471 | Bound-variable hypothesis ... |
| 0mpo0 7472 | A mapping operation with e... |
| mpo0v 7473 | A mapping operation with e... |
| mpo0 7474 | A mapping operation with e... |
| oprab4 7475 | Two ways to state the doma... |
| cbvoprab1 7476 | Rule used to change first ... |
| cbvoprab2 7477 | Change the second bound va... |
| cbvoprab12 7478 | Rule used to change first ... |
| cbvoprab12v 7479 | Rule used to change first ... |
| cbvoprab3 7480 | Rule used to change the th... |
| cbvoprab3v 7481 | Rule used to change the th... |
| cbvmpox 7482 | Rule to change the bound v... |
| cbvmpo 7483 | Rule to change the bound v... |
| cbvmpov 7484 | Rule to change the bound v... |
| elimdelov 7485 | Eliminate a hypothesis whi... |
| brif1 7486 | Move a relation inside and... |
| ovif 7487 | Move a conditional outside... |
| ovif2 7488 | Move a conditional outside... |
| ovif12 7489 | Move a conditional outside... |
| ifov 7490 | Move a conditional outside... |
| ifmpt2v 7491 | Move a conditional inside ... |
| dmoprab 7492 | The domain of an operation... |
| dmoprabss 7493 | The domain of an operation... |
| rnoprab 7494 | The range of an operation ... |
| rnoprab2 7495 | The range of a restricted ... |
| reldmoprab 7496 | The domain of an operation... |
| oprabss 7497 | Structure of an operation ... |
| eloprabga 7498 | The law of concretion for ... |
| eloprabg 7499 | The law of concretion for ... |
| ssoprab2i 7500 | Inference of operation cla... |
| mpov 7501 | Operation with universal d... |
| mpomptx 7502 | Express a two-argument fun... |
| mpompt 7503 | Express a two-argument fun... |
| mpodifsnif 7504 | A mapping with two argumen... |
| mposnif 7505 | A mapping with two argumen... |
| fconstmpo 7506 | Representation of a consta... |
| resoprab 7507 | Restriction of an operatio... |
| resoprab2 7508 | Restriction of an operator... |
| resmpo 7509 | Restriction of the mapping... |
| funoprabg 7510 | "At most one" is a suffici... |
| funoprab 7511 | "At most one" is a suffici... |
| fnoprabg 7512 | Functionality and domain o... |
| mpofun 7513 | The maps-to notation for a... |
| fnoprab 7514 | Functionality and domain o... |
| ffnov 7515 | An operation maps to a cla... |
| fovcld 7516 | Closure law for an operati... |
| fovcl 7517 | Closure law for an operati... |
| eqfnov 7518 | Equality of two operations... |
| eqfnov2 7519 | Two operators with the sam... |
| fnov 7520 | Representation of a functi... |
| mpo2eqb 7521 | Bidirectional equality the... |
| rnmpo 7522 | The range of an operation ... |
| reldmmpo 7523 | The domain of an operation... |
| elrnmpog 7524 | Membership in the range of... |
| elrnmpo 7525 | Membership in the range of... |
| elimampo 7526 | Membership in the image of... |
| elrnmpores 7527 | Membership in the range of... |
| ralrnmpo 7528 | A restricted quantifier ov... |
| rexrnmpo 7529 | A restricted quantifier ov... |
| ovid 7530 | The value of an operation ... |
| ovidig 7531 | The value of an operation ... |
| ovidi 7532 | The value of an operation ... |
| ov 7533 | The value of an operation ... |
| ovigg 7534 | The value of an operation ... |
| ovig 7535 | The value of an operation ... |
| ovmpt4g 7536 | Value of a function given ... |
| ovmpos 7537 | Value of a function given ... |
| ov2gf 7538 | The value of an operation ... |
| ovmpodxf 7539 | Value of an operation give... |
| ovmpodx 7540 | Value of an operation give... |
| ovmpod 7541 | Value of an operation give... |
| ovmpox 7542 | The value of an operation ... |
| ovmpoga 7543 | Value of an operation give... |
| ovmpoa 7544 | Value of an operation give... |
| ovmpodf 7545 | Alternate deduction versio... |
| ovmpodv 7546 | Alternate deduction versio... |
| ovmpodv2 7547 | Alternate deduction versio... |
| ovmpog 7548 | Value of an operation give... |
| ovmpo 7549 | Value of an operation give... |
| ovmpot 7550 | The value of an operation ... |
| fvmpopr2d 7551 | Value of an operation give... |
| ov3 7552 | The value of an operation ... |
| ov6g 7553 | The value of an operation ... |
| ovg 7554 | The value of an operation ... |
| ovres 7555 | The value of a restricted ... |
| ovresd 7556 | Lemma for converting metri... |
| oprres 7557 | The restriction of an oper... |
| oprssov 7558 | The value of a member of t... |
| fovcdm 7559 | An operation's value belon... |
| fovcdmda 7560 | An operation's value belon... |
| fovcdmd 7561 | An operation's value belon... |
| fnrnov 7562 | The range of an operation ... |
| foov 7563 | An onto mapping of an oper... |
| fnovrn 7564 | An operation's value belon... |
| ovelrn 7565 | A member of an operation's... |
| funimassov 7566 | Membership relation for th... |
| ovelimab 7567 | Operation value in an imag... |
| ovima0 7568 | An operation value is a me... |
| ovconst2 7569 | The value of a constant op... |
| oprssdm 7570 | Domain of closure of an op... |
| nssdmovg 7571 | The value of an operation ... |
| ndmovg 7572 | The value of an operation ... |
| ndmov 7573 | The value of an operation ... |
| ndmovcl 7574 | The closure of an operatio... |
| ndmovrcl 7575 | Reverse closure law, when ... |
| ndmovcom 7576 | Any operation is commutati... |
| ndmovass 7577 | Any operation is associati... |
| ndmovdistr 7578 | Any operation is distribut... |
| ndmovord 7579 | Elimination of redundant a... |
| ndmovordi 7580 | Elimination of redundant a... |
| caovclg 7581 | Convert an operation closu... |
| caovcld 7582 | Convert an operation closu... |
| caovcl 7583 | Convert an operation closu... |
| caovcomg 7584 | Convert an operation commu... |
| caovcomd 7585 | Convert an operation commu... |
| caovcom 7586 | Convert an operation commu... |
| caovassg 7587 | Convert an operation assoc... |
| caovassd 7588 | Convert an operation assoc... |
| caovass 7589 | Convert an operation assoc... |
| caovcang 7590 | Convert an operation cance... |
| caovcand 7591 | Convert an operation cance... |
| caovcanrd 7592 | Commute the arguments of a... |
| caovcan 7593 | Convert an operation cance... |
| caovordig 7594 | Convert an operation order... |
| caovordid 7595 | Convert an operation order... |
| caovordg 7596 | Convert an operation order... |
| caovordd 7597 | Convert an operation order... |
| caovord2d 7598 | Operation ordering law wit... |
| caovord3d 7599 | Ordering law. (Contribute... |
| caovord 7600 | Convert an operation order... |
| caovord2 7601 | Operation ordering law wit... |
| caovord3 7602 | Ordering law. (Contribute... |
| caovdig 7603 | Convert an operation distr... |
| caovdid 7604 | Convert an operation distr... |
| caovdir2d 7605 | Convert an operation distr... |
| caovdirg 7606 | Convert an operation rever... |
| caovdird 7607 | Convert an operation distr... |
| caovdi 7608 | Convert an operation distr... |
| caov32d 7609 | Rearrange arguments in a c... |
| caov12d 7610 | Rearrange arguments in a c... |
| caov31d 7611 | Rearrange arguments in a c... |
| caov13d 7612 | Rearrange arguments in a c... |
| caov4d 7613 | Rearrange arguments in a c... |
| caov411d 7614 | Rearrange arguments in a c... |
| caov42d 7615 | Rearrange arguments in a c... |
| caov32 7616 | Rearrange arguments in a c... |
| caov12 7617 | Rearrange arguments in a c... |
| caov31 7618 | Rearrange arguments in a c... |
| caov13 7619 | Rearrange arguments in a c... |
| caov4 7620 | Rearrange arguments in a c... |
| caov411 7621 | Rearrange arguments in a c... |
| caov42 7622 | Rearrange arguments in a c... |
| caovdir 7623 | Reverse distributive law. ... |
| caovdilem 7624 | Lemma used by real number ... |
| caovlem2 7625 | Lemma used in real number ... |
| caovmo 7626 | Uniqueness of inverse elem... |
| imaeqexov 7627 | Substitute an operation va... |
| imaeqalov 7628 | Substitute an operation va... |
| mpondm0 7629 | The value of an operation ... |
| elmpocl 7630 | If a two-parameter class i... |
| elmpocl1 7631 | If a two-parameter class i... |
| elmpocl2 7632 | If a two-parameter class i... |
| elovmpod 7633 | Utility lemma for two-para... |
| elovmpo 7634 | Utility lemma for two-para... |
| elovmporab 7635 | Implications for the value... |
| elovmporab1w 7636 | Implications for the value... |
| elovmporab1 7637 | Implications for the value... |
| 2mpo0 7638 | If the operation value of ... |
| relmptopab 7639 | Any function to sets of or... |
| f1ocnvd 7640 | Describe an implicit one-t... |
| f1od 7641 | Describe an implicit one-t... |
| f1ocnv2d 7642 | Describe an implicit one-t... |
| f1o2d 7643 | Describe an implicit one-t... |
| f1opw2 7644 | A one-to-one mapping induc... |
| f1opw 7645 | A one-to-one mapping induc... |
| elovmpt3imp 7646 | If the value of a function... |
| ovmpt3rab1 7647 | The value of an operation ... |
| ovmpt3rabdm 7648 | If the value of a function... |
| elovmpt3rab1 7649 | Implications for the value... |
| elovmpt3rab 7650 | Implications for the value... |
| ofeqd 7655 | Equality theorem for funct... |
| ofeq 7656 | Equality theorem for funct... |
| ofreq 7657 | Equality theorem for funct... |
| ofexg 7658 | A function operation restr... |
| nfof 7659 | Hypothesis builder for fun... |
| nfofr 7660 | Hypothesis builder for fun... |
| ofrfvalg 7661 | Value of a relation applie... |
| offval 7662 | Value of an operation appl... |
| ofrfval 7663 | Value of a relation applie... |
| ofval 7664 | Evaluate a function operat... |
| ofrval 7665 | Exhibit a function relatio... |
| offn 7666 | The function operation pro... |
| offun 7667 | The function operation pro... |
| offval2f 7668 | The function operation exp... |
| ofmresval 7669 | Value of a restriction of ... |
| fnfvof 7670 | Function value of a pointw... |
| off 7671 | The function operation pro... |
| ofres 7672 | Restrict the operands of a... |
| offval2 7673 | The function operation exp... |
| ofrfval2 7674 | The function relation acti... |
| offvalfv 7675 | The function operation exp... |
| ofmpteq 7676 | Value of a pointwise opera... |
| coof 7677 | The composition of a _homo... |
| ofco 7678 | The composition of a funct... |
| offveq 7679 | Convert an identity of the... |
| offveqb 7680 | Equivalent expressions for... |
| ofc1 7681 | Left operation by a consta... |
| ofc2 7682 | Right operation by a const... |
| ofc12 7683 | Function operation on two ... |
| caofref 7684 | Transfer a reflexive law t... |
| caofinvl 7685 | Transfer a left inverse la... |
| caofid0l 7686 | Transfer a left identity l... |
| caofid0r 7687 | Transfer a right identity ... |
| caofid1 7688 | Transfer a right absorptio... |
| caofid2 7689 | Transfer a right absorptio... |
| caofcom 7690 | Transfer a commutative law... |
| caofidlcan 7691 | Transfer a cancellation/id... |
| caofrss 7692 | Transfer a relation subset... |
| caofass 7693 | Transfer an associative la... |
| caoftrn 7694 | Transfer a transitivity la... |
| caofdi 7695 | Transfer a distributive la... |
| caofdir 7696 | Transfer a reverse distrib... |
| caonncan 7697 | Transfer ~ nncan -shaped l... |
| relrpss 7700 | The proper subset relation... |
| brrpssg 7701 | The proper subset relation... |
| brrpss 7702 | The proper subset relation... |
| porpss 7703 | Every class is partially o... |
| sorpss 7704 | Express strict ordering un... |
| sorpssi 7705 | Property of a chain of set... |
| sorpssun 7706 | A chain of sets is closed ... |
| sorpssin 7707 | A chain of sets is closed ... |
| sorpssuni 7708 | In a chain of sets, a maxi... |
| sorpssint 7709 | In a chain of sets, a mini... |
| sorpsscmpl 7710 | The componentwise compleme... |
| zfun 7712 | Axiom of Union expressed w... |
| axun2 7713 | A variant of the Axiom of ... |
| uniex2 7714 | The Axiom of Union using t... |
| vuniex 7715 | The union of a setvar is a... |
| uniexg 7716 | The ZF Axiom of Union in c... |
| uniex 7717 | The Axiom of Union in clas... |
| uniexd 7718 | Deduction version of the Z... |
| unexg 7719 | The union of two sets is a... |
| unex 7720 | The union of two sets is a... |
| unexOLD 7721 | Obsolete version of ~ unex... |
| tpex 7722 | An unordered triple of cla... |
| unexb 7723 | Existence of union is equi... |
| unexbOLD 7724 | Obsolete version of ~ unex... |
| unexgOLD 7725 | Obsolete version of ~ unex... |
| xpexg 7726 | The Cartesian product of t... |
| xpexd 7727 | The Cartesian product of t... |
| 3xpexg 7728 | The Cartesian product of t... |
| xpex 7729 | The Cartesian product of t... |
| unexd 7730 | The union of two sets is a... |
| sqxpexg 7731 | The Cartesian square of a ... |
| abnexg 7732 | Sufficient condition for a... |
| abnex 7733 | Sufficient condition for a... |
| snnex 7734 | The class of all singleton... |
| pwnex 7735 | The class of all power set... |
| difex2 7736 | If the subtrahend of a cla... |
| difsnexi 7737 | If the difference of a cla... |
| uniuni 7738 | Expression for double unio... |
| uniexr 7739 | Converse of the Axiom of U... |
| uniexb 7740 | The Axiom of Union and its... |
| pwexr 7741 | Converse of the Axiom of P... |
| pwexb 7742 | The Axiom of Power Sets an... |
| elpwpwel 7743 | A class belongs to a doubl... |
| eldifpw 7744 | Membership in a power clas... |
| elpwun 7745 | Membership in the power cl... |
| pwuncl 7746 | Power classes are closed u... |
| iunpw 7747 | An indexed union of a powe... |
| fr3nr 7748 | A well-founded relation ha... |
| epne3 7749 | A well-founded class conta... |
| dfwe2 7750 | Alternate definition of we... |
| epweon 7751 | The membership relation we... |
| epweonALT 7752 | Alternate proof of ~ epweo... |
| ordon 7753 | The class of all ordinal n... |
| onprc 7754 | No set contains all ordina... |
| ssorduni 7755 | The union of a class of or... |
| ssonuni 7756 | The union of a set of ordi... |
| ssonunii 7757 | The union of a set of ordi... |
| ordeleqon 7758 | A way to express the ordin... |
| ordsson 7759 | Any ordinal class is a sub... |
| dford5 7760 | A class is ordinal iff it ... |
| onss 7761 | An ordinal number is a sub... |
| predon 7762 | The predecessor of an ordi... |
| ssonprc 7763 | Two ways of saying a class... |
| onuni 7764 | The union of an ordinal nu... |
| orduni 7765 | The union of an ordinal cl... |
| onint 7766 | The intersection (infimum)... |
| onint0 7767 | The intersection of a clas... |
| onssmin 7768 | A nonempty class of ordina... |
| onminesb 7769 | If a property is true for ... |
| onminsb 7770 | If a property is true for ... |
| oninton 7771 | The intersection of a none... |
| onintrab 7772 | The intersection of a clas... |
| onintrab2 7773 | An existence condition equ... |
| onnmin 7774 | No member of a set of ordi... |
| onnminsb 7775 | An ordinal number smaller ... |
| oneqmin 7776 | A way to show that an ordi... |
| uniordint 7777 | The union of a set of ordi... |
| onminex 7778 | If a wff is true for an or... |
| sucon 7779 | The class of all ordinal n... |
| sucexb 7780 | A successor exists iff its... |
| sucexg 7781 | The successor of a set is ... |
| sucex 7782 | The successor of a set is ... |
| onmindif2 7783 | The minimum of a class of ... |
| ordsuci 7784 | The successor of an ordina... |
| sucexeloni 7785 | If the successor of an ord... |
| sucexeloniOLD 7786 | Obsolete version of ~ suce... |
| onsuc 7787 | The successor of an ordina... |
| ordsuc 7788 | A class is ordinal if and ... |
| ordsucOLD 7789 | Obsolete version of ~ ords... |
| ordpwsuc 7790 | The collection of ordinals... |
| onpwsuc 7791 | The collection of ordinal ... |
| onsucb 7792 | A class is an ordinal numb... |
| ordsucss 7793 | The successor of an elemen... |
| onpsssuc 7794 | An ordinal number is a pro... |
| ordelsuc 7795 | A set belongs to an ordina... |
| onsucmin 7796 | The successor of an ordina... |
| ordsucelsuc 7797 | Membership is inherited by... |
| ordsucsssuc 7798 | The subclass relationship ... |
| ordsucuniel 7799 | Given an element ` A ` of ... |
| ordsucun 7800 | The successor of the maxim... |
| ordunpr 7801 | The maximum of two ordinal... |
| ordunel 7802 | The maximum of two ordinal... |
| onsucuni 7803 | A class of ordinal numbers... |
| ordsucuni 7804 | An ordinal class is a subc... |
| orduniorsuc 7805 | An ordinal class is either... |
| unon 7806 | The class of all ordinal n... |
| ordunisuc 7807 | An ordinal class is equal ... |
| orduniss2 7808 | The union of the ordinal s... |
| onsucuni2 7809 | A successor ordinal is the... |
| 0elsuc 7810 | The successor of an ordina... |
| limon 7811 | The class of ordinal numbe... |
| onuniorsuc 7812 | An ordinal number is eithe... |
| onssi 7813 | An ordinal number is a sub... |
| onsuci 7814 | The successor of an ordina... |
| onuniorsuciOLD 7815 | Obsolete version of ~ onun... |
| onuninsuci 7816 | An ordinal is equal to its... |
| onsucssi 7817 | A set belongs to an ordina... |
| nlimsucg 7818 | A successor is not a limit... |
| orduninsuc 7819 | An ordinal class is equal ... |
| ordunisuc2 7820 | An ordinal equal to its un... |
| ordzsl 7821 | An ordinal is zero, a succ... |
| onzsl 7822 | An ordinal number is zero,... |
| dflim3 7823 | An alternate definition of... |
| dflim4 7824 | An alternate definition of... |
| limsuc 7825 | The successor of a member ... |
| limsssuc 7826 | A class includes a limit o... |
| nlimon 7827 | Two ways to express the cl... |
| limuni3 7828 | The union of a nonempty cl... |
| tfi 7829 | The Principle of Transfini... |
| tfisg 7830 | A closed form of ~ tfis . ... |
| tfis 7831 | Transfinite Induction Sche... |
| tfis2f 7832 | Transfinite Induction Sche... |
| tfis2 7833 | Transfinite Induction Sche... |
| tfis3 7834 | Transfinite Induction Sche... |
| tfisi 7835 | A transfinite induction sc... |
| tfinds 7836 | Principle of Transfinite I... |
| tfindsg 7837 | Transfinite Induction (inf... |
| tfindsg2 7838 | Transfinite Induction (inf... |
| tfindes 7839 | Transfinite Induction with... |
| tfinds2 7840 | Transfinite Induction (inf... |
| tfinds3 7841 | Principle of Transfinite I... |
| dfom2 7844 | An alternate definition of... |
| elom 7845 | Membership in omega. The ... |
| omsson 7846 | Omega is a subset of ` On ... |
| limomss 7847 | The class of natural numbe... |
| nnon 7848 | A natural number is an ord... |
| nnoni 7849 | A natural number is an ord... |
| nnord 7850 | A natural number is ordina... |
| trom 7851 | The class of finite ordina... |
| ordom 7852 | The class of finite ordina... |
| elnn 7853 | A member of a natural numb... |
| omon 7854 | The class of natural numbe... |
| omelon2 7855 | Omega is an ordinal number... |
| nnlim 7856 | A natural number is not a ... |
| omssnlim 7857 | The class of natural numbe... |
| limom 7858 | Omega is a limit ordinal. ... |
| peano2b 7859 | A class belongs to omega i... |
| nnsuc 7860 | A nonzero natural number i... |
| omsucne 7861 | A natural number is not th... |
| ssnlim 7862 | An ordinal subclass of non... |
| omsinds 7863 | Strong (or "total") induct... |
| omun 7864 | The union of two finite or... |
| peano1 7865 | Zero is a natural number. ... |
| peano2 7866 | The successor of any natur... |
| peano3 7867 | The successor of any natur... |
| peano4 7868 | Two natural numbers are eq... |
| peano5 7869 | The induction postulate: a... |
| nn0suc 7870 | A natural number is either... |
| find 7871 | The Principle of Finite In... |
| finds 7872 | Principle of Finite Induct... |
| findsg 7873 | Principle of Finite Induct... |
| finds2 7874 | Principle of Finite Induct... |
| finds1 7875 | Principle of Finite Induct... |
| findes 7876 | Finite induction with expl... |
| dmexg 7877 | The domain of a set is a s... |
| rnexg 7878 | The range of a set is a se... |
| dmexd 7879 | The domain of a set is a s... |
| fndmexd 7880 | If a function is a set, it... |
| dmfex 7881 | If a mapping is a set, its... |
| fndmexb 7882 | The domain of a function i... |
| fdmexb 7883 | The domain of a function i... |
| dmfexALT 7884 | Alternate proof of ~ dmfex... |
| dmex 7885 | The domain of a set is a s... |
| rnex 7886 | The range of a set is a se... |
| iprc 7887 | The identity function is a... |
| resiexg 7888 | The existence of a restric... |
| imaexg 7889 | The image of a set is a se... |
| imaex 7890 | The image of a set is a se... |
| rnexd 7891 | The range of a set is a se... |
| imaexd 7892 | The image of a set is a se... |
| exse2 7893 | Any set relation is set-li... |
| xpexr 7894 | If a Cartesian product is ... |
| xpexr2 7895 | If a nonempty Cartesian pr... |
| xpexcnv 7896 | A condition where the conv... |
| soex 7897 | If the relation in a stric... |
| elxp4 7898 | Membership in a Cartesian ... |
| elxp5 7899 | Membership in a Cartesian ... |
| cnvexg 7900 | The converse of a set is a... |
| cnvex 7901 | The converse of a set is a... |
| relcnvexb 7902 | A relation is a set iff it... |
| f1oexrnex 7903 | If the range of a 1-1 onto... |
| f1oexbi 7904 | There is a one-to-one onto... |
| coexg 7905 | The composition of two set... |
| coex 7906 | The composition of two set... |
| coexd 7907 | The composition of two set... |
| funcnvuni 7908 | The union of a chain (with... |
| fun11uni 7909 | The union of a chain (with... |
| resf1extb 7910 | Extension of an injection ... |
| resf1ext2b 7911 | Extension of an injection ... |
| fex2 7912 | A function with bounded do... |
| fabexd 7913 | Existence of a set of func... |
| fabexg 7914 | Existence of a set of func... |
| fabexgOLD 7915 | Obsolete version of ~ fabe... |
| fabex 7916 | Existence of a set of func... |
| mapex 7917 | The class of all functions... |
| f1oabexg 7918 | The class of all 1-1-onto ... |
| f1oabexgOLD 7919 | Obsolete version of ~ f1oa... |
| fiunlem 7920 | Lemma for ~ fiun and ~ f1i... |
| fiun 7921 | The union of a chain (with... |
| f1iun 7922 | The union of a chain (with... |
| fviunfun 7923 | The function value of an i... |
| ffoss 7924 | Relationship between a map... |
| f11o 7925 | Relationship between one-t... |
| resfunexgALT 7926 | Alternate proof of ~ resfu... |
| cofunexg 7927 | Existence of a composition... |
| cofunex2g 7928 | Existence of a composition... |
| fnexALT 7929 | Alternate proof of ~ fnex ... |
| funexw 7930 | Weak version of ~ funex th... |
| mptexw 7931 | Weak version of ~ mptex th... |
| funrnex 7932 | If the domain of a functio... |
| zfrep6 7933 | A version of the Axiom of ... |
| focdmex 7934 | If the domain of an onto f... |
| f1dmex 7935 | If the codomain of a one-t... |
| f1ovv 7936 | The codomain/range of a 1-... |
| fvclex 7937 | Existence of the class of ... |
| fvresex 7938 | Existence of the class of ... |
| abrexexg 7939 | Existence of a class abstr... |
| abrexexgOLD 7940 | Obsolete version of ~ abre... |
| abrexex 7941 | Existence of a class abstr... |
| iunexg 7942 | The existence of an indexe... |
| abrexex2g 7943 | Existence of an existentia... |
| opabex3d 7944 | Existence of an ordered pa... |
| opabex3rd 7945 | Existence of an ordered pa... |
| opabex3 7946 | Existence of an ordered pa... |
| iunex 7947 | The existence of an indexe... |
| abrexex2 7948 | Existence of an existentia... |
| abexssex 7949 | Existence of a class abstr... |
| abexex 7950 | A condition where a class ... |
| f1oweALT 7951 | Alternate proof of ~ f1owe... |
| wemoiso 7952 | Thus, there is at most one... |
| wemoiso2 7953 | Thus, there is at most one... |
| oprabexd 7954 | Existence of an operator a... |
| oprabex 7955 | Existence of an operation ... |
| oprabex3 7956 | Existence of an operation ... |
| oprabrexex2 7957 | Existence of an existentia... |
| ab2rexex 7958 | Existence of a class abstr... |
| ab2rexex2 7959 | Existence of an existentia... |
| xpexgALT 7960 | Alternate proof of ~ xpexg... |
| offval3 7961 | General value of ` ( F oF ... |
| offres 7962 | Pointwise combination comm... |
| ofmres 7963 | Equivalent expressions for... |
| ofmresex 7964 | Existence of a restriction... |
| mptcnfimad 7965 | The converse of a mapping ... |
| 1stval 7970 | The value of the function ... |
| 2ndval 7971 | The value of the function ... |
| 1stnpr 7972 | Value of the first-member ... |
| 2ndnpr 7973 | Value of the second-member... |
| 1st0 7974 | The value of the first-mem... |
| 2nd0 7975 | The value of the second-me... |
| op1st 7976 | Extract the first member o... |
| op2nd 7977 | Extract the second member ... |
| op1std 7978 | Extract the first member o... |
| op2ndd 7979 | Extract the second member ... |
| op1stg 7980 | Extract the first member o... |
| op2ndg 7981 | Extract the second member ... |
| ot1stg 7982 | Extract the first member o... |
| ot2ndg 7983 | Extract the second member ... |
| ot3rdg 7984 | Extract the third member o... |
| 1stval2 7985 | Alternate value of the fun... |
| 2ndval2 7986 | Alternate value of the fun... |
| oteqimp 7987 | The components of an order... |
| fo1st 7988 | The ` 1st ` function maps ... |
| fo2nd 7989 | The ` 2nd ` function maps ... |
| br1steqg 7990 | Uniqueness condition for t... |
| br2ndeqg 7991 | Uniqueness condition for t... |
| f1stres 7992 | Mapping of a restriction o... |
| f2ndres 7993 | Mapping of a restriction o... |
| fo1stres 7994 | Onto mapping of a restrict... |
| fo2ndres 7995 | Onto mapping of a restrict... |
| 1st2val 7996 | Value of an alternate defi... |
| 2nd2val 7997 | Value of an alternate defi... |
| 1stcof 7998 | Composition of the first m... |
| 2ndcof 7999 | Composition of the second ... |
| xp1st 8000 | Location of the first elem... |
| xp2nd 8001 | Location of the second ele... |
| elxp6 8002 | Membership in a Cartesian ... |
| elxp7 8003 | Membership in a Cartesian ... |
| eqopi 8004 | Equality with an ordered p... |
| xp2 8005 | Representation of Cartesia... |
| unielxp 8006 | The membership relation fo... |
| 1st2nd2 8007 | Reconstruction of a member... |
| 1st2ndb 8008 | Reconstruction of an order... |
| xpopth 8009 | An ordered pair theorem fo... |
| eqop 8010 | Two ways to express equali... |
| eqop2 8011 | Two ways to express equali... |
| op1steq 8012 | Two ways of expressing tha... |
| opreuopreu 8013 | There is a unique ordered ... |
| el2xptp 8014 | A member of a nested Carte... |
| el2xptp0 8015 | A member of a nested Carte... |
| el2xpss 8016 | Version of ~ elrel for tri... |
| 2nd1st 8017 | Swap the members of an ord... |
| 1st2nd 8018 | Reconstruction of a member... |
| 1stdm 8019 | The first ordered pair com... |
| 2ndrn 8020 | The second ordered pair co... |
| 1st2ndbr 8021 | Express an element of a re... |
| releldm2 8022 | Two ways of expressing mem... |
| reldm 8023 | An expression for the doma... |
| releldmdifi 8024 | One way of expressing memb... |
| funfv1st2nd 8025 | The function value for the... |
| funelss 8026 | If the first component of ... |
| funeldmdif 8027 | Two ways of expressing mem... |
| sbcopeq1a 8028 | Equality theorem for subst... |
| csbopeq1a 8029 | Equality theorem for subst... |
| sbcoteq1a 8030 | Equality theorem for subst... |
| dfopab2 8031 | A way to define an ordered... |
| dfoprab3s 8032 | A way to define an operati... |
| dfoprab3 8033 | Operation class abstractio... |
| dfoprab4 8034 | Operation class abstractio... |
| dfoprab4f 8035 | Operation class abstractio... |
| opabex2 8036 | Condition for an operation... |
| opabn1stprc 8037 | An ordered-pair class abst... |
| opiota 8038 | The property of a uniquely... |
| cnvoprab 8039 | The converse of a class ab... |
| dfxp3 8040 | Define the Cartesian produ... |
| elopabi 8041 | A consequence of membershi... |
| eloprabi 8042 | A consequence of membershi... |
| mpomptsx 8043 | Express a two-argument fun... |
| mpompts 8044 | Express a two-argument fun... |
| dmmpossx 8045 | The domain of a mapping is... |
| fmpox 8046 | Functionality, domain and ... |
| fmpo 8047 | Functionality, domain and ... |
| fnmpo 8048 | Functionality and domain o... |
| fnmpoi 8049 | Functionality and domain o... |
| dmmpo 8050 | Domain of a class given by... |
| ovmpoelrn 8051 | An operation's value belon... |
| dmmpoga 8052 | Domain of an operation giv... |
| dmmpog 8053 | Domain of an operation giv... |
| mpoexxg 8054 | Existence of an operation ... |
| mpoexg 8055 | Existence of an operation ... |
| mpoexga 8056 | If the domain of an operat... |
| mpoexw 8057 | Weak version of ~ mpoex th... |
| mpoex 8058 | If the domain of an operat... |
| mptmpoopabbrd 8059 | The operation value of a f... |
| mptmpoopabbrdOLD 8060 | Obsolete version of ~ mptm... |
| mptmpoopabovd 8061 | The operation value of a f... |
| mptmpoopabbrdOLDOLD 8062 | Obsolete version of ~ mptm... |
| mptmpoopabovdOLD 8063 | Obsolete version of ~ mptm... |
| el2mpocsbcl 8064 | If the operation value of ... |
| el2mpocl 8065 | If the operation value of ... |
| fnmpoovd 8066 | A function with a Cartesia... |
| offval22 8067 | The function operation exp... |
| brovpreldm 8068 | If a binary relation holds... |
| bropopvvv 8069 | If a binary relation holds... |
| bropfvvvvlem 8070 | Lemma for ~ bropfvvvv . (... |
| bropfvvvv 8071 | If a binary relation holds... |
| ovmptss 8072 | If all the values of the m... |
| relmpoopab 8073 | Any function to sets of or... |
| fmpoco 8074 | Composition of two functio... |
| oprabco 8075 | Composition of a function ... |
| oprab2co 8076 | Composition of operator ab... |
| df1st2 8077 | An alternate possible defi... |
| df2nd2 8078 | An alternate possible defi... |
| 1stconst 8079 | The mapping of a restricti... |
| 2ndconst 8080 | The mapping of a restricti... |
| dfmpo 8081 | Alternate definition for t... |
| mposn 8082 | An operation (in maps-to n... |
| curry1 8083 | Composition with ` ``' ( 2... |
| curry1val 8084 | The value of a curried fun... |
| curry1f 8085 | Functionality of a curried... |
| curry2 8086 | Composition with ` ``' ( 1... |
| curry2f 8087 | Functionality of a curried... |
| curry2val 8088 | The value of a curried fun... |
| cnvf1olem 8089 | Lemma for ~ cnvf1o . (Con... |
| cnvf1o 8090 | Describe a function that m... |
| fparlem1 8091 | Lemma for ~ fpar . (Contr... |
| fparlem2 8092 | Lemma for ~ fpar . (Contr... |
| fparlem3 8093 | Lemma for ~ fpar . (Contr... |
| fparlem4 8094 | Lemma for ~ fpar . (Contr... |
| fpar 8095 | Merge two functions in par... |
| fsplit 8096 | A function that can be use... |
| fsplitfpar 8097 | Merge two functions with a... |
| offsplitfpar 8098 | Express the function opera... |
| f2ndf 8099 | The ` 2nd ` (second compon... |
| fo2ndf 8100 | The ` 2nd ` (second compon... |
| f1o2ndf1 8101 | The ` 2nd ` (second compon... |
| opco1 8102 | Value of an operation prec... |
| opco2 8103 | Value of an operation prec... |
| opco1i 8104 | Inference form of ~ opco1 ... |
| frxp 8105 | A lexicographical ordering... |
| xporderlem 8106 | Lemma for lexicographical ... |
| poxp 8107 | A lexicographical ordering... |
| soxp 8108 | A lexicographical ordering... |
| wexp 8109 | A lexicographical ordering... |
| fnwelem 8110 | Lemma for ~ fnwe . (Contr... |
| fnwe 8111 | A variant on lexicographic... |
| fnse 8112 | Condition for the well-ord... |
| fvproj 8113 | Value of a function on ord... |
| fimaproj 8114 | Image of a cartesian produ... |
| ralxpes 8115 | A version of ~ ralxp with ... |
| ralxp3f 8116 | Restricted for all over a ... |
| ralxp3 8117 | Restricted for all over a ... |
| ralxp3es 8118 | Restricted for-all over a ... |
| frpoins3xpg 8119 | Special case of founded pa... |
| frpoins3xp3g 8120 | Special case of founded pa... |
| xpord2lem 8121 | Lemma for Cartesian produc... |
| poxp2 8122 | Another way of partially o... |
| frxp2 8123 | Another way of giving a we... |
| xpord2pred 8124 | Calculate the predecessor ... |
| sexp2 8125 | Condition for the relation... |
| xpord2indlem 8126 | Induction over the Cartesi... |
| xpord2ind 8127 | Induction over the Cartesi... |
| xpord3lem 8128 | Lemma for triple ordering.... |
| poxp3 8129 | Triple Cartesian product p... |
| frxp3 8130 | Give well-foundedness over... |
| xpord3pred 8131 | Calculate the predecsessor... |
| sexp3 8132 | Show that the triple order... |
| xpord3inddlem 8133 | Induction over the triple ... |
| xpord3indd 8134 | Induction over the triple ... |
| xpord3ind 8135 | Induction over the triple ... |
| orderseqlem 8136 | Lemma for ~ poseq and ~ so... |
| poseq 8137 | A partial ordering of ordi... |
| soseq 8138 | A linear ordering of ordin... |
| suppval 8141 | The value of the operation... |
| supp0prc 8142 | The support of a class is ... |
| suppvalbr 8143 | The value of the operation... |
| supp0 8144 | The support of the empty s... |
| suppval1 8145 | The value of the operation... |
| suppvalfng 8146 | The value of the operation... |
| suppvalfn 8147 | The value of the operation... |
| elsuppfng 8148 | An element of the support ... |
| elsuppfn 8149 | An element of the support ... |
| fvdifsupp 8150 | Function value is zero out... |
| cnvimadfsn 8151 | The support of functions "... |
| suppimacnvss 8152 | The support of functions "... |
| suppimacnv 8153 | Support sets of functions ... |
| fsuppeq 8154 | Two ways of writing the su... |
| fsuppeqg 8155 | Version of ~ fsuppeq avoid... |
| suppssdm 8156 | The support of a function ... |
| suppsnop 8157 | The support of a singleton... |
| snopsuppss 8158 | The support of a singleton... |
| fvn0elsupp 8159 | If the function value for ... |
| fvn0elsuppb 8160 | The function value for a g... |
| rexsupp 8161 | Existential quantification... |
| ressuppss 8162 | The support of the restric... |
| suppun 8163 | The support of a class/fun... |
| ressuppssdif 8164 | The support of the restric... |
| mptsuppdifd 8165 | The support of a function ... |
| mptsuppd 8166 | The support of a function ... |
| extmptsuppeq 8167 | The support of an extended... |
| suppfnss 8168 | The support of a function ... |
| funsssuppss 8169 | The support of a function ... |
| fnsuppres 8170 | Two ways to express restri... |
| fnsuppeq0 8171 | The support of a function ... |
| fczsupp0 8172 | The support of a constant ... |
| suppss 8173 | Show that the support of a... |
| suppssr 8174 | A function is zero outside... |
| suppssrg 8175 | A function is zero outside... |
| suppssov1 8176 | Formula building theorem f... |
| suppssov2 8177 | Formula building theorem f... |
| suppssof1 8178 | Formula building theorem f... |
| suppss2 8179 | Show that the support of a... |
| suppsssn 8180 | Show that the support of a... |
| suppssfv 8181 | Formula building theorem f... |
| suppofssd 8182 | Condition for the support ... |
| suppofss1d 8183 | Condition for the support ... |
| suppofss2d 8184 | Condition for the support ... |
| suppco 8185 | The support of the composi... |
| suppcoss 8186 | The support of the composi... |
| supp0cosupp0 8187 | The support of the composi... |
| imacosupp 8188 | The image of the support o... |
| opeliunxp2f 8189 | Membership in a union of C... |
| mpoxeldm 8190 | If there is an element of ... |
| mpoxneldm 8191 | If the first argument of a... |
| mpoxopn0yelv 8192 | If there is an element of ... |
| mpoxopynvov0g 8193 | If the second argument of ... |
| mpoxopxnop0 8194 | If the first argument of a... |
| mpoxopx0ov0 8195 | If the first argument of a... |
| mpoxopxprcov0 8196 | If the components of the f... |
| mpoxopynvov0 8197 | If the second argument of ... |
| mpoxopoveq 8198 | Value of an operation give... |
| mpoxopovel 8199 | Element of the value of an... |
| mpoxopoveqd 8200 | Value of an operation give... |
| brovex 8201 | A binary relation of the v... |
| brovmpoex 8202 | A binary relation of the v... |
| sprmpod 8203 | The extension of a binary ... |
| tposss 8206 | Subset theorem for transpo... |
| tposeq 8207 | Equality theorem for trans... |
| tposeqd 8208 | Equality theorem for trans... |
| tposssxp 8209 | The transposition is a sub... |
| reltpos 8210 | The transposition is a rel... |
| brtpos2 8211 | Value of the transposition... |
| brtpos0 8212 | The behavior of ` tpos ` w... |
| reldmtpos 8213 | Necessary and sufficient c... |
| brtpos 8214 | The transposition swaps ar... |
| ottpos 8215 | The transposition swaps th... |
| relbrtpos 8216 | The transposition swaps ar... |
| dmtpos 8217 | The domain of ` tpos F ` w... |
| rntpos 8218 | The range of ` tpos F ` wh... |
| tposexg 8219 | The transposition of a set... |
| ovtpos 8220 | The transposition swaps th... |
| tposfun 8221 | The transposition of a fun... |
| dftpos2 8222 | Alternate definition of ` ... |
| dftpos3 8223 | Alternate definition of ` ... |
| dftpos4 8224 | Alternate definition of ` ... |
| tpostpos 8225 | Value of the double transp... |
| tpostpos2 8226 | Value of the double transp... |
| tposfn2 8227 | The domain of a transposit... |
| tposfo2 8228 | Condition for a surjective... |
| tposf2 8229 | The domain and codomain of... |
| tposf12 8230 | Condition for an injective... |
| tposf1o2 8231 | Condition of a bijective t... |
| tposfo 8232 | The domain and codomain/ra... |
| tposf 8233 | The domain and codomain of... |
| tposfn 8234 | Functionality of a transpo... |
| tpos0 8235 | Transposition of the empty... |
| tposco 8236 | Transposition of a composi... |
| tpossym 8237 | Two ways to say a function... |
| tposeqi 8238 | Equality theorem for trans... |
| tposex 8239 | A transposition is a set. ... |
| nftpos 8240 | Hypothesis builder for tra... |
| tposoprab 8241 | Transposition of a class o... |
| tposmpo 8242 | Transposition of a two-arg... |
| tposconst 8243 | The transposition of a con... |
| mpocurryd 8248 | The currying of an operati... |
| mpocurryvald 8249 | The value of a curried ope... |
| fvmpocurryd 8250 | The value of the value of ... |
| pwuninel2 8253 | Proof of ~ pwuninel under ... |
| pwuninel 8254 | The powerclass of the unio... |
| undefval 8255 | Value of the undefined val... |
| undefnel2 8256 | The undefined value genera... |
| undefnel 8257 | The undefined value genera... |
| undefne0 8258 | The undefined value genera... |
| frecseq123 8261 | Equality theorem for the w... |
| nffrecs 8262 | Bound-variable hypothesis ... |
| csbfrecsg 8263 | Move class substitution in... |
| fpr3g 8264 | Functions defined by well-... |
| frrlem1 8265 | Lemma for well-founded rec... |
| frrlem2 8266 | Lemma for well-founded rec... |
| frrlem3 8267 | Lemma for well-founded rec... |
| frrlem4 8268 | Lemma for well-founded rec... |
| frrlem5 8269 | Lemma for well-founded rec... |
| frrlem6 8270 | Lemma for well-founded rec... |
| frrlem7 8271 | Lemma for well-founded rec... |
| frrlem8 8272 | Lemma for well-founded rec... |
| frrlem9 8273 | Lemma for well-founded rec... |
| frrlem10 8274 | Lemma for well-founded rec... |
| frrlem11 8275 | Lemma for well-founded rec... |
| frrlem12 8276 | Lemma for well-founded rec... |
| frrlem13 8277 | Lemma for well-founded rec... |
| frrlem14 8278 | Lemma for well-founded rec... |
| fprlem1 8279 | Lemma for well-founded rec... |
| fprlem2 8280 | Lemma for well-founded rec... |
| fpr2a 8281 | Weak version of ~ fpr2 whi... |
| fpr1 8282 | Law of well-founded recurs... |
| fpr2 8283 | Law of well-founded recurs... |
| fpr3 8284 | Law of well-founded recurs... |
| frrrel 8285 | Show without using the axi... |
| frrdmss 8286 | Show without using the axi... |
| frrdmcl 8287 | Show without using the axi... |
| fprfung 8288 | A "function" defined by we... |
| fprresex 8289 | The restriction of a funct... |
| wrecseq123 8292 | General equality theorem f... |
| nfwrecs 8293 | Bound-variable hypothesis ... |
| wrecseq1 8294 | Equality theorem for the w... |
| wrecseq2 8295 | Equality theorem for the w... |
| wrecseq3 8296 | Equality theorem for the w... |
| csbwrecsg 8297 | Move class substitution in... |
| wfr3g 8298 | Functions defined by well-... |
| wfrrel 8299 | The well-ordered recursion... |
| wfrdmss 8300 | The domain of the well-ord... |
| wfrdmcl 8301 | The predecessor class of a... |
| wfrfun 8302 | The "function" generated b... |
| wfrresex 8303 | Show without using the axi... |
| wfr2a 8304 | A weak version of ~ wfr2 w... |
| wfr1 8305 | The Principle of Well-Orde... |
| wfr2 8306 | The Principle of Well-Orde... |
| wfr3 8307 | The principle of Well-Orde... |
| iunon 8308 | The indexed union of a set... |
| iinon 8309 | The nonempty indexed inter... |
| onfununi 8310 | A property of functions on... |
| onovuni 8311 | A variant of ~ onfununi fo... |
| onoviun 8312 | A variant of ~ onovuni wit... |
| onnseq 8313 | There are no length ` _om ... |
| dfsmo2 8316 | Alternate definition of a ... |
| issmo 8317 | Conditions for which ` A `... |
| issmo2 8318 | Alternate definition of a ... |
| smoeq 8319 | Equality theorem for stric... |
| smodm 8320 | The domain of a strictly m... |
| smores 8321 | A strictly monotone functi... |
| smores3 8322 | A strictly monotone functi... |
| smores2 8323 | A strictly monotone ordina... |
| smodm2 8324 | The domain of a strictly m... |
| smofvon2 8325 | The function values of a s... |
| iordsmo 8326 | The identity relation rest... |
| smo0 8327 | The null set is a strictly... |
| smofvon 8328 | If ` B ` is a strictly mon... |
| smoel 8329 | If ` x ` is less than ` y ... |
| smoiun 8330 | The value of a strictly mo... |
| smoiso 8331 | If ` F ` is an isomorphism... |
| smoel2 8332 | A strictly monotone ordina... |
| smo11 8333 | A strictly monotone ordina... |
| smoord 8334 | A strictly monotone ordina... |
| smoword 8335 | A strictly monotone ordina... |
| smogt 8336 | A strictly monotone ordina... |
| smocdmdom 8337 | The codomain of a strictly... |
| smoiso2 8338 | The strictly monotone ordi... |
| dfrecs3 8341 | The old definition of tran... |
| recseq 8342 | Equality theorem for ` rec... |
| nfrecs 8343 | Bound-variable hypothesis ... |
| tfrlem1 8344 | A technical lemma for tran... |
| tfrlem3a 8345 | Lemma for transfinite recu... |
| tfrlem3 8346 | Lemma for transfinite recu... |
| tfrlem4 8347 | Lemma for transfinite recu... |
| tfrlem5 8348 | Lemma for transfinite recu... |
| recsfval 8349 | Lemma for transfinite recu... |
| tfrlem6 8350 | Lemma for transfinite recu... |
| tfrlem7 8351 | Lemma for transfinite recu... |
| tfrlem8 8352 | Lemma for transfinite recu... |
| tfrlem9 8353 | Lemma for transfinite recu... |
| tfrlem9a 8354 | Lemma for transfinite recu... |
| tfrlem10 8355 | Lemma for transfinite recu... |
| tfrlem11 8356 | Lemma for transfinite recu... |
| tfrlem12 8357 | Lemma for transfinite recu... |
| tfrlem13 8358 | Lemma for transfinite recu... |
| tfrlem14 8359 | Lemma for transfinite recu... |
| tfrlem15 8360 | Lemma for transfinite recu... |
| tfrlem16 8361 | Lemma for finite recursion... |
| tfr1a 8362 | A weak version of ~ tfr1 w... |
| tfr2a 8363 | A weak version of ~ tfr2 w... |
| tfr2b 8364 | Without assuming ~ ax-rep ... |
| tfr1 8365 | Principle of Transfinite R... |
| tfr2 8366 | Principle of Transfinite R... |
| tfr3 8367 | Principle of Transfinite R... |
| tfr1ALT 8368 | Alternate proof of ~ tfr1 ... |
| tfr2ALT 8369 | Alternate proof of ~ tfr2 ... |
| tfr3ALT 8370 | Alternate proof of ~ tfr3 ... |
| recsfnon 8371 | Strong transfinite recursi... |
| recsval 8372 | Strong transfinite recursi... |
| tz7.44lem1 8373 | The ordered pair abstracti... |
| tz7.44-1 8374 | The value of ` F ` at ` (/... |
| tz7.44-2 8375 | The value of ` F ` at a su... |
| tz7.44-3 8376 | The value of ` F ` at a li... |
| rdgeq1 8379 | Equality theorem for the r... |
| rdgeq2 8380 | Equality theorem for the r... |
| rdgeq12 8381 | Equality theorem for the r... |
| nfrdg 8382 | Bound-variable hypothesis ... |
| rdglem1 8383 | Lemma used with the recurs... |
| rdgfun 8384 | The recursive definition g... |
| rdgdmlim 8385 | The domain of the recursiv... |
| rdgfnon 8386 | The recursive definition g... |
| rdgvalg 8387 | Value of the recursive def... |
| rdgval 8388 | Value of the recursive def... |
| rdg0 8389 | The initial value of the r... |
| rdgseg 8390 | The initial segments of th... |
| rdgsucg 8391 | The value of the recursive... |
| rdgsuc 8392 | The value of the recursive... |
| rdglimg 8393 | The value of the recursive... |
| rdglim 8394 | The value of the recursive... |
| rdg0g 8395 | The initial value of the r... |
| rdgsucmptf 8396 | The value of the recursive... |
| rdgsucmptnf 8397 | The value of the recursive... |
| rdgsucmpt2 8398 | This version of ~ rdgsucmp... |
| rdgsucmpt 8399 | The value of the recursive... |
| rdglim2 8400 | The value of the recursive... |
| rdglim2a 8401 | The value of the recursive... |
| rdg0n 8402 | If ` A ` is a proper class... |
| frfnom 8403 | The function generated by ... |
| fr0g 8404 | The initial value resultin... |
| frsuc 8405 | The successor value result... |
| frsucmpt 8406 | The successor value result... |
| frsucmptn 8407 | The value of the finite re... |
| frsucmpt2 8408 | The successor value result... |
| tz7.48lem 8409 | A way of showing an ordina... |
| tz7.48-2 8410 | Proposition 7.48(2) of [Ta... |
| tz7.48-1 8411 | Proposition 7.48(1) of [Ta... |
| tz7.48-3 8412 | Proposition 7.48(3) of [Ta... |
| tz7.49 8413 | Proposition 7.49 of [Takeu... |
| tz7.49c 8414 | Corollary of Proposition 7... |
| seqomlem0 8417 | Lemma for ` seqom ` . Cha... |
| seqomlem1 8418 | Lemma for ` seqom ` . The... |
| seqomlem2 8419 | Lemma for ` seqom ` . (Co... |
| seqomlem3 8420 | Lemma for ` seqom ` . (Co... |
| seqomlem4 8421 | Lemma for ` seqom ` . (Co... |
| seqomeq12 8422 | Equality theorem for ` seq... |
| fnseqom 8423 | An index-aware recursive d... |
| seqom0g 8424 | Value of an index-aware re... |
| seqomsuc 8425 | Value of an index-aware re... |
| omsucelsucb 8426 | Membership is inherited by... |
| df1o2 8441 | Expanded value of the ordi... |
| df2o3 8442 | Expanded value of the ordi... |
| df2o2 8443 | Expanded value of the ordi... |
| 1oex 8444 | Ordinal 1 is a set. (Cont... |
| 2oex 8445 | ` 2o ` is a set. (Contrib... |
| 1on 8446 | Ordinal 1 is an ordinal nu... |
| 2on 8447 | Ordinal 2 is an ordinal nu... |
| 2on0 8448 | Ordinal two is not zero. ... |
| ord3 8449 | Ordinal 3 is an ordinal cl... |
| 3on 8450 | Ordinal 3 is an ordinal nu... |
| 4on 8451 | Ordinal 4 is an ordinal nu... |
| 1n0 8452 | Ordinal one is not equal t... |
| nlim1 8453 | 1 is not a limit ordinal. ... |
| nlim2 8454 | 2 is not a limit ordinal. ... |
| xp01disj 8455 | Cartesian products with th... |
| xp01disjl 8456 | Cartesian products with th... |
| ordgt0ge1 8457 | Two ways to express that a... |
| ordge1n0 8458 | An ordinal greater than or... |
| el1o 8459 | Membership in ordinal one.... |
| ord1eln01 8460 | An ordinal that is not 0 o... |
| ord2eln012 8461 | An ordinal that is not 0, ... |
| 1ellim 8462 | A limit ordinal contains 1... |
| 2ellim 8463 | A limit ordinal contains 2... |
| dif1o 8464 | Two ways to say that ` A `... |
| ondif1 8465 | Two ways to say that ` A `... |
| ondif2 8466 | Two ways to say that ` A `... |
| 2oconcl 8467 | Closure of the pair swappi... |
| 0lt1o 8468 | Ordinal zero is less than ... |
| dif20el 8469 | An ordinal greater than on... |
| 0we1 8470 | The empty set is a well-or... |
| brwitnlem 8471 | Lemma for relations which ... |
| fnoa 8472 | Functionality and domain o... |
| fnom 8473 | Functionality and domain o... |
| fnoe 8474 | Functionality and domain o... |
| oav 8475 | Value of ordinal addition.... |
| omv 8476 | Value of ordinal multiplic... |
| oe0lem 8477 | A helper lemma for ~ oe0 a... |
| oev 8478 | Value of ordinal exponenti... |
| oevn0 8479 | Value of ordinal exponenti... |
| oa0 8480 | Addition with zero. Propo... |
| om0 8481 | Ordinal multiplication wit... |
| oe0m 8482 | Value of zero raised to an... |
| om0x 8483 | Ordinal multiplication wit... |
| oe0m0 8484 | Ordinal exponentiation wit... |
| oe0m1 8485 | Ordinal exponentiation wit... |
| oe0 8486 | Ordinal exponentiation wit... |
| oev2 8487 | Alternate value of ordinal... |
| oasuc 8488 | Addition with successor. ... |
| oesuclem 8489 | Lemma for ~ oesuc . (Cont... |
| omsuc 8490 | Multiplication with succes... |
| oesuc 8491 | Ordinal exponentiation wit... |
| onasuc 8492 | Addition with successor. ... |
| onmsuc 8493 | Multiplication with succes... |
| onesuc 8494 | Exponentiation with a succ... |
| oa1suc 8495 | Addition with 1 is same as... |
| oalim 8496 | Ordinal addition with a li... |
| omlim 8497 | Ordinal multiplication wit... |
| oelim 8498 | Ordinal exponentiation wit... |
| oacl 8499 | Closure law for ordinal ad... |
| omcl 8500 | Closure law for ordinal mu... |
| oecl 8501 | Closure law for ordinal ex... |
| oa0r 8502 | Ordinal addition with zero... |
| om0r 8503 | Ordinal multiplication wit... |
| o1p1e2 8504 | 1 + 1 = 2 for ordinal numb... |
| o2p2e4 8505 | 2 + 2 = 4 for ordinal numb... |
| om1 8506 | Ordinal multiplication wit... |
| om1r 8507 | Ordinal multiplication wit... |
| oe1 8508 | Ordinal exponentiation wit... |
| oe1m 8509 | Ordinal exponentiation wit... |
| oaordi 8510 | Ordering property of ordin... |
| oaord 8511 | Ordering property of ordin... |
| oacan 8512 | Left cancellation law for ... |
| oaword 8513 | Weak ordering property of ... |
| oawordri 8514 | Weak ordering property of ... |
| oaord1 8515 | An ordinal is less than it... |
| oaword1 8516 | An ordinal is less than or... |
| oaword2 8517 | An ordinal is less than or... |
| oawordeulem 8518 | Lemma for ~ oawordex . (C... |
| oawordeu 8519 | Existence theorem for weak... |
| oawordexr 8520 | Existence theorem for weak... |
| oawordex 8521 | Existence theorem for weak... |
| oaordex 8522 | Existence theorem for orde... |
| oa00 8523 | An ordinal sum is zero iff... |
| oalimcl 8524 | The ordinal sum with a lim... |
| oaass 8525 | Ordinal addition is associ... |
| oarec 8526 | Recursive definition of or... |
| oaf1o 8527 | Left addition by a constan... |
| oacomf1olem 8528 | Lemma for ~ oacomf1o . (C... |
| oacomf1o 8529 | Define a bijection from ` ... |
| omordi 8530 | Ordering property of ordin... |
| omord2 8531 | Ordering property of ordin... |
| omord 8532 | Ordering property of ordin... |
| omcan 8533 | Left cancellation law for ... |
| omword 8534 | Weak ordering property of ... |
| omwordi 8535 | Weak ordering property of ... |
| omwordri 8536 | Weak ordering property of ... |
| omword1 8537 | An ordinal is less than or... |
| omword2 8538 | An ordinal is less than or... |
| om00 8539 | The product of two ordinal... |
| om00el 8540 | The product of two nonzero... |
| omordlim 8541 | Ordering involving the pro... |
| omlimcl 8542 | The product of any nonzero... |
| odi 8543 | Distributive law for ordin... |
| omass 8544 | Multiplication of ordinal ... |
| oneo 8545 | If an ordinal number is ev... |
| omeulem1 8546 | Lemma for ~ omeu : existen... |
| omeulem2 8547 | Lemma for ~ omeu : uniquen... |
| omopth2 8548 | An ordered pair-like theor... |
| omeu 8549 | The division algorithm for... |
| oen0 8550 | Ordinal exponentiation wit... |
| oeordi 8551 | Ordering law for ordinal e... |
| oeord 8552 | Ordering property of ordin... |
| oecan 8553 | Left cancellation law for ... |
| oeword 8554 | Weak ordering property of ... |
| oewordi 8555 | Weak ordering property of ... |
| oewordri 8556 | Weak ordering property of ... |
| oeworde 8557 | Ordinal exponentiation com... |
| oeordsuc 8558 | Ordering property of ordin... |
| oelim2 8559 | Ordinal exponentiation wit... |
| oeoalem 8560 | Lemma for ~ oeoa . (Contr... |
| oeoa 8561 | Sum of exponents law for o... |
| oeoelem 8562 | Lemma for ~ oeoe . (Contr... |
| oeoe 8563 | Product of exponents law f... |
| oelimcl 8564 | The ordinal exponential wi... |
| oeeulem 8565 | Lemma for ~ oeeu . (Contr... |
| oeeui 8566 | The division algorithm for... |
| oeeu 8567 | The division algorithm for... |
| nna0 8568 | Addition with zero. Theor... |
| nnm0 8569 | Multiplication with zero. ... |
| nnasuc 8570 | Addition with successor. ... |
| nnmsuc 8571 | Multiplication with succes... |
| nnesuc 8572 | Exponentiation with a succ... |
| nna0r 8573 | Addition to zero. Remark ... |
| nnm0r 8574 | Multiplication with zero. ... |
| nnacl 8575 | Closure of addition of nat... |
| nnmcl 8576 | Closure of multiplication ... |
| nnecl 8577 | Closure of exponentiation ... |
| nnacli 8578 | ` _om ` is closed under ad... |
| nnmcli 8579 | ` _om ` is closed under mu... |
| nnarcl 8580 | Reverse closure law for ad... |
| nnacom 8581 | Addition of natural number... |
| nnaordi 8582 | Ordering property of addit... |
| nnaord 8583 | Ordering property of addit... |
| nnaordr 8584 | Ordering property of addit... |
| nnawordi 8585 | Adding to both sides of an... |
| nnaass 8586 | Addition of natural number... |
| nndi 8587 | Distributive law for natur... |
| nnmass 8588 | Multiplication of natural ... |
| nnmsucr 8589 | Multiplication with succes... |
| nnmcom 8590 | Multiplication of natural ... |
| nnaword 8591 | Weak ordering property of ... |
| nnacan 8592 | Cancellation law for addit... |
| nnaword1 8593 | Weak ordering property of ... |
| nnaword2 8594 | Weak ordering property of ... |
| nnmordi 8595 | Ordering property of multi... |
| nnmord 8596 | Ordering property of multi... |
| nnmword 8597 | Weak ordering property of ... |
| nnmcan 8598 | Cancellation law for multi... |
| nnmwordi 8599 | Weak ordering property of ... |
| nnmwordri 8600 | Weak ordering property of ... |
| nnawordex 8601 | Equivalence for weak order... |
| nnaordex 8602 | Equivalence for ordering. ... |
| nnaordex2 8603 | Equivalence for ordering. ... |
| 1onn 8604 | The ordinal 1 is a natural... |
| 1onnALT 8605 | Shorter proof of ~ 1onn us... |
| 2onn 8606 | The ordinal 2 is a natural... |
| 2onnALT 8607 | Shorter proof of ~ 2onn us... |
| 3onn 8608 | The ordinal 3 is a natural... |
| 4onn 8609 | The ordinal 4 is a natural... |
| 1one2o 8610 | Ordinal one is not ordinal... |
| oaabslem 8611 | Lemma for ~ oaabs . (Cont... |
| oaabs 8612 | Ordinal addition absorbs a... |
| oaabs2 8613 | The absorption law ~ oaabs... |
| omabslem 8614 | Lemma for ~ omabs . (Cont... |
| omabs 8615 | Ordinal multiplication is ... |
| nnm1 8616 | Multiply an element of ` _... |
| nnm2 8617 | Multiply an element of ` _... |
| nn2m 8618 | Multiply an element of ` _... |
| nnneo 8619 | If a natural number is eve... |
| nneob 8620 | A natural number is even i... |
| omsmolem 8621 | Lemma for ~ omsmo . (Cont... |
| omsmo 8622 | A strictly monotonic ordin... |
| omopthlem1 8623 | Lemma for ~ omopthi . (Co... |
| omopthlem2 8624 | Lemma for ~ omopthi . (Co... |
| omopthi 8625 | An ordered pair theorem fo... |
| omopth 8626 | An ordered pair theorem fo... |
| nnasmo 8627 | There is at most one left ... |
| eldifsucnn 8628 | Condition for membership i... |
| on2recsfn 8631 | Show that double recursion... |
| on2recsov 8632 | Calculate the value of the... |
| on2ind 8633 | Double induction over ordi... |
| on3ind 8634 | Triple induction over ordi... |
| coflton 8635 | Cofinality theorem for ord... |
| cofon1 8636 | Cofinality theorem for ord... |
| cofon2 8637 | Cofinality theorem for ord... |
| cofonr 8638 | Inverse cofinality law for... |
| naddfn 8639 | Natural addition is a func... |
| naddcllem 8640 | Lemma for ordinal addition... |
| naddcl 8641 | Closure law for natural ad... |
| naddov 8642 | The value of natural addit... |
| naddov2 8643 | Alternate expression for n... |
| naddov3 8644 | Alternate expression for n... |
| naddf 8645 | Function statement for nat... |
| naddcom 8646 | Natural addition commutes.... |
| naddrid 8647 | Ordinal zero is the additi... |
| naddlid 8648 | Ordinal zero is the additi... |
| naddssim 8649 | Ordinal less-than-or-equal... |
| naddelim 8650 | Ordinal less-than is prese... |
| naddel1 8651 | Ordinal less-than is not a... |
| naddel2 8652 | Ordinal less-than is not a... |
| naddss1 8653 | Ordinal less-than-or-equal... |
| naddss2 8654 | Ordinal less-than-or-equal... |
| naddword1 8655 | Weak-ordering principle fo... |
| naddword2 8656 | Weak-ordering principle fo... |
| naddunif 8657 | Uniformity theorem for nat... |
| naddasslem1 8658 | Lemma for ~ naddass . Exp... |
| naddasslem2 8659 | Lemma for ~ naddass . Exp... |
| naddass 8660 | Natural ordinal addition i... |
| nadd32 8661 | Commutative/associative la... |
| nadd4 8662 | Rearragement of terms in a... |
| nadd42 8663 | Rearragement of terms in a... |
| naddel12 8664 | Natural addition to both s... |
| naddsuc2 8665 | Natural addition with succ... |
| naddoa 8666 | Natural addition of a natu... |
| omnaddcl 8667 | The naturals are closed un... |
| dfer2 8672 | Alternate definition of eq... |
| dfec2 8674 | Alternate definition of ` ... |
| ecexg 8675 | An equivalence class modul... |
| ecexr 8676 | A nonempty equivalence cla... |
| ereq1 8678 | Equality theorem for equiv... |
| ereq2 8679 | Equality theorem for equiv... |
| errel 8680 | An equivalence relation is... |
| erdm 8681 | The domain of an equivalen... |
| ercl 8682 | Elementhood in the field o... |
| ersym 8683 | An equivalence relation is... |
| ercl2 8684 | Elementhood in the field o... |
| ersymb 8685 | An equivalence relation is... |
| ertr 8686 | An equivalence relation is... |
| ertrd 8687 | A transitivity relation fo... |
| ertr2d 8688 | A transitivity relation fo... |
| ertr3d 8689 | A transitivity relation fo... |
| ertr4d 8690 | A transitivity relation fo... |
| erref 8691 | An equivalence relation is... |
| ercnv 8692 | The converse of an equival... |
| errn 8693 | The range and domain of an... |
| erssxp 8694 | An equivalence relation is... |
| erex 8695 | An equivalence relation is... |
| erexb 8696 | An equivalence relation is... |
| iserd 8697 | A reflexive, symmetric, tr... |
| iseri 8698 | A reflexive, symmetric, tr... |
| iseriALT 8699 | Alternate proof of ~ iseri... |
| brinxper 8700 | Conditions for a reflexive... |
| brdifun 8701 | Evaluate the incomparabili... |
| swoer 8702 | Incomparability under a st... |
| swoord1 8703 | The incomparability equiva... |
| swoord2 8704 | The incomparability equiva... |
| swoso 8705 | If the incomparability rel... |
| eqerlem 8706 | Lemma for ~ eqer . (Contr... |
| eqer 8707 | Equivalence relation invol... |
| ider 8708 | The identity relation is a... |
| 0er 8709 | The empty set is an equiva... |
| eceq1 8710 | Equality theorem for equiv... |
| eceq1d 8711 | Equality theorem for equiv... |
| eceq2 8712 | Equality theorem for equiv... |
| eceq2i 8713 | Equality theorem for the `... |
| eceq2d 8714 | Equality theorem for the `... |
| elecg 8715 | Membership in an equivalen... |
| ecref 8716 | All elements are in their ... |
| elec 8717 | Membership in an equivalen... |
| relelec 8718 | Membership in an equivalen... |
| elecres 8719 | Elementhood in the restric... |
| elecreseq 8720 | The restricted coset of ` ... |
| elecex 8721 | Condition for a coset to b... |
| ecss 8722 | An equivalence class is a ... |
| ecdmn0 8723 | A representative of a none... |
| ereldm 8724 | Equality of equivalence cl... |
| erth 8725 | Basic property of equivale... |
| erth2 8726 | Basic property of equivale... |
| erthi 8727 | Basic property of equivale... |
| erdisj 8728 | Equivalence classes do not... |
| ecidsn 8729 | An equivalence class modul... |
| qseq1 8730 | Equality theorem for quoti... |
| qseq2 8731 | Equality theorem for quoti... |
| qseq2i 8732 | Equality theorem for quoti... |
| qseq1d 8733 | Equality theorem for quoti... |
| qseq2d 8734 | Equality theorem for quoti... |
| qseq12 8735 | Equality theorem for quoti... |
| 0qs 8736 | Quotient set with the empt... |
| elqsg 8737 | Closed form of ~ elqs . (... |
| elqs 8738 | Membership in a quotient s... |
| elqsi 8739 | Membership in a quotient s... |
| elqsecl 8740 | Membership in a quotient s... |
| ecelqs 8741 | Membership of an equivalen... |
| ecelqsw 8742 | Membership of an equivalen... |
| ecelqsi 8743 | Membership of an equivalen... |
| ecopqsi 8744 | "Closure" law for equivale... |
| qsexg 8745 | A quotient set exists. (C... |
| qsex 8746 | A quotient set exists. (C... |
| uniqs 8747 | The union of a quotient se... |
| uniqsw 8748 | The union of a quotient se... |
| qsss 8749 | A quotient set is a set of... |
| uniqs2 8750 | The union of a quotient se... |
| snec 8751 | The singleton of an equiva... |
| ecqs 8752 | Equivalence class in terms... |
| ecid 8753 | A set is equal to its cose... |
| qsid 8754 | A set is equal to its quot... |
| ectocld 8755 | Implicit substitution of c... |
| ectocl 8756 | Implicit substitution of c... |
| elqsn0 8757 | A quotient set does not co... |
| ecelqsdm 8758 | Membership of an equivalen... |
| ecelqsdmb 8759 | ` R ` -coset of ` B ` in a... |
| eceldmqs 8760 | ` R ` -coset in its domain... |
| xpider 8761 | A Cartesian square is an e... |
| iiner 8762 | The intersection of a none... |
| riiner 8763 | The relative intersection ... |
| erinxp 8764 | A restricted equivalence r... |
| ecinxp 8765 | Restrict the relation in a... |
| qsinxp 8766 | Restrict the equivalence r... |
| qsdisj 8767 | Members of a quotient set ... |
| qsdisj2 8768 | A quotient set is a disjoi... |
| qsel 8769 | If an element of a quotien... |
| uniinqs 8770 | Class union distributes ov... |
| qliftlem 8771 | Lemma for theorems about a... |
| qliftrel 8772 | ` F ` , a function lift, i... |
| qliftel 8773 | Elementhood in the relatio... |
| qliftel1 8774 | Elementhood in the relatio... |
| qliftfun 8775 | The function ` F ` is the ... |
| qliftfund 8776 | The function ` F ` is the ... |
| qliftfuns 8777 | The function ` F ` is the ... |
| qliftf 8778 | The domain and codomain of... |
| qliftval 8779 | The value of the function ... |
| ecoptocl 8780 | Implicit substitution of c... |
| 2ecoptocl 8781 | Implicit substitution of c... |
| 3ecoptocl 8782 | Implicit substitution of c... |
| brecop 8783 | Binary relation on a quoti... |
| brecop2 8784 | Binary relation on a quoti... |
| eroveu 8785 | Lemma for ~ erov and ~ ero... |
| erovlem 8786 | Lemma for ~ erov and ~ ero... |
| erov 8787 | The value of an operation ... |
| eroprf 8788 | Functionality of an operat... |
| erov2 8789 | The value of an operation ... |
| eroprf2 8790 | Functionality of an operat... |
| ecopoveq 8791 | This is the first of sever... |
| ecopovsym 8792 | Assuming the operation ` F... |
| ecopovtrn 8793 | Assuming that operation ` ... |
| ecopover 8794 | Assuming that operation ` ... |
| eceqoveq 8795 | Equality of equivalence re... |
| ecovcom 8796 | Lemma used to transfer a c... |
| ecovass 8797 | Lemma used to transfer an ... |
| ecovdi 8798 | Lemma used to transfer a d... |
| mapprc 8803 | When ` A ` is a proper cla... |
| pmex 8804 | The class of all partial f... |
| mapexOLD 8805 | Obsolete version of ~ mape... |
| fnmap 8806 | Set exponentiation has a u... |
| fnpm 8807 | Partial function exponenti... |
| reldmmap 8808 | Set exponentiation is a we... |
| mapvalg 8809 | The value of set exponenti... |
| pmvalg 8810 | The value of the partial m... |
| mapval 8811 | The value of set exponenti... |
| elmapg 8812 | Membership relation for se... |
| elmapd 8813 | Deduction form of ~ elmapg... |
| elmapdd 8814 | Deduction associated with ... |
| mapdm0 8815 | The empty set is the only ... |
| elpmg 8816 | The predicate "is a partia... |
| elpm2g 8817 | The predicate "is a partia... |
| elpm2r 8818 | Sufficient condition for b... |
| elpmi 8819 | A partial function is a fu... |
| pmfun 8820 | A partial function is a fu... |
| elmapex 8821 | Eliminate antecedent for m... |
| elmapi 8822 | A mapping is a function, f... |
| mapfset 8823 | If ` B ` is a set, the val... |
| mapssfset 8824 | The value of the set expon... |
| mapfoss 8825 | The value of the set expon... |
| fsetsspwxp 8826 | The class of all functions... |
| fset0 8827 | The set of functions from ... |
| fsetdmprc0 8828 | The set of functions with ... |
| fsetex 8829 | The set of functions betwe... |
| f1setex 8830 | The set of injections betw... |
| fosetex 8831 | The set of surjections bet... |
| f1osetex 8832 | The set of bijections betw... |
| fsetfcdm 8833 | The class of functions wit... |
| fsetfocdm 8834 | The class of functions wit... |
| fsetprcnex 8835 | The class of all functions... |
| fsetcdmex 8836 | The class of all functions... |
| fsetexb 8837 | The class of all functions... |
| elmapfn 8838 | A mapping is a function wi... |
| elmapfun 8839 | A mapping is always a func... |
| elmapssres 8840 | A restricted mapping is a ... |
| fpmg 8841 | A total function is a part... |
| pmss12g 8842 | Subset relation for the se... |
| pmresg 8843 | Elementhood of a restricte... |
| elmap 8844 | Membership relation for se... |
| mapval2 8845 | Alternate expression for t... |
| elpm 8846 | The predicate "is a partia... |
| elpm2 8847 | The predicate "is a partia... |
| fpm 8848 | A total function is a part... |
| mapsspm 8849 | Set exponentiation is a su... |
| pmsspw 8850 | Partial maps are a subset ... |
| mapsspw 8851 | Set exponentiation is a su... |
| mapfvd 8852 | The value of a function th... |
| elmapresaun 8853 | ~ fresaun transposed to ma... |
| fvmptmap 8854 | Special case of ~ fvmpt fo... |
| map0e 8855 | Set exponentiation with an... |
| map0b 8856 | Set exponentiation with an... |
| map0g 8857 | Set exponentiation is empt... |
| 0map0sn0 8858 | The set of mappings of the... |
| mapsnd 8859 | The value of set exponenti... |
| map0 8860 | Set exponentiation is empt... |
| mapsn 8861 | The value of set exponenti... |
| mapss 8862 | Subset inheritance for set... |
| fdiagfn 8863 | Functionality of the diago... |
| fvdiagfn 8864 | Functionality of the diago... |
| mapsnconst 8865 | Every singleton map is a c... |
| mapsncnv 8866 | Expression for the inverse... |
| mapsnf1o2 8867 | Explicit bijection between... |
| mapsnf1o3 8868 | Explicit bijection in the ... |
| ralxpmap 8869 | Quantification over functi... |
| dfixp 8872 | Eliminate the expression `... |
| ixpsnval 8873 | The value of an infinite C... |
| elixp2 8874 | Membership in an infinite ... |
| fvixp 8875 | Projection of a factor of ... |
| ixpfn 8876 | A nuple is a function. (C... |
| elixp 8877 | Membership in an infinite ... |
| elixpconst 8878 | Membership in an infinite ... |
| ixpconstg 8879 | Infinite Cartesian product... |
| ixpconst 8880 | Infinite Cartesian product... |
| ixpeq1 8881 | Equality theorem for infin... |
| ixpeq1d 8882 | Equality theorem for infin... |
| ss2ixp 8883 | Subclass theorem for infin... |
| ixpeq2 8884 | Equality theorem for infin... |
| ixpeq2dva 8885 | Equality theorem for infin... |
| ixpeq2dv 8886 | Equality theorem for infin... |
| cbvixp 8887 | Change bound variable in a... |
| cbvixpv 8888 | Change bound variable in a... |
| nfixpw 8889 | Bound-variable hypothesis ... |
| nfixp 8890 | Bound-variable hypothesis ... |
| nfixp1 8891 | The index variable in an i... |
| ixpprc 8892 | A cartesian product of pro... |
| ixpf 8893 | A member of an infinite Ca... |
| uniixp 8894 | The union of an infinite C... |
| ixpexg 8895 | The existence of an infini... |
| ixpin 8896 | The intersection of two in... |
| ixpiin 8897 | The indexed intersection o... |
| ixpint 8898 | The intersection of a coll... |
| ixp0x 8899 | An infinite Cartesian prod... |
| ixpssmap2g 8900 | An infinite Cartesian prod... |
| ixpssmapg 8901 | An infinite Cartesian prod... |
| 0elixp 8902 | Membership of the empty se... |
| ixpn0 8903 | The infinite Cartesian pro... |
| ixp0 8904 | The infinite Cartesian pro... |
| ixpssmap 8905 | An infinite Cartesian prod... |
| resixp 8906 | Restriction of an element ... |
| undifixp 8907 | Union of two projections o... |
| mptelixpg 8908 | Condition for an explicit ... |
| resixpfo 8909 | Restriction of elements of... |
| elixpsn 8910 | Membership in a class of s... |
| ixpsnf1o 8911 | A bijection between a clas... |
| mapsnf1o 8912 | A bijection between a set ... |
| boxriin 8913 | A rectangular subset of a ... |
| boxcutc 8914 | The relative complement of... |
| relen 8923 | Equinumerosity is a relati... |
| reldom 8924 | Dominance is a relation. ... |
| relsdom 8925 | Strict dominance is a rela... |
| encv 8926 | If two classes are equinum... |
| breng 8927 | Equinumerosity relation. ... |
| bren 8928 | Equinumerosity relation. ... |
| brdom2g 8929 | Dominance relation. This ... |
| brdomg 8930 | Dominance relation. (Cont... |
| brdomi 8931 | Dominance relation. (Cont... |
| brdom 8932 | Dominance relation. (Cont... |
| domen 8933 | Dominance in terms of equi... |
| domeng 8934 | Dominance in terms of equi... |
| ctex 8935 | A countable set is a set. ... |
| f1oen4g 8936 | The domain and range of a ... |
| f1dom4g 8937 | The domain of a one-to-one... |
| f1oen3g 8938 | The domain and range of a ... |
| f1dom3g 8939 | The domain of a one-to-one... |
| f1oen2g 8940 | The domain and range of a ... |
| f1dom2g 8941 | The domain of a one-to-one... |
| f1oeng 8942 | The domain and range of a ... |
| f1domg 8943 | The domain of a one-to-one... |
| f1oen 8944 | The domain and range of a ... |
| f1dom 8945 | The domain of a one-to-one... |
| brsdom 8946 | Strict dominance relation,... |
| isfi 8947 | Express " ` A ` is finite"... |
| enssdom 8948 | Equinumerosity implies dom... |
| dfdom2 8949 | Alternate definition of do... |
| endom 8950 | Equinumerosity implies dom... |
| sdomdom 8951 | Strict dominance implies d... |
| sdomnen 8952 | Strict dominance implies n... |
| brdom2 8953 | Dominance in terms of stri... |
| bren2 8954 | Equinumerosity expressed i... |
| enrefg 8955 | Equinumerosity is reflexiv... |
| enref 8956 | Equinumerosity is reflexiv... |
| eqeng 8957 | Equality implies equinumer... |
| domrefg 8958 | Dominance is reflexive. (... |
| en2d 8959 | Equinumerosity inference f... |
| en3d 8960 | Equinumerosity inference f... |
| en2i 8961 | Equinumerosity inference f... |
| en3i 8962 | Equinumerosity inference f... |
| dom2lem 8963 | A mapping (first hypothesi... |
| dom2d 8964 | A mapping (first hypothesi... |
| dom3d 8965 | A mapping (first hypothesi... |
| dom2 8966 | A mapping (first hypothesi... |
| dom3 8967 | A mapping (first hypothesi... |
| idssen 8968 | Equality implies equinumer... |
| domssl 8969 | If ` A ` is a subset of ` ... |
| domssr 8970 | If ` C ` is a superset of ... |
| ssdomg 8971 | A set dominates its subset... |
| ener 8972 | Equinumerosity is an equiv... |
| ensymb 8973 | Symmetry of equinumerosity... |
| ensym 8974 | Symmetry of equinumerosity... |
| ensymi 8975 | Symmetry of equinumerosity... |
| ensymd 8976 | Symmetry of equinumerosity... |
| entr 8977 | Transitivity of equinumero... |
| domtr 8978 | Transitivity of dominance ... |
| entri 8979 | A chained equinumerosity i... |
| entr2i 8980 | A chained equinumerosity i... |
| entr3i 8981 | A chained equinumerosity i... |
| entr4i 8982 | A chained equinumerosity i... |
| endomtr 8983 | Transitivity of equinumero... |
| domentr 8984 | Transitivity of dominance ... |
| f1imaeng 8985 | If a function is one-to-on... |
| f1imaen2g 8986 | If a function is one-to-on... |
| f1imaen3g 8987 | If a set function is one-t... |
| f1imaen 8988 | If a function is one-to-on... |
| en0 8989 | The empty set is equinumer... |
| en0ALT 8990 | Shorter proof of ~ en0 , d... |
| en0r 8991 | The empty set is equinumer... |
| ensn1 8992 | A singleton is equinumerou... |
| ensn1g 8993 | A singleton is equinumerou... |
| enpr1g 8994 | ` { A , A } ` has only one... |
| en1 8995 | A set is equinumerous to o... |
| en1b 8996 | A set is equinumerous to o... |
| reuen1 8997 | Two ways to express "exact... |
| euen1 8998 | Two ways to express "exact... |
| euen1b 8999 | Two ways to express " ` A ... |
| en1uniel 9000 | A singleton contains its s... |
| 2dom 9001 | A set that dominates ordin... |
| fundmen 9002 | A function is equinumerous... |
| fundmeng 9003 | A function is equinumerous... |
| cnven 9004 | A relational set is equinu... |
| cnvct 9005 | If a set is countable, so ... |
| fndmeng 9006 | A function is equinumerate... |
| mapsnend 9007 | Set exponentiation to a si... |
| mapsnen 9008 | Set exponentiation to a si... |
| snmapen 9009 | Set exponentiation: a sing... |
| snmapen1 9010 | Set exponentiation: a sing... |
| map1 9011 | Set exponentiation: ordina... |
| en2sn 9012 | Two singletons are equinum... |
| 0fi 9013 | The empty set is finite. ... |
| snfi 9014 | A singleton is finite. (C... |
| snfiOLD 9015 | Obsolete version of ~ snfi... |
| fiprc 9016 | The class of finite sets i... |
| unen 9017 | Equinumerosity of union of... |
| enrefnn 9018 | Equinumerosity is reflexiv... |
| en2prd 9019 | Two unordered pairs are eq... |
| enpr2d 9020 | A pair with distinct eleme... |
| enpr2dOLD 9021 | Obsolete version of ~ enpr... |
| ssct 9022 | Any subset of a countable ... |
| difsnen 9023 | All decrements of a set ar... |
| domdifsn 9024 | Dominance over a set with ... |
| xpsnen 9025 | A set is equinumerous to i... |
| xpsneng 9026 | A set is equinumerous to i... |
| xp1en 9027 | One times a cardinal numbe... |
| endisj 9028 | Any two sets are equinumer... |
| undom 9029 | Dominance law for union. ... |
| xpcomf1o 9030 | The canonical bijection fr... |
| xpcomco 9031 | Composition with the bijec... |
| xpcomen 9032 | Commutative law for equinu... |
| xpcomeng 9033 | Commutative law for equinu... |
| xpsnen2g 9034 | A set is equinumerous to i... |
| xpassen 9035 | Associative law for equinu... |
| xpdom2 9036 | Dominance law for Cartesia... |
| xpdom2g 9037 | Dominance law for Cartesia... |
| xpdom1g 9038 | Dominance law for Cartesia... |
| xpdom3 9039 | A set is dominated by its ... |
| xpdom1 9040 | Dominance law for Cartesia... |
| domunsncan 9041 | A singleton cancellation l... |
| omxpenlem 9042 | Lemma for ~ omxpen . (Con... |
| omxpen 9043 | The cardinal and ordinal p... |
| omf1o 9044 | Construct an explicit bije... |
| pw2f1olem 9045 | Lemma for ~ pw2f1o . (Con... |
| pw2f1o 9046 | The power set of a set is ... |
| pw2eng 9047 | The power set of a set is ... |
| pw2en 9048 | The power set of a set is ... |
| fopwdom 9049 | Covering implies injection... |
| enfixsn 9050 | Given two equipollent sets... |
| sbthlem1 9051 | Lemma for ~ sbth . (Contr... |
| sbthlem2 9052 | Lemma for ~ sbth . (Contr... |
| sbthlem3 9053 | Lemma for ~ sbth . (Contr... |
| sbthlem4 9054 | Lemma for ~ sbth . (Contr... |
| sbthlem5 9055 | Lemma for ~ sbth . (Contr... |
| sbthlem6 9056 | Lemma for ~ sbth . (Contr... |
| sbthlem7 9057 | Lemma for ~ sbth . (Contr... |
| sbthlem8 9058 | Lemma for ~ sbth . (Contr... |
| sbthlem9 9059 | Lemma for ~ sbth . (Contr... |
| sbthlem10 9060 | Lemma for ~ sbth . (Contr... |
| sbth 9061 | Schroeder-Bernstein Theore... |
| sbthb 9062 | Schroeder-Bernstein Theore... |
| sbthcl 9063 | Schroeder-Bernstein Theore... |
| dfsdom2 9064 | Alternate definition of st... |
| brsdom2 9065 | Alternate definition of st... |
| sdomnsym 9066 | Strict dominance is asymme... |
| domnsym 9067 | Theorem 22(i) of [Suppes] ... |
| 0domg 9068 | Any set dominates the empt... |
| dom0 9069 | A set dominated by the emp... |
| 0sdomg 9070 | A set strictly dominates t... |
| 0dom 9071 | Any set dominates the empt... |
| 0sdom 9072 | A set strictly dominates t... |
| sdom0 9073 | The empty set does not str... |
| sdomdomtr 9074 | Transitivity of strict dom... |
| sdomentr 9075 | Transitivity of strict dom... |
| domsdomtr 9076 | Transitivity of dominance ... |
| ensdomtr 9077 | Transitivity of equinumero... |
| sdomirr 9078 | Strict dominance is irrefl... |
| sdomtr 9079 | Strict dominance is transi... |
| sdomn2lp 9080 | Strict dominance has no 2-... |
| enen1 9081 | Equality-like theorem for ... |
| enen2 9082 | Equality-like theorem for ... |
| domen1 9083 | Equality-like theorem for ... |
| domen2 9084 | Equality-like theorem for ... |
| sdomen1 9085 | Equality-like theorem for ... |
| sdomen2 9086 | Equality-like theorem for ... |
| domtriord 9087 | Dominance is trichotomous ... |
| sdomel 9088 | For ordinals, strict domin... |
| sdomdif 9089 | The difference of a set fr... |
| onsdominel 9090 | An ordinal with more eleme... |
| domunsn 9091 | Dominance over a set with ... |
| fodomr 9092 | There exists a mapping fro... |
| pwdom 9093 | Injection of sets implies ... |
| canth2 9094 | Cantor's Theorem. No set ... |
| canth2g 9095 | Cantor's theorem with the ... |
| 2pwuninel 9096 | The power set of the power... |
| 2pwne 9097 | No set equals the power se... |
| disjen 9098 | A stronger form of ~ pwuni... |
| disjenex 9099 | Existence version of ~ dis... |
| domss2 9100 | A corollary of ~ disjenex ... |
| domssex2 9101 | A corollary of ~ disjenex ... |
| domssex 9102 | Weakening of ~ domssex2 to... |
| xpf1o 9103 | Construct a bijection on a... |
| xpen 9104 | Equinumerosity law for Car... |
| mapen 9105 | Two set exponentiations ar... |
| mapdom1 9106 | Order-preserving property ... |
| mapxpen 9107 | Equinumerosity law for dou... |
| xpmapenlem 9108 | Lemma for ~ xpmapen . (Co... |
| xpmapen 9109 | Equinumerosity law for set... |
| mapunen 9110 | Equinumerosity law for set... |
| map2xp 9111 | A cardinal power with expo... |
| mapdom2 9112 | Order-preserving property ... |
| mapdom3 9113 | Set exponentiation dominat... |
| pwen 9114 | If two sets are equinumero... |
| ssenen 9115 | Equinumerosity of equinume... |
| limenpsi 9116 | A limit ordinal is equinum... |
| limensuci 9117 | A limit ordinal is equinum... |
| limensuc 9118 | A limit ordinal is equinum... |
| infensuc 9119 | Any infinite ordinal is eq... |
| dif1enlem 9120 | Lemma for ~ rexdif1en and ... |
| dif1enlemOLD 9121 | Obsolete version of ~ dif1... |
| rexdif1en 9122 | If a set is equinumerous t... |
| rexdif1enOLD 9123 | Obsolete version of ~ rexd... |
| dif1en 9124 | If a set ` A ` is equinume... |
| dif1ennn 9125 | If a set ` A ` is equinume... |
| dif1enOLD 9126 | Obsolete version of ~ dif1... |
| findcard 9127 | Schema for induction on th... |
| findcard2 9128 | Schema for induction on th... |
| findcard2s 9129 | Variation of ~ findcard2 r... |
| findcard2d 9130 | Deduction version of ~ fin... |
| nnfi 9131 | Natural numbers are finite... |
| pssnn 9132 | A proper subset of a natur... |
| ssnnfi 9133 | A subset of a natural numb... |
| 0finOLD 9134 | Obsolete version of ~ 0fi ... |
| unfi 9135 | The union of two finite se... |
| unfid 9136 | The union of two finite se... |
| ssfi 9137 | A subset of a finite set i... |
| ssfiALT 9138 | Shorter proof of ~ ssfi us... |
| diffi 9139 | If ` A ` is finite, ` ( A ... |
| cnvfi 9140 | If a set is finite, its co... |
| pwssfi 9141 | Every element of the power... |
| fnfi 9142 | A version of ~ fnex for fi... |
| f1oenfi 9143 | If the domain of a one-to-... |
| f1oenfirn 9144 | If the range of a one-to-o... |
| f1domfi 9145 | If the codomain of a one-t... |
| f1domfi2 9146 | If the domain of a one-to-... |
| enreffi 9147 | Equinumerosity is reflexiv... |
| ensymfib 9148 | Symmetry of equinumerosity... |
| entrfil 9149 | Transitivity of equinumero... |
| enfii 9150 | A set equinumerous to a fi... |
| enfi 9151 | Equinumerous sets have the... |
| enfiALT 9152 | Shorter proof of ~ enfi us... |
| domfi 9153 | A set dominated by a finit... |
| entrfi 9154 | Transitivity of equinumero... |
| entrfir 9155 | Transitivity of equinumero... |
| domtrfil 9156 | Transitivity of dominance ... |
| domtrfi 9157 | Transitivity of dominance ... |
| domtrfir 9158 | Transitivity of dominance ... |
| f1imaenfi 9159 | If a function is one-to-on... |
| ssdomfi 9160 | A finite set dominates its... |
| ssdomfi2 9161 | A set dominates its finite... |
| sbthfilem 9162 | Lemma for ~ sbthfi . (Con... |
| sbthfi 9163 | Schroeder-Bernstein Theore... |
| domnsymfi 9164 | If a set dominates a finit... |
| sdomdomtrfi 9165 | Transitivity of strict dom... |
| domsdomtrfi 9166 | Transitivity of dominance ... |
| sucdom2 9167 | Strict dominance of a set ... |
| phplem1 9168 | Lemma for Pigeonhole Princ... |
| phplem2 9169 | Lemma for Pigeonhole Princ... |
| nneneq 9170 | Two equinumerous natural n... |
| php 9171 | Pigeonhole Principle. A n... |
| php2 9172 | Corollary of Pigeonhole Pr... |
| php3 9173 | Corollary of Pigeonhole Pr... |
| php4 9174 | Corollary of the Pigeonhol... |
| php5 9175 | Corollary of the Pigeonhol... |
| phpeqd 9176 | Corollary of the Pigeonhol... |
| nndomog 9177 | Cardinal ordering agrees w... |
| onomeneq 9178 | An ordinal number equinume... |
| onfin 9179 | An ordinal number is finit... |
| onfin2 9180 | A set is a natural number ... |
| nndomo 9181 | Cardinal ordering agrees w... |
| nnsdomo 9182 | Cardinal ordering agrees w... |
| sucdom 9183 | Strict dominance of a set ... |
| snnen2o 9184 | A singleton ` { A } ` is n... |
| 0sdom1dom 9185 | Strict dominance over 0 is... |
| 0sdom1domALT 9186 | Alternate proof of ~ 0sdom... |
| 1sdom2 9187 | Ordinal 1 is strictly domi... |
| 1sdom2ALT 9188 | Alternate proof of ~ 1sdom... |
| sdom1 9189 | A set has less than one me... |
| sdom1OLD 9190 | Obsolete version of ~ sdom... |
| modom 9191 | Two ways to express "at mo... |
| modom2 9192 | Two ways to express "at mo... |
| rex2dom 9193 | A set that has at least 2 ... |
| 1sdom2dom 9194 | Strict dominance over 1 is... |
| 1sdom 9195 | A set that strictly domina... |
| 1sdomOLD 9196 | Obsolete version of ~ 1sdo... |
| unxpdomlem1 9197 | Lemma for ~ unxpdom . (Tr... |
| unxpdomlem2 9198 | Lemma for ~ unxpdom . (Co... |
| unxpdomlem3 9199 | Lemma for ~ unxpdom . (Co... |
| unxpdom 9200 | Cartesian product dominate... |
| unxpdom2 9201 | Corollary of ~ unxpdom . ... |
| sucxpdom 9202 | Cartesian product dominate... |
| pssinf 9203 | A set equinumerous to a pr... |
| fisseneq 9204 | A finite set is equal to i... |
| ominf 9205 | The set of natural numbers... |
| ominfOLD 9206 | Obsolete version of ~ omin... |
| isinf 9207 | Any set that is not finite... |
| isinfOLD 9208 | Obsolete version of ~ isin... |
| fineqvlem 9209 | Lemma for ~ fineqv . (Con... |
| fineqv 9210 | If the Axiom of Infinity i... |
| xpfir 9211 | The components of a nonemp... |
| ssfid 9212 | A subset of a finite set i... |
| infi 9213 | The intersection of two se... |
| rabfi 9214 | A restricted class built f... |
| finresfin 9215 | The restriction of a finit... |
| f1finf1o 9216 | Any injection from one fin... |
| f1finf1oOLD 9217 | Obsolete version of ~ f1fi... |
| nfielex 9218 | If a class is not finite, ... |
| en1eqsn 9219 | A set with one element is ... |
| en1eqsnOLD 9220 | Obsolete version of ~ en1e... |
| en1eqsnbi 9221 | A set containing an elemen... |
| dif1ennnALT 9222 | Alternate proof of ~ dif1e... |
| enp1ilem 9223 | Lemma for uses of ~ enp1i ... |
| enp1i 9224 | Proof induction for ~ en2 ... |
| enp1iOLD 9225 | Obsolete version of ~ enp1... |
| en2 9226 | A set equinumerous to ordi... |
| en3 9227 | A set equinumerous to ordi... |
| en4 9228 | A set equinumerous to ordi... |
| findcard3 9229 | Schema for strong inductio... |
| findcard3OLD 9230 | Obsolete version of ~ find... |
| ac6sfi 9231 | A version of ~ ac6s for fi... |
| frfi 9232 | A partial order is well-fo... |
| fimax2g 9233 | A finite set has a maximum... |
| fimaxg 9234 | A finite set has a maximum... |
| fisupg 9235 | Lemma showing existence an... |
| wofi 9236 | A total order on a finite ... |
| ordunifi 9237 | The maximum of a finite co... |
| nnunifi 9238 | The union (supremum) of a ... |
| unblem1 9239 | Lemma for ~ unbnn . After... |
| unblem2 9240 | Lemma for ~ unbnn . The v... |
| unblem3 9241 | Lemma for ~ unbnn . The v... |
| unblem4 9242 | Lemma for ~ unbnn . The f... |
| unbnn 9243 | Any unbounded subset of na... |
| unbnn2 9244 | Version of ~ unbnn that do... |
| isfinite2 9245 | Any set strictly dominated... |
| nnsdomg 9246 | Omega strictly dominates a... |
| nnsdomgOLD 9247 | Obsolete version of ~ nnsd... |
| isfiniteg 9248 | A set is finite iff it is ... |
| infsdomnn 9249 | An infinite set strictly d... |
| infsdomnnOLD 9250 | Obsolete version of ~ infs... |
| infn0 9251 | An infinite set is not emp... |
| infn0ALT 9252 | Shorter proof of ~ infn0 u... |
| fin2inf 9253 | This (useless) theorem, wh... |
| unfilem1 9254 | Lemma for proving that the... |
| unfilem2 9255 | Lemma for proving that the... |
| unfilem3 9256 | Lemma for proving that the... |
| unfir 9257 | If a union is finite, the ... |
| unfib 9258 | A union is finite if and o... |
| unfi2 9259 | The union of two finite se... |
| difinf 9260 | An infinite set ` A ` minu... |
| fodomfi 9261 | An onto function implies d... |
| fofi 9262 | If an onto function has a ... |
| f1fi 9263 | If a 1-to-1 function has a... |
| imafi 9264 | Images of finite sets are ... |
| imafiOLD 9265 | Obsolete version of ~ imaf... |
| pwfir 9266 | If the power set of a set ... |
| pwfilem 9267 | Lemma for ~ pwfi . (Contr... |
| pwfi 9268 | The power set of a finite ... |
| xpfi 9269 | The Cartesian product of t... |
| xpfiOLD 9270 | Obsolete version of ~ xpfi... |
| 3xpfi 9271 | The Cartesian product of t... |
| domunfican 9272 | A finite set union cancell... |
| infcntss 9273 | Every infinite set has a d... |
| prfi 9274 | An unordered pair is finit... |
| prfiALT 9275 | Shorter proof of ~ prfi us... |
| tpfi 9276 | An unordered triple is fin... |
| fiint 9277 | Equivalent ways of stating... |
| fiintOLD 9278 | Obsolete version of ~ fiin... |
| fodomfir 9279 | There exists a mapping fro... |
| fodomfib 9280 | Equivalence of an onto map... |
| fodomfiOLD 9281 | Obsolete version of ~ fodo... |
| fodomfibOLD 9282 | Obsolete version of ~ fodo... |
| fofinf1o 9283 | Any surjection from one fi... |
| rneqdmfinf1o 9284 | Any function from a finite... |
| fidomdm 9285 | Any finite set dominates i... |
| dmfi 9286 | The domain of a finite set... |
| fundmfibi 9287 | A function is finite if an... |
| resfnfinfin 9288 | The restriction of a funct... |
| residfi 9289 | A restricted identity func... |
| cnvfiALT 9290 | Shorter proof of ~ cnvfi u... |
| rnfi 9291 | The range of a finite set ... |
| f1dmvrnfibi 9292 | A one-to-one function whos... |
| f1vrnfibi 9293 | A one-to-one function whic... |
| iunfi 9294 | The finite union of finite... |
| unifi 9295 | The finite union of finite... |
| unifi2 9296 | The finite union of finite... |
| infssuni 9297 | If an infinite set ` A ` i... |
| unirnffid 9298 | The union of the range of ... |
| mapfi 9299 | Set exponentiation of fini... |
| ixpfi 9300 | A Cartesian product of fin... |
| ixpfi2 9301 | A Cartesian product of fin... |
| mptfi 9302 | A finite mapping set is fi... |
| abrexfi 9303 | An image set from a finite... |
| cnvimamptfin 9304 | A preimage of a mapping wi... |
| elfpw 9305 | Membership in a class of f... |
| unifpw 9306 | A set is the union of its ... |
| f1opwfi 9307 | A one-to-one mapping induc... |
| fissuni 9308 | A finite subset of a union... |
| fipreima 9309 | Given a finite subset ` A ... |
| finsschain 9310 | A finite subset of the uni... |
| indexfi 9311 | If for every element of a ... |
| relfsupp 9314 | The property of a function... |
| relprcnfsupp 9315 | A proper class is never fi... |
| isfsupp 9316 | The property of a class to... |
| isfsuppd 9317 | Deduction form of ~ isfsup... |
| funisfsupp 9318 | The property of a function... |
| fsuppimp 9319 | Implications of a class be... |
| fsuppimpd 9320 | A finitely supported funct... |
| fsuppfund 9321 | A finitely supported funct... |
| fisuppfi 9322 | A function on a finite set... |
| fidmfisupp 9323 | A function with a finite d... |
| finnzfsuppd 9324 | If a function is zero outs... |
| fdmfisuppfi 9325 | The support of a function ... |
| fdmfifsupp 9326 | A function with a finite d... |
| fsuppmptdm 9327 | A mapping with a finite do... |
| fndmfisuppfi 9328 | The support of a function ... |
| fndmfifsupp 9329 | A function with a finite d... |
| suppeqfsuppbi 9330 | If two functions have the ... |
| suppssfifsupp 9331 | If the support of a functi... |
| fsuppsssupp 9332 | If the support of a functi... |
| fsuppsssuppgd 9333 | If the support of a functi... |
| fsuppss 9334 | A subset of a finitely sup... |
| fsuppssov1 9335 | Formula building theorem f... |
| fsuppxpfi 9336 | The cartesian product of t... |
| fczfsuppd 9337 | A constant function with v... |
| fsuppun 9338 | The union of two finitely ... |
| fsuppunfi 9339 | The union of the support o... |
| fsuppunbi 9340 | If the union of two classe... |
| 0fsupp 9341 | The empty set is a finitel... |
| snopfsupp 9342 | A singleton containing an ... |
| funsnfsupp 9343 | Finite support for a funct... |
| fsuppres 9344 | The restriction of a finit... |
| fmptssfisupp 9345 | The restriction of a mappi... |
| ressuppfi 9346 | If the support of the rest... |
| resfsupp 9347 | If the restriction of a fu... |
| resfifsupp 9348 | The restriction of a funct... |
| ffsuppbi 9349 | Two ways of saying that a ... |
| fsuppmptif 9350 | A function mapping an argu... |
| sniffsupp 9351 | A function mapping all but... |
| fsuppcolem 9352 | Lemma for ~ fsuppco . For... |
| fsuppco 9353 | The composition of a 1-1 f... |
| fsuppco2 9354 | The composition of a funct... |
| fsuppcor 9355 | The composition of a funct... |
| mapfienlem1 9356 | Lemma 1 for ~ mapfien . (... |
| mapfienlem2 9357 | Lemma 2 for ~ mapfien . (... |
| mapfienlem3 9358 | Lemma 3 for ~ mapfien . (... |
| mapfien 9359 | A bijection of the base se... |
| mapfien2 9360 | Equinumerousity relation f... |
| fival 9363 | The set of all the finite ... |
| elfi 9364 | Specific properties of an ... |
| elfi2 9365 | The empty intersection nee... |
| elfir 9366 | Sufficient condition for a... |
| intrnfi 9367 | Sufficient condition for t... |
| iinfi 9368 | An indexed intersection of... |
| inelfi 9369 | The intersection of two se... |
| ssfii 9370 | Any element of a set ` A `... |
| fi0 9371 | The set of finite intersec... |
| fieq0 9372 | A set is empty iff the cla... |
| fiin 9373 | The elements of ` ( fi `` ... |
| dffi2 9374 | The set of finite intersec... |
| fiss 9375 | Subset relationship for fu... |
| inficl 9376 | A set which is closed unde... |
| fipwuni 9377 | The set of finite intersec... |
| fisn 9378 | A singleton is closed unde... |
| fiuni 9379 | The union of the finite in... |
| fipwss 9380 | If a set is a family of su... |
| elfiun 9381 | A finite intersection of e... |
| dffi3 9382 | The set of finite intersec... |
| fifo 9383 | Describe a surjection from... |
| marypha1lem 9384 | Core induction for Philip ... |
| marypha1 9385 | (Philip) Hall's marriage t... |
| marypha2lem1 9386 | Lemma for ~ marypha2 . Pr... |
| marypha2lem2 9387 | Lemma for ~ marypha2 . Pr... |
| marypha2lem3 9388 | Lemma for ~ marypha2 . Pr... |
| marypha2lem4 9389 | Lemma for ~ marypha2 . Pr... |
| marypha2 9390 | Version of ~ marypha1 usin... |
| dfsup2 9395 | Quantifier-free definition... |
| supeq1 9396 | Equality theorem for supre... |
| supeq1d 9397 | Equality deduction for sup... |
| supeq1i 9398 | Equality inference for sup... |
| supeq2 9399 | Equality theorem for supre... |
| supeq3 9400 | Equality theorem for supre... |
| supeq123d 9401 | Equality deduction for sup... |
| nfsup 9402 | Hypothesis builder for sup... |
| supmo 9403 | Any class ` B ` has at mos... |
| supexd 9404 | A supremum is a set. (Con... |
| supeu 9405 | A supremum is unique. Sim... |
| supval2 9406 | Alternate expression for t... |
| eqsup 9407 | Sufficient condition for a... |
| eqsupd 9408 | Sufficient condition for a... |
| supcl 9409 | A supremum belongs to its ... |
| supub 9410 | A supremum is an upper bou... |
| suplub 9411 | A supremum is the least up... |
| suplub2 9412 | Bidirectional form of ~ su... |
| supnub 9413 | An upper bound is not less... |
| supssd 9414 | Inequality deduction for s... |
| supex 9415 | A supremum is a set. (Con... |
| sup00 9416 | The supremum under an empt... |
| sup0riota 9417 | The supremum of an empty s... |
| sup0 9418 | The supremum of an empty s... |
| supmax 9419 | The greatest element of a ... |
| fisup2g 9420 | A finite set satisfies the... |
| fisupcl 9421 | A nonempty finite set cont... |
| supgtoreq 9422 | The supremum of a finite s... |
| suppr 9423 | The supremum of a pair. (... |
| supsn 9424 | The supremum of a singleto... |
| supisolem 9425 | Lemma for ~ supiso . (Con... |
| supisoex 9426 | Lemma for ~ supiso . (Con... |
| supiso 9427 | Image of a supremum under ... |
| infeq1 9428 | Equality theorem for infim... |
| infeq1d 9429 | Equality deduction for inf... |
| infeq1i 9430 | Equality inference for inf... |
| infeq2 9431 | Equality theorem for infim... |
| infeq3 9432 | Equality theorem for infim... |
| infeq123d 9433 | Equality deduction for inf... |
| nfinf 9434 | Hypothesis builder for inf... |
| infexd 9435 | An infimum is a set. (Con... |
| eqinf 9436 | Sufficient condition for a... |
| eqinfd 9437 | Sufficient condition for a... |
| infval 9438 | Alternate expression for t... |
| infcllem 9439 | Lemma for ~ infcl , ~ infl... |
| infcl 9440 | An infimum belongs to its ... |
| inflb 9441 | An infimum is a lower boun... |
| infglb 9442 | An infimum is the greatest... |
| infglbb 9443 | Bidirectional form of ~ in... |
| infnlb 9444 | A lower bound is not great... |
| infssd 9445 | Inequality deduction for i... |
| infex 9446 | An infimum is a set. (Con... |
| infmin 9447 | The smallest element of a ... |
| infmo 9448 | Any class ` B ` has at mos... |
| infeu 9449 | An infimum is unique. (Co... |
| fimin2g 9450 | A finite set has a minimum... |
| fiming 9451 | A finite set has a minimum... |
| fiinfg 9452 | Lemma showing existence an... |
| fiinf2g 9453 | A finite set satisfies the... |
| fiinfcl 9454 | A nonempty finite set cont... |
| infltoreq 9455 | The infimum of a finite se... |
| infpr 9456 | The infimum of a pair. (C... |
| infsupprpr 9457 | The infimum of a proper pa... |
| infsn 9458 | The infimum of a singleton... |
| inf00 9459 | The infimum regarding an e... |
| infempty 9460 | The infimum of an empty se... |
| infiso 9461 | Image of an infimum under ... |
| dfoi 9464 | Rewrite ~ df-oi with abbre... |
| oieq1 9465 | Equality theorem for ordin... |
| oieq2 9466 | Equality theorem for ordin... |
| nfoi 9467 | Hypothesis builder for ord... |
| ordiso2 9468 | Generalize ~ ordiso to pro... |
| ordiso 9469 | Order-isomorphic ordinal n... |
| ordtypecbv 9470 | Lemma for ~ ordtype . (Co... |
| ordtypelem1 9471 | Lemma for ~ ordtype . (Co... |
| ordtypelem2 9472 | Lemma for ~ ordtype . (Co... |
| ordtypelem3 9473 | Lemma for ~ ordtype . (Co... |
| ordtypelem4 9474 | Lemma for ~ ordtype . (Co... |
| ordtypelem5 9475 | Lemma for ~ ordtype . (Co... |
| ordtypelem6 9476 | Lemma for ~ ordtype . (Co... |
| ordtypelem7 9477 | Lemma for ~ ordtype . ` ra... |
| ordtypelem8 9478 | Lemma for ~ ordtype . (Co... |
| ordtypelem9 9479 | Lemma for ~ ordtype . Eit... |
| ordtypelem10 9480 | Lemma for ~ ordtype . Usi... |
| oi0 9481 | Definition of the ordinal ... |
| oicl 9482 | The order type of the well... |
| oif 9483 | The order isomorphism of t... |
| oiiso2 9484 | The order isomorphism of t... |
| ordtype 9485 | For any set-like well-orde... |
| oiiniseg 9486 | ` ran F ` is an initial se... |
| ordtype2 9487 | For any set-like well-orde... |
| oiexg 9488 | The order isomorphism on a... |
| oion 9489 | The order type of the well... |
| oiiso 9490 | The order isomorphism of t... |
| oien 9491 | The order type of a well-o... |
| oieu 9492 | Uniqueness of the unique o... |
| oismo 9493 | When ` A ` is a subclass o... |
| oiid 9494 | The order type of an ordin... |
| hartogslem1 9495 | Lemma for ~ hartogs . (Co... |
| hartogslem2 9496 | Lemma for ~ hartogs . (Co... |
| hartogs 9497 | The class of ordinals domi... |
| wofib 9498 | The only sets which are we... |
| wemaplem1 9499 | Value of the lexicographic... |
| wemaplem2 9500 | Lemma for ~ wemapso . Tra... |
| wemaplem3 9501 | Lemma for ~ wemapso . Tra... |
| wemappo 9502 | Construct lexicographic or... |
| wemapsolem 9503 | Lemma for ~ wemapso . (Co... |
| wemapso 9504 | Construct lexicographic or... |
| wemapso2lem 9505 | Lemma for ~ wemapso2 . (C... |
| wemapso2 9506 | An alternative to having a... |
| card2on 9507 | The alternate definition o... |
| card2inf 9508 | The alternate definition o... |
| harf 9511 | Functionality of the Harto... |
| harcl 9512 | Values of the Hartogs func... |
| harval 9513 | Function value of the Hart... |
| elharval 9514 | The Hartogs number of a se... |
| harndom 9515 | The Hartogs number of a se... |
| harword 9516 | Weak ordering property of ... |
| relwdom 9519 | Weak dominance is a relati... |
| brwdom 9520 | Property of weak dominance... |
| brwdomi 9521 | Property of weak dominance... |
| brwdomn0 9522 | Weak dominance over nonemp... |
| 0wdom 9523 | Any set weakly dominates t... |
| fowdom 9524 | An onto function implies w... |
| wdomref 9525 | Reflexivity of weak domina... |
| brwdom2 9526 | Alternate characterization... |
| domwdom 9527 | Weak dominance is implied ... |
| wdomtr 9528 | Transitivity of weak domin... |
| wdomen1 9529 | Equality-like theorem for ... |
| wdomen2 9530 | Equality-like theorem for ... |
| wdompwdom 9531 | Weak dominance strengthens... |
| canthwdom 9532 | Cantor's Theorem, stated u... |
| wdom2d 9533 | Deduce weak dominance from... |
| wdomd 9534 | Deduce weak dominance from... |
| brwdom3 9535 | Condition for weak dominan... |
| brwdom3i 9536 | Weak dominance implies exi... |
| unwdomg 9537 | Weak dominance of a (disjo... |
| xpwdomg 9538 | Weak dominance of a Cartes... |
| wdomima2g 9539 | A set is weakly dominant o... |
| wdomimag 9540 | A set is weakly dominant o... |
| unxpwdom2 9541 | Lemma for ~ unxpwdom . (C... |
| unxpwdom 9542 | If a Cartesian product is ... |
| ixpiunwdom 9543 | Describe an onto function ... |
| harwdom 9544 | The value of the Hartogs f... |
| axreg2 9546 | Axiom of Regularity expres... |
| zfregcl 9547 | The Axiom of Regularity wi... |
| zfreg 9548 | The Axiom of Regularity us... |
| elirrv 9549 | The membership relation is... |
| elirr 9550 | No class is a member of it... |
| elneq 9551 | A class is not equal to an... |
| nelaneq 9552 | A class is not an element ... |
| epinid0 9553 | The membership relation an... |
| sucprcreg 9554 | A class is equal to its su... |
| ruv 9555 | The Russell class is equal... |
| ruALT 9556 | Alternate proof of ~ ru , ... |
| disjcsn 9557 | A class is disjoint from i... |
| zfregfr 9558 | The membership relation is... |
| en2lp 9559 | No class has 2-cycle membe... |
| elnanel 9560 | Two classes are not elemen... |
| cnvepnep 9561 | The membership (epsilon) r... |
| epnsym 9562 | The membership (epsilon) r... |
| elnotel 9563 | A class cannot be an eleme... |
| elnel 9564 | A class cannot be an eleme... |
| en3lplem1 9565 | Lemma for ~ en3lp . (Cont... |
| en3lplem2 9566 | Lemma for ~ en3lp . (Cont... |
| en3lp 9567 | No class has 3-cycle membe... |
| preleqg 9568 | Equality of two unordered ... |
| preleq 9569 | Equality of two unordered ... |
| preleqALT 9570 | Alternate proof of ~ prele... |
| opthreg 9571 | Theorem for alternate repr... |
| suc11reg 9572 | The successor operation be... |
| dford2 9573 | Assuming ~ ax-reg , an ord... |
| inf0 9574 | Existence of ` _om ` impli... |
| inf1 9575 | Variation of Axiom of Infi... |
| inf2 9576 | Variation of Axiom of Infi... |
| inf3lema 9577 | Lemma for our Axiom of Inf... |
| inf3lemb 9578 | Lemma for our Axiom of Inf... |
| inf3lemc 9579 | Lemma for our Axiom of Inf... |
| inf3lemd 9580 | Lemma for our Axiom of Inf... |
| inf3lem1 9581 | Lemma for our Axiom of Inf... |
| inf3lem2 9582 | Lemma for our Axiom of Inf... |
| inf3lem3 9583 | Lemma for our Axiom of Inf... |
| inf3lem4 9584 | Lemma for our Axiom of Inf... |
| inf3lem5 9585 | Lemma for our Axiom of Inf... |
| inf3lem6 9586 | Lemma for our Axiom of Inf... |
| inf3lem7 9587 | Lemma for our Axiom of Inf... |
| inf3 9588 | Our Axiom of Infinity ~ ax... |
| infeq5i 9589 | Half of ~ infeq5 . (Contr... |
| infeq5 9590 | The statement "there exist... |
| zfinf 9592 | Axiom of Infinity expresse... |
| axinf2 9593 | A standard version of Axio... |
| zfinf2 9595 | A standard version of the ... |
| omex 9596 | The existence of omega (th... |
| axinf 9597 | The first version of the A... |
| inf5 9598 | The statement "there exist... |
| omelon 9599 | Omega is an ordinal number... |
| dfom3 9600 | The class of natural numbe... |
| elom3 9601 | A simplification of ~ elom... |
| dfom4 9602 | A simplification of ~ df-o... |
| dfom5 9603 | ` _om ` is the smallest li... |
| oancom 9604 | Ordinal addition is not co... |
| isfinite 9605 | A set is finite iff it is ... |
| fict 9606 | A finite set is countable ... |
| nnsdom 9607 | A natural number is strict... |
| omenps 9608 | Omega is equinumerous to a... |
| omensuc 9609 | The set of natural numbers... |
| infdifsn 9610 | Removing a singleton from ... |
| infdiffi 9611 | Removing a finite set from... |
| unbnn3 9612 | Any unbounded subset of na... |
| noinfep 9613 | Using the Axiom of Regular... |
| cantnffval 9616 | The value of the Cantor no... |
| cantnfdm 9617 | The domain of the Cantor n... |
| cantnfvalf 9618 | Lemma for ~ cantnf . The ... |
| cantnfs 9619 | Elementhood in the set of ... |
| cantnfcl 9620 | Basic properties of the or... |
| cantnfval 9621 | The value of the Cantor no... |
| cantnfval2 9622 | Alternate expression for t... |
| cantnfsuc 9623 | The value of the recursive... |
| cantnfle 9624 | A lower bound on the ` CNF... |
| cantnflt 9625 | An upper bound on the part... |
| cantnflt2 9626 | An upper bound on the ` CN... |
| cantnff 9627 | The ` CNF ` function is a ... |
| cantnf0 9628 | The value of the zero func... |
| cantnfrescl 9629 | A function is finitely sup... |
| cantnfres 9630 | The ` CNF ` function respe... |
| cantnfp1lem1 9631 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem2 9632 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem3 9633 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1 9634 | If ` F ` is created by add... |
| oemapso 9635 | The relation ` T ` is a st... |
| oemapval 9636 | Value of the relation ` T ... |
| oemapvali 9637 | If ` F < G ` , then there ... |
| cantnflem1a 9638 | Lemma for ~ cantnf . (Con... |
| cantnflem1b 9639 | Lemma for ~ cantnf . (Con... |
| cantnflem1c 9640 | Lemma for ~ cantnf . (Con... |
| cantnflem1d 9641 | Lemma for ~ cantnf . (Con... |
| cantnflem1 9642 | Lemma for ~ cantnf . This... |
| cantnflem2 9643 | Lemma for ~ cantnf . (Con... |
| cantnflem3 9644 | Lemma for ~ cantnf . Here... |
| cantnflem4 9645 | Lemma for ~ cantnf . Comp... |
| cantnf 9646 | The Cantor Normal Form the... |
| oemapwe 9647 | The lexicographic order on... |
| cantnffval2 9648 | An alternate definition of... |
| cantnff1o 9649 | Simplify the isomorphism o... |
| wemapwe 9650 | Construct lexicographic or... |
| oef1o 9651 | A bijection of the base se... |
| cnfcomlem 9652 | Lemma for ~ cnfcom . (Con... |
| cnfcom 9653 | Any ordinal ` B ` is equin... |
| cnfcom2lem 9654 | Lemma for ~ cnfcom2 . (Co... |
| cnfcom2 9655 | Any nonzero ordinal ` B ` ... |
| cnfcom3lem 9656 | Lemma for ~ cnfcom3 . (Co... |
| cnfcom3 9657 | Any infinite ordinal ` B `... |
| cnfcom3clem 9658 | Lemma for ~ cnfcom3c . (C... |
| cnfcom3c 9659 | Wrap the construction of ~... |
| ttrcleq 9662 | Equality theorem for trans... |
| nfttrcld 9663 | Bound variable hypothesis ... |
| nfttrcl 9664 | Bound variable hypothesis ... |
| relttrcl 9665 | The transitive closure of ... |
| brttrcl 9666 | Characterization of elemen... |
| brttrcl2 9667 | Characterization of elemen... |
| ssttrcl 9668 | If ` R ` is a relation, th... |
| ttrcltr 9669 | The transitive closure of ... |
| ttrclresv 9670 | The transitive closure of ... |
| ttrclco 9671 | Composition law for the tr... |
| cottrcl 9672 | Composition law for the tr... |
| ttrclss 9673 | If ` R ` is a subclass of ... |
| dmttrcl 9674 | The domain of a transitive... |
| rnttrcl 9675 | The range of a transitive ... |
| ttrclexg 9676 | If ` R ` is a set, then so... |
| dfttrcl2 9677 | When ` R ` is a set and a ... |
| ttrclselem1 9678 | Lemma for ~ ttrclse . Sho... |
| ttrclselem2 9679 | Lemma for ~ ttrclse . Sho... |
| ttrclse 9680 | If ` R ` is set-like over ... |
| trcl 9681 | For any set ` A ` , show t... |
| tz9.1 9682 | Every set has a transitive... |
| tz9.1c 9683 | Alternate expression for t... |
| epfrs 9684 | The strong form of the Axi... |
| zfregs 9685 | The strong form of the Axi... |
| zfregs2 9686 | Alternate strong form of t... |
| setind 9687 | Set (epsilon) induction. ... |
| setind2 9688 | Set (epsilon) induction, s... |
| tcvalg 9691 | Value of the transitive cl... |
| tcid 9692 | Defining property of the t... |
| tctr 9693 | Defining property of the t... |
| tcmin 9694 | Defining property of the t... |
| tc2 9695 | A variant of the definitio... |
| tcsni 9696 | The transitive closure of ... |
| tcss 9697 | The transitive closure fun... |
| tcel 9698 | The transitive closure fun... |
| tcidm 9699 | The transitive closure fun... |
| tc0 9700 | The transitive closure of ... |
| tc00 9701 | The transitive closure is ... |
| frmin 9702 | Every (possibly proper) su... |
| frind 9703 | A subclass of a well-found... |
| frinsg 9704 | Well-Founded Induction Sch... |
| frins 9705 | Well-Founded Induction Sch... |
| frins2f 9706 | Well-Founded Induction sch... |
| frins2 9707 | Well-Founded Induction sch... |
| frins3 9708 | Well-Founded Induction sch... |
| frr3g 9709 | Functions defined by well-... |
| frrlem15 9710 | Lemma for general well-fou... |
| frrlem16 9711 | Lemma for general well-fou... |
| frr1 9712 | Law of general well-founde... |
| frr2 9713 | Law of general well-founde... |
| frr3 9714 | Law of general well-founde... |
| r1funlim 9719 | The cumulative hierarchy o... |
| r1fnon 9720 | The cumulative hierarchy o... |
| r10 9721 | Value of the cumulative hi... |
| r1sucg 9722 | Value of the cumulative hi... |
| r1suc 9723 | Value of the cumulative hi... |
| r1limg 9724 | Value of the cumulative hi... |
| r1lim 9725 | Value of the cumulative hi... |
| r1fin 9726 | The first ` _om ` levels o... |
| r1sdom 9727 | Each stage in the cumulati... |
| r111 9728 | The cumulative hierarchy i... |
| r1tr 9729 | The cumulative hierarchy o... |
| r1tr2 9730 | The union of a cumulative ... |
| r1ordg 9731 | Ordering relation for the ... |
| r1ord3g 9732 | Ordering relation for the ... |
| r1ord 9733 | Ordering relation for the ... |
| r1ord2 9734 | Ordering relation for the ... |
| r1ord3 9735 | Ordering relation for the ... |
| r1sssuc 9736 | The value of the cumulativ... |
| r1pwss 9737 | Each set of the cumulative... |
| r1sscl 9738 | Each set of the cumulative... |
| r1val1 9739 | The value of the cumulativ... |
| tz9.12lem1 9740 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem2 9741 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem3 9742 | Lemma for ~ tz9.12 . (Con... |
| tz9.12 9743 | A set is well-founded if a... |
| tz9.13 9744 | Every set is well-founded,... |
| tz9.13g 9745 | Every set is well-founded,... |
| rankwflemb 9746 | Two ways of saying a set i... |
| rankf 9747 | The domain and codomain of... |
| rankon 9748 | The rank of a set is an or... |
| r1elwf 9749 | Any member of the cumulati... |
| rankvalb 9750 | Value of the rank function... |
| rankr1ai 9751 | One direction of ~ rankr1a... |
| rankvaln 9752 | Value of the rank function... |
| rankidb 9753 | Identity law for the rank ... |
| rankdmr1 9754 | A rank is a member of the ... |
| rankr1ag 9755 | A version of ~ rankr1a tha... |
| rankr1bg 9756 | A relationship between ran... |
| r1rankidb 9757 | Any set is a subset of the... |
| r1elssi 9758 | The range of the ` R1 ` fu... |
| r1elss 9759 | The range of the ` R1 ` fu... |
| pwwf 9760 | A power set is well-founde... |
| sswf 9761 | A subset of a well-founded... |
| snwf 9762 | A singleton is well-founde... |
| unwf 9763 | A binary union is well-fou... |
| prwf 9764 | An unordered pair is well-... |
| opwf 9765 | An ordered pair is well-fo... |
| unir1 9766 | The cumulative hierarchy o... |
| jech9.3 9767 | Every set belongs to some ... |
| rankwflem 9768 | Every set is well-founded,... |
| rankval 9769 | Value of the rank function... |
| rankvalg 9770 | Value of the rank function... |
| rankval2 9771 | Value of an alternate defi... |
| uniwf 9772 | A union is well-founded if... |
| rankr1clem 9773 | Lemma for ~ rankr1c . (Co... |
| rankr1c 9774 | A relationship between the... |
| rankidn 9775 | A relationship between the... |
| rankpwi 9776 | The rank of a power set. ... |
| rankelb 9777 | The membership relation is... |
| wfelirr 9778 | A well-founded set is not ... |
| rankval3b 9779 | The value of the rank func... |
| ranksnb 9780 | The rank of a singleton. ... |
| rankonidlem 9781 | Lemma for ~ rankonid . (C... |
| rankonid 9782 | The rank of an ordinal num... |
| onwf 9783 | The ordinals are all well-... |
| onssr1 9784 | Initial segments of the or... |
| rankr1g 9785 | A relationship between the... |
| rankid 9786 | Identity law for the rank ... |
| rankr1 9787 | A relationship between the... |
| ssrankr1 9788 | A relationship between an ... |
| rankr1a 9789 | A relationship between ran... |
| r1val2 9790 | The value of the cumulativ... |
| r1val3 9791 | The value of the cumulativ... |
| rankel 9792 | The membership relation is... |
| rankval3 9793 | The value of the rank func... |
| bndrank 9794 | Any class whose elements h... |
| unbndrank 9795 | The elements of a proper c... |
| rankpw 9796 | The rank of a power set. ... |
| ranklim 9797 | The rank of a set belongs ... |
| r1pw 9798 | A stronger property of ` R... |
| r1pwALT 9799 | Alternate shorter proof of... |
| r1pwcl 9800 | The cumulative hierarchy o... |
| rankssb 9801 | The subset relation is inh... |
| rankss 9802 | The subset relation is inh... |
| rankunb 9803 | The rank of the union of t... |
| rankprb 9804 | The rank of an unordered p... |
| rankopb 9805 | The rank of an ordered pai... |
| rankuni2b 9806 | The value of the rank func... |
| ranksn 9807 | The rank of a singleton. ... |
| rankuni2 9808 | The rank of a union. Part... |
| rankun 9809 | The rank of the union of t... |
| rankpr 9810 | The rank of an unordered p... |
| rankop 9811 | The rank of an ordered pai... |
| r1rankid 9812 | Any set is a subset of the... |
| rankeq0b 9813 | A set is empty iff its ran... |
| rankeq0 9814 | A set is empty iff its ran... |
| rankr1id 9815 | The rank of the hierarchy ... |
| rankuni 9816 | The rank of a union. Part... |
| rankr1b 9817 | A relationship between ran... |
| ranksuc 9818 | The rank of a successor. ... |
| rankuniss 9819 | Upper bound of the rank of... |
| rankval4 9820 | The rank of a set is the s... |
| rankbnd 9821 | The rank of a set is bound... |
| rankbnd2 9822 | The rank of a set is bound... |
| rankc1 9823 | A relationship that can be... |
| rankc2 9824 | A relationship that can be... |
| rankelun 9825 | Rank membership is inherit... |
| rankelpr 9826 | Rank membership is inherit... |
| rankelop 9827 | Rank membership is inherit... |
| rankxpl 9828 | A lower bound on the rank ... |
| rankxpu 9829 | An upper bound on the rank... |
| rankfu 9830 | An upper bound on the rank... |
| rankmapu 9831 | An upper bound on the rank... |
| rankxplim 9832 | The rank of a Cartesian pr... |
| rankxplim2 9833 | If the rank of a Cartesian... |
| rankxplim3 9834 | The rank of a Cartesian pr... |
| rankxpsuc 9835 | The rank of a Cartesian pr... |
| tcwf 9836 | The transitive closure fun... |
| tcrank 9837 | This theorem expresses two... |
| scottex 9838 | Scott's trick collects all... |
| scott0 9839 | Scott's trick collects all... |
| scottexs 9840 | Theorem scheme version of ... |
| scott0s 9841 | Theorem scheme version of ... |
| cplem1 9842 | Lemma for the Collection P... |
| cplem2 9843 | Lemma for the Collection P... |
| cp 9844 | Collection Principle. Thi... |
| bnd 9845 | A very strong generalizati... |
| bnd2 9846 | A variant of the Boundedne... |
| kardex 9847 | The collection of all sets... |
| karden 9848 | If we allow the Axiom of R... |
| htalem 9849 | Lemma for defining an emul... |
| hta 9850 | A ZFC emulation of Hilbert... |
| djueq12 9857 | Equality theorem for disjo... |
| djueq1 9858 | Equality theorem for disjo... |
| djueq2 9859 | Equality theorem for disjo... |
| nfdju 9860 | Bound-variable hypothesis ... |
| djuex 9861 | The disjoint union of sets... |
| djuexb 9862 | The disjoint union of two ... |
| djulcl 9863 | Left closure of disjoint u... |
| djurcl 9864 | Right closure of disjoint ... |
| djulf1o 9865 | The left injection functio... |
| djurf1o 9866 | The right injection functi... |
| inlresf 9867 | The left injection restric... |
| inlresf1 9868 | The left injection restric... |
| inrresf 9869 | The right injection restri... |
| inrresf1 9870 | The right injection restri... |
| djuin 9871 | The images of any classes ... |
| djur 9872 | A member of a disjoint uni... |
| djuss 9873 | A disjoint union is a subc... |
| djuunxp 9874 | The union of a disjoint un... |
| djuexALT 9875 | Alternate proof of ~ djuex... |
| eldju1st 9876 | The first component of an ... |
| eldju2ndl 9877 | The second component of an... |
| eldju2ndr 9878 | The second component of an... |
| djuun 9879 | The disjoint union of two ... |
| 1stinl 9880 | The first component of the... |
| 2ndinl 9881 | The second component of th... |
| 1stinr 9882 | The first component of the... |
| 2ndinr 9883 | The second component of th... |
| updjudhf 9884 | The mapping of an element ... |
| updjudhcoinlf 9885 | The composition of the map... |
| updjudhcoinrg 9886 | The composition of the map... |
| updjud 9887 | Universal property of the ... |
| cardf2 9896 | The cardinality function i... |
| cardon 9897 | The cardinal number of a s... |
| isnum2 9898 | A way to express well-orde... |
| isnumi 9899 | A set equinumerous to an o... |
| ennum 9900 | Equinumerous sets are equi... |
| finnum 9901 | Every finite set is numera... |
| onenon 9902 | Every ordinal number is nu... |
| tskwe 9903 | A Tarski set is well-order... |
| xpnum 9904 | The cartesian product of n... |
| cardval3 9905 | An alternate definition of... |
| cardid2 9906 | Any numerable set is equin... |
| isnum3 9907 | A set is numerable iff it ... |
| oncardval 9908 | The value of the cardinal ... |
| oncardid 9909 | Any ordinal number is equi... |
| cardonle 9910 | The cardinal of an ordinal... |
| card0 9911 | The cardinality of the emp... |
| cardidm 9912 | The cardinality function i... |
| oncard 9913 | A set is a cardinal number... |
| ficardom 9914 | The cardinal number of a f... |
| ficardid 9915 | A finite set is equinumero... |
| cardnn 9916 | The cardinality of a natur... |
| cardnueq0 9917 | The empty set is the only ... |
| cardne 9918 | No member of a cardinal nu... |
| carden2a 9919 | If two sets have equal non... |
| carden2b 9920 | If two sets are equinumero... |
| card1 9921 | A set has cardinality one ... |
| cardsn 9922 | A singleton has cardinalit... |
| carddomi2 9923 | Two sets have the dominanc... |
| sdomsdomcardi 9924 | A set strictly dominates i... |
| cardlim 9925 | An infinite cardinal is a ... |
| cardsdomelir 9926 | A cardinal strictly domina... |
| cardsdomel 9927 | A cardinal strictly domina... |
| iscard 9928 | Two ways to express the pr... |
| iscard2 9929 | Two ways to express the pr... |
| carddom2 9930 | Two numerable sets have th... |
| harcard 9931 | The class of ordinal numbe... |
| cardprclem 9932 | Lemma for ~ cardprc . (Co... |
| cardprc 9933 | The class of all cardinal ... |
| carduni 9934 | The union of a set of card... |
| cardiun 9935 | The indexed union of a set... |
| cardennn 9936 | If ` A ` is equinumerous t... |
| cardsucinf 9937 | The cardinality of the suc... |
| cardsucnn 9938 | The cardinality of the suc... |
| cardom 9939 | The set of natural numbers... |
| carden2 9940 | Two numerable sets are equ... |
| cardsdom2 9941 | A numerable set is strictl... |
| domtri2 9942 | Trichotomy of dominance fo... |
| nnsdomel 9943 | Strict dominance and eleme... |
| cardval2 9944 | An alternate version of th... |
| isinffi 9945 | An infinite set contains s... |
| fidomtri 9946 | Trichotomy of dominance wi... |
| fidomtri2 9947 | Trichotomy of dominance wi... |
| harsdom 9948 | The Hartogs number of a we... |
| onsdom 9949 | Any well-orderable set is ... |
| harval2 9950 | An alternate expression fo... |
| harsucnn 9951 | The next cardinal after a ... |
| cardmin2 9952 | The smallest ordinal that ... |
| pm54.43lem 9953 | In Theorem *54.43 of [Whit... |
| pm54.43 9954 | Theorem *54.43 of [Whitehe... |
| enpr2 9955 | An unordered pair with dis... |
| pr2nelemOLD 9956 | Obsolete version of ~ enpr... |
| pr2ne 9957 | If an unordered pair has t... |
| pr2neOLD 9958 | Obsolete version of ~ pr2n... |
| prdom2 9959 | An unordered pair has at m... |
| en2eqpr 9960 | Building a set with two el... |
| en2eleq 9961 | Express a set of pair card... |
| en2other2 9962 | Taking the other element t... |
| dif1card 9963 | The cardinality of a nonem... |
| leweon 9964 | Lexicographical order is a... |
| r0weon 9965 | A set-like well-ordering o... |
| infxpenlem 9966 | Lemma for ~ infxpen . (Co... |
| infxpen 9967 | Every infinite ordinal is ... |
| xpomen 9968 | The Cartesian product of o... |
| xpct 9969 | The cartesian product of t... |
| infxpidm2 9970 | Every infinite well-ordera... |
| infxpenc 9971 | A canonical version of ~ i... |
| infxpenc2lem1 9972 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem2 9973 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem3 9974 | Lemma for ~ infxpenc2 . (... |
| infxpenc2 9975 | Existence form of ~ infxpe... |
| iunmapdisj 9976 | The union ` U_ n e. C ( A ... |
| fseqenlem1 9977 | Lemma for ~ fseqen . (Con... |
| fseqenlem2 9978 | Lemma for ~ fseqen . (Con... |
| fseqdom 9979 | One half of ~ fseqen . (C... |
| fseqen 9980 | A set that is equinumerous... |
| infpwfidom 9981 | The collection of finite s... |
| dfac8alem 9982 | Lemma for ~ dfac8a . If t... |
| dfac8a 9983 | Numeration theorem: every ... |
| dfac8b 9984 | The well-ordering theorem:... |
| dfac8clem 9985 | Lemma for ~ dfac8c . (Con... |
| dfac8c 9986 | If the union of a set is w... |
| ac10ct 9987 | A proof of the well-orderi... |
| ween 9988 | A set is numerable iff it ... |
| ac5num 9989 | A version of ~ ac5b with t... |
| ondomen 9990 | If a set is dominated by a... |
| numdom 9991 | A set dominated by a numer... |
| ssnum 9992 | A subset of a numerable se... |
| onssnum 9993 | All subsets of the ordinal... |
| indcardi 9994 | Indirect strong induction ... |
| acnrcl 9995 | Reverse closure for the ch... |
| acneq 9996 | Equality theorem for the c... |
| isacn 9997 | The property of being a ch... |
| acni 9998 | The property of being a ch... |
| acni2 9999 | The property of being a ch... |
| acni3 10000 | The property of being a ch... |
| acnlem 10001 | Construct a mapping satisf... |
| numacn 10002 | A well-orderable set has c... |
| finacn 10003 | Every set has finite choic... |
| acndom 10004 | A set with long choice seq... |
| acnnum 10005 | A set ` X ` which has choi... |
| acnen 10006 | The class of choice sets o... |
| acndom2 10007 | A set smaller than one wit... |
| acnen2 10008 | The class of sets with cho... |
| fodomacn 10009 | A version of ~ fodom that ... |
| fodomnum 10010 | A version of ~ fodom that ... |
| fonum 10011 | A surjection maps numerabl... |
| numwdom 10012 | A surjection maps numerabl... |
| fodomfi2 10013 | Onto functions define domi... |
| wdomfil 10014 | Weak dominance agrees with... |
| infpwfien 10015 | Any infinite well-orderabl... |
| inffien 10016 | The set of finite intersec... |
| wdomnumr 10017 | Weak dominance agrees with... |
| alephfnon 10018 | The aleph function is a fu... |
| aleph0 10019 | The first infinite cardina... |
| alephlim 10020 | Value of the aleph functio... |
| alephsuc 10021 | Value of the aleph functio... |
| alephon 10022 | An aleph is an ordinal num... |
| alephcard 10023 | Every aleph is a cardinal ... |
| alephnbtwn 10024 | No cardinal can be sandwic... |
| alephnbtwn2 10025 | No set has equinumerosity ... |
| alephordilem1 10026 | Lemma for ~ alephordi . (... |
| alephordi 10027 | Strict ordering property o... |
| alephord 10028 | Ordering property of the a... |
| alephord2 10029 | Ordering property of the a... |
| alephord2i 10030 | Ordering property of the a... |
| alephord3 10031 | Ordering property of the a... |
| alephsucdom 10032 | A set dominated by an alep... |
| alephsuc2 10033 | An alternate representatio... |
| alephdom 10034 | Relationship between inclu... |
| alephgeom 10035 | Every aleph is greater tha... |
| alephislim 10036 | Every aleph is a limit ord... |
| aleph11 10037 | The aleph function is one-... |
| alephf1 10038 | The aleph function is a on... |
| alephsdom 10039 | If an ordinal is smaller t... |
| alephdom2 10040 | A dominated initial ordina... |
| alephle 10041 | The argument of the aleph ... |
| cardaleph 10042 | Given any transfinite card... |
| cardalephex 10043 | Every transfinite cardinal... |
| infenaleph 10044 | An infinite numerable set ... |
| isinfcard 10045 | Two ways to express the pr... |
| iscard3 10046 | Two ways to express the pr... |
| cardnum 10047 | Two ways to express the cl... |
| alephinit 10048 | An infinite initial ordina... |
| carduniima 10049 | The union of the image of ... |
| cardinfima 10050 | If a mapping to cardinals ... |
| alephiso 10051 | Aleph is an order isomorph... |
| alephprc 10052 | The class of all transfini... |
| alephsson 10053 | The class of transfinite c... |
| unialeph 10054 | The union of the class of ... |
| alephsmo 10055 | The aleph function is stri... |
| alephf1ALT 10056 | Alternate proof of ~ aleph... |
| alephfplem1 10057 | Lemma for ~ alephfp . (Co... |
| alephfplem2 10058 | Lemma for ~ alephfp . (Co... |
| alephfplem3 10059 | Lemma for ~ alephfp . (Co... |
| alephfplem4 10060 | Lemma for ~ alephfp . (Co... |
| alephfp 10061 | The aleph function has a f... |
| alephfp2 10062 | The aleph function has at ... |
| alephval3 10063 | An alternate way to expres... |
| alephsucpw2 10064 | The power set of an aleph ... |
| mappwen 10065 | Power rule for cardinal ar... |
| finnisoeu 10066 | A finite totally ordered s... |
| iunfictbso 10067 | Countability of a countabl... |
| aceq1 10070 | Equivalence of two version... |
| aceq0 10071 | Equivalence of two version... |
| aceq2 10072 | Equivalence of two version... |
| aceq3lem 10073 | Lemma for ~ dfac3 . (Cont... |
| dfac3 10074 | Equivalence of two version... |
| dfac4 10075 | Equivalence of two version... |
| dfac5lem1 10076 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem2 10077 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem3 10078 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4 10079 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem5 10080 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4OLD 10081 | Obsolete version of ~ dfac... |
| dfac5 10082 | Equivalence of two version... |
| dfac2a 10083 | Our Axiom of Choice (in th... |
| dfac2b 10084 | Axiom of Choice (first for... |
| dfac2 10085 | Axiom of Choice (first for... |
| dfac7 10086 | Equivalence of the Axiom o... |
| dfac0 10087 | Equivalence of two version... |
| dfac1 10088 | Equivalence of two version... |
| dfac8 10089 | A proof of the equivalency... |
| dfac9 10090 | Equivalence of the axiom o... |
| dfac10 10091 | Axiom of Choice equivalent... |
| dfac10c 10092 | Axiom of Choice equivalent... |
| dfac10b 10093 | Axiom of Choice equivalent... |
| acacni 10094 | A choice equivalent: every... |
| dfacacn 10095 | A choice equivalent: every... |
| dfac13 10096 | The axiom of choice holds ... |
| dfac12lem1 10097 | Lemma for ~ dfac12 . (Con... |
| dfac12lem2 10098 | Lemma for ~ dfac12 . (Con... |
| dfac12lem3 10099 | Lemma for ~ dfac12 . (Con... |
| dfac12r 10100 | The axiom of choice holds ... |
| dfac12k 10101 | Equivalence of ~ dfac12 an... |
| dfac12a 10102 | The axiom of choice holds ... |
| dfac12 10103 | The axiom of choice holds ... |
| kmlem1 10104 | Lemma for 5-quantifier AC ... |
| kmlem2 10105 | Lemma for 5-quantifier AC ... |
| kmlem3 10106 | Lemma for 5-quantifier AC ... |
| kmlem4 10107 | Lemma for 5-quantifier AC ... |
| kmlem5 10108 | Lemma for 5-quantifier AC ... |
| kmlem6 10109 | Lemma for 5-quantifier AC ... |
| kmlem7 10110 | Lemma for 5-quantifier AC ... |
| kmlem8 10111 | Lemma for 5-quantifier AC ... |
| kmlem9 10112 | Lemma for 5-quantifier AC ... |
| kmlem10 10113 | Lemma for 5-quantifier AC ... |
| kmlem11 10114 | Lemma for 5-quantifier AC ... |
| kmlem12 10115 | Lemma for 5-quantifier AC ... |
| kmlem13 10116 | Lemma for 5-quantifier AC ... |
| kmlem14 10117 | Lemma for 5-quantifier AC ... |
| kmlem15 10118 | Lemma for 5-quantifier AC ... |
| kmlem16 10119 | Lemma for 5-quantifier AC ... |
| dfackm 10120 | Equivalence of the Axiom o... |
| undjudom 10121 | Cardinal addition dominate... |
| endjudisj 10122 | Equinumerosity of a disjoi... |
| djuen 10123 | Disjoint unions of equinum... |
| djuenun 10124 | Disjoint union is equinume... |
| dju1en 10125 | Cardinal addition with car... |
| dju1dif 10126 | Adding and subtracting one... |
| dju1p1e2 10127 | 1+1=2 for cardinal number ... |
| dju1p1e2ALT 10128 | Alternate proof of ~ dju1p... |
| dju0en 10129 | Cardinal addition with car... |
| xp2dju 10130 | Two times a cardinal numbe... |
| djucomen 10131 | Commutative law for cardin... |
| djuassen 10132 | Associative law for cardin... |
| xpdjuen 10133 | Cardinal multiplication di... |
| mapdjuen 10134 | Sum of exponents law for c... |
| pwdjuen 10135 | Sum of exponents law for c... |
| djudom1 10136 | Ordering law for cardinal ... |
| djudom2 10137 | Ordering law for cardinal ... |
| djudoml 10138 | A set is dominated by its ... |
| djuxpdom 10139 | Cartesian product dominate... |
| djufi 10140 | The disjoint union of two ... |
| cdainflem 10141 | Any partition of omega int... |
| djuinf 10142 | A set is infinite iff the ... |
| infdju1 10143 | An infinite set is equinum... |
| pwdju1 10144 | The sum of a powerset with... |
| pwdjuidm 10145 | If the natural numbers inj... |
| djulepw 10146 | If ` A ` is idempotent und... |
| onadju 10147 | The cardinal and ordinal s... |
| cardadju 10148 | The cardinal sum is equinu... |
| djunum 10149 | The disjoint union of two ... |
| unnum 10150 | The union of two numerable... |
| nnadju 10151 | The cardinal and ordinal s... |
| nnadjuALT 10152 | Shorter proof of ~ nnadju ... |
| ficardadju 10153 | The disjoint union of fini... |
| ficardun 10154 | The cardinality of the uni... |
| ficardun2 10155 | The cardinality of the uni... |
| pwsdompw 10156 | Lemma for ~ domtriom . Th... |
| unctb 10157 | The union of two countable... |
| infdjuabs 10158 | Absorption law for additio... |
| infunabs 10159 | An infinite set is equinum... |
| infdju 10160 | The sum of two cardinal nu... |
| infdif 10161 | The cardinality of an infi... |
| infdif2 10162 | Cardinality ordering for a... |
| infxpdom 10163 | Dominance law for multipli... |
| infxpabs 10164 | Absorption law for multipl... |
| infunsdom1 10165 | The union of two sets that... |
| infunsdom 10166 | The union of two sets that... |
| infxp 10167 | Absorption law for multipl... |
| pwdjudom 10168 | A property of dominance ov... |
| infpss 10169 | Every infinite set has an ... |
| infmap2 10170 | An exponentiation law for ... |
| ackbij2lem1 10171 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem1 10172 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem2 10173 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem3 10174 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem4 10175 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem5 10176 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem6 10177 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem7 10178 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem8 10179 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem9 10180 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem10 10181 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem11 10182 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem12 10183 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem13 10184 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem14 10185 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem15 10186 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem16 10187 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem17 10188 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem18 10189 | Lemma for ~ ackbij1 . (Co... |
| ackbij1 10190 | The Ackermann bijection, p... |
| ackbij1b 10191 | The Ackermann bijection, p... |
| ackbij2lem2 10192 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem3 10193 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem4 10194 | Lemma for ~ ackbij2 . (Co... |
| ackbij2 10195 | The Ackermann bijection, p... |
| r1om 10196 | The set of hereditarily fi... |
| fictb 10197 | A set is countable iff its... |
| cflem 10198 | A lemma used to simplify c... |
| cflemOLD 10199 | Obsolete version of ~ cfle... |
| cfval 10200 | Value of the cofinality fu... |
| cff 10201 | Cofinality is a function o... |
| cfub 10202 | An upper bound on cofinali... |
| cflm 10203 | Value of the cofinality fu... |
| cf0 10204 | Value of the cofinality fu... |
| cardcf 10205 | Cofinality is a cardinal n... |
| cflecard 10206 | Cofinality is bounded by t... |
| cfle 10207 | Cofinality is bounded by i... |
| cfon 10208 | The cofinality of any set ... |
| cfeq0 10209 | Only the ordinal zero has ... |
| cfsuc 10210 | Value of the cofinality fu... |
| cff1 10211 | There is always a map from... |
| cfflb 10212 | If there is a cofinal map ... |
| cfval2 10213 | Another expression for the... |
| coflim 10214 | A simpler expression for t... |
| cflim3 10215 | Another expression for the... |
| cflim2 10216 | The cofinality function is... |
| cfom 10217 | Value of the cofinality fu... |
| cfss 10218 | There is a cofinal subset ... |
| cfslb 10219 | Any cofinal subset of ` A ... |
| cfslbn 10220 | Any subset of ` A ` smalle... |
| cfslb2n 10221 | Any small collection of sm... |
| cofsmo 10222 | Any cofinal map implies th... |
| cfsmolem 10223 | Lemma for ~ cfsmo . (Cont... |
| cfsmo 10224 | The map in ~ cff1 can be a... |
| cfcoflem 10225 | Lemma for ~ cfcof , showin... |
| coftr 10226 | If there is a cofinal map ... |
| cfcof 10227 | If there is a cofinal map ... |
| cfidm 10228 | The cofinality function is... |
| alephsing 10229 | The cofinality of a limit ... |
| sornom 10230 | The range of a single-step... |
| isfin1a 10245 | Definition of a Ia-finite ... |
| fin1ai 10246 | Property of a Ia-finite se... |
| isfin2 10247 | Definition of a II-finite ... |
| fin2i 10248 | Property of a II-finite se... |
| isfin3 10249 | Definition of a III-finite... |
| isfin4 10250 | Definition of a IV-finite ... |
| fin4i 10251 | Infer that a set is IV-inf... |
| isfin5 10252 | Definition of a V-finite s... |
| isfin6 10253 | Definition of a VI-finite ... |
| isfin7 10254 | Definition of a VII-finite... |
| sdom2en01 10255 | A set with less than two e... |
| infpssrlem1 10256 | Lemma for ~ infpssr . (Co... |
| infpssrlem2 10257 | Lemma for ~ infpssr . (Co... |
| infpssrlem3 10258 | Lemma for ~ infpssr . (Co... |
| infpssrlem4 10259 | Lemma for ~ infpssr . (Co... |
| infpssrlem5 10260 | Lemma for ~ infpssr . (Co... |
| infpssr 10261 | Dedekind infinity implies ... |
| fin4en1 10262 | Dedekind finite is a cardi... |
| ssfin4 10263 | Dedekind finite sets have ... |
| domfin4 10264 | A set dominated by a Dedek... |
| ominf4 10265 | ` _om ` is Dedekind infini... |
| infpssALT 10266 | Alternate proof of ~ infps... |
| isfin4-2 10267 | Alternate definition of IV... |
| isfin4p1 10268 | Alternate definition of IV... |
| fin23lem7 10269 | Lemma for ~ isfin2-2 . Th... |
| fin23lem11 10270 | Lemma for ~ isfin2-2 . (C... |
| fin2i2 10271 | A II-finite set contains m... |
| isfin2-2 10272 | ` Fin2 ` expressed in term... |
| ssfin2 10273 | A subset of a II-finite se... |
| enfin2i 10274 | II-finiteness is a cardina... |
| fin23lem24 10275 | Lemma for ~ fin23 . In a ... |
| fincssdom 10276 | In a chain of finite sets,... |
| fin23lem25 10277 | Lemma for ~ fin23 . In a ... |
| fin23lem26 10278 | Lemma for ~ fin23lem22 . ... |
| fin23lem23 10279 | Lemma for ~ fin23lem22 . ... |
| fin23lem22 10280 | Lemma for ~ fin23 but coul... |
| fin23lem27 10281 | The mapping constructed in... |
| isfin3ds 10282 | Property of a III-finite s... |
| ssfin3ds 10283 | A subset of a III-finite s... |
| fin23lem12 10284 | The beginning of the proof... |
| fin23lem13 10285 | Lemma for ~ fin23 . Each ... |
| fin23lem14 10286 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem15 10287 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem16 10288 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem19 10289 | Lemma for ~ fin23 . The f... |
| fin23lem20 10290 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem17 10291 | Lemma for ~ fin23 . By ? ... |
| fin23lem21 10292 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem28 10293 | Lemma for ~ fin23 . The r... |
| fin23lem29 10294 | Lemma for ~ fin23 . The r... |
| fin23lem30 10295 | Lemma for ~ fin23 . The r... |
| fin23lem31 10296 | Lemma for ~ fin23 . The r... |
| fin23lem32 10297 | Lemma for ~ fin23 . Wrap ... |
| fin23lem33 10298 | Lemma for ~ fin23 . Disch... |
| fin23lem34 10299 | Lemma for ~ fin23 . Estab... |
| fin23lem35 10300 | Lemma for ~ fin23 . Stric... |
| fin23lem36 10301 | Lemma for ~ fin23 . Weak ... |
| fin23lem38 10302 | Lemma for ~ fin23 . The c... |
| fin23lem39 10303 | Lemma for ~ fin23 . Thus,... |
| fin23lem40 10304 | Lemma for ~ fin23 . ` Fin2... |
| fin23lem41 10305 | Lemma for ~ fin23 . A set... |
| isf32lem1 10306 | Lemma for ~ isfin3-2 . De... |
| isf32lem2 10307 | Lemma for ~ isfin3-2 . No... |
| isf32lem3 10308 | Lemma for ~ isfin3-2 . Be... |
| isf32lem4 10309 | Lemma for ~ isfin3-2 . Be... |
| isf32lem5 10310 | Lemma for ~ isfin3-2 . Th... |
| isf32lem6 10311 | Lemma for ~ isfin3-2 . Ea... |
| isf32lem7 10312 | Lemma for ~ isfin3-2 . Di... |
| isf32lem8 10313 | Lemma for ~ isfin3-2 . K ... |
| isf32lem9 10314 | Lemma for ~ isfin3-2 . Co... |
| isf32lem10 10315 | Lemma for isfin3-2 . Writ... |
| isf32lem11 10316 | Lemma for ~ isfin3-2 . Re... |
| isf32lem12 10317 | Lemma for ~ isfin3-2 . (C... |
| isfin32i 10318 | One half of ~ isfin3-2 . ... |
| isf33lem 10319 | Lemma for ~ isfin3-3 . (C... |
| isfin3-2 10320 | Weakly Dedekind-infinite s... |
| isfin3-3 10321 | Weakly Dedekind-infinite s... |
| fin33i 10322 | Inference from ~ isfin3-3 ... |
| compsscnvlem 10323 | Lemma for ~ compsscnv . (... |
| compsscnv 10324 | Complementation on a power... |
| isf34lem1 10325 | Lemma for ~ isfin3-4 . (C... |
| isf34lem2 10326 | Lemma for ~ isfin3-4 . (C... |
| compssiso 10327 | Complementation is an anti... |
| isf34lem3 10328 | Lemma for ~ isfin3-4 . (C... |
| compss 10329 | Express image under of the... |
| isf34lem4 10330 | Lemma for ~ isfin3-4 . (C... |
| isf34lem5 10331 | Lemma for ~ isfin3-4 . (C... |
| isf34lem7 10332 | Lemma for ~ isfin3-4 . (C... |
| isf34lem6 10333 | Lemma for ~ isfin3-4 . (C... |
| fin34i 10334 | Inference from ~ isfin3-4 ... |
| isfin3-4 10335 | Weakly Dedekind-infinite s... |
| fin11a 10336 | Every I-finite set is Ia-f... |
| enfin1ai 10337 | Ia-finiteness is a cardina... |
| isfin1-2 10338 | A set is finite in the usu... |
| isfin1-3 10339 | A set is I-finite iff ever... |
| isfin1-4 10340 | A set is I-finite iff ever... |
| dffin1-5 10341 | Compact quantifier-free ve... |
| fin23 10342 | Every II-finite set (every... |
| fin34 10343 | Every III-finite set is IV... |
| isfin5-2 10344 | Alternate definition of V-... |
| fin45 10345 | Every IV-finite set is V-f... |
| fin56 10346 | Every V-finite set is VI-f... |
| fin17 10347 | Every I-finite set is VII-... |
| fin67 10348 | Every VI-finite set is VII... |
| isfin7-2 10349 | A set is VII-finite iff it... |
| fin71num 10350 | A well-orderable set is VI... |
| dffin7-2 10351 | Class form of ~ isfin7-2 .... |
| dfacfin7 10352 | Axiom of Choice equivalent... |
| fin1a2lem1 10353 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem2 10354 | Lemma for ~ fin1a2 . The ... |
| fin1a2lem3 10355 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem4 10356 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem5 10357 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem6 10358 | Lemma for ~ fin1a2 . Esta... |
| fin1a2lem7 10359 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem8 10360 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem9 10361 | Lemma for ~ fin1a2 . In a... |
| fin1a2lem10 10362 | Lemma for ~ fin1a2 . A no... |
| fin1a2lem11 10363 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem12 10364 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem13 10365 | Lemma for ~ fin1a2 . (Con... |
| fin12 10366 | Weak theorem which skips I... |
| fin1a2s 10367 | An II-infinite set can hav... |
| fin1a2 10368 | Every Ia-finite set is II-... |
| itunifval 10369 | Function value of iterated... |
| itunifn 10370 | Functionality of the itera... |
| ituni0 10371 | A zero-fold iterated union... |
| itunisuc 10372 | Successor iterated union. ... |
| itunitc1 10373 | Each union iterate is a me... |
| itunitc 10374 | The union of all union ite... |
| ituniiun 10375 | Unwrap an iterated union f... |
| hsmexlem7 10376 | Lemma for ~ hsmex . Prope... |
| hsmexlem8 10377 | Lemma for ~ hsmex . Prope... |
| hsmexlem9 10378 | Lemma for ~ hsmex . Prope... |
| hsmexlem1 10379 | Lemma for ~ hsmex . Bound... |
| hsmexlem2 10380 | Lemma for ~ hsmex . Bound... |
| hsmexlem3 10381 | Lemma for ~ hsmex . Clear... |
| hsmexlem4 10382 | Lemma for ~ hsmex . The c... |
| hsmexlem5 10383 | Lemma for ~ hsmex . Combi... |
| hsmexlem6 10384 | Lemma for ~ hsmex . (Cont... |
| hsmex 10385 | The collection of heredita... |
| hsmex2 10386 | The set of hereditary size... |
| hsmex3 10387 | The set of hereditary size... |
| axcc2lem 10389 | Lemma for ~ axcc2 . (Cont... |
| axcc2 10390 | A possibly more useful ver... |
| axcc3 10391 | A possibly more useful ver... |
| axcc4 10392 | A version of ~ axcc3 that ... |
| acncc 10393 | An ~ ax-cc equivalent: eve... |
| axcc4dom 10394 | Relax the constraint on ~ ... |
| domtriomlem 10395 | Lemma for ~ domtriom . (C... |
| domtriom 10396 | Trichotomy of equinumerosi... |
| fin41 10397 | Under countable choice, th... |
| dominf 10398 | A nonempty set that is a s... |
| dcomex 10400 | The Axiom of Dependent Cho... |
| axdc2lem 10401 | Lemma for ~ axdc2 . We co... |
| axdc2 10402 | An apparent strengthening ... |
| axdc3lem 10403 | The class ` S ` of finite ... |
| axdc3lem2 10404 | Lemma for ~ axdc3 . We ha... |
| axdc3lem3 10405 | Simple substitution lemma ... |
| axdc3lem4 10406 | Lemma for ~ axdc3 . We ha... |
| axdc3 10407 | Dependent Choice. Axiom D... |
| axdc4lem 10408 | Lemma for ~ axdc4 . (Cont... |
| axdc4 10409 | A more general version of ... |
| axcclem 10410 | Lemma for ~ axcc . (Contr... |
| axcc 10411 | Although CC can be proven ... |
| zfac 10413 | Axiom of Choice expressed ... |
| ac2 10414 | Axiom of Choice equivalent... |
| ac3 10415 | Axiom of Choice using abbr... |
| axac3 10417 | This theorem asserts that ... |
| ackm 10418 | A remarkable equivalent to... |
| axac2 10419 | Derive ~ ax-ac2 from ~ ax-... |
| axac 10420 | Derive ~ ax-ac from ~ ax-a... |
| axaci 10421 | Apply a choice equivalent.... |
| cardeqv 10422 | All sets are well-orderabl... |
| numth3 10423 | All sets are well-orderabl... |
| numth2 10424 | Numeration theorem: any se... |
| numth 10425 | Numeration theorem: every ... |
| ac7 10426 | An Axiom of Choice equival... |
| ac7g 10427 | An Axiom of Choice equival... |
| ac4 10428 | Equivalent of Axiom of Cho... |
| ac4c 10429 | Equivalent of Axiom of Cho... |
| ac5 10430 | An Axiom of Choice equival... |
| ac5b 10431 | Equivalent of Axiom of Cho... |
| ac6num 10432 | A version of ~ ac6 which t... |
| ac6 10433 | Equivalent of Axiom of Cho... |
| ac6c4 10434 | Equivalent of Axiom of Cho... |
| ac6c5 10435 | Equivalent of Axiom of Cho... |
| ac9 10436 | An Axiom of Choice equival... |
| ac6s 10437 | Equivalent of Axiom of Cho... |
| ac6n 10438 | Equivalent of Axiom of Cho... |
| ac6s2 10439 | Generalization of the Axio... |
| ac6s3 10440 | Generalization of the Axio... |
| ac6sg 10441 | ~ ac6s with sethood as ant... |
| ac6sf 10442 | Version of ~ ac6 with boun... |
| ac6s4 10443 | Generalization of the Axio... |
| ac6s5 10444 | Generalization of the Axio... |
| ac8 10445 | An Axiom of Choice equival... |
| ac9s 10446 | An Axiom of Choice equival... |
| numthcor 10447 | Any set is strictly domina... |
| weth 10448 | Well-ordering theorem: any... |
| zorn2lem1 10449 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem2 10450 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem3 10451 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem4 10452 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem5 10453 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem6 10454 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem7 10455 | Lemma for ~ zorn2 . (Cont... |
| zorn2g 10456 | Zorn's Lemma of [Monk1] p.... |
| zorng 10457 | Zorn's Lemma. If the unio... |
| zornn0g 10458 | Variant of Zorn's lemma ~ ... |
| zorn2 10459 | Zorn's Lemma of [Monk1] p.... |
| zorn 10460 | Zorn's Lemma. If the unio... |
| zornn0 10461 | Variant of Zorn's lemma ~ ... |
| ttukeylem1 10462 | Lemma for ~ ttukey . Expa... |
| ttukeylem2 10463 | Lemma for ~ ttukey . A pr... |
| ttukeylem3 10464 | Lemma for ~ ttukey . (Con... |
| ttukeylem4 10465 | Lemma for ~ ttukey . (Con... |
| ttukeylem5 10466 | Lemma for ~ ttukey . The ... |
| ttukeylem6 10467 | Lemma for ~ ttukey . (Con... |
| ttukeylem7 10468 | Lemma for ~ ttukey . (Con... |
| ttukey2g 10469 | The Teichmüller-Tukey... |
| ttukeyg 10470 | The Teichmüller-Tukey... |
| ttukey 10471 | The Teichmüller-Tukey... |
| axdclem 10472 | Lemma for ~ axdc . (Contr... |
| axdclem2 10473 | Lemma for ~ axdc . Using ... |
| axdc 10474 | This theorem derives ~ ax-... |
| fodomg 10475 | An onto function implies d... |
| fodom 10476 | An onto function implies d... |
| dmct 10477 | The domain of a countable ... |
| rnct 10478 | The range of a countable s... |
| fodomb 10479 | Equivalence of an onto map... |
| wdomac 10480 | When assuming AC, weak and... |
| brdom3 10481 | Equivalence to a dominance... |
| brdom5 10482 | An equivalence to a domina... |
| brdom4 10483 | An equivalence to a domina... |
| brdom7disj 10484 | An equivalence to a domina... |
| brdom6disj 10485 | An equivalence to a domina... |
| fin71ac 10486 | Once we allow AC, the "str... |
| imadomg 10487 | An image of a function und... |
| fimact 10488 | The image by a function of... |
| fnrndomg 10489 | The range of a function is... |
| fnct 10490 | If the domain of a functio... |
| mptct 10491 | A countable mapping set is... |
| iunfo 10492 | Existence of an onto funct... |
| iundom2g 10493 | An upper bound for the car... |
| iundomg 10494 | An upper bound for the car... |
| iundom 10495 | An upper bound for the car... |
| unidom 10496 | An upper bound for the car... |
| uniimadom 10497 | An upper bound for the car... |
| uniimadomf 10498 | An upper bound for the car... |
| cardval 10499 | The value of the cardinal ... |
| cardid 10500 | Any set is equinumerous to... |
| cardidg 10501 | Any set is equinumerous to... |
| cardidd 10502 | Any set is equinumerous to... |
| cardf 10503 | The cardinality function i... |
| carden 10504 | Two sets are equinumerous ... |
| cardeq0 10505 | Only the empty set has car... |
| unsnen 10506 | Equinumerosity of a set wi... |
| carddom 10507 | Two sets have the dominanc... |
| cardsdom 10508 | Two sets have the strict d... |
| domtri 10509 | Trichotomy law for dominan... |
| entric 10510 | Trichotomy of equinumerosi... |
| entri2 10511 | Trichotomy of dominance an... |
| entri3 10512 | Trichotomy of dominance. ... |
| sdomsdomcard 10513 | A set strictly dominates i... |
| canth3 10514 | Cantor's theorem in terms ... |
| infxpidm 10515 | Every infinite class is eq... |
| ondomon 10516 | The class of ordinals domi... |
| cardmin 10517 | The smallest ordinal that ... |
| ficard 10518 | A set is finite iff its ca... |
| infinf 10519 | Equivalence between two in... |
| unirnfdomd 10520 | The union of the range of ... |
| konigthlem 10521 | Lemma for ~ konigth . (Co... |
| konigth 10522 | Konig's Theorem. If ` m (... |
| alephsucpw 10523 | The power set of an aleph ... |
| aleph1 10524 | The set exponentiation of ... |
| alephval2 10525 | An alternate way to expres... |
| dominfac 10526 | A nonempty set that is a s... |
| iunctb 10527 | The countable union of cou... |
| unictb 10528 | The countable union of cou... |
| infmap 10529 | An exponentiation law for ... |
| alephadd 10530 | The sum of two alephs is t... |
| alephmul 10531 | The product of two alephs ... |
| alephexp1 10532 | An exponentiation law for ... |
| alephsuc3 10533 | An alternate representatio... |
| alephexp2 10534 | An expression equinumerous... |
| alephreg 10535 | A successor aleph is regul... |
| pwcfsdom 10536 | A corollary of Konig's The... |
| cfpwsdom 10537 | A corollary of Konig's The... |
| alephom 10538 | From ~ canth2 , we know th... |
| smobeth 10539 | The beth function is stric... |
| nd1 10540 | A lemma for proving condit... |
| nd2 10541 | A lemma for proving condit... |
| nd3 10542 | A lemma for proving condit... |
| nd4 10543 | A lemma for proving condit... |
| axextnd 10544 | A version of the Axiom of ... |
| axrepndlem1 10545 | Lemma for the Axiom of Rep... |
| axrepndlem2 10546 | Lemma for the Axiom of Rep... |
| axrepnd 10547 | A version of the Axiom of ... |
| axunndlem1 10548 | Lemma for the Axiom of Uni... |
| axunnd 10549 | A version of the Axiom of ... |
| axpowndlem1 10550 | Lemma for the Axiom of Pow... |
| axpowndlem2 10551 | Lemma for the Axiom of Pow... |
| axpowndlem3 10552 | Lemma for the Axiom of Pow... |
| axpowndlem4 10553 | Lemma for the Axiom of Pow... |
| axpownd 10554 | A version of the Axiom of ... |
| axregndlem1 10555 | Lemma for the Axiom of Reg... |
| axregndlem2 10556 | Lemma for the Axiom of Reg... |
| axregnd 10557 | A version of the Axiom of ... |
| axinfndlem1 10558 | Lemma for the Axiom of Inf... |
| axinfnd 10559 | A version of the Axiom of ... |
| axacndlem1 10560 | Lemma for the Axiom of Cho... |
| axacndlem2 10561 | Lemma for the Axiom of Cho... |
| axacndlem3 10562 | Lemma for the Axiom of Cho... |
| axacndlem4 10563 | Lemma for the Axiom of Cho... |
| axacndlem5 10564 | Lemma for the Axiom of Cho... |
| axacnd 10565 | A version of the Axiom of ... |
| zfcndext 10566 | Axiom of Extensionality ~ ... |
| zfcndrep 10567 | Axiom of Replacement ~ ax-... |
| zfcndun 10568 | Axiom of Union ~ ax-un , r... |
| zfcndpow 10569 | Axiom of Power Sets ~ ax-p... |
| zfcndreg 10570 | Axiom of Regularity ~ ax-r... |
| zfcndinf 10571 | Axiom of Infinity ~ ax-inf... |
| zfcndac 10572 | Axiom of Choice ~ ax-ac , ... |
| elgch 10575 | Elementhood in the collect... |
| fingch 10576 | A finite set is a GCH-set.... |
| gchi 10577 | The only GCH-sets which ha... |
| gchen1 10578 | If ` A <_ B < ~P A ` , and... |
| gchen2 10579 | If ` A < B <_ ~P A ` , and... |
| gchor 10580 | If ` A <_ B <_ ~P A ` , an... |
| engch 10581 | The property of being a GC... |
| gchdomtri 10582 | Under certain conditions, ... |
| fpwwe2cbv 10583 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem1 10584 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem2 10585 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem3 10586 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem4 10587 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem5 10588 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem6 10589 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem7 10590 | Lemma for ~ fpwwe2 . Show... |
| fpwwe2lem8 10591 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem9 10592 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem10 10593 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem11 10594 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem12 10595 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2 10596 | Given any function ` F ` f... |
| fpwwecbv 10597 | Lemma for ~ fpwwe . (Cont... |
| fpwwelem 10598 | Lemma for ~ fpwwe . (Cont... |
| fpwwe 10599 | Given any function ` F ` f... |
| canth4 10600 | An "effective" form of Can... |
| canthnumlem 10601 | Lemma for ~ canthnum . (C... |
| canthnum 10602 | The set of well-orderable ... |
| canthwelem 10603 | Lemma for ~ canthwe . (Co... |
| canthwe 10604 | The set of well-orders of ... |
| canthp1lem1 10605 | Lemma for ~ canthp1 . (Co... |
| canthp1lem2 10606 | Lemma for ~ canthp1 . (Co... |
| canthp1 10607 | A slightly stronger form o... |
| finngch 10608 | The exclusion of finite se... |
| gchdju1 10609 | An infinite GCH-set is ide... |
| gchinf 10610 | An infinite GCH-set is Ded... |
| pwfseqlem1 10611 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem2 10612 | Lemma for ~ pwfseq . (Con... |
| pwfseqlem3 10613 | Lemma for ~ pwfseq . Usin... |
| pwfseqlem4a 10614 | Lemma for ~ pwfseqlem4 . ... |
| pwfseqlem4 10615 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem5 10616 | Lemma for ~ pwfseq . Alth... |
| pwfseq 10617 | The powerset of a Dedekind... |
| pwxpndom2 10618 | The powerset of a Dedekind... |
| pwxpndom 10619 | The powerset of a Dedekind... |
| pwdjundom 10620 | The powerset of a Dedekind... |
| gchdjuidm 10621 | An infinite GCH-set is ide... |
| gchxpidm 10622 | An infinite GCH-set is ide... |
| gchpwdom 10623 | A relationship between dom... |
| gchaleph 10624 | If ` ( aleph `` A ) ` is a... |
| gchaleph2 10625 | If ` ( aleph `` A ) ` and ... |
| hargch 10626 | If ` A + ~~ ~P A ` , then ... |
| alephgch 10627 | If ` ( aleph `` suc A ) ` ... |
| gch2 10628 | It is sufficient to requir... |
| gch3 10629 | An equivalent formulation ... |
| gch-kn 10630 | The equivalence of two ver... |
| gchaclem 10631 | Lemma for ~ gchac (obsolet... |
| gchhar 10632 | A "local" form of ~ gchac ... |
| gchacg 10633 | A "local" form of ~ gchac ... |
| gchac 10634 | The Generalized Continuum ... |
| elwina 10639 | Conditions of weak inacces... |
| elina 10640 | Conditions of strong inacc... |
| winaon 10641 | A weakly inaccessible card... |
| inawinalem 10642 | Lemma for ~ inawina . (Co... |
| inawina 10643 | Every strongly inaccessibl... |
| omina 10644 | ` _om ` is a strongly inac... |
| winacard 10645 | A weakly inaccessible card... |
| winainflem 10646 | A weakly inaccessible card... |
| winainf 10647 | A weakly inaccessible card... |
| winalim 10648 | A weakly inaccessible card... |
| winalim2 10649 | A nontrivial weakly inacce... |
| winafp 10650 | A nontrivial weakly inacce... |
| winafpi 10651 | This theorem, which states... |
| gchina 10652 | Assuming the GCH, weakly a... |
| iswun 10657 | Properties of a weak unive... |
| wuntr 10658 | A weak universe is transit... |
| wununi 10659 | A weak universe is closed ... |
| wunpw 10660 | A weak universe is closed ... |
| wunelss 10661 | The elements of a weak uni... |
| wunpr 10662 | A weak universe is closed ... |
| wunun 10663 | A weak universe is closed ... |
| wuntp 10664 | A weak universe is closed ... |
| wunss 10665 | A weak universe is closed ... |
| wunin 10666 | A weak universe is closed ... |
| wundif 10667 | A weak universe is closed ... |
| wunint 10668 | A weak universe is closed ... |
| wunsn 10669 | A weak universe is closed ... |
| wunsuc 10670 | A weak universe is closed ... |
| wun0 10671 | A weak universe contains t... |
| wunr1om 10672 | A weak universe is infinit... |
| wunom 10673 | A weak universe contains a... |
| wunfi 10674 | A weak universe contains a... |
| wunop 10675 | A weak universe is closed ... |
| wunot 10676 | A weak universe is closed ... |
| wunxp 10677 | A weak universe is closed ... |
| wunpm 10678 | A weak universe is closed ... |
| wunmap 10679 | A weak universe is closed ... |
| wunf 10680 | A weak universe is closed ... |
| wundm 10681 | A weak universe is closed ... |
| wunrn 10682 | A weak universe is closed ... |
| wuncnv 10683 | A weak universe is closed ... |
| wunres 10684 | A weak universe is closed ... |
| wunfv 10685 | A weak universe is closed ... |
| wunco 10686 | A weak universe is closed ... |
| wuntpos 10687 | A weak universe is closed ... |
| intwun 10688 | The intersection of a coll... |
| r1limwun 10689 | Each limit stage in the cu... |
| r1wunlim 10690 | The weak universes in the ... |
| wunex2 10691 | Construct a weak universe ... |
| wunex 10692 | Construct a weak universe ... |
| uniwun 10693 | Every set is contained in ... |
| wunex3 10694 | Construct a weak universe ... |
| wuncval 10695 | Value of the weak universe... |
| wuncid 10696 | The weak universe closure ... |
| wunccl 10697 | The weak universe closure ... |
| wuncss 10698 | The weak universe closure ... |
| wuncidm 10699 | The weak universe closure ... |
| wuncval2 10700 | Our earlier expression for... |
| eltskg 10703 | Properties of a Tarski cla... |
| eltsk2g 10704 | Properties of a Tarski cla... |
| tskpwss 10705 | First axiom of a Tarski cl... |
| tskpw 10706 | Second axiom of a Tarski c... |
| tsken 10707 | Third axiom of a Tarski cl... |
| 0tsk 10708 | The empty set is a (transi... |
| tsksdom 10709 | An element of a Tarski cla... |
| tskssel 10710 | A part of a Tarski class s... |
| tskss 10711 | The subsets of an element ... |
| tskin 10712 | The intersection of two el... |
| tsksn 10713 | A singleton of an element ... |
| tsktrss 10714 | A transitive element of a ... |
| tsksuc 10715 | If an element of a Tarski ... |
| tsk0 10716 | A nonempty Tarski class co... |
| tsk1 10717 | One is an element of a non... |
| tsk2 10718 | Two is an element of a non... |
| 2domtsk 10719 | If a Tarski class is not e... |
| tskr1om 10720 | A nonempty Tarski class is... |
| tskr1om2 10721 | A nonempty Tarski class co... |
| tskinf 10722 | A nonempty Tarski class is... |
| tskpr 10723 | If ` A ` and ` B ` are mem... |
| tskop 10724 | If ` A ` and ` B ` are mem... |
| tskxpss 10725 | A Cartesian product of two... |
| tskwe2 10726 | A Tarski class is well-ord... |
| inttsk 10727 | The intersection of a coll... |
| inar1 10728 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omALT 10729 | Alternate proof of ~ r1om ... |
| rankcf 10730 | Any set must be at least a... |
| inatsk 10731 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omtsk 10732 | The set of hereditarily fi... |
| tskord 10733 | A Tarski class contains al... |
| tskcard 10734 | An even more direct relati... |
| r1tskina 10735 | There is a direct relation... |
| tskuni 10736 | The union of an element of... |
| tskwun 10737 | A nonempty transitive Tars... |
| tskint 10738 | The intersection of an ele... |
| tskun 10739 | The union of two elements ... |
| tskxp 10740 | The Cartesian product of t... |
| tskmap 10741 | Set exponentiation is an e... |
| tskurn 10742 | A transitive Tarski class ... |
| elgrug 10745 | Properties of a Grothendie... |
| grutr 10746 | A Grothendieck universe is... |
| gruelss 10747 | A Grothendieck universe is... |
| grupw 10748 | A Grothendieck universe co... |
| gruss 10749 | Any subset of an element o... |
| grupr 10750 | A Grothendieck universe co... |
| gruurn 10751 | A Grothendieck universe co... |
| gruiun 10752 | If ` B ( x ) ` is a family... |
| gruuni 10753 | A Grothendieck universe co... |
| grurn 10754 | A Grothendieck universe co... |
| gruima 10755 | A Grothendieck universe co... |
| gruel 10756 | Any element of an element ... |
| grusn 10757 | A Grothendieck universe co... |
| gruop 10758 | A Grothendieck universe co... |
| gruun 10759 | A Grothendieck universe co... |
| gruxp 10760 | A Grothendieck universe co... |
| grumap 10761 | A Grothendieck universe co... |
| gruixp 10762 | A Grothendieck universe co... |
| gruiin 10763 | A Grothendieck universe co... |
| gruf 10764 | A Grothendieck universe co... |
| gruen 10765 | A Grothendieck universe co... |
| gruwun 10766 | A nonempty Grothendieck un... |
| intgru 10767 | The intersection of a fami... |
| ingru 10768 | The intersection of a univ... |
| wfgru 10769 | The wellfounded part of a ... |
| grudomon 10770 | Each ordinal that is compa... |
| gruina 10771 | If a Grothendieck universe... |
| grur1a 10772 | A characterization of Grot... |
| grur1 10773 | A characterization of Grot... |
| grutsk1 10774 | Grothendieck universes are... |
| grutsk 10775 | Grothendieck universes are... |
| axgroth5 10777 | The Tarski-Grothendieck ax... |
| axgroth2 10778 | Alternate version of the T... |
| grothpw 10779 | Derive the Axiom of Power ... |
| grothpwex 10780 | Derive the Axiom of Power ... |
| axgroth6 10781 | The Tarski-Grothendieck ax... |
| grothomex 10782 | The Tarski-Grothendieck Ax... |
| grothac 10783 | The Tarski-Grothendieck Ax... |
| axgroth3 10784 | Alternate version of the T... |
| axgroth4 10785 | Alternate version of the T... |
| grothprimlem 10786 | Lemma for ~ grothprim . E... |
| grothprim 10787 | The Tarski-Grothendieck Ax... |
| grothtsk 10788 | The Tarski-Grothendieck Ax... |
| inaprc 10789 | An equivalent to the Tarsk... |
| tskmval 10792 | Value of our tarski map. ... |
| tskmid 10793 | The set ` A ` is an elemen... |
| tskmcl 10794 | A Tarski class that contai... |
| sstskm 10795 | Being a part of ` ( tarski... |
| eltskm 10796 | Belonging to ` ( tarskiMap... |
| elni 10829 | Membership in the class of... |
| elni2 10830 | Membership in the class of... |
| pinn 10831 | A positive integer is a na... |
| pion 10832 | A positive integer is an o... |
| piord 10833 | A positive integer is ordi... |
| niex 10834 | The class of positive inte... |
| 0npi 10835 | The empty set is not a pos... |
| 1pi 10836 | Ordinal 'one' is a positiv... |
| addpiord 10837 | Positive integer addition ... |
| mulpiord 10838 | Positive integer multiplic... |
| mulidpi 10839 | 1 is an identity element f... |
| ltpiord 10840 | Positive integer 'less tha... |
| ltsopi 10841 | Positive integer 'less tha... |
| ltrelpi 10842 | Positive integer 'less tha... |
| dmaddpi 10843 | Domain of addition on posi... |
| dmmulpi 10844 | Domain of multiplication o... |
| addclpi 10845 | Closure of addition of pos... |
| mulclpi 10846 | Closure of multiplication ... |
| addcompi 10847 | Addition of positive integ... |
| addasspi 10848 | Addition of positive integ... |
| mulcompi 10849 | Multiplication of positive... |
| mulasspi 10850 | Multiplication of positive... |
| distrpi 10851 | Multiplication of positive... |
| addcanpi 10852 | Addition cancellation law ... |
| mulcanpi 10853 | Multiplication cancellatio... |
| addnidpi 10854 | There is no identity eleme... |
| ltexpi 10855 | Ordering on positive integ... |
| ltapi 10856 | Ordering property of addit... |
| ltmpi 10857 | Ordering property of multi... |
| 1lt2pi 10858 | One is less than two (one ... |
| nlt1pi 10859 | No positive integer is les... |
| indpi 10860 | Principle of Finite Induct... |
| enqbreq 10872 | Equivalence relation for p... |
| enqbreq2 10873 | Equivalence relation for p... |
| enqer 10874 | The equivalence relation f... |
| enqex 10875 | The equivalence relation f... |
| nqex 10876 | The class of positive frac... |
| 0nnq 10877 | The empty set is not a pos... |
| elpqn 10878 | Each positive fraction is ... |
| ltrelnq 10879 | Positive fraction 'less th... |
| pinq 10880 | The representatives of pos... |
| 1nq 10881 | The positive fraction 'one... |
| nqereu 10882 | There is a unique element ... |
| nqerf 10883 | Corollary of ~ nqereu : th... |
| nqercl 10884 | Corollary of ~ nqereu : cl... |
| nqerrel 10885 | Any member of ` ( N. X. N.... |
| nqerid 10886 | Corollary of ~ nqereu : th... |
| enqeq 10887 | Corollary of ~ nqereu : if... |
| nqereq 10888 | The function ` /Q ` acts a... |
| addpipq2 10889 | Addition of positive fract... |
| addpipq 10890 | Addition of positive fract... |
| addpqnq 10891 | Addition of positive fract... |
| mulpipq2 10892 | Multiplication of positive... |
| mulpipq 10893 | Multiplication of positive... |
| mulpqnq 10894 | Multiplication of positive... |
| ordpipq 10895 | Ordering of positive fract... |
| ordpinq 10896 | Ordering of positive fract... |
| addpqf 10897 | Closure of addition on pos... |
| addclnq 10898 | Closure of addition on pos... |
| mulpqf 10899 | Closure of multiplication ... |
| mulclnq 10900 | Closure of multiplication ... |
| addnqf 10901 | Domain of addition on posi... |
| mulnqf 10902 | Domain of multiplication o... |
| addcompq 10903 | Addition of positive fract... |
| addcomnq 10904 | Addition of positive fract... |
| mulcompq 10905 | Multiplication of positive... |
| mulcomnq 10906 | Multiplication of positive... |
| adderpqlem 10907 | Lemma for ~ adderpq . (Co... |
| mulerpqlem 10908 | Lemma for ~ mulerpq . (Co... |
| adderpq 10909 | Addition is compatible wit... |
| mulerpq 10910 | Multiplication is compatib... |
| addassnq 10911 | Addition of positive fract... |
| mulassnq 10912 | Multiplication of positive... |
| mulcanenq 10913 | Lemma for distributive law... |
| distrnq 10914 | Multiplication of positive... |
| 1nqenq 10915 | The equivalence class of r... |
| mulidnq 10916 | Multiplication identity el... |
| recmulnq 10917 | Relationship between recip... |
| recidnq 10918 | A positive fraction times ... |
| recclnq 10919 | Closure law for positive f... |
| recrecnq 10920 | Reciprocal of reciprocal o... |
| dmrecnq 10921 | Domain of reciprocal on po... |
| ltsonq 10922 | 'Less than' is a strict or... |
| lterpq 10923 | Compatibility of ordering ... |
| ltanq 10924 | Ordering property of addit... |
| ltmnq 10925 | Ordering property of multi... |
| 1lt2nq 10926 | One is less than two (one ... |
| ltaddnq 10927 | The sum of two fractions i... |
| ltexnq 10928 | Ordering on positive fract... |
| halfnq 10929 | One-half of any positive f... |
| nsmallnq 10930 | The is no smallest positiv... |
| ltbtwnnq 10931 | There exists a number betw... |
| ltrnq 10932 | Ordering property of recip... |
| archnq 10933 | For any fraction, there is... |
| npex 10939 | The class of positive real... |
| elnp 10940 | Membership in positive rea... |
| elnpi 10941 | Membership in positive rea... |
| prn0 10942 | A positive real is not emp... |
| prpssnq 10943 | A positive real is a subse... |
| elprnq 10944 | A positive real is a set o... |
| 0npr 10945 | The empty set is not a pos... |
| prcdnq 10946 | A positive real is closed ... |
| prub 10947 | A positive fraction not in... |
| prnmax 10948 | A positive real has no lar... |
| npomex 10949 | A simplifying observation,... |
| prnmadd 10950 | A positive real has no lar... |
| ltrelpr 10951 | Positive real 'less than' ... |
| genpv 10952 | Value of general operation... |
| genpelv 10953 | Membership in value of gen... |
| genpprecl 10954 | Pre-closure law for genera... |
| genpdm 10955 | Domain of general operatio... |
| genpn0 10956 | The result of an operation... |
| genpss 10957 | The result of an operation... |
| genpnnp 10958 | The result of an operation... |
| genpcd 10959 | Downward closure of an ope... |
| genpnmax 10960 | An operation on positive r... |
| genpcl 10961 | Closure of an operation on... |
| genpass 10962 | Associativity of an operat... |
| plpv 10963 | Value of addition on posit... |
| mpv 10964 | Value of multiplication on... |
| dmplp 10965 | Domain of addition on posi... |
| dmmp 10966 | Domain of multiplication o... |
| nqpr 10967 | The canonical embedding of... |
| 1pr 10968 | The positive real number '... |
| addclprlem1 10969 | Lemma to prove downward cl... |
| addclprlem2 10970 | Lemma to prove downward cl... |
| addclpr 10971 | Closure of addition on pos... |
| mulclprlem 10972 | Lemma to prove downward cl... |
| mulclpr 10973 | Closure of multiplication ... |
| addcompr 10974 | Addition of positive reals... |
| addasspr 10975 | Addition of positive reals... |
| mulcompr 10976 | Multiplication of positive... |
| mulasspr 10977 | Multiplication of positive... |
| distrlem1pr 10978 | Lemma for distributive law... |
| distrlem4pr 10979 | Lemma for distributive law... |
| distrlem5pr 10980 | Lemma for distributive law... |
| distrpr 10981 | Multiplication of positive... |
| 1idpr 10982 | 1 is an identity element f... |
| ltprord 10983 | Positive real 'less than' ... |
| psslinpr 10984 | Proper subset is a linear ... |
| ltsopr 10985 | Positive real 'less than' ... |
| prlem934 10986 | Lemma 9-3.4 of [Gleason] p... |
| ltaddpr 10987 | The sum of two positive re... |
| ltaddpr2 10988 | The sum of two positive re... |
| ltexprlem1 10989 | Lemma for Proposition 9-3.... |
| ltexprlem2 10990 | Lemma for Proposition 9-3.... |
| ltexprlem3 10991 | Lemma for Proposition 9-3.... |
| ltexprlem4 10992 | Lemma for Proposition 9-3.... |
| ltexprlem5 10993 | Lemma for Proposition 9-3.... |
| ltexprlem6 10994 | Lemma for Proposition 9-3.... |
| ltexprlem7 10995 | Lemma for Proposition 9-3.... |
| ltexpri 10996 | Proposition 9-3.5(iv) of [... |
| ltaprlem 10997 | Lemma for Proposition 9-3.... |
| ltapr 10998 | Ordering property of addit... |
| addcanpr 10999 | Addition cancellation law ... |
| prlem936 11000 | Lemma 9-3.6 of [Gleason] p... |
| reclem2pr 11001 | Lemma for Proposition 9-3.... |
| reclem3pr 11002 | Lemma for Proposition 9-3.... |
| reclem4pr 11003 | Lemma for Proposition 9-3.... |
| recexpr 11004 | The reciprocal of a positi... |
| suplem1pr 11005 | The union of a nonempty, b... |
| suplem2pr 11006 | The union of a set of posi... |
| supexpr 11007 | The union of a nonempty, b... |
| enrer 11016 | The equivalence relation f... |
| nrex1 11017 | The class of signed reals ... |
| enrbreq 11018 | Equivalence relation for s... |
| enreceq 11019 | Equivalence class equality... |
| enrex 11020 | The equivalence relation f... |
| ltrelsr 11021 | Signed real 'less than' is... |
| addcmpblnr 11022 | Lemma showing compatibilit... |
| mulcmpblnrlem 11023 | Lemma used in lemma showin... |
| mulcmpblnr 11024 | Lemma showing compatibilit... |
| prsrlem1 11025 | Decomposing signed reals i... |
| addsrmo 11026 | There is at most one resul... |
| mulsrmo 11027 | There is at most one resul... |
| addsrpr 11028 | Addition of signed reals i... |
| mulsrpr 11029 | Multiplication of signed r... |
| ltsrpr 11030 | Ordering of signed reals i... |
| gt0srpr 11031 | Greater than zero in terms... |
| 0nsr 11032 | The empty set is not a sig... |
| 0r 11033 | The constant ` 0R ` is a s... |
| 1sr 11034 | The constant ` 1R ` is a s... |
| m1r 11035 | The constant ` -1R ` is a ... |
| addclsr 11036 | Closure of addition on sig... |
| mulclsr 11037 | Closure of multiplication ... |
| dmaddsr 11038 | Domain of addition on sign... |
| dmmulsr 11039 | Domain of multiplication o... |
| addcomsr 11040 | Addition of signed reals i... |
| addasssr 11041 | Addition of signed reals i... |
| mulcomsr 11042 | Multiplication of signed r... |
| mulasssr 11043 | Multiplication of signed r... |
| distrsr 11044 | Multiplication of signed r... |
| m1p1sr 11045 | Minus one plus one is zero... |
| m1m1sr 11046 | Minus one times minus one ... |
| ltsosr 11047 | Signed real 'less than' is... |
| 0lt1sr 11048 | 0 is less than 1 for signe... |
| 1ne0sr 11049 | 1 and 0 are distinct for s... |
| 0idsr 11050 | The signed real number 0 i... |
| 1idsr 11051 | 1 is an identity element f... |
| 00sr 11052 | A signed real times 0 is 0... |
| ltasr 11053 | Ordering property of addit... |
| pn0sr 11054 | A signed real plus its neg... |
| negexsr 11055 | Existence of negative sign... |
| recexsrlem 11056 | The reciprocal of a positi... |
| addgt0sr 11057 | The sum of two positive si... |
| mulgt0sr 11058 | The product of two positiv... |
| sqgt0sr 11059 | The square of a nonzero si... |
| recexsr 11060 | The reciprocal of a nonzer... |
| mappsrpr 11061 | Mapping from positive sign... |
| ltpsrpr 11062 | Mapping of order from posi... |
| map2psrpr 11063 | Equivalence for positive s... |
| supsrlem 11064 | Lemma for supremum theorem... |
| supsr 11065 | A nonempty, bounded set of... |
| opelcn 11082 | Ordered pair membership in... |
| opelreal 11083 | Ordered pair membership in... |
| elreal 11084 | Membership in class of rea... |
| elreal2 11085 | Ordered pair membership in... |
| 0ncn 11086 | The empty set is not a com... |
| ltrelre 11087 | 'Less than' is a relation ... |
| addcnsr 11088 | Addition of complex number... |
| mulcnsr 11089 | Multiplication of complex ... |
| eqresr 11090 | Equality of real numbers i... |
| addresr 11091 | Addition of real numbers i... |
| mulresr 11092 | Multiplication of real num... |
| ltresr 11093 | Ordering of real subset of... |
| ltresr2 11094 | Ordering of real subset of... |
| dfcnqs 11095 | Technical trick to permit ... |
| addcnsrec 11096 | Technical trick to permit ... |
| mulcnsrec 11097 | Technical trick to permit ... |
| axaddf 11098 | Addition is an operation o... |
| axmulf 11099 | Multiplication is an opera... |
| axcnex 11100 | The complex numbers form a... |
| axresscn 11101 | The real numbers are a sub... |
| ax1cn 11102 | 1 is a complex number. Ax... |
| axicn 11103 | ` _i ` is a complex number... |
| axaddcl 11104 | Closure law for addition o... |
| axaddrcl 11105 | Closure law for addition i... |
| axmulcl 11106 | Closure law for multiplica... |
| axmulrcl 11107 | Closure law for multiplica... |
| axmulcom 11108 | Multiplication of complex ... |
| axaddass 11109 | Addition of complex number... |
| axmulass 11110 | Multiplication of complex ... |
| axdistr 11111 | Distributive law for compl... |
| axi2m1 11112 | i-squared equals -1 (expre... |
| ax1ne0 11113 | 1 and 0 are distinct. Axi... |
| ax1rid 11114 | ` 1 ` is an identity eleme... |
| axrnegex 11115 | Existence of negative of r... |
| axrrecex 11116 | Existence of reciprocal of... |
| axcnre 11117 | A complex number can be ex... |
| axpre-lttri 11118 | Ordering on reals satisfie... |
| axpre-lttrn 11119 | Ordering on reals is trans... |
| axpre-ltadd 11120 | Ordering property of addit... |
| axpre-mulgt0 11121 | The product of two positiv... |
| axpre-sup 11122 | A nonempty, bounded-above ... |
| wuncn 11123 | A weak universe containing... |
| cnex 11149 | Alias for ~ ax-cnex . See... |
| addcl 11150 | Alias for ~ ax-addcl , for... |
| readdcl 11151 | Alias for ~ ax-addrcl , fo... |
| mulcl 11152 | Alias for ~ ax-mulcl , for... |
| remulcl 11153 | Alias for ~ ax-mulrcl , fo... |
| mulcom 11154 | Alias for ~ ax-mulcom , fo... |
| addass 11155 | Alias for ~ ax-addass , fo... |
| mulass 11156 | Alias for ~ ax-mulass , fo... |
| adddi 11157 | Alias for ~ ax-distr , for... |
| recn 11158 | A real number is a complex... |
| reex 11159 | The real numbers form a se... |
| reelprrecn 11160 | Reals are a subset of the ... |
| cnelprrecn 11161 | Complex numbers are a subs... |
| mpoaddf 11162 | Addition is an operation o... |
| mpomulf 11163 | Multiplication is an opera... |
| elimne0 11164 | Hypothesis for weak deduct... |
| adddir 11165 | Distributive law for compl... |
| 0cn 11166 | Zero is a complex number. ... |
| 0cnd 11167 | Zero is a complex number, ... |
| c0ex 11168 | Zero is a set. (Contribut... |
| 1cnd 11169 | One is a complex number, d... |
| 1ex 11170 | One is a set. (Contribute... |
| cnre 11171 | Alias for ~ ax-cnre , for ... |
| mulrid 11172 | The number 1 is an identit... |
| mullid 11173 | Identity law for multiplic... |
| 1re 11174 | The number 1 is real. Thi... |
| 1red 11175 | The number 1 is real, dedu... |
| 0re 11176 | The number 0 is real. Rem... |
| 0red 11177 | The number 0 is real, dedu... |
| mulridi 11178 | Identity law for multiplic... |
| mullidi 11179 | Identity law for multiplic... |
| addcli 11180 | Closure law for addition. ... |
| mulcli 11181 | Closure law for multiplica... |
| mulcomi 11182 | Commutative law for multip... |
| mulcomli 11183 | Commutative law for multip... |
| addassi 11184 | Associative law for additi... |
| mulassi 11185 | Associative law for multip... |
| adddii 11186 | Distributive law (left-dis... |
| adddiri 11187 | Distributive law (right-di... |
| recni 11188 | A real number is a complex... |
| readdcli 11189 | Closure law for addition o... |
| remulcli 11190 | Closure law for multiplica... |
| mulridd 11191 | Identity law for multiplic... |
| mullidd 11192 | Identity law for multiplic... |
| addcld 11193 | Closure law for addition. ... |
| mulcld 11194 | Closure law for multiplica... |
| mulcomd 11195 | Commutative law for multip... |
| addassd 11196 | Associative law for additi... |
| mulassd 11197 | Associative law for multip... |
| adddid 11198 | Distributive law (left-dis... |
| adddird 11199 | Distributive law (right-di... |
| adddirp1d 11200 | Distributive law, plus 1 v... |
| joinlmuladdmuld 11201 | Join AB+CB into (A+C) on L... |
| recnd 11202 | Deduction from real number... |
| readdcld 11203 | Closure law for addition o... |
| remulcld 11204 | Closure law for multiplica... |
| pnfnre 11215 | Plus infinity is not a rea... |
| pnfnre2 11216 | Plus infinity is not a rea... |
| mnfnre 11217 | Minus infinity is not a re... |
| ressxr 11218 | The standard reals are a s... |
| rexpssxrxp 11219 | The Cartesian product of s... |
| rexr 11220 | A standard real is an exte... |
| 0xr 11221 | Zero is an extended real. ... |
| renepnf 11222 | No (finite) real equals pl... |
| renemnf 11223 | No real equals minus infin... |
| rexrd 11224 | A standard real is an exte... |
| renepnfd 11225 | No (finite) real equals pl... |
| renemnfd 11226 | No real equals minus infin... |
| pnfex 11227 | Plus infinity exists. (Co... |
| pnfxr 11228 | Plus infinity belongs to t... |
| pnfnemnf 11229 | Plus and minus infinity ar... |
| mnfnepnf 11230 | Minus and plus infinity ar... |
| mnfxr 11231 | Minus infinity belongs to ... |
| rexri 11232 | A standard real is an exte... |
| 1xr 11233 | ` 1 ` is an extended real ... |
| renfdisj 11234 | The reals and the infiniti... |
| ltrelxr 11235 | "Less than" is a relation ... |
| ltrel 11236 | "Less than" is a relation.... |
| lerelxr 11237 | "Less than or equal to" is... |
| lerel 11238 | "Less than or equal to" is... |
| xrlenlt 11239 | "Less than or equal to" ex... |
| xrlenltd 11240 | "Less than or equal to" ex... |
| xrltnle 11241 | "Less than" expressed in t... |
| xrnltled 11242 | "Not less than" implies "l... |
| ssxr 11243 | The three (non-exclusive) ... |
| ltxrlt 11244 | The standard less-than ` <... |
| axlttri 11245 | Ordering on reals satisfie... |
| axlttrn 11246 | Ordering on reals is trans... |
| axltadd 11247 | Ordering property of addit... |
| axmulgt0 11248 | The product of two positiv... |
| axsup 11249 | A nonempty, bounded-above ... |
| lttr 11250 | Alias for ~ axlttrn , for ... |
| mulgt0 11251 | The product of two positiv... |
| lenlt 11252 | 'Less than or equal to' ex... |
| ltnle 11253 | 'Less than' expressed in t... |
| ltso 11254 | 'Less than' is a strict or... |
| gtso 11255 | 'Greater than' is a strict... |
| lttri2 11256 | Consequence of trichotomy.... |
| lttri3 11257 | Trichotomy law for 'less t... |
| lttri4 11258 | Trichotomy law for 'less t... |
| letri3 11259 | Trichotomy law. (Contribu... |
| leloe 11260 | 'Less than or equal to' ex... |
| eqlelt 11261 | Equality in terms of 'less... |
| ltle 11262 | 'Less than' implies 'less ... |
| leltne 11263 | 'Less than or equal to' im... |
| lelttr 11264 | Transitive law. (Contribu... |
| leltletr 11265 | Transitive law, weaker for... |
| ltletr 11266 | Transitive law. (Contribu... |
| ltleletr 11267 | Transitive law, weaker for... |
| letr 11268 | Transitive law. (Contribu... |
| ltnr 11269 | 'Less than' is irreflexive... |
| leid 11270 | 'Less than or equal to' is... |
| ltne 11271 | 'Less than' implies not eq... |
| ltnsym 11272 | 'Less than' is not symmetr... |
| ltnsym2 11273 | 'Less than' is antisymmetr... |
| letric 11274 | Trichotomy law. (Contribu... |
| ltlen 11275 | 'Less than' expressed in t... |
| eqle 11276 | Equality implies 'less tha... |
| eqled 11277 | Equality implies 'less tha... |
| ltadd2 11278 | Addition to both sides of ... |
| ne0gt0 11279 | A nonzero nonnegative numb... |
| lecasei 11280 | Ordering elimination by ca... |
| lelttric 11281 | Trichotomy law. (Contribu... |
| ltlecasei 11282 | Ordering elimination by ca... |
| ltnri 11283 | 'Less than' is irreflexive... |
| eqlei 11284 | Equality implies 'less tha... |
| eqlei2 11285 | Equality implies 'less tha... |
| gtneii 11286 | 'Less than' implies not eq... |
| ltneii 11287 | 'Greater than' implies not... |
| lttri2i 11288 | Consequence of trichotomy.... |
| lttri3i 11289 | Consequence of trichotomy.... |
| letri3i 11290 | Consequence of trichotomy.... |
| leloei 11291 | 'Less than or equal to' in... |
| ltleni 11292 | 'Less than' expressed in t... |
| ltnsymi 11293 | 'Less than' is not symmetr... |
| lenlti 11294 | 'Less than or equal to' in... |
| ltnlei 11295 | 'Less than' in terms of 'l... |
| ltlei 11296 | 'Less than' implies 'less ... |
| ltleii 11297 | 'Less than' implies 'less ... |
| ltnei 11298 | 'Less than' implies not eq... |
| letrii 11299 | Trichotomy law for 'less t... |
| lttri 11300 | 'Less than' is transitive.... |
| lelttri 11301 | 'Less than or equal to', '... |
| ltletri 11302 | 'Less than', 'less than or... |
| letri 11303 | 'Less than or equal to' is... |
| le2tri3i 11304 | Extended trichotomy law fo... |
| ltadd2i 11305 | Addition to both sides of ... |
| mulgt0i 11306 | The product of two positiv... |
| mulgt0ii 11307 | The product of two positiv... |
| ltnrd 11308 | 'Less than' is irreflexive... |
| gtned 11309 | 'Less than' implies not eq... |
| ltned 11310 | 'Greater than' implies not... |
| ne0gt0d 11311 | A nonzero nonnegative numb... |
| lttrid 11312 | Ordering on reals satisfie... |
| lttri2d 11313 | Consequence of trichotomy.... |
| lttri3d 11314 | Consequence of trichotomy.... |
| lttri4d 11315 | Trichotomy law for 'less t... |
| letri3d 11316 | Consequence of trichotomy.... |
| leloed 11317 | 'Less than or equal to' in... |
| eqleltd 11318 | Equality in terms of 'less... |
| ltlend 11319 | 'Less than' expressed in t... |
| lenltd 11320 | 'Less than or equal to' in... |
| ltnled 11321 | 'Less than' in terms of 'l... |
| ltled 11322 | 'Less than' implies 'less ... |
| ltnsymd 11323 | 'Less than' implies 'less ... |
| nltled 11324 | 'Not less than ' implies '... |
| lensymd 11325 | 'Less than or equal to' im... |
| letrid 11326 | Trichotomy law for 'less t... |
| leltned 11327 | 'Less than or equal to' im... |
| leneltd 11328 | 'Less than or equal to' an... |
| mulgt0d 11329 | The product of two positiv... |
| ltadd2d 11330 | Addition to both sides of ... |
| letrd 11331 | Transitive law deduction f... |
| lelttrd 11332 | Transitive law deduction f... |
| ltadd2dd 11333 | Addition to both sides of ... |
| ltletrd 11334 | Transitive law deduction f... |
| lttrd 11335 | Transitive law deduction f... |
| lelttrdi 11336 | If a number is less than a... |
| dedekind 11337 | The Dedekind cut theorem. ... |
| dedekindle 11338 | The Dedekind cut theorem, ... |
| mul12 11339 | Commutative/associative la... |
| mul32 11340 | Commutative/associative la... |
| mul31 11341 | Commutative/associative la... |
| mul4 11342 | Rearrangement of 4 factors... |
| mul4r 11343 | Rearrangement of 4 factors... |
| muladd11 11344 | A simple product of sums e... |
| 1p1times 11345 | Two times a number. (Cont... |
| peano2cn 11346 | A theorem for complex numb... |
| peano2re 11347 | A theorem for reals analog... |
| readdcan 11348 | Cancellation law for addit... |
| 00id 11349 | ` 0 ` is its own additive ... |
| mul02lem1 11350 | Lemma for ~ mul02 . If an... |
| mul02lem2 11351 | Lemma for ~ mul02 . Zero ... |
| mul02 11352 | Multiplication by ` 0 ` . ... |
| mul01 11353 | Multiplication by ` 0 ` . ... |
| addrid 11354 | ` 0 ` is an additive ident... |
| cnegex 11355 | Existence of the negative ... |
| cnegex2 11356 | Existence of a left invers... |
| addlid 11357 | ` 0 ` is a left identity f... |
| addcan 11358 | Cancellation law for addit... |
| addcan2 11359 | Cancellation law for addit... |
| addcom 11360 | Addition commutes. This u... |
| addridi 11361 | ` 0 ` is an additive ident... |
| addlidi 11362 | ` 0 ` is a left identity f... |
| mul02i 11363 | Multiplication by 0. Theo... |
| mul01i 11364 | Multiplication by ` 0 ` . ... |
| addcomi 11365 | Addition commutes. Based ... |
| addcomli 11366 | Addition commutes. (Contr... |
| addcani 11367 | Cancellation law for addit... |
| addcan2i 11368 | Cancellation law for addit... |
| mul12i 11369 | Commutative/associative la... |
| mul32i 11370 | Commutative/associative la... |
| mul4i 11371 | Rearrangement of 4 factors... |
| mul02d 11372 | Multiplication by 0. Theo... |
| mul01d 11373 | Multiplication by ` 0 ` . ... |
| addridd 11374 | ` 0 ` is an additive ident... |
| addlidd 11375 | ` 0 ` is a left identity f... |
| addcomd 11376 | Addition commutes. Based ... |
| addcand 11377 | Cancellation law for addit... |
| addcan2d 11378 | Cancellation law for addit... |
| addcanad 11379 | Cancelling a term on the l... |
| addcan2ad 11380 | Cancelling a term on the r... |
| addneintrd 11381 | Introducing a term on the ... |
| addneintr2d 11382 | Introducing a term on the ... |
| mul12d 11383 | Commutative/associative la... |
| mul32d 11384 | Commutative/associative la... |
| mul31d 11385 | Commutative/associative la... |
| mul4d 11386 | Rearrangement of 4 factors... |
| muladd11r 11387 | A simple product of sums e... |
| comraddd 11388 | Commute RHS addition, in d... |
| comraddi 11389 | Commute RHS addition. See... |
| ltaddneg 11390 | Adding a negative number t... |
| ltaddnegr 11391 | Adding a negative number t... |
| add12 11392 | Commutative/associative la... |
| add32 11393 | Commutative/associative la... |
| add32r 11394 | Commutative/associative la... |
| add4 11395 | Rearrangement of 4 terms i... |
| add42 11396 | Rearrangement of 4 terms i... |
| add12i 11397 | Commutative/associative la... |
| add32i 11398 | Commutative/associative la... |
| add4i 11399 | Rearrangement of 4 terms i... |
| add42i 11400 | Rearrangement of 4 terms i... |
| add12d 11401 | Commutative/associative la... |
| add32d 11402 | Commutative/associative la... |
| add4d 11403 | Rearrangement of 4 terms i... |
| add42d 11404 | Rearrangement of 4 terms i... |
| 0cnALT 11409 | Alternate proof of ~ 0cn w... |
| 0cnALT2 11410 | Alternate proof of ~ 0cnAL... |
| negeu 11411 | Existential uniqueness of ... |
| subval 11412 | Value of subtraction, whic... |
| negeq 11413 | Equality theorem for negat... |
| negeqi 11414 | Equality inference for neg... |
| negeqd 11415 | Equality deduction for neg... |
| nfnegd 11416 | Deduction version of ~ nfn... |
| nfneg 11417 | Bound-variable hypothesis ... |
| csbnegg 11418 | Move class substitution in... |
| negex 11419 | A negative is a set. (Con... |
| subcl 11420 | Closure law for subtractio... |
| negcl 11421 | Closure law for negative. ... |
| negicn 11422 | ` -u _i ` is a complex num... |
| subf 11423 | Subtraction is an operatio... |
| subadd 11424 | Relationship between subtr... |
| subadd2 11425 | Relationship between subtr... |
| subsub23 11426 | Swap subtrahend and result... |
| pncan 11427 | Cancellation law for subtr... |
| pncan2 11428 | Cancellation law for subtr... |
| pncan3 11429 | Subtraction and addition o... |
| npcan 11430 | Cancellation law for subtr... |
| addsubass 11431 | Associative-type law for a... |
| addsub 11432 | Law for addition and subtr... |
| subadd23 11433 | Commutative/associative la... |
| addsub12 11434 | Commutative/associative la... |
| 2addsub 11435 | Law for subtraction and ad... |
| addsubeq4 11436 | Relation between sums and ... |
| pncan3oi 11437 | Subtraction and addition o... |
| mvrraddi 11438 | Move the right term in a s... |
| mvrladdi 11439 | Move the left term in a su... |
| mvlladdi 11440 | Move the left term in a su... |
| subid 11441 | Subtraction of a number fr... |
| subid1 11442 | Identity law for subtracti... |
| npncan 11443 | Cancellation law for subtr... |
| nppcan 11444 | Cancellation law for subtr... |
| nnpcan 11445 | Cancellation law for subtr... |
| nppcan3 11446 | Cancellation law for subtr... |
| subcan2 11447 | Cancellation law for subtr... |
| subeq0 11448 | If the difference between ... |
| npncan2 11449 | Cancellation law for subtr... |
| subsub2 11450 | Law for double subtraction... |
| nncan 11451 | Cancellation law for subtr... |
| subsub 11452 | Law for double subtraction... |
| nppcan2 11453 | Cancellation law for subtr... |
| subsub3 11454 | Law for double subtraction... |
| subsub4 11455 | Law for double subtraction... |
| sub32 11456 | Swap the second and third ... |
| nnncan 11457 | Cancellation law for subtr... |
| nnncan1 11458 | Cancellation law for subtr... |
| nnncan2 11459 | Cancellation law for subtr... |
| npncan3 11460 | Cancellation law for subtr... |
| pnpcan 11461 | Cancellation law for mixed... |
| pnpcan2 11462 | Cancellation law for mixed... |
| pnncan 11463 | Cancellation law for mixed... |
| ppncan 11464 | Cancellation law for mixed... |
| addsub4 11465 | Rearrangement of 4 terms i... |
| subadd4 11466 | Rearrangement of 4 terms i... |
| sub4 11467 | Rearrangement of 4 terms i... |
| neg0 11468 | Minus 0 equals 0. (Contri... |
| negid 11469 | Addition of a number and i... |
| negsub 11470 | Relationship between subtr... |
| subneg 11471 | Relationship between subtr... |
| negneg 11472 | A number is equal to the n... |
| neg11 11473 | Negative is one-to-one. (... |
| negcon1 11474 | Negative contraposition la... |
| negcon2 11475 | Negative contraposition la... |
| negeq0 11476 | A number is zero iff its n... |
| subcan 11477 | Cancellation law for subtr... |
| negsubdi 11478 | Distribution of negative o... |
| negdi 11479 | Distribution of negative o... |
| negdi2 11480 | Distribution of negative o... |
| negsubdi2 11481 | Distribution of negative o... |
| neg2sub 11482 | Relationship between subtr... |
| renegcli 11483 | Closure law for negative o... |
| resubcli 11484 | Closure law for subtractio... |
| renegcl 11485 | Closure law for negative o... |
| resubcl 11486 | Closure law for subtractio... |
| negreb 11487 | The negative of a real is ... |
| peano2cnm 11488 | "Reverse" second Peano pos... |
| peano2rem 11489 | "Reverse" second Peano pos... |
| negcli 11490 | Closure law for negative. ... |
| negidi 11491 | Addition of a number and i... |
| negnegi 11492 | A number is equal to the n... |
| subidi 11493 | Subtraction of a number fr... |
| subid1i 11494 | Identity law for subtracti... |
| negne0bi 11495 | A number is nonzero iff it... |
| negrebi 11496 | The negative of a real is ... |
| negne0i 11497 | The negative of a nonzero ... |
| subcli 11498 | Closure law for subtractio... |
| pncan3i 11499 | Subtraction and addition o... |
| negsubi 11500 | Relationship between subtr... |
| subnegi 11501 | Relationship between subtr... |
| subeq0i 11502 | If the difference between ... |
| neg11i 11503 | Negative is one-to-one. (... |
| negcon1i 11504 | Negative contraposition la... |
| negcon2i 11505 | Negative contraposition la... |
| negdii 11506 | Distribution of negative o... |
| negsubdii 11507 | Distribution of negative o... |
| negsubdi2i 11508 | Distribution of negative o... |
| subaddi 11509 | Relationship between subtr... |
| subadd2i 11510 | Relationship between subtr... |
| subaddrii 11511 | Relationship between subtr... |
| subsub23i 11512 | Swap subtrahend and result... |
| addsubassi 11513 | Associative-type law for s... |
| addsubi 11514 | Law for subtraction and ad... |
| subcani 11515 | Cancellation law for subtr... |
| subcan2i 11516 | Cancellation law for subtr... |
| pnncani 11517 | Cancellation law for mixed... |
| addsub4i 11518 | Rearrangement of 4 terms i... |
| 0reALT 11519 | Alternate proof of ~ 0re .... |
| negcld 11520 | Closure law for negative. ... |
| subidd 11521 | Subtraction of a number fr... |
| subid1d 11522 | Identity law for subtracti... |
| negidd 11523 | Addition of a number and i... |
| negnegd 11524 | A number is equal to the n... |
| negeq0d 11525 | A number is zero iff its n... |
| negne0bd 11526 | A number is nonzero iff it... |
| negcon1d 11527 | Contraposition law for una... |
| negcon1ad 11528 | Contraposition law for una... |
| neg11ad 11529 | The negatives of two compl... |
| negned 11530 | If two complex numbers are... |
| negne0d 11531 | The negative of a nonzero ... |
| negrebd 11532 | The negative of a real is ... |
| subcld 11533 | Closure law for subtractio... |
| pncand 11534 | Cancellation law for subtr... |
| pncan2d 11535 | Cancellation law for subtr... |
| pncan3d 11536 | Subtraction and addition o... |
| npcand 11537 | Cancellation law for subtr... |
| nncand 11538 | Cancellation law for subtr... |
| negsubd 11539 | Relationship between subtr... |
| subnegd 11540 | Relationship between subtr... |
| subeq0d 11541 | If the difference between ... |
| subne0d 11542 | Two unequal numbers have n... |
| subeq0ad 11543 | The difference of two comp... |
| subne0ad 11544 | If the difference of two c... |
| neg11d 11545 | If the difference between ... |
| negdid 11546 | Distribution of negative o... |
| negdi2d 11547 | Distribution of negative o... |
| negsubdid 11548 | Distribution of negative o... |
| negsubdi2d 11549 | Distribution of negative o... |
| neg2subd 11550 | Relationship between subtr... |
| subaddd 11551 | Relationship between subtr... |
| subadd2d 11552 | Relationship between subtr... |
| addsubassd 11553 | Associative-type law for s... |
| addsubd 11554 | Law for subtraction and ad... |
| subadd23d 11555 | Commutative/associative la... |
| addsub12d 11556 | Commutative/associative la... |
| npncand 11557 | Cancellation law for subtr... |
| nppcand 11558 | Cancellation law for subtr... |
| nppcan2d 11559 | Cancellation law for subtr... |
| nppcan3d 11560 | Cancellation law for subtr... |
| subsubd 11561 | Law for double subtraction... |
| subsub2d 11562 | Law for double subtraction... |
| subsub3d 11563 | Law for double subtraction... |
| subsub4d 11564 | Law for double subtraction... |
| sub32d 11565 | Swap the second and third ... |
| nnncand 11566 | Cancellation law for subtr... |
| nnncan1d 11567 | Cancellation law for subtr... |
| nnncan2d 11568 | Cancellation law for subtr... |
| npncan3d 11569 | Cancellation law for subtr... |
| pnpcand 11570 | Cancellation law for mixed... |
| pnpcan2d 11571 | Cancellation law for mixed... |
| pnncand 11572 | Cancellation law for mixed... |
| ppncand 11573 | Cancellation law for mixed... |
| subcand 11574 | Cancellation law for subtr... |
| subcan2d 11575 | Cancellation law for subtr... |
| subcanad 11576 | Cancellation law for subtr... |
| subneintrd 11577 | Introducing subtraction on... |
| subcan2ad 11578 | Cancellation law for subtr... |
| subneintr2d 11579 | Introducing subtraction on... |
| addsub4d 11580 | Rearrangement of 4 terms i... |
| subadd4d 11581 | Rearrangement of 4 terms i... |
| sub4d 11582 | Rearrangement of 4 terms i... |
| 2addsubd 11583 | Law for subtraction and ad... |
| addsubeq4d 11584 | Relation between sums and ... |
| subsubadd23 11585 | Swap the second and the th... |
| addsubsub23 11586 | Swap the second and the th... |
| subeqxfrd 11587 | Transfer two terms of a su... |
| mvlraddd 11588 | Move the right term in a s... |
| mvlladdd 11589 | Move the left term in a su... |
| mvrraddd 11590 | Move the right term in a s... |
| mvrladdd 11591 | Move the left term in a su... |
| assraddsubd 11592 | Associate RHS addition-sub... |
| subaddeqd 11593 | Transfer two terms of a su... |
| addlsub 11594 | Left-subtraction: Subtrac... |
| addrsub 11595 | Right-subtraction: Subtra... |
| subexsub 11596 | A subtraction law: Exchan... |
| addid0 11597 | If adding a number to a an... |
| addn0nid 11598 | Adding a nonzero number to... |
| pnpncand 11599 | Addition/subtraction cance... |
| subeqrev 11600 | Reverse the order of subtr... |
| addeq0 11601 | Two complex numbers add up... |
| pncan1 11602 | Cancellation law for addit... |
| npcan1 11603 | Cancellation law for subtr... |
| subeq0bd 11604 | If two complex numbers are... |
| renegcld 11605 | Closure law for negative o... |
| resubcld 11606 | Closure law for subtractio... |
| negn0 11607 | The image under negation o... |
| negf1o 11608 | Negation is an isomorphism... |
| kcnktkm1cn 11609 | k times k minus 1 is a com... |
| muladd 11610 | Product of two sums. (Con... |
| subdi 11611 | Distribution of multiplica... |
| subdir 11612 | Distribution of multiplica... |
| ine0 11613 | The imaginary unit ` _i ` ... |
| mulneg1 11614 | Product with negative is n... |
| mulneg2 11615 | The product with a negativ... |
| mulneg12 11616 | Swap the negative sign in ... |
| mul2neg 11617 | Product of two negatives. ... |
| submul2 11618 | Convert a subtraction to a... |
| mulm1 11619 | Product with minus one is ... |
| addneg1mul 11620 | Addition with product with... |
| mulsub 11621 | Product of two differences... |
| mulsub2 11622 | Swap the order of subtract... |
| mulm1i 11623 | Product with minus one is ... |
| mulneg1i 11624 | Product with negative is n... |
| mulneg2i 11625 | Product with negative is n... |
| mul2negi 11626 | Product of two negatives. ... |
| subdii 11627 | Distribution of multiplica... |
| subdiri 11628 | Distribution of multiplica... |
| muladdi 11629 | Product of two sums. (Con... |
| mulm1d 11630 | Product with minus one is ... |
| mulneg1d 11631 | Product with negative is n... |
| mulneg2d 11632 | Product with negative is n... |
| mul2negd 11633 | Product of two negatives. ... |
| subdid 11634 | Distribution of multiplica... |
| subdird 11635 | Distribution of multiplica... |
| muladdd 11636 | Product of two sums. (Con... |
| mulsubd 11637 | Product of two differences... |
| muls1d 11638 | Multiplication by one minu... |
| mulsubfacd 11639 | Multiplication followed by... |
| addmulsub 11640 | The product of a sum and a... |
| subaddmulsub 11641 | The difference with a prod... |
| mulsubaddmulsub 11642 | A special difference of a ... |
| gt0ne0 11643 | Positive implies nonzero. ... |
| lt0ne0 11644 | A number which is less tha... |
| ltadd1 11645 | Addition to both sides of ... |
| leadd1 11646 | Addition to both sides of ... |
| leadd2 11647 | Addition to both sides of ... |
| ltsubadd 11648 | 'Less than' relationship b... |
| ltsubadd2 11649 | 'Less than' relationship b... |
| lesubadd 11650 | 'Less than or equal to' re... |
| lesubadd2 11651 | 'Less than or equal to' re... |
| ltaddsub 11652 | 'Less than' relationship b... |
| ltaddsub2 11653 | 'Less than' relationship b... |
| leaddsub 11654 | 'Less than or equal to' re... |
| leaddsub2 11655 | 'Less than or equal to' re... |
| suble 11656 | Swap subtrahends in an ine... |
| lesub 11657 | Swap subtrahends in an ine... |
| ltsub23 11658 | 'Less than' relationship b... |
| ltsub13 11659 | 'Less than' relationship b... |
| le2add 11660 | Adding both sides of two '... |
| ltleadd 11661 | Adding both sides of two o... |
| leltadd 11662 | Adding both sides of two o... |
| lt2add 11663 | Adding both sides of two '... |
| addgt0 11664 | The sum of 2 positive numb... |
| addgegt0 11665 | The sum of nonnegative and... |
| addgtge0 11666 | The sum of nonnegative and... |
| addge0 11667 | The sum of 2 nonnegative n... |
| ltaddpos 11668 | Adding a positive number t... |
| ltaddpos2 11669 | Adding a positive number t... |
| ltsubpos 11670 | Subtracting a positive num... |
| posdif 11671 | Comparison of two numbers ... |
| lesub1 11672 | Subtraction from both side... |
| lesub2 11673 | Subtraction of both sides ... |
| ltsub1 11674 | Subtraction from both side... |
| ltsub2 11675 | Subtraction of both sides ... |
| lt2sub 11676 | Subtracting both sides of ... |
| le2sub 11677 | Subtracting both sides of ... |
| ltneg 11678 | Negative of both sides of ... |
| ltnegcon1 11679 | Contraposition of negative... |
| ltnegcon2 11680 | Contraposition of negative... |
| leneg 11681 | Negative of both sides of ... |
| lenegcon1 11682 | Contraposition of negative... |
| lenegcon2 11683 | Contraposition of negative... |
| lt0neg1 11684 | Comparison of a number and... |
| lt0neg2 11685 | Comparison of a number and... |
| le0neg1 11686 | Comparison of a number and... |
| le0neg2 11687 | Comparison of a number and... |
| addge01 11688 | A number is less than or e... |
| addge02 11689 | A number is less than or e... |
| add20 11690 | Two nonnegative numbers ar... |
| subge0 11691 | Nonnegative subtraction. ... |
| suble0 11692 | Nonpositive subtraction. ... |
| leaddle0 11693 | The sum of a real number a... |
| subge02 11694 | Nonnegative subtraction. ... |
| lesub0 11695 | Lemma to show a nonnegativ... |
| mulge0 11696 | The product of two nonnega... |
| mullt0 11697 | The product of two negativ... |
| msqgt0 11698 | A nonzero square is positi... |
| msqge0 11699 | A square is nonnegative. ... |
| 0lt1 11700 | 0 is less than 1. Theorem... |
| 0le1 11701 | 0 is less than or equal to... |
| relin01 11702 | An interval law for less t... |
| ltordlem 11703 | Lemma for ~ ltord1 . (Con... |
| ltord1 11704 | Infer an ordering relation... |
| leord1 11705 | Infer an ordering relation... |
| eqord1 11706 | A strictly increasing real... |
| ltord2 11707 | Infer an ordering relation... |
| leord2 11708 | Infer an ordering relation... |
| eqord2 11709 | A strictly decreasing real... |
| wloglei 11710 | Form of ~ wlogle where bot... |
| wlogle 11711 | If the predicate ` ch ( x ... |
| leidi 11712 | 'Less than or equal to' is... |
| gt0ne0i 11713 | Positive means nonzero (us... |
| gt0ne0ii 11714 | Positive implies nonzero. ... |
| msqgt0i 11715 | A nonzero square is positi... |
| msqge0i 11716 | A square is nonnegative. ... |
| addgt0i 11717 | Addition of 2 positive num... |
| addge0i 11718 | Addition of 2 nonnegative ... |
| addgegt0i 11719 | Addition of nonnegative an... |
| addgt0ii 11720 | Addition of 2 positive num... |
| add20i 11721 | Two nonnegative numbers ar... |
| ltnegi 11722 | Negative of both sides of ... |
| lenegi 11723 | Negative of both sides of ... |
| ltnegcon2i 11724 | Contraposition of negative... |
| mulge0i 11725 | The product of two nonnega... |
| lesub0i 11726 | Lemma to show a nonnegativ... |
| ltaddposi 11727 | Adding a positive number t... |
| posdifi 11728 | Comparison of two numbers ... |
| ltnegcon1i 11729 | Contraposition of negative... |
| lenegcon1i 11730 | Contraposition of negative... |
| subge0i 11731 | Nonnegative subtraction. ... |
| ltadd1i 11732 | Addition to both sides of ... |
| leadd1i 11733 | Addition to both sides of ... |
| leadd2i 11734 | Addition to both sides of ... |
| ltsubaddi 11735 | 'Less than' relationship b... |
| lesubaddi 11736 | 'Less than or equal to' re... |
| ltsubadd2i 11737 | 'Less than' relationship b... |
| lesubadd2i 11738 | 'Less than or equal to' re... |
| ltaddsubi 11739 | 'Less than' relationship b... |
| lt2addi 11740 | Adding both side of two in... |
| le2addi 11741 | Adding both side of two in... |
| gt0ne0d 11742 | Positive implies nonzero. ... |
| lt0ne0d 11743 | Something less than zero i... |
| leidd 11744 | 'Less than or equal to' is... |
| msqgt0d 11745 | A nonzero square is positi... |
| msqge0d 11746 | A square is nonnegative. ... |
| lt0neg1d 11747 | Comparison of a number and... |
| lt0neg2d 11748 | Comparison of a number and... |
| le0neg1d 11749 | Comparison of a number and... |
| le0neg2d 11750 | Comparison of a number and... |
| addgegt0d 11751 | Addition of nonnegative an... |
| addgtge0d 11752 | Addition of positive and n... |
| addgt0d 11753 | Addition of 2 positive num... |
| addge0d 11754 | Addition of 2 nonnegative ... |
| mulge0d 11755 | The product of two nonnega... |
| ltnegd 11756 | Negative of both sides of ... |
| lenegd 11757 | Negative of both sides of ... |
| ltnegcon1d 11758 | Contraposition of negative... |
| ltnegcon2d 11759 | Contraposition of negative... |
| lenegcon1d 11760 | Contraposition of negative... |
| lenegcon2d 11761 | Contraposition of negative... |
| ltaddposd 11762 | Adding a positive number t... |
| ltaddpos2d 11763 | Adding a positive number t... |
| ltsubposd 11764 | Subtracting a positive num... |
| posdifd 11765 | Comparison of two numbers ... |
| addge01d 11766 | A number is less than or e... |
| addge02d 11767 | A number is less than or e... |
| subge0d 11768 | Nonnegative subtraction. ... |
| suble0d 11769 | Nonpositive subtraction. ... |
| subge02d 11770 | Nonnegative subtraction. ... |
| ltadd1d 11771 | Addition to both sides of ... |
| leadd1d 11772 | Addition to both sides of ... |
| leadd2d 11773 | Addition to both sides of ... |
| ltsubaddd 11774 | 'Less than' relationship b... |
| lesubaddd 11775 | 'Less than or equal to' re... |
| ltsubadd2d 11776 | 'Less than' relationship b... |
| lesubadd2d 11777 | 'Less than or equal to' re... |
| ltaddsubd 11778 | 'Less than' relationship b... |
| ltaddsub2d 11779 | 'Less than' relationship b... |
| leaddsub2d 11780 | 'Less than or equal to' re... |
| subled 11781 | Swap subtrahends in an ine... |
| lesubd 11782 | Swap subtrahends in an ine... |
| ltsub23d 11783 | 'Less than' relationship b... |
| ltsub13d 11784 | 'Less than' relationship b... |
| lesub1d 11785 | Subtraction from both side... |
| lesub2d 11786 | Subtraction of both sides ... |
| ltsub1d 11787 | Subtraction from both side... |
| ltsub2d 11788 | Subtraction of both sides ... |
| ltadd1dd 11789 | Addition to both sides of ... |
| ltsub1dd 11790 | Subtraction from both side... |
| ltsub2dd 11791 | Subtraction of both sides ... |
| leadd1dd 11792 | Addition to both sides of ... |
| leadd2dd 11793 | Addition to both sides of ... |
| lesub1dd 11794 | Subtraction from both side... |
| lesub2dd 11795 | Subtraction of both sides ... |
| lesub3d 11796 | The result of subtracting ... |
| le2addd 11797 | Adding both side of two in... |
| le2subd 11798 | Subtracting both sides of ... |
| ltleaddd 11799 | Adding both sides of two o... |
| leltaddd 11800 | Adding both sides of two o... |
| lt2addd 11801 | Adding both side of two in... |
| lt2subd 11802 | Subtracting both sides of ... |
| possumd 11803 | Condition for a positive s... |
| sublt0d 11804 | When a subtraction gives a... |
| ltaddsublt 11805 | Addition and subtraction o... |
| 1le1 11806 | One is less than or equal ... |
| ixi 11807 | ` _i ` times itself is min... |
| recextlem1 11808 | Lemma for ~ recex . (Cont... |
| recextlem2 11809 | Lemma for ~ recex . (Cont... |
| recex 11810 | Existence of reciprocal of... |
| mulcand 11811 | Cancellation law for multi... |
| mulcan2d 11812 | Cancellation law for multi... |
| mulcanad 11813 | Cancellation of a nonzero ... |
| mulcan2ad 11814 | Cancellation of a nonzero ... |
| mulcan 11815 | Cancellation law for multi... |
| mulcan2 11816 | Cancellation law for multi... |
| mulcani 11817 | Cancellation law for multi... |
| mul0or 11818 | If a product is zero, one ... |
| mulne0b 11819 | The product of two nonzero... |
| mulne0 11820 | The product of two nonzero... |
| mulne0i 11821 | The product of two nonzero... |
| muleqadd 11822 | Property of numbers whose ... |
| receu 11823 | Existential uniqueness of ... |
| mulnzcnf 11824 | Multiplication maps nonzer... |
| mul0ori 11825 | If a product is zero, one ... |
| mul0ord 11826 | If a product is zero, one ... |
| msq0i 11827 | A number is zero iff its s... |
| msq0d 11828 | A number is zero iff its s... |
| mulne0bd 11829 | The product of two nonzero... |
| mulne0d 11830 | The product of two nonzero... |
| mulcan1g 11831 | A generalized form of the ... |
| mulcan2g 11832 | A generalized form of the ... |
| mulne0bad 11833 | A factor of a nonzero comp... |
| mulne0bbd 11834 | A factor of a nonzero comp... |
| 1div0 11837 | You can't divide by zero, ... |
| 1div0OLD 11838 | Obsolete version of ~ 1div... |
| divval 11839 | Value of division: if ` A ... |
| divmul 11840 | Relationship between divis... |
| divmul2 11841 | Relationship between divis... |
| divmul3 11842 | Relationship between divis... |
| divcl 11843 | Closure law for division. ... |
| reccl 11844 | Closure law for reciprocal... |
| divcan2 11845 | A cancellation law for div... |
| divcan1 11846 | A cancellation law for div... |
| diveq0 11847 | A ratio is zero iff the nu... |
| divne0b 11848 | The ratio of nonzero numbe... |
| divne0 11849 | The ratio of nonzero numbe... |
| recne0 11850 | The reciprocal of a nonzer... |
| recid 11851 | Multiplication of a number... |
| recid2 11852 | Multiplication of a number... |
| divrec 11853 | Relationship between divis... |
| divrec2 11854 | Relationship between divis... |
| divass 11855 | An associative law for div... |
| div23 11856 | A commutative/associative ... |
| div32 11857 | A commutative/associative ... |
| div13 11858 | A commutative/associative ... |
| div12 11859 | A commutative/associative ... |
| divmulass 11860 | An associative law for div... |
| divmulasscom 11861 | An associative/commutative... |
| divdir 11862 | Distribution of division o... |
| divcan3 11863 | A cancellation law for div... |
| divcan4 11864 | A cancellation law for div... |
| div11 11865 | One-to-one relationship fo... |
| div11OLD 11866 | Obsolete version of ~ div1... |
| diveq1 11867 | Equality in terms of unit ... |
| divid 11868 | A number divided by itself... |
| dividOLD 11869 | Obsolete version of ~ divi... |
| div0 11870 | Division into zero is zero... |
| div0OLD 11871 | Obsolete version of ~ div0... |
| div1 11872 | A number divided by 1 is i... |
| 1div1e1 11873 | 1 divided by 1 is 1. (Con... |
| divneg 11874 | Move negative sign inside ... |
| muldivdir 11875 | Distribution of division o... |
| divsubdir 11876 | Distribution of division o... |
| subdivcomb1 11877 | Bring a term in a subtract... |
| subdivcomb2 11878 | Bring a term in a subtract... |
| recrec 11879 | A number is equal to the r... |
| rec11 11880 | Reciprocal is one-to-one. ... |
| rec11r 11881 | Mutual reciprocals. (Cont... |
| divmuldiv 11882 | Multiplication of two rati... |
| divdivdiv 11883 | Division of two ratios. T... |
| divcan5 11884 | Cancellation of common fac... |
| divmul13 11885 | Swap the denominators in t... |
| divmul24 11886 | Swap the numerators in the... |
| divmuleq 11887 | Cross-multiply in an equal... |
| recdiv 11888 | The reciprocal of a ratio.... |
| divcan6 11889 | Cancellation of inverted f... |
| divdiv32 11890 | Swap denominators in a div... |
| divcan7 11891 | Cancel equal divisors in a... |
| dmdcan 11892 | Cancellation law for divis... |
| divdiv1 11893 | Division into a fraction. ... |
| divdiv2 11894 | Division by a fraction. (... |
| recdiv2 11895 | Division into a reciprocal... |
| ddcan 11896 | Cancellation in a double d... |
| divadddiv 11897 | Addition of two ratios. T... |
| divsubdiv 11898 | Subtraction of two ratios.... |
| conjmul 11899 | Two numbers whose reciproc... |
| rereccl 11900 | Closure law for reciprocal... |
| redivcl 11901 | Closure law for division o... |
| eqneg 11902 | A number equal to its nega... |
| eqnegd 11903 | A complex number equals it... |
| eqnegad 11904 | If a complex number equals... |
| div2neg 11905 | Quotient of two negatives.... |
| divneg2 11906 | Move negative sign inside ... |
| recclzi 11907 | Closure law for reciprocal... |
| recne0zi 11908 | The reciprocal of a nonzer... |
| recidzi 11909 | Multiplication of a number... |
| div1i 11910 | A number divided by 1 is i... |
| eqnegi 11911 | A number equal to its nega... |
| reccli 11912 | Closure law for reciprocal... |
| recidi 11913 | Multiplication of a number... |
| recreci 11914 | A number is equal to the r... |
| dividi 11915 | A number divided by itself... |
| div0i 11916 | Division into zero is zero... |
| divclzi 11917 | Closure law for division. ... |
| divcan1zi 11918 | A cancellation law for div... |
| divcan2zi 11919 | A cancellation law for div... |
| divreczi 11920 | Relationship between divis... |
| divcan3zi 11921 | A cancellation law for div... |
| divcan4zi 11922 | A cancellation law for div... |
| rec11i 11923 | Reciprocal is one-to-one. ... |
| divcli 11924 | Closure law for division. ... |
| divcan2i 11925 | A cancellation law for div... |
| divcan1i 11926 | A cancellation law for div... |
| divreci 11927 | Relationship between divis... |
| divcan3i 11928 | A cancellation law for div... |
| divcan4i 11929 | A cancellation law for div... |
| divne0i 11930 | The ratio of nonzero numbe... |
| rec11ii 11931 | Reciprocal is one-to-one. ... |
| divasszi 11932 | An associative law for div... |
| divmulzi 11933 | Relationship between divis... |
| divdirzi 11934 | Distribution of division o... |
| divdiv23zi 11935 | Swap denominators in a div... |
| divmuli 11936 | Relationship between divis... |
| divdiv32i 11937 | Swap denominators in a div... |
| divassi 11938 | An associative law for div... |
| divdiri 11939 | Distribution of division o... |
| div23i 11940 | A commutative/associative ... |
| div11i 11941 | One-to-one relationship fo... |
| divmuldivi 11942 | Multiplication of two rati... |
| divmul13i 11943 | Swap denominators of two r... |
| divadddivi 11944 | Addition of two ratios. T... |
| divdivdivi 11945 | Division of two ratios. T... |
| rerecclzi 11946 | Closure law for reciprocal... |
| rereccli 11947 | Closure law for reciprocal... |
| redivclzi 11948 | Closure law for division o... |
| redivcli 11949 | Closure law for division o... |
| div1d 11950 | A number divided by 1 is i... |
| reccld 11951 | Closure law for reciprocal... |
| recne0d 11952 | The reciprocal of a nonzer... |
| recidd 11953 | Multiplication of a number... |
| recid2d 11954 | Multiplication of a number... |
| recrecd 11955 | A number is equal to the r... |
| dividd 11956 | A number divided by itself... |
| div0d 11957 | Division into zero is zero... |
| divcld 11958 | Closure law for division. ... |
| divcan1d 11959 | A cancellation law for div... |
| divcan2d 11960 | A cancellation law for div... |
| divrecd 11961 | Relationship between divis... |
| divrec2d 11962 | Relationship between divis... |
| divcan3d 11963 | A cancellation law for div... |
| divcan4d 11964 | A cancellation law for div... |
| diveq0d 11965 | A ratio is zero iff the nu... |
| diveq1d 11966 | Equality in terms of unit ... |
| diveq1ad 11967 | The quotient of two comple... |
| diveq0ad 11968 | A fraction of complex numb... |
| divne1d 11969 | If two complex numbers are... |
| divne0bd 11970 | A ratio is zero iff the nu... |
| divnegd 11971 | Move negative sign inside ... |
| divneg2d 11972 | Move negative sign inside ... |
| div2negd 11973 | Quotient of two negatives.... |
| divne0d 11974 | The ratio of nonzero numbe... |
| recdivd 11975 | The reciprocal of a ratio.... |
| recdiv2d 11976 | Division into a reciprocal... |
| divcan6d 11977 | Cancellation of inverted f... |
| ddcand 11978 | Cancellation in a double d... |
| rec11d 11979 | Reciprocal is one-to-one. ... |
| divmuld 11980 | Relationship between divis... |
| div32d 11981 | A commutative/associative ... |
| div13d 11982 | A commutative/associative ... |
| divdiv32d 11983 | Swap denominators in a div... |
| divcan5d 11984 | Cancellation of common fac... |
| divcan5rd 11985 | Cancellation of common fac... |
| divcan7d 11986 | Cancel equal divisors in a... |
| dmdcand 11987 | Cancellation law for divis... |
| dmdcan2d 11988 | Cancellation law for divis... |
| divdiv1d 11989 | Division into a fraction. ... |
| divdiv2d 11990 | Division by a fraction. (... |
| divmul2d 11991 | Relationship between divis... |
| divmul3d 11992 | Relationship between divis... |
| divassd 11993 | An associative law for div... |
| div12d 11994 | A commutative/associative ... |
| div23d 11995 | A commutative/associative ... |
| divdird 11996 | Distribution of division o... |
| divsubdird 11997 | Distribution of division o... |
| div11d 11998 | One-to-one relationship fo... |
| divmuldivd 11999 | Multiplication of two rati... |
| divmul13d 12000 | Swap denominators of two r... |
| divmul24d 12001 | Swap the numerators in the... |
| divadddivd 12002 | Addition of two ratios. T... |
| divsubdivd 12003 | Subtraction of two ratios.... |
| divmuleqd 12004 | Cross-multiply in an equal... |
| divdivdivd 12005 | Division of two ratios. T... |
| diveq1bd 12006 | If two complex numbers are... |
| div2sub 12007 | Swap the order of subtract... |
| div2subd 12008 | Swap subtrahend and minuen... |
| rereccld 12009 | Closure law for reciprocal... |
| redivcld 12010 | Closure law for division o... |
| subrecd 12011 | Subtraction of reciprocals... |
| subrec 12012 | Subtraction of reciprocals... |
| subreci 12013 | Subtraction of reciprocals... |
| mvllmuld 12014 | Move the left term in a pr... |
| mvllmuli 12015 | Move the left term in a pr... |
| ldiv 12016 | Left-division. (Contribut... |
| rdiv 12017 | Right-division. (Contribu... |
| mdiv 12018 | A division law. (Contribu... |
| lineq 12019 | Solution of a (scalar) lin... |
| elimgt0 12020 | Hypothesis for weak deduct... |
| elimge0 12021 | Hypothesis for weak deduct... |
| ltp1 12022 | A number is less than itse... |
| lep1 12023 | A number is less than or e... |
| ltm1 12024 | A number minus 1 is less t... |
| lem1 12025 | A number minus 1 is less t... |
| letrp1 12026 | A transitive property of '... |
| p1le 12027 | A transitive property of p... |
| recgt0 12028 | The reciprocal of a positi... |
| prodgt0 12029 | Infer that a multiplicand ... |
| prodgt02 12030 | Infer that a multiplier is... |
| ltmul1a 12031 | Lemma for ~ ltmul1 . Mult... |
| ltmul1 12032 | Multiplication of both sid... |
| ltmul2 12033 | Multiplication of both sid... |
| lemul1 12034 | Multiplication of both sid... |
| lemul2 12035 | Multiplication of both sid... |
| lemul1a 12036 | Multiplication of both sid... |
| lemul2a 12037 | Multiplication of both sid... |
| ltmul12a 12038 | Comparison of product of t... |
| lemul12b 12039 | Comparison of product of t... |
| lemul12a 12040 | Comparison of product of t... |
| mulgt1OLD 12041 | Obsolete version of ~ mulg... |
| ltmulgt11 12042 | Multiplication by a number... |
| ltmulgt12 12043 | Multiplication by a number... |
| mulgt1 12044 | The product of two numbers... |
| lemulge11 12045 | Multiplication by a number... |
| lemulge12 12046 | Multiplication by a number... |
| ltdiv1 12047 | Division of both sides of ... |
| lediv1 12048 | Division of both sides of ... |
| gt0div 12049 | Division of a positive num... |
| ge0div 12050 | Division of a nonnegative ... |
| divgt0 12051 | The ratio of two positive ... |
| divge0 12052 | The ratio of nonnegative a... |
| mulge0b 12053 | A condition for multiplica... |
| mulle0b 12054 | A condition for multiplica... |
| mulsuble0b 12055 | A condition for multiplica... |
| ltmuldiv 12056 | 'Less than' relationship b... |
| ltmuldiv2 12057 | 'Less than' relationship b... |
| ltdivmul 12058 | 'Less than' relationship b... |
| ledivmul 12059 | 'Less than or equal to' re... |
| ltdivmul2 12060 | 'Less than' relationship b... |
| lt2mul2div 12061 | 'Less than' relationship b... |
| ledivmul2 12062 | 'Less than or equal to' re... |
| lemuldiv 12063 | 'Less than or equal' relat... |
| lemuldiv2 12064 | 'Less than or equal' relat... |
| ltrec 12065 | The reciprocal of both sid... |
| lerec 12066 | The reciprocal of both sid... |
| lt2msq1 12067 | Lemma for ~ lt2msq . (Con... |
| lt2msq 12068 | Two nonnegative numbers co... |
| ltdiv2 12069 | Division of a positive num... |
| ltrec1 12070 | Reciprocal swap in a 'less... |
| lerec2 12071 | Reciprocal swap in a 'less... |
| ledivdiv 12072 | Invert ratios of positive ... |
| lediv2 12073 | Division of a positive num... |
| ltdiv23 12074 | Swap denominator with othe... |
| lediv23 12075 | Swap denominator with othe... |
| lediv12a 12076 | Comparison of ratio of two... |
| lediv2a 12077 | Division of both sides of ... |
| reclt1 12078 | The reciprocal of a positi... |
| recgt1 12079 | The reciprocal of a positi... |
| recgt1i 12080 | The reciprocal of a number... |
| recp1lt1 12081 | Construct a number less th... |
| recreclt 12082 | Given a positive number ` ... |
| le2msq 12083 | The square function on non... |
| msq11 12084 | The square of a nonnegativ... |
| ledivp1 12085 | "Less than or equal to" an... |
| squeeze0 12086 | If a nonnegative number is... |
| ltp1i 12087 | A number is less than itse... |
| recgt0i 12088 | The reciprocal of a positi... |
| recgt0ii 12089 | The reciprocal of a positi... |
| prodgt0i 12090 | Infer that a multiplicand ... |
| divgt0i 12091 | The ratio of two positive ... |
| divge0i 12092 | The ratio of nonnegative a... |
| ltreci 12093 | The reciprocal of both sid... |
| lereci 12094 | The reciprocal of both sid... |
| lt2msqi 12095 | The square function on non... |
| le2msqi 12096 | The square function on non... |
| msq11i 12097 | The square of a nonnegativ... |
| divgt0i2i 12098 | The ratio of two positive ... |
| ltrecii 12099 | The reciprocal of both sid... |
| divgt0ii 12100 | The ratio of two positive ... |
| ltmul1i 12101 | Multiplication of both sid... |
| ltdiv1i 12102 | Division of both sides of ... |
| ltmuldivi 12103 | 'Less than' relationship b... |
| ltmul2i 12104 | Multiplication of both sid... |
| lemul1i 12105 | Multiplication of both sid... |
| lemul2i 12106 | Multiplication of both sid... |
| ltdiv23i 12107 | Swap denominator with othe... |
| ledivp1i 12108 | "Less than or equal to" an... |
| ltdivp1i 12109 | Less-than and division rel... |
| ltdiv23ii 12110 | Swap denominator with othe... |
| ltmul1ii 12111 | Multiplication of both sid... |
| ltdiv1ii 12112 | Division of both sides of ... |
| ltp1d 12113 | A number is less than itse... |
| lep1d 12114 | A number is less than or e... |
| ltm1d 12115 | A number minus 1 is less t... |
| lem1d 12116 | A number minus 1 is less t... |
| recgt0d 12117 | The reciprocal of a positi... |
| divgt0d 12118 | The ratio of two positive ... |
| mulgt1d 12119 | The product of two numbers... |
| lemulge11d 12120 | Multiplication by a number... |
| lemulge12d 12121 | Multiplication by a number... |
| lemul1ad 12122 | Multiplication of both sid... |
| lemul2ad 12123 | Multiplication of both sid... |
| ltmul12ad 12124 | Comparison of product of t... |
| lemul12ad 12125 | Comparison of product of t... |
| lemul12bd 12126 | Comparison of product of t... |
| fimaxre 12127 | A finite set of real numbe... |
| fimaxre2 12128 | A nonempty finite set of r... |
| fimaxre3 12129 | A nonempty finite set of r... |
| fiminre 12130 | A nonempty finite set of r... |
| fiminre2 12131 | A nonempty finite set of r... |
| negfi 12132 | The negation of a finite s... |
| lbreu 12133 | If a set of reals contains... |
| lbcl 12134 | If a set of reals contains... |
| lble 12135 | If a set of reals contains... |
| lbinf 12136 | If a set of reals contains... |
| lbinfcl 12137 | If a set of reals contains... |
| lbinfle 12138 | If a set of reals contains... |
| sup2 12139 | A nonempty, bounded-above ... |
| sup3 12140 | A version of the completen... |
| infm3lem 12141 | Lemma for ~ infm3 . (Cont... |
| infm3 12142 | The completeness axiom for... |
| suprcl 12143 | Closure of supremum of a n... |
| suprub 12144 | A member of a nonempty bou... |
| suprubd 12145 | Natural deduction form of ... |
| suprcld 12146 | Natural deduction form of ... |
| suprlub 12147 | The supremum of a nonempty... |
| suprnub 12148 | An upper bound is not less... |
| suprleub 12149 | The supremum of a nonempty... |
| supaddc 12150 | The supremum function dist... |
| supadd 12151 | The supremum function dist... |
| supmul1 12152 | The supremum function dist... |
| supmullem1 12153 | Lemma for ~ supmul . (Con... |
| supmullem2 12154 | Lemma for ~ supmul . (Con... |
| supmul 12155 | The supremum function dist... |
| sup3ii 12156 | A version of the completen... |
| suprclii 12157 | Closure of supremum of a n... |
| suprubii 12158 | A member of a nonempty bou... |
| suprlubii 12159 | The supremum of a nonempty... |
| suprnubii 12160 | An upper bound is not less... |
| suprleubii 12161 | The supremum of a nonempty... |
| riotaneg 12162 | The negative of the unique... |
| negiso 12163 | Negation is an order anti-... |
| dfinfre 12164 | The infimum of a set of re... |
| infrecl 12165 | Closure of infimum of a no... |
| infrenegsup 12166 | The infimum of a set of re... |
| infregelb 12167 | Any lower bound of a nonem... |
| infrelb 12168 | If a nonempty set of real ... |
| infrefilb 12169 | The infimum of a finite se... |
| supfirege 12170 | The supremum of a finite s... |
| neg1cn 12171 | -1 is a complex number. (... |
| neg1rr 12172 | -1 is a real number. (Con... |
| neg1ne0 12173 | -1 is nonzero. (Contribut... |
| neg1lt0 12174 | -1 is less than 0. (Contr... |
| negneg1e1 12175 | ` -u -u 1 ` is 1. (Contri... |
| inelr 12176 | The imaginary unit ` _i ` ... |
| rimul 12177 | A real number times the im... |
| cru 12178 | The representation of comp... |
| crne0 12179 | The real representation of... |
| creur 12180 | The real part of a complex... |
| creui 12181 | The imaginary part of a co... |
| cju 12182 | The complex conjugate of a... |
| ofsubeq0 12183 | Function analogue of ~ sub... |
| ofnegsub 12184 | Function analogue of ~ neg... |
| ofsubge0 12185 | Function analogue of ~ sub... |
| nnexALT 12188 | Alternate proof of ~ nnex ... |
| peano5nni 12189 | Peano's inductive postulat... |
| nnssre 12190 | The positive integers are ... |
| nnsscn 12191 | The positive integers are ... |
| nnex 12192 | The set of positive intege... |
| nnre 12193 | A positive integer is a re... |
| nncn 12194 | A positive integer is a co... |
| nnrei 12195 | A positive integer is a re... |
| nncni 12196 | A positive integer is a co... |
| 1nn 12197 | Peano postulate: 1 is a po... |
| peano2nn 12198 | Peano postulate: a success... |
| dfnn2 12199 | Alternate definition of th... |
| dfnn3 12200 | Alternate definition of th... |
| nnred 12201 | A positive integer is a re... |
| nncnd 12202 | A positive integer is a co... |
| peano2nnd 12203 | Peano postulate: a success... |
| nnind 12204 | Principle of Mathematical ... |
| nnindALT 12205 | Principle of Mathematical ... |
| nnindd 12206 | Principle of Mathematical ... |
| nn1m1nn 12207 | Every positive integer is ... |
| nn1suc 12208 | If a statement holds for 1... |
| nnaddcl 12209 | Closure of addition of pos... |
| nnmulcl 12210 | Closure of multiplication ... |
| nnmulcli 12211 | Closure of multiplication ... |
| nnmtmip 12212 | "Minus times minus is plus... |
| nn2ge 12213 | There exists a positive in... |
| nnge1 12214 | A positive integer is one ... |
| nngt1ne1 12215 | A positive integer is grea... |
| nnle1eq1 12216 | A positive integer is less... |
| nngt0 12217 | A positive integer is posi... |
| nnnlt1 12218 | A positive integer is not ... |
| nnnle0 12219 | A positive integer is not ... |
| nnne0 12220 | A positive integer is nonz... |
| nnneneg 12221 | No positive integer is equ... |
| 0nnn 12222 | Zero is not a positive int... |
| 0nnnALT 12223 | Alternate proof of ~ 0nnn ... |
| nnne0ALT 12224 | Alternate version of ~ nnn... |
| nngt0i 12225 | A positive integer is posi... |
| nnne0i 12226 | A positive integer is nonz... |
| nndivre 12227 | The quotient of a real and... |
| nnrecre 12228 | The reciprocal of a positi... |
| nnrecgt0 12229 | The reciprocal of a positi... |
| nnsub 12230 | Subtraction of positive in... |
| nnsubi 12231 | Subtraction of positive in... |
| nndiv 12232 | Two ways to express " ` A ... |
| nndivtr 12233 | Transitive property of div... |
| nnge1d 12234 | A positive integer is one ... |
| nngt0d 12235 | A positive integer is posi... |
| nnne0d 12236 | A positive integer is nonz... |
| nnrecred 12237 | The reciprocal of a positi... |
| nnaddcld 12238 | Closure of addition of pos... |
| nnmulcld 12239 | Closure of multiplication ... |
| nndivred 12240 | A positive integer is one ... |
| 0ne1 12257 | Zero is different from one... |
| 1m1e0 12258 | One minus one equals zero.... |
| 2nn 12259 | 2 is a positive integer. ... |
| 2re 12260 | The number 2 is real. (Co... |
| 2cn 12261 | The number 2 is a complex ... |
| 2cnALT 12262 | Alternate proof of ~ 2cn .... |
| 2ex 12263 | The number 2 is a set. (C... |
| 2cnd 12264 | The number 2 is a complex ... |
| 3nn 12265 | 3 is a positive integer. ... |
| 3re 12266 | The number 3 is real. (Co... |
| 3cn 12267 | The number 3 is a complex ... |
| 3ex 12268 | The number 3 is a set. (C... |
| 4nn 12269 | 4 is a positive integer. ... |
| 4re 12270 | The number 4 is real. (Co... |
| 4cn 12271 | The number 4 is a complex ... |
| 5nn 12272 | 5 is a positive integer. ... |
| 5re 12273 | The number 5 is real. (Co... |
| 5cn 12274 | The number 5 is a complex ... |
| 6nn 12275 | 6 is a positive integer. ... |
| 6re 12276 | The number 6 is real. (Co... |
| 6cn 12277 | The number 6 is a complex ... |
| 7nn 12278 | 7 is a positive integer. ... |
| 7re 12279 | The number 7 is real. (Co... |
| 7cn 12280 | The number 7 is a complex ... |
| 8nn 12281 | 8 is a positive integer. ... |
| 8re 12282 | The number 8 is real. (Co... |
| 8cn 12283 | The number 8 is a complex ... |
| 9nn 12284 | 9 is a positive integer. ... |
| 9re 12285 | The number 9 is real. (Co... |
| 9cn 12286 | The number 9 is a complex ... |
| 0le0 12287 | Zero is nonnegative. (Con... |
| 0le2 12288 | The number 0 is less than ... |
| 2pos 12289 | The number 2 is positive. ... |
| 2ne0 12290 | The number 2 is nonzero. ... |
| 3pos 12291 | The number 3 is positive. ... |
| 3ne0 12292 | The number 3 is nonzero. ... |
| 4pos 12293 | The number 4 is positive. ... |
| 4ne0 12294 | The number 4 is nonzero. ... |
| 5pos 12295 | The number 5 is positive. ... |
| 6pos 12296 | The number 6 is positive. ... |
| 7pos 12297 | The number 7 is positive. ... |
| 8pos 12298 | The number 8 is positive. ... |
| 9pos 12299 | The number 9 is positive. ... |
| 1pneg1e0 12300 | ` 1 + -u 1 ` is 0. (Contr... |
| 0m0e0 12301 | 0 minus 0 equals 0. (Cont... |
| 1m0e1 12302 | 1 - 0 = 1. (Contributed b... |
| 0p1e1 12303 | 0 + 1 = 1. (Contributed b... |
| fv0p1e1 12304 | Function value at ` N + 1 ... |
| 1p0e1 12305 | 1 + 0 = 1. (Contributed b... |
| 1p1e2 12306 | 1 + 1 = 2. (Contributed b... |
| 2m1e1 12307 | 2 - 1 = 1. The result is ... |
| 1e2m1 12308 | 1 = 2 - 1. (Contributed b... |
| 3m1e2 12309 | 3 - 1 = 2. (Contributed b... |
| 4m1e3 12310 | 4 - 1 = 3. (Contributed b... |
| 5m1e4 12311 | 5 - 1 = 4. (Contributed b... |
| 6m1e5 12312 | 6 - 1 = 5. (Contributed b... |
| 7m1e6 12313 | 7 - 1 = 6. (Contributed b... |
| 8m1e7 12314 | 8 - 1 = 7. (Contributed b... |
| 9m1e8 12315 | 9 - 1 = 8. (Contributed b... |
| 2p2e4 12316 | Two plus two equals four. ... |
| 2times 12317 | Two times a number. (Cont... |
| times2 12318 | A number times 2. (Contri... |
| 2timesi 12319 | Two times a number. (Cont... |
| times2i 12320 | A number times 2. (Contri... |
| 2txmxeqx 12321 | Two times a complex number... |
| 2div2e1 12322 | 2 divided by 2 is 1. (Con... |
| 2p1e3 12323 | 2 + 1 = 3. (Contributed b... |
| 1p2e3 12324 | 1 + 2 = 3. For a shorter ... |
| 1p2e3ALT 12325 | Alternate proof of ~ 1p2e3... |
| 3p1e4 12326 | 3 + 1 = 4. (Contributed b... |
| 4p1e5 12327 | 4 + 1 = 5. (Contributed b... |
| 5p1e6 12328 | 5 + 1 = 6. (Contributed b... |
| 6p1e7 12329 | 6 + 1 = 7. (Contributed b... |
| 7p1e8 12330 | 7 + 1 = 8. (Contributed b... |
| 8p1e9 12331 | 8 + 1 = 9. (Contributed b... |
| 3p2e5 12332 | 3 + 2 = 5. (Contributed b... |
| 3p3e6 12333 | 3 + 3 = 6. (Contributed b... |
| 4p2e6 12334 | 4 + 2 = 6. (Contributed b... |
| 4p3e7 12335 | 4 + 3 = 7. (Contributed b... |
| 4p4e8 12336 | 4 + 4 = 8. (Contributed b... |
| 5p2e7 12337 | 5 + 2 = 7. (Contributed b... |
| 5p3e8 12338 | 5 + 3 = 8. (Contributed b... |
| 5p4e9 12339 | 5 + 4 = 9. (Contributed b... |
| 6p2e8 12340 | 6 + 2 = 8. (Contributed b... |
| 6p3e9 12341 | 6 + 3 = 9. (Contributed b... |
| 7p2e9 12342 | 7 + 2 = 9. (Contributed b... |
| 1t1e1 12343 | 1 times 1 equals 1. (Cont... |
| 2t1e2 12344 | 2 times 1 equals 2. (Cont... |
| 2t2e4 12345 | 2 times 2 equals 4. (Cont... |
| 3t1e3 12346 | 3 times 1 equals 3. (Cont... |
| 3t2e6 12347 | 3 times 2 equals 6. (Cont... |
| 3t3e9 12348 | 3 times 3 equals 9. (Cont... |
| 4t2e8 12349 | 4 times 2 equals 8. (Cont... |
| 2t0e0 12350 | 2 times 0 equals 0. (Cont... |
| 4d2e2 12351 | One half of four is two. ... |
| 1lt2 12352 | 1 is less than 2. (Contri... |
| 2lt3 12353 | 2 is less than 3. (Contri... |
| 1lt3 12354 | 1 is less than 3. (Contri... |
| 3lt4 12355 | 3 is less than 4. (Contri... |
| 2lt4 12356 | 2 is less than 4. (Contri... |
| 1lt4 12357 | 1 is less than 4. (Contri... |
| 4lt5 12358 | 4 is less than 5. (Contri... |
| 3lt5 12359 | 3 is less than 5. (Contri... |
| 2lt5 12360 | 2 is less than 5. (Contri... |
| 1lt5 12361 | 1 is less than 5. (Contri... |
| 5lt6 12362 | 5 is less than 6. (Contri... |
| 4lt6 12363 | 4 is less than 6. (Contri... |
| 3lt6 12364 | 3 is less than 6. (Contri... |
| 2lt6 12365 | 2 is less than 6. (Contri... |
| 1lt6 12366 | 1 is less than 6. (Contri... |
| 6lt7 12367 | 6 is less than 7. (Contri... |
| 5lt7 12368 | 5 is less than 7. (Contri... |
| 4lt7 12369 | 4 is less than 7. (Contri... |
| 3lt7 12370 | 3 is less than 7. (Contri... |
| 2lt7 12371 | 2 is less than 7. (Contri... |
| 1lt7 12372 | 1 is less than 7. (Contri... |
| 7lt8 12373 | 7 is less than 8. (Contri... |
| 6lt8 12374 | 6 is less than 8. (Contri... |
| 5lt8 12375 | 5 is less than 8. (Contri... |
| 4lt8 12376 | 4 is less than 8. (Contri... |
| 3lt8 12377 | 3 is less than 8. (Contri... |
| 2lt8 12378 | 2 is less than 8. (Contri... |
| 1lt8 12379 | 1 is less than 8. (Contri... |
| 8lt9 12380 | 8 is less than 9. (Contri... |
| 7lt9 12381 | 7 is less than 9. (Contri... |
| 6lt9 12382 | 6 is less than 9. (Contri... |
| 5lt9 12383 | 5 is less than 9. (Contri... |
| 4lt9 12384 | 4 is less than 9. (Contri... |
| 3lt9 12385 | 3 is less than 9. (Contri... |
| 2lt9 12386 | 2 is less than 9. (Contri... |
| 1lt9 12387 | 1 is less than 9. (Contri... |
| 0ne2 12388 | 0 is not equal to 2. (Con... |
| 1ne2 12389 | 1 is not equal to 2. (Con... |
| 1le2 12390 | 1 is less than or equal to... |
| 2cnne0 12391 | 2 is a nonzero complex num... |
| 2rene0 12392 | 2 is a nonzero real number... |
| 1le3 12393 | 1 is less than or equal to... |
| neg1mulneg1e1 12394 | ` -u 1 x. -u 1 ` is 1. (C... |
| halfre 12395 | One-half is real. (Contri... |
| halfcn 12396 | One-half is a complex numb... |
| halfgt0 12397 | One-half is greater than z... |
| halfge0 12398 | One-half is not negative. ... |
| halflt1 12399 | One-half is less than one.... |
| 2halves 12400 | Two halves make a whole. ... |
| 1mhlfehlf 12401 | Prove that 1 - 1/2 = 1/2. ... |
| 8th4div3 12402 | An eighth of four thirds i... |
| halfthird 12403 | Half minus a third. (Cont... |
| halfpm6th 12404 | One half plus or minus one... |
| it0e0 12405 | i times 0 equals 0. (Cont... |
| 2mulicn 12406 | ` ( 2 x. _i ) e. CC ` . (... |
| 2muline0 12407 | ` ( 2 x. _i ) =/= 0 ` . (... |
| halfcl 12408 | Closure of half of a numbe... |
| rehalfcl 12409 | Real closure of half. (Co... |
| half0 12410 | Half of a number is zero i... |
| halfpos2 12411 | A number is positive iff i... |
| halfpos 12412 | A positive number is great... |
| halfnneg2 12413 | A number is nonnegative if... |
| halfaddsubcl 12414 | Closure of half-sum and ha... |
| halfaddsub 12415 | Sum and difference of half... |
| subhalfhalf 12416 | Subtracting the half of a ... |
| lt2halves 12417 | A sum is less than the who... |
| addltmul 12418 | Sum is less than product f... |
| nominpos 12419 | There is no smallest posit... |
| avglt1 12420 | Ordering property for aver... |
| avglt2 12421 | Ordering property for aver... |
| avgle1 12422 | Ordering property for aver... |
| avgle2 12423 | Ordering property for aver... |
| avgle 12424 | The average of two numbers... |
| 2timesd 12425 | Two times a number. (Cont... |
| times2d 12426 | A number times 2. (Contri... |
| halfcld 12427 | Closure of half of a numbe... |
| 2halvesd 12428 | Two halves make a whole. ... |
| rehalfcld 12429 | Real closure of half. (Co... |
| lt2halvesd 12430 | A sum is less than the who... |
| rehalfcli 12431 | Half a real number is real... |
| lt2addmuld 12432 | If two real numbers are le... |
| add1p1 12433 | Adding two times 1 to a nu... |
| sub1m1 12434 | Subtracting two times 1 fr... |
| cnm2m1cnm3 12435 | Subtracting 2 and afterwar... |
| xp1d2m1eqxm1d2 12436 | A complex number increased... |
| div4p1lem1div2 12437 | An integer greater than 5,... |
| nnunb 12438 | The set of positive intege... |
| arch 12439 | Archimedean property of re... |
| nnrecl 12440 | There exists a positive in... |
| bndndx 12441 | A bounded real sequence ` ... |
| elnn0 12444 | Nonnegative integers expre... |
| nnssnn0 12445 | Positive naturals are a su... |
| nn0ssre 12446 | Nonnegative integers are a... |
| nn0sscn 12447 | Nonnegative integers are a... |
| nn0ex 12448 | The set of nonnegative int... |
| nnnn0 12449 | A positive integer is a no... |
| nnnn0i 12450 | A positive integer is a no... |
| nn0re 12451 | A nonnegative integer is a... |
| nn0cn 12452 | A nonnegative integer is a... |
| nn0rei 12453 | A nonnegative integer is a... |
| nn0cni 12454 | A nonnegative integer is a... |
| dfn2 12455 | The set of positive intege... |
| elnnne0 12456 | The positive integer prope... |
| 0nn0 12457 | 0 is a nonnegative integer... |
| 1nn0 12458 | 1 is a nonnegative integer... |
| 2nn0 12459 | 2 is a nonnegative integer... |
| 3nn0 12460 | 3 is a nonnegative integer... |
| 4nn0 12461 | 4 is a nonnegative integer... |
| 5nn0 12462 | 5 is a nonnegative integer... |
| 6nn0 12463 | 6 is a nonnegative integer... |
| 7nn0 12464 | 7 is a nonnegative integer... |
| 8nn0 12465 | 8 is a nonnegative integer... |
| 9nn0 12466 | 9 is a nonnegative integer... |
| nn0ge0 12467 | A nonnegative integer is g... |
| nn0nlt0 12468 | A nonnegative integer is n... |
| nn0ge0i 12469 | Nonnegative integers are n... |
| nn0le0eq0 12470 | A nonnegative integer is l... |
| nn0p1gt0 12471 | A nonnegative integer incr... |
| nnnn0addcl 12472 | A positive integer plus a ... |
| nn0nnaddcl 12473 | A nonnegative integer plus... |
| 0mnnnnn0 12474 | The result of subtracting ... |
| un0addcl 12475 | If ` S ` is closed under a... |
| un0mulcl 12476 | If ` S ` is closed under m... |
| nn0addcl 12477 | Closure of addition of non... |
| nn0mulcl 12478 | Closure of multiplication ... |
| nn0addcli 12479 | Closure of addition of non... |
| nn0mulcli 12480 | Closure of multiplication ... |
| nn0p1nn 12481 | A nonnegative integer plus... |
| peano2nn0 12482 | Second Peano postulate for... |
| nnm1nn0 12483 | A positive integer minus 1... |
| elnn0nn 12484 | The nonnegative integer pr... |
| elnnnn0 12485 | The positive integer prope... |
| elnnnn0b 12486 | The positive integer prope... |
| elnnnn0c 12487 | The positive integer prope... |
| nn0addge1 12488 | A number is less than or e... |
| nn0addge2 12489 | A number is less than or e... |
| nn0addge1i 12490 | A number is less than or e... |
| nn0addge2i 12491 | A number is less than or e... |
| nn0sub 12492 | Subtraction of nonnegative... |
| ltsubnn0 12493 | Subtracting a nonnegative ... |
| nn0negleid 12494 | A nonnegative integer is g... |
| difgtsumgt 12495 | If the difference of a rea... |
| nn0le2x 12496 | A nonnegative integer is l... |
| nn0le2xi 12497 | A nonnegative integer is l... |
| nn0lele2xi 12498 | 'Less than or equal to' im... |
| fcdmnn0supp 12499 | Two ways to write the supp... |
| fcdmnn0fsupp 12500 | A function into ` NN0 ` is... |
| fcdmnn0suppg 12501 | Version of ~ fcdmnn0supp a... |
| fcdmnn0fsuppg 12502 | Version of ~ fcdmnn0fsupp ... |
| nnnn0d 12503 | A positive integer is a no... |
| nn0red 12504 | A nonnegative integer is a... |
| nn0cnd 12505 | A nonnegative integer is a... |
| nn0ge0d 12506 | A nonnegative integer is g... |
| nn0addcld 12507 | Closure of addition of non... |
| nn0mulcld 12508 | Closure of multiplication ... |
| nn0readdcl 12509 | Closure law for addition o... |
| nn0n0n1ge2 12510 | A nonnegative integer whic... |
| nn0n0n1ge2b 12511 | A nonnegative integer is n... |
| nn0ge2m1nn 12512 | If a nonnegative integer i... |
| nn0ge2m1nn0 12513 | If a nonnegative integer i... |
| nn0nndivcl 12514 | Closure law for dividing o... |
| elxnn0 12517 | An extended nonnegative in... |
| nn0ssxnn0 12518 | The standard nonnegative i... |
| nn0xnn0 12519 | A standard nonnegative int... |
| xnn0xr 12520 | An extended nonnegative in... |
| 0xnn0 12521 | Zero is an extended nonneg... |
| pnf0xnn0 12522 | Positive infinity is an ex... |
| nn0nepnf 12523 | No standard nonnegative in... |
| nn0xnn0d 12524 | A standard nonnegative int... |
| nn0nepnfd 12525 | No standard nonnegative in... |
| xnn0nemnf 12526 | No extended nonnegative in... |
| xnn0xrnemnf 12527 | The extended nonnegative i... |
| xnn0nnn0pnf 12528 | An extended nonnegative in... |
| elz 12531 | Membership in the set of i... |
| nnnegz 12532 | The negative of a positive... |
| zre 12533 | An integer is a real. (Co... |
| zcn 12534 | An integer is a complex nu... |
| zrei 12535 | An integer is a real numbe... |
| zssre 12536 | The integers are a subset ... |
| zsscn 12537 | The integers are a subset ... |
| zex 12538 | The set of integers exists... |
| elnnz 12539 | Positive integer property ... |
| 0z 12540 | Zero is an integer. (Cont... |
| 0zd 12541 | Zero is an integer, deduct... |
| elnn0z 12542 | Nonnegative integer proper... |
| elznn0nn 12543 | Integer property expressed... |
| elznn0 12544 | Integer property expressed... |
| elznn 12545 | Integer property expressed... |
| zle0orge1 12546 | There is no integer in the... |
| elz2 12547 | Membership in the set of i... |
| dfz2 12548 | Alternative definition of ... |
| zexALT 12549 | Alternate proof of ~ zex .... |
| nnz 12550 | A positive integer is an i... |
| nnssz 12551 | Positive integers are a su... |
| nn0ssz 12552 | Nonnegative integers are a... |
| nnzOLD 12553 | Obsolete version of ~ nnz ... |
| nn0z 12554 | A nonnegative integer is a... |
| nn0zd 12555 | A nonnegative integer is a... |
| nnzd 12556 | A positive integer is an i... |
| nnzi 12557 | A positive integer is an i... |
| nn0zi 12558 | A nonnegative integer is a... |
| elnnz1 12559 | Positive integer property ... |
| znnnlt1 12560 | An integer is not a positi... |
| nnzrab 12561 | Positive integers expresse... |
| nn0zrab 12562 | Nonnegative integers expre... |
| 1z 12563 | One is an integer. (Contr... |
| 1zzd 12564 | One is an integer, deducti... |
| 2z 12565 | 2 is an integer. (Contrib... |
| 3z 12566 | 3 is an integer. (Contrib... |
| 4z 12567 | 4 is an integer. (Contrib... |
| znegcl 12568 | Closure law for negative i... |
| neg1z 12569 | -1 is an integer. (Contri... |
| znegclb 12570 | A complex number is an int... |
| nn0negz 12571 | The negative of a nonnegat... |
| nn0negzi 12572 | The negative of a nonnegat... |
| zaddcl 12573 | Closure of addition of int... |
| peano2z 12574 | Second Peano postulate gen... |
| zsubcl 12575 | Closure of subtraction of ... |
| peano2zm 12576 | "Reverse" second Peano pos... |
| zletr 12577 | Transitive law of ordering... |
| zrevaddcl 12578 | Reverse closure law for ad... |
| znnsub 12579 | The positive difference of... |
| znn0sub 12580 | The nonnegative difference... |
| nzadd 12581 | The sum of a real number n... |
| zmulcl 12582 | Closure of multiplication ... |
| zltp1le 12583 | Integer ordering relation.... |
| zleltp1 12584 | Integer ordering relation.... |
| zlem1lt 12585 | Integer ordering relation.... |
| zltlem1 12586 | Integer ordering relation.... |
| zltlem1d 12587 | Integer ordering relation,... |
| zgt0ge1 12588 | An integer greater than ` ... |
| nnleltp1 12589 | Positive integer ordering ... |
| nnltp1le 12590 | Positive integer ordering ... |
| nnaddm1cl 12591 | Closure of addition of pos... |
| nn0ltp1le 12592 | Nonnegative integer orderi... |
| nn0leltp1 12593 | Nonnegative integer orderi... |
| nn0ltlem1 12594 | Nonnegative integer orderi... |
| nn0sub2 12595 | Subtraction of nonnegative... |
| nn0lt10b 12596 | A nonnegative integer less... |
| nn0lt2 12597 | A nonnegative integer less... |
| nn0le2is012 12598 | A nonnegative integer whic... |
| nn0lem1lt 12599 | Nonnegative integer orderi... |
| nnlem1lt 12600 | Positive integer ordering ... |
| nnltlem1 12601 | Positive integer ordering ... |
| nnm1ge0 12602 | A positive integer decreas... |
| nn0ge0div 12603 | Division of a nonnegative ... |
| zdiv 12604 | Two ways to express " ` M ... |
| zdivadd 12605 | Property of divisibility: ... |
| zdivmul 12606 | Property of divisibility: ... |
| zextle 12607 | An extensionality-like pro... |
| zextlt 12608 | An extensionality-like pro... |
| recnz 12609 | The reciprocal of a number... |
| btwnnz 12610 | A number between an intege... |
| gtndiv 12611 | A larger number does not d... |
| halfnz 12612 | One-half is not an integer... |
| 3halfnz 12613 | Three halves is not an int... |
| suprzcl 12614 | The supremum of a bounded-... |
| prime 12615 | Two ways to express " ` A ... |
| msqznn 12616 | The square of a nonzero in... |
| zneo 12617 | No even integer equals an ... |
| nneo 12618 | A positive integer is even... |
| nneoi 12619 | A positive integer is even... |
| zeo 12620 | An integer is even or odd.... |
| zeo2 12621 | An integer is even or odd ... |
| peano2uz2 12622 | Second Peano postulate for... |
| peano5uzi 12623 | Peano's inductive postulat... |
| peano5uzti 12624 | Peano's inductive postulat... |
| dfuzi 12625 | An expression for the uppe... |
| uzind 12626 | Induction on the upper int... |
| uzind2 12627 | Induction on the upper int... |
| uzind3 12628 | Induction on the upper int... |
| nn0ind 12629 | Principle of Mathematical ... |
| nn0indALT 12630 | Principle of Mathematical ... |
| nn0indd 12631 | Principle of Mathematical ... |
| fzind 12632 | Induction on the integers ... |
| fnn0ind 12633 | Induction on the integers ... |
| nn0ind-raph 12634 | Principle of Mathematical ... |
| zindd 12635 | Principle of Mathematical ... |
| fzindd 12636 | Induction on the integers ... |
| btwnz 12637 | Any real number can be san... |
| zred 12638 | An integer is a real numbe... |
| zcnd 12639 | An integer is a complex nu... |
| znegcld 12640 | Closure law for negative i... |
| peano2zd 12641 | Deduction from second Pean... |
| zaddcld 12642 | Closure of addition of int... |
| zsubcld 12643 | Closure of subtraction of ... |
| zmulcld 12644 | Closure of multiplication ... |
| znnn0nn 12645 | The negative of a negative... |
| zadd2cl 12646 | Increasing an integer by 2... |
| zriotaneg 12647 | The negative of the unique... |
| suprfinzcl 12648 | The supremum of a nonempty... |
| 9p1e10 12651 | 9 + 1 = 10. (Contributed ... |
| dfdec10 12652 | Version of the definition ... |
| decex 12653 | A decimal number is a set.... |
| deceq1 12654 | Equality theorem for the d... |
| deceq2 12655 | Equality theorem for the d... |
| deceq1i 12656 | Equality theorem for the d... |
| deceq2i 12657 | Equality theorem for the d... |
| deceq12i 12658 | Equality theorem for the d... |
| numnncl 12659 | Closure for a numeral (wit... |
| num0u 12660 | Add a zero in the units pl... |
| num0h 12661 | Add a zero in the higher p... |
| numcl 12662 | Closure for a decimal inte... |
| numsuc 12663 | The successor of a decimal... |
| deccl 12664 | Closure for a numeral. (C... |
| 10nn 12665 | 10 is a positive integer. ... |
| 10pos 12666 | The number 10 is positive.... |
| 10nn0 12667 | 10 is a nonnegative intege... |
| 10re 12668 | The number 10 is real. (C... |
| decnncl 12669 | Closure for a numeral. (C... |
| dec0u 12670 | Add a zero in the units pl... |
| dec0h 12671 | Add a zero in the higher p... |
| numnncl2 12672 | Closure for a decimal inte... |
| decnncl2 12673 | Closure for a decimal inte... |
| numlt 12674 | Comparing two decimal inte... |
| numltc 12675 | Comparing two decimal inte... |
| le9lt10 12676 | A "decimal digit" (i.e. a ... |
| declt 12677 | Comparing two decimal inte... |
| decltc 12678 | Comparing two decimal inte... |
| declth 12679 | Comparing two decimal inte... |
| decsuc 12680 | The successor of a decimal... |
| 3declth 12681 | Comparing two decimal inte... |
| 3decltc 12682 | Comparing two decimal inte... |
| decle 12683 | Comparing two decimal inte... |
| decleh 12684 | Comparing two decimal inte... |
| declei 12685 | Comparing a digit to a dec... |
| numlti 12686 | Comparing a digit to a dec... |
| declti 12687 | Comparing a digit to a dec... |
| decltdi 12688 | Comparing a digit to a dec... |
| numsucc 12689 | The successor of a decimal... |
| decsucc 12690 | The successor of a decimal... |
| 1e0p1 12691 | The successor of zero. (C... |
| dec10p 12692 | Ten plus an integer. (Con... |
| numma 12693 | Perform a multiply-add of ... |
| nummac 12694 | Perform a multiply-add of ... |
| numma2c 12695 | Perform a multiply-add of ... |
| numadd 12696 | Add two decimal integers `... |
| numaddc 12697 | Add two decimal integers `... |
| nummul1c 12698 | The product of a decimal i... |
| nummul2c 12699 | The product of a decimal i... |
| decma 12700 | Perform a multiply-add of ... |
| decmac 12701 | Perform a multiply-add of ... |
| decma2c 12702 | Perform a multiply-add of ... |
| decadd 12703 | Add two numerals ` M ` and... |
| decaddc 12704 | Add two numerals ` M ` and... |
| decaddc2 12705 | Add two numerals ` M ` and... |
| decrmanc 12706 | Perform a multiply-add of ... |
| decrmac 12707 | Perform a multiply-add of ... |
| decaddm10 12708 | The sum of two multiples o... |
| decaddi 12709 | Add two numerals ` M ` and... |
| decaddci 12710 | Add two numerals ` M ` and... |
| decaddci2 12711 | Add two numerals ` M ` and... |
| decsubi 12712 | Difference between a numer... |
| decmul1 12713 | The product of a numeral w... |
| decmul1c 12714 | The product of a numeral w... |
| decmul2c 12715 | The product of a numeral w... |
| decmulnc 12716 | The product of a numeral w... |
| 11multnc 12717 | The product of 11 (as nume... |
| decmul10add 12718 | A multiplication of a numb... |
| 6p5lem 12719 | Lemma for ~ 6p5e11 and rel... |
| 5p5e10 12720 | 5 + 5 = 10. (Contributed ... |
| 6p4e10 12721 | 6 + 4 = 10. (Contributed ... |
| 6p5e11 12722 | 6 + 5 = 11. (Contributed ... |
| 6p6e12 12723 | 6 + 6 = 12. (Contributed ... |
| 7p3e10 12724 | 7 + 3 = 10. (Contributed ... |
| 7p4e11 12725 | 7 + 4 = 11. (Contributed ... |
| 7p5e12 12726 | 7 + 5 = 12. (Contributed ... |
| 7p6e13 12727 | 7 + 6 = 13. (Contributed ... |
| 7p7e14 12728 | 7 + 7 = 14. (Contributed ... |
| 8p2e10 12729 | 8 + 2 = 10. (Contributed ... |
| 8p3e11 12730 | 8 + 3 = 11. (Contributed ... |
| 8p4e12 12731 | 8 + 4 = 12. (Contributed ... |
| 8p5e13 12732 | 8 + 5 = 13. (Contributed ... |
| 8p6e14 12733 | 8 + 6 = 14. (Contributed ... |
| 8p7e15 12734 | 8 + 7 = 15. (Contributed ... |
| 8p8e16 12735 | 8 + 8 = 16. (Contributed ... |
| 9p2e11 12736 | 9 + 2 = 11. (Contributed ... |
| 9p3e12 12737 | 9 + 3 = 12. (Contributed ... |
| 9p4e13 12738 | 9 + 4 = 13. (Contributed ... |
| 9p5e14 12739 | 9 + 5 = 14. (Contributed ... |
| 9p6e15 12740 | 9 + 6 = 15. (Contributed ... |
| 9p7e16 12741 | 9 + 7 = 16. (Contributed ... |
| 9p8e17 12742 | 9 + 8 = 17. (Contributed ... |
| 9p9e18 12743 | 9 + 9 = 18. (Contributed ... |
| 10p10e20 12744 | 10 + 10 = 20. (Contribute... |
| 10m1e9 12745 | 10 - 1 = 9. (Contributed ... |
| 4t3lem 12746 | Lemma for ~ 4t3e12 and rel... |
| 4t3e12 12747 | 4 times 3 equals 12. (Con... |
| 4t4e16 12748 | 4 times 4 equals 16. (Con... |
| 5t2e10 12749 | 5 times 2 equals 10. (Con... |
| 5t3e15 12750 | 5 times 3 equals 15. (Con... |
| 5t4e20 12751 | 5 times 4 equals 20. (Con... |
| 5t5e25 12752 | 5 times 5 equals 25. (Con... |
| 6t2e12 12753 | 6 times 2 equals 12. (Con... |
| 6t3e18 12754 | 6 times 3 equals 18. (Con... |
| 6t4e24 12755 | 6 times 4 equals 24. (Con... |
| 6t5e30 12756 | 6 times 5 equals 30. (Con... |
| 6t6e36 12757 | 6 times 6 equals 36. (Con... |
| 7t2e14 12758 | 7 times 2 equals 14. (Con... |
| 7t3e21 12759 | 7 times 3 equals 21. (Con... |
| 7t4e28 12760 | 7 times 4 equals 28. (Con... |
| 7t5e35 12761 | 7 times 5 equals 35. (Con... |
| 7t6e42 12762 | 7 times 6 equals 42. (Con... |
| 7t7e49 12763 | 7 times 7 equals 49. (Con... |
| 8t2e16 12764 | 8 times 2 equals 16. (Con... |
| 8t3e24 12765 | 8 times 3 equals 24. (Con... |
| 8t4e32 12766 | 8 times 4 equals 32. (Con... |
| 8t5e40 12767 | 8 times 5 equals 40. (Con... |
| 8t6e48 12768 | 8 times 6 equals 48. (Con... |
| 8t7e56 12769 | 8 times 7 equals 56. (Con... |
| 8t8e64 12770 | 8 times 8 equals 64. (Con... |
| 9t2e18 12771 | 9 times 2 equals 18. (Con... |
| 9t3e27 12772 | 9 times 3 equals 27. (Con... |
| 9t4e36 12773 | 9 times 4 equals 36. (Con... |
| 9t5e45 12774 | 9 times 5 equals 45. (Con... |
| 9t6e54 12775 | 9 times 6 equals 54. (Con... |
| 9t7e63 12776 | 9 times 7 equals 63. (Con... |
| 9t8e72 12777 | 9 times 8 equals 72. (Con... |
| 9t9e81 12778 | 9 times 9 equals 81. (Con... |
| 9t11e99 12779 | 9 times 11 equals 99. (Co... |
| 9lt10 12780 | 9 is less than 10. (Contr... |
| 8lt10 12781 | 8 is less than 10. (Contr... |
| 7lt10 12782 | 7 is less than 10. (Contr... |
| 6lt10 12783 | 6 is less than 10. (Contr... |
| 5lt10 12784 | 5 is less than 10. (Contr... |
| 4lt10 12785 | 4 is less than 10. (Contr... |
| 3lt10 12786 | 3 is less than 10. (Contr... |
| 2lt10 12787 | 2 is less than 10. (Contr... |
| 1lt10 12788 | 1 is less than 10. (Contr... |
| decbin0 12789 | Decompose base 4 into base... |
| decbin2 12790 | Decompose base 4 into base... |
| decbin3 12791 | Decompose base 4 into base... |
| 5recm6rec 12792 | One fifth minus one sixth.... |
| uzval 12795 | The value of the upper int... |
| uzf 12796 | The domain and codomain of... |
| eluz1 12797 | Membership in the upper se... |
| eluzel2 12798 | Implication of membership ... |
| eluz2 12799 | Membership in an upper set... |
| eluzmn 12800 | Membership in an earlier u... |
| eluz1i 12801 | Membership in an upper set... |
| eluzuzle 12802 | An integer in an upper set... |
| eluzelz 12803 | A member of an upper set o... |
| eluzelre 12804 | A member of an upper set o... |
| eluzelcn 12805 | A member of an upper set o... |
| eluzle 12806 | Implication of membership ... |
| eluz 12807 | Membership in an upper set... |
| uzid 12808 | Membership of the least me... |
| uzidd 12809 | Membership of the least me... |
| uzn0 12810 | The upper integers are all... |
| uztrn 12811 | Transitive law for sets of... |
| uztrn2 12812 | Transitive law for sets of... |
| uzneg 12813 | Contraposition law for upp... |
| uzssz 12814 | An upper set of integers i... |
| uzssre 12815 | An upper set of integers i... |
| uzss 12816 | Subset relationship for tw... |
| uztric 12817 | Totality of the ordering r... |
| uz11 12818 | The upper integers functio... |
| eluzp1m1 12819 | Membership in the next upp... |
| eluzp1l 12820 | Strict ordering implied by... |
| eluzp1p1 12821 | Membership in the next upp... |
| eluzadd 12822 | Membership in a later uppe... |
| eluzsub 12823 | Membership in an earlier u... |
| eluzaddi 12824 | Membership in a later uppe... |
| eluzaddiOLD 12825 | Obsolete version of ~ eluz... |
| eluzsubi 12826 | Membership in an earlier u... |
| eluzsubiOLD 12827 | Obsolete version of ~ eluz... |
| eluzaddOLD 12828 | Obsolete version of ~ eluz... |
| eluzsubOLD 12829 | Obsolete version of ~ eluz... |
| subeluzsub 12830 | Membership of a difference... |
| uzm1 12831 | Choices for an element of ... |
| uznn0sub 12832 | The nonnegative difference... |
| uzin 12833 | Intersection of two upper ... |
| uzp1 12834 | Choices for an element of ... |
| nn0uz 12835 | Nonnegative integers expre... |
| nnuz 12836 | Positive integers expresse... |
| elnnuz 12837 | A positive integer express... |
| elnn0uz 12838 | A nonnegative integer expr... |
| 1eluzge0 12839 | 1 is an integer greater th... |
| 2eluzge0 12840 | 2 is an integer greater th... |
| 2eluzge1 12841 | 2 is an integer greater th... |
| 5eluz3 12842 | 5 is an integer greater th... |
| uzuzle23 12843 | An integer greater than or... |
| uzuzle24 12844 | An integer greater than or... |
| uzuzle34 12845 | An integer greater than or... |
| uzuzle35 12846 | An integer greater than or... |
| eluz2nn 12847 | An integer greater than or... |
| eluz3nn 12848 | An integer greater than or... |
| eluz4nn 12849 | An integer greater than or... |
| eluz5nn 12850 | An integer greater than or... |
| eluzge2nn0 12851 | If an integer is greater t... |
| eluz2n0 12852 | An integer greater than or... |
| uz3m2nn 12853 | An integer greater than or... |
| uznnssnn 12854 | The upper integers startin... |
| raluz 12855 | Restricted universal quant... |
| raluz2 12856 | Restricted universal quant... |
| rexuz 12857 | Restricted existential qua... |
| rexuz2 12858 | Restricted existential qua... |
| 2rexuz 12859 | Double existential quantif... |
| peano2uz 12860 | Second Peano postulate for... |
| peano2uzs 12861 | Second Peano postulate for... |
| peano2uzr 12862 | Reversed second Peano axio... |
| uzaddcl 12863 | Addition closure law for a... |
| nn0pzuz 12864 | The sum of a nonnegative i... |
| uzind4 12865 | Induction on the upper set... |
| uzind4ALT 12866 | Induction on the upper set... |
| uzind4s 12867 | Induction on the upper set... |
| uzind4s2 12868 | Induction on the upper set... |
| uzind4i 12869 | Induction on the upper int... |
| uzwo 12870 | Well-ordering principle: a... |
| uzwo2 12871 | Well-ordering principle: a... |
| nnwo 12872 | Well-ordering principle: a... |
| nnwof 12873 | Well-ordering principle: a... |
| nnwos 12874 | Well-ordering principle: a... |
| indstr 12875 | Strong Mathematical Induct... |
| eluznn0 12876 | Membership in a nonnegativ... |
| eluznn 12877 | Membership in a positive u... |
| eluz2b1 12878 | Two ways to say "an intege... |
| eluz2gt1 12879 | An integer greater than or... |
| eluz2b2 12880 | Two ways to say "an intege... |
| eluz2b3 12881 | Two ways to say "an intege... |
| uz2m1nn 12882 | One less than an integer g... |
| 1nuz2 12883 | 1 is not in ` ( ZZ>= `` 2 ... |
| elnn1uz2 12884 | A positive integer is eith... |
| uz2mulcl 12885 | Closure of multiplication ... |
| indstr2 12886 | Strong Mathematical Induct... |
| uzinfi 12887 | Extract the lower bound of... |
| nninf 12888 | The infimum of the set of ... |
| nn0inf 12889 | The infimum of the set of ... |
| infssuzle 12890 | The infimum of a subset of... |
| infssuzcl 12891 | The infimum of a subset of... |
| ublbneg 12892 | The image under negation o... |
| eqreznegel 12893 | Two ways to express the im... |
| supminf 12894 | The supremum of a bounded-... |
| lbzbi 12895 | If a set of reals is bound... |
| zsupss 12896 | Any nonempty bounded subse... |
| suprzcl2 12897 | The supremum of a bounded-... |
| suprzub 12898 | The supremum of a bounded-... |
| uzsupss 12899 | Any bounded subset of an u... |
| nn01to3 12900 | A (nonnegative) integer be... |
| nn0ge2m1nnALT 12901 | Alternate proof of ~ nn0ge... |
| uzwo3 12902 | Well-ordering principle: a... |
| zmin 12903 | There is a unique smallest... |
| zmax 12904 | There is a unique largest ... |
| zbtwnre 12905 | There is a unique integer ... |
| rebtwnz 12906 | There is a unique greatest... |
| elq 12909 | Membership in the set of r... |
| qmulz 12910 | If ` A ` is rational, then... |
| znq 12911 | The ratio of an integer an... |
| qre 12912 | A rational number is a rea... |
| zq 12913 | An integer is a rational n... |
| qred 12914 | A rational number is a rea... |
| zssq 12915 | The integers are a subset ... |
| nn0ssq 12916 | The nonnegative integers a... |
| nnssq 12917 | The positive integers are ... |
| qssre 12918 | The rationals are a subset... |
| qsscn 12919 | The rationals are a subset... |
| qex 12920 | The set of rational number... |
| nnq 12921 | A positive integer is rati... |
| qcn 12922 | A rational number is a com... |
| qexALT 12923 | Alternate proof of ~ qex .... |
| qaddcl 12924 | Closure of addition of rat... |
| qnegcl 12925 | Closure law for the negati... |
| qmulcl 12926 | Closure of multiplication ... |
| qsubcl 12927 | Closure of subtraction of ... |
| qreccl 12928 | Closure of reciprocal of r... |
| qdivcl 12929 | Closure of division of rat... |
| qrevaddcl 12930 | Reverse closure law for ad... |
| nnrecq 12931 | The reciprocal of a positi... |
| irradd 12932 | The sum of an irrational n... |
| irrmul 12933 | The product of an irration... |
| elpq 12934 | A positive rational is the... |
| elpqb 12935 | A class is a positive rati... |
| rpnnen1lem2 12936 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem1 12937 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem3 12938 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem4 12939 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem5 12940 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem6 12941 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1 12942 | One half of ~ rpnnen , whe... |
| reexALT 12943 | Alternate proof of ~ reex ... |
| cnref1o 12944 | There is a natural one-to-... |
| cnexALT 12945 | The set of complex numbers... |
| xrex 12946 | The set of extended reals ... |
| mpoaddex 12947 | The addition operation is ... |
| addex 12948 | The addition operation is ... |
| mpomulex 12949 | The multiplication operati... |
| mulex 12950 | The multiplication operati... |
| elrp 12953 | Membership in the set of p... |
| elrpii 12954 | Membership in the set of p... |
| 1rp 12955 | 1 is a positive real. (Co... |
| 2rp 12956 | 2 is a positive real. (Co... |
| 3rp 12957 | 3 is a positive real. (Co... |
| 5rp 12958 | 5 is a positive real. (Co... |
| rpssre 12959 | The positive reals are a s... |
| rpre 12960 | A positive real is a real.... |
| rpxr 12961 | A positive real is an exte... |
| rpcn 12962 | A positive real is a compl... |
| nnrp 12963 | A positive integer is a po... |
| rpgt0 12964 | A positive real is greater... |
| rpge0 12965 | A positive real is greater... |
| rpregt0 12966 | A positive real is a posit... |
| rprege0 12967 | A positive real is a nonne... |
| rpne0 12968 | A positive real is nonzero... |
| rprene0 12969 | A positive real is a nonze... |
| rpcnne0 12970 | A positive real is a nonze... |
| neglt 12971 | The negative of a positive... |
| rpcndif0 12972 | A positive real number is ... |
| ralrp 12973 | Quantification over positi... |
| rexrp 12974 | Quantification over positi... |
| rpaddcl 12975 | Closure law for addition o... |
| rpmulcl 12976 | Closure law for multiplica... |
| rpmtmip 12977 | "Minus times minus is plus... |
| rpdivcl 12978 | Closure law for division o... |
| rpreccl 12979 | Closure law for reciprocat... |
| rphalfcl 12980 | Closure law for half of a ... |
| rpgecl 12981 | A number greater than or e... |
| rphalflt 12982 | Half of a positive real is... |
| rerpdivcl 12983 | Closure law for division o... |
| ge0p1rp 12984 | A nonnegative number plus ... |
| rpneg 12985 | Either a nonzero real or i... |
| negelrp 12986 | Elementhood of a negation ... |
| negelrpd 12987 | The negation of a negative... |
| 0nrp 12988 | Zero is not a positive rea... |
| ltsubrp 12989 | Subtracting a positive rea... |
| ltaddrp 12990 | Adding a positive number t... |
| difrp 12991 | Two ways to say one number... |
| elrpd 12992 | Membership in the set of p... |
| nnrpd 12993 | A positive integer is a po... |
| zgt1rpn0n1 12994 | An integer greater than 1 ... |
| rpred 12995 | A positive real is a real.... |
| rpxrd 12996 | A positive real is an exte... |
| rpcnd 12997 | A positive real is a compl... |
| rpgt0d 12998 | A positive real is greater... |
| rpge0d 12999 | A positive real is greater... |
| rpne0d 13000 | A positive real is nonzero... |
| rpregt0d 13001 | A positive real is real an... |
| rprege0d 13002 | A positive real is real an... |
| rprene0d 13003 | A positive real is a nonze... |
| rpcnne0d 13004 | A positive real is a nonze... |
| rpreccld 13005 | Closure law for reciprocat... |
| rprecred 13006 | Closure law for reciprocat... |
| rphalfcld 13007 | Closure law for half of a ... |
| reclt1d 13008 | The reciprocal of a positi... |
| recgt1d 13009 | The reciprocal of a positi... |
| rpaddcld 13010 | Closure law for addition o... |
| rpmulcld 13011 | Closure law for multiplica... |
| rpdivcld 13012 | Closure law for division o... |
| ltrecd 13013 | The reciprocal of both sid... |
| lerecd 13014 | The reciprocal of both sid... |
| ltrec1d 13015 | Reciprocal swap in a 'less... |
| lerec2d 13016 | Reciprocal swap in a 'less... |
| lediv2ad 13017 | Division of both sides of ... |
| ltdiv2d 13018 | Division of a positive num... |
| lediv2d 13019 | Division of a positive num... |
| ledivdivd 13020 | Invert ratios of positive ... |
| divge1 13021 | The ratio of a number over... |
| divlt1lt 13022 | A real number divided by a... |
| divle1le 13023 | A real number divided by a... |
| ledivge1le 13024 | If a number is less than o... |
| ge0p1rpd 13025 | A nonnegative number plus ... |
| rerpdivcld 13026 | Closure law for division o... |
| ltsubrpd 13027 | Subtracting a positive rea... |
| ltaddrpd 13028 | Adding a positive number t... |
| ltaddrp2d 13029 | Adding a positive number t... |
| ltmulgt11d 13030 | Multiplication by a number... |
| ltmulgt12d 13031 | Multiplication by a number... |
| gt0divd 13032 | Division of a positive num... |
| ge0divd 13033 | Division of a nonnegative ... |
| rpgecld 13034 | A number greater than or e... |
| divge0d 13035 | The ratio of nonnegative a... |
| ltmul1d 13036 | The ratio of nonnegative a... |
| ltmul2d 13037 | Multiplication of both sid... |
| lemul1d 13038 | Multiplication of both sid... |
| lemul2d 13039 | Multiplication of both sid... |
| ltdiv1d 13040 | Division of both sides of ... |
| lediv1d 13041 | Division of both sides of ... |
| ltmuldivd 13042 | 'Less than' relationship b... |
| ltmuldiv2d 13043 | 'Less than' relationship b... |
| lemuldivd 13044 | 'Less than or equal to' re... |
| lemuldiv2d 13045 | 'Less than or equal to' re... |
| ltdivmuld 13046 | 'Less than' relationship b... |
| ltdivmul2d 13047 | 'Less than' relationship b... |
| ledivmuld 13048 | 'Less than or equal to' re... |
| ledivmul2d 13049 | 'Less than or equal to' re... |
| ltmul1dd 13050 | The ratio of nonnegative a... |
| ltmul2dd 13051 | Multiplication of both sid... |
| ltdiv1dd 13052 | Division of both sides of ... |
| lediv1dd 13053 | Division of both sides of ... |
| lediv12ad 13054 | Comparison of ratio of two... |
| mul2lt0rlt0 13055 | If the result of a multipl... |
| mul2lt0rgt0 13056 | If the result of a multipl... |
| mul2lt0llt0 13057 | If the result of a multipl... |
| mul2lt0lgt0 13058 | If the result of a multipl... |
| mul2lt0bi 13059 | If the result of a multipl... |
| prodge0rd 13060 | Infer that a multiplicand ... |
| prodge0ld 13061 | Infer that a multiplier is... |
| ltdiv23d 13062 | Swap denominator with othe... |
| lediv23d 13063 | Swap denominator with othe... |
| lt2mul2divd 13064 | The ratio of nonnegative a... |
| nnledivrp 13065 | Division of a positive int... |
| nn0ledivnn 13066 | Division of a nonnegative ... |
| addlelt 13067 | If the sum of a real numbe... |
| ge2halflem1 13068 | Half of an integer greater... |
| ltxr 13075 | The 'less than' binary rel... |
| elxr 13076 | Membership in the set of e... |
| xrnemnf 13077 | An extended real other tha... |
| xrnepnf 13078 | An extended real other tha... |
| xrltnr 13079 | The extended real 'less th... |
| ltpnf 13080 | Any (finite) real is less ... |
| ltpnfd 13081 | Any (finite) real is less ... |
| 0ltpnf 13082 | Zero is less than plus inf... |
| mnflt 13083 | Minus infinity is less tha... |
| mnfltd 13084 | Minus infinity is less tha... |
| mnflt0 13085 | Minus infinity is less tha... |
| mnfltpnf 13086 | Minus infinity is less tha... |
| mnfltxr 13087 | Minus infinity is less tha... |
| pnfnlt 13088 | No extended real is greate... |
| nltmnf 13089 | No extended real is less t... |
| pnfge 13090 | Plus infinity is an upper ... |
| pnfged 13091 | Plus infinity is an upper ... |
| xnn0n0n1ge2b 13092 | An extended nonnegative in... |
| 0lepnf 13093 | 0 less than or equal to po... |
| xnn0ge0 13094 | An extended nonnegative in... |
| mnfle 13095 | Minus infinity is less tha... |
| mnfled 13096 | Minus infinity is less tha... |
| xrltnsym 13097 | Ordering on the extended r... |
| xrltnsym2 13098 | 'Less than' is antisymmetr... |
| xrlttri 13099 | Ordering on the extended r... |
| xrlttr 13100 | Ordering on the extended r... |
| xrltso 13101 | 'Less than' is a strict or... |
| xrlttri2 13102 | Trichotomy law for 'less t... |
| xrlttri3 13103 | Trichotomy law for 'less t... |
| xrleloe 13104 | 'Less than or equal' expre... |
| xrleltne 13105 | 'Less than or equal to' im... |
| xrltlen 13106 | 'Less than' expressed in t... |
| dfle2 13107 | Alternative definition of ... |
| dflt2 13108 | Alternative definition of ... |
| xrltle 13109 | 'Less than' implies 'less ... |
| xrltled 13110 | 'Less than' implies 'less ... |
| xrleid 13111 | 'Less than or equal to' is... |
| xrleidd 13112 | 'Less than or equal to' is... |
| xrletri 13113 | Trichotomy law for extende... |
| xrletri3 13114 | Trichotomy law for extende... |
| xrletrid 13115 | Trichotomy law for extende... |
| xrlelttr 13116 | Transitive law for orderin... |
| xrltletr 13117 | Transitive law for orderin... |
| xrletr 13118 | Transitive law for orderin... |
| xrlttrd 13119 | Transitive law for orderin... |
| xrlelttrd 13120 | Transitive law for orderin... |
| xrltletrd 13121 | Transitive law for orderin... |
| xrletrd 13122 | Transitive law for orderin... |
| xrltne 13123 | 'Less than' implies not eq... |
| nltpnft 13124 | An extended real is not le... |
| xgepnf 13125 | An extended real which is ... |
| ngtmnft 13126 | An extended real is not gr... |
| xlemnf 13127 | An extended real which is ... |
| xrrebnd 13128 | An extended real is real i... |
| xrre 13129 | A way of proving that an e... |
| xrre2 13130 | An extended real between t... |
| xrre3 13131 | A way of proving that an e... |
| ge0gtmnf 13132 | A nonnegative extended rea... |
| ge0nemnf 13133 | A nonnegative extended rea... |
| xrrege0 13134 | A nonnegative extended rea... |
| xrmax1 13135 | An extended real is less t... |
| xrmax2 13136 | An extended real is less t... |
| xrmin1 13137 | The minimum of two extende... |
| xrmin2 13138 | The minimum of two extende... |
| xrmaxeq 13139 | The maximum of two extende... |
| xrmineq 13140 | The minimum of two extende... |
| xrmaxlt 13141 | Two ways of saying the max... |
| xrltmin 13142 | Two ways of saying an exte... |
| xrmaxle 13143 | Two ways of saying the max... |
| xrlemin 13144 | Two ways of saying a numbe... |
| max1 13145 | A number is less than or e... |
| max1ALT 13146 | A number is less than or e... |
| max2 13147 | A number is less than or e... |
| 2resupmax 13148 | The supremum of two real n... |
| min1 13149 | The minimum of two numbers... |
| min2 13150 | The minimum of two numbers... |
| maxle 13151 | Two ways of saying the max... |
| lemin 13152 | Two ways of saying a numbe... |
| maxlt 13153 | Two ways of saying the max... |
| ltmin 13154 | Two ways of saying a numbe... |
| lemaxle 13155 | A real number which is les... |
| max0sub 13156 | Decompose a real number in... |
| ifle 13157 | An if statement transforms... |
| z2ge 13158 | There exists an integer gr... |
| qbtwnre 13159 | The rational numbers are d... |
| qbtwnxr 13160 | The rational numbers are d... |
| qsqueeze 13161 | If a nonnegative real is l... |
| qextltlem 13162 | Lemma for ~ qextlt and qex... |
| qextlt 13163 | An extensionality-like pro... |
| qextle 13164 | An extensionality-like pro... |
| xralrple 13165 | Show that ` A ` is less th... |
| alrple 13166 | Show that ` A ` is less th... |
| xnegeq 13167 | Equality of two extended n... |
| xnegex 13168 | A negative extended real e... |
| xnegpnf 13169 | Minus ` +oo ` . Remark of... |
| xnegmnf 13170 | Minus ` -oo ` . Remark of... |
| rexneg 13171 | Minus a real number. Rema... |
| xneg0 13172 | The negative of zero. (Co... |
| xnegcl 13173 | Closure of extended real n... |
| xnegneg 13174 | Extended real version of ~... |
| xneg11 13175 | Extended real version of ~... |
| xltnegi 13176 | Forward direction of ~ xlt... |
| xltneg 13177 | Extended real version of ~... |
| xleneg 13178 | Extended real version of ~... |
| xlt0neg1 13179 | Extended real version of ~... |
| xlt0neg2 13180 | Extended real version of ~... |
| xle0neg1 13181 | Extended real version of ~... |
| xle0neg2 13182 | Extended real version of ~... |
| xaddval 13183 | Value of the extended real... |
| xaddf 13184 | The extended real addition... |
| xmulval 13185 | Value of the extended real... |
| xaddpnf1 13186 | Addition of positive infin... |
| xaddpnf2 13187 | Addition of positive infin... |
| xaddmnf1 13188 | Addition of negative infin... |
| xaddmnf2 13189 | Addition of negative infin... |
| pnfaddmnf 13190 | Addition of positive and n... |
| mnfaddpnf 13191 | Addition of negative and p... |
| rexadd 13192 | The extended real addition... |
| rexsub 13193 | Extended real subtraction ... |
| rexaddd 13194 | The extended real addition... |
| xnn0xaddcl 13195 | The extended nonnegative i... |
| xaddnemnf 13196 | Closure of extended real a... |
| xaddnepnf 13197 | Closure of extended real a... |
| xnegid 13198 | Extended real version of ~... |
| xaddcl 13199 | The extended real addition... |
| xaddcom 13200 | The extended real addition... |
| xaddrid 13201 | Extended real version of ~... |
| xaddlid 13202 | Extended real version of ~... |
| xaddridd 13203 | ` 0 ` is a right identity ... |
| xnn0lem1lt 13204 | Extended nonnegative integ... |
| xnn0lenn0nn0 13205 | An extended nonnegative in... |
| xnn0le2is012 13206 | An extended nonnegative in... |
| xnn0xadd0 13207 | The sum of two extended no... |
| xnegdi 13208 | Extended real version of ~... |
| xaddass 13209 | Associativity of extended ... |
| xaddass2 13210 | Associativity of extended ... |
| xpncan 13211 | Extended real version of ~... |
| xnpcan 13212 | Extended real version of ~... |
| xleadd1a 13213 | Extended real version of ~... |
| xleadd2a 13214 | Commuted form of ~ xleadd1... |
| xleadd1 13215 | Weakened version of ~ xlea... |
| xltadd1 13216 | Extended real version of ~... |
| xltadd2 13217 | Extended real version of ~... |
| xaddge0 13218 | The sum of nonnegative ext... |
| xle2add 13219 | Extended real version of ~... |
| xlt2add 13220 | Extended real version of ~... |
| xsubge0 13221 | Extended real version of ~... |
| xposdif 13222 | Extended real version of ~... |
| xlesubadd 13223 | Under certain conditions, ... |
| xmullem 13224 | Lemma for ~ rexmul . (Con... |
| xmullem2 13225 | Lemma for ~ xmulneg1 . (C... |
| xmulcom 13226 | Extended real multiplicati... |
| xmul01 13227 | Extended real version of ~... |
| xmul02 13228 | Extended real version of ~... |
| xmulneg1 13229 | Extended real version of ~... |
| xmulneg2 13230 | Extended real version of ~... |
| rexmul 13231 | The extended real multipli... |
| xmulf 13232 | The extended real multipli... |
| xmulcl 13233 | Closure of extended real m... |
| xmulpnf1 13234 | Multiplication by plus inf... |
| xmulpnf2 13235 | Multiplication by plus inf... |
| xmulmnf1 13236 | Multiplication by minus in... |
| xmulmnf2 13237 | Multiplication by minus in... |
| xmulpnf1n 13238 | Multiplication by plus inf... |
| xmulrid 13239 | Extended real version of ~... |
| xmullid 13240 | Extended real version of ~... |
| xmulm1 13241 | Extended real version of ~... |
| xmulasslem2 13242 | Lemma for ~ xmulass . (Co... |
| xmulgt0 13243 | Extended real version of ~... |
| xmulge0 13244 | Extended real version of ~... |
| xmulasslem 13245 | Lemma for ~ xmulass . (Co... |
| xmulasslem3 13246 | Lemma for ~ xmulass . (Co... |
| xmulass 13247 | Associativity of the exten... |
| xlemul1a 13248 | Extended real version of ~... |
| xlemul2a 13249 | Extended real version of ~... |
| xlemul1 13250 | Extended real version of ~... |
| xlemul2 13251 | Extended real version of ~... |
| xltmul1 13252 | Extended real version of ~... |
| xltmul2 13253 | Extended real version of ~... |
| xadddilem 13254 | Lemma for ~ xadddi . (Con... |
| xadddi 13255 | Distributive property for ... |
| xadddir 13256 | Commuted version of ~ xadd... |
| xadddi2 13257 | The assumption that the mu... |
| xadddi2r 13258 | Commuted version of ~ xadd... |
| x2times 13259 | Extended real version of ~... |
| xnegcld 13260 | Closure of extended real n... |
| xaddcld 13261 | The extended real addition... |
| xmulcld 13262 | Closure of extended real m... |
| xadd4d 13263 | Rearrangement of 4 terms i... |
| xnn0add4d 13264 | Rearrangement of 4 terms i... |
| xrsupexmnf 13265 | Adding minus infinity to a... |
| xrinfmexpnf 13266 | Adding plus infinity to a ... |
| xrsupsslem 13267 | Lemma for ~ xrsupss . (Co... |
| xrinfmsslem 13268 | Lemma for ~ xrinfmss . (C... |
| xrsupss 13269 | Any subset of extended rea... |
| xrinfmss 13270 | Any subset of extended rea... |
| xrinfmss2 13271 | Any subset of extended rea... |
| xrub 13272 | By quantifying only over r... |
| supxr 13273 | The supremum of a set of e... |
| supxr2 13274 | The supremum of a set of e... |
| supxrcl 13275 | The supremum of an arbitra... |
| supxrun 13276 | The supremum of the union ... |
| supxrmnf 13277 | Adding minus infinity to a... |
| supxrpnf 13278 | The supremum of a set of e... |
| supxrunb1 13279 | The supremum of an unbound... |
| supxrunb2 13280 | The supremum of an unbound... |
| supxrbnd1 13281 | The supremum of a bounded-... |
| supxrbnd2 13282 | The supremum of a bounded-... |
| xrsup0 13283 | The supremum of an empty s... |
| supxrub 13284 | A member of a set of exten... |
| supxrlub 13285 | The supremum of a set of e... |
| supxrleub 13286 | The supremum of a set of e... |
| supxrre 13287 | The real and extended real... |
| supxrbnd 13288 | The supremum of a bounded-... |
| supxrgtmnf 13289 | The supremum of a nonempty... |
| supxrre1 13290 | The supremum of a nonempty... |
| supxrre2 13291 | The supremum of a nonempty... |
| supxrss 13292 | Smaller sets of extended r... |
| xrsupssd 13293 | Inequality deduction for s... |
| infxrcl 13294 | The infimum of an arbitrar... |
| infxrlb 13295 | A member of a set of exten... |
| infxrgelb 13296 | The infimum of a set of ex... |
| infxrre 13297 | The real and extended real... |
| infxrmnf 13298 | The infinimum of a set of ... |
| xrinf0 13299 | The infimum of the empty s... |
| infxrss 13300 | Larger sets of extended re... |
| reltre 13301 | For all real numbers there... |
| rpltrp 13302 | For all positive real numb... |
| reltxrnmnf 13303 | For all extended real numb... |
| infmremnf 13304 | The infimum of the reals i... |
| infmrp1 13305 | The infimum of the positiv... |
| ixxval 13314 | Value of the interval func... |
| elixx1 13315 | Membership in an interval ... |
| ixxf 13316 | The set of intervals of ex... |
| ixxex 13317 | The set of intervals of ex... |
| ixxssxr 13318 | The set of intervals of ex... |
| elixx3g 13319 | Membership in a set of ope... |
| ixxssixx 13320 | An interval is a subset of... |
| ixxdisj 13321 | Split an interval into dis... |
| ixxun 13322 | Split an interval into two... |
| ixxin 13323 | Intersection of two interv... |
| ixxss1 13324 | Subset relationship for in... |
| ixxss2 13325 | Subset relationship for in... |
| ixxss12 13326 | Subset relationship for in... |
| ixxub 13327 | Extract the upper bound of... |
| ixxlb 13328 | Extract the lower bound of... |
| iooex 13329 | The set of open intervals ... |
| iooval 13330 | Value of the open interval... |
| ioo0 13331 | An empty open interval of ... |
| ioon0 13332 | An open interval of extend... |
| ndmioo 13333 | The open interval function... |
| iooid 13334 | An open interval with iden... |
| elioo3g 13335 | Membership in a set of ope... |
| elioore 13336 | A member of an open interv... |
| lbioo 13337 | An open interval does not ... |
| ubioo 13338 | An open interval does not ... |
| iooval2 13339 | Value of the open interval... |
| iooin 13340 | Intersection of two open i... |
| iooss1 13341 | Subset relationship for op... |
| iooss2 13342 | Subset relationship for op... |
| iocval 13343 | Value of the open-below, c... |
| icoval 13344 | Value of the closed-below,... |
| iccval 13345 | Value of the closed interv... |
| elioo1 13346 | Membership in an open inte... |
| elioo2 13347 | Membership in an open inte... |
| elioc1 13348 | Membership in an open-belo... |
| elico1 13349 | Membership in a closed-bel... |
| elicc1 13350 | Membership in a closed int... |
| iccid 13351 | A closed interval with ide... |
| ico0 13352 | An empty open interval of ... |
| ioc0 13353 | An empty open interval of ... |
| icc0 13354 | An empty closed interval o... |
| dfrp2 13355 | Alternate definition of th... |
| elicod 13356 | Membership in a left-close... |
| icogelb 13357 | An element of a left-close... |
| icogelbd 13358 | An element of a left-close... |
| elicore 13359 | A member of a left-closed ... |
| ubioc1 13360 | The upper bound belongs to... |
| lbico1 13361 | The lower bound belongs to... |
| iccleub 13362 | An element of a closed int... |
| iccgelb 13363 | An element of a closed int... |
| elioo5 13364 | Membership in an open inte... |
| eliooxr 13365 | A nonempty open interval s... |
| eliooord 13366 | Ordering implied by a memb... |
| elioo4g 13367 | Membership in an open inte... |
| ioossre 13368 | An open interval is a set ... |
| ioosscn 13369 | An open interval is a set ... |
| elioc2 13370 | Membership in an open-belo... |
| elico2 13371 | Membership in a closed-bel... |
| elicc2 13372 | Membership in a closed rea... |
| elicc2i 13373 | Inference for membership i... |
| elicc4 13374 | Membership in a closed rea... |
| iccss 13375 | Condition for a closed int... |
| iccssioo 13376 | Condition for a closed int... |
| icossico 13377 | Condition for a closed-bel... |
| iccss2 13378 | Condition for a closed int... |
| iccssico 13379 | Condition for a closed int... |
| iccssioo2 13380 | Condition for a closed int... |
| iccssico2 13381 | Condition for a closed int... |
| icossico2d 13382 | Condition for a closed-bel... |
| ioomax 13383 | The open interval from min... |
| iccmax 13384 | The closed interval from m... |
| ioopos 13385 | The set of positive reals ... |
| ioorp 13386 | The set of positive reals ... |
| iooshf 13387 | Shift the arguments of the... |
| iocssre 13388 | A closed-above interval wi... |
| icossre 13389 | A closed-below interval wi... |
| iccssre 13390 | A closed real interval is ... |
| iccssxr 13391 | A closed interval is a set... |
| iocssxr 13392 | An open-below, closed-abov... |
| icossxr 13393 | A closed-below, open-above... |
| ioossicc 13394 | An open interval is a subs... |
| iccssred 13395 | A closed real interval is ... |
| eliccxr 13396 | A member of a closed inter... |
| icossicc 13397 | A closed-below, open-above... |
| iocssicc 13398 | A closed-above, open-below... |
| ioossico 13399 | An open interval is a subs... |
| iocssioo 13400 | Condition for a closed int... |
| icossioo 13401 | Condition for a closed int... |
| ioossioo 13402 | Condition for an open inte... |
| iccsupr 13403 | A nonempty subset of a clo... |
| elioopnf 13404 | Membership in an unbounded... |
| elioomnf 13405 | Membership in an unbounded... |
| elicopnf 13406 | Membership in a closed unb... |
| repos 13407 | Two ways of saying that a ... |
| ioof 13408 | The set of open intervals ... |
| iccf 13409 | The set of closed interval... |
| unirnioo 13410 | The union of the range of ... |
| dfioo2 13411 | Alternate definition of th... |
| ioorebas 13412 | Open intervals are element... |
| xrge0neqmnf 13413 | A nonnegative extended rea... |
| xrge0nre 13414 | An extended real which is ... |
| elrege0 13415 | The predicate "is a nonneg... |
| nn0rp0 13416 | A nonnegative integer is a... |
| rge0ssre 13417 | Nonnegative real numbers a... |
| elxrge0 13418 | Elementhood in the set of ... |
| 0e0icopnf 13419 | 0 is a member of ` ( 0 [,)... |
| 0e0iccpnf 13420 | 0 is a member of ` ( 0 [,]... |
| ge0addcl 13421 | The nonnegative reals are ... |
| ge0mulcl 13422 | The nonnegative reals are ... |
| ge0xaddcl 13423 | The nonnegative reals are ... |
| ge0xmulcl 13424 | The nonnegative extended r... |
| lbicc2 13425 | The lower bound of a close... |
| ubicc2 13426 | The upper bound of a close... |
| elicc01 13427 | Membership in the closed r... |
| elunitrn 13428 | The closed unit interval i... |
| elunitcn 13429 | The closed unit interval i... |
| 0elunit 13430 | Zero is an element of the ... |
| 1elunit 13431 | One is an element of the c... |
| iooneg 13432 | Membership in a negated op... |
| iccneg 13433 | Membership in a negated cl... |
| icoshft 13434 | A shifted real is a member... |
| icoshftf1o 13435 | Shifting a closed-below, o... |
| icoun 13436 | The union of two adjacent ... |
| icodisj 13437 | Adjacent left-closed right... |
| ioounsn 13438 | The union of an open inter... |
| snunioo 13439 | The closure of one end of ... |
| snunico 13440 | The closure of the open en... |
| snunioc 13441 | The closure of the open en... |
| prunioo 13442 | The closure of an open rea... |
| ioodisj 13443 | If the upper bound of one ... |
| ioojoin 13444 | Join two open intervals to... |
| difreicc 13445 | The class difference of ` ... |
| iccsplit 13446 | Split a closed interval in... |
| iccshftr 13447 | Membership in a shifted in... |
| iccshftri 13448 | Membership in a shifted in... |
| iccshftl 13449 | Membership in a shifted in... |
| iccshftli 13450 | Membership in a shifted in... |
| iccdil 13451 | Membership in a dilated in... |
| iccdili 13452 | Membership in a dilated in... |
| icccntr 13453 | Membership in a contracted... |
| icccntri 13454 | Membership in a contracted... |
| divelunit 13455 | A condition for a ratio to... |
| lincmb01cmp 13456 | A linear combination of tw... |
| iccf1o 13457 | Describe a bijection from ... |
| iccen 13458 | Any nontrivial closed inte... |
| xov1plusxeqvd 13459 | A complex number ` X ` is ... |
| unitssre 13460 | ` ( 0 [,] 1 ) ` is a subse... |
| unitsscn 13461 | The closed unit interval i... |
| supicc 13462 | Supremum of a bounded set ... |
| supiccub 13463 | The supremum of a bounded ... |
| supicclub 13464 | The supremum of a bounded ... |
| supicclub2 13465 | The supremum of a bounded ... |
| zltaddlt1le 13466 | The sum of an integer and ... |
| xnn0xrge0 13467 | An extended nonnegative in... |
| fzval 13470 | The value of a finite set ... |
| fzval2 13471 | An alternative way of expr... |
| fzf 13472 | Establish the domain and c... |
| elfz1 13473 | Membership in a finite set... |
| elfz 13474 | Membership in a finite set... |
| elfz2 13475 | Membership in a finite set... |
| elfzd 13476 | Membership in a finite set... |
| elfz5 13477 | Membership in a finite set... |
| elfz4 13478 | Membership in a finite set... |
| elfzuzb 13479 | Membership in a finite set... |
| eluzfz 13480 | Membership in a finite set... |
| elfzuz 13481 | A member of a finite set o... |
| elfzuz3 13482 | Membership in a finite set... |
| elfzel2 13483 | Membership in a finite set... |
| elfzel1 13484 | Membership in a finite set... |
| elfzelz 13485 | A member of a finite set o... |
| elfzelzd 13486 | A member of a finite set o... |
| fzssz 13487 | A finite sequence of integ... |
| elfzle1 13488 | A member of a finite set o... |
| elfzle2 13489 | A member of a finite set o... |
| elfzuz2 13490 | Implication of membership ... |
| elfzle3 13491 | Membership in a finite set... |
| eluzfz1 13492 | Membership in a finite set... |
| eluzfz2 13493 | Membership in a finite set... |
| eluzfz2b 13494 | Membership in a finite set... |
| elfz3 13495 | Membership in a finite set... |
| elfz1eq 13496 | Membership in a finite set... |
| elfzubelfz 13497 | If there is a member in a ... |
| peano2fzr 13498 | A Peano-postulate-like the... |
| fzn0 13499 | Properties of a finite int... |
| fz0 13500 | A finite set of sequential... |
| fzn 13501 | A finite set of sequential... |
| fzen 13502 | A shifted finite set of se... |
| fz1n 13503 | A 1-based finite set of se... |
| 0nelfz1 13504 | 0 is not an element of a f... |
| 0fz1 13505 | Two ways to say a finite 1... |
| fz10 13506 | There are no integers betw... |
| uzsubsubfz 13507 | Membership of an integer g... |
| uzsubsubfz1 13508 | Membership of an integer g... |
| ige3m2fz 13509 | Membership of an integer g... |
| fzsplit2 13510 | Split a finite interval of... |
| fzsplit 13511 | Split a finite interval of... |
| fzdisj 13512 | Condition for two finite i... |
| fz01en 13513 | 0-based and 1-based finite... |
| elfznn 13514 | A member of a finite set o... |
| elfz1end 13515 | A nonempty finite range of... |
| fz1ssnn 13516 | A finite set of positive i... |
| fznn0sub 13517 | Subtraction closure for a ... |
| fzmmmeqm 13518 | Subtracting the difference... |
| fzaddel 13519 | Membership of a sum in a f... |
| fzadd2 13520 | Membership of a sum in a f... |
| fzsubel 13521 | Membership of a difference... |
| fzopth 13522 | A finite set of sequential... |
| fzass4 13523 | Two ways to express a nond... |
| fzss1 13524 | Subset relationship for fi... |
| fzss2 13525 | Subset relationship for fi... |
| fzssuz 13526 | A finite set of sequential... |
| fzsn 13527 | A finite interval of integ... |
| fzssp1 13528 | Subset relationship for fi... |
| fzssnn 13529 | Finite sets of sequential ... |
| ssfzunsnext 13530 | A subset of a finite seque... |
| ssfzunsn 13531 | A subset of a finite seque... |
| fzsuc 13532 | Join a successor to the en... |
| fzpred 13533 | Join a predecessor to the ... |
| fzpreddisj 13534 | A finite set of sequential... |
| elfzp1 13535 | Append an element to a fin... |
| fzp1ss 13536 | Subset relationship for fi... |
| fzelp1 13537 | Membership in a set of seq... |
| fzp1elp1 13538 | Add one to an element of a... |
| fznatpl1 13539 | Shift membership in a fini... |
| fzpr 13540 | A finite interval of integ... |
| fztp 13541 | A finite interval of integ... |
| fz12pr 13542 | An integer range between 1... |
| fzsuc2 13543 | Join a successor to the en... |
| fzp1disj 13544 | ` ( M ... ( N + 1 ) ) ` is... |
| fzdifsuc 13545 | Remove a successor from th... |
| fzprval 13546 | Two ways of defining the f... |
| fztpval 13547 | Two ways of defining the f... |
| fzrev 13548 | Reversal of start and end ... |
| fzrev2 13549 | Reversal of start and end ... |
| fzrev2i 13550 | Reversal of start and end ... |
| fzrev3 13551 | The "complement" of a memb... |
| fzrev3i 13552 | The "complement" of a memb... |
| fznn 13553 | Finite set of sequential i... |
| elfz1b 13554 | Membership in a 1-based fi... |
| elfz1uz 13555 | Membership in a 1-based fi... |
| elfzm11 13556 | Membership in a finite set... |
| uzsplit 13557 | Express an upper integer s... |
| uzdisj 13558 | The first ` N ` elements o... |
| fseq1p1m1 13559 | Add/remove an item to/from... |
| fseq1m1p1 13560 | Add/remove an item to/from... |
| fz1sbc 13561 | Quantification over a one-... |
| elfzp1b 13562 | An integer is a member of ... |
| elfzm1b 13563 | An integer is a member of ... |
| elfzp12 13564 | Options for membership in ... |
| fzne1 13565 | Elementhood in a finite se... |
| fzdif1 13566 | Split the first element of... |
| fz0dif1 13567 | Split the first element of... |
| fzm1 13568 | Choices for an element of ... |
| fzneuz 13569 | No finite set of sequentia... |
| fznuz 13570 | Disjointness of the upper ... |
| uznfz 13571 | Disjointness of the upper ... |
| fzp1nel 13572 | One plus the upper bound o... |
| fzrevral 13573 | Reversal of scanning order... |
| fzrevral2 13574 | Reversal of scanning order... |
| fzrevral3 13575 | Reversal of scanning order... |
| fzshftral 13576 | Shift the scanning order i... |
| ige2m1fz1 13577 | Membership of an integer g... |
| ige2m1fz 13578 | Membership in a 0-based fi... |
| elfz2nn0 13579 | Membership in a finite set... |
| fznn0 13580 | Characterization of a fini... |
| elfznn0 13581 | A member of a finite set o... |
| elfz3nn0 13582 | The upper bound of a nonem... |
| fz0ssnn0 13583 | Finite sets of sequential ... |
| fz1ssfz0 13584 | Subset relationship for fi... |
| 0elfz 13585 | 0 is an element of a finit... |
| nn0fz0 13586 | A nonnegative integer is a... |
| elfz0add 13587 | An element of a finite set... |
| fz0sn 13588 | An integer range from 0 to... |
| fz0tp 13589 | An integer range from 0 to... |
| fz0to3un2pr 13590 | An integer range from 0 to... |
| fz0to4untppr 13591 | An integer range from 0 to... |
| fz0to5un2tp 13592 | An integer range from 0 to... |
| elfz0ubfz0 13593 | An element of a finite set... |
| elfz0fzfz0 13594 | A member of a finite set o... |
| fz0fzelfz0 13595 | If a member of a finite se... |
| fznn0sub2 13596 | Subtraction closure for a ... |
| uzsubfz0 13597 | Membership of an integer g... |
| fz0fzdiffz0 13598 | The difference of an integ... |
| elfzmlbm 13599 | Subtracting the lower boun... |
| elfzmlbp 13600 | Subtracting the lower boun... |
| fzctr 13601 | Lemma for theorems about t... |
| difelfzle 13602 | The difference of two inte... |
| difelfznle 13603 | The difference of two inte... |
| nn0split 13604 | Express the set of nonnega... |
| nn0disj 13605 | The first ` N + 1 ` elemen... |
| fz0sn0fz1 13606 | A finite set of sequential... |
| fvffz0 13607 | The function value of a fu... |
| 1fv 13608 | A function on a singleton.... |
| 4fvwrd4 13609 | The first four function va... |
| 2ffzeq 13610 | Two functions over 0-based... |
| preduz 13611 | The value of the predecess... |
| prednn 13612 | The value of the predecess... |
| prednn0 13613 | The value of the predecess... |
| predfz 13614 | Calculate the predecessor ... |
| fzof 13617 | Functionality of the half-... |
| elfzoel1 13618 | Reverse closure for half-o... |
| elfzoel2 13619 | Reverse closure for half-o... |
| elfzoelz 13620 | Reverse closure for half-o... |
| fzoval 13621 | Value of the half-open int... |
| elfzo 13622 | Membership in a half-open ... |
| elfzo2 13623 | Membership in a half-open ... |
| elfzouz 13624 | Membership in a half-open ... |
| nelfzo 13625 | An integer not being a mem... |
| fzolb 13626 | The left endpoint of a hal... |
| fzolb2 13627 | The left endpoint of a hal... |
| elfzole1 13628 | A member in a half-open in... |
| elfzolt2 13629 | A member in a half-open in... |
| elfzolt3 13630 | Membership in a half-open ... |
| elfzolt2b 13631 | A member in a half-open in... |
| elfzolt3b 13632 | Membership in a half-open ... |
| elfzop1le2 13633 | A member in a half-open in... |
| fzonel 13634 | A half-open range does not... |
| elfzouz2 13635 | The upper bound of a half-... |
| elfzofz 13636 | A half-open range is conta... |
| elfzo3 13637 | Express membership in a ha... |
| fzon0 13638 | A half-open integer interv... |
| fzossfz 13639 | A half-open range is conta... |
| fzossz 13640 | A half-open integer interv... |
| fzon 13641 | A half-open set of sequent... |
| fzo0n 13642 | A half-open range of nonne... |
| fzonlt0 13643 | A half-open integer range ... |
| fzo0 13644 | Half-open sets with equal ... |
| fzonnsub 13645 | If ` K < N ` then ` N - K ... |
| fzonnsub2 13646 | If ` M < N ` then ` N - M ... |
| fzoss1 13647 | Subset relationship for ha... |
| fzoss2 13648 | Subset relationship for ha... |
| fzossrbm1 13649 | Subset of a half-open rang... |
| fzo0ss1 13650 | Subset relationship for ha... |
| fzossnn0 13651 | A half-open integer range ... |
| fzospliti 13652 | One direction of splitting... |
| fzosplit 13653 | Split a half-open integer ... |
| fzodisj 13654 | Abutting half-open integer... |
| fzouzsplit 13655 | Split an upper integer set... |
| fzouzdisj 13656 | A half-open integer range ... |
| fzoun 13657 | A half-open integer range ... |
| fzodisjsn 13658 | A half-open integer range ... |
| prinfzo0 13659 | The intersection of a half... |
| lbfzo0 13660 | An integer is strictly gre... |
| elfzo0 13661 | Membership in a half-open ... |
| elfzo0z 13662 | Membership in a half-open ... |
| nn0p1elfzo 13663 | A nonnegative integer incr... |
| elfzo0le 13664 | A member in a half-open ra... |
| elfzolem1 13665 | A member in a half-open in... |
| elfzo0subge1 13666 | The difference of the uppe... |
| elfzo0suble 13667 | The difference of the uppe... |
| elfzonn0 13668 | A member of a half-open ra... |
| fzonmapblen 13669 | The result of subtracting ... |
| fzofzim 13670 | If a nonnegative integer i... |
| fz1fzo0m1 13671 | Translation of one between... |
| fzossnn 13672 | Half-open integer ranges s... |
| elfzo1 13673 | Membership in a half-open ... |
| fzo1lb 13674 | 1 is the left endpoint of ... |
| 1elfzo1 13675 | 1 is in a half-open range ... |
| fzo1fzo0n0 13676 | An integer between 1 and a... |
| fzo0n0 13677 | A half-open integer range ... |
| fzoaddel 13678 | Translate membership in a ... |
| fzo0addel 13679 | Translate membership in a ... |
| fzo0addelr 13680 | Translate membership in a ... |
| fzoaddel2 13681 | Translate membership in a ... |
| elfzoextl 13682 | Membership of an integer i... |
| elfzoext 13683 | Membership of an integer i... |
| elincfzoext 13684 | Membership of an increased... |
| fzosubel 13685 | Translate membership in a ... |
| fzosubel2 13686 | Membership in a translated... |
| fzosubel3 13687 | Membership in a translated... |
| eluzgtdifelfzo 13688 | Membership of the differen... |
| ige2m2fzo 13689 | Membership of an integer g... |
| fzocatel 13690 | Translate membership in a ... |
| ubmelfzo 13691 | If an integer in a 1-based... |
| elfzodifsumelfzo 13692 | If an integer is in a half... |
| elfzom1elp1fzo 13693 | Membership of an integer i... |
| elfzom1elfzo 13694 | Membership in a half-open ... |
| fzval3 13695 | Expressing a closed intege... |
| fz0add1fz1 13696 | Translate membership in a ... |
| fzosn 13697 | Expressing a singleton as ... |
| elfzomin 13698 | Membership of an integer i... |
| zpnn0elfzo 13699 | Membership of an integer i... |
| zpnn0elfzo1 13700 | Membership of an integer i... |
| fzosplitsnm1 13701 | Removing a singleton from ... |
| elfzonlteqm1 13702 | If an element of a half-op... |
| fzonn0p1 13703 | A nonnegative integer is a... |
| fzossfzop1 13704 | A half-open range of nonne... |
| fzonn0p1p1 13705 | If a nonnegative integer i... |
| elfzom1p1elfzo 13706 | Increasing an element of a... |
| fzo0ssnn0 13707 | Half-open integer ranges s... |
| fzo01 13708 | Expressing the singleton o... |
| fzo12sn 13709 | A 1-based half-open intege... |
| fzo13pr 13710 | A 1-based half-open intege... |
| fzo0to2pr 13711 | A half-open integer range ... |
| fz01pr 13712 | An integer range between 0... |
| fzo0to3tp 13713 | A half-open integer range ... |
| fzo0to42pr 13714 | A half-open integer range ... |
| fzo1to4tp 13715 | A half-open integer range ... |
| fzo0sn0fzo1 13716 | A half-open range of nonne... |
| elfzo0l 13717 | A member of a half-open ra... |
| fzoend 13718 | The endpoint of a half-ope... |
| fzo0end 13719 | The endpoint of a zero-bas... |
| ssfzo12 13720 | Subset relationship for ha... |
| ssfzoulel 13721 | If a half-open integer ran... |
| ssfzo12bi 13722 | Subset relationship for ha... |
| fzoopth 13723 | A half-open integer range ... |
| ubmelm1fzo 13724 | The result of subtracting ... |
| fzofzp1 13725 | If a point is in a half-op... |
| fzofzp1b 13726 | If a point is in a half-op... |
| elfzom1b 13727 | An integer is a member of ... |
| elfzom1elp1fzo1 13728 | Membership of a nonnegativ... |
| elfzo1elm1fzo0 13729 | Membership of a positive i... |
| elfzonelfzo 13730 | If an element of a half-op... |
| fzonfzoufzol 13731 | If an element of a half-op... |
| elfzomelpfzo 13732 | An integer increased by an... |
| elfznelfzo 13733 | A value in a finite set of... |
| elfznelfzob 13734 | A value in a finite set of... |
| peano2fzor 13735 | A Peano-postulate-like the... |
| fzosplitsn 13736 | Extending a half-open rang... |
| fzosplitpr 13737 | Extending a half-open inte... |
| fzosplitprm1 13738 | Extending a half-open inte... |
| fzosplitsni 13739 | Membership in a half-open ... |
| fzisfzounsn 13740 | A finite interval of integ... |
| elfzr 13741 | A member of a finite inter... |
| elfzlmr 13742 | A member of a finite inter... |
| elfz0lmr 13743 | A member of a finite inter... |
| fzostep1 13744 | Two possibilities for a nu... |
| fzoshftral 13745 | Shift the scanning order i... |
| fzind2 13746 | Induction on the integers ... |
| fvinim0ffz 13747 | The function values for th... |
| injresinjlem 13748 | Lemma for ~ injresinj . (... |
| injresinj 13749 | A function whose restricti... |
| subfzo0 13750 | The difference between two... |
| fvf1tp 13751 | Values of a one-to-one fun... |
| flval 13756 | Value of the floor (greate... |
| flcl 13757 | The floor (greatest intege... |
| reflcl 13758 | The floor (greatest intege... |
| fllelt 13759 | A basic property of the fl... |
| flcld 13760 | The floor (greatest intege... |
| flle 13761 | A basic property of the fl... |
| flltp1 13762 | A basic property of the fl... |
| fllep1 13763 | A basic property of the fl... |
| fraclt1 13764 | The fractional part of a r... |
| fracle1 13765 | The fractional part of a r... |
| fracge0 13766 | The fractional part of a r... |
| flge 13767 | The floor function value i... |
| fllt 13768 | The floor function value i... |
| flflp1 13769 | Move floor function betwee... |
| flid 13770 | An integer is its own floo... |
| flidm 13771 | The floor function is idem... |
| flidz 13772 | A real number equals its f... |
| flltnz 13773 | The floor of a non-integer... |
| flwordi 13774 | Ordering relation for the ... |
| flword2 13775 | Ordering relation for the ... |
| flval2 13776 | An alternate way to define... |
| flval3 13777 | An alternate way to define... |
| flbi 13778 | A condition equivalent to ... |
| flbi2 13779 | A condition equivalent to ... |
| adddivflid 13780 | The floor of a sum of an i... |
| ico01fl0 13781 | The floor of a real number... |
| flge0nn0 13782 | The floor of a number grea... |
| flge1nn 13783 | The floor of a number grea... |
| fldivnn0 13784 | The floor function of a di... |
| refldivcl 13785 | The floor function of a di... |
| divfl0 13786 | The floor of a fraction is... |
| fladdz 13787 | An integer can be moved in... |
| flzadd 13788 | An integer can be moved in... |
| flmulnn0 13789 | Move a nonnegative integer... |
| btwnzge0 13790 | A real bounded between an ... |
| 2tnp1ge0ge0 13791 | Two times an integer plus ... |
| flhalf 13792 | Ordering relation for the ... |
| fldivle 13793 | The floor function of a di... |
| fldivnn0le 13794 | The floor function of a di... |
| flltdivnn0lt 13795 | The floor function of a di... |
| ltdifltdiv 13796 | If the dividend of a divis... |
| fldiv4p1lem1div2 13797 | The floor of an integer eq... |
| fldiv4lem1div2uz2 13798 | The floor of an integer gr... |
| fldiv4lem1div2 13799 | The floor of a positive in... |
| ceilval 13800 | The value of the ceiling f... |
| dfceil2 13801 | Alternative definition of ... |
| ceilval2 13802 | The value of the ceiling f... |
| ceicl 13803 | The ceiling function retur... |
| ceilcl 13804 | Closure of the ceiling fun... |
| ceilcld 13805 | Closure of the ceiling fun... |
| ceige 13806 | The ceiling of a real numb... |
| ceilge 13807 | The ceiling of a real numb... |
| ceilged 13808 | The ceiling of a real numb... |
| ceim1l 13809 | One less than the ceiling ... |
| ceilm1lt 13810 | One less than the ceiling ... |
| ceile 13811 | The ceiling of a real numb... |
| ceille 13812 | The ceiling of a real numb... |
| ceilid 13813 | An integer is its own ceil... |
| ceilidz 13814 | A real number equals its c... |
| flleceil 13815 | The floor of a real number... |
| fleqceilz 13816 | A real number is an intege... |
| quoremz 13817 | Quotient and remainder of ... |
| quoremnn0 13818 | Quotient and remainder of ... |
| quoremnn0ALT 13819 | Alternate proof of ~ quore... |
| intfrac2 13820 | Decompose a real into inte... |
| intfracq 13821 | Decompose a rational numbe... |
| fldiv 13822 | Cancellation of the embedd... |
| fldiv2 13823 | Cancellation of an embedde... |
| fznnfl 13824 | Finite set of sequential i... |
| uzsup 13825 | An upper set of integers i... |
| ioopnfsup 13826 | An upper set of reals is u... |
| icopnfsup 13827 | An upper set of reals is u... |
| rpsup 13828 | The positive reals are unb... |
| resup 13829 | The real numbers are unbou... |
| xrsup 13830 | The extended real numbers ... |
| modval 13833 | The value of the modulo op... |
| modvalr 13834 | The value of the modulo op... |
| modcl 13835 | Closure law for the modulo... |
| flpmodeq 13836 | Partition of a division in... |
| modcld 13837 | Closure law for the modulo... |
| mod0 13838 | ` A mod B ` is zero iff ` ... |
| mulmod0 13839 | The product of an integer ... |
| negmod0 13840 | ` A ` is divisible by ` B ... |
| modge0 13841 | The modulo operation is no... |
| modlt 13842 | The modulo operation is le... |
| modelico 13843 | Modular reduction produces... |
| moddiffl 13844 | Value of the modulo operat... |
| moddifz 13845 | The modulo operation diffe... |
| modfrac 13846 | The fractional part of a n... |
| flmod 13847 | The floor function express... |
| intfrac 13848 | Break a number into its in... |
| zmod10 13849 | An integer modulo 1 is 0. ... |
| zmod1congr 13850 | Two arbitrary integers are... |
| modmulnn 13851 | Move a positive integer in... |
| modvalp1 13852 | The value of the modulo op... |
| zmodcl 13853 | Closure law for the modulo... |
| zmodcld 13854 | Closure law for the modulo... |
| zmodfz 13855 | An integer mod ` B ` lies ... |
| zmodfzo 13856 | An integer mod ` B ` lies ... |
| zmodfzp1 13857 | An integer mod ` B ` lies ... |
| modid 13858 | Identity law for modulo. ... |
| modid0 13859 | A positive real number mod... |
| modid2 13860 | Identity law for modulo. ... |
| zmodid2 13861 | Identity law for modulo re... |
| zmodidfzo 13862 | Identity law for modulo re... |
| zmodidfzoimp 13863 | Identity law for modulo re... |
| 0mod 13864 | Special case: 0 modulo a p... |
| 1mod 13865 | Special case: 1 modulo a r... |
| modabs 13866 | Absorption law for modulo.... |
| modabs2 13867 | Absorption law for modulo.... |
| modcyc 13868 | The modulo operation is pe... |
| modcyc2 13869 | The modulo operation is pe... |
| modadd1 13870 | Addition property of the m... |
| modaddb 13871 | Addition property of the m... |
| modaddid 13872 | The sums of two nonnegativ... |
| modaddabs 13873 | Absorption law for modulo.... |
| modaddmod 13874 | The sum of a real number m... |
| muladdmodid 13875 | The sum of a positive real... |
| mulp1mod1 13876 | The product of an integer ... |
| muladdmod 13877 | A real number is the sum o... |
| modmuladd 13878 | Decomposition of an intege... |
| modmuladdim 13879 | Implication of a decomposi... |
| modmuladdnn0 13880 | Implication of a decomposi... |
| negmod 13881 | The negation of a number m... |
| m1modnnsub1 13882 | Minus one modulo a positiv... |
| m1modge3gt1 13883 | Minus one modulo an intege... |
| addmodid 13884 | The sum of a positive inte... |
| addmodidr 13885 | The sum of a positive inte... |
| modadd2mod 13886 | The sum of a real number m... |
| modm1p1mod0 13887 | If a real number modulo a ... |
| modltm1p1mod 13888 | If a real number modulo a ... |
| modmul1 13889 | Multiplication property of... |
| modmul12d 13890 | Multiplication property of... |
| modnegd 13891 | Negation property of the m... |
| modadd12d 13892 | Additive property of the m... |
| modsub12d 13893 | Subtraction property of th... |
| modsubmod 13894 | The difference of a real n... |
| modsubmodmod 13895 | The difference of a real n... |
| 2txmodxeq0 13896 | Two times a positive real ... |
| 2submod 13897 | If a real number is betwee... |
| modifeq2int 13898 | If a nonnegative integer i... |
| modaddmodup 13899 | The sum of an integer modu... |
| modaddmodlo 13900 | The sum of an integer modu... |
| modmulmod 13901 | The product of a real numb... |
| modmulmodr 13902 | The product of an integer ... |
| modaddmulmod 13903 | The sum of a real number a... |
| moddi 13904 | Distribute multiplication ... |
| modsubdir 13905 | Distribute the modulo oper... |
| modeqmodmin 13906 | A real number equals the d... |
| modirr 13907 | A number modulo an irratio... |
| modfzo0difsn 13908 | For a number within a half... |
| modsumfzodifsn 13909 | The sum of a number within... |
| modlteq 13910 | Two nonnegative integers l... |
| addmodlteq 13911 | Two nonnegative integers l... |
| om2uz0i 13912 | The mapping ` G ` is a one... |
| om2uzsuci 13913 | The value of ` G ` (see ~ ... |
| om2uzuzi 13914 | The value ` G ` (see ~ om2... |
| om2uzlti 13915 | Less-than relation for ` G... |
| om2uzlt2i 13916 | The mapping ` G ` (see ~ o... |
| om2uzrani 13917 | Range of ` G ` (see ~ om2u... |
| om2uzf1oi 13918 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzisoi 13919 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzoi 13920 | An alternative definition ... |
| om2uzrdg 13921 | A helper lemma for the val... |
| uzrdglem 13922 | A helper lemma for the val... |
| uzrdgfni 13923 | The recursive definition g... |
| uzrdg0i 13924 | Initial value of a recursi... |
| uzrdgsuci 13925 | Successor value of a recur... |
| ltweuz 13926 | ` < ` is a well-founded re... |
| ltwenn 13927 | Less than well-orders the ... |
| ltwefz 13928 | Less than well-orders a se... |
| uzenom 13929 | An upper integer set is de... |
| uzinf 13930 | An upper integer set is in... |
| nnnfi 13931 | The set of positive intege... |
| uzrdgxfr 13932 | Transfer the value of the ... |
| fzennn 13933 | The cardinality of a finit... |
| fzen2 13934 | The cardinality of a finit... |
| cardfz 13935 | The cardinality of a finit... |
| hashgf1o 13936 | ` G ` maps ` _om ` one-to-... |
| fzfi 13937 | A finite interval of integ... |
| fzfid 13938 | Commonly used special case... |
| fzofi 13939 | Half-open integer sets are... |
| fsequb 13940 | The values of a finite rea... |
| fsequb2 13941 | The values of a finite rea... |
| fseqsupcl 13942 | The values of a finite rea... |
| fseqsupubi 13943 | The values of a finite rea... |
| nn0ennn 13944 | The nonnegative integers a... |
| nnenom 13945 | The set of positive intege... |
| nnct 13946 | ` NN ` is countable. (Con... |
| uzindi 13947 | Indirect strong induction ... |
| axdc4uzlem 13948 | Lemma for ~ axdc4uz . (Co... |
| axdc4uz 13949 | A version of ~ axdc4 that ... |
| ssnn0fi 13950 | A subset of the nonnegativ... |
| rabssnn0fi 13951 | A subset of the nonnegativ... |
| uzsinds 13952 | Strong (or "total") induct... |
| nnsinds 13953 | Strong (or "total") induct... |
| nn0sinds 13954 | Strong (or "total") induct... |
| fsuppmapnn0fiublem 13955 | Lemma for ~ fsuppmapnn0fiu... |
| fsuppmapnn0fiub 13956 | If all functions of a fini... |
| fsuppmapnn0fiubex 13957 | If all functions of a fini... |
| fsuppmapnn0fiub0 13958 | If all functions of a fini... |
| suppssfz 13959 | Condition for a function o... |
| fsuppmapnn0ub 13960 | If a function over the non... |
| fsuppmapnn0fz 13961 | If a function over the non... |
| mptnn0fsupp 13962 | A mapping from the nonnega... |
| mptnn0fsuppd 13963 | A mapping from the nonnega... |
| mptnn0fsuppr 13964 | A finitely supported mappi... |
| f13idfv 13965 | A one-to-one function with... |
| seqex 13968 | Existence of the sequence ... |
| seqeq1 13969 | Equality theorem for the s... |
| seqeq2 13970 | Equality theorem for the s... |
| seqeq3 13971 | Equality theorem for the s... |
| seqeq1d 13972 | Equality deduction for the... |
| seqeq2d 13973 | Equality deduction for the... |
| seqeq3d 13974 | Equality deduction for the... |
| seqeq123d 13975 | Equality deduction for the... |
| nfseq 13976 | Hypothesis builder for the... |
| seqval 13977 | Value of the sequence buil... |
| seqfn 13978 | The sequence builder funct... |
| seq1 13979 | Value of the sequence buil... |
| seq1i 13980 | Value of the sequence buil... |
| seqp1 13981 | Value of the sequence buil... |
| seqexw 13982 | Weak version of ~ seqex th... |
| seqp1d 13983 | Value of the sequence buil... |
| seqm1 13984 | Value of the sequence buil... |
| seqcl2 13985 | Closure properties of the ... |
| seqf2 13986 | Range of the recursive seq... |
| seqcl 13987 | Closure properties of the ... |
| seqf 13988 | Range of the recursive seq... |
| seqfveq2 13989 | Equality of sequences. (C... |
| seqfeq2 13990 | Equality of sequences. (C... |
| seqfveq 13991 | Equality of sequences. (C... |
| seqfeq 13992 | Equality of sequences. (C... |
| seqshft2 13993 | Shifting the index set of ... |
| seqres 13994 | Restricting its characteri... |
| serf 13995 | An infinite series of comp... |
| serfre 13996 | An infinite series of real... |
| monoord 13997 | Ordering relation for a mo... |
| monoord2 13998 | Ordering relation for a mo... |
| sermono 13999 | The partial sums in an inf... |
| seqsplit 14000 | Split a sequence into two ... |
| seq1p 14001 | Removing the first term fr... |
| seqcaopr3 14002 | Lemma for ~ seqcaopr2 . (... |
| seqcaopr2 14003 | The sum of two infinite se... |
| seqcaopr 14004 | The sum of two infinite se... |
| seqf1olem2a 14005 | Lemma for ~ seqf1o . (Con... |
| seqf1olem1 14006 | Lemma for ~ seqf1o . (Con... |
| seqf1olem2 14007 | Lemma for ~ seqf1o . (Con... |
| seqf1o 14008 | Rearrange a sum via an arb... |
| seradd 14009 | The sum of two infinite se... |
| sersub 14010 | The difference of two infi... |
| seqid3 14011 | A sequence that consists e... |
| seqid 14012 | Discarding the first few t... |
| seqid2 14013 | The last few partial sums ... |
| seqhomo 14014 | Apply a homomorphism to a ... |
| seqz 14015 | If the operation ` .+ ` ha... |
| seqfeq4 14016 | Equality of series under d... |
| seqfeq3 14017 | Equality of series under d... |
| seqdistr 14018 | The distributive property ... |
| ser0 14019 | The value of the partial s... |
| ser0f 14020 | A zero-valued infinite ser... |
| serge0 14021 | A finite sum of nonnegativ... |
| serle 14022 | Comparison of partial sums... |
| ser1const 14023 | Value of the partial serie... |
| seqof 14024 | Distribute function operat... |
| seqof2 14025 | Distribute function operat... |
| expval 14028 | Value of exponentiation to... |
| expnnval 14029 | Value of exponentiation to... |
| exp0 14030 | Value of a complex number ... |
| 0exp0e1 14031 | The zeroth power of zero e... |
| exp1 14032 | Value of a complex number ... |
| expp1 14033 | Value of a complex number ... |
| expneg 14034 | Value of a complex number ... |
| expneg2 14035 | Value of a complex number ... |
| expn1 14036 | A complex number raised to... |
| expcllem 14037 | Lemma for proving nonnegat... |
| expcl2lem 14038 | Lemma for proving integer ... |
| nnexpcl 14039 | Closure of exponentiation ... |
| nn0expcl 14040 | Closure of exponentiation ... |
| zexpcl 14041 | Closure of exponentiation ... |
| qexpcl 14042 | Closure of exponentiation ... |
| reexpcl 14043 | Closure of exponentiation ... |
| expcl 14044 | Closure law for nonnegativ... |
| rpexpcl 14045 | Closure law for integer ex... |
| qexpclz 14046 | Closure of integer exponen... |
| reexpclz 14047 | Closure of integer exponen... |
| expclzlem 14048 | Lemma for ~ expclz . (Con... |
| expclz 14049 | Closure law for integer ex... |
| m1expcl2 14050 | Closure of integer exponen... |
| m1expcl 14051 | Closure of exponentiation ... |
| zexpcld 14052 | Closure of exponentiation ... |
| nn0expcli 14053 | Closure of exponentiation ... |
| nn0sqcl 14054 | The square of a nonnegativ... |
| expm1t 14055 | Exponentiation in terms of... |
| 1exp 14056 | Value of 1 raised to an in... |
| expeq0 14057 | A positive integer power i... |
| expne0 14058 | A positive integer power i... |
| expne0i 14059 | An integer power is nonzer... |
| expgt0 14060 | A positive real raised to ... |
| expnegz 14061 | Value of a nonzero complex... |
| 0exp 14062 | Value of zero raised to a ... |
| expge0 14063 | A nonnegative real raised ... |
| expge1 14064 | A real greater than or equ... |
| expgt1 14065 | A real greater than 1 rais... |
| mulexp 14066 | Nonnegative integer expone... |
| mulexpz 14067 | Integer exponentiation of ... |
| exprec 14068 | Integer exponentiation of ... |
| expadd 14069 | Sum of exponents law for n... |
| expaddzlem 14070 | Lemma for ~ expaddz . (Co... |
| expaddz 14071 | Sum of exponents law for i... |
| expmul 14072 | Product of exponents law f... |
| expmulz 14073 | Product of exponents law f... |
| m1expeven 14074 | Exponentiation of negative... |
| expsub 14075 | Exponent subtraction law f... |
| expp1z 14076 | Value of a nonzero complex... |
| expm1 14077 | Value of a nonzero complex... |
| expdiv 14078 | Nonnegative integer expone... |
| sqval 14079 | Value of the square of a c... |
| sqneg 14080 | The square of the negative... |
| sqnegd 14081 | The square of the negative... |
| sqsubswap 14082 | Swap the order of subtract... |
| sqcl 14083 | Closure of square. (Contr... |
| sqmul 14084 | Distribution of squaring o... |
| sqeq0 14085 | A complex number is zero i... |
| sqdiv 14086 | Distribution of squaring o... |
| sqdivid 14087 | The square of a nonzero co... |
| sqne0 14088 | A complex number is nonzer... |
| resqcl 14089 | Closure of squaring in rea... |
| resqcld 14090 | Closure of squaring in rea... |
| sqgt0 14091 | The square of a nonzero re... |
| sqn0rp 14092 | The square of a nonzero re... |
| nnsqcl 14093 | The positive naturals are ... |
| zsqcl 14094 | Integers are closed under ... |
| qsqcl 14095 | The square of a rational i... |
| sq11 14096 | The square function is one... |
| nn0sq11 14097 | The square function is one... |
| lt2sq 14098 | The square function is inc... |
| le2sq 14099 | The square function is non... |
| le2sq2 14100 | The square function is non... |
| sqge0 14101 | The square of a real is no... |
| sqge0d 14102 | The square of a real is no... |
| zsqcl2 14103 | The square of an integer i... |
| 0expd 14104 | Value of zero raised to a ... |
| exp0d 14105 | Value of a complex number ... |
| exp1d 14106 | Value of a complex number ... |
| expeq0d 14107 | If a positive integer powe... |
| sqvald 14108 | Value of square. Inferenc... |
| sqcld 14109 | Closure of square. (Contr... |
| sqeq0d 14110 | A number is zero iff its s... |
| expcld 14111 | Closure law for nonnegativ... |
| expp1d 14112 | Value of a complex number ... |
| expaddd 14113 | Sum of exponents law for n... |
| expmuld 14114 | Product of exponents law f... |
| sqrecd 14115 | Square of reciprocal is re... |
| expclzd 14116 | Closure law for integer ex... |
| expne0d 14117 | A nonnegative integer powe... |
| expnegd 14118 | Value of a nonzero complex... |
| exprecd 14119 | An integer power of a reci... |
| expp1zd 14120 | Value of a nonzero complex... |
| expm1d 14121 | Value of a nonzero complex... |
| expsubd 14122 | Exponent subtraction law f... |
| sqmuld 14123 | Distribution of squaring o... |
| sqdivd 14124 | Distribution of squaring o... |
| expdivd 14125 | Nonnegative integer expone... |
| mulexpd 14126 | Nonnegative integer expone... |
| znsqcld 14127 | The square of a nonzero in... |
| reexpcld 14128 | Closure of exponentiation ... |
| expge0d 14129 | A nonnegative real raised ... |
| expge1d 14130 | A real greater than or equ... |
| ltexp2a 14131 | Exponent ordering relation... |
| expmordi 14132 | Base ordering relationship... |
| rpexpmord 14133 | Base ordering relationship... |
| expcan 14134 | Cancellation law for integ... |
| ltexp2 14135 | Strict ordering law for ex... |
| leexp2 14136 | Ordering law for exponenti... |
| leexp2a 14137 | Weak ordering relationship... |
| ltexp2r 14138 | The integer powers of a fi... |
| leexp2r 14139 | Weak ordering relationship... |
| leexp1a 14140 | Weak base ordering relatio... |
| leexp1ad 14141 | Weak base ordering relatio... |
| exple1 14142 | A real between 0 and 1 inc... |
| expubnd 14143 | An upper bound on ` A ^ N ... |
| sumsqeq0 14144 | The sum of two squres of r... |
| sqvali 14145 | Value of square. Inferenc... |
| sqcli 14146 | Closure of square. (Contr... |
| sqeq0i 14147 | A complex number is zero i... |
| sqrecii 14148 | The square of a reciprocal... |
| sqmuli 14149 | Distribution of squaring o... |
| sqdivi 14150 | Distribution of squaring o... |
| resqcli 14151 | Closure of square in reals... |
| sqgt0i 14152 | The square of a nonzero re... |
| sqge0i 14153 | The square of a real is no... |
| lt2sqi 14154 | The square function on non... |
| le2sqi 14155 | The square function on non... |
| sq11i 14156 | The square function is one... |
| sq0 14157 | The square of 0 is 0. (Co... |
| sq0i 14158 | If a number is zero, then ... |
| sq0id 14159 | If a number is zero, then ... |
| sq1 14160 | The square of 1 is 1. (Co... |
| neg1sqe1 14161 | The square of ` -u 1 ` is ... |
| sq2 14162 | The square of 2 is 4. (Co... |
| sq3 14163 | The square of 3 is 9. (Co... |
| sq4e2t8 14164 | The square of 4 is 2 times... |
| cu2 14165 | The cube of 2 is 8. (Cont... |
| irec 14166 | The reciprocal of ` _i ` .... |
| i2 14167 | ` _i ` squared. (Contribu... |
| i3 14168 | ` _i ` cubed. (Contribute... |
| i4 14169 | ` _i ` to the fourth power... |
| nnlesq 14170 | A positive integer is less... |
| zzlesq 14171 | An integer is less than or... |
| iexpcyc 14172 | Taking ` _i ` to the ` K `... |
| expnass 14173 | A counterexample showing t... |
| sqlecan 14174 | Cancel one factor of a squ... |
| subsq 14175 | Factor the difference of t... |
| subsq2 14176 | Express the difference of ... |
| binom2i 14177 | The square of a binomial. ... |
| subsqi 14178 | Factor the difference of t... |
| sqeqori 14179 | The squares of two complex... |
| subsq0i 14180 | The two solutions to the d... |
| sqeqor 14181 | The squares of two complex... |
| binom2 14182 | The square of a binomial. ... |
| binom2d 14183 | Deduction form of ~ binom2... |
| binom21 14184 | Special case of ~ binom2 w... |
| binom2sub 14185 | Expand the square of a sub... |
| binom2sub1 14186 | Special case of ~ binom2su... |
| binom2subi 14187 | Expand the square of a sub... |
| mulbinom2 14188 | The square of a binomial w... |
| binom3 14189 | The cube of a binomial. (... |
| sq01 14190 | If a complex number equals... |
| zesq 14191 | An integer is even iff its... |
| nnesq 14192 | A positive integer is even... |
| crreczi 14193 | Reciprocal of a complex nu... |
| bernneq 14194 | Bernoulli's inequality, du... |
| bernneq2 14195 | Variation of Bernoulli's i... |
| bernneq3 14196 | A corollary of ~ bernneq .... |
| expnbnd 14197 | Exponentiation with a base... |
| expnlbnd 14198 | The reciprocal of exponent... |
| expnlbnd2 14199 | The reciprocal of exponent... |
| expmulnbnd 14200 | Exponentiation with a base... |
| digit2 14201 | Two ways to express the ` ... |
| digit1 14202 | Two ways to express the ` ... |
| modexp 14203 | Exponentiation property of... |
| discr1 14204 | A nonnegative quadratic fo... |
| discr 14205 | If a quadratic polynomial ... |
| expnngt1 14206 | If an integer power with a... |
| expnngt1b 14207 | An integer power with an i... |
| sqoddm1div8 14208 | A squared odd number minus... |
| nnsqcld 14209 | The naturals are closed un... |
| nnexpcld 14210 | Closure of exponentiation ... |
| nn0expcld 14211 | Closure of exponentiation ... |
| rpexpcld 14212 | Closure law for exponentia... |
| ltexp2rd 14213 | The power of a positive nu... |
| reexpclzd 14214 | Closure of exponentiation ... |
| sqgt0d 14215 | The square of a nonzero re... |
| ltexp2d 14216 | Ordering relationship for ... |
| leexp2d 14217 | Ordering law for exponenti... |
| expcand 14218 | Ordering relationship for ... |
| leexp2ad 14219 | Ordering relationship for ... |
| leexp2rd 14220 | Ordering relationship for ... |
| lt2sqd 14221 | The square function on non... |
| le2sqd 14222 | The square function on non... |
| sq11d 14223 | The square function is one... |
| ltexp1d 14224 | Elevating to a positive po... |
| ltexp1dd 14225 | Raising both sides of 'les... |
| exp11nnd 14226 | The function elevating non... |
| mulsubdivbinom2 14227 | The square of a binomial w... |
| muldivbinom2 14228 | The square of a binomial w... |
| sq10 14229 | The square of 10 is 100. ... |
| sq10e99m1 14230 | The square of 10 is 99 plu... |
| 3dec 14231 | A "decimal constructor" wh... |
| nn0le2msqi 14232 | The square function on non... |
| nn0opthlem1 14233 | A rather pretty lemma for ... |
| nn0opthlem2 14234 | Lemma for ~ nn0opthi . (C... |
| nn0opthi 14235 | An ordered pair theorem fo... |
| nn0opth2i 14236 | An ordered pair theorem fo... |
| nn0opth2 14237 | An ordered pair theorem fo... |
| facnn 14240 | Value of the factorial fun... |
| fac0 14241 | The factorial of 0. (Cont... |
| fac1 14242 | The factorial of 1. (Cont... |
| facp1 14243 | The factorial of a success... |
| fac2 14244 | The factorial of 2. (Cont... |
| fac3 14245 | The factorial of 3. (Cont... |
| fac4 14246 | The factorial of 4. (Cont... |
| facnn2 14247 | Value of the factorial fun... |
| faccl 14248 | Closure of the factorial f... |
| faccld 14249 | Closure of the factorial f... |
| facmapnn 14250 | The factorial function res... |
| facne0 14251 | The factorial function is ... |
| facdiv 14252 | A positive integer divides... |
| facndiv 14253 | No positive integer (great... |
| facwordi 14254 | Ordering property of facto... |
| faclbnd 14255 | A lower bound for the fact... |
| faclbnd2 14256 | A lower bound for the fact... |
| faclbnd3 14257 | A lower bound for the fact... |
| faclbnd4lem1 14258 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4lem2 14259 | Lemma for ~ faclbnd4 . Us... |
| faclbnd4lem3 14260 | Lemma for ~ faclbnd4 . Th... |
| faclbnd4lem4 14261 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4 14262 | Variant of ~ faclbnd5 prov... |
| faclbnd5 14263 | The factorial function gro... |
| faclbnd6 14264 | Geometric lower bound for ... |
| facubnd 14265 | An upper bound for the fac... |
| facavg 14266 | The product of two factori... |
| bcval 14269 | Value of the binomial coef... |
| bcval2 14270 | Value of the binomial coef... |
| bcval3 14271 | Value of the binomial coef... |
| bcval4 14272 | Value of the binomial coef... |
| bcrpcl 14273 | Closure of the binomial co... |
| bccmpl 14274 | "Complementing" its second... |
| bcn0 14275 | ` N ` choose 0 is 1. Rema... |
| bc0k 14276 | The binomial coefficient "... |
| bcnn 14277 | ` N ` choose ` N ` is 1. ... |
| bcn1 14278 | Binomial coefficient: ` N ... |
| bcnp1n 14279 | Binomial coefficient: ` N ... |
| bcm1k 14280 | The proportion of one bino... |
| bcp1n 14281 | The proportion of one bino... |
| bcp1nk 14282 | The proportion of one bino... |
| bcval5 14283 | Write out the top and bott... |
| bcn2 14284 | Binomial coefficient: ` N ... |
| bcp1m1 14285 | Compute the binomial coeff... |
| bcpasc 14286 | Pascal's rule for the bino... |
| bccl 14287 | A binomial coefficient, in... |
| bccl2 14288 | A binomial coefficient, in... |
| bcn2m1 14289 | Compute the binomial coeff... |
| bcn2p1 14290 | Compute the binomial coeff... |
| permnn 14291 | The number of permutations... |
| bcnm1 14292 | The binomial coefficient o... |
| 4bc3eq4 14293 | The value of four choose t... |
| 4bc2eq6 14294 | The value of four choose t... |
| hashkf 14297 | The finite part of the siz... |
| hashgval 14298 | The value of the ` # ` fun... |
| hashginv 14299 | The converse of ` G ` maps... |
| hashinf 14300 | The value of the ` # ` fun... |
| hashbnd 14301 | If ` A ` has size bounded ... |
| hashfxnn0 14302 | The size function is a fun... |
| hashf 14303 | The size function maps all... |
| hashxnn0 14304 | The value of the hash func... |
| hashresfn 14305 | Restriction of the domain ... |
| dmhashres 14306 | Restriction of the domain ... |
| hashnn0pnf 14307 | The value of the hash func... |
| hashnnn0genn0 14308 | If the size of a set is no... |
| hashnemnf 14309 | The size of a set is never... |
| hashv01gt1 14310 | The size of a set is eithe... |
| hashfz1 14311 | The set ` ( 1 ... N ) ` ha... |
| hashen 14312 | Two finite sets have the s... |
| hasheni 14313 | Equinumerous sets have the... |
| hasheqf1o 14314 | The size of two finite set... |
| fiinfnf1o 14315 | There is no bijection betw... |
| hasheqf1oi 14316 | The size of two sets is eq... |
| hashf1rn 14317 | The size of a finite set w... |
| hasheqf1od 14318 | The size of two sets is eq... |
| fz1eqb 14319 | Two possibly-empty 1-based... |
| hashcard 14320 | The size function of the c... |
| hashcl 14321 | Closure of the ` # ` funct... |
| hashxrcl 14322 | Extended real closure of t... |
| hashclb 14323 | Reverse closure of the ` #... |
| nfile 14324 | The size of any infinite s... |
| hashvnfin 14325 | A set of finite size is a ... |
| hashnfinnn0 14326 | The size of an infinite se... |
| isfinite4 14327 | A finite set is equinumero... |
| hasheq0 14328 | Two ways of saying a set i... |
| hashneq0 14329 | Two ways of saying a set i... |
| hashgt0n0 14330 | If the size of a set is gr... |
| hashnncl 14331 | Positive natural closure o... |
| hash0 14332 | The empty set has size zer... |
| hashelne0d 14333 | A set with an element has ... |
| hashsng 14334 | The size of a singleton. ... |
| hashen1 14335 | A set has size 1 if and on... |
| hash1elsn 14336 | A set of size 1 with a kno... |
| hashrabrsn 14337 | The size of a restricted c... |
| hashrabsn01 14338 | The size of a restricted c... |
| hashrabsn1 14339 | If the size of a restricte... |
| hashfn 14340 | A function is equinumerous... |
| fseq1hash 14341 | The value of the size func... |
| hashgadd 14342 | ` G ` maps ordinal additio... |
| hashgval2 14343 | A short expression for the... |
| hashdom 14344 | Dominance relation for the... |
| hashdomi 14345 | Non-strict order relation ... |
| hashsdom 14346 | Strict dominance relation ... |
| hashun 14347 | The size of the union of d... |
| hashun2 14348 | The size of the union of f... |
| hashun3 14349 | The size of the union of f... |
| hashinfxadd 14350 | The extended real addition... |
| hashunx 14351 | The size of the union of d... |
| hashge0 14352 | The cardinality of a set i... |
| hashgt0 14353 | The cardinality of a nonem... |
| hashge1 14354 | The cardinality of a nonem... |
| 1elfz0hash 14355 | 1 is an element of the fin... |
| hashnn0n0nn 14356 | If a nonnegative integer i... |
| hashunsng 14357 | The size of the union of a... |
| hashunsngx 14358 | The size of the union of a... |
| hashunsnggt 14359 | The size of a set is great... |
| hashprg 14360 | The size of an unordered p... |
| elprchashprn2 14361 | If one element of an unord... |
| hashprb 14362 | The size of an unordered p... |
| hashprdifel 14363 | The elements of an unorder... |
| prhash2ex 14364 | There is (at least) one se... |
| hashle00 14365 | If the size of a set is le... |
| hashgt0elex 14366 | If the size of a set is gr... |
| hashgt0elexb 14367 | The size of a set is great... |
| hashp1i 14368 | Size of a finite ordinal. ... |
| hash1 14369 | Size of a finite ordinal. ... |
| hash2 14370 | Size of a finite ordinal. ... |
| hash3 14371 | Size of a finite ordinal. ... |
| hash4 14372 | Size of a finite ordinal. ... |
| pr0hash2ex 14373 | There is (at least) one se... |
| hashss 14374 | The size of a subset is le... |
| prsshashgt1 14375 | The size of a superset of ... |
| hashin 14376 | The size of the intersecti... |
| hashssdif 14377 | The size of the difference... |
| hashdif 14378 | The size of the difference... |
| hashdifsn 14379 | The size of the difference... |
| hashdifpr 14380 | The size of the difference... |
| hashsn01 14381 | The size of a singleton is... |
| hashsnle1 14382 | The size of a singleton is... |
| hashsnlei 14383 | Get an upper bound on a co... |
| hash1snb 14384 | The size of a set is 1 if ... |
| euhash1 14385 | The size of a set is 1 in ... |
| hash1n0 14386 | If the size of a set is 1 ... |
| hashgt12el 14387 | In a set with more than on... |
| hashgt12el2 14388 | In a set with more than on... |
| hashgt23el 14389 | A set with more than two e... |
| hashunlei 14390 | Get an upper bound on a co... |
| hashsslei 14391 | Get an upper bound on a co... |
| hashfz 14392 | Value of the numeric cardi... |
| fzsdom2 14393 | Condition for finite range... |
| hashfzo 14394 | Cardinality of a half-open... |
| hashfzo0 14395 | Cardinality of a half-open... |
| hashfzp1 14396 | Value of the numeric cardi... |
| hashfz0 14397 | Value of the numeric cardi... |
| hashxplem 14398 | Lemma for ~ hashxp . (Con... |
| hashxp 14399 | The size of the Cartesian ... |
| hashmap 14400 | The size of the set expone... |
| hashpw 14401 | The size of the power set ... |
| hashfun 14402 | A finite set is a function... |
| hashres 14403 | The number of elements of ... |
| hashreshashfun 14404 | The number of elements of ... |
| hashimarn 14405 | The size of the image of a... |
| hashimarni 14406 | If the size of the image o... |
| hashfundm 14407 | The size of a set function... |
| hashf1dmrn 14408 | The size of the domain of ... |
| hashf1dmcdm 14409 | The size of the domain of ... |
| resunimafz0 14410 | TODO-AV: Revise using ` F... |
| fnfz0hash 14411 | The size of a function on ... |
| ffz0hash 14412 | The size of a function on ... |
| fnfz0hashnn0 14413 | The size of a function on ... |
| ffzo0hash 14414 | The size of a function on ... |
| fnfzo0hash 14415 | The size of a function on ... |
| fnfzo0hashnn0 14416 | The value of the size func... |
| hashbclem 14417 | Lemma for ~ hashbc : induc... |
| hashbc 14418 | The binomial coefficient c... |
| hashfacen 14419 | The number of bijections b... |
| hashf1lem1 14420 | Lemma for ~ hashf1 . (Con... |
| hashf1lem2 14421 | Lemma for ~ hashf1 . (Con... |
| hashf1 14422 | The permutation number ` |... |
| hashfac 14423 | A factorial counts the num... |
| leiso 14424 | Two ways to write a strict... |
| leisorel 14425 | Version of ~ isorel for st... |
| fz1isolem 14426 | Lemma for ~ fz1iso . (Con... |
| fz1iso 14427 | Any finite ordered set has... |
| ishashinf 14428 | Any set that is not finite... |
| seqcoll 14429 | The function ` F ` contain... |
| seqcoll2 14430 | The function ` F ` contain... |
| phphashd 14431 | Corollary of the Pigeonhol... |
| phphashrd 14432 | Corollary of the Pigeonhol... |
| hashprlei 14433 | An unordered pair has at m... |
| hash2pr 14434 | A set of size two is an un... |
| hash2prde 14435 | A set of size two is an un... |
| hash2exprb 14436 | A set of size two is an un... |
| hash2prb 14437 | A set of size two is a pro... |
| prprrab 14438 | The set of proper pairs of... |
| nehash2 14439 | The cardinality of a set w... |
| hash2prd 14440 | A set of size two is an un... |
| hash2pwpr 14441 | If the size of a subset of... |
| hashle2pr 14442 | A nonempty set of size les... |
| hashle2prv 14443 | A nonempty subset of a pow... |
| pr2pwpr 14444 | The set of subsets of a pa... |
| hashge2el2dif 14445 | A set with size at least 2... |
| hashge2el2difr 14446 | A set with at least 2 diff... |
| hashge2el2difb 14447 | A set has size at least 2 ... |
| hashdmpropge2 14448 | The size of the domain of ... |
| hashtplei 14449 | An unordered triple has at... |
| hashtpg 14450 | The size of an unordered t... |
| hash7g 14451 | The size of an unordered s... |
| hashge3el3dif 14452 | A set with size at least 3... |
| elss2prb 14453 | An element of the set of s... |
| hash2sspr 14454 | A subset of size two is an... |
| exprelprel 14455 | If there is an element of ... |
| hash3tr 14456 | A set of size three is an ... |
| hash1to3 14457 | If the size of a set is be... |
| hash3tpde 14458 | A set of size three is an ... |
| hash3tpexb 14459 | A set of size three is an ... |
| hash3tpb 14460 | A set of size three is a p... |
| tpf1ofv0 14461 | The value of a one-to-one ... |
| tpf1ofv1 14462 | The value of a one-to-one ... |
| tpf1ofv2 14463 | The value of a one-to-one ... |
| tpf 14464 | A function into a (proper)... |
| tpfo 14465 | A function onto a (proper)... |
| tpf1o 14466 | A bijection onto a (proper... |
| fundmge2nop0 14467 | A function with a domain c... |
| fundmge2nop 14468 | A function with a domain c... |
| fun2dmnop0 14469 | A function with a domain c... |
| fun2dmnop 14470 | A function with a domain c... |
| hashdifsnp1 14471 | If the size of a set is a ... |
| fi1uzind 14472 | Properties of an ordered p... |
| brfi1uzind 14473 | Properties of a binary rel... |
| brfi1ind 14474 | Properties of a binary rel... |
| brfi1indALT 14475 | Alternate proof of ~ brfi1... |
| opfi1uzind 14476 | Properties of an ordered p... |
| opfi1ind 14477 | Properties of an ordered p... |
| iswrd 14480 | Property of being a word o... |
| wrdval 14481 | Value of the set of words ... |
| iswrdi 14482 | A zero-based sequence is a... |
| wrdf 14483 | A word is a zero-based seq... |
| wrdfd 14484 | A word is a zero-based seq... |
| iswrdb 14485 | A word over an alphabet is... |
| wrddm 14486 | The indices of a word (i.e... |
| sswrd 14487 | The set of words respects ... |
| snopiswrd 14488 | A singleton of an ordered ... |
| wrdexg 14489 | The set of words over a se... |
| wrdexb 14490 | The set of words over a se... |
| wrdexi 14491 | The set of words over a se... |
| wrdsymbcl 14492 | A symbol within a word ove... |
| wrdfn 14493 | A word is a function with ... |
| wrdv 14494 | A word over an alphabet is... |
| wrdlndm 14495 | The length of a word is no... |
| iswrdsymb 14496 | An arbitrary word is a wor... |
| wrdfin 14497 | A word is a finite set. (... |
| lencl 14498 | The length of a word is a ... |
| lennncl 14499 | The length of a nonempty w... |
| wrdffz 14500 | A word is a function from ... |
| wrdeq 14501 | Equality theorem for the s... |
| wrdeqi 14502 | Equality theorem for the s... |
| iswrddm0 14503 | A function with empty doma... |
| wrd0 14504 | The empty set is a word (t... |
| 0wrd0 14505 | The empty word is the only... |
| ffz0iswrd 14506 | A sequence with zero-based... |
| wrdsymb 14507 | A word is a word over the ... |
| nfwrd 14508 | Hypothesis builder for ` W... |
| csbwrdg 14509 | Class substitution for the... |
| wrdnval 14510 | Words of a fixed length ar... |
| wrdmap 14511 | Words as a mapping. (Cont... |
| hashwrdn 14512 | If there is only a finite ... |
| wrdnfi 14513 | If there is only a finite ... |
| wrdsymb0 14514 | A symbol at a position "ou... |
| wrdlenge1n0 14515 | A word with length at leas... |
| len0nnbi 14516 | The length of a word is a ... |
| wrdlenge2n0 14517 | A word with length at leas... |
| wrdsymb1 14518 | The first symbol of a none... |
| wrdlen1 14519 | A word of length 1 starts ... |
| fstwrdne 14520 | The first symbol of a none... |
| fstwrdne0 14521 | The first symbol of a none... |
| eqwrd 14522 | Two words are equal iff th... |
| elovmpowrd 14523 | Implications for the value... |
| elovmptnn0wrd 14524 | Implications for the value... |
| wrdred1 14525 | A word truncated by a symb... |
| wrdred1hash 14526 | The length of a word trunc... |
| lsw 14529 | Extract the last symbol of... |
| lsw0 14530 | The last symbol of an empt... |
| lsw0g 14531 | The last symbol of an empt... |
| lsw1 14532 | The last symbol of a word ... |
| lswcl 14533 | Closure of the last symbol... |
| lswlgt0cl 14534 | The last symbol of a nonem... |
| ccatfn 14537 | The concatenation operator... |
| ccatfval 14538 | Value of the concatenation... |
| ccatcl 14539 | The concatenation of two w... |
| ccatlen 14540 | The length of a concatenat... |
| ccat0 14541 | The concatenation of two w... |
| ccatval1 14542 | Value of a symbol in the l... |
| ccatval2 14543 | Value of a symbol in the r... |
| ccatval3 14544 | Value of a symbol in the r... |
| elfzelfzccat 14545 | An element of a finite set... |
| ccatvalfn 14546 | The concatenation of two w... |
| ccatsymb 14547 | The symbol at a given posi... |
| ccatfv0 14548 | The first symbol of a conc... |
| ccatval1lsw 14549 | The last symbol of the lef... |
| ccatval21sw 14550 | The first symbol of the ri... |
| ccatlid 14551 | Concatenation of a word by... |
| ccatrid 14552 | Concatenation of a word by... |
| ccatass 14553 | Associative law for concat... |
| ccatrn 14554 | The range of a concatenate... |
| ccatidid 14555 | Concatenation of the empty... |
| lswccatn0lsw 14556 | The last symbol of a word ... |
| lswccat0lsw 14557 | The last symbol of a word ... |
| ccatalpha 14558 | A concatenation of two arb... |
| ccatrcl1 14559 | Reverse closure of a conca... |
| ids1 14562 | Identity function protecti... |
| s1val 14563 | Value of a singleton word.... |
| s1rn 14564 | The range of a singleton w... |
| s1eq 14565 | Equality theorem for a sin... |
| s1eqd 14566 | Equality theorem for a sin... |
| s1cl 14567 | A singleton word is a word... |
| s1cld 14568 | A singleton word is a word... |
| s1prc 14569 | Value of a singleton word ... |
| s1cli 14570 | A singleton word is a word... |
| s1len 14571 | Length of a singleton word... |
| s1nz 14572 | A singleton word is not th... |
| s1dm 14573 | The domain of a singleton ... |
| s1dmALT 14574 | Alternate version of ~ s1d... |
| s1fv 14575 | Sole symbol of a singleton... |
| lsws1 14576 | The last symbol of a singl... |
| eqs1 14577 | A word of length 1 is a si... |
| wrdl1exs1 14578 | A word of length 1 is a si... |
| wrdl1s1 14579 | A word of length 1 is a si... |
| s111 14580 | The singleton word functio... |
| ccatws1cl 14581 | The concatenation of a wor... |
| ccatws1clv 14582 | The concatenation of a wor... |
| ccat2s1cl 14583 | The concatenation of two s... |
| ccats1alpha 14584 | A concatenation of a word ... |
| ccatws1len 14585 | The length of the concaten... |
| ccatws1lenp1b 14586 | The length of a word is ` ... |
| wrdlenccats1lenm1 14587 | The length of a word is th... |
| ccat2s1len 14588 | The length of the concaten... |
| ccatw2s1cl 14589 | The concatenation of a wor... |
| ccatw2s1len 14590 | The length of the concaten... |
| ccats1val1 14591 | Value of a symbol in the l... |
| ccats1val2 14592 | Value of the symbol concat... |
| ccat1st1st 14593 | The first symbol of a word... |
| ccat2s1p1 14594 | Extract the first of two c... |
| ccat2s1p2 14595 | Extract the second of two ... |
| ccatw2s1ass 14596 | Associative law for a conc... |
| ccatws1n0 14597 | The concatenation of a wor... |
| ccatws1ls 14598 | The last symbol of the con... |
| lswccats1 14599 | The last symbol of a word ... |
| lswccats1fst 14600 | The last symbol of a nonem... |
| ccatw2s1p1 14601 | Extract the symbol of the ... |
| ccatw2s1p2 14602 | Extract the second of two ... |
| ccat2s1fvw 14603 | Extract a symbol of a word... |
| ccat2s1fst 14604 | The first symbol of the co... |
| swrdnznd 14607 | The value of a subword ope... |
| swrdval 14608 | Value of a subword. (Cont... |
| swrd00 14609 | A zero length substring. ... |
| swrdcl 14610 | Closure of the subword ext... |
| swrdval2 14611 | Value of the subword extra... |
| swrdlen 14612 | Length of an extracted sub... |
| swrdfv 14613 | A symbol in an extracted s... |
| swrdfv0 14614 | The first symbol in an ext... |
| swrdf 14615 | A subword of a word is a f... |
| swrdvalfn 14616 | Value of the subword extra... |
| swrdrn 14617 | The range of a subword of ... |
| swrdlend 14618 | The value of the subword e... |
| swrdnd 14619 | The value of the subword e... |
| swrdnd2 14620 | Value of the subword extra... |
| swrdnnn0nd 14621 | The value of a subword ope... |
| swrdnd0 14622 | The value of a subword ope... |
| swrd0 14623 | A subword of an empty set ... |
| swrdrlen 14624 | Length of a right-anchored... |
| swrdlen2 14625 | Length of an extracted sub... |
| swrdfv2 14626 | A symbol in an extracted s... |
| swrdwrdsymb 14627 | A subword is a word over t... |
| swrdsb0eq 14628 | Two subwords with the same... |
| swrdsbslen 14629 | Two subwords with the same... |
| swrdspsleq 14630 | Two words have a common su... |
| swrds1 14631 | Extract a single symbol fr... |
| swrdlsw 14632 | Extract the last single sy... |
| ccatswrd 14633 | Joining two adjacent subwo... |
| swrdccat2 14634 | Recover the right half of ... |
| pfxnndmnd 14637 | The value of a prefix oper... |
| pfxval 14638 | Value of a prefix operatio... |
| pfx00 14639 | The zero length prefix is ... |
| pfx0 14640 | A prefix of an empty set i... |
| pfxval0 14641 | Value of a prefix operatio... |
| pfxcl 14642 | Closure of the prefix extr... |
| pfxmpt 14643 | Value of the prefix extrac... |
| pfxres 14644 | Value of the subword extra... |
| pfxf 14645 | A prefix of a word is a fu... |
| pfxfn 14646 | Value of the prefix extrac... |
| pfxfv 14647 | A symbol in a prefix of a ... |
| pfxlen 14648 | Length of a prefix. (Cont... |
| pfxid 14649 | A word is a prefix of itse... |
| pfxrn 14650 | The range of a prefix of a... |
| pfxn0 14651 | A prefix consisting of at ... |
| pfxnd 14652 | The value of a prefix oper... |
| pfxnd0 14653 | The value of a prefix oper... |
| pfxwrdsymb 14654 | A prefix of a word is a wo... |
| addlenrevpfx 14655 | The sum of the lengths of ... |
| addlenpfx 14656 | The sum of the lengths of ... |
| pfxfv0 14657 | The first symbol of a pref... |
| pfxtrcfv 14658 | A symbol in a word truncat... |
| pfxtrcfv0 14659 | The first symbol in a word... |
| pfxfvlsw 14660 | The last symbol in a nonem... |
| pfxeq 14661 | The prefixes of two words ... |
| pfxtrcfvl 14662 | The last symbol in a word ... |
| pfxsuffeqwrdeq 14663 | Two words are equal if and... |
| pfxsuff1eqwrdeq 14664 | Two (nonempty) words are e... |
| disjwrdpfx 14665 | Sets of words are disjoint... |
| ccatpfx 14666 | Concatenating a prefix wit... |
| pfxccat1 14667 | Recover the left half of a... |
| pfx1 14668 | The prefix of length one o... |
| swrdswrdlem 14669 | Lemma for ~ swrdswrd . (C... |
| swrdswrd 14670 | A subword of a subword is ... |
| pfxswrd 14671 | A prefix of a subword is a... |
| swrdpfx 14672 | A subword of a prefix is a... |
| pfxpfx 14673 | A prefix of a prefix is a ... |
| pfxpfxid 14674 | A prefix of a prefix with ... |
| pfxcctswrd 14675 | The concatenation of the p... |
| lenpfxcctswrd 14676 | The length of the concaten... |
| lenrevpfxcctswrd 14677 | The length of the concaten... |
| pfxlswccat 14678 | Reconstruct a nonempty wor... |
| ccats1pfxeq 14679 | The last symbol of a word ... |
| ccats1pfxeqrex 14680 | There exists a symbol such... |
| ccatopth 14681 | An ~ opth -like theorem fo... |
| ccatopth2 14682 | An ~ opth -like theorem fo... |
| ccatlcan 14683 | Concatenation of words is ... |
| ccatrcan 14684 | Concatenation of words is ... |
| wrdeqs1cat 14685 | Decompose a nonempty word ... |
| cats1un 14686 | Express a word with an ext... |
| wrdind 14687 | Perform induction over the... |
| wrd2ind 14688 | Perform induction over the... |
| swrdccatfn 14689 | The subword of a concatena... |
| swrdccatin1 14690 | The subword of a concatena... |
| pfxccatin12lem4 14691 | Lemma 4 for ~ pfxccatin12 ... |
| pfxccatin12lem2a 14692 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem1 14693 | Lemma 1 for ~ pfxccatin12 ... |
| swrdccatin2 14694 | The subword of a concatena... |
| pfxccatin12lem2c 14695 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem2 14696 | Lemma 2 for ~ pfxccatin12 ... |
| pfxccatin12lem3 14697 | Lemma 3 for ~ pfxccatin12 ... |
| pfxccatin12 14698 | The subword of a concatena... |
| pfxccat3 14699 | The subword of a concatena... |
| swrdccat 14700 | The subword of a concatena... |
| pfxccatpfx1 14701 | A prefix of a concatenatio... |
| pfxccatpfx2 14702 | A prefix of a concatenatio... |
| pfxccat3a 14703 | A prefix of a concatenatio... |
| swrdccat3blem 14704 | Lemma for ~ swrdccat3b . ... |
| swrdccat3b 14705 | A suffix of a concatenatio... |
| pfxccatid 14706 | A prefix of a concatenatio... |
| ccats1pfxeqbi 14707 | A word is a prefix of a wo... |
| swrdccatin1d 14708 | The subword of a concatena... |
| swrdccatin2d 14709 | The subword of a concatena... |
| pfxccatin12d 14710 | The subword of a concatena... |
| reuccatpfxs1lem 14711 | Lemma for ~ reuccatpfxs1 .... |
| reuccatpfxs1 14712 | There is a unique word hav... |
| reuccatpfxs1v 14713 | There is a unique word hav... |
| splval 14716 | Value of the substring rep... |
| splcl 14717 | Closure of the substring r... |
| splid 14718 | Splicing a subword for the... |
| spllen 14719 | The length of a splice. (... |
| splfv1 14720 | Symbols to the left of a s... |
| splfv2a 14721 | Symbols within the replace... |
| splval2 14722 | Value of a splice, assumin... |
| revval 14725 | Value of the word reversin... |
| revcl 14726 | The reverse of a word is a... |
| revlen 14727 | The reverse of a word has ... |
| revfv 14728 | Reverse of a word at a poi... |
| rev0 14729 | The empty word is its own ... |
| revs1 14730 | Singleton words are their ... |
| revccat 14731 | Antiautomorphic property o... |
| revrev 14732 | Reversal is an involution ... |
| reps 14735 | Construct a function mappi... |
| repsundef 14736 | A function mapping a half-... |
| repsconst 14737 | Construct a function mappi... |
| repsf 14738 | The constructed function m... |
| repswsymb 14739 | The symbols of a "repeated... |
| repsw 14740 | A function mapping a half-... |
| repswlen 14741 | The length of a "repeated ... |
| repsw0 14742 | The "repeated symbol word"... |
| repsdf2 14743 | Alternative definition of ... |
| repswsymball 14744 | All the symbols of a "repe... |
| repswsymballbi 14745 | A word is a "repeated symb... |
| repswfsts 14746 | The first symbol of a none... |
| repswlsw 14747 | The last symbol of a nonem... |
| repsw1 14748 | The "repeated symbol word"... |
| repswswrd 14749 | A subword of a "repeated s... |
| repswpfx 14750 | A prefix of a repeated sym... |
| repswccat 14751 | The concatenation of two "... |
| repswrevw 14752 | The reverse of a "repeated... |
| cshfn 14755 | Perform a cyclical shift f... |
| cshword 14756 | Perform a cyclical shift f... |
| cshnz 14757 | A cyclical shift is the em... |
| 0csh0 14758 | Cyclically shifting an emp... |
| cshw0 14759 | A word cyclically shifted ... |
| cshwmodn 14760 | Cyclically shifting a word... |
| cshwsublen 14761 | Cyclically shifting a word... |
| cshwn 14762 | A word cyclically shifted ... |
| cshwcl 14763 | A cyclically shifted word ... |
| cshwlen 14764 | The length of a cyclically... |
| cshwf 14765 | A cyclically shifted word ... |
| cshwfn 14766 | A cyclically shifted word ... |
| cshwrn 14767 | The range of a cyclically ... |
| cshwidxmod 14768 | The symbol at a given inde... |
| cshwidxmodr 14769 | The symbol at a given inde... |
| cshwidx0mod 14770 | The symbol at index 0 of a... |
| cshwidx0 14771 | The symbol at index 0 of a... |
| cshwidxm1 14772 | The symbol at index ((n-N)... |
| cshwidxm 14773 | The symbol at index (n-N) ... |
| cshwidxn 14774 | The symbol at index (n-1) ... |
| cshf1 14775 | Cyclically shifting a word... |
| cshinj 14776 | If a word is injectiv (reg... |
| repswcshw 14777 | A cyclically shifted "repe... |
| 2cshw 14778 | Cyclically shifting a word... |
| 2cshwid 14779 | Cyclically shifting a word... |
| lswcshw 14780 | The last symbol of a word ... |
| 2cshwcom 14781 | Cyclically shifting a word... |
| cshwleneq 14782 | If the results of cyclical... |
| 3cshw 14783 | Cyclically shifting a word... |
| cshweqdif2 14784 | If cyclically shifting two... |
| cshweqdifid 14785 | If cyclically shifting a w... |
| cshweqrep 14786 | If cyclically shifting a w... |
| cshw1 14787 | If cyclically shifting a w... |
| cshw1repsw 14788 | If cyclically shifting a w... |
| cshwsexa 14789 | The class of (different!) ... |
| cshwsexaOLD 14790 | Obsolete version of ~ cshw... |
| 2cshwcshw 14791 | If a word is a cyclically ... |
| scshwfzeqfzo 14792 | For a nonempty word the se... |
| cshwcshid 14793 | A cyclically shifted word ... |
| cshwcsh2id 14794 | A cyclically shifted word ... |
| cshimadifsn 14795 | The image of a cyclically ... |
| cshimadifsn0 14796 | The image of a cyclically ... |
| wrdco 14797 | Mapping a word by a functi... |
| lenco 14798 | Length of a mapped word is... |
| s1co 14799 | Mapping of a singleton wor... |
| revco 14800 | Mapping of words (i.e., a ... |
| ccatco 14801 | Mapping of words commutes ... |
| cshco 14802 | Mapping of words commutes ... |
| swrdco 14803 | Mapping of words commutes ... |
| pfxco 14804 | Mapping of words commutes ... |
| lswco 14805 | Mapping of (nonempty) word... |
| repsco 14806 | Mapping of words commutes ... |
| cats1cld 14821 | Closure of concatenation w... |
| cats1co 14822 | Closure of concatenation w... |
| cats1cli 14823 | Closure of concatenation w... |
| cats1fvn 14824 | The last symbol of a conca... |
| cats1fv 14825 | A symbol other than the la... |
| cats1len 14826 | The length of concatenatio... |
| cats1cat 14827 | Closure of concatenation w... |
| cats2cat 14828 | Closure of concatenation o... |
| s2eqd 14829 | Equality theorem for a dou... |
| s3eqd 14830 | Equality theorem for a len... |
| s4eqd 14831 | Equality theorem for a len... |
| s5eqd 14832 | Equality theorem for a len... |
| s6eqd 14833 | Equality theorem for a len... |
| s7eqd 14834 | Equality theorem for a len... |
| s8eqd 14835 | Equality theorem for a len... |
| s3eq2 14836 | Equality theorem for a len... |
| s2cld 14837 | A doubleton word is a word... |
| s3cld 14838 | A length 3 string is a wor... |
| s4cld 14839 | A length 4 string is a wor... |
| s5cld 14840 | A length 5 string is a wor... |
| s6cld 14841 | A length 6 string is a wor... |
| s7cld 14842 | A length 7 string is a wor... |
| s8cld 14843 | A length 7 string is a wor... |
| s2cl 14844 | A doubleton word is a word... |
| s3cl 14845 | A length 3 string is a wor... |
| s2cli 14846 | A doubleton word is a word... |
| s3cli 14847 | A length 3 string is a wor... |
| s4cli 14848 | A length 4 string is a wor... |
| s5cli 14849 | A length 5 string is a wor... |
| s6cli 14850 | A length 6 string is a wor... |
| s7cli 14851 | A length 7 string is a wor... |
| s8cli 14852 | A length 8 string is a wor... |
| s2fv0 14853 | Extract the first symbol f... |
| s2fv1 14854 | Extract the second symbol ... |
| s2len 14855 | The length of a doubleton ... |
| s2dm 14856 | The domain of a doubleton ... |
| s3fv0 14857 | Extract the first symbol f... |
| s3fv1 14858 | Extract the second symbol ... |
| s3fv2 14859 | Extract the third symbol f... |
| s3len 14860 | The length of a length 3 s... |
| s4fv0 14861 | Extract the first symbol f... |
| s4fv1 14862 | Extract the second symbol ... |
| s4fv2 14863 | Extract the third symbol f... |
| s4fv3 14864 | Extract the fourth symbol ... |
| s4len 14865 | The length of a length 4 s... |
| s5len 14866 | The length of a length 5 s... |
| s6len 14867 | The length of a length 6 s... |
| s7len 14868 | The length of a length 7 s... |
| s8len 14869 | The length of a length 8 s... |
| lsws2 14870 | The last symbol of a doubl... |
| lsws3 14871 | The last symbol of a 3 let... |
| lsws4 14872 | The last symbol of a 4 let... |
| s2prop 14873 | A length 2 word is an unor... |
| s2dmALT 14874 | Alternate version of ~ s2d... |
| s3tpop 14875 | A length 3 word is an unor... |
| s4prop 14876 | A length 4 word is a union... |
| s3fn 14877 | A length 3 word is a funct... |
| funcnvs1 14878 | The converse of a singleto... |
| funcnvs2 14879 | The converse of a length 2... |
| funcnvs3 14880 | The converse of a length 3... |
| funcnvs4 14881 | The converse of a length 4... |
| s2f1o 14882 | A length 2 word with mutua... |
| f1oun2prg 14883 | A union of unordered pairs... |
| s4f1o 14884 | A length 4 word with mutua... |
| s4dom 14885 | The domain of a length 4 w... |
| s2co 14886 | Mapping a doubleton word b... |
| s3co 14887 | Mapping a length 3 string ... |
| s0s1 14888 | Concatenation of fixed len... |
| s1s2 14889 | Concatenation of fixed len... |
| s1s3 14890 | Concatenation of fixed len... |
| s1s4 14891 | Concatenation of fixed len... |
| s1s5 14892 | Concatenation of fixed len... |
| s1s6 14893 | Concatenation of fixed len... |
| s1s7 14894 | Concatenation of fixed len... |
| s2s2 14895 | Concatenation of fixed len... |
| s4s2 14896 | Concatenation of fixed len... |
| s4s3 14897 | Concatenation of fixed len... |
| s4s4 14898 | Concatenation of fixed len... |
| s3s4 14899 | Concatenation of fixed len... |
| s2s5 14900 | Concatenation of fixed len... |
| s5s2 14901 | Concatenation of fixed len... |
| s2eq2s1eq 14902 | Two length 2 words are equ... |
| s2eq2seq 14903 | Two length 2 words are equ... |
| s3eqs2s1eq 14904 | Two length 3 words are equ... |
| s3eq3seq 14905 | Two length 3 words are equ... |
| swrds2 14906 | Extract two adjacent symbo... |
| swrds2m 14907 | Extract two adjacent symbo... |
| wrdlen2i 14908 | Implications of a word of ... |
| wrd2pr2op 14909 | A word of length two repre... |
| wrdlen2 14910 | A word of length two. (Co... |
| wrdlen2s2 14911 | A word of length two as do... |
| wrdl2exs2 14912 | A word of length two is a ... |
| pfx2 14913 | A prefix of length two. (... |
| wrd3tpop 14914 | A word of length three rep... |
| wrdlen3s3 14915 | A word of length three as ... |
| repsw2 14916 | The "repeated symbol word"... |
| repsw3 14917 | The "repeated symbol word"... |
| swrd2lsw 14918 | Extract the last two symbo... |
| 2swrd2eqwrdeq 14919 | Two words of length at lea... |
| ccatw2s1ccatws2 14920 | The concatenation of a wor... |
| ccat2s1fvwALT 14921 | Alternate proof of ~ ccat2... |
| wwlktovf 14922 | Lemma 1 for ~ wrd2f1tovbij... |
| wwlktovf1 14923 | Lemma 2 for ~ wrd2f1tovbij... |
| wwlktovfo 14924 | Lemma 3 for ~ wrd2f1tovbij... |
| wwlktovf1o 14925 | Lemma 4 for ~ wrd2f1tovbij... |
| wrd2f1tovbij 14926 | There is a bijection betwe... |
| eqwrds3 14927 | A word is equal with a len... |
| wrdl3s3 14928 | A word of length 3 is a le... |
| s2rn 14929 | Range of a length 2 string... |
| s3rn 14930 | Range of a length 3 string... |
| s7rn 14931 | Range of a length 7 string... |
| s7f1o 14932 | A length 7 word with mutua... |
| s3sndisj 14933 | The singletons consisting ... |
| s3iunsndisj 14934 | The union of singletons co... |
| ofccat 14935 | Letterwise operations on w... |
| ofs1 14936 | Letterwise operations on a... |
| ofs2 14937 | Letterwise operations on a... |
| coss12d 14938 | Subset deduction for compo... |
| trrelssd 14939 | The composition of subclas... |
| xpcogend 14940 | The most interesting case ... |
| xpcoidgend 14941 | If two classes are not dis... |
| cotr2g 14942 | Two ways of saying that th... |
| cotr2 14943 | Two ways of saying a relat... |
| cotr3 14944 | Two ways of saying a relat... |
| coemptyd 14945 | Deduction about compositio... |
| xptrrel 14946 | The cross product is alway... |
| 0trrel 14947 | The empty class is a trans... |
| cleq1lem 14948 | Equality implies bijection... |
| cleq1 14949 | Equality of relations impl... |
| clsslem 14950 | The closure of a subclass ... |
| trcleq1 14955 | Equality of relations impl... |
| trclsslem 14956 | The transitive closure (as... |
| trcleq2lem 14957 | Equality implies bijection... |
| cvbtrcl 14958 | Change of bound variable i... |
| trcleq12lem 14959 | Equality implies bijection... |
| trclexlem 14960 | Existence of relation impl... |
| trclublem 14961 | If a relation exists then ... |
| trclubi 14962 | The Cartesian product of t... |
| trclubgi 14963 | The union with the Cartesi... |
| trclub 14964 | The Cartesian product of t... |
| trclubg 14965 | The union with the Cartesi... |
| trclfv 14966 | The transitive closure of ... |
| brintclab 14967 | Two ways to express a bina... |
| brtrclfv 14968 | Two ways of expressing the... |
| brcnvtrclfv 14969 | Two ways of expressing the... |
| brtrclfvcnv 14970 | Two ways of expressing the... |
| brcnvtrclfvcnv 14971 | Two ways of expressing the... |
| trclfvss 14972 | The transitive closure (as... |
| trclfvub 14973 | The transitive closure of ... |
| trclfvlb 14974 | The transitive closure of ... |
| trclfvcotr 14975 | The transitive closure of ... |
| trclfvlb2 14976 | The transitive closure of ... |
| trclfvlb3 14977 | The transitive closure of ... |
| cotrtrclfv 14978 | The transitive closure of ... |
| trclidm 14979 | The transitive closure of ... |
| trclun 14980 | Transitive closure of a un... |
| trclfvg 14981 | The value of the transitiv... |
| trclfvcotrg 14982 | The value of the transitiv... |
| reltrclfv 14983 | The transitive closure of ... |
| dmtrclfv 14984 | The domain of the transiti... |
| reldmrelexp 14987 | The domain of the repeated... |
| relexp0g 14988 | A relation composed zero t... |
| relexp0 14989 | A relation composed zero t... |
| relexp0d 14990 | A relation composed zero t... |
| relexpsucnnr 14991 | A reduction for relation e... |
| relexp1g 14992 | A relation composed once i... |
| dfid5 14993 | Identity relation is equal... |
| dfid6 14994 | Identity relation expresse... |
| relexp1d 14995 | A relation composed once i... |
| relexpsucnnl 14996 | A reduction for relation e... |
| relexpsucl 14997 | A reduction for relation e... |
| relexpsucr 14998 | A reduction for relation e... |
| relexpsucrd 14999 | A reduction for relation e... |
| relexpsucld 15000 | A reduction for relation e... |
| relexpcnv 15001 | Commutation of converse an... |
| relexpcnvd 15002 | Commutation of converse an... |
| relexp0rel 15003 | The exponentiation of a cl... |
| relexprelg 15004 | The exponentiation of a cl... |
| relexprel 15005 | The exponentiation of a re... |
| relexpreld 15006 | The exponentiation of a re... |
| relexpnndm 15007 | The domain of an exponenti... |
| relexpdmg 15008 | The domain of an exponenti... |
| relexpdm 15009 | The domain of an exponenti... |
| relexpdmd 15010 | The domain of an exponenti... |
| relexpnnrn 15011 | The range of an exponentia... |
| relexprng 15012 | The range of an exponentia... |
| relexprn 15013 | The range of an exponentia... |
| relexprnd 15014 | The range of an exponentia... |
| relexpfld 15015 | The field of an exponentia... |
| relexpfldd 15016 | The field of an exponentia... |
| relexpaddnn 15017 | Relation composition becom... |
| relexpuzrel 15018 | The exponentiation of a cl... |
| relexpaddg 15019 | Relation composition becom... |
| relexpaddd 15020 | Relation composition becom... |
| rtrclreclem1 15023 | The reflexive, transitive ... |
| dfrtrclrec2 15024 | If two elements are connec... |
| rtrclreclem2 15025 | The reflexive, transitive ... |
| rtrclreclem3 15026 | The reflexive, transitive ... |
| rtrclreclem4 15027 | The reflexive, transitive ... |
| dfrtrcl2 15028 | The two definitions ` t* `... |
| relexpindlem 15029 | Principle of transitive in... |
| relexpind 15030 | Principle of transitive in... |
| rtrclind 15031 | Principle of transitive in... |
| shftlem 15034 | Two ways to write a shifte... |
| shftuz 15035 | A shift of the upper integ... |
| shftfval 15036 | The value of the sequence ... |
| shftdm 15037 | Domain of a relation shift... |
| shftfib 15038 | Value of a fiber of the re... |
| shftfn 15039 | Functionality and domain o... |
| shftval 15040 | Value of a sequence shifte... |
| shftval2 15041 | Value of a sequence shifte... |
| shftval3 15042 | Value of a sequence shifte... |
| shftval4 15043 | Value of a sequence shifte... |
| shftval5 15044 | Value of a shifted sequenc... |
| shftf 15045 | Functionality of a shifted... |
| 2shfti 15046 | Composite shift operations... |
| shftidt2 15047 | Identity law for the shift... |
| shftidt 15048 | Identity law for the shift... |
| shftcan1 15049 | Cancellation law for the s... |
| shftcan2 15050 | Cancellation law for the s... |
| seqshft 15051 | Shifting the index set of ... |
| sgnval 15054 | Value of the signum functi... |
| sgn0 15055 | The signum of 0 is 0. (Co... |
| sgnp 15056 | The signum of a positive e... |
| sgnrrp 15057 | The signum of a positive r... |
| sgn1 15058 | The signum of 1 is 1. (Co... |
| sgnpnf 15059 | The signum of ` +oo ` is 1... |
| sgnn 15060 | The signum of a negative e... |
| sgnmnf 15061 | The signum of ` -oo ` is -... |
| cjval 15068 | The value of the conjugate... |
| cjth 15069 | The defining property of t... |
| cjf 15070 | Domain and codomain of the... |
| cjcl 15071 | The conjugate of a complex... |
| reval 15072 | The value of the real part... |
| imval 15073 | The value of the imaginary... |
| imre 15074 | The imaginary part of a co... |
| reim 15075 | The real part of a complex... |
| recl 15076 | The real part of a complex... |
| imcl 15077 | The imaginary part of a co... |
| ref 15078 | Domain and codomain of the... |
| imf 15079 | Domain and codomain of the... |
| crre 15080 | The real part of a complex... |
| crim 15081 | The real part of a complex... |
| replim 15082 | Reconstruct a complex numb... |
| remim 15083 | Value of the conjugate of ... |
| reim0 15084 | The imaginary part of a re... |
| reim0b 15085 | A number is real iff its i... |
| rereb 15086 | A number is real iff it eq... |
| mulre 15087 | A product with a nonzero r... |
| rere 15088 | A real number equals its r... |
| cjreb 15089 | A number is real iff it eq... |
| recj 15090 | Real part of a complex con... |
| reneg 15091 | Real part of negative. (C... |
| readd 15092 | Real part distributes over... |
| resub 15093 | Real part distributes over... |
| remullem 15094 | Lemma for ~ remul , ~ immu... |
| remul 15095 | Real part of a product. (... |
| remul2 15096 | Real part of a product. (... |
| rediv 15097 | Real part of a division. ... |
| imcj 15098 | Imaginary part of a comple... |
| imneg 15099 | The imaginary part of a ne... |
| imadd 15100 | Imaginary part distributes... |
| imsub 15101 | Imaginary part distributes... |
| immul 15102 | Imaginary part of a produc... |
| immul2 15103 | Imaginary part of a produc... |
| imdiv 15104 | Imaginary part of a divisi... |
| cjre 15105 | A real number equals its c... |
| cjcj 15106 | The conjugate of the conju... |
| cjadd 15107 | Complex conjugate distribu... |
| cjmul 15108 | Complex conjugate distribu... |
| ipcnval 15109 | Standard inner product on ... |
| cjmulrcl 15110 | A complex number times its... |
| cjmulval 15111 | A complex number times its... |
| cjmulge0 15112 | A complex number times its... |
| cjneg 15113 | Complex conjugate of negat... |
| addcj 15114 | A number plus its conjugat... |
| cjsub 15115 | Complex conjugate distribu... |
| cjexp 15116 | Complex conjugate of posit... |
| imval2 15117 | The imaginary part of a nu... |
| re0 15118 | The real part of zero. (C... |
| im0 15119 | The imaginary part of zero... |
| re1 15120 | The real part of one. (Co... |
| im1 15121 | The imaginary part of one.... |
| rei 15122 | The real part of ` _i ` . ... |
| imi 15123 | The imaginary part of ` _i... |
| cj0 15124 | The conjugate of zero. (C... |
| cji 15125 | The complex conjugate of t... |
| cjreim 15126 | The conjugate of a represe... |
| cjreim2 15127 | The conjugate of the repre... |
| cj11 15128 | Complex conjugate is a one... |
| cjne0 15129 | A number is nonzero iff it... |
| cjdiv 15130 | Complex conjugate distribu... |
| cnrecnv 15131 | The inverse to the canonic... |
| sqeqd 15132 | A deduction for showing tw... |
| recli 15133 | The real part of a complex... |
| imcli 15134 | The imaginary part of a co... |
| cjcli 15135 | Closure law for complex co... |
| replimi 15136 | Construct a complex number... |
| cjcji 15137 | The conjugate of the conju... |
| reim0bi 15138 | A number is real iff its i... |
| rerebi 15139 | A real number equals its r... |
| cjrebi 15140 | A number is real iff it eq... |
| recji 15141 | Real part of a complex con... |
| imcji 15142 | Imaginary part of a comple... |
| cjmulrcli 15143 | A complex number times its... |
| cjmulvali 15144 | A complex number times its... |
| cjmulge0i 15145 | A complex number times its... |
| renegi 15146 | Real part of negative. (C... |
| imnegi 15147 | Imaginary part of negative... |
| cjnegi 15148 | Complex conjugate of negat... |
| addcji 15149 | A number plus its conjugat... |
| readdi 15150 | Real part distributes over... |
| imaddi 15151 | Imaginary part distributes... |
| remuli 15152 | Real part of a product. (... |
| immuli 15153 | Imaginary part of a produc... |
| cjaddi 15154 | Complex conjugate distribu... |
| cjmuli 15155 | Complex conjugate distribu... |
| ipcni 15156 | Standard inner product on ... |
| cjdivi 15157 | Complex conjugate distribu... |
| crrei 15158 | The real part of a complex... |
| crimi 15159 | The imaginary part of a co... |
| recld 15160 | The real part of a complex... |
| imcld 15161 | The imaginary part of a co... |
| cjcld 15162 | Closure law for complex co... |
| replimd 15163 | Construct a complex number... |
| remimd 15164 | Value of the conjugate of ... |
| cjcjd 15165 | The conjugate of the conju... |
| reim0bd 15166 | A number is real iff its i... |
| rerebd 15167 | A real number equals its r... |
| cjrebd 15168 | A number is real iff it eq... |
| cjne0d 15169 | A number is nonzero iff it... |
| recjd 15170 | Real part of a complex con... |
| imcjd 15171 | Imaginary part of a comple... |
| cjmulrcld 15172 | A complex number times its... |
| cjmulvald 15173 | A complex number times its... |
| cjmulge0d 15174 | A complex number times its... |
| renegd 15175 | Real part of negative. (C... |
| imnegd 15176 | Imaginary part of negative... |
| cjnegd 15177 | Complex conjugate of negat... |
| addcjd 15178 | A number plus its conjugat... |
| cjexpd 15179 | Complex conjugate of posit... |
| readdd 15180 | Real part distributes over... |
| imaddd 15181 | Imaginary part distributes... |
| resubd 15182 | Real part distributes over... |
| imsubd 15183 | Imaginary part distributes... |
| remuld 15184 | Real part of a product. (... |
| immuld 15185 | Imaginary part of a produc... |
| cjaddd 15186 | Complex conjugate distribu... |
| cjmuld 15187 | Complex conjugate distribu... |
| ipcnd 15188 | Standard inner product on ... |
| cjdivd 15189 | Complex conjugate distribu... |
| rered 15190 | A real number equals its r... |
| reim0d 15191 | The imaginary part of a re... |
| cjred 15192 | A real number equals its c... |
| remul2d 15193 | Real part of a product. (... |
| immul2d 15194 | Imaginary part of a produc... |
| redivd 15195 | Real part of a division. ... |
| imdivd 15196 | Imaginary part of a divisi... |
| crred 15197 | The real part of a complex... |
| crimd 15198 | The imaginary part of a co... |
| sqrtval 15203 | Value of square root funct... |
| absval 15204 | The absolute value (modulu... |
| rennim 15205 | A real number does not lie... |
| cnpart 15206 | The specification of restr... |
| sqrt0 15207 | The square root of zero is... |
| 01sqrexlem1 15208 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem2 15209 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem3 15210 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem4 15211 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem5 15212 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem6 15213 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem7 15214 | Lemma for ~ 01sqrex . (Co... |
| 01sqrex 15215 | Existence of a square root... |
| resqrex 15216 | Existence of a square root... |
| sqrmo 15217 | Uniqueness for the square ... |
| resqreu 15218 | Existence and uniqueness f... |
| resqrtcl 15219 | Closure of the square root... |
| resqrtthlem 15220 | Lemma for ~ resqrtth . (C... |
| resqrtth 15221 | Square root theorem over t... |
| remsqsqrt 15222 | Square of square root. (C... |
| sqrtge0 15223 | The square root function i... |
| sqrtgt0 15224 | The square root function i... |
| sqrtmul 15225 | Square root distributes ov... |
| sqrtle 15226 | Square root is monotonic. ... |
| sqrtlt 15227 | Square root is strictly mo... |
| sqrt11 15228 | The square root function i... |
| sqrt00 15229 | A square root is zero iff ... |
| rpsqrtcl 15230 | The square root of a posit... |
| sqrtdiv 15231 | Square root distributes ov... |
| sqrtneglem 15232 | The square root of a negat... |
| sqrtneg 15233 | The square root of a negat... |
| sqrtsq2 15234 | Relationship between squar... |
| sqrtsq 15235 | Square root of square. (C... |
| sqrtmsq 15236 | Square root of square. (C... |
| sqrt1 15237 | The square root of 1 is 1.... |
| sqrt4 15238 | The square root of 4 is 2.... |
| sqrt9 15239 | The square root of 9 is 3.... |
| sqrt2gt1lt2 15240 | The square root of 2 is bo... |
| sqrtm1 15241 | The imaginary unit is the ... |
| nn0sqeq1 15242 | A natural number with squa... |
| absneg 15243 | Absolute value of the nega... |
| abscl 15244 | Real closure of absolute v... |
| abscj 15245 | The absolute value of a nu... |
| absvalsq 15246 | Square of value of absolut... |
| absvalsq2 15247 | Square of value of absolut... |
| sqabsadd 15248 | Square of absolute value o... |
| sqabssub 15249 | Square of absolute value o... |
| absval2 15250 | Value of absolute value fu... |
| abs0 15251 | The absolute value of 0. ... |
| absi 15252 | The absolute value of the ... |
| absge0 15253 | Absolute value is nonnegat... |
| absrpcl 15254 | The absolute value of a no... |
| abs00 15255 | The absolute value of a nu... |
| abs00ad 15256 | A complex number is zero i... |
| abs00bd 15257 | If a complex number is zer... |
| absreimsq 15258 | Square of the absolute val... |
| absreim 15259 | Absolute value of a number... |
| absmul 15260 | Absolute value distributes... |
| absdiv 15261 | Absolute value distributes... |
| absid 15262 | A nonnegative number is it... |
| abs1 15263 | The absolute value of one ... |
| absnid 15264 | For a negative number, its... |
| leabs 15265 | A real number is less than... |
| absor 15266 | The absolute value of a re... |
| absre 15267 | Absolute value of a real n... |
| absresq 15268 | Square of the absolute val... |
| absmod0 15269 | ` A ` is divisible by ` B ... |
| absexp 15270 | Absolute value of positive... |
| absexpz 15271 | Absolute value of integer ... |
| abssq 15272 | Square can be moved in and... |
| sqabs 15273 | The squares of two reals a... |
| absrele 15274 | The absolute value of a co... |
| absimle 15275 | The absolute value of a co... |
| max0add 15276 | The sum of the positive an... |
| absz 15277 | A real number is an intege... |
| nn0abscl 15278 | The absolute value of an i... |
| zabscl 15279 | The absolute value of an i... |
| zabs0b 15280 | An integer has an absolute... |
| abslt 15281 | Absolute value and 'less t... |
| absle 15282 | Absolute value and 'less t... |
| abssubne0 15283 | If the absolute value of a... |
| absdiflt 15284 | The absolute value of a di... |
| absdifle 15285 | The absolute value of a di... |
| elicc4abs 15286 | Membership in a symmetric ... |
| lenegsq 15287 | Comparison to a nonnegativ... |
| releabs 15288 | The real part of a number ... |
| recval 15289 | Reciprocal expressed with ... |
| absidm 15290 | The absolute value functio... |
| absgt0 15291 | The absolute value of a no... |
| nnabscl 15292 | The absolute value of a no... |
| abssub 15293 | Swapping order of subtract... |
| abssubge0 15294 | Absolute value of a nonneg... |
| abssuble0 15295 | Absolute value of a nonpos... |
| absmax 15296 | The maximum of two numbers... |
| abstri 15297 | Triangle inequality for ab... |
| abs3dif 15298 | Absolute value of differen... |
| abs2dif 15299 | Difference of absolute val... |
| abs2dif2 15300 | Difference of absolute val... |
| abs2difabs 15301 | Absolute value of differen... |
| abs1m 15302 | For any complex number, th... |
| recan 15303 | Cancellation law involving... |
| absf 15304 | Mapping domain and codomai... |
| abs3lem 15305 | Lemma involving absolute v... |
| abslem2 15306 | Lemma involving absolute v... |
| rddif 15307 | The difference between a r... |
| absrdbnd 15308 | Bound on the absolute valu... |
| fzomaxdiflem 15309 | Lemma for ~ fzomaxdif . (... |
| fzomaxdif 15310 | A bound on the separation ... |
| uzin2 15311 | The upper integers are clo... |
| rexanuz 15312 | Combine two different uppe... |
| rexanre 15313 | Combine two different uppe... |
| rexfiuz 15314 | Combine finitely many diff... |
| rexuz3 15315 | Restrict the base of the u... |
| rexanuz2 15316 | Combine two different uppe... |
| r19.29uz 15317 | A version of ~ 19.29 for u... |
| r19.2uz 15318 | A version of ~ r19.2z for ... |
| rexuzre 15319 | Convert an upper real quan... |
| rexico 15320 | Restrict the base of an up... |
| cau3lem 15321 | Lemma for ~ cau3 . (Contr... |
| cau3 15322 | Convert between three-quan... |
| cau4 15323 | Change the base of a Cauch... |
| caubnd2 15324 | A Cauchy sequence of compl... |
| caubnd 15325 | A Cauchy sequence of compl... |
| sqreulem 15326 | Lemma for ~ sqreu : write ... |
| sqreu 15327 | Existence and uniqueness f... |
| sqrtcl 15328 | Closure of the square root... |
| sqrtthlem 15329 | Lemma for ~ sqrtth . (Con... |
| sqrtf 15330 | Mapping domain and codomai... |
| sqrtth 15331 | Square root theorem over t... |
| sqrtrege0 15332 | The square root function m... |
| eqsqrtor 15333 | Solve an equation containi... |
| eqsqrtd 15334 | A deduction for showing th... |
| eqsqrt2d 15335 | A deduction for showing th... |
| amgm2 15336 | Arithmetic-geometric mean ... |
| sqrtthi 15337 | Square root theorem. Theo... |
| sqrtcli 15338 | The square root of a nonne... |
| sqrtgt0i 15339 | The square root of a posit... |
| sqrtmsqi 15340 | Square root of square. (C... |
| sqrtsqi 15341 | Square root of square. (C... |
| sqsqrti 15342 | Square of square root. (C... |
| sqrtge0i 15343 | The square root of a nonne... |
| absidi 15344 | A nonnegative number is it... |
| absnidi 15345 | A negative number is the n... |
| leabsi 15346 | A real number is less than... |
| absori 15347 | The absolute value of a re... |
| absrei 15348 | Absolute value of a real n... |
| sqrtpclii 15349 | The square root of a posit... |
| sqrtgt0ii 15350 | The square root of a posit... |
| sqrt11i 15351 | The square root function i... |
| sqrtmuli 15352 | Square root distributes ov... |
| sqrtmulii 15353 | Square root distributes ov... |
| sqrtmsq2i 15354 | Relationship between squar... |
| sqrtlei 15355 | Square root is monotonic. ... |
| sqrtlti 15356 | Square root is strictly mo... |
| abslti 15357 | Absolute value and 'less t... |
| abslei 15358 | Absolute value and 'less t... |
| cnsqrt00 15359 | A square root of a complex... |
| absvalsqi 15360 | Square of value of absolut... |
| absvalsq2i 15361 | Square of value of absolut... |
| abscli 15362 | Real closure of absolute v... |
| absge0i 15363 | Absolute value is nonnegat... |
| absval2i 15364 | Value of absolute value fu... |
| abs00i 15365 | The absolute value of a nu... |
| absgt0i 15366 | The absolute value of a no... |
| absnegi 15367 | Absolute value of negative... |
| abscji 15368 | The absolute value of a nu... |
| releabsi 15369 | The real part of a number ... |
| abssubi 15370 | Swapping order of subtract... |
| absmuli 15371 | Absolute value distributes... |
| sqabsaddi 15372 | Square of absolute value o... |
| sqabssubi 15373 | Square of absolute value o... |
| absdivzi 15374 | Absolute value distributes... |
| abstrii 15375 | Triangle inequality for ab... |
| abs3difi 15376 | Absolute value of differen... |
| abs3lemi 15377 | Lemma involving absolute v... |
| rpsqrtcld 15378 | The square root of a posit... |
| sqrtgt0d 15379 | The square root of a posit... |
| absnidd 15380 | A negative number is the n... |
| leabsd 15381 | A real number is less than... |
| absord 15382 | The absolute value of a re... |
| absred 15383 | Absolute value of a real n... |
| resqrtcld 15384 | The square root of a nonne... |
| sqrtmsqd 15385 | Square root of square. (C... |
| sqrtsqd 15386 | Square root of square. (C... |
| sqrtge0d 15387 | The square root of a nonne... |
| sqrtnegd 15388 | The square root of a negat... |
| absidd 15389 | A nonnegative number is it... |
| sqrtdivd 15390 | Square root distributes ov... |
| sqrtmuld 15391 | Square root distributes ov... |
| sqrtsq2d 15392 | Relationship between squar... |
| sqrtled 15393 | Square root is monotonic. ... |
| sqrtltd 15394 | Square root is strictly mo... |
| sqr11d 15395 | The square root function i... |
| nn0absid 15396 | A nonnegative integer is i... |
| nn0absidi 15397 | A nonnegative integer is i... |
| absltd 15398 | Absolute value and 'less t... |
| absled 15399 | Absolute value and 'less t... |
| abssubge0d 15400 | Absolute value of a nonneg... |
| abssuble0d 15401 | Absolute value of a nonpos... |
| absdifltd 15402 | The absolute value of a di... |
| absdifled 15403 | The absolute value of a di... |
| icodiamlt 15404 | Two elements in a half-ope... |
| abscld 15405 | Real closure of absolute v... |
| sqrtcld 15406 | Closure of the square root... |
| sqrtrege0d 15407 | The real part of the squar... |
| sqsqrtd 15408 | Square root theorem. Theo... |
| msqsqrtd 15409 | Square root theorem. Theo... |
| sqr00d 15410 | A square root is zero iff ... |
| absvalsqd 15411 | Square of value of absolut... |
| absvalsq2d 15412 | Square of value of absolut... |
| absge0d 15413 | Absolute value is nonnegat... |
| absval2d 15414 | Value of absolute value fu... |
| abs00d 15415 | The absolute value of a nu... |
| absne0d 15416 | The absolute value of a nu... |
| absrpcld 15417 | The absolute value of a no... |
| absnegd 15418 | Absolute value of negative... |
| abscjd 15419 | The absolute value of a nu... |
| releabsd 15420 | The real part of a number ... |
| absexpd 15421 | Absolute value of positive... |
| abssubd 15422 | Swapping order of subtract... |
| absmuld 15423 | Absolute value distributes... |
| absdivd 15424 | Absolute value distributes... |
| abstrid 15425 | Triangle inequality for ab... |
| abs2difd 15426 | Difference of absolute val... |
| abs2dif2d 15427 | Difference of absolute val... |
| abs2difabsd 15428 | Absolute value of differen... |
| abs3difd 15429 | Absolute value of differen... |
| abs3lemd 15430 | Lemma involving absolute v... |
| reusq0 15431 | A complex number is the sq... |
| bhmafibid1cn 15432 | The Brahmagupta-Fibonacci ... |
| bhmafibid2cn 15433 | The Brahmagupta-Fibonacci ... |
| bhmafibid1 15434 | The Brahmagupta-Fibonacci ... |
| bhmafibid2 15435 | The Brahmagupta-Fibonacci ... |
| limsupgord 15438 | Ordering property of the s... |
| limsupcl 15439 | Closure of the superior li... |
| limsupval 15440 | The superior limit of an i... |
| limsupgf 15441 | Closure of the superior li... |
| limsupgval 15442 | Value of the superior limi... |
| limsupgle 15443 | The defining property of t... |
| limsuple 15444 | The defining property of t... |
| limsuplt 15445 | The defining property of t... |
| limsupval2 15446 | The superior limit, relati... |
| limsupgre 15447 | If a sequence of real numb... |
| limsupbnd1 15448 | If a sequence is eventuall... |
| limsupbnd2 15449 | If a sequence is eventuall... |
| climrel 15458 | The limit relation is a re... |
| rlimrel 15459 | The limit relation is a re... |
| clim 15460 | Express the predicate: Th... |
| rlim 15461 | Express the predicate: Th... |
| rlim2 15462 | Rewrite ~ rlim for a mappi... |
| rlim2lt 15463 | Use strictly less-than in ... |
| rlim3 15464 | Restrict the range of the ... |
| climcl 15465 | Closure of the limit of a ... |
| rlimpm 15466 | Closure of a function with... |
| rlimf 15467 | Closure of a function with... |
| rlimss 15468 | Domain closure of a functi... |
| rlimcl 15469 | Closure of the limit of a ... |
| clim2 15470 | Express the predicate: Th... |
| clim2c 15471 | Express the predicate ` F ... |
| clim0 15472 | Express the predicate ` F ... |
| clim0c 15473 | Express the predicate ` F ... |
| rlim0 15474 | Express the predicate ` B ... |
| rlim0lt 15475 | Use strictly less-than in ... |
| climi 15476 | Convergence of a sequence ... |
| climi2 15477 | Convergence of a sequence ... |
| climi0 15478 | Convergence of a sequence ... |
| rlimi 15479 | Convergence at infinity of... |
| rlimi2 15480 | Convergence at infinity of... |
| ello1 15481 | Elementhood in the set of ... |
| ello12 15482 | Elementhood in the set of ... |
| ello12r 15483 | Sufficient condition for e... |
| lo1f 15484 | An eventually upper bounde... |
| lo1dm 15485 | An eventually upper bounde... |
| lo1bdd 15486 | The defining property of a... |
| ello1mpt 15487 | Elementhood in the set of ... |
| ello1mpt2 15488 | Elementhood in the set of ... |
| ello1d 15489 | Sufficient condition for e... |
| lo1bdd2 15490 | If an eventually bounded f... |
| lo1bddrp 15491 | Refine ~ o1bdd2 to give a ... |
| elo1 15492 | Elementhood in the set of ... |
| elo12 15493 | Elementhood in the set of ... |
| elo12r 15494 | Sufficient condition for e... |
| o1f 15495 | An eventually bounded func... |
| o1dm 15496 | An eventually bounded func... |
| o1bdd 15497 | The defining property of a... |
| lo1o1 15498 | A function is eventually b... |
| lo1o12 15499 | A function is eventually b... |
| elo1mpt 15500 | Elementhood in the set of ... |
| elo1mpt2 15501 | Elementhood in the set of ... |
| elo1d 15502 | Sufficient condition for e... |
| o1lo1 15503 | A real function is eventua... |
| o1lo12 15504 | A lower bounded real funct... |
| o1lo1d 15505 | A real eventually bounded ... |
| icco1 15506 | Derive eventual boundednes... |
| o1bdd2 15507 | If an eventually bounded f... |
| o1bddrp 15508 | Refine ~ o1bdd2 to give a ... |
| climconst 15509 | An (eventually) constant s... |
| rlimconst 15510 | A constant sequence conver... |
| rlimclim1 15511 | Forward direction of ~ rli... |
| rlimclim 15512 | A sequence on an upper int... |
| climrlim2 15513 | Produce a real limit from ... |
| climconst2 15514 | A constant sequence conver... |
| climz 15515 | The zero sequence converge... |
| rlimuni 15516 | A real function whose doma... |
| rlimdm 15517 | Two ways to express that a... |
| climuni 15518 | An infinite sequence of co... |
| fclim 15519 | The limit relation is func... |
| climdm 15520 | Two ways to express that a... |
| climeu 15521 | An infinite sequence of co... |
| climreu 15522 | An infinite sequence of co... |
| climmo 15523 | An infinite sequence of co... |
| rlimres 15524 | The restriction of a funct... |
| lo1res 15525 | The restriction of an even... |
| o1res 15526 | The restriction of an even... |
| rlimres2 15527 | The restriction of a funct... |
| lo1res2 15528 | The restriction of a funct... |
| o1res2 15529 | The restriction of a funct... |
| lo1resb 15530 | The restriction of a funct... |
| rlimresb 15531 | The restriction of a funct... |
| o1resb 15532 | The restriction of a funct... |
| climeq 15533 | Two functions that are eve... |
| lo1eq 15534 | Two functions that are eve... |
| rlimeq 15535 | Two functions that are eve... |
| o1eq 15536 | Two functions that are eve... |
| climmpt 15537 | Exhibit a function ` G ` w... |
| 2clim 15538 | If two sequences converge ... |
| climmpt2 15539 | Relate an integer limit on... |
| climshftlem 15540 | A shifted function converg... |
| climres 15541 | A function restricted to u... |
| climshft 15542 | A shifted function converg... |
| serclim0 15543 | The zero series converges ... |
| rlimcld2 15544 | If ` D ` is a closed set i... |
| rlimrege0 15545 | The limit of a sequence of... |
| rlimrecl 15546 | The limit of a real sequen... |
| rlimge0 15547 | The limit of a sequence of... |
| climshft2 15548 | A shifted function converg... |
| climrecl 15549 | The limit of a convergent ... |
| climge0 15550 | A nonnegative sequence con... |
| climabs0 15551 | Convergence to zero of the... |
| o1co 15552 | Sufficient condition for t... |
| o1compt 15553 | Sufficient condition for t... |
| rlimcn1 15554 | Image of a limit under a c... |
| rlimcn1b 15555 | Image of a limit under a c... |
| rlimcn3 15556 | Image of a limit under a c... |
| rlimcn2 15557 | Image of a limit under a c... |
| climcn1 15558 | Image of a limit under a c... |
| climcn2 15559 | Image of a limit under a c... |
| addcn2 15560 | Complex number addition is... |
| subcn2 15561 | Complex number subtraction... |
| mulcn2 15562 | Complex number multiplicat... |
| reccn2 15563 | The reciprocal function is... |
| cn1lem 15564 | A sufficient condition for... |
| abscn2 15565 | The absolute value functio... |
| cjcn2 15566 | The complex conjugate func... |
| recn2 15567 | The real part function is ... |
| imcn2 15568 | The imaginary part functio... |
| climcn1lem 15569 | The limit of a continuous ... |
| climabs 15570 | Limit of the absolute valu... |
| climcj 15571 | Limit of the complex conju... |
| climre 15572 | Limit of the real part of ... |
| climim 15573 | Limit of the imaginary par... |
| rlimmptrcl 15574 | Reverse closure for a real... |
| rlimabs 15575 | Limit of the absolute valu... |
| rlimcj 15576 | Limit of the complex conju... |
| rlimre 15577 | Limit of the real part of ... |
| rlimim 15578 | Limit of the imaginary par... |
| o1of2 15579 | Show that a binary operati... |
| o1add 15580 | The sum of two eventually ... |
| o1mul 15581 | The product of two eventua... |
| o1sub 15582 | The difference of two even... |
| rlimo1 15583 | Any function with a finite... |
| rlimdmo1 15584 | A convergent function is e... |
| o1rlimmul 15585 | The product of an eventual... |
| o1const 15586 | A constant function is eve... |
| lo1const 15587 | A constant function is eve... |
| lo1mptrcl 15588 | Reverse closure for an eve... |
| o1mptrcl 15589 | Reverse closure for an eve... |
| o1add2 15590 | The sum of two eventually ... |
| o1mul2 15591 | The product of two eventua... |
| o1sub2 15592 | The product of two eventua... |
| lo1add 15593 | The sum of two eventually ... |
| lo1mul 15594 | The product of an eventual... |
| lo1mul2 15595 | The product of an eventual... |
| o1dif 15596 | If the difference of two f... |
| lo1sub 15597 | The difference of an event... |
| climadd 15598 | Limit of the sum of two co... |
| climmul 15599 | Limit of the product of tw... |
| climsub 15600 | Limit of the difference of... |
| climaddc1 15601 | Limit of a constant ` C ` ... |
| climaddc2 15602 | Limit of a constant ` C ` ... |
| climmulc2 15603 | Limit of a sequence multip... |
| climsubc1 15604 | Limit of a constant ` C ` ... |
| climsubc2 15605 | Limit of a constant ` C ` ... |
| climle 15606 | Comparison of the limits o... |
| climsqz 15607 | Convergence of a sequence ... |
| climsqz2 15608 | Convergence of a sequence ... |
| rlimadd 15609 | Limit of the sum of two co... |
| rlimsub 15610 | Limit of the difference of... |
| rlimmul 15611 | Limit of the product of tw... |
| rlimdiv 15612 | Limit of the quotient of t... |
| rlimneg 15613 | Limit of the negative of a... |
| rlimle 15614 | Comparison of the limits o... |
| rlimsqzlem 15615 | Lemma for ~ rlimsqz and ~ ... |
| rlimsqz 15616 | Convergence of a sequence ... |
| rlimsqz2 15617 | Convergence of a sequence ... |
| lo1le 15618 | Transfer eventual upper bo... |
| o1le 15619 | Transfer eventual boundedn... |
| rlimno1 15620 | A function whose inverse c... |
| clim2ser 15621 | The limit of an infinite s... |
| clim2ser2 15622 | The limit of an infinite s... |
| iserex 15623 | An infinite series converg... |
| isermulc2 15624 | Multiplication of an infin... |
| climlec2 15625 | Comparison of a constant t... |
| iserle 15626 | Comparison of the limits o... |
| iserge0 15627 | The limit of an infinite s... |
| climub 15628 | The limit of a monotonic s... |
| climserle 15629 | The partial sums of a conv... |
| isershft 15630 | Index shift of the limit o... |
| isercolllem1 15631 | Lemma for ~ isercoll . (C... |
| isercolllem2 15632 | Lemma for ~ isercoll . (C... |
| isercolllem3 15633 | Lemma for ~ isercoll . (C... |
| isercoll 15634 | Rearrange an infinite seri... |
| isercoll2 15635 | Generalize ~ isercoll so t... |
| climsup 15636 | A bounded monotonic sequen... |
| climcau 15637 | A converging sequence of c... |
| climbdd 15638 | A converging sequence of c... |
| caucvgrlem 15639 | Lemma for ~ caurcvgr . (C... |
| caurcvgr 15640 | A Cauchy sequence of real ... |
| caucvgrlem2 15641 | Lemma for ~ caucvgr . (Co... |
| caucvgr 15642 | A Cauchy sequence of compl... |
| caurcvg 15643 | A Cauchy sequence of real ... |
| caurcvg2 15644 | A Cauchy sequence of real ... |
| caucvg 15645 | A Cauchy sequence of compl... |
| caucvgb 15646 | A function is convergent i... |
| serf0 15647 | If an infinite series conv... |
| iseraltlem1 15648 | Lemma for ~ iseralt . A d... |
| iseraltlem2 15649 | Lemma for ~ iseralt . The... |
| iseraltlem3 15650 | Lemma for ~ iseralt . Fro... |
| iseralt 15651 | The alternating series tes... |
| sumex 15654 | A sum is a set. (Contribu... |
| sumeq1 15655 | Equality theorem for a sum... |
| nfsum1 15656 | Bound-variable hypothesis ... |
| nfsum 15657 | Bound-variable hypothesis ... |
| sumeq2w 15658 | Equality theorem for sum, ... |
| sumeq2ii 15659 | Equality theorem for sum, ... |
| sumeq2 15660 | Equality theorem for sum. ... |
| cbvsum 15661 | Change bound variable in a... |
| cbvsumv 15662 | Change bound variable in a... |
| sumeq1i 15663 | Equality inference for sum... |
| sumeq2i 15664 | Equality inference for sum... |
| sumeq12i 15665 | Equality inference for sum... |
| sumeq1d 15666 | Equality deduction for sum... |
| sumeq2d 15667 | Equality deduction for sum... |
| sumeq2dv 15668 | Equality deduction for sum... |
| sumeq2sdv 15669 | Equality deduction for sum... |
| sumeq2sdvOLD 15670 | Obsolete version of ~ sume... |
| 2sumeq2dv 15671 | Equality deduction for dou... |
| sumeq12dv 15672 | Equality deduction for sum... |
| sumeq12rdv 15673 | Equality deduction for sum... |
| sum2id 15674 | The second class argument ... |
| sumfc 15675 | A lemma to facilitate conv... |
| fz1f1o 15676 | A lemma for working with f... |
| sumrblem 15677 | Lemma for ~ sumrb . (Cont... |
| fsumcvg 15678 | The sequence of partial su... |
| sumrb 15679 | Rebase the starting point ... |
| summolem3 15680 | Lemma for ~ summo . (Cont... |
| summolem2a 15681 | Lemma for ~ summo . (Cont... |
| summolem2 15682 | Lemma for ~ summo . (Cont... |
| summo 15683 | A sum has at most one limi... |
| zsum 15684 | Series sum with index set ... |
| isum 15685 | Series sum with an upper i... |
| fsum 15686 | The value of a sum over a ... |
| sum0 15687 | Any sum over the empty set... |
| sumz 15688 | Any sum of zero over a sum... |
| fsumf1o 15689 | Re-index a finite sum usin... |
| sumss 15690 | Change the index set to a ... |
| fsumss 15691 | Change the index set to a ... |
| sumss2 15692 | Change the index set of a ... |
| fsumcvg2 15693 | The sequence of partial su... |
| fsumsers 15694 | Special case of series sum... |
| fsumcvg3 15695 | A finite sum is convergent... |
| fsumser 15696 | A finite sum expressed in ... |
| fsumcl2lem 15697 | - Lemma for finite sum clo... |
| fsumcllem 15698 | - Lemma for finite sum clo... |
| fsumcl 15699 | Closure of a finite sum of... |
| fsumrecl 15700 | Closure of a finite sum of... |
| fsumzcl 15701 | Closure of a finite sum of... |
| fsumnn0cl 15702 | Closure of a finite sum of... |
| fsumrpcl 15703 | Closure of a finite sum of... |
| fsumclf 15704 | Closure of a finite sum of... |
| fsumzcl2 15705 | A finite sum with integer ... |
| fsumadd 15706 | The sum of two finite sums... |
| fsumsplit 15707 | Split a sum into two parts... |
| fsumsplitf 15708 | Split a sum into two parts... |
| sumsnf 15709 | A sum of a singleton is th... |
| fsumsplitsn 15710 | Separate out a term in a f... |
| fsumsplit1 15711 | Separate out a term in a f... |
| sumsn 15712 | A sum of a singleton is th... |
| fsum1 15713 | The finite sum of ` A ( k ... |
| sumpr 15714 | A sum over a pair is the s... |
| sumtp 15715 | A sum over a triple is the... |
| sumsns 15716 | A sum of a singleton is th... |
| fsumm1 15717 | Separate out the last term... |
| fzosump1 15718 | Separate out the last term... |
| fsum1p 15719 | Separate out the first ter... |
| fsummsnunz 15720 | A finite sum all of whose ... |
| fsumsplitsnun 15721 | Separate out a term in a f... |
| fsump1 15722 | The addition of the next t... |
| isumclim 15723 | An infinite sum equals the... |
| isumclim2 15724 | A converging series conver... |
| isumclim3 15725 | The sequence of partial fi... |
| sumnul 15726 | The sum of a non-convergen... |
| isumcl 15727 | The sum of a converging in... |
| isummulc2 15728 | An infinite sum multiplied... |
| isummulc1 15729 | An infinite sum multiplied... |
| isumdivc 15730 | An infinite sum divided by... |
| isumrecl 15731 | The sum of a converging in... |
| isumge0 15732 | An infinite sum of nonnega... |
| isumadd 15733 | Addition of infinite sums.... |
| sumsplit 15734 | Split a sum into two parts... |
| fsump1i 15735 | Optimized version of ~ fsu... |
| fsum2dlem 15736 | Lemma for ~ fsum2d - induc... |
| fsum2d 15737 | Write a double sum as a su... |
| fsumxp 15738 | Combine two sums into a si... |
| fsumcnv 15739 | Transform a region of summ... |
| fsumcom2 15740 | Interchange order of summa... |
| fsumcom 15741 | Interchange order of summa... |
| fsum0diaglem 15742 | Lemma for ~ fsum0diag . (... |
| fsum0diag 15743 | Two ways to express "the s... |
| mptfzshft 15744 | 1-1 onto function in maps-... |
| fsumrev 15745 | Reversal of a finite sum. ... |
| fsumshft 15746 | Index shift of a finite su... |
| fsumshftm 15747 | Negative index shift of a ... |
| fsumrev2 15748 | Reversal of a finite sum. ... |
| fsum0diag2 15749 | Two ways to express "the s... |
| fsummulc2 15750 | A finite sum multiplied by... |
| fsummulc1 15751 | A finite sum multiplied by... |
| fsumdivc 15752 | A finite sum divided by a ... |
| fsumneg 15753 | Negation of a finite sum. ... |
| fsumsub 15754 | Split a finite sum over a ... |
| fsum2mul 15755 | Separate the nested sum of... |
| fsumconst 15756 | The sum of constant terms ... |
| fsumdifsnconst 15757 | The sum of constant terms ... |
| modfsummodslem1 15758 | Lemma 1 for ~ modfsummods ... |
| modfsummods 15759 | Induction step for ~ modfs... |
| modfsummod 15760 | A finite sum modulo a posi... |
| fsumge0 15761 | If all of the terms of a f... |
| fsumless 15762 | A shorter sum of nonnegati... |
| fsumge1 15763 | A sum of nonnegative numbe... |
| fsum00 15764 | A sum of nonnegative numbe... |
| fsumle 15765 | If all of the terms of fin... |
| fsumlt 15766 | If every term in one finit... |
| fsumabs 15767 | Generalized triangle inequ... |
| telfsumo 15768 | Sum of a telescoping serie... |
| telfsumo2 15769 | Sum of a telescoping serie... |
| telfsum 15770 | Sum of a telescoping serie... |
| telfsum2 15771 | Sum of a telescoping serie... |
| fsumparts 15772 | Summation by parts. (Cont... |
| fsumrelem 15773 | Lemma for ~ fsumre , ~ fsu... |
| fsumre 15774 | The real part of a sum. (... |
| fsumim 15775 | The imaginary part of a su... |
| fsumcj 15776 | The complex conjugate of a... |
| fsumrlim 15777 | Limit of a finite sum of c... |
| fsumo1 15778 | The finite sum of eventual... |
| o1fsum 15779 | If ` A ( k ) ` is O(1), th... |
| seqabs 15780 | Generalized triangle inequ... |
| iserabs 15781 | Generalized triangle inequ... |
| cvgcmp 15782 | A comparison test for conv... |
| cvgcmpub 15783 | An upper bound for the lim... |
| cvgcmpce 15784 | A comparison test for conv... |
| abscvgcvg 15785 | An absolutely convergent s... |
| climfsum 15786 | Limit of a finite sum of c... |
| fsumiun 15787 | Sum over a disjoint indexe... |
| hashiun 15788 | The cardinality of a disjo... |
| hash2iun 15789 | The cardinality of a neste... |
| hash2iun1dif1 15790 | The cardinality of a neste... |
| hashrabrex 15791 | The number of elements in ... |
| hashuni 15792 | The cardinality of a disjo... |
| qshash 15793 | The cardinality of a set w... |
| ackbijnn 15794 | Translate the Ackermann bi... |
| binomlem 15795 | Lemma for ~ binom (binomia... |
| binom 15796 | The binomial theorem: ` ( ... |
| binom1p 15797 | Special case of the binomi... |
| binom11 15798 | Special case of the binomi... |
| binom1dif 15799 | A summation for the differ... |
| bcxmaslem1 15800 | Lemma for ~ bcxmas . (Con... |
| bcxmas 15801 | Parallel summation (Christ... |
| incexclem 15802 | Lemma for ~ incexc . (Con... |
| incexc 15803 | The inclusion/exclusion pr... |
| incexc2 15804 | The inclusion/exclusion pr... |
| isumshft 15805 | Index shift of an infinite... |
| isumsplit 15806 | Split off the first ` N ` ... |
| isum1p 15807 | The infinite sum of a conv... |
| isumnn0nn 15808 | Sum from 0 to infinity in ... |
| isumrpcl 15809 | The infinite sum of positi... |
| isumle 15810 | Comparison of two infinite... |
| isumless 15811 | A finite sum of nonnegativ... |
| isumsup2 15812 | An infinite sum of nonnega... |
| isumsup 15813 | An infinite sum of nonnega... |
| isumltss 15814 | A partial sum of a series ... |
| climcndslem1 15815 | Lemma for ~ climcnds : bou... |
| climcndslem2 15816 | Lemma for ~ climcnds : bou... |
| climcnds 15817 | The Cauchy condensation te... |
| divrcnv 15818 | The sequence of reciprocal... |
| divcnv 15819 | The sequence of reciprocal... |
| flo1 15820 | The floor function satisfi... |
| divcnvshft 15821 | Limit of a ratio function.... |
| supcvg 15822 | Extract a sequence ` f ` i... |
| infcvgaux1i 15823 | Auxiliary theorem for appl... |
| infcvgaux2i 15824 | Auxiliary theorem for appl... |
| harmonic 15825 | The harmonic series ` H ` ... |
| arisum 15826 | Arithmetic series sum of t... |
| arisum2 15827 | Arithmetic series sum of t... |
| trireciplem 15828 | Lemma for ~ trirecip . Sh... |
| trirecip 15829 | The sum of the reciprocals... |
| expcnv 15830 | A sequence of powers of a ... |
| explecnv 15831 | A sequence of terms conver... |
| geoserg 15832 | The value of the finite ge... |
| geoser 15833 | The value of the finite ge... |
| pwdif 15834 | The difference of two numb... |
| pwm1geoser 15835 | The n-th power of a number... |
| geolim 15836 | The partial sums in the in... |
| geolim2 15837 | The partial sums in the ge... |
| georeclim 15838 | The limit of a geometric s... |
| geo2sum 15839 | The value of the finite ge... |
| geo2sum2 15840 | The value of the finite ge... |
| geo2lim 15841 | The value of the infinite ... |
| geomulcvg 15842 | The geometric series conve... |
| geoisum 15843 | The infinite sum of ` 1 + ... |
| geoisumr 15844 | The infinite sum of recipr... |
| geoisum1 15845 | The infinite sum of ` A ^ ... |
| geoisum1c 15846 | The infinite sum of ` A x.... |
| 0.999... 15847 | The recurring decimal 0.99... |
| geoihalfsum 15848 | Prove that the infinite ge... |
| cvgrat 15849 | Ratio test for convergence... |
| mertenslem1 15850 | Lemma for ~ mertens . (Co... |
| mertenslem2 15851 | Lemma for ~ mertens . (Co... |
| mertens 15852 | Mertens' theorem. If ` A ... |
| prodf 15853 | An infinite product of com... |
| clim2prod 15854 | The limit of an infinite p... |
| clim2div 15855 | The limit of an infinite p... |
| prodfmul 15856 | The product of two infinit... |
| prodf1 15857 | The value of the partial p... |
| prodf1f 15858 | A one-valued infinite prod... |
| prodfclim1 15859 | The constant one product c... |
| prodfn0 15860 | No term of a nonzero infin... |
| prodfrec 15861 | The reciprocal of an infin... |
| prodfdiv 15862 | The quotient of two infini... |
| ntrivcvg 15863 | A non-trivially converging... |
| ntrivcvgn0 15864 | A product that converges t... |
| ntrivcvgfvn0 15865 | Any value of a product seq... |
| ntrivcvgtail 15866 | A tail of a non-trivially ... |
| ntrivcvgmullem 15867 | Lemma for ~ ntrivcvgmul . ... |
| ntrivcvgmul 15868 | The product of two non-tri... |
| prodex 15871 | A product is a set. (Cont... |
| prodeq1f 15872 | Equality theorem for a pro... |
| prodeq1 15873 | Equality theorem for a pro... |
| nfcprod1 15874 | Bound-variable hypothesis ... |
| nfcprod 15875 | Bound-variable hypothesis ... |
| prodeq2w 15876 | Equality theorem for produ... |
| prodeq2ii 15877 | Equality theorem for produ... |
| prodeq2 15878 | Equality theorem for produ... |
| cbvprod 15879 | Change bound variable in a... |
| cbvprodv 15880 | Change bound variable in a... |
| cbvprodi 15881 | Change bound variable in a... |
| prodeq1i 15882 | Equality inference for pro... |
| prodeq1iOLD 15883 | Obsolete version of ~ prod... |
| prodeq2i 15884 | Equality inference for pro... |
| prodeq12i 15885 | Equality inference for pro... |
| prodeq1d 15886 | Equality deduction for pro... |
| prodeq2d 15887 | Equality deduction for pro... |
| prodeq2dv 15888 | Equality deduction for pro... |
| prodeq2sdv 15889 | Equality deduction for pro... |
| prodeq2sdvOLD 15890 | Obsolete version of ~ prod... |
| 2cprodeq2dv 15891 | Equality deduction for dou... |
| prodeq12dv 15892 | Equality deduction for pro... |
| prodeq12rdv 15893 | Equality deduction for pro... |
| prod2id 15894 | The second class argument ... |
| prodrblem 15895 | Lemma for ~ prodrb . (Con... |
| fprodcvg 15896 | The sequence of partial pr... |
| prodrblem2 15897 | Lemma for ~ prodrb . (Con... |
| prodrb 15898 | Rebase the starting point ... |
| prodmolem3 15899 | Lemma for ~ prodmo . (Con... |
| prodmolem2a 15900 | Lemma for ~ prodmo . (Con... |
| prodmolem2 15901 | Lemma for ~ prodmo . (Con... |
| prodmo 15902 | A product has at most one ... |
| zprod 15903 | Series product with index ... |
| iprod 15904 | Series product with an upp... |
| zprodn0 15905 | Nonzero series product wit... |
| iprodn0 15906 | Nonzero series product wit... |
| fprod 15907 | The value of a product ove... |
| fprodntriv 15908 | A non-triviality lemma for... |
| prod0 15909 | A product over the empty s... |
| prod1 15910 | Any product of one over a ... |
| prodfc 15911 | A lemma to facilitate conv... |
| fprodf1o 15912 | Re-index a finite product ... |
| prodss 15913 | Change the index set to a ... |
| fprodss 15914 | Change the index set to a ... |
| fprodser 15915 | A finite product expressed... |
| fprodcl2lem 15916 | Finite product closure lem... |
| fprodcllem 15917 | Finite product closure lem... |
| fprodcl 15918 | Closure of a finite produc... |
| fprodrecl 15919 | Closure of a finite produc... |
| fprodzcl 15920 | Closure of a finite produc... |
| fprodnncl 15921 | Closure of a finite produc... |
| fprodrpcl 15922 | Closure of a finite produc... |
| fprodnn0cl 15923 | Closure of a finite produc... |
| fprodcllemf 15924 | Finite product closure lem... |
| fprodreclf 15925 | Closure of a finite produc... |
| fprodmul 15926 | The product of two finite ... |
| fproddiv 15927 | The quotient of two finite... |
| prodsn 15928 | A product of a singleton i... |
| fprod1 15929 | A finite product of only o... |
| prodsnf 15930 | A product of a singleton i... |
| climprod1 15931 | The limit of a product ove... |
| fprodsplit 15932 | Split a finite product int... |
| fprodm1 15933 | Separate out the last term... |
| fprod1p 15934 | Separate out the first ter... |
| fprodp1 15935 | Multiply in the last term ... |
| fprodm1s 15936 | Separate out the last term... |
| fprodp1s 15937 | Multiply in the last term ... |
| prodsns 15938 | A product of the singleton... |
| fprodfac 15939 | Factorial using product no... |
| fprodabs 15940 | The absolute value of a fi... |
| fprodeq0 15941 | Any finite product contain... |
| fprodshft 15942 | Shift the index of a finit... |
| fprodrev 15943 | Reversal of a finite produ... |
| fprodconst 15944 | The product of constant te... |
| fprodn0 15945 | A finite product of nonzer... |
| fprod2dlem 15946 | Lemma for ~ fprod2d - indu... |
| fprod2d 15947 | Write a double product as ... |
| fprodxp 15948 | Combine two products into ... |
| fprodcnv 15949 | Transform a product region... |
| fprodcom2 15950 | Interchange order of multi... |
| fprodcom 15951 | Interchange product order.... |
| fprod0diag 15952 | Two ways to express "the p... |
| fproddivf 15953 | The quotient of two finite... |
| fprodsplitf 15954 | Split a finite product int... |
| fprodsplitsn 15955 | Separate out a term in a f... |
| fprodsplit1f 15956 | Separate out a term in a f... |
| fprodn0f 15957 | A finite product of nonzer... |
| fprodclf 15958 | Closure of a finite produc... |
| fprodge0 15959 | If all the terms of a fini... |
| fprodeq0g 15960 | Any finite product contain... |
| fprodge1 15961 | If all of the terms of a f... |
| fprodle 15962 | If all the terms of two fi... |
| fprodmodd 15963 | If all factors of two fini... |
| iprodclim 15964 | An infinite product equals... |
| iprodclim2 15965 | A converging product conve... |
| iprodclim3 15966 | The sequence of partial fi... |
| iprodcl 15967 | The product of a non-trivi... |
| iprodrecl 15968 | The product of a non-trivi... |
| iprodmul 15969 | Multiplication of infinite... |
| risefacval 15974 | The value of the rising fa... |
| fallfacval 15975 | The value of the falling f... |
| risefacval2 15976 | One-based value of rising ... |
| fallfacval2 15977 | One-based value of falling... |
| fallfacval3 15978 | A product representation o... |
| risefaccllem 15979 | Lemma for rising factorial... |
| fallfaccllem 15980 | Lemma for falling factoria... |
| risefaccl 15981 | Closure law for rising fac... |
| fallfaccl 15982 | Closure law for falling fa... |
| rerisefaccl 15983 | Closure law for rising fac... |
| refallfaccl 15984 | Closure law for falling fa... |
| nnrisefaccl 15985 | Closure law for rising fac... |
| zrisefaccl 15986 | Closure law for rising fac... |
| zfallfaccl 15987 | Closure law for falling fa... |
| nn0risefaccl 15988 | Closure law for rising fac... |
| rprisefaccl 15989 | Closure law for rising fac... |
| risefallfac 15990 | A relationship between ris... |
| fallrisefac 15991 | A relationship between fal... |
| risefall0lem 15992 | Lemma for ~ risefac0 and ~... |
| risefac0 15993 | The value of the rising fa... |
| fallfac0 15994 | The value of the falling f... |
| risefacp1 15995 | The value of the rising fa... |
| fallfacp1 15996 | The value of the falling f... |
| risefacp1d 15997 | The value of the rising fa... |
| fallfacp1d 15998 | The value of the falling f... |
| risefac1 15999 | The value of rising factor... |
| fallfac1 16000 | The value of falling facto... |
| risefacfac 16001 | Relate rising factorial to... |
| fallfacfwd 16002 | The forward difference of ... |
| 0fallfac 16003 | The value of the zero fall... |
| 0risefac 16004 | The value of the zero risi... |
| binomfallfaclem1 16005 | Lemma for ~ binomfallfac .... |
| binomfallfaclem2 16006 | Lemma for ~ binomfallfac .... |
| binomfallfac 16007 | A version of the binomial ... |
| binomrisefac 16008 | A version of the binomial ... |
| fallfacval4 16009 | Represent the falling fact... |
| bcfallfac 16010 | Binomial coefficient in te... |
| fallfacfac 16011 | Relate falling factorial t... |
| bpolylem 16014 | Lemma for ~ bpolyval . (C... |
| bpolyval 16015 | The value of the Bernoulli... |
| bpoly0 16016 | The value of the Bernoulli... |
| bpoly1 16017 | The value of the Bernoulli... |
| bpolycl 16018 | Closure law for Bernoulli ... |
| bpolysum 16019 | A sum for Bernoulli polyno... |
| bpolydiflem 16020 | Lemma for ~ bpolydif . (C... |
| bpolydif 16021 | Calculate the difference b... |
| fsumkthpow 16022 | A closed-form expression f... |
| bpoly2 16023 | The Bernoulli polynomials ... |
| bpoly3 16024 | The Bernoulli polynomials ... |
| bpoly4 16025 | The Bernoulli polynomials ... |
| fsumcube 16026 | Express the sum of cubes i... |
| eftcl 16039 | Closure of a term in the s... |
| reeftcl 16040 | The terms of the series ex... |
| eftabs 16041 | The absolute value of a te... |
| eftval 16042 | The value of a term in the... |
| efcllem 16043 | Lemma for ~ efcl . The se... |
| ef0lem 16044 | The series defining the ex... |
| efval 16045 | Value of the exponential f... |
| esum 16046 | Value of Euler's constant ... |
| eff 16047 | Domain and codomain of the... |
| efcl 16048 | Closure law for the expone... |
| efcld 16049 | Closure law for the expone... |
| efval2 16050 | Value of the exponential f... |
| efcvg 16051 | The series that defines th... |
| efcvgfsum 16052 | Exponential function conve... |
| reefcl 16053 | The exponential function i... |
| reefcld 16054 | The exponential function i... |
| ere 16055 | Euler's constant ` _e ` = ... |
| ege2le3 16056 | Lemma for ~ egt2lt3 . (Co... |
| ef0 16057 | Value of the exponential f... |
| efcj 16058 | The exponential of a compl... |
| efaddlem 16059 | Lemma for ~ efadd (exponen... |
| efadd 16060 | Sum of exponents law for e... |
| fprodefsum 16061 | Move the exponential funct... |
| efcan 16062 | Cancellation law for expon... |
| efne0d 16063 | The exponential of a compl... |
| efne0 16064 | The exponential of a compl... |
| efne0OLD 16065 | Obsolete version of ~ efne... |
| 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... |
| addmulmodb 16235 | An integer plus a product ... |
| dvds0lem 16236 | A lemma to assist theorems... |
| dvds1lem 16237 | A lemma to assist theorems... |
| dvds2lem 16238 | A lemma to assist theorems... |
| iddvds 16239 | An integer divides itself.... |
| 1dvds 16240 | 1 divides any integer. Th... |
| dvds0 16241 | Any integer divides 0. Th... |
| negdvdsb 16242 | An integer divides another... |
| dvdsnegb 16243 | An integer divides another... |
| absdvdsb 16244 | An integer divides another... |
| dvdsabsb 16245 | An integer divides another... |
| 0dvds 16246 | Only 0 is divisible by 0. ... |
| dvdsmul1 16247 | An integer divides a multi... |
| dvdsmul2 16248 | An integer divides a multi... |
| iddvdsexp 16249 | An integer divides a posit... |
| muldvds1 16250 | If a product divides an in... |
| muldvds2 16251 | If a product divides an in... |
| dvdscmul 16252 | Multiplication by a consta... |
| dvdsmulc 16253 | Multiplication by a consta... |
| dvdscmulr 16254 | Cancellation law for the d... |
| dvdsmulcr 16255 | Cancellation law for the d... |
| summodnegmod 16256 | The sum of two integers mo... |
| difmod0 16257 | The difference of two inte... |
| modmulconst 16258 | Constant multiplication in... |
| dvds2ln 16259 | If an integer divides each... |
| dvds2add 16260 | If an integer divides each... |
| dvds2sub 16261 | If an integer divides each... |
| dvds2addd 16262 | Deduction form of ~ dvds2a... |
| dvds2subd 16263 | Deduction form of ~ dvds2s... |
| dvdstr 16264 | The divides relation is tr... |
| dvdstrd 16265 | The divides relation is tr... |
| dvdsmultr1 16266 | If an integer divides anot... |
| dvdsmultr1d 16267 | Deduction form of ~ dvdsmu... |
| dvdsmultr2 16268 | If an integer divides anot... |
| dvdsmultr2d 16269 | Deduction form of ~ dvdsmu... |
| ordvdsmul 16270 | If an integer divides eith... |
| dvdssub2 16271 | If an integer divides a di... |
| dvdsadd 16272 | An integer divides another... |
| dvdsaddr 16273 | An integer divides another... |
| dvdssub 16274 | An integer divides another... |
| dvdssubr 16275 | An integer divides another... |
| dvdsadd2b 16276 | Adding a multiple of the b... |
| dvdsaddre2b 16277 | Adding a multiple of the b... |
| fsumdvds 16278 | If every term in a sum is ... |
| dvdslelem 16279 | Lemma for ~ dvdsle . (Con... |
| dvdsle 16280 | The divisors of a positive... |
| dvdsleabs 16281 | The divisors of a nonzero ... |
| dvdsleabs2 16282 | Transfer divisibility to a... |
| dvdsabseq 16283 | If two integers divide eac... |
| dvdseq 16284 | If two nonnegative integer... |
| divconjdvds 16285 | If a nonzero integer ` M `... |
| dvdsdivcl 16286 | The complement of a diviso... |
| dvdsflip 16287 | An involution of the divis... |
| dvdsssfz1 16288 | The set of divisors of a n... |
| dvds1 16289 | The only nonnegative integ... |
| alzdvds 16290 | Only 0 is divisible by all... |
| dvdsext 16291 | Poset extensionality for d... |
| fzm1ndvds 16292 | No number between ` 1 ` an... |
| fzo0dvdseq 16293 | Zero is the only one of th... |
| fzocongeq 16294 | Two different elements of ... |
| addmodlteqALT 16295 | Two nonnegative integers l... |
| dvdsfac 16296 | A positive integer divides... |
| dvdsexp2im 16297 | If an integer divides anot... |
| dvdsexp 16298 | A power divides a power wi... |
| dvdsmod 16299 | Any number ` K ` whose mod... |
| mulmoddvds 16300 | If an integer is divisible... |
| 3dvds 16301 | A rule for divisibility by... |
| 3dvdsdec 16302 | A decimal number is divisi... |
| 3dvds2dec 16303 | A decimal number is divisi... |
| fprodfvdvdsd 16304 | A finite product of intege... |
| fproddvdsd 16305 | A finite product of intege... |
| evenelz 16306 | An even number is an integ... |
| zeo3 16307 | An integer is even or odd.... |
| zeo4 16308 | An integer is even or odd ... |
| zeneo 16309 | No even integer equals an ... |
| odd2np1lem 16310 | Lemma for ~ odd2np1 . (Co... |
| odd2np1 16311 | An integer is odd iff it i... |
| even2n 16312 | An integer is even iff it ... |
| oddm1even 16313 | An integer is odd iff its ... |
| oddp1even 16314 | An integer is odd iff its ... |
| oexpneg 16315 | The exponential of the neg... |
| mod2eq0even 16316 | An integer is 0 modulo 2 i... |
| mod2eq1n2dvds 16317 | An integer is 1 modulo 2 i... |
| oddnn02np1 16318 | A nonnegative integer is o... |
| oddge22np1 16319 | An integer greater than on... |
| evennn02n 16320 | A nonnegative integer is e... |
| evennn2n 16321 | A positive integer is even... |
| 2tp1odd 16322 | A number which is twice an... |
| mulsucdiv2z 16323 | An integer multiplied with... |
| sqoddm1div8z 16324 | A squared odd number minus... |
| 2teven 16325 | A number which is twice an... |
| zeo5 16326 | An integer is either even ... |
| evend2 16327 | An integer is even iff its... |
| oddp1d2 16328 | An integer is odd iff its ... |
| zob 16329 | Alternate characterization... |
| oddm1d2 16330 | An integer is odd iff its ... |
| ltoddhalfle 16331 | An integer is less than ha... |
| halfleoddlt 16332 | An integer is greater than... |
| opoe 16333 | The sum of two odds is eve... |
| omoe 16334 | The difference of two odds... |
| opeo 16335 | The sum of an odd and an e... |
| omeo 16336 | The difference of an odd a... |
| z0even 16337 | 2 divides 0. That means 0... |
| n2dvds1 16338 | 2 does not divide 1. That... |
| n2dvdsm1 16339 | 2 does not divide -1. Tha... |
| z2even 16340 | 2 divides 2. That means 2... |
| n2dvds3 16341 | 2 does not divide 3. That... |
| z4even 16342 | 2 divides 4. That means 4... |
| 4dvdseven 16343 | An integer which is divisi... |
| m1expe 16344 | Exponentiation of -1 by an... |
| m1expo 16345 | Exponentiation of -1 by an... |
| m1exp1 16346 | Exponentiation of negative... |
| nn0enne 16347 | A positive integer is an e... |
| nn0ehalf 16348 | The half of an even nonneg... |
| nnehalf 16349 | The half of an even positi... |
| nn0onn 16350 | An odd nonnegative integer... |
| nn0o1gt2 16351 | An odd nonnegative integer... |
| nno 16352 | An alternate characterizat... |
| nn0o 16353 | An alternate characterizat... |
| nn0ob 16354 | Alternate characterization... |
| nn0oddm1d2 16355 | A positive integer is odd ... |
| nnoddm1d2 16356 | A positive integer is odd ... |
| sumeven 16357 | If every term in a sum is ... |
| sumodd 16358 | If every term in a sum is ... |
| evensumodd 16359 | If every term in a sum wit... |
| oddsumodd 16360 | If every term in a sum wit... |
| pwp1fsum 16361 | The n-th power of a number... |
| oddpwp1fsum 16362 | An odd power of a number i... |
| divalglem0 16363 | Lemma for ~ divalg . (Con... |
| divalglem1 16364 | Lemma for ~ divalg . (Con... |
| divalglem2 16365 | Lemma for ~ divalg . (Con... |
| divalglem4 16366 | Lemma for ~ divalg . (Con... |
| divalglem5 16367 | Lemma for ~ divalg . (Con... |
| divalglem6 16368 | Lemma for ~ divalg . (Con... |
| divalglem7 16369 | Lemma for ~ divalg . (Con... |
| divalglem8 16370 | Lemma for ~ divalg . (Con... |
| divalglem9 16371 | Lemma for ~ divalg . (Con... |
| divalglem10 16372 | Lemma for ~ divalg . (Con... |
| divalg 16373 | The division algorithm (th... |
| divalgb 16374 | Express the division algor... |
| divalg2 16375 | The division algorithm (th... |
| divalgmod 16376 | The result of the ` mod ` ... |
| divalgmodcl 16377 | The result of the ` mod ` ... |
| modremain 16378 | The result of the modulo o... |
| ndvdssub 16379 | Corollary of the division ... |
| ndvdsadd 16380 | Corollary of the division ... |
| ndvdsp1 16381 | Special case of ~ ndvdsadd... |
| ndvdsi 16382 | A quick test for non-divis... |
| 5ndvds3 16383 | 5 does not divide 3. (Con... |
| 5ndvds6 16384 | 5 does not divide 6. (Con... |
| flodddiv4 16385 | The floor of an odd intege... |
| fldivndvdslt 16386 | The floor of an integer di... |
| flodddiv4lt 16387 | The floor of an odd number... |
| flodddiv4t2lthalf 16388 | The floor of an odd number... |
| bitsfval 16393 | Expand the definition of t... |
| bitsval 16394 | Expand the definition of t... |
| bitsval2 16395 | Expand the definition of t... |
| bitsss 16396 | The set of bits of an inte... |
| bitsf 16397 | The ` bits ` function is a... |
| bits0 16398 | Value of the zeroth bit. ... |
| bits0e 16399 | The zeroth bit of an even ... |
| bits0o 16400 | The zeroth bit of an odd n... |
| bitsp1 16401 | The ` M + 1 ` -th bit of `... |
| bitsp1e 16402 | The ` M + 1 ` -th bit of `... |
| bitsp1o 16403 | The ` M + 1 ` -th bit of `... |
| bitsfzolem 16404 | Lemma for ~ bitsfzo . (Co... |
| bitsfzo 16405 | The bits of a number are a... |
| bitsmod 16406 | Truncating the bit sequenc... |
| bitsfi 16407 | Every number is associated... |
| bitscmp 16408 | The bit complement of ` N ... |
| 0bits 16409 | The bits of zero. (Contri... |
| m1bits 16410 | The bits of negative one. ... |
| bitsinv1lem 16411 | Lemma for ~ bitsinv1 . (C... |
| bitsinv1 16412 | There is an explicit inver... |
| bitsinv2 16413 | There is an explicit inver... |
| bitsf1ocnv 16414 | The ` bits ` function rest... |
| bitsf1o 16415 | The ` bits ` function rest... |
| bitsf1 16416 | The ` bits ` function is a... |
| 2ebits 16417 | The bits of a power of two... |
| bitsinv 16418 | The inverse of the ` bits ... |
| bitsinvp1 16419 | Recursive definition of th... |
| sadadd2lem2 16420 | The core of the proof of ~... |
| sadfval 16422 | Define the addition of two... |
| sadcf 16423 | The carry sequence is a se... |
| sadc0 16424 | The initial element of the... |
| sadcp1 16425 | The carry sequence (which ... |
| sadval 16426 | The full adder sequence is... |
| sadcaddlem 16427 | Lemma for ~ sadcadd . (Co... |
| sadcadd 16428 | Non-recursive definition o... |
| sadadd2lem 16429 | Lemma for ~ sadadd2 . (Co... |
| sadadd2 16430 | Sum of initial segments of... |
| sadadd3 16431 | Sum of initial segments of... |
| sadcl 16432 | The sum of two sequences i... |
| sadcom 16433 | The adder sequence functio... |
| saddisjlem 16434 | Lemma for ~ sadadd . (Con... |
| saddisj 16435 | The sum of disjoint sequen... |
| sadaddlem 16436 | Lemma for ~ sadadd . (Con... |
| sadadd 16437 | For sequences that corresp... |
| sadid1 16438 | The adder sequence functio... |
| sadid2 16439 | The adder sequence functio... |
| sadasslem 16440 | Lemma for ~ sadass . (Con... |
| sadass 16441 | Sequence addition is assoc... |
| sadeq 16442 | Any element of a sequence ... |
| bitsres 16443 | Restrict the bits of a num... |
| bitsuz 16444 | The bits of a number are a... |
| bitsshft 16445 | Shifting a bit sequence to... |
| smufval 16447 | The multiplication of two ... |
| smupf 16448 | The sequence of partial su... |
| smup0 16449 | The initial element of the... |
| smupp1 16450 | The initial element of the... |
| smuval 16451 | Define the addition of two... |
| smuval2 16452 | The partial sum sequence s... |
| smupvallem 16453 | If ` A ` only has elements... |
| smucl 16454 | The product of two sequenc... |
| smu01lem 16455 | Lemma for ~ smu01 and ~ sm... |
| smu01 16456 | Multiplication of a sequen... |
| smu02 16457 | Multiplication of a sequen... |
| smupval 16458 | Rewrite the elements of th... |
| smup1 16459 | Rewrite ~ smupp1 using onl... |
| smueqlem 16460 | Any element of a sequence ... |
| smueq 16461 | Any element of a sequence ... |
| smumullem 16462 | Lemma for ~ smumul . (Con... |
| smumul 16463 | For sequences that corresp... |
| gcdval 16466 | The value of the ` gcd ` o... |
| gcd0val 16467 | The value, by convention, ... |
| gcdn0val 16468 | The value of the ` gcd ` o... |
| gcdcllem1 16469 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem2 16470 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem3 16471 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdn0cl 16472 | Closure of the ` gcd ` ope... |
| gcddvds 16473 | The gcd of two integers di... |
| dvdslegcd 16474 | An integer which divides b... |
| nndvdslegcd 16475 | A positive integer which d... |
| gcdcl 16476 | Closure of the ` gcd ` ope... |
| gcdnncl 16477 | Closure of the ` gcd ` ope... |
| gcdcld 16478 | Closure of the ` gcd ` ope... |
| gcd2n0cl 16479 | Closure of the ` gcd ` ope... |
| zeqzmulgcd 16480 | An integer is the product ... |
| divgcdz 16481 | An integer divided by the ... |
| gcdf 16482 | Domain and codomain of the... |
| gcdcom 16483 | The ` gcd ` operator is co... |
| gcdcomd 16484 | The ` gcd ` operator is co... |
| divgcdnn 16485 | A positive integer divided... |
| divgcdnnr 16486 | A positive integer divided... |
| gcdeq0 16487 | The gcd of two integers is... |
| gcdn0gt0 16488 | The gcd of two integers is... |
| gcd0id 16489 | The gcd of 0 and an intege... |
| gcdid0 16490 | The gcd of an integer and ... |
| nn0gcdid0 16491 | The gcd of a nonnegative i... |
| gcdneg 16492 | Negating one operand of th... |
| neggcd 16493 | Negating one operand of th... |
| gcdaddmlem 16494 | Lemma for ~ gcdaddm . (Co... |
| gcdaddm 16495 | Adding a multiple of one o... |
| gcdadd 16496 | The GCD of two numbers is ... |
| gcdid 16497 | The gcd of a number and it... |
| gcd1 16498 | The gcd of a number with 1... |
| gcdabs1 16499 | ` gcd ` of the absolute va... |
| gcdabs2 16500 | ` gcd ` of the absolute va... |
| gcdabs 16501 | The gcd of two integers is... |
| modgcd 16502 | The gcd remains unchanged ... |
| 1gcd 16503 | The GCD of one and an inte... |
| gcdmultipled 16504 | The greatest common diviso... |
| gcdmultiplez 16505 | The GCD of a multiple of a... |
| gcdmultiple 16506 | The GCD of a multiple of a... |
| dvdsgcdidd 16507 | The greatest common diviso... |
| 6gcd4e2 16508 | The greatest common diviso... |
| bezoutlem1 16509 | Lemma for ~ bezout . (Con... |
| bezoutlem2 16510 | Lemma for ~ bezout . (Con... |
| bezoutlem3 16511 | Lemma for ~ bezout . (Con... |
| bezoutlem4 16512 | Lemma for ~ bezout . (Con... |
| bezout 16513 | Bézout's identity: ... |
| dvdsgcd 16514 | An integer which divides e... |
| dvdsgcdb 16515 | Biconditional form of ~ dv... |
| dfgcd2 16516 | Alternate definition of th... |
| gcdass 16517 | Associative law for ` gcd ... |
| mulgcd 16518 | Distribute multiplication ... |
| absmulgcd 16519 | Distribute absolute value ... |
| mulgcdr 16520 | Reverse distribution law f... |
| gcddiv 16521 | Division law for GCD. (Con... |
| gcdzeq 16522 | A positive integer ` A ` i... |
| gcdeq 16523 | ` A ` is equal to its gcd ... |
| dvdssqim 16524 | Unidirectional form of ~ d... |
| dvdsexpim 16525 | If two numbers are divisib... |
| dvdsmulgcd 16526 | A divisibility equivalent ... |
| rpmulgcd 16527 | If ` K ` and ` M ` are rel... |
| rplpwr 16528 | If ` A ` and ` B ` are rel... |
| rprpwr 16529 | If ` A ` and ` B ` are rel... |
| rppwr 16530 | If ` A ` and ` B ` are rel... |
| nn0rppwr 16531 | If ` A ` and ` B ` are rel... |
| sqgcd 16532 | Square distributes over gc... |
| expgcd 16533 | Exponentiation distributes... |
| nn0expgcd 16534 | Exponentiation distributes... |
| zexpgcd 16535 | Exponentiation distributes... |
| dvdssqlem 16536 | Lemma for ~ dvdssq . (Con... |
| dvdssq 16537 | Two numbers are divisible ... |
| bezoutr 16538 | Partial converse to ~ bezo... |
| bezoutr1 16539 | Converse of ~ bezout for w... |
| nn0seqcvgd 16540 | A strictly-decreasing nonn... |
| seq1st 16541 | A sequence whose iteration... |
| algr0 16542 | The value of the algorithm... |
| algrf 16543 | An algorithm is a step fun... |
| algrp1 16544 | The value of the algorithm... |
| alginv 16545 | If ` I ` is an invariant o... |
| algcvg 16546 | One way to prove that an a... |
| algcvgblem 16547 | Lemma for ~ algcvgb . (Co... |
| algcvgb 16548 | Two ways of expressing tha... |
| algcvga 16549 | The countdown function ` C... |
| algfx 16550 | If ` F ` reaches a fixed p... |
| eucalgval2 16551 | The value of the step func... |
| eucalgval 16552 | Euclid's Algorithm ~ eucal... |
| eucalgf 16553 | Domain and codomain of the... |
| eucalginv 16554 | The invariant of the step ... |
| eucalglt 16555 | The second member of the s... |
| eucalgcvga 16556 | Once Euclid's Algorithm ha... |
| eucalg 16557 | Euclid's Algorithm compute... |
| lcmval 16562 | Value of the ` lcm ` opera... |
| lcmcom 16563 | The ` lcm ` operator is co... |
| lcm0val 16564 | The value, by convention, ... |
| lcmn0val 16565 | The value of the ` lcm ` o... |
| lcmcllem 16566 | Lemma for ~ lcmn0cl and ~ ... |
| lcmn0cl 16567 | Closure of the ` lcm ` ope... |
| dvdslcm 16568 | The lcm of two integers is... |
| lcmledvds 16569 | A positive integer which b... |
| lcmeq0 16570 | The lcm of two integers is... |
| lcmcl 16571 | Closure of the ` lcm ` ope... |
| gcddvdslcm 16572 | The greatest common diviso... |
| lcmneg 16573 | Negating one operand of th... |
| neglcm 16574 | Negating one operand of th... |
| lcmabs 16575 | The lcm of two integers is... |
| lcmgcdlem 16576 | Lemma for ~ lcmgcd and ~ l... |
| lcmgcd 16577 | The product of two numbers... |
| lcmdvds 16578 | The lcm of two integers di... |
| lcmid 16579 | The lcm of an integer and ... |
| lcm1 16580 | The lcm of an integer and ... |
| lcmgcdnn 16581 | The product of two positiv... |
| lcmgcdeq 16582 | Two integers' absolute val... |
| lcmdvdsb 16583 | Biconditional form of ~ lc... |
| lcmass 16584 | Associative law for ` lcm ... |
| 3lcm2e6woprm 16585 | The least common multiple ... |
| 6lcm4e12 16586 | The least common multiple ... |
| absproddvds 16587 | The absolute value of the ... |
| absprodnn 16588 | The absolute value of the ... |
| fissn0dvds 16589 | For each finite subset of ... |
| fissn0dvdsn0 16590 | For each finite subset of ... |
| lcmfval 16591 | Value of the ` _lcm ` func... |
| lcmf0val 16592 | The value, by convention, ... |
| lcmfn0val 16593 | The value of the ` _lcm ` ... |
| lcmfnnval 16594 | The value of the ` _lcm ` ... |
| lcmfcllem 16595 | Lemma for ~ lcmfn0cl and ~... |
| lcmfn0cl 16596 | Closure of the ` _lcm ` fu... |
| lcmfpr 16597 | The value of the ` _lcm ` ... |
| lcmfcl 16598 | Closure of the ` _lcm ` fu... |
| lcmfnncl 16599 | Closure of the ` _lcm ` fu... |
| lcmfeq0b 16600 | The least common multiple ... |
| dvdslcmf 16601 | The least common multiple ... |
| lcmfledvds 16602 | A positive integer which i... |
| lcmf 16603 | Characterization of the le... |
| lcmf0 16604 | The least common multiple ... |
| lcmfsn 16605 | The least common multiple ... |
| lcmftp 16606 | The least common multiple ... |
| lcmfunsnlem1 16607 | Lemma for ~ lcmfdvds and ~... |
| lcmfunsnlem2lem1 16608 | Lemma 1 for ~ lcmfunsnlem2... |
| lcmfunsnlem2lem2 16609 | Lemma 2 for ~ lcmfunsnlem2... |
| lcmfunsnlem2 16610 | Lemma for ~ lcmfunsn and ~... |
| lcmfunsnlem 16611 | Lemma for ~ lcmfdvds and ~... |
| lcmfdvds 16612 | The least common multiple ... |
| lcmfdvdsb 16613 | Biconditional form of ~ lc... |
| lcmfunsn 16614 | The ` _lcm ` function for ... |
| lcmfun 16615 | The ` _lcm ` function for ... |
| lcmfass 16616 | Associative law for the ` ... |
| lcmf2a3a4e12 16617 | The least common multiple ... |
| lcmflefac 16618 | The least common multiple ... |
| coprmgcdb 16619 | Two positive integers are ... |
| ncoprmgcdne1b 16620 | Two positive integers are ... |
| ncoprmgcdgt1b 16621 | Two positive integers are ... |
| coprmdvds1 16622 | If two positive integers a... |
| coprmdvds 16623 | Euclid's Lemma (see ProofW... |
| coprmdvds2 16624 | If an integer is divisible... |
| mulgcddvds 16625 | One half of ~ rpmulgcd2 , ... |
| rpmulgcd2 16626 | If ` M ` is relatively pri... |
| qredeq 16627 | Two equal reduced fraction... |
| qredeu 16628 | Every rational number has ... |
| rpmul 16629 | If ` K ` is relatively pri... |
| rpdvds 16630 | If ` K ` is relatively pri... |
| coprmprod 16631 | The product of the element... |
| coprmproddvdslem 16632 | Lemma for ~ coprmproddvds ... |
| coprmproddvds 16633 | If a positive integer is d... |
| congr 16634 | Definition of congruence b... |
| divgcdcoprm0 16635 | Integers divided by gcd ar... |
| divgcdcoprmex 16636 | Integers divided by gcd ar... |
| cncongr1 16637 | One direction of the bicon... |
| cncongr2 16638 | The other direction of the... |
| cncongr 16639 | Cancellability of Congruen... |
| cncongrcoprm 16640 | Corollary 1 of Cancellabil... |
| isprm 16643 | The predicate "is a prime ... |
| prmnn 16644 | A prime number is a positi... |
| prmz 16645 | A prime number is an integ... |
| prmssnn 16646 | The prime numbers are a su... |
| prmex 16647 | The set of prime numbers e... |
| 0nprm 16648 | 0 is not a prime number. ... |
| 1nprm 16649 | 1 is not a prime number. ... |
| 1idssfct 16650 | The positive divisors of a... |
| isprm2lem 16651 | Lemma for ~ isprm2 . (Con... |
| isprm2 16652 | The predicate "is a prime ... |
| isprm3 16653 | The predicate "is a prime ... |
| isprm4 16654 | The predicate "is a prime ... |
| prmind2 16655 | A variation on ~ prmind as... |
| prmind 16656 | Perform induction over the... |
| dvdsprime 16657 | If ` M ` divides a prime, ... |
| nprm 16658 | A product of two integers ... |
| nprmi 16659 | An inference for composite... |
| dvdsnprmd 16660 | If a number is divisible b... |
| prm2orodd 16661 | A prime number is either 2... |
| 2prm 16662 | 2 is a prime number. (Con... |
| 2mulprm 16663 | A multiple of two is prime... |
| 3prm 16664 | 3 is a prime number. (Con... |
| 4nprm 16665 | 4 is not a prime number. ... |
| prmuz2 16666 | A prime number is an integ... |
| prmgt1 16667 | A prime number is an integ... |
| prmm2nn0 16668 | Subtracting 2 from a prime... |
| oddprmgt2 16669 | An odd prime is greater th... |
| oddprmge3 16670 | An odd prime is greater th... |
| ge2nprmge4 16671 | A composite integer greate... |
| sqnprm 16672 | A square is never prime. ... |
| dvdsprm 16673 | An integer greater than or... |
| exprmfct 16674 | Every integer greater than... |
| prmdvdsfz 16675 | Each integer greater than ... |
| nprmdvds1 16676 | No prime number divides 1.... |
| isprm5 16677 | One need only check prime ... |
| isprm7 16678 | One need only check prime ... |
| maxprmfct 16679 | The set of prime factors o... |
| divgcdodd 16680 | Either ` A / ( A gcd B ) `... |
| coprm 16681 | A prime number either divi... |
| prmrp 16682 | Unequal prime numbers are ... |
| euclemma 16683 | Euclid's lemma. A prime n... |
| isprm6 16684 | A number is prime iff it s... |
| prmdvdsexp 16685 | A prime divides a positive... |
| prmdvdsexpb 16686 | A prime divides a positive... |
| prmdvdsexpr 16687 | If a prime divides a nonne... |
| prmdvdssq 16688 | Condition for a prime divi... |
| prmexpb 16689 | Two positive prime powers ... |
| prmfac1 16690 | The factorial of a number ... |
| dvdszzq 16691 | Divisibility for an intege... |
| rpexp 16692 | If two numbers ` A ` and `... |
| rpexp1i 16693 | Relative primality passes ... |
| rpexp12i 16694 | Relative primality passes ... |
| prmndvdsfaclt 16695 | A prime number does not di... |
| prmdvdsbc 16696 | Condition for a prime numb... |
| prmdvdsncoprmbd 16697 | Two positive integers are ... |
| ncoprmlnprm 16698 | If two positive integers a... |
| cncongrprm 16699 | Corollary 2 of Cancellabil... |
| isevengcd2 16700 | The predicate "is an even ... |
| isoddgcd1 16701 | The predicate "is an odd n... |
| 3lcm2e6 16702 | The least common multiple ... |
| qnumval 16707 | Value of the canonical num... |
| qdenval 16708 | Value of the canonical den... |
| qnumdencl 16709 | Lemma for ~ qnumcl and ~ q... |
| qnumcl 16710 | The canonical numerator of... |
| qdencl 16711 | The canonical denominator ... |
| fnum 16712 | Canonical numerator define... |
| fden 16713 | Canonical denominator defi... |
| qnumdenbi 16714 | Two numbers are the canoni... |
| qnumdencoprm 16715 | The canonical representati... |
| qeqnumdivden 16716 | Recover a rational number ... |
| qmuldeneqnum 16717 | Multiplying a rational by ... |
| divnumden 16718 | Calculate the reduced form... |
| divdenle 16719 | Reducing a quotient never ... |
| qnumgt0 16720 | A rational is positive iff... |
| qgt0numnn 16721 | A rational is positive iff... |
| nn0gcdsq 16722 | Squaring commutes with GCD... |
| zgcdsq 16723 | ~ nn0gcdsq extended to int... |
| numdensq 16724 | Squaring a rational square... |
| numsq 16725 | Square commutes with canon... |
| densq 16726 | Square commutes with canon... |
| qden1elz 16727 | A rational is an integer i... |
| zsqrtelqelz 16728 | If an integer has a ration... |
| nonsq 16729 | Any integer strictly betwe... |
| numdenexp 16730 | Elevating a rational numbe... |
| numexp 16731 | Elevating to a nonnegative... |
| denexp 16732 | Elevating to a nonnegative... |
| phival 16737 | Value of the Euler ` phi `... |
| phicl2 16738 | Bounds and closure for the... |
| phicl 16739 | Closure for the value of t... |
| phibndlem 16740 | Lemma for ~ phibnd . (Con... |
| phibnd 16741 | A slightly tighter bound o... |
| phicld 16742 | Closure for the value of t... |
| phi1 16743 | Value of the Euler ` phi `... |
| dfphi2 16744 | Alternate definition of th... |
| hashdvds 16745 | The number of numbers in a... |
| phiprmpw 16746 | Value of the Euler ` phi `... |
| phiprm 16747 | Value of the Euler ` phi `... |
| crth 16748 | The Chinese Remainder Theo... |
| phimullem 16749 | Lemma for ~ phimul . (Con... |
| phimul 16750 | The Euler ` phi ` function... |
| eulerthlem1 16751 | Lemma for ~ eulerth . (Co... |
| eulerthlem2 16752 | Lemma for ~ eulerth . (Co... |
| eulerth 16753 | Euler's theorem, a general... |
| fermltl 16754 | Fermat's little theorem. ... |
| prmdiv 16755 | Show an explicit expressio... |
| prmdiveq 16756 | The modular inverse of ` A... |
| prmdivdiv 16757 | The (modular) inverse of t... |
| hashgcdlem 16758 | A correspondence between e... |
| dvdsfi 16759 | A natural number has finit... |
| hashgcdeq 16760 | Number of initial positive... |
| phisum 16761 | The divisor sum identity o... |
| odzval 16762 | Value of the order functio... |
| odzcllem 16763 | - Lemma for ~ odzcl , show... |
| odzcl 16764 | The order of a group eleme... |
| odzid 16765 | Any element raised to the ... |
| odzdvds 16766 | The only powers of ` A ` t... |
| odzphi 16767 | The order of any group ele... |
| modprm1div 16768 | A prime number divides an ... |
| m1dvdsndvds 16769 | If an integer minus 1 is d... |
| modprminv 16770 | Show an explicit expressio... |
| modprminveq 16771 | The modular inverse of ` A... |
| vfermltl 16772 | Variant of Fermat's little... |
| vfermltlALT 16773 | Alternate proof of ~ vferm... |
| powm2modprm 16774 | If an integer minus 1 is d... |
| reumodprminv 16775 | For any prime number and f... |
| modprm0 16776 | For two positive integers ... |
| nnnn0modprm0 16777 | For a positive integer and... |
| modprmn0modprm0 16778 | For an integer not being 0... |
| coprimeprodsq 16779 | If three numbers are copri... |
| coprimeprodsq2 16780 | If three numbers are copri... |
| oddprm 16781 | A prime not equal to ` 2 `... |
| nnoddn2prm 16782 | A prime not equal to ` 2 `... |
| oddn2prm 16783 | A prime not equal to ` 2 `... |
| nnoddn2prmb 16784 | A number is a prime number... |
| prm23lt5 16785 | A prime less than 5 is eit... |
| prm23ge5 16786 | A prime is either 2 or 3 o... |
| pythagtriplem1 16787 | Lemma for ~ pythagtrip . ... |
| pythagtriplem2 16788 | Lemma for ~ pythagtrip . ... |
| pythagtriplem3 16789 | Lemma for ~ pythagtrip . ... |
| pythagtriplem4 16790 | Lemma for ~ pythagtrip . ... |
| pythagtriplem10 16791 | Lemma for ~ pythagtrip . ... |
| pythagtriplem6 16792 | Lemma for ~ pythagtrip . ... |
| pythagtriplem7 16793 | Lemma for ~ pythagtrip . ... |
| pythagtriplem8 16794 | Lemma for ~ pythagtrip . ... |
| pythagtriplem9 16795 | Lemma for ~ pythagtrip . ... |
| pythagtriplem11 16796 | Lemma for ~ pythagtrip . ... |
| pythagtriplem12 16797 | Lemma for ~ pythagtrip . ... |
| pythagtriplem13 16798 | Lemma for ~ pythagtrip . ... |
| pythagtriplem14 16799 | Lemma for ~ pythagtrip . ... |
| pythagtriplem15 16800 | Lemma for ~ pythagtrip . ... |
| pythagtriplem16 16801 | Lemma for ~ pythagtrip . ... |
| pythagtriplem17 16802 | Lemma for ~ pythagtrip . ... |
| pythagtriplem18 16803 | Lemma for ~ pythagtrip . ... |
| pythagtriplem19 16804 | Lemma for ~ pythagtrip . ... |
| pythagtrip 16805 | Parameterize the Pythagore... |
| iserodd 16806 | Collect the odd terms in a... |
| pclem 16809 | - Lemma for the prime powe... |
| pcprecl 16810 | Closure of the prime power... |
| pcprendvds 16811 | Non-divisibility property ... |
| pcprendvds2 16812 | Non-divisibility property ... |
| pcpre1 16813 | Value of the prime power p... |
| pcpremul 16814 | Multiplicative property of... |
| pcval 16815 | The value of the prime pow... |
| pceulem 16816 | Lemma for ~ pceu . (Contr... |
| pceu 16817 | Uniqueness for the prime p... |
| pczpre 16818 | Connect the prime count pr... |
| pczcl 16819 | Closure of the prime power... |
| pccl 16820 | Closure of the prime power... |
| pccld 16821 | Closure of the prime power... |
| pcmul 16822 | Multiplication property of... |
| pcdiv 16823 | Division property of the p... |
| pcqmul 16824 | Multiplication property of... |
| pc0 16825 | The value of the prime pow... |
| pc1 16826 | Value of the prime count f... |
| pcqcl 16827 | Closure of the general pri... |
| pcqdiv 16828 | Division property of the p... |
| pcrec 16829 | Prime power of a reciproca... |
| pcexp 16830 | Prime power of an exponent... |
| pcxnn0cl 16831 | Extended nonnegative integ... |
| pcxcl 16832 | Extended real closure of t... |
| pcge0 16833 | The prime count of an inte... |
| pczdvds 16834 | Defining property of the p... |
| pcdvds 16835 | Defining property of the p... |
| pczndvds 16836 | Defining property of the p... |
| pcndvds 16837 | Defining property of the p... |
| pczndvds2 16838 | The remainder after dividi... |
| pcndvds2 16839 | The remainder after dividi... |
| pcdvdsb 16840 | ` P ^ A ` divides ` N ` if... |
| pcelnn 16841 | There are a positive numbe... |
| pceq0 16842 | There are zero powers of a... |
| pcidlem 16843 | The prime count of a prime... |
| pcid 16844 | The prime count of a prime... |
| pcneg 16845 | The prime count of a negat... |
| pcabs 16846 | The prime count of an abso... |
| pcdvdstr 16847 | The prime count increases ... |
| pcgcd1 16848 | The prime count of a GCD i... |
| pcgcd 16849 | The prime count of a GCD i... |
| pc2dvds 16850 | A characterization of divi... |
| pc11 16851 | The prime count function, ... |
| pcz 16852 | The prime count function c... |
| pcprmpw2 16853 | Self-referential expressio... |
| pcprmpw 16854 | Self-referential expressio... |
| dvdsprmpweq 16855 | If a positive integer divi... |
| dvdsprmpweqnn 16856 | If an integer greater than... |
| dvdsprmpweqle 16857 | If a positive integer divi... |
| difsqpwdvds 16858 | If the difference of two s... |
| pcaddlem 16859 | Lemma for ~ pcadd . The o... |
| pcadd 16860 | An inequality for the prim... |
| pcadd2 16861 | The inequality of ~ pcadd ... |
| pcmptcl 16862 | Closure for the prime powe... |
| pcmpt 16863 | Construct a function with ... |
| pcmpt2 16864 | Dividing two prime count m... |
| pcmptdvds 16865 | The partial products of th... |
| pcprod 16866 | The product of the primes ... |
| sumhash 16867 | The sum of 1 over a set is... |
| fldivp1 16868 | The difference between the... |
| pcfaclem 16869 | Lemma for ~ pcfac . (Cont... |
| pcfac 16870 | Calculate the prime count ... |
| pcbc 16871 | Calculate the prime count ... |
| qexpz 16872 | If a power of a rational n... |
| expnprm 16873 | A second or higher power o... |
| oddprmdvds 16874 | Every positive integer whi... |
| prmpwdvds 16875 | A relation involving divis... |
| pockthlem 16876 | Lemma for ~ pockthg . (Co... |
| pockthg 16877 | The generalized Pocklingto... |
| pockthi 16878 | Pocklington's theorem, whi... |
| unbenlem 16879 | Lemma for ~ unben . (Cont... |
| unben 16880 | An unbounded set of positi... |
| infpnlem1 16881 | Lemma for ~ infpn . The s... |
| infpnlem2 16882 | Lemma for ~ infpn . For a... |
| infpn 16883 | There exist infinitely man... |
| infpn2 16884 | There exist infinitely man... |
| prmunb 16885 | The primes are unbounded. ... |
| prminf 16886 | There are an infinite numb... |
| prmreclem1 16887 | Lemma for ~ prmrec . Prop... |
| prmreclem2 16888 | Lemma for ~ prmrec . Ther... |
| prmreclem3 16889 | Lemma for ~ prmrec . The ... |
| prmreclem4 16890 | Lemma for ~ prmrec . Show... |
| prmreclem5 16891 | Lemma for ~ prmrec . Here... |
| prmreclem6 16892 | Lemma for ~ prmrec . If t... |
| prmrec 16893 | The sum of the reciprocals... |
| 1arithlem1 16894 | Lemma for ~ 1arith . (Con... |
| 1arithlem2 16895 | Lemma for ~ 1arith . (Con... |
| 1arithlem3 16896 | Lemma for ~ 1arith . (Con... |
| 1arithlem4 16897 | Lemma for ~ 1arith . (Con... |
| 1arith 16898 | Fundamental theorem of ari... |
| 1arith2 16899 | Fundamental theorem of ari... |
| elgz 16902 | Elementhood in the gaussia... |
| gzcn 16903 | A gaussian integer is a co... |
| zgz 16904 | An integer is a gaussian i... |
| igz 16905 | ` _i ` is a gaussian integ... |
| gznegcl 16906 | The gaussian integers are ... |
| gzcjcl 16907 | The gaussian integers are ... |
| gzaddcl 16908 | The gaussian integers are ... |
| gzmulcl 16909 | The gaussian integers are ... |
| gzreim 16910 | Construct a gaussian integ... |
| gzsubcl 16911 | The gaussian integers are ... |
| gzabssqcl 16912 | The squared norm of a gaus... |
| 4sqlem5 16913 | Lemma for ~ 4sq . (Contri... |
| 4sqlem6 16914 | Lemma for ~ 4sq . (Contri... |
| 4sqlem7 16915 | Lemma for ~ 4sq . (Contri... |
| 4sqlem8 16916 | Lemma for ~ 4sq . (Contri... |
| 4sqlem9 16917 | Lemma for ~ 4sq . (Contri... |
| 4sqlem10 16918 | Lemma for ~ 4sq . (Contri... |
| 4sqlem1 16919 | Lemma for ~ 4sq . The set... |
| 4sqlem2 16920 | Lemma for ~ 4sq . Change ... |
| 4sqlem3 16921 | Lemma for ~ 4sq . Suffici... |
| 4sqlem4a 16922 | Lemma for ~ 4sqlem4 . (Co... |
| 4sqlem4 16923 | Lemma for ~ 4sq . We can ... |
| mul4sqlem 16924 | Lemma for ~ mul4sq : algeb... |
| mul4sq 16925 | Euler's four-square identi... |
| 4sqlem11 16926 | Lemma for ~ 4sq . Use the... |
| 4sqlem12 16927 | Lemma for ~ 4sq . For any... |
| 4sqlem13 16928 | Lemma for ~ 4sq . (Contri... |
| 4sqlem14 16929 | Lemma for ~ 4sq . (Contri... |
| 4sqlem15 16930 | Lemma for ~ 4sq . (Contri... |
| 4sqlem16 16931 | Lemma for ~ 4sq . (Contri... |
| 4sqlem17 16932 | Lemma for ~ 4sq . (Contri... |
| 4sqlem18 16933 | Lemma for ~ 4sq . Inducti... |
| 4sqlem19 16934 | Lemma for ~ 4sq . The pro... |
| 4sq 16935 | Lagrange's four-square the... |
| vdwapfval 16942 | Define the arithmetic prog... |
| vdwapf 16943 | The arithmetic progression... |
| vdwapval 16944 | Value of the arithmetic pr... |
| vdwapun 16945 | Remove the first element o... |
| vdwapid1 16946 | The first element of an ar... |
| vdwap0 16947 | Value of a length-1 arithm... |
| vdwap1 16948 | Value of a length-1 arithm... |
| vdwmc 16949 | The predicate " The ` <. R... |
| vdwmc2 16950 | Expand out the definition ... |
| vdwpc 16951 | The predicate " The colori... |
| vdwlem1 16952 | Lemma for ~ vdw . (Contri... |
| vdwlem2 16953 | Lemma for ~ vdw . (Contri... |
| vdwlem3 16954 | Lemma for ~ vdw . (Contri... |
| vdwlem4 16955 | Lemma for ~ vdw . (Contri... |
| vdwlem5 16956 | Lemma for ~ vdw . (Contri... |
| vdwlem6 16957 | Lemma for ~ vdw . (Contri... |
| vdwlem7 16958 | Lemma for ~ vdw . (Contri... |
| vdwlem8 16959 | Lemma for ~ vdw . (Contri... |
| vdwlem9 16960 | Lemma for ~ vdw . (Contri... |
| vdwlem10 16961 | Lemma for ~ vdw . Set up ... |
| vdwlem11 16962 | Lemma for ~ vdw . (Contri... |
| vdwlem12 16963 | Lemma for ~ vdw . ` K = 2 ... |
| vdwlem13 16964 | Lemma for ~ vdw . Main in... |
| vdw 16965 | Van der Waerden's theorem.... |
| vdwnnlem1 16966 | Corollary of ~ vdw , and l... |
| vdwnnlem2 16967 | Lemma for ~ vdwnn . The s... |
| vdwnnlem3 16968 | Lemma for ~ vdwnn . (Cont... |
| vdwnn 16969 | Van der Waerden's theorem,... |
| ramtlecl 16971 | The set ` T ` of numbers w... |
| hashbcval 16973 | Value of the "binomial set... |
| hashbccl 16974 | The binomial set is a fini... |
| hashbcss 16975 | Subset relation for the bi... |
| hashbc0 16976 | The set of subsets of size... |
| hashbc2 16977 | The size of the binomial s... |
| 0hashbc 16978 | There are no subsets of th... |
| ramval 16979 | The value of the Ramsey nu... |
| ramcl2lem 16980 | Lemma for extended real cl... |
| ramtcl 16981 | The Ramsey number has the ... |
| ramtcl2 16982 | The Ramsey number is an in... |
| ramtub 16983 | The Ramsey number is a low... |
| ramub 16984 | The Ramsey number is a low... |
| ramub2 16985 | It is sufficient to check ... |
| rami 16986 | The defining property of a... |
| ramcl2 16987 | The Ramsey number is eithe... |
| ramxrcl 16988 | The Ramsey number is an ex... |
| ramubcl 16989 | If the Ramsey number is up... |
| ramlb 16990 | Establish a lower bound on... |
| 0ram 16991 | The Ramsey number when ` M... |
| 0ram2 16992 | The Ramsey number when ` M... |
| ram0 16993 | The Ramsey number when ` R... |
| 0ramcl 16994 | Lemma for ~ ramcl : Exist... |
| ramz2 16995 | The Ramsey number when ` F... |
| ramz 16996 | The Ramsey number when ` F... |
| ramub1lem1 16997 | Lemma for ~ ramub1 . (Con... |
| ramub1lem2 16998 | Lemma for ~ ramub1 . (Con... |
| ramub1 16999 | Inductive step for Ramsey'... |
| ramcl 17000 | Ramsey's theorem: the Rams... |
| ramsey 17001 | Ramsey's theorem with the ... |
| prmoval 17004 | Value of the primorial fun... |
| prmocl 17005 | Closure of the primorial f... |
| prmone0 17006 | The primorial function is ... |
| prmo0 17007 | The primorial of 0. (Cont... |
| prmo1 17008 | The primorial of 1. (Cont... |
| prmop1 17009 | The primorial of a success... |
| prmonn2 17010 | Value of the primorial fun... |
| prmo2 17011 | The primorial of 2. (Cont... |
| prmo3 17012 | The primorial of 3. (Cont... |
| prmdvdsprmo 17013 | The primorial of a number ... |
| prmdvdsprmop 17014 | The primorial of a number ... |
| fvprmselelfz 17015 | The value of the prime sel... |
| fvprmselgcd1 17016 | The greatest common diviso... |
| prmolefac 17017 | The primorial of a positiv... |
| prmodvdslcmf 17018 | The primorial of a nonnega... |
| prmolelcmf 17019 | The primorial of a positiv... |
| prmgaplem1 17020 | Lemma for ~ prmgap : The ... |
| prmgaplem2 17021 | Lemma for ~ prmgap : The ... |
| prmgaplcmlem1 17022 | Lemma for ~ prmgaplcm : T... |
| prmgaplcmlem2 17023 | Lemma for ~ prmgaplcm : T... |
| prmgaplem3 17024 | Lemma for ~ prmgap . (Con... |
| prmgaplem4 17025 | Lemma for ~ prmgap . (Con... |
| prmgaplem5 17026 | Lemma for ~ prmgap : for e... |
| prmgaplem6 17027 | Lemma for ~ prmgap : for e... |
| prmgaplem7 17028 | Lemma for ~ prmgap . (Con... |
| prmgaplem8 17029 | Lemma for ~ prmgap . (Con... |
| prmgap 17030 | The prime gap theorem: for... |
| prmgaplcm 17031 | Alternate proof of ~ prmga... |
| prmgapprmolem 17032 | Lemma for ~ prmgapprmo : ... |
| prmgapprmo 17033 | Alternate proof of ~ prmga... |
| dec2dvds 17034 | Divisibility by two is obv... |
| dec5dvds 17035 | Divisibility by five is ob... |
| dec5dvds2 17036 | Divisibility by five is ob... |
| dec5nprm 17037 | A decimal number greater t... |
| dec2nprm 17038 | A decimal number greater t... |
| modxai 17039 | Add exponents in a power m... |
| mod2xi 17040 | Double exponents in a powe... |
| modxp1i 17041 | Add one to an exponent in ... |
| mod2xnegi 17042 | Version of ~ mod2xi with a... |
| modsubi 17043 | Subtract from within a mod... |
| gcdi 17044 | Calculate a GCD via Euclid... |
| gcdmodi 17045 | Calculate a GCD via Euclid... |
| numexp0 17046 | Calculate an integer power... |
| numexp1 17047 | Calculate an integer power... |
| numexpp1 17048 | Calculate an integer power... |
| numexp2x 17049 | Double an integer power. ... |
| decsplit0b 17050 | Split a decimal number int... |
| decsplit0 17051 | Split a decimal number int... |
| decsplit1 17052 | Split a decimal number int... |
| decsplit 17053 | Split a decimal number int... |
| karatsuba 17054 | The Karatsuba multiplicati... |
| 2exp4 17055 | Two to the fourth power is... |
| 2exp5 17056 | Two to the fifth power is ... |
| 2exp6 17057 | Two to the sixth power is ... |
| 2exp7 17058 | Two to the seventh power i... |
| 2exp8 17059 | Two to the eighth power is... |
| 2exp11 17060 | Two to the eleventh power ... |
| 2exp16 17061 | Two to the sixteenth power... |
| 3exp3 17062 | Three to the third power i... |
| 2expltfac 17063 | The factorial grows faster... |
| cshwsidrepsw 17064 | If cyclically shifting a w... |
| cshwsidrepswmod0 17065 | If cyclically shifting a w... |
| cshwshashlem1 17066 | If cyclically shifting a w... |
| cshwshashlem2 17067 | If cyclically shifting a w... |
| cshwshashlem3 17068 | If cyclically shifting a w... |
| cshwsdisj 17069 | The singletons resulting b... |
| cshwsiun 17070 | The set of (different!) wo... |
| cshwsex 17071 | The class of (different!) ... |
| cshws0 17072 | The size of the set of (di... |
| cshwrepswhash1 17073 | The size of the set of (di... |
| cshwshashnsame 17074 | If a word (not consisting ... |
| cshwshash 17075 | If a word has a length bei... |
| prmlem0 17076 | Lemma for ~ prmlem1 and ~ ... |
| prmlem1a 17077 | A quick proof skeleton to ... |
| prmlem1 17078 | A quick proof skeleton to ... |
| 5prm 17079 | 5 is a prime number. (Con... |
| 6nprm 17080 | 6 is not a prime number. ... |
| 7prm 17081 | 7 is a prime number. (Con... |
| 8nprm 17082 | 8 is not a prime number. ... |
| 9nprm 17083 | 9 is not a prime number. ... |
| 10nprm 17084 | 10 is not a prime number. ... |
| 11prm 17085 | 11 is a prime number. (Co... |
| 13prm 17086 | 13 is a prime number. (Co... |
| 17prm 17087 | 17 is a prime number. (Co... |
| 19prm 17088 | 19 is a prime number. (Co... |
| 23prm 17089 | 23 is a prime number. (Co... |
| prmlem2 17090 | Our last proving session g... |
| 37prm 17091 | 37 is a prime number. (Co... |
| 43prm 17092 | 43 is a prime number. (Co... |
| 83prm 17093 | 83 is a prime number. (Co... |
| 139prm 17094 | 139 is a prime number. (C... |
| 163prm 17095 | 163 is a prime number. (C... |
| 317prm 17096 | 317 is a prime number. (C... |
| 631prm 17097 | 631 is a prime number. (C... |
| prmo4 17098 | The primorial of 4. (Cont... |
| prmo5 17099 | The primorial of 5. (Cont... |
| prmo6 17100 | The primorial of 6. (Cont... |
| 1259lem1 17101 | Lemma for ~ 1259prm . Cal... |
| 1259lem2 17102 | Lemma for ~ 1259prm . Cal... |
| 1259lem3 17103 | Lemma for ~ 1259prm . Cal... |
| 1259lem4 17104 | Lemma for ~ 1259prm . Cal... |
| 1259lem5 17105 | Lemma for ~ 1259prm . Cal... |
| 1259prm 17106 | 1259 is a prime number. (... |
| 2503lem1 17107 | Lemma for ~ 2503prm . Cal... |
| 2503lem2 17108 | Lemma for ~ 2503prm . Cal... |
| 2503lem3 17109 | Lemma for ~ 2503prm . Cal... |
| 2503prm 17110 | 2503 is a prime number. (... |
| 4001lem1 17111 | Lemma for ~ 4001prm . Cal... |
| 4001lem2 17112 | Lemma for ~ 4001prm . Cal... |
| 4001lem3 17113 | Lemma for ~ 4001prm . Cal... |
| 4001lem4 17114 | Lemma for ~ 4001prm . Cal... |
| 4001prm 17115 | 4001 is a prime number. (... |
| brstruct 17118 | The structure relation is ... |
| isstruct2 17119 | The property of being a st... |
| structex 17120 | A structure is a set. (Co... |
| structn0fun 17121 | A structure without the em... |
| isstruct 17122 | The property of being a st... |
| structcnvcnv 17123 | Two ways to express the re... |
| structfung 17124 | The converse of the conver... |
| structfun 17125 | Convert between two kinds ... |
| structfn 17126 | Convert between two kinds ... |
| strleun 17127 | Combine two structures int... |
| strle1 17128 | Make a structure from a si... |
| strle2 17129 | Make a structure from a pa... |
| strle3 17130 | Make a structure from a tr... |
| sbcie2s 17131 | A special version of class... |
| sbcie3s 17132 | A special version of class... |
| reldmsets 17135 | The structure override ope... |
| setsvalg 17136 | Value of the structure rep... |
| setsval 17137 | Value of the structure rep... |
| fvsetsid 17138 | The value of the structure... |
| fsets 17139 | The structure replacement ... |
| setsdm 17140 | The domain of a structure ... |
| setsfun 17141 | A structure with replaceme... |
| setsfun0 17142 | A structure with replaceme... |
| setsn0fun 17143 | The value of the structure... |
| setsstruct2 17144 | An extensible structure wi... |
| setsexstruct2 17145 | An extensible structure wi... |
| setsstruct 17146 | An extensible structure wi... |
| wunsets 17147 | Closure of structure repla... |
| setsres 17148 | The structure replacement ... |
| setsabs 17149 | Replacing the same compone... |
| setscom 17150 | Different components can b... |
| sloteq 17153 | Equality theorem for the `... |
| slotfn 17154 | A slot is a function on se... |
| strfvnd 17155 | Deduction version of ~ str... |
| strfvn 17156 | Value of a structure compo... |
| strfvss 17157 | A structure component extr... |
| wunstr 17158 | Closure of a structure ind... |
| str0 17159 | All components of the empt... |
| strfvi 17160 | Structure slot extractors ... |
| fveqprc 17161 | Lemma for showing the equa... |
| oveqprc 17162 | Lemma for showing the equa... |
| wunndx 17165 | Closure of the index extra... |
| ndxarg 17166 | Get the numeric argument f... |
| ndxid 17167 | A structure component extr... |
| strndxid 17168 | The value of a structure c... |
| setsidvald 17169 | Value of the structure rep... |
| strfvd 17170 | Deduction version of ~ str... |
| strfv2d 17171 | Deduction version of ~ str... |
| strfv2 17172 | A variation on ~ strfv to ... |
| strfv 17173 | Extract a structure compon... |
| strfv3 17174 | Variant on ~ strfv for lar... |
| strssd 17175 | Deduction version of ~ str... |
| strss 17176 | Propagate component extrac... |
| setsid 17177 | Value of the structure rep... |
| setsnid 17178 | Value of the structure rep... |
| baseval 17181 | Value of the base set extr... |
| baseid 17182 | Utility theorem: index-ind... |
| basfn 17183 | The base set extractor is ... |
| base0 17184 | The base set of the empty ... |
| elbasfv 17185 | Utility theorem: reverse c... |
| elbasov 17186 | Utility theorem: reverse c... |
| strov2rcl 17187 | Partial reverse closure fo... |
| basendx 17188 | Index value of the base se... |
| basendxnn 17189 | The index value of the bas... |
| basndxelwund 17190 | The index of the base set ... |
| basprssdmsets 17191 | The pair of the base index... |
| opelstrbas 17192 | The base set of a structur... |
| 1strstr 17193 | A constructed one-slot str... |
| 1strbas 17194 | The base set of a construc... |
| 1strwunbndx 17195 | A constructed one-slot str... |
| 1strwun 17196 | A constructed one-slot str... |
| 2strstr 17197 | A constructed two-slot str... |
| 2strbas 17198 | The base set of a construc... |
| 2strop 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... |
| ressbasssg 17207 | The base set of a restrict... |
| ressbas2 17208 | Base set of a structure re... |
| ressbasss 17209 | The base set of a restrict... |
| ressbasssOLD 17210 | Obsolete version of ~ ress... |
| ressbasss2 17211 | The base set of a restrict... |
| resseqnbas 17212 | The components of an exten... |
| ress0 17213 | All restrictions of the nu... |
| ressid 17214 | Behavior of trivial restri... |
| ressinbas 17215 | Restriction only cares abo... |
| ressval3d 17216 | Value of structure restric... |
| ressress 17217 | Restriction composition la... |
| ressabs 17218 | Restriction absorption law... |
| wunress 17219 | Closure of structure restr... |
| plusgndx 17246 | Index value of the ~ df-pl... |
| plusgid 17247 | Utility theorem: index-ind... |
| plusgndxnn 17248 | The index of the slot for ... |
| basendxltplusgndx 17249 | The index of the slot for ... |
| basendxnplusgndx 17250 | The slot for the base set ... |
| grpstr 17251 | A constructed group is a s... |
| grpbase 17252 | The base set of a construc... |
| grpplusg 17253 | The operation of a constru... |
| ressplusg 17254 | ` +g ` is unaffected by re... |
| grpbasex 17255 | The base of an explicitly ... |
| grpplusgx 17256 | The operation of an explic... |
| mulrndx 17257 | Index value of the ~ df-mu... |
| mulridx 17258 | Utility theorem: index-ind... |
| basendxnmulrndx 17259 | The slot for the base set ... |
| plusgndxnmulrndx 17260 | The slot for the group (ad... |
| rngstr 17261 | A constructed ring is a st... |
| rngbase 17262 | The base set of a construc... |
| rngplusg 17263 | The additive operation of ... |
| rngmulr 17264 | The multiplicative operati... |
| starvndx 17265 | Index value of the ~ df-st... |
| starvid 17266 | Utility theorem: index-ind... |
| starvndxnbasendx 17267 | The slot for the involutio... |
| starvndxnplusgndx 17268 | The slot for the involutio... |
| starvndxnmulrndx 17269 | The slot for the involutio... |
| ressmulr 17270 | ` .r ` is unaffected by re... |
| ressstarv 17271 | ` *r ` is unaffected by re... |
| srngstr 17272 | A constructed star ring is... |
| srngbase 17273 | The base set of a construc... |
| srngplusg 17274 | The addition operation of ... |
| srngmulr 17275 | The multiplication operati... |
| srnginvl 17276 | The involution function of... |
| scandx 17277 | Index value of the ~ df-sc... |
| scaid 17278 | Utility theorem: index-ind... |
| scandxnbasendx 17279 | The slot for the scalar is... |
| scandxnplusgndx 17280 | The slot for the scalar fi... |
| scandxnmulrndx 17281 | The slot for the scalar fi... |
| vscandx 17282 | Index value of the ~ df-vs... |
| vscaid 17283 | Utility theorem: index-ind... |
| vscandxnbasendx 17284 | The slot for the scalar pr... |
| vscandxnplusgndx 17285 | The slot for the scalar pr... |
| vscandxnmulrndx 17286 | The slot for the scalar pr... |
| vscandxnscandx 17287 | The slot for the scalar pr... |
| lmodstr 17288 | A constructed left module ... |
| lmodbase 17289 | The base set of a construc... |
| lmodplusg 17290 | The additive operation of ... |
| lmodsca 17291 | The set of scalars of a co... |
| lmodvsca 17292 | The scalar product operati... |
| ipndx 17293 | Index value of the ~ df-ip... |
| ipid 17294 | Utility theorem: index-ind... |
| ipndxnbasendx 17295 | The slot for the inner pro... |
| ipndxnplusgndx 17296 | The slot for the inner pro... |
| ipndxnmulrndx 17297 | The slot for the inner pro... |
| slotsdifipndx 17298 | The slot for the scalar is... |
| ipsstr 17299 | Lemma to shorten proofs of... |
| ipsbase 17300 | The base set of a construc... |
| ipsaddg 17301 | The additive operation of ... |
| ipsmulr 17302 | The multiplicative operati... |
| ipssca 17303 | The set of scalars of a co... |
| ipsvsca 17304 | The scalar product operati... |
| ipsip 17305 | The multiplicative operati... |
| resssca 17306 | ` Scalar ` is unaffected b... |
| ressvsca 17307 | ` .s ` is unaffected by re... |
| ressip 17308 | The inner product is unaff... |
| phlstr 17309 | A constructed pre-Hilbert ... |
| phlbase 17310 | The base set of a construc... |
| phlplusg 17311 | The additive operation of ... |
| phlsca 17312 | The ring of scalars of a c... |
| phlvsca 17313 | The scalar product operati... |
| phlip 17314 | The inner product (Hermiti... |
| tsetndx 17315 | Index value of the ~ df-ts... |
| tsetid 17316 | Utility theorem: index-ind... |
| tsetndxnn 17317 | The index of the slot for ... |
| basendxlttsetndx 17318 | The index of the slot for ... |
| tsetndxnbasendx 17319 | The slot for the topology ... |
| tsetndxnplusgndx 17320 | The slot for the topology ... |
| tsetndxnmulrndx 17321 | The slot for the topology ... |
| tsetndxnstarvndx 17322 | The slot for the topology ... |
| slotstnscsi 17323 | The slots ` Scalar ` , ` .... |
| topgrpstr 17324 | A constructed topological ... |
| topgrpbas 17325 | The base set of a construc... |
| topgrpplusg 17326 | The additive operation of ... |
| topgrptset 17327 | The topology of a construc... |
| resstset 17328 | ` TopSet ` is unaffected b... |
| plendx 17329 | Index value of the ~ df-pl... |
| pleid 17330 | Utility theorem: self-refe... |
| plendxnn 17331 | The index value of the ord... |
| basendxltplendx 17332 | The index value of the ` B... |
| plendxnbasendx 17333 | The slot for the order is ... |
| plendxnplusgndx 17334 | The slot for the "less tha... |
| plendxnmulrndx 17335 | The slot for the "less tha... |
| plendxnscandx 17336 | The slot for the "less tha... |
| plendxnvscandx 17337 | The slot for the "less tha... |
| slotsdifplendx 17338 | The index of the slot for ... |
| otpsstr 17339 | Functionality of a topolog... |
| otpsbas 17340 | The base set of a topologi... |
| otpstset 17341 | The open sets of a topolog... |
| otpsle 17342 | The order of a topological... |
| ressle 17343 | ` le ` is unaffected by re... |
| ocndx 17344 | Index value of the ~ df-oc... |
| ocid 17345 | Utility theorem: index-ind... |
| basendxnocndx 17346 | The slot for the orthocomp... |
| plendxnocndx 17347 | The slot for the orthocomp... |
| dsndx 17348 | Index value of the ~ df-ds... |
| dsid 17349 | Utility theorem: index-ind... |
| dsndxnn 17350 | The index of the slot for ... |
| basendxltdsndx 17351 | The index of the slot for ... |
| dsndxnbasendx 17352 | The slot for the distance ... |
| dsndxnplusgndx 17353 | The slot for the distance ... |
| dsndxnmulrndx 17354 | The slot for the distance ... |
| slotsdnscsi 17355 | The slots ` Scalar ` , ` .... |
| dsndxntsetndx 17356 | The slot for the distance ... |
| slotsdifdsndx 17357 | The index of the slot for ... |
| unifndx 17358 | Index value of the ~ df-un... |
| unifid 17359 | Utility theorem: index-ind... |
| unifndxnn 17360 | The index of the slot for ... |
| basendxltunifndx 17361 | The index of the slot for ... |
| unifndxnbasendx 17362 | The slot for the uniform s... |
| unifndxntsetndx 17363 | The slot for the uniform s... |
| slotsdifunifndx 17364 | The index of the slot for ... |
| ressunif 17365 | ` UnifSet ` is unaffected ... |
| odrngstr 17366 | Functionality of an ordere... |
| odrngbas 17367 | The base set of an ordered... |
| odrngplusg 17368 | The addition operation of ... |
| odrngmulr 17369 | The multiplication operati... |
| odrngtset 17370 | The open sets of an ordere... |
| odrngle 17371 | The order of an ordered me... |
| odrngds 17372 | The metric of an ordered m... |
| ressds 17373 | ` dist ` is unaffected by ... |
| homndx 17374 | Index value of the ~ df-ho... |
| homid 17375 | Utility theorem: index-ind... |
| ccondx 17376 | Index value of the ~ df-cc... |
| ccoid 17377 | Utility theorem: index-ind... |
| slotsbhcdif 17378 | The slots ` Base ` , ` Hom... |
| slotsdifplendx2 17379 | The index of the slot for ... |
| slotsdifocndx 17380 | The index of the slot for ... |
| resshom 17381 | ` Hom ` is unaffected by r... |
| ressco 17382 | ` comp ` is unaffected by ... |
| restfn 17387 | The subspace topology oper... |
| topnfn 17388 | The topology extractor fun... |
| restval 17389 | The subspace topology indu... |
| elrest 17390 | The predicate "is an open ... |
| elrestr 17391 | Sufficient condition for b... |
| 0rest 17392 | Value of the structure res... |
| restid2 17393 | The subspace topology over... |
| restsspw 17394 | The subspace topology is a... |
| firest 17395 | The finite intersections o... |
| restid 17396 | The subspace topology of t... |
| topnval 17397 | Value of the topology extr... |
| topnid 17398 | Value of the topology extr... |
| topnpropd 17399 | The topology extractor fun... |
| reldmprds 17411 | The structure product is a... |
| prdsbasex 17413 | Lemma for structure produc... |
| imasvalstr 17414 | An image structure value i... |
| prdsvalstr 17415 | Structure product value is... |
| prdsbaslem 17416 | Lemma for ~ prdsbas and si... |
| prdsvallem 17417 | Lemma for ~ prdsval . (Co... |
| prdsval 17418 | Value of the structure pro... |
| prdssca 17419 | Scalar ring of a structure... |
| prdsbas 17420 | Base set of a structure pr... |
| prdsplusg 17421 | Addition in a structure pr... |
| prdsmulr 17422 | Multiplication in a struct... |
| prdsvsca 17423 | Scalar multiplication in a... |
| prdsip 17424 | Inner product in a structu... |
| prdsle 17425 | Structure product weak ord... |
| prdsless 17426 | Closure of the order relat... |
| prdsds 17427 | Structure product distance... |
| prdsdsfn 17428 | Structure product distance... |
| prdstset 17429 | Structure product topology... |
| prdshom 17430 | Structure product hom-sets... |
| prdsco 17431 | Structure product composit... |
| prdsbas2 17432 | The base set of a structur... |
| prdsbasmpt 17433 | A constructed tuple is a p... |
| prdsbasfn 17434 | Points in the structure pr... |
| prdsbasprj 17435 | Each point in a structure ... |
| prdsplusgval 17436 | Value of a componentwise s... |
| prdsplusgfval 17437 | Value of a structure produ... |
| prdsmulrval 17438 | Value of a componentwise r... |
| prdsmulrfval 17439 | Value of a structure produ... |
| prdsleval 17440 | Value of the product order... |
| prdsdsval 17441 | Value of the metric in a s... |
| prdsvscaval 17442 | Scalar multiplication in a... |
| prdsvscafval 17443 | Scalar multiplication of a... |
| prdsbas3 17444 | The base set of an indexed... |
| prdsbasmpt2 17445 | A constructed tuple is a p... |
| prdsbascl 17446 | An element of the base has... |
| prdsdsval2 17447 | Value of the metric in a s... |
| prdsdsval3 17448 | Value of the metric in a s... |
| pwsval 17449 | Value of a structure power... |
| pwsbas 17450 | Base set of a structure po... |
| pwselbasb 17451 | Membership in the base set... |
| pwselbas 17452 | An element of a structure ... |
| pwsplusgval 17453 | Value of addition in a str... |
| pwsmulrval 17454 | Value of multiplication in... |
| pwsle 17455 | Ordering in a structure po... |
| pwsleval 17456 | Ordering in a structure po... |
| pwsvscafval 17457 | Scalar multiplication in a... |
| pwsvscaval 17458 | Scalar multiplication of a... |
| pwssca 17459 | The ring of scalars of a s... |
| pwsdiagel 17460 | Membership of diagonal ele... |
| pwssnf1o 17461 | Triviality of singleton po... |
| imasval 17474 | Value of an image structur... |
| imasbas 17475 | The base set of an image s... |
| imasds 17476 | The distance function of a... |
| imasdsfn 17477 | The distance function is a... |
| imasdsval 17478 | The distance function of a... |
| imasdsval2 17479 | The distance function of a... |
| imasplusg 17480 | The group operation in an ... |
| imasmulr 17481 | The ring multiplication in... |
| imassca 17482 | The scalar field of an ima... |
| imasvsca 17483 | The scalar multiplication ... |
| imasip 17484 | The inner product of an im... |
| imastset 17485 | The topology of an image s... |
| imasle 17486 | The ordering of an image s... |
| f1ocpbllem 17487 | Lemma for ~ f1ocpbl . (Co... |
| f1ocpbl 17488 | An injection is compatible... |
| f1ovscpbl 17489 | An injection is compatible... |
| f1olecpbl 17490 | An injection is compatible... |
| imasaddfnlem 17491 | The image structure operat... |
| imasaddvallem 17492 | The operation of an image ... |
| imasaddflem 17493 | The image set operations a... |
| imasaddfn 17494 | The image structure's grou... |
| imasaddval 17495 | The value of an image stru... |
| imasaddf 17496 | The image structure's grou... |
| imasmulfn 17497 | The image structure's ring... |
| imasmulval 17498 | The value of an image stru... |
| imasmulf 17499 | The image structure's ring... |
| imasvscafn 17500 | The image structure's scal... |
| imasvscaval 17501 | The value of an image stru... |
| imasvscaf 17502 | The image structure's scal... |
| imasless 17503 | The order relation defined... |
| imasleval 17504 | The value of the image str... |
| qusval 17505 | Value of a quotient struct... |
| quslem 17506 | The function in ~ qusval i... |
| qusin 17507 | Restrict the equivalence r... |
| qusbas 17508 | Base set of a quotient str... |
| quss 17509 | The scalar field of a quot... |
| divsfval 17510 | Value of the function in ~... |
| ercpbllem 17511 | Lemma for ~ ercpbl . (Con... |
| ercpbl 17512 | Translate the function com... |
| erlecpbl 17513 | Translate the relation com... |
| qusaddvallem 17514 | Value of an operation defi... |
| qusaddflem 17515 | The operation of a quotien... |
| qusaddval 17516 | The addition in a quotient... |
| qusaddf 17517 | The addition in a quotient... |
| qusmulval 17518 | The multiplication in a qu... |
| qusmulf 17519 | The multiplication in a qu... |
| fnpr2o 17520 | Function with a domain of ... |
| fnpr2ob 17521 | Biconditional version of ~... |
| fvpr0o 17522 | The value of a function wi... |
| fvpr1o 17523 | The value of a function wi... |
| fvprif 17524 | The value of the pair func... |
| xpsfrnel 17525 | Elementhood in the target ... |
| xpsfeq 17526 | A function on ` 2o ` is de... |
| xpsfrnel2 17527 | Elementhood in the target ... |
| xpscf 17528 | Equivalent condition for t... |
| xpsfval 17529 | The value of the function ... |
| xpsff1o 17530 | The function appearing in ... |
| xpsfrn 17531 | A short expression for the... |
| xpsff1o2 17532 | The function appearing in ... |
| xpsval 17533 | Value of the binary struct... |
| xpsrnbas 17534 | The indexed structure prod... |
| xpsbas 17535 | The base set of the binary... |
| xpsaddlem 17536 | Lemma for ~ xpsadd and ~ x... |
| xpsadd 17537 | Value of the addition oper... |
| xpsmul 17538 | Value of the multiplicatio... |
| xpssca 17539 | Value of the scalar field ... |
| xpsvsca 17540 | Value of the scalar multip... |
| xpsless 17541 | Closure of the ordering in... |
| xpsle 17542 | Value of the ordering in a... |
| ismre 17551 | Property of being a Moore ... |
| fnmre 17552 | The Moore collection gener... |
| mresspw 17553 | A Moore collection is a su... |
| mress 17554 | A Moore-closed subset is a... |
| mre1cl 17555 | In any Moore collection th... |
| mreintcl 17556 | A nonempty collection of c... |
| mreiincl 17557 | A nonempty indexed interse... |
| mrerintcl 17558 | The relative intersection ... |
| mreriincl 17559 | The relative intersection ... |
| mreincl 17560 | Two closed sets have a clo... |
| mreuni 17561 | Since the entire base set ... |
| mreunirn 17562 | Two ways to express the no... |
| ismred 17563 | Properties that determine ... |
| ismred2 17564 | Properties that determine ... |
| mremre 17565 | The Moore collections of s... |
| submre 17566 | The subcollection of a clo... |
| mrcflem 17567 | The domain and codomain of... |
| fnmrc 17568 | Moore-closure is a well-be... |
| mrcfval 17569 | Value of the function expr... |
| mrcf 17570 | The Moore closure is a fun... |
| mrcval 17571 | Evaluation of the Moore cl... |
| mrccl 17572 | The Moore closure of a set... |
| mrcsncl 17573 | The Moore closure of a sin... |
| mrcid 17574 | The closure of a closed se... |
| mrcssv 17575 | The closure of a set is a ... |
| mrcidb 17576 | A set is closed iff it is ... |
| mrcss 17577 | Closure preserves subset o... |
| mrcssid 17578 | The closure of a set is a ... |
| mrcidb2 17579 | A set is closed iff it con... |
| mrcidm 17580 | The closure operation is i... |
| mrcsscl 17581 | The closure is the minimal... |
| mrcuni 17582 | Idempotence of closure und... |
| mrcun 17583 | Idempotence of closure und... |
| mrcssvd 17584 | The Moore closure of a set... |
| mrcssd 17585 | Moore closure preserves su... |
| mrcssidd 17586 | A set is contained in its ... |
| mrcidmd 17587 | Moore closure is idempoten... |
| mressmrcd 17588 | In a Moore system, if a se... |
| submrc 17589 | In a closure system which ... |
| mrieqvlemd 17590 | In a Moore system, if ` Y ... |
| mrisval 17591 | Value of the set of indepe... |
| ismri 17592 | Criterion for a set to be ... |
| ismri2 17593 | Criterion for a subset of ... |
| ismri2d 17594 | Criterion for a subset of ... |
| ismri2dd 17595 | Definition of independence... |
| mriss 17596 | An independent set of a Mo... |
| mrissd 17597 | An independent set of a Mo... |
| ismri2dad 17598 | Consequence of a set in a ... |
| mrieqvd 17599 | In a Moore system, a set i... |
| mrieqv2d 17600 | In a Moore system, a set i... |
| mrissmrcd 17601 | In a Moore system, if an i... |
| mrissmrid 17602 | In a Moore system, subsets... |
| mreexd 17603 | In a Moore system, the clo... |
| mreexmrid 17604 | In a Moore system whose cl... |
| mreexexlemd 17605 | This lemma is used to gene... |
| mreexexlem2d 17606 | Used in ~ mreexexlem4d to ... |
| mreexexlem3d 17607 | Base case of the induction... |
| mreexexlem4d 17608 | Induction step of the indu... |
| mreexexd 17609 | Exchange-type theorem. In... |
| mreexdomd 17610 | In a Moore system whose cl... |
| mreexfidimd 17611 | In a Moore system whose cl... |
| isacs 17612 | A set is an algebraic clos... |
| acsmre 17613 | Algebraic closure systems ... |
| isacs2 17614 | In the definition of an al... |
| acsfiel 17615 | A set is closed in an alge... |
| acsfiel2 17616 | A set is closed in an alge... |
| acsmred 17617 | An algebraic closure syste... |
| isacs1i 17618 | A closure system determine... |
| mreacs 17619 | Algebraicity is a composab... |
| acsfn 17620 | Algebraicity of a conditio... |
| acsfn0 17621 | Algebraicity of a point cl... |
| acsfn1 17622 | Algebraicity of a one-argu... |
| acsfn1c 17623 | Algebraicity of a one-argu... |
| acsfn2 17624 | Algebraicity of a two-argu... |
| iscat 17633 | The predicate "is a catego... |
| iscatd 17634 | Properties that determine ... |
| catidex 17635 | Each object in a category ... |
| catideu 17636 | Each object in a category ... |
| cidfval 17637 | Each object in a category ... |
| cidval 17638 | Each object in a category ... |
| cidffn 17639 | The identity arrow constru... |
| cidfn 17640 | The identity arrow operato... |
| catidd 17641 | Deduce the identity arrow ... |
| iscatd2 17642 | Version of ~ iscatd with a... |
| catidcl 17643 | Each object in a category ... |
| catlid 17644 | Left identity property of ... |
| catrid 17645 | Right identity property of... |
| catcocl 17646 | Closure of a composition a... |
| catass 17647 | Associativity of compositi... |
| catcone0 17648 | Composition of non-empty h... |
| 0catg 17649 | Any structure with an empt... |
| 0cat 17650 | The empty set is a categor... |
| homffval 17651 | Value of the functionalize... |
| fnhomeqhomf 17652 | If the Hom-set operation i... |
| homfval 17653 | Value of the functionalize... |
| homffn 17654 | The functionalized Hom-set... |
| homfeq 17655 | Condition for two categori... |
| homfeqd 17656 | If two structures have the... |
| homfeqbas 17657 | Deduce equality of base se... |
| homfeqval 17658 | Value of the functionalize... |
| comfffval 17659 | Value of the functionalize... |
| comffval 17660 | Value of the functionalize... |
| comfval 17661 | Value of the functionalize... |
| comfffval2 17662 | Value of the functionalize... |
| comffval2 17663 | Value of the functionalize... |
| comfval2 17664 | Value of the functionalize... |
| comfffn 17665 | The functionalized composi... |
| comffn 17666 | The functionalized composi... |
| comfeq 17667 | Condition for two categori... |
| comfeqd 17668 | Condition for two categori... |
| comfeqval 17669 | Equality of two compositio... |
| catpropd 17670 | Two structures with the sa... |
| cidpropd 17671 | Two structures with the sa... |
| oppcval 17674 | Value of the opposite cate... |
| oppchomfval 17675 | Hom-sets of the opposite c... |
| oppchom 17676 | Hom-sets of the opposite c... |
| oppccofval 17677 | Composition in the opposit... |
| oppcco 17678 | Composition in the opposit... |
| oppcbas 17679 | Base set of an opposite ca... |
| oppccatid 17680 | Lemma for ~ oppccat . (Co... |
| oppchomf 17681 | Hom-sets of the opposite c... |
| oppcid 17682 | Identity function of an op... |
| oppccat 17683 | An opposite category is a ... |
| 2oppcbas 17684 | The double opposite catego... |
| 2oppchomf 17685 | The double opposite catego... |
| 2oppccomf 17686 | The double opposite catego... |
| oppchomfpropd 17687 | If two categories have the... |
| oppccomfpropd 17688 | If two categories have the... |
| oppccatf 17689 | ` oppCat ` restricted to `... |
| monfval 17694 | Definition of a monomorphi... |
| ismon 17695 | Definition of a monomorphi... |
| ismon2 17696 | Write out the monomorphism... |
| monhom 17697 | A monomorphism is a morphi... |
| moni 17698 | Property of a monomorphism... |
| monpropd 17699 | If two categories have the... |
| oppcmon 17700 | A monomorphism in the oppo... |
| oppcepi 17701 | An epimorphism in the oppo... |
| isepi 17702 | Definition of an epimorphi... |
| isepi2 17703 | Write out the epimorphism ... |
| epihom 17704 | An epimorphism is a morphi... |
| epii 17705 | Property of an epimorphism... |
| sectffval 17712 | Value of the section opera... |
| sectfval 17713 | Value of the section relat... |
| sectss 17714 | The section relation is a ... |
| issect 17715 | The property " ` F ` is a ... |
| issect2 17716 | Property of being a sectio... |
| sectcan 17717 | If ` G ` is a section of `... |
| sectco 17718 | Composition of two section... |
| isofval 17719 | Function value of the func... |
| invffval 17720 | Value of the inverse relat... |
| invfval 17721 | Value of the inverse relat... |
| isinv 17722 | Value of the inverse relat... |
| invss 17723 | The inverse relation is a ... |
| invsym 17724 | The inverse relation is sy... |
| invsym2 17725 | The inverse relation is sy... |
| invfun 17726 | The inverse relation is a ... |
| isoval 17727 | The isomorphisms are the d... |
| inviso1 17728 | If ` G ` is an inverse to ... |
| inviso2 17729 | If ` G ` is an inverse to ... |
| invf 17730 | The inverse relation is a ... |
| invf1o 17731 | The inverse relation is a ... |
| invinv 17732 | The inverse of the inverse... |
| invco 17733 | The composition of two iso... |
| dfiso2 17734 | Alternate definition of an... |
| dfiso3 17735 | Alternate definition of an... |
| inveq 17736 | If there are two inverses ... |
| isofn 17737 | The function value of the ... |
| isohom 17738 | An isomorphism is a homomo... |
| isoco 17739 | The composition of two iso... |
| oppcsect 17740 | A section in the opposite ... |
| oppcsect2 17741 | A section in the opposite ... |
| oppcinv 17742 | An inverse in the opposite... |
| oppciso 17743 | An isomorphism in the oppo... |
| sectmon 17744 | If ` F ` is a section of `... |
| monsect 17745 | If ` F ` is a monomorphism... |
| sectepi 17746 | If ` F ` is a section of `... |
| episect 17747 | If ` F ` is an epimorphism... |
| sectid 17748 | The identity is a section ... |
| invid 17749 | The inverse of the identit... |
| idiso 17750 | The identity is an isomorp... |
| idinv 17751 | The inverse of the identit... |
| invisoinvl 17752 | The inverse of an isomorph... |
| invisoinvr 17753 | The inverse of an isomorph... |
| invcoisoid 17754 | The inverse of an isomorph... |
| isocoinvid 17755 | The inverse of an isomorph... |
| rcaninv 17756 | Right cancellation of an i... |
| cicfval 17759 | The set of isomorphic obje... |
| brcic 17760 | The relation "is isomorphi... |
| cic 17761 | Objects ` X ` and ` Y ` in... |
| brcici 17762 | Prove that two objects are... |
| cicref 17763 | Isomorphism is reflexive. ... |
| ciclcl 17764 | Isomorphism implies the le... |
| cicrcl 17765 | Isomorphism implies the ri... |
| cicsym 17766 | Isomorphism is symmetric. ... |
| cictr 17767 | Isomorphism is transitive.... |
| cicer 17768 | Isomorphism is an equivale... |
| sscrel 17775 | The subcategory subset rel... |
| brssc 17776 | The subcategory subset rel... |
| sscpwex 17777 | An analogue of ~ pwex for ... |
| subcrcl 17778 | Reverse closure for the su... |
| sscfn1 17779 | The subcategory subset rel... |
| sscfn2 17780 | The subcategory subset rel... |
| ssclem 17781 | Lemma for ~ ssc1 and simil... |
| isssc 17782 | Value of the subcategory s... |
| ssc1 17783 | Infer subset relation on o... |
| ssc2 17784 | Infer subset relation on m... |
| sscres 17785 | Any function restricted to... |
| sscid 17786 | The subcategory subset rel... |
| ssctr 17787 | The subcategory subset rel... |
| ssceq 17788 | The subcategory subset rel... |
| rescval 17789 | Value of the category rest... |
| rescval2 17790 | Value of the category rest... |
| rescbas 17791 | Base set of the category r... |
| reschom 17792 | Hom-sets of the category r... |
| reschomf 17793 | Hom-sets of the category r... |
| rescco 17794 | Composition in the categor... |
| rescabs 17795 | Restriction absorption law... |
| rescabs2 17796 | Restriction absorption law... |
| issubc 17797 | Elementhood in the set of ... |
| issubc2 17798 | Elementhood in the set of ... |
| 0ssc 17799 | For any category ` C ` , t... |
| 0subcat 17800 | For any category ` C ` , t... |
| catsubcat 17801 | For any category ` C ` , `... |
| subcssc 17802 | An element in the set of s... |
| subcfn 17803 | An element in the set of s... |
| subcss1 17804 | The objects of a subcatego... |
| subcss2 17805 | The morphisms of a subcate... |
| subcidcl 17806 | The identity of the origin... |
| subccocl 17807 | A subcategory is closed un... |
| subccatid 17808 | A subcategory is a categor... |
| subcid 17809 | The identity in a subcateg... |
| subccat 17810 | A subcategory is a categor... |
| issubc3 17811 | Alternate definition of a ... |
| fullsubc 17812 | The full subcategory gener... |
| fullresc 17813 | The category formed by str... |
| resscat 17814 | A category restricted to a... |
| subsubc 17815 | A subcategory of a subcate... |
| relfunc 17824 | The set of functors is a r... |
| funcrcl 17825 | Reverse closure for a func... |
| isfunc 17826 | Value of the set of functo... |
| isfuncd 17827 | Deduce that an operation i... |
| funcf1 17828 | The object part of a funct... |
| funcixp 17829 | The morphism part of a fun... |
| funcf2 17830 | The morphism part of a fun... |
| funcfn2 17831 | The morphism part of a fun... |
| funcid 17832 | A functor maps each identi... |
| funcco 17833 | A functor maps composition... |
| funcsect 17834 | The image of a section und... |
| funcinv 17835 | The image of an inverse un... |
| funciso 17836 | The image of an isomorphis... |
| funcoppc 17837 | A functor on categories yi... |
| idfuval 17838 | Value of the identity func... |
| idfu2nd 17839 | Value of the morphism part... |
| idfu2 17840 | Value of the morphism part... |
| idfu1st 17841 | Value of the object part o... |
| idfu1 17842 | Value of the object part o... |
| idfucl 17843 | The identity functor is a ... |
| cofuval 17844 | Value of the composition o... |
| cofu1st 17845 | Value of the object part o... |
| cofu1 17846 | Value of the object part o... |
| cofu2nd 17847 | Value of the morphism part... |
| cofu2 17848 | Value of the morphism part... |
| cofuval2 17849 | Value of the composition o... |
| cofucl 17850 | The composition of two fun... |
| cofuass 17851 | Functor composition is ass... |
| cofulid 17852 | The identity functor is a ... |
| cofurid 17853 | The identity functor is a ... |
| resfval 17854 | Value of the functor restr... |
| resfval2 17855 | Value of the functor restr... |
| resf1st 17856 | Value of the functor restr... |
| resf2nd 17857 | Value of the functor restr... |
| funcres 17858 | A functor restricted to a ... |
| funcres2b 17859 | Condition for a functor to... |
| funcres2 17860 | A functor into a restricte... |
| idfusubc0 17861 | The identity functor for a... |
| idfusubc 17862 | The identity functor for a... |
| wunfunc 17863 | A weak universe is closed ... |
| funcpropd 17864 | If two categories have the... |
| funcres2c 17865 | Condition for a functor to... |
| fullfunc 17870 | A full functor is a functo... |
| fthfunc 17871 | A faithful functor is a fu... |
| relfull 17872 | The set of full functors i... |
| relfth 17873 | The set of faithful functo... |
| isfull 17874 | Value of the set of full f... |
| isfull2 17875 | Equivalent condition for a... |
| fullfo 17876 | The morphism map of a full... |
| fulli 17877 | The morphism map of a full... |
| isfth 17878 | Value of the set of faithf... |
| isfth2 17879 | Equivalent condition for a... |
| isffth2 17880 | A fully faithful functor i... |
| fthf1 17881 | The morphism map of a fait... |
| fthi 17882 | The morphism map of a fait... |
| ffthf1o 17883 | The morphism map of a full... |
| fullpropd 17884 | If two categories have the... |
| fthpropd 17885 | If two categories have the... |
| fulloppc 17886 | The opposite functor of a ... |
| fthoppc 17887 | The opposite functor of a ... |
| ffthoppc 17888 | The opposite functor of a ... |
| fthsect 17889 | A faithful functor reflect... |
| fthinv 17890 | A faithful functor reflect... |
| fthmon 17891 | A faithful functor reflect... |
| fthepi 17892 | A faithful functor reflect... |
| ffthiso 17893 | A fully faithful functor r... |
| fthres2b 17894 | Condition for a faithful f... |
| fthres2c 17895 | Condition for a faithful f... |
| fthres2 17896 | A faithful functor into a ... |
| idffth 17897 | The identity functor is a ... |
| cofull 17898 | The composition of two ful... |
| cofth 17899 | The composition of two fai... |
| coffth 17900 | The composition of two ful... |
| rescfth 17901 | The inclusion functor from... |
| ressffth 17902 | The inclusion functor from... |
| fullres2c 17903 | Condition for a full funct... |
| ffthres2c 17904 | Condition for a fully fait... |
| inclfusubc 17905 | The "inclusion functor" fr... |
| fnfuc 17910 | The ` FuncCat ` operation ... |
| natfval 17911 | Value of the function givi... |
| isnat 17912 | Property of being a natura... |
| isnat2 17913 | Property of being a natura... |
| natffn 17914 | The natural transformation... |
| natrcl 17915 | Reverse closure for a natu... |
| nat1st2nd 17916 | Rewrite the natural transf... |
| natixp 17917 | A natural transformation i... |
| natcl 17918 | A component of a natural t... |
| natfn 17919 | A natural transformation i... |
| nati 17920 | Naturality property of a n... |
| wunnat 17921 | A weak universe is closed ... |
| catstr 17922 | A category structure is a ... |
| fucval 17923 | Value of the functor categ... |
| fuccofval 17924 | Value of the functor categ... |
| fucbas 17925 | The objects of the functor... |
| fuchom 17926 | The morphisms in the funct... |
| fucco 17927 | Value of the composition o... |
| fuccoval 17928 | Value of the functor categ... |
| fuccocl 17929 | The composition of two nat... |
| fucidcl 17930 | The identity natural trans... |
| fuclid 17931 | Left identity of natural t... |
| fucrid 17932 | Right identity of natural ... |
| fucass 17933 | Associativity of natural t... |
| fuccatid 17934 | The functor category is a ... |
| fuccat 17935 | The functor category is a ... |
| fucid 17936 | The identity morphism in t... |
| fucsect 17937 | Two natural transformation... |
| fucinv 17938 | Two natural transformation... |
| invfuc 17939 | If ` V ( x ) ` is an inver... |
| fuciso 17940 | A natural transformation i... |
| natpropd 17941 | If two categories have the... |
| fucpropd 17942 | If two categories have the... |
| initofn 17949 | ` InitO ` is a function on... |
| termofn 17950 | ` TermO ` is a function on... |
| zeroofn 17951 | ` ZeroO ` is a function on... |
| initorcl 17952 | Reverse closure for an ini... |
| termorcl 17953 | Reverse closure for a term... |
| zeroorcl 17954 | Reverse closure for a zero... |
| initoval 17955 | The value of the initial o... |
| termoval 17956 | The value of the terminal ... |
| zerooval 17957 | The value of the zero obje... |
| isinito 17958 | The predicate "is an initi... |
| istermo 17959 | The predicate "is a termin... |
| iszeroo 17960 | The predicate "is a zero o... |
| isinitoi 17961 | Implication of a class bei... |
| istermoi 17962 | Implication of a class bei... |
| initoid 17963 | For an initial object, the... |
| termoid 17964 | For a terminal object, the... |
| dfinito2 17965 | An initial object is a ter... |
| dftermo2 17966 | A terminal object is an in... |
| dfinito3 17967 | An alternate definition of... |
| dftermo3 17968 | An alternate definition of... |
| initoo 17969 | An initial object is an ob... |
| termoo 17970 | A terminal object is an ob... |
| iszeroi 17971 | Implication of a class bei... |
| 2initoinv 17972 | Morphisms between two init... |
| initoeu1 17973 | Initial objects are essent... |
| initoeu1w 17974 | Initial objects are essent... |
| initoeu2lem0 17975 | Lemma 0 for ~ initoeu2 . ... |
| initoeu2lem1 17976 | Lemma 1 for ~ initoeu2 . ... |
| initoeu2lem2 17977 | Lemma 2 for ~ initoeu2 . ... |
| initoeu2 17978 | Initial objects are essent... |
| 2termoinv 17979 | Morphisms between two term... |
| termoeu1 17980 | Terminal objects are essen... |
| termoeu1w 17981 | Terminal objects are essen... |
| homarcl 17990 | Reverse closure for an arr... |
| homafval 17991 | Value of the disjointified... |
| homaf 17992 | Functionality of the disjo... |
| homaval 17993 | Value of the disjointified... |
| elhoma 17994 | Value of the disjointified... |
| elhomai 17995 | Produce an arrow from a mo... |
| elhomai2 17996 | Produce an arrow from a mo... |
| homarcl2 17997 | Reverse closure for the do... |
| homarel 17998 | An arrow is an ordered pai... |
| homa1 17999 | The first component of an ... |
| homahom2 18000 | The second component of an... |
| homahom 18001 | The second component of an... |
| homadm 18002 | The domain of an arrow wit... |
| homacd 18003 | The codomain of an arrow w... |
| homadmcd 18004 | Decompose an arrow into do... |
| arwval 18005 | The set of arrows is the u... |
| arwrcl 18006 | The first component of an ... |
| arwhoma 18007 | An arrow is contained in t... |
| homarw 18008 | A hom-set is a subset of t... |
| arwdm 18009 | The domain of an arrow is ... |
| arwcd 18010 | The codomain of an arrow i... |
| dmaf 18011 | The domain function is a f... |
| cdaf 18012 | The codomain function is a... |
| arwhom 18013 | The second component of an... |
| arwdmcd 18014 | Decompose an arrow into do... |
| idafval 18019 | Value of the identity arro... |
| idaval 18020 | Value of the identity arro... |
| ida2 18021 | Morphism part of the ident... |
| idahom 18022 | Domain and codomain of the... |
| idadm 18023 | Domain of the identity arr... |
| idacd 18024 | Codomain of the identity a... |
| idaf 18025 | The identity arrow functio... |
| coafval 18026 | The value of the compositi... |
| eldmcoa 18027 | A pair ` <. G , F >. ` is ... |
| dmcoass 18028 | The domain of composition ... |
| homdmcoa 18029 | If ` F : X --> Y ` and ` G... |
| coaval 18030 | Value of composition for c... |
| coa2 18031 | The morphism part of arrow... |
| coahom 18032 | The composition of two com... |
| coapm 18033 | Composition of arrows is a... |
| arwlid 18034 | Left identity of a categor... |
| arwrid 18035 | Right identity of a catego... |
| arwass 18036 | Associativity of compositi... |
| setcval 18039 | Value of the category of s... |
| setcbas 18040 | Set of objects of the cate... |
| setchomfval 18041 | Set of arrows of the categ... |
| setchom 18042 | Set of arrows of the categ... |
| elsetchom 18043 | A morphism of sets is a fu... |
| setccofval 18044 | Composition in the categor... |
| setcco 18045 | Composition in the categor... |
| setccatid 18046 | Lemma for ~ setccat . (Co... |
| setccat 18047 | The category of sets is a ... |
| setcid 18048 | The identity arrow in the ... |
| setcmon 18049 | A monomorphism of sets is ... |
| setcepi 18050 | An epimorphism of sets is ... |
| setcsect 18051 | A section in the category ... |
| setcinv 18052 | An inverse in the category... |
| setciso 18053 | An isomorphism in the cate... |
| resssetc 18054 | The restriction of the cat... |
| funcsetcres2 18055 | A functor into a smaller c... |
| setc2obas 18056 | ` (/) ` and ` 1o ` are dis... |
| setc2ohom 18057 | ` ( SetCat `` 2o ) ` is a ... |
| cat1lem 18058 | The category of sets in a ... |
| cat1 18059 | The definition of category... |
| catcval 18062 | Value of the category of c... |
| catcbas 18063 | Set of objects of the cate... |
| catchomfval 18064 | Set of arrows of the categ... |
| catchom 18065 | Set of arrows of the categ... |
| catccofval 18066 | Composition in the categor... |
| catcco 18067 | Composition in the categor... |
| catccatid 18068 | Lemma for ~ catccat . (Co... |
| catcid 18069 | The identity arrow in the ... |
| catccat 18070 | The category of categories... |
| resscatc 18071 | The restriction of the cat... |
| catcisolem 18072 | Lemma for ~ catciso . (Co... |
| catciso 18073 | A functor is an isomorphis... |
| catcbascl 18074 | An element of the base set... |
| catcslotelcl 18075 | A slot entry of an element... |
| catcbaselcl 18076 | The base set of an element... |
| catchomcl 18077 | The Hom-set of an element ... |
| catcccocl 18078 | The composition operation ... |
| catcoppccl 18079 | The category of categories... |
| catcfuccl 18080 | The category of categories... |
| fncnvimaeqv 18081 | The inverse images of the ... |
| bascnvimaeqv 18082 | The inverse image of the u... |
| estrcval 18085 | Value of the category of e... |
| estrcbas 18086 | Set of objects of the cate... |
| estrchomfval 18087 | Set of morphisms ("arrows"... |
| estrchom 18088 | The morphisms between exte... |
| elestrchom 18089 | A morphism between extensi... |
| estrccofval 18090 | Composition in the categor... |
| estrcco 18091 | Composition in the categor... |
| estrcbasbas 18092 | An element of the base set... |
| estrccatid 18093 | Lemma for ~ estrccat . (C... |
| estrccat 18094 | The category of extensible... |
| estrcid 18095 | The identity arrow in the ... |
| estrchomfn 18096 | The Hom-set operation in t... |
| estrchomfeqhom 18097 | The functionalized Hom-set... |
| estrreslem1 18098 | Lemma 1 for ~ estrres . (... |
| estrreslem2 18099 | Lemma 2 for ~ estrres . (... |
| estrres 18100 | Any restriction of a categ... |
| funcestrcsetclem1 18101 | Lemma 1 for ~ funcestrcset... |
| funcestrcsetclem2 18102 | Lemma 2 for ~ funcestrcset... |
| funcestrcsetclem3 18103 | Lemma 3 for ~ funcestrcset... |
| funcestrcsetclem4 18104 | Lemma 4 for ~ funcestrcset... |
| funcestrcsetclem5 18105 | Lemma 5 for ~ funcestrcset... |
| funcestrcsetclem6 18106 | Lemma 6 for ~ funcestrcset... |
| funcestrcsetclem7 18107 | Lemma 7 for ~ funcestrcset... |
| funcestrcsetclem8 18108 | Lemma 8 for ~ funcestrcset... |
| funcestrcsetclem9 18109 | Lemma 9 for ~ funcestrcset... |
| funcestrcsetc 18110 | The "natural forgetful fun... |
| fthestrcsetc 18111 | The "natural forgetful fun... |
| fullestrcsetc 18112 | The "natural forgetful fun... |
| equivestrcsetc 18113 | The "natural forgetful fun... |
| setc1strwun 18114 | A constructed one-slot str... |
| funcsetcestrclem1 18115 | Lemma 1 for ~ funcsetcestr... |
| funcsetcestrclem2 18116 | Lemma 2 for ~ funcsetcestr... |
| funcsetcestrclem3 18117 | Lemma 3 for ~ funcsetcestr... |
| embedsetcestrclem 18118 | Lemma for ~ embedsetcestrc... |
| funcsetcestrclem4 18119 | Lemma 4 for ~ funcsetcestr... |
| funcsetcestrclem5 18120 | Lemma 5 for ~ funcsetcestr... |
| funcsetcestrclem6 18121 | Lemma 6 for ~ funcsetcestr... |
| funcsetcestrclem7 18122 | Lemma 7 for ~ funcsetcestr... |
| funcsetcestrclem8 18123 | Lemma 8 for ~ funcsetcestr... |
| funcsetcestrclem9 18124 | Lemma 9 for ~ funcsetcestr... |
| funcsetcestrc 18125 | The "embedding functor" fr... |
| fthsetcestrc 18126 | The "embedding functor" fr... |
| fullsetcestrc 18127 | The "embedding functor" fr... |
| embedsetcestrc 18128 | The "embedding functor" fr... |
| fnxpc 18137 | The binary product of cate... |
| xpcval 18138 | Value of the binary produc... |
| xpcbas 18139 | Set of objects of the bina... |
| xpchomfval 18140 | Set of morphisms of the bi... |
| xpchom 18141 | Set of morphisms of the bi... |
| relxpchom 18142 | A hom-set in the binary pr... |
| xpccofval 18143 | Value of composition in th... |
| xpcco 18144 | Value of composition in th... |
| xpcco1st 18145 | Value of composition in th... |
| xpcco2nd 18146 | Value of composition in th... |
| xpchom2 18147 | Value of the set of morphi... |
| xpcco2 18148 | Value of composition in th... |
| xpccatid 18149 | The product of two categor... |
| xpcid 18150 | The identity morphism in t... |
| xpccat 18151 | The product of two categor... |
| 1stfval 18152 | Value of the first project... |
| 1stf1 18153 | Value of the first project... |
| 1stf2 18154 | Value of the first project... |
| 2ndfval 18155 | Value of the first project... |
| 2ndf1 18156 | Value of the first project... |
| 2ndf2 18157 | Value of the first project... |
| 1stfcl 18158 | The first projection funct... |
| 2ndfcl 18159 | The second projection func... |
| prfval 18160 | Value of the pairing funct... |
| prf1 18161 | Value of the pairing funct... |
| prf2fval 18162 | Value of the pairing funct... |
| prf2 18163 | Value of the pairing funct... |
| prfcl 18164 | The pairing of functors ` ... |
| prf1st 18165 | Cancellation of pairing wi... |
| prf2nd 18166 | Cancellation of pairing wi... |
| 1st2ndprf 18167 | Break a functor into a pro... |
| catcxpccl 18168 | The category of categories... |
| xpcpropd 18169 | If two categories have the... |
| evlfval 18178 | Value of the evaluation fu... |
| evlf2 18179 | Value of the evaluation fu... |
| evlf2val 18180 | Value of the evaluation na... |
| evlf1 18181 | Value of the evaluation fu... |
| evlfcllem 18182 | Lemma for ~ evlfcl . (Con... |
| evlfcl 18183 | The evaluation functor is ... |
| curfval 18184 | Value of the curry functor... |
| curf1fval 18185 | Value of the object part o... |
| curf1 18186 | Value of the object part o... |
| curf11 18187 | Value of the double evalua... |
| curf12 18188 | The partially evaluated cu... |
| curf1cl 18189 | The partially evaluated cu... |
| curf2 18190 | Value of the curry functor... |
| curf2val 18191 | Value of a component of th... |
| curf2cl 18192 | The curry functor at a mor... |
| curfcl 18193 | The curry functor of a fun... |
| curfpropd 18194 | If two categories have the... |
| uncfval 18195 | Value of the uncurry funct... |
| uncfcl 18196 | The uncurry operation take... |
| uncf1 18197 | Value of the uncurry funct... |
| uncf2 18198 | Value of the uncurry funct... |
| curfuncf 18199 | Cancellation of curry with... |
| uncfcurf 18200 | Cancellation of uncurry wi... |
| diagval 18201 | Define the diagonal functo... |
| diagcl 18202 | The diagonal functor is a ... |
| diag1cl 18203 | The constant functor of ` ... |
| diag11 18204 | Value of the constant func... |
| diag12 18205 | Value of the constant func... |
| diag2 18206 | Value of the diagonal func... |
| diag2cl 18207 | The diagonal functor at a ... |
| curf2ndf 18208 | As shown in ~ diagval , th... |
| hofval 18213 | Value of the Hom functor, ... |
| hof1fval 18214 | The object part of the Hom... |
| hof1 18215 | The object part of the Hom... |
| hof2fval 18216 | The morphism part of the H... |
| hof2val 18217 | The morphism part of the H... |
| hof2 18218 | The morphism part of the H... |
| hofcllem 18219 | Lemma for ~ hofcl . (Cont... |
| hofcl 18220 | Closure of the Hom functor... |
| oppchofcl 18221 | Closure of the opposite Ho... |
| yonval 18222 | Value of the Yoneda embedd... |
| yoncl 18223 | The Yoneda embedding is a ... |
| yon1cl 18224 | The Yoneda embedding at an... |
| yon11 18225 | Value of the Yoneda embedd... |
| yon12 18226 | Value of the Yoneda embedd... |
| yon2 18227 | Value of the Yoneda embedd... |
| hofpropd 18228 | If two categories have the... |
| yonpropd 18229 | If two categories have the... |
| oppcyon 18230 | Value of the opposite Yone... |
| oyoncl 18231 | The opposite Yoneda embedd... |
| oyon1cl 18232 | The opposite Yoneda embedd... |
| yonedalem1 18233 | Lemma for ~ yoneda . (Con... |
| yonedalem21 18234 | Lemma for ~ yoneda . (Con... |
| yonedalem3a 18235 | Lemma for ~ yoneda . (Con... |
| yonedalem4a 18236 | Lemma for ~ yoneda . (Con... |
| yonedalem4b 18237 | Lemma for ~ yoneda . (Con... |
| yonedalem4c 18238 | Lemma for ~ yoneda . (Con... |
| yonedalem22 18239 | Lemma for ~ yoneda . (Con... |
| yonedalem3b 18240 | Lemma for ~ yoneda . (Con... |
| yonedalem3 18241 | Lemma for ~ yoneda . (Con... |
| yonedainv 18242 | The Yoneda Lemma with expl... |
| yonffthlem 18243 | Lemma for ~ yonffth . (Co... |
| yoneda 18244 | The Yoneda Lemma. There i... |
| yonffth 18245 | The Yoneda Lemma. The Yon... |
| yoniso 18246 | If the codomain is recover... |
| oduval 18249 | Value of an order dual str... |
| oduleval 18250 | Value of the less-equal re... |
| oduleg 18251 | Truth of the less-equal re... |
| odubas 18252 | Base set of an order dual ... |
| isprs 18257 | Property of being a preord... |
| prslem 18258 | Lemma for ~ prsref and ~ p... |
| prsref 18259 | "Less than or equal to" is... |
| prstr 18260 | "Less than or equal to" is... |
| oduprs 18261 | Being a proset is a self-d... |
| isdrs 18262 | Property of being a direct... |
| drsdir 18263 | Direction of a directed se... |
| drsprs 18264 | A directed set is a proset... |
| drsbn0 18265 | The base of a directed set... |
| drsdirfi 18266 | Any _finite_ number of ele... |
| isdrs2 18267 | Directed sets may be defin... |
| ispos 18275 | The predicate "is a poset"... |
| ispos2 18276 | A poset is an antisymmetri... |
| posprs 18277 | A poset is a proset. (Con... |
| posi 18278 | Lemma for poset properties... |
| posref 18279 | A poset ordering is reflex... |
| posasymb 18280 | A poset ordering is asymme... |
| postr 18281 | A poset ordering is transi... |
| 0pos 18282 | Technical lemma to simplif... |
| isposd 18283 | Properties that determine ... |
| isposi 18284 | Properties that determine ... |
| isposix 18285 | Properties that determine ... |
| pospropd 18286 | Posethood is determined on... |
| odupos 18287 | Being a poset is a self-du... |
| oduposb 18288 | Being a poset is a self-du... |
| pltfval 18290 | Value of the less-than rel... |
| pltval 18291 | Less-than relation. ( ~ d... |
| pltle 18292 | "Less than" implies "less ... |
| pltne 18293 | The "less than" relation i... |
| pltirr 18294 | The "less than" relation i... |
| pleval2i 18295 | One direction of ~ pleval2... |
| pleval2 18296 | "Less than or equal to" in... |
| pltnle 18297 | "Less than" implies not co... |
| pltval3 18298 | Alternate expression for t... |
| pltnlt 18299 | The less-than relation imp... |
| pltn2lp 18300 | The less-than relation has... |
| plttr 18301 | The less-than relation is ... |
| pltletr 18302 | Transitive law for chained... |
| plelttr 18303 | Transitive law for chained... |
| pospo 18304 | Write a poset structure in... |
| lubfval 18309 | Value of the least upper b... |
| lubdm 18310 | Domain of the least upper ... |
| lubfun 18311 | The LUB is a function. (C... |
| lubeldm 18312 | Member of the domain of th... |
| lubelss 18313 | A member of the domain of ... |
| lubeu 18314 | Unique existence proper of... |
| lubval 18315 | Value of the least upper b... |
| lubcl 18316 | The least upper bound func... |
| lubprop 18317 | Properties of greatest low... |
| luble 18318 | The greatest lower bound i... |
| lublecllem 18319 | Lemma for ~ lublecl and ~ ... |
| lublecl 18320 | The set of all elements le... |
| lubid 18321 | The LUB of elements less t... |
| glbfval 18322 | Value of the greatest lowe... |
| glbdm 18323 | Domain of the greatest low... |
| glbfun 18324 | The GLB is a function. (C... |
| glbeldm 18325 | Member of the domain of th... |
| glbelss 18326 | A member of the domain of ... |
| glbeu 18327 | Unique existence proper of... |
| glbval 18328 | Value of the greatest lowe... |
| glbcl 18329 | The least upper bound func... |
| glbprop 18330 | Properties of greatest low... |
| glble 18331 | The greatest lower bound i... |
| joinfval 18332 | Value of join function for... |
| joinfval2 18333 | Value of join function for... |
| joindm 18334 | Domain of join function fo... |
| joindef 18335 | Two ways to say that a joi... |
| joinval 18336 | Join value. Since both si... |
| joincl 18337 | Closure of join of element... |
| joindmss 18338 | Subset property of domain ... |
| joinval2lem 18339 | Lemma for ~ joinval2 and ~... |
| joinval2 18340 | Value of join for a poset ... |
| joineu 18341 | Uniqueness of join of elem... |
| joinlem 18342 | Lemma for join properties.... |
| lejoin1 18343 | A join's first argument is... |
| lejoin2 18344 | A join's second argument i... |
| joinle 18345 | A join is less than or equ... |
| meetfval 18346 | Value of meet function for... |
| meetfval2 18347 | Value of meet function for... |
| meetdm 18348 | Domain of meet function fo... |
| meetdef 18349 | Two ways to say that a mee... |
| meetval 18350 | Meet value. Since both si... |
| meetcl 18351 | Closure of meet of element... |
| meetdmss 18352 | Subset property of domain ... |
| meetval2lem 18353 | Lemma for ~ meetval2 and ~... |
| meetval2 18354 | Value of meet for a poset ... |
| meeteu 18355 | Uniqueness of meet of elem... |
| meetlem 18356 | Lemma for meet properties.... |
| lemeet1 18357 | A meet's first argument is... |
| lemeet2 18358 | A meet's second argument i... |
| meetle 18359 | A meet is less than or equ... |
| joincomALT 18360 | The join of a poset is com... |
| joincom 18361 | The join of a poset is com... |
| meetcomALT 18362 | The meet of a poset is com... |
| meetcom 18363 | The meet of a poset is com... |
| join0 18364 | Lemma for ~ odumeet . (Co... |
| meet0 18365 | Lemma for ~ odujoin . (Co... |
| odulub 18366 | Least upper bounds in a du... |
| odujoin 18367 | Joins in a dual order are ... |
| oduglb 18368 | Greatest lower bounds in a... |
| odumeet 18369 | Meets in a dual order are ... |
| poslubmo 18370 | Least upper bounds in a po... |
| posglbmo 18371 | Greatest lower bounds in a... |
| poslubd 18372 | Properties which determine... |
| poslubdg 18373 | Properties which determine... |
| posglbdg 18374 | Properties which determine... |
| istos 18377 | The predicate "is a toset"... |
| tosso 18378 | Write the totally ordered ... |
| tospos 18379 | A Toset is a Poset. (Cont... |
| tleile 18380 | In a Toset, any two elemen... |
| tltnle 18381 | In a Toset, "less than" is... |
| p0val 18386 | Value of poset zero. (Con... |
| p1val 18387 | Value of poset zero. (Con... |
| p0le 18388 | Any element is less than o... |
| ple1 18389 | Any element is less than o... |
| islat 18392 | The predicate "is a lattic... |
| odulatb 18393 | Being a lattice is self-du... |
| odulat 18394 | Being a lattice is self-du... |
| latcl2 18395 | The join and meet of any t... |
| latlem 18396 | Lemma for lattice properti... |
| latpos 18397 | A lattice is a poset. (Co... |
| latjcl 18398 | Closure of join operation ... |
| latmcl 18399 | Closure of meet operation ... |
| latref 18400 | A lattice ordering is refl... |
| latasymb 18401 | A lattice ordering is asym... |
| latasym 18402 | A lattice ordering is asym... |
| lattr 18403 | A lattice ordering is tran... |
| latasymd 18404 | Deduce equality from latti... |
| lattrd 18405 | A lattice ordering is tran... |
| latjcom 18406 | The join of a lattice comm... |
| latlej1 18407 | A join's first argument is... |
| latlej2 18408 | A join's second argument i... |
| latjle12 18409 | A join is less than or equ... |
| latleeqj1 18410 | "Less than or equal to" in... |
| latleeqj2 18411 | "Less than or equal to" in... |
| latjlej1 18412 | Add join to both sides of ... |
| latjlej2 18413 | Add join to both sides of ... |
| latjlej12 18414 | Add join to both sides of ... |
| latnlej 18415 | An idiom to express that a... |
| latnlej1l 18416 | An idiom to express that a... |
| latnlej1r 18417 | An idiom to express that a... |
| latnlej2 18418 | An idiom to express that a... |
| latnlej2l 18419 | An idiom to express that a... |
| latnlej2r 18420 | An idiom to express that a... |
| latjidm 18421 | Lattice join is idempotent... |
| latmcom 18422 | The join of a lattice comm... |
| latmle1 18423 | A meet is less than or equ... |
| latmle2 18424 | A meet is less than or equ... |
| latlem12 18425 | An element is less than or... |
| latleeqm1 18426 | "Less than or equal to" in... |
| latleeqm2 18427 | "Less than or equal to" in... |
| latmlem1 18428 | Add meet to both sides of ... |
| latmlem2 18429 | Add meet to both sides of ... |
| latmlem12 18430 | Add join to both sides of ... |
| latnlemlt 18431 | Negation of "less than or ... |
| latnle 18432 | Equivalent expressions for... |
| latmidm 18433 | Lattice meet is idempotent... |
| latabs1 18434 | Lattice absorption law. F... |
| latabs2 18435 | Lattice absorption law. F... |
| latledi 18436 | An ortholattice is distrib... |
| latmlej11 18437 | Ordering of a meet and joi... |
| latmlej12 18438 | Ordering of a meet and joi... |
| latmlej21 18439 | Ordering of a meet and joi... |
| latmlej22 18440 | Ordering of a meet and joi... |
| lubsn 18441 | The least upper bound of a... |
| latjass 18442 | Lattice join is associativ... |
| latj12 18443 | Swap 1st and 2nd members o... |
| latj32 18444 | Swap 2nd and 3rd members o... |
| latj13 18445 | Swap 1st and 3rd members o... |
| latj31 18446 | Swap 2nd and 3rd members o... |
| latjrot 18447 | Rotate lattice join of 3 c... |
| latj4 18448 | Rearrangement of lattice j... |
| latj4rot 18449 | Rotate lattice join of 4 c... |
| latjjdi 18450 | Lattice join distributes o... |
| latjjdir 18451 | Lattice join distributes o... |
| mod1ile 18452 | The weak direction of the ... |
| mod2ile 18453 | The weak direction of the ... |
| latmass 18454 | Lattice meet is associativ... |
| latdisdlem 18455 | Lemma for ~ latdisd . (Co... |
| latdisd 18456 | In a lattice, joins distri... |
| isclat 18459 | The predicate "is a comple... |
| clatpos 18460 | A complete lattice is a po... |
| clatlem 18461 | Lemma for properties of a ... |
| clatlubcl 18462 | Any subset of the base set... |
| clatlubcl2 18463 | Any subset of the base set... |
| clatglbcl 18464 | Any subset of the base set... |
| clatglbcl2 18465 | Any subset of the base set... |
| oduclatb 18466 | Being a complete lattice i... |
| clatl 18467 | A complete lattice is a la... |
| isglbd 18468 | Properties that determine ... |
| lublem 18469 | Lemma for the least upper ... |
| lubub 18470 | The LUB of a complete latt... |
| lubl 18471 | The LUB of a complete latt... |
| lubss 18472 | Subset law for least upper... |
| lubel 18473 | An element of a set is les... |
| lubun 18474 | The LUB of a union. (Cont... |
| clatglb 18475 | Properties of greatest low... |
| clatglble 18476 | The greatest lower bound i... |
| clatleglb 18477 | Two ways of expressing "le... |
| clatglbss 18478 | Subset law for greatest lo... |
| isdlat 18481 | Property of being a distri... |
| dlatmjdi 18482 | In a distributive lattice,... |
| dlatl 18483 | A distributive lattice is ... |
| odudlatb 18484 | The dual of a distributive... |
| dlatjmdi 18485 | In a distributive lattice,... |
| ipostr 18488 | The structure of ~ df-ipo ... |
| ipoval 18489 | Value of the inclusion pos... |
| ipobas 18490 | Base set of the inclusion ... |
| ipolerval 18491 | Relation of the inclusion ... |
| ipotset 18492 | Topology of the inclusion ... |
| ipole 18493 | Weak order condition of th... |
| ipolt 18494 | Strict order condition of ... |
| ipopos 18495 | The inclusion poset on a f... |
| isipodrs 18496 | Condition for a family of ... |
| ipodrscl 18497 | Direction by inclusion as ... |
| ipodrsfi 18498 | Finite upper bound propert... |
| fpwipodrs 18499 | The finite subsets of any ... |
| ipodrsima 18500 | The monotone image of a di... |
| isacs3lem 18501 | An algebraic closure syste... |
| acsdrsel 18502 | An algebraic closure syste... |
| isacs4lem 18503 | In a closure system in whi... |
| isacs5lem 18504 | If closure commutes with d... |
| acsdrscl 18505 | In an algebraic closure sy... |
| acsficl 18506 | A closure in an algebraic ... |
| isacs5 18507 | A closure system is algebr... |
| isacs4 18508 | A closure system is algebr... |
| isacs3 18509 | A closure system is algebr... |
| acsficld 18510 | In an algebraic closure sy... |
| acsficl2d 18511 | In an algebraic closure sy... |
| acsfiindd 18512 | In an algebraic closure sy... |
| acsmapd 18513 | In an algebraic closure sy... |
| acsmap2d 18514 | In an algebraic closure sy... |
| acsinfd 18515 | In an algebraic closure sy... |
| acsdomd 18516 | In an algebraic closure sy... |
| acsinfdimd 18517 | In an algebraic closure sy... |
| acsexdimd 18518 | In an algebraic closure sy... |
| mrelatglb 18519 | Greatest lower bounds in a... |
| mrelatglb0 18520 | The empty intersection in ... |
| mrelatlub 18521 | Least upper bounds in a Mo... |
| mreclatBAD 18522 | A Moore space is a complet... |
| isps 18527 | The predicate "is a poset"... |
| psrel 18528 | A poset is a relation. (C... |
| psref2 18529 | A poset is antisymmetric a... |
| pstr2 18530 | A poset is transitive. (C... |
| pslem 18531 | Lemma for ~ psref and othe... |
| psdmrn 18532 | The domain and range of a ... |
| psref 18533 | A poset is reflexive. (Co... |
| psrn 18534 | The range of a poset equal... |
| psasym 18535 | A poset is antisymmetric. ... |
| pstr 18536 | A poset is transitive. (C... |
| cnvps 18537 | The converse of a poset is... |
| cnvpsb 18538 | The converse of a poset is... |
| psss 18539 | Any subset of a partially ... |
| psssdm2 18540 | Field of a subposet. (Con... |
| psssdm 18541 | Field of a subposet. (Con... |
| istsr 18542 | The predicate is a toset. ... |
| istsr2 18543 | The predicate is a toset. ... |
| tsrlin 18544 | A toset is a linear order.... |
| tsrlemax 18545 | Two ways of saying a numbe... |
| tsrps 18546 | A toset is a poset. (Cont... |
| cnvtsr 18547 | The converse of a toset is... |
| tsrss 18548 | Any subset of a totally or... |
| ledm 18549 | The domain of ` <_ ` is ` ... |
| lern 18550 | The range of ` <_ ` is ` R... |
| lefld 18551 | The field of the 'less or ... |
| letsr 18552 | The "less than or equal to... |
| isdir 18557 | A condition for a relation... |
| reldir 18558 | A direction is a relation.... |
| dirdm 18559 | A direction's domain is eq... |
| dirref 18560 | A direction is reflexive. ... |
| dirtr 18561 | A direction is transitive.... |
| dirge 18562 | For any two elements of a ... |
| tsrdir 18563 | A totally ordered set is a... |
| ismgm 18568 | The predicate "is a magma"... |
| ismgmn0 18569 | The predicate "is a magma"... |
| mgmcl 18570 | Closure of the operation o... |
| isnmgm 18571 | A condition for a structur... |
| mgmsscl 18572 | If the base set of a magma... |
| plusffval 18573 | The group addition operati... |
| plusfval 18574 | The group addition operati... |
| plusfeq 18575 | If the addition operation ... |
| plusffn 18576 | The group addition operati... |
| mgmplusf 18577 | The group addition functio... |
| mgmpropd 18578 | If two structures have the... |
| ismgmd 18579 | Deduce a magma from its pr... |
| issstrmgm 18580 | Characterize a substructur... |
| intopsn 18581 | The internal operation for... |
| mgmb1mgm1 18582 | The only magma with a base... |
| mgm0 18583 | Any set with an empty base... |
| mgm0b 18584 | The structure with an empt... |
| mgm1 18585 | The structure with one ele... |
| opifismgm 18586 | A structure with a group a... |
| mgmidmo 18587 | A two-sided identity eleme... |
| grpidval 18588 | The value of the identity ... |
| grpidpropd 18589 | If two structures have the... |
| fn0g 18590 | The group zero extractor i... |
| 0g0 18591 | The identity element funct... |
| ismgmid 18592 | The identity element of a ... |
| mgmidcl 18593 | The identity element of a ... |
| mgmlrid 18594 | The identity element of a ... |
| ismgmid2 18595 | Show that a given element ... |
| lidrideqd 18596 | If there is a left and rig... |
| lidrididd 18597 | If there is a left and rig... |
| grpidd 18598 | Deduce the identity elemen... |
| mgmidsssn0 18599 | Property of the set of ide... |
| grpinvalem 18600 | Lemma for ~ grpinva . (Co... |
| grpinva 18601 | Deduce right inverse from ... |
| grprida 18602 | Deduce right identity from... |
| gsumvalx 18603 | Expand out the substitutio... |
| gsumval 18604 | Expand out the substitutio... |
| gsumpropd 18605 | The group sum depends only... |
| gsumpropd2lem 18606 | Lemma for ~ gsumpropd2 . ... |
| gsumpropd2 18607 | A stronger version of ~ gs... |
| gsummgmpropd 18608 | A stronger version of ~ gs... |
| gsumress 18609 | The group sum in a substru... |
| gsumval1 18610 | Value of the group sum ope... |
| gsum0 18611 | Value of the empty group s... |
| gsumval2a 18612 | Value of the group sum ope... |
| gsumval2 18613 | Value of the group sum ope... |
| gsumsplit1r 18614 | Splitting off the rightmos... |
| gsumprval 18615 | Value of the group sum ope... |
| gsumpr12val 18616 | Value of the group sum ope... |
| mgmhmrcl 18621 | Reverse closure of a magma... |
| submgmrcl 18622 | Reverse closure for submag... |
| ismgmhm 18623 | Property of a magma homomo... |
| mgmhmf 18624 | A magma homomorphism is a ... |
| mgmhmpropd 18625 | Magma homomorphism depends... |
| mgmhmlin 18626 | A magma homomorphism prese... |
| mgmhmf1o 18627 | A magma homomorphism is bi... |
| idmgmhm 18628 | The identity homomorphism ... |
| issubmgm 18629 | Expand definition of a sub... |
| issubmgm2 18630 | Submagmas are subsets that... |
| rabsubmgmd 18631 | Deduction for proving that... |
| submgmss 18632 | Submagmas are subsets of t... |
| submgmid 18633 | Every magma is trivially a... |
| submgmcl 18634 | Submagmas are closed under... |
| submgmmgm 18635 | Submagmas are themselves m... |
| submgmbas 18636 | The base set of a submagma... |
| subsubmgm 18637 | A submagma of a submagma i... |
| resmgmhm 18638 | Restriction of a magma hom... |
| resmgmhm2 18639 | One direction of ~ resmgmh... |
| resmgmhm2b 18640 | Restriction of the codomai... |
| mgmhmco 18641 | The composition of magma h... |
| mgmhmima 18642 | The homomorphic image of a... |
| mgmhmeql 18643 | The equalizer of two magma... |
| submgmacs 18644 | Submagmas are an algebraic... |
| issgrp 18647 | The predicate "is a semigr... |
| issgrpv 18648 | The predicate "is a semigr... |
| issgrpn0 18649 | The predicate "is a semigr... |
| isnsgrp 18650 | A condition for a structur... |
| sgrpmgm 18651 | A semigroup is a magma. (... |
| sgrpass 18652 | A semigroup operation is a... |
| sgrpcl 18653 | Closure of the operation o... |
| sgrp0 18654 | Any set with an empty base... |
| sgrp0b 18655 | The structure with an empt... |
| sgrp1 18656 | The structure with one ele... |
| issgrpd 18657 | Deduce a semigroup from it... |
| sgrppropd 18658 | If two structures are sets... |
| prdsplusgsgrpcl 18659 | Structure product pointwis... |
| prdssgrpd 18660 | The product of a family of... |
| ismnddef 18663 | The predicate "is a monoid... |
| ismnd 18664 | The predicate "is a monoid... |
| isnmnd 18665 | A condition for a structur... |
| sgrpidmnd 18666 | A semigroup with an identi... |
| mndsgrp 18667 | A monoid is a semigroup. ... |
| mndmgm 18668 | A monoid is a magma. (Con... |
| mndcl 18669 | Closure of the operation o... |
| mndass 18670 | A monoid operation is asso... |
| mndid 18671 | A monoid has a two-sided i... |
| mndideu 18672 | The two-sided identity ele... |
| mnd32g 18673 | Commutative/associative la... |
| mnd12g 18674 | Commutative/associative la... |
| mnd4g 18675 | Commutative/associative la... |
| mndidcl 18676 | The identity element of a ... |
| mndbn0 18677 | The base set of a monoid i... |
| hashfinmndnn 18678 | A finite monoid has positi... |
| mndplusf 18679 | The group addition operati... |
| mndlrid 18680 | A monoid's identity elemen... |
| mndlid 18681 | The identity element of a ... |
| mndrid 18682 | The identity element of a ... |
| ismndd 18683 | Deduce a monoid from its p... |
| mndpfo 18684 | The addition operation of ... |
| mndfo 18685 | The addition operation of ... |
| mndpropd 18686 | If two structures have the... |
| mndprop 18687 | If two structures have the... |
| issubmnd 18688 | Characterize a submonoid b... |
| ress0g 18689 | ` 0g ` is unaffected by re... |
| submnd0 18690 | The zero of a submonoid is... |
| mndinvmod 18691 | Uniqueness of an inverse e... |
| mndpsuppss 18692 | The support of a mapping o... |
| mndpsuppfi 18693 | The support of a mapping o... |
| mndpfsupp 18694 | A mapping of a scalar mult... |
| prdsplusgcl 18695 | Structure product pointwis... |
| prdsidlem 18696 | Characterization of identi... |
| prdsmndd 18697 | The product of a family of... |
| prds0g 18698 | The identity in a product ... |
| pwsmnd 18699 | The structure power of a m... |
| pws0g 18700 | The identity in a structur... |
| imasmnd2 18701 | The image structure of a m... |
| imasmnd 18702 | The image structure of a m... |
| imasmndf1 18703 | The image of a monoid unde... |
| xpsmnd 18704 | The binary product of mono... |
| xpsmnd0 18705 | The identity element of a ... |
| mnd1 18706 | The (smallest) structure r... |
| mnd1id 18707 | The singleton element of a... |
| ismhm 18712 | Property of a monoid homom... |
| ismhmd 18713 | Deduction version of ~ ism... |
| mhmrcl1 18714 | Reverse closure of a monoi... |
| mhmrcl2 18715 | Reverse closure of a monoi... |
| mhmf 18716 | A monoid homomorphism is a... |
| ismhm0 18717 | Property of a monoid homom... |
| mhmismgmhm 18718 | Each monoid homomorphism i... |
| mhmpropd 18719 | Monoid homomorphism depend... |
| mhmlin 18720 | A monoid homomorphism comm... |
| mhm0 18721 | A monoid homomorphism pres... |
| idmhm 18722 | The identity homomorphism ... |
| mhmf1o 18723 | A monoid homomorphism is b... |
| mndvcl 18724 | Tuple-wise additive closur... |
| mndvass 18725 | Tuple-wise associativity i... |
| mndvlid 18726 | Tuple-wise left identity i... |
| mndvrid 18727 | Tuple-wise right identity ... |
| mhmvlin 18728 | Tuple extension of monoid ... |
| submrcl 18729 | Reverse closure for submon... |
| issubm 18730 | Expand definition of a sub... |
| issubm2 18731 | Submonoids are subsets tha... |
| issubmndb 18732 | The submonoid predicate. ... |
| issubmd 18733 | Deduction for proving a su... |
| mndissubm 18734 | If the base set of a monoi... |
| resmndismnd 18735 | If the base set of a monoi... |
| submss 18736 | Submonoids are subsets of ... |
| submid 18737 | Every monoid is trivially ... |
| subm0cl 18738 | Submonoids contain zero. ... |
| submcl 18739 | Submonoids are closed unde... |
| submmnd 18740 | Submonoids are themselves ... |
| submbas 18741 | The base set of a submonoi... |
| subm0 18742 | Submonoids have the same i... |
| subsubm 18743 | A submonoid of a submonoid... |
| 0subm 18744 | The zero submonoid of an a... |
| insubm 18745 | The intersection of two su... |
| 0mhm 18746 | The constant zero linear f... |
| resmhm 18747 | Restriction of a monoid ho... |
| resmhm2 18748 | One direction of ~ resmhm2... |
| resmhm2b 18749 | Restriction of the codomai... |
| mhmco 18750 | The composition of monoid ... |
| mhmimalem 18751 | Lemma for ~ mhmima and sim... |
| mhmima 18752 | The homomorphic image of a... |
| mhmeql 18753 | The equalizer of two monoi... |
| submacs 18754 | Submonoids are an algebrai... |
| mndind 18755 | Induction in a monoid. In... |
| prdspjmhm 18756 | A projection from a produc... |
| pwspjmhm 18757 | A projection from a struct... |
| pwsdiagmhm 18758 | Diagonal monoid homomorphi... |
| pwsco1mhm 18759 | Right composition with a f... |
| pwsco2mhm 18760 | Left composition with a mo... |
| gsumvallem2 18761 | Lemma for properties of th... |
| gsumsubm 18762 | Evaluate a group sum in a ... |
| gsumz 18763 | Value of a group sum over ... |
| gsumwsubmcl 18764 | Closure of the composite i... |
| gsumws1 18765 | A singleton composite reco... |
| gsumwcl 18766 | Closure of the composite o... |
| gsumsgrpccat 18767 | Homomorphic property of no... |
| gsumccat 18768 | Homomorphic property of co... |
| gsumws2 18769 | Valuation of a pair in a m... |
| gsumccatsn 18770 | Homomorphic property of co... |
| gsumspl 18771 | The primary purpose of the... |
| gsumwmhm 18772 | Behavior of homomorphisms ... |
| gsumwspan 18773 | The submonoid generated by... |
| frmdval 18778 | Value of the free monoid c... |
| frmdbas 18779 | The base set of a free mon... |
| frmdelbas 18780 | An element of the base set... |
| frmdplusg 18781 | The monoid operation of a ... |
| frmdadd 18782 | Value of the monoid operat... |
| vrmdfval 18783 | The canonical injection fr... |
| vrmdval 18784 | The value of the generatin... |
| vrmdf 18785 | The mapping from the index... |
| frmdmnd 18786 | A free monoid is a monoid.... |
| frmd0 18787 | The identity of the free m... |
| frmdsssubm 18788 | The set of words taking va... |
| frmdgsum 18789 | Any word in a free monoid ... |
| frmdss2 18790 | A subset of generators is ... |
| frmdup1 18791 | Any assignment of the gene... |
| frmdup2 18792 | The evaluation map has the... |
| frmdup3lem 18793 | Lemma for ~ frmdup3 . (Co... |
| frmdup3 18794 | Universal property of the ... |
| efmnd 18797 | The monoid of endofunction... |
| efmndbas 18798 | The base set of the monoid... |
| efmndbasabf 18799 | The base set of the monoid... |
| elefmndbas 18800 | Two ways of saying a funct... |
| elefmndbas2 18801 | Two ways of saying a funct... |
| efmndbasf 18802 | Elements in the monoid of ... |
| efmndhash 18803 | The monoid of endofunction... |
| efmndbasfi 18804 | The monoid of endofunction... |
| efmndfv 18805 | The function value of an e... |
| efmndtset 18806 | The topology of the monoid... |
| efmndplusg 18807 | The group operation of a m... |
| efmndov 18808 | The value of the group ope... |
| efmndcl 18809 | The group operation of the... |
| efmndtopn 18810 | The topology of the monoid... |
| symggrplem 18811 | Lemma for ~ symggrp and ~ ... |
| efmndmgm 18812 | The monoid of endofunction... |
| efmndsgrp 18813 | The monoid of endofunction... |
| ielefmnd 18814 | The identity function rest... |
| efmndid 18815 | The identity function rest... |
| efmndmnd 18816 | The monoid of endofunction... |
| efmnd0nmnd 18817 | Even the monoid of endofun... |
| efmndbas0 18818 | The base set of the monoid... |
| efmnd1hash 18819 | The monoid of endofunction... |
| efmnd1bas 18820 | The monoid of endofunction... |
| efmnd2hash 18821 | The monoid of endofunction... |
| submefmnd 18822 | If the base set of a monoi... |
| sursubmefmnd 18823 | The set of surjective endo... |
| injsubmefmnd 18824 | The set of injective endof... |
| idressubmefmnd 18825 | The singleton containing o... |
| idresefmnd 18826 | The structure with the sin... |
| smndex1ibas 18827 | The modulo function ` I ` ... |
| smndex1iidm 18828 | The modulo function ` I ` ... |
| smndex1gbas 18829 | The constant functions ` (... |
| smndex1gid 18830 | The composition of a const... |
| smndex1igid 18831 | The composition of the mod... |
| smndex1basss 18832 | The modulo function ` I ` ... |
| smndex1bas 18833 | The base set of the monoid... |
| smndex1mgm 18834 | The monoid of endofunction... |
| smndex1sgrp 18835 | The monoid of endofunction... |
| smndex1mndlem 18836 | Lemma for ~ smndex1mnd and... |
| smndex1mnd 18837 | The monoid of endofunction... |
| smndex1id 18838 | The modulo function ` I ` ... |
| smndex1n0mnd 18839 | The identity of the monoid... |
| nsmndex1 18840 | The base set ` B ` of the ... |
| smndex2dbas 18841 | The doubling function ` D ... |
| smndex2dnrinv 18842 | The doubling function ` D ... |
| smndex2hbas 18843 | The halving functions ` H ... |
| smndex2dlinvh 18844 | The halving functions ` H ... |
| mgm2nsgrplem1 18845 | Lemma 1 for ~ mgm2nsgrp : ... |
| mgm2nsgrplem2 18846 | Lemma 2 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem3 18847 | Lemma 3 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem4 18848 | Lemma 4 for ~ mgm2nsgrp : ... |
| mgm2nsgrp 18849 | A small magma (with two el... |
| sgrp2nmndlem1 18850 | Lemma 1 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem2 18851 | Lemma 2 for ~ sgrp2nmnd . ... |
| sgrp2nmndlem3 18852 | Lemma 3 for ~ sgrp2nmnd . ... |
| sgrp2rid2 18853 | A small semigroup (with tw... |
| sgrp2rid2ex 18854 | A small semigroup (with tw... |
| sgrp2nmndlem4 18855 | Lemma 4 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem5 18856 | Lemma 5 for ~ sgrp2nmnd : ... |
| sgrp2nmnd 18857 | A small semigroup (with tw... |
| mgmnsgrpex 18858 | There is a magma which is ... |
| sgrpnmndex 18859 | There is a semigroup which... |
| sgrpssmgm 18860 | The class of all semigroup... |
| mndsssgrp 18861 | The class of all monoids i... |
| pwmndgplus 18862 | The operation of the monoi... |
| pwmndid 18863 | The identity of the monoid... |
| pwmnd 18864 | The power set of a class `... |
| isgrp 18871 | The predicate "is a group"... |
| grpmnd 18872 | A group is a monoid. (Con... |
| grpcl 18873 | Closure of the operation o... |
| grpass 18874 | A group operation is assoc... |
| grpinvex 18875 | Every member of a group ha... |
| grpideu 18876 | The two-sided identity ele... |
| grpassd 18877 | A group operation is assoc... |
| grpmndd 18878 | A group is a monoid. (Con... |
| grpcld 18879 | Closure of the operation o... |
| grpplusf 18880 | The group addition operati... |
| grpplusfo 18881 | The group addition operati... |
| resgrpplusfrn 18882 | The underlying set of a gr... |
| grppropd 18883 | If two structures have the... |
| grpprop 18884 | If two structures have the... |
| grppropstr 18885 | Generalize a specific 2-el... |
| grpss 18886 | Show that a structure exte... |
| isgrpd2e 18887 | Deduce a group from its pr... |
| isgrpd2 18888 | Deduce a group from its pr... |
| isgrpde 18889 | Deduce a group from its pr... |
| isgrpd 18890 | Deduce a group from its pr... |
| isgrpi 18891 | Properties that determine ... |
| grpsgrp 18892 | A group is a semigroup. (... |
| grpmgmd 18893 | A group is a magma, deduct... |
| dfgrp2 18894 | Alternate definition of a ... |
| dfgrp2e 18895 | Alternate definition of a ... |
| isgrpix 18896 | Properties that determine ... |
| grpidcl 18897 | The identity element of a ... |
| grpbn0 18898 | The base set of a group is... |
| grplid 18899 | The identity element of a ... |
| grprid 18900 | The identity element of a ... |
| grplidd 18901 | The identity element of a ... |
| grpridd 18902 | The identity element of a ... |
| grpn0 18903 | A group is not empty. (Co... |
| hashfingrpnn 18904 | A finite group has positiv... |
| grprcan 18905 | Right cancellation law for... |
| grpinveu 18906 | The left inverse element o... |
| grpid 18907 | Two ways of saying that an... |
| isgrpid2 18908 | Properties showing that an... |
| grpidd2 18909 | Deduce the identity elemen... |
| grpinvfval 18910 | The inverse function of a ... |
| grpinvfvalALT 18911 | Shorter proof of ~ grpinvf... |
| grpinvval 18912 | The inverse of a group ele... |
| grpinvfn 18913 | Functionality of the group... |
| grpinvfvi 18914 | The group inverse function... |
| grpsubfval 18915 | Group subtraction (divisio... |
| grpsubfvalALT 18916 | Shorter proof of ~ grpsubf... |
| grpsubval 18917 | Group subtraction (divisio... |
| grpinvf 18918 | The group inversion operat... |
| grpinvcl 18919 | A group element's inverse ... |
| grpinvcld 18920 | A group element's inverse ... |
| grplinv 18921 | The left inverse of a grou... |
| grprinv 18922 | The right inverse of a gro... |
| grpinvid1 18923 | The inverse of a group ele... |
| grpinvid2 18924 | The inverse of a group ele... |
| isgrpinv 18925 | Properties showing that a ... |
| grplinvd 18926 | The left inverse of a grou... |
| grprinvd 18927 | The right inverse of a gro... |
| grplrinv 18928 | In a group, every member h... |
| grpidinv2 18929 | A group's properties using... |
| grpidinv 18930 | A group has a left and rig... |
| grpinvid 18931 | The inverse of the identit... |
| grplcan 18932 | Left cancellation law for ... |
| grpasscan1 18933 | An associative cancellatio... |
| grpasscan2 18934 | An associative cancellatio... |
| grpidrcan 18935 | If right adding an element... |
| grpidlcan 18936 | If left adding an element ... |
| grpinvinv 18937 | Double inverse law for gro... |
| grpinvcnv 18938 | The group inverse is its o... |
| grpinv11 18939 | The group inverse is one-t... |
| grpinv11OLD 18940 | Obsolete version of ~ grpi... |
| grpinvf1o 18941 | The group inverse is a one... |
| grpinvnz 18942 | The inverse of a nonzero g... |
| grpinvnzcl 18943 | The inverse of a nonzero g... |
| grpsubinv 18944 | Subtraction of an inverse.... |
| grplmulf1o 18945 | Left multiplication by a g... |
| grpraddf1o 18946 | Right addition by a group ... |
| grpinvpropd 18947 | If two structures have the... |
| grpidssd 18948 | If the base set of a group... |
| grpinvssd 18949 | If the base set of a group... |
| grpinvadd 18950 | The inverse of the group o... |
| grpsubf 18951 | Functionality of group sub... |
| grpsubcl 18952 | Closure of group subtracti... |
| grpsubrcan 18953 | Right cancellation law for... |
| grpinvsub 18954 | Inverse of a group subtrac... |
| grpinvval2 18955 | A ~ df-neg -like equation ... |
| grpsubid 18956 | Subtraction of a group ele... |
| grpsubid1 18957 | Subtraction of the identit... |
| grpsubeq0 18958 | If the difference between ... |
| grpsubadd0sub 18959 | Subtraction expressed as a... |
| grpsubadd 18960 | Relationship between group... |
| grpsubsub 18961 | Double group subtraction. ... |
| grpaddsubass 18962 | Associative-type law for g... |
| grppncan 18963 | Cancellation law for subtr... |
| grpnpcan 18964 | Cancellation law for subtr... |
| grpsubsub4 18965 | Double group subtraction (... |
| grppnpcan2 18966 | Cancellation law for mixed... |
| grpnpncan 18967 | Cancellation law for group... |
| grpnpncan0 18968 | Cancellation law for group... |
| grpnnncan2 18969 | Cancellation law for group... |
| dfgrp3lem 18970 | Lemma for ~ dfgrp3 . (Con... |
| dfgrp3 18971 | Alternate definition of a ... |
| dfgrp3e 18972 | Alternate definition of a ... |
| grplactfval 18973 | The left group action of e... |
| grplactval 18974 | The value of the left grou... |
| grplactcnv 18975 | The left group action of e... |
| grplactf1o 18976 | The left group action of e... |
| grpsubpropd 18977 | Weak property deduction fo... |
| grpsubpropd2 18978 | Strong property deduction ... |
| grp1 18979 | The (smallest) structure r... |
| grp1inv 18980 | The inverse function of th... |
| prdsinvlem 18981 | Characterization of invers... |
| prdsgrpd 18982 | The product of a family of... |
| prdsinvgd 18983 | Negation in a product of g... |
| pwsgrp 18984 | A structure power of a gro... |
| pwsinvg 18985 | Negation in a group power.... |
| pwssub 18986 | Subtraction in a group pow... |
| imasgrp2 18987 | The image structure of a g... |
| imasgrp 18988 | The image structure of a g... |
| imasgrpf1 18989 | The image of a group under... |
| qusgrp2 18990 | Prove that a quotient stru... |
| xpsgrp 18991 | The binary product of grou... |
| xpsinv 18992 | Value of the negation oper... |
| xpsgrpsub 18993 | Value of the subtraction o... |
| mhmlem 18994 | Lemma for ~ mhmmnd and ~ g... |
| mhmid 18995 | A surjective monoid morphi... |
| mhmmnd 18996 | The image of a monoid ` G ... |
| mhmfmhm 18997 | The function fulfilling th... |
| ghmgrp 18998 | The image of a group ` G `... |
| mulgfval 19001 | Group multiple (exponentia... |
| mulgfvalALT 19002 | Shorter proof of ~ mulgfva... |
| mulgval 19003 | Value of the group multipl... |
| mulgfn 19004 | Functionality of the group... |
| mulgfvi 19005 | The group multiple operati... |
| mulg0 19006 | Group multiple (exponentia... |
| mulgnn 19007 | Group multiple (exponentia... |
| ressmulgnn 19008 | Values for the group multi... |
| ressmulgnn0 19009 | Values for the group multi... |
| ressmulgnnd 19010 | Values for the group multi... |
| mulgnngsum 19011 | Group multiple (exponentia... |
| mulgnn0gsum 19012 | Group multiple (exponentia... |
| mulg1 19013 | Group multiple (exponentia... |
| mulgnnp1 19014 | Group multiple (exponentia... |
| mulg2 19015 | Group multiple (exponentia... |
| mulgnegnn 19016 | Group multiple (exponentia... |
| mulgnn0p1 19017 | Group multiple (exponentia... |
| mulgnnsubcl 19018 | Closure of the group multi... |
| mulgnn0subcl 19019 | Closure of the group multi... |
| mulgsubcl 19020 | Closure of the group multi... |
| mulgnncl 19021 | Closure of the group multi... |
| mulgnn0cl 19022 | Closure of the group multi... |
| mulgcl 19023 | Closure of the group multi... |
| mulgneg 19024 | Group multiple (exponentia... |
| mulgnegneg 19025 | The inverse of a negative ... |
| mulgm1 19026 | Group multiple (exponentia... |
| mulgnn0cld 19027 | Closure of the group multi... |
| mulgcld 19028 | Deduction associated with ... |
| mulgaddcomlem 19029 | Lemma for ~ mulgaddcom . ... |
| mulgaddcom 19030 | The group multiple operato... |
| mulginvcom 19031 | The group multiple operato... |
| mulginvinv 19032 | The group multiple operato... |
| mulgnn0z 19033 | A group multiple of the id... |
| mulgz 19034 | A group multiple of the id... |
| mulgnndir 19035 | Sum of group multiples, fo... |
| mulgnn0dir 19036 | Sum of group multiples, ge... |
| mulgdirlem 19037 | Lemma for ~ mulgdir . (Co... |
| mulgdir 19038 | Sum of group multiples, ge... |
| mulgp1 19039 | Group multiple (exponentia... |
| mulgneg2 19040 | Group multiple (exponentia... |
| mulgnnass 19041 | Product of group multiples... |
| mulgnn0ass 19042 | Product of group multiples... |
| mulgass 19043 | Product of group multiples... |
| mulgassr 19044 | Reversed product of group ... |
| mulgmodid 19045 | Casting out multiples of t... |
| mulgsubdir 19046 | Distribution of group mult... |
| mhmmulg 19047 | A homomorphism of monoids ... |
| mulgpropd 19048 | Two structures with the sa... |
| submmulgcl 19049 | Closure of the group multi... |
| submmulg 19050 | A group multiple is the sa... |
| pwsmulg 19051 | Value of a group multiple ... |
| issubg 19058 | The subgroup predicate. (... |
| subgss 19059 | A subgroup is a subset. (... |
| subgid 19060 | A group is a subgroup of i... |
| subggrp 19061 | A subgroup is a group. (C... |
| subgbas 19062 | The base of the restricted... |
| subgrcl 19063 | Reverse closure for the su... |
| subg0 19064 | A subgroup of a group must... |
| subginv 19065 | The inverse of an element ... |
| subg0cl 19066 | The group identity is an e... |
| subginvcl 19067 | The inverse of an element ... |
| subgcl 19068 | A subgroup is closed under... |
| subgsubcl 19069 | A subgroup is closed under... |
| subgsub 19070 | The subtraction of element... |
| subgmulgcl 19071 | Closure of the group multi... |
| subgmulg 19072 | A group multiple is the sa... |
| issubg2 19073 | Characterize the subgroups... |
| issubgrpd2 19074 | Prove a subgroup by closur... |
| issubgrpd 19075 | Prove a subgroup by closur... |
| issubg3 19076 | A subgroup is a symmetric ... |
| issubg4 19077 | A subgroup is a nonempty s... |
| grpissubg 19078 | If the base set of a group... |
| resgrpisgrp 19079 | If the base set of a group... |
| subgsubm 19080 | A subgroup is a submonoid.... |
| subsubg 19081 | A subgroup of a subgroup i... |
| subgint 19082 | The intersection of a none... |
| 0subg 19083 | The zero subgroup of an ar... |
| 0subgOLD 19084 | Obsolete version of ~ 0sub... |
| trivsubgd 19085 | The only subgroup of a tri... |
| trivsubgsnd 19086 | The only subgroup of a tri... |
| isnsg 19087 | Property of being a normal... |
| isnsg2 19088 | Weaken the condition of ~ ... |
| nsgbi 19089 | Defining property of a nor... |
| nsgsubg 19090 | A normal subgroup is a sub... |
| nsgconj 19091 | The conjugation of an elem... |
| isnsg3 19092 | A subgroup is normal iff t... |
| subgacs 19093 | Subgroups are an algebraic... |
| nsgacs 19094 | Normal subgroups form an a... |
| elnmz 19095 | Elementhood in the normali... |
| nmzbi 19096 | Defining property of the n... |
| nmzsubg 19097 | The normalizer N_G(S) of a... |
| ssnmz 19098 | A subgroup is a subset of ... |
| isnsg4 19099 | A subgroup is normal iff i... |
| nmznsg 19100 | Any subgroup is a normal s... |
| 0nsg 19101 | The zero subgroup is norma... |
| nsgid 19102 | The whole group is a norma... |
| 0idnsgd 19103 | The whole group and the ze... |
| trivnsgd 19104 | The only normal subgroup o... |
| triv1nsgd 19105 | A trivial group has exactl... |
| 1nsgtrivd 19106 | A group with exactly one n... |
| releqg 19107 | The left coset equivalence... |
| eqgfval 19108 | Value of the subgroup left... |
| eqgval 19109 | Value of the subgroup left... |
| eqger 19110 | The subgroup coset equival... |
| eqglact 19111 | A left coset can be expres... |
| eqgid 19112 | The left coset containing ... |
| eqgen 19113 | Each coset is equipotent t... |
| eqgcpbl 19114 | The subgroup coset equival... |
| eqg0el 19115 | Equivalence class of a quo... |
| quselbas 19116 | Membership in the base set... |
| quseccl0 19117 | Closure of the quotient ma... |
| qusgrp 19118 | If ` Y ` is a normal subgr... |
| quseccl 19119 | Closure of the quotient ma... |
| qusadd 19120 | Value of the group operati... |
| qus0 19121 | Value of the group identit... |
| qusinv 19122 | Value of the group inverse... |
| qussub 19123 | Value of the group subtrac... |
| ecqusaddd 19124 | Addition of equivalence cl... |
| ecqusaddcl 19125 | Closure of the addition in... |
| lagsubg2 19126 | Lagrange's theorem for fin... |
| lagsubg 19127 | Lagrange's theorem for Gro... |
| eqg0subg 19128 | The coset equivalence rela... |
| eqg0subgecsn 19129 | The equivalence classes mo... |
| qus0subgbas 19130 | The base set of a quotient... |
| qus0subgadd 19131 | The addition in a quotient... |
| cycsubmel 19132 | Characterization of an ele... |
| cycsubmcl 19133 | The set of nonnegative int... |
| cycsubm 19134 | The set of nonnegative int... |
| cyccom 19135 | Condition for an operation... |
| cycsubmcom 19136 | The operation of a monoid ... |
| cycsubggend 19137 | The cyclic subgroup genera... |
| cycsubgcl 19138 | The set of integer powers ... |
| cycsubgss 19139 | The cyclic subgroup genera... |
| cycsubg 19140 | The cyclic group generated... |
| cycsubgcld 19141 | The cyclic subgroup genera... |
| cycsubg2 19142 | The subgroup generated by ... |
| cycsubg2cl 19143 | Any multiple of an element... |
| reldmghm 19146 | Lemma for group homomorphi... |
| isghm 19147 | Property of being a homomo... |
| isghmOLD 19148 | Obsolete version of ~ isgh... |
| isghm3 19149 | Property of a group homomo... |
| ghmgrp1 19150 | A group homomorphism is on... |
| ghmgrp2 19151 | A group homomorphism is on... |
| ghmf 19152 | A group homomorphism is a ... |
| ghmlin 19153 | A homomorphism of groups i... |
| ghmid 19154 | A homomorphism of groups p... |
| ghminv 19155 | A homomorphism of groups p... |
| ghmsub 19156 | Linearity of subtraction t... |
| isghmd 19157 | Deduction for a group homo... |
| ghmmhm 19158 | A group homomorphism is a ... |
| ghmmhmb 19159 | Group homomorphisms and mo... |
| ghmmulg 19160 | A group homomorphism prese... |
| ghmrn 19161 | The range of a homomorphis... |
| 0ghm 19162 | The constant zero linear f... |
| idghm 19163 | The identity homomorphism ... |
| resghm 19164 | Restriction of a homomorph... |
| resghm2 19165 | One direction of ~ resghm2... |
| resghm2b 19166 | Restriction of the codomai... |
| ghmghmrn 19167 | A group homomorphism from ... |
| ghmco 19168 | The composition of group h... |
| ghmima 19169 | The image of a subgroup un... |
| ghmpreima 19170 | The inverse image of a sub... |
| ghmeql 19171 | The equalizer of two group... |
| ghmnsgima 19172 | The image of a normal subg... |
| ghmnsgpreima 19173 | The inverse image of a nor... |
| ghmker 19174 | The kernel of a homomorphi... |
| ghmeqker 19175 | Two source points map to t... |
| pwsdiagghm 19176 | Diagonal homomorphism into... |
| f1ghm0to0 19177 | If a group homomorphism ` ... |
| ghmf1 19178 | Two ways of saying a group... |
| kerf1ghm 19179 | A group homomorphism ` F `... |
| ghmf1o 19180 | A bijective group homomorp... |
| conjghm 19181 | Conjugation is an automorp... |
| conjsubg 19182 | A conjugated subgroup is a... |
| conjsubgen 19183 | A conjugated subgroup is e... |
| conjnmz 19184 | A subgroup is unchanged un... |
| conjnmzb 19185 | Alternative condition for ... |
| conjnsg 19186 | A normal subgroup is uncha... |
| qusghm 19187 | If ` Y ` is a normal subgr... |
| ghmpropd 19188 | Group homomorphism depends... |
| gimfn 19193 | The group isomorphism func... |
| isgim 19194 | An isomorphism of groups i... |
| gimf1o 19195 | An isomorphism of groups i... |
| gimghm 19196 | An isomorphism of groups i... |
| isgim2 19197 | A group isomorphism is a h... |
| subggim 19198 | Behavior of subgroups unde... |
| gimcnv 19199 | The converse of a group is... |
| gimco 19200 | The composition of group i... |
| gim0to0 19201 | A group isomorphism maps t... |
| brgic 19202 | The relation "is isomorphi... |
| brgici 19203 | Prove isomorphic by an exp... |
| gicref 19204 | Isomorphism is reflexive. ... |
| giclcl 19205 | Isomorphism implies the le... |
| gicrcl 19206 | Isomorphism implies the ri... |
| gicsym 19207 | Isomorphism is symmetric. ... |
| gictr 19208 | Isomorphism is transitive.... |
| gicer 19209 | Isomorphism is an equivale... |
| gicen 19210 | Isomorphic groups have equ... |
| gicsubgen 19211 | A less trivial example of ... |
| ghmqusnsglem1 19212 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsglem2 19213 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsg 19214 | The mapping ` H ` induced ... |
| ghmquskerlem1 19215 | Lemma for ~ ghmqusker . (... |
| ghmquskerco 19216 | In the case of theorem ~ g... |
| ghmquskerlem2 19217 | Lemma for ~ ghmqusker . (... |
| ghmquskerlem3 19218 | The mapping ` H ` induced ... |
| ghmqusker 19219 | A surjective group homomor... |
| gicqusker 19220 | The image ` H ` of a group... |
| isga 19223 | The predicate "is a (left)... |
| gagrp 19224 | The left argument of a gro... |
| gaset 19225 | The right argument of a gr... |
| gagrpid 19226 | The identity of the group ... |
| gaf 19227 | The mapping of the group a... |
| gafo 19228 | A group action is onto its... |
| gaass 19229 | An "associative" property ... |
| ga0 19230 | The action of a group on t... |
| gaid 19231 | The trivial action of a gr... |
| subgga 19232 | A subgroup acts on its par... |
| gass 19233 | A subset of a group action... |
| gasubg 19234 | The restriction of a group... |
| gaid2 19235 | A group operation is a lef... |
| galcan 19236 | The action of a particular... |
| gacan 19237 | Group inverses cancel in a... |
| gapm 19238 | The action of a particular... |
| gaorb 19239 | The orbit equivalence rela... |
| gaorber 19240 | The orbit equivalence rela... |
| gastacl 19241 | The stabilizer subgroup in... |
| gastacos 19242 | Write the coset relation f... |
| orbstafun 19243 | Existence and uniqueness f... |
| orbstaval 19244 | Value of the function at a... |
| orbsta 19245 | The Orbit-Stabilizer theor... |
| orbsta2 19246 | Relation between the size ... |
| cntrval 19251 | Substitute definition of t... |
| cntzfval 19252 | First level substitution f... |
| cntzval 19253 | Definition substitution fo... |
| elcntz 19254 | Elementhood in the central... |
| cntzel 19255 | Membership in a centralize... |
| cntzsnval 19256 | Special substitution for t... |
| elcntzsn 19257 | Value of the centralizer o... |
| sscntz 19258 | A centralizer expression f... |
| cntzrcl 19259 | Reverse closure for elemen... |
| cntzssv 19260 | The centralizer is uncondi... |
| cntzi 19261 | Membership in a centralize... |
| elcntr 19262 | Elementhood in the center ... |
| cntrss 19263 | The center is a subset of ... |
| cntri 19264 | Defining property of the c... |
| resscntz 19265 | Centralizer in a substruct... |
| cntzsgrpcl 19266 | Centralizers are closed un... |
| cntz2ss 19267 | Centralizers reverse the s... |
| cntzrec 19268 | Reciprocity relationship f... |
| cntziinsn 19269 | Express any centralizer as... |
| cntzsubm 19270 | Centralizers in a monoid a... |
| cntzsubg 19271 | Centralizers in a group ar... |
| cntzidss 19272 | If the elements of ` S ` c... |
| cntzmhm 19273 | Centralizers in a monoid a... |
| cntzmhm2 19274 | Centralizers in a monoid a... |
| cntrsubgnsg 19275 | A central subgroup is norm... |
| cntrnsg 19276 | The center of a group is a... |
| oppgval 19279 | Value of the opposite grou... |
| oppgplusfval 19280 | Value of the addition oper... |
| oppgplus 19281 | Value of the addition oper... |
| setsplusg 19282 | The other components of an... |
| oppgbas 19283 | Base set of an opposite gr... |
| oppgtset 19284 | Topology of an opposite gr... |
| oppgtopn 19285 | Topology of an opposite gr... |
| oppgmnd 19286 | The opposite of a monoid i... |
| oppgmndb 19287 | Bidirectional form of ~ op... |
| oppgid 19288 | Zero in a monoid is a symm... |
| oppggrp 19289 | The opposite of a group is... |
| oppggrpb 19290 | Bidirectional form of ~ op... |
| oppginv 19291 | Inverses in a group are a ... |
| invoppggim 19292 | The inverse is an antiauto... |
| oppggic 19293 | Every group is (naturally)... |
| oppgsubm 19294 | Being a submonoid is a sym... |
| oppgsubg 19295 | Being a subgroup is a symm... |
| oppgcntz 19296 | A centralizer in a group i... |
| oppgcntr 19297 | The center of a group is t... |
| gsumwrev 19298 | A sum in an opposite monoi... |
| symgval 19301 | The value of the symmetric... |
| symgbas 19302 | The base set of the symmet... |
| elsymgbas2 19303 | Two ways of saying a funct... |
| elsymgbas 19304 | Two ways of saying a funct... |
| symgbasf1o 19305 | Elements in the symmetric ... |
| symgbasf 19306 | A permutation (element of ... |
| symgbasmap 19307 | A permutation (element of ... |
| symghash 19308 | The symmetric group on ` n... |
| symgbasfi 19309 | The symmetric group on a f... |
| symgfv 19310 | The function value of a pe... |
| symgfvne 19311 | The function values of a p... |
| symgressbas 19312 | The symmetric group on ` A... |
| symgplusg 19313 | The group operation of a s... |
| symgov 19314 | The value of the group ope... |
| symgcl 19315 | The group operation of the... |
| idresperm 19316 | The identity function rest... |
| symgmov1 19317 | For a permutation of a set... |
| symgmov2 19318 | For a permutation of a set... |
| symgbas0 19319 | The base set of the symmet... |
| symg1hash 19320 | The symmetric group on a s... |
| symg1bas 19321 | The symmetric group on a s... |
| symg2hash 19322 | The symmetric group on a (... |
| symg2bas 19323 | The symmetric group on a p... |
| 0symgefmndeq 19324 | The symmetric group on the... |
| snsymgefmndeq 19325 | The symmetric group on a s... |
| symgpssefmnd 19326 | For a set ` A ` with more ... |
| symgvalstruct 19327 | The value of the symmetric... |
| symgsubmefmnd 19328 | The symmetric group on a s... |
| symgtset 19329 | The topology of the symmet... |
| symggrp 19330 | The symmetric group on a s... |
| symgid 19331 | The group identity element... |
| symginv 19332 | The group inverse in the s... |
| symgsubmefmndALT 19333 | The symmetric group on a s... |
| galactghm 19334 | The currying of a group ac... |
| lactghmga 19335 | The converse of ~ galactgh... |
| symgtopn 19336 | The topology of the symmet... |
| symgga 19337 | The symmetric group induce... |
| pgrpsubgsymgbi 19338 | Every permutation group is... |
| pgrpsubgsymg 19339 | Every permutation group is... |
| idressubgsymg 19340 | The singleton containing o... |
| idrespermg 19341 | The structure with the sin... |
| cayleylem1 19342 | Lemma for ~ cayley . (Con... |
| cayleylem2 19343 | Lemma for ~ cayley . (Con... |
| cayley 19344 | Cayley's Theorem (construc... |
| cayleyth 19345 | Cayley's Theorem (existenc... |
| symgfix2 19346 | If a permutation does not ... |
| symgextf 19347 | The extension of a permuta... |
| symgextfv 19348 | The function value of the ... |
| symgextfve 19349 | The function value of the ... |
| symgextf1lem 19350 | Lemma for ~ symgextf1 . (... |
| symgextf1 19351 | The extension of a permuta... |
| symgextfo 19352 | The extension of a permuta... |
| symgextf1o 19353 | The extension of a permuta... |
| symgextsymg 19354 | The extension of a permuta... |
| symgextres 19355 | The restriction of the ext... |
| gsumccatsymgsn 19356 | Homomorphic property of co... |
| gsmsymgrfixlem1 19357 | Lemma 1 for ~ gsmsymgrfix ... |
| gsmsymgrfix 19358 | The composition of permuta... |
| fvcosymgeq 19359 | The values of two composit... |
| gsmsymgreqlem1 19360 | Lemma 1 for ~ gsmsymgreq .... |
| gsmsymgreqlem2 19361 | Lemma 2 for ~ gsmsymgreq .... |
| gsmsymgreq 19362 | Two combination of permuta... |
| symgfixelq 19363 | A permutation of a set fix... |
| symgfixels 19364 | The restriction of a permu... |
| symgfixelsi 19365 | The restriction of a permu... |
| symgfixf 19366 | The mapping of a permutati... |
| symgfixf1 19367 | The mapping of a permutati... |
| symgfixfolem1 19368 | Lemma 1 for ~ symgfixfo . ... |
| symgfixfo 19369 | The mapping of a permutati... |
| symgfixf1o 19370 | The mapping of a permutati... |
| f1omvdmvd 19373 | A permutation of any class... |
| f1omvdcnv 19374 | A permutation and its inve... |
| mvdco 19375 | Composing two permutations... |
| f1omvdconj 19376 | Conjugation of a permutati... |
| f1otrspeq 19377 | A transposition is charact... |
| f1omvdco2 19378 | If exactly one of two perm... |
| f1omvdco3 19379 | If a point is moved by exa... |
| pmtrfval 19380 | The function generating tr... |
| pmtrval 19381 | A generated transposition,... |
| pmtrfv 19382 | General value of mapping a... |
| pmtrprfv 19383 | In a transposition of two ... |
| pmtrprfv3 19384 | In a transposition of two ... |
| pmtrf 19385 | Functionality of a transpo... |
| pmtrmvd 19386 | A transposition moves prec... |
| pmtrrn 19387 | Transposing two points giv... |
| pmtrfrn 19388 | A transposition (as a kind... |
| pmtrffv 19389 | Mapping of a point under a... |
| pmtrrn2 19390 | For any transposition ther... |
| pmtrfinv 19391 | A transposition function i... |
| pmtrfmvdn0 19392 | A transposition moves at l... |
| pmtrff1o 19393 | A transposition function i... |
| pmtrfcnv 19394 | A transposition function i... |
| pmtrfb 19395 | An intrinsic characterizat... |
| pmtrfconj 19396 | Any conjugate of a transpo... |
| symgsssg 19397 | The symmetric group has su... |
| symgfisg 19398 | The symmetric group has a ... |
| symgtrf 19399 | Transpositions are element... |
| symggen 19400 | The span of the transposit... |
| symggen2 19401 | A finite permutation group... |
| symgtrinv 19402 | To invert a permutation re... |
| pmtr3ncomlem1 19403 | Lemma 1 for ~ pmtr3ncom . ... |
| pmtr3ncomlem2 19404 | Lemma 2 for ~ pmtr3ncom . ... |
| pmtr3ncom 19405 | Transpositions over sets w... |
| pmtrdifellem1 19406 | Lemma 1 for ~ pmtrdifel . ... |
| pmtrdifellem2 19407 | Lemma 2 for ~ pmtrdifel . ... |
| pmtrdifellem3 19408 | Lemma 3 for ~ pmtrdifel . ... |
| pmtrdifellem4 19409 | Lemma 4 for ~ pmtrdifel . ... |
| pmtrdifel 19410 | A transposition of element... |
| pmtrdifwrdellem1 19411 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem2 19412 | Lemma 2 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem3 19413 | Lemma 3 for ~ pmtrdifwrdel... |
| pmtrdifwrdel2lem1 19414 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdel 19415 | A sequence of transpositio... |
| pmtrdifwrdel2 19416 | A sequence of transpositio... |
| pmtrprfval 19417 | The transpositions on a pa... |
| pmtrprfvalrn 19418 | The range of the transposi... |
| psgnunilem1 19423 | Lemma for ~ psgnuni . Giv... |
| psgnunilem5 19424 | Lemma for ~ psgnuni . It ... |
| psgnunilem2 19425 | Lemma for ~ psgnuni . Ind... |
| psgnunilem3 19426 | Lemma for ~ psgnuni . Any... |
| psgnunilem4 19427 | Lemma for ~ psgnuni . An ... |
| m1expaddsub 19428 | Addition and subtraction o... |
| psgnuni 19429 | If the same permutation ca... |
| psgnfval 19430 | Function definition of the... |
| psgnfn 19431 | Functionality and domain o... |
| psgndmsubg 19432 | The finitary permutations ... |
| psgneldm 19433 | Property of being a finita... |
| psgneldm2 19434 | The finitary permutations ... |
| psgneldm2i 19435 | A sequence of transpositio... |
| psgneu 19436 | A finitary permutation has... |
| psgnval 19437 | Value of the permutation s... |
| psgnvali 19438 | A finitary permutation has... |
| psgnvalii 19439 | Any representation of a pe... |
| psgnpmtr 19440 | All transpositions are odd... |
| psgn0fv0 19441 | The permutation sign funct... |
| sygbasnfpfi 19442 | The class of non-fixed poi... |
| psgnfvalfi 19443 | Function definition of the... |
| psgnvalfi 19444 | Value of the permutation s... |
| psgnran 19445 | The range of the permutati... |
| gsmtrcl 19446 | The group sum of transposi... |
| psgnfitr 19447 | A permutation of a finite ... |
| psgnfieu 19448 | A permutation of a finite ... |
| pmtrsn 19449 | The value of the transposi... |
| psgnsn 19450 | The permutation sign funct... |
| psgnprfval 19451 | The permutation sign funct... |
| psgnprfval1 19452 | The permutation sign of th... |
| psgnprfval2 19453 | The permutation sign of th... |
| odfval 19462 | Value of the order functio... |
| odfvalALT 19463 | Shorter proof of ~ odfval ... |
| odval 19464 | Second substitution for th... |
| odlem1 19465 | The group element order is... |
| odcl 19466 | The order of a group eleme... |
| odf 19467 | Functionality of the group... |
| odid 19468 | Any element to the power o... |
| odlem2 19469 | Any positive annihilator o... |
| odmodnn0 19470 | Reduce the argument of a g... |
| mndodconglem 19471 | Lemma for ~ mndodcong . (... |
| mndodcong 19472 | If two multipliers are con... |
| mndodcongi 19473 | If two multipliers are con... |
| oddvdsnn0 19474 | The only multiples of ` A ... |
| odnncl 19475 | If a nonzero multiple of a... |
| odmod 19476 | Reduce the argument of a g... |
| oddvds 19477 | The only multiples of ` A ... |
| oddvdsi 19478 | Any group element is annih... |
| odcong 19479 | If two multipliers are con... |
| odeq 19480 | The ~ oddvds property uniq... |
| odval2 19481 | A non-conditional definiti... |
| odcld 19482 | The order of a group eleme... |
| odm1inv 19483 | The (order-1)th multiple o... |
| odmulgid 19484 | A relationship between the... |
| odmulg2 19485 | The order of a multiple di... |
| odmulg 19486 | Relationship between the o... |
| odmulgeq 19487 | A multiple of a point of f... |
| odbezout 19488 | If ` N ` is coprime to the... |
| od1 19489 | The order of the group ide... |
| odeq1 19490 | The group identity is the ... |
| odinv 19491 | The order of the inverse o... |
| odf1 19492 | The multiples of an elemen... |
| odinf 19493 | The multiples of an elemen... |
| dfod2 19494 | An alternative definition ... |
| odcl2 19495 | The order of an element of... |
| oddvds2 19496 | The order of an element of... |
| finodsubmsubg 19497 | A submonoid whose elements... |
| 0subgALT 19498 | A shorter proof of ~ 0subg... |
| submod 19499 | The order of an element is... |
| subgod 19500 | The order of an element is... |
| odsubdvds 19501 | The order of an element of... |
| odf1o1 19502 | An element with zero order... |
| odf1o2 19503 | An element with nonzero or... |
| odhash 19504 | An element of zero order g... |
| odhash2 19505 | If an element has nonzero ... |
| odhash3 19506 | An element which generates... |
| odngen 19507 | A cyclic subgroup of size ... |
| gexval 19508 | Value of the exponent of a... |
| gexlem1 19509 | The group element order is... |
| gexcl 19510 | The exponent of a group is... |
| gexid 19511 | Any element to the power o... |
| gexlem2 19512 | Any positive annihilator o... |
| gexdvdsi 19513 | Any group element is annih... |
| gexdvds 19514 | The only ` N ` that annihi... |
| gexdvds2 19515 | An integer divides the gro... |
| gexod 19516 | Any group element is annih... |
| gexcl3 19517 | If the order of every grou... |
| gexnnod 19518 | Every group element has fi... |
| gexcl2 19519 | The exponent of a finite g... |
| gexdvds3 19520 | The exponent of a finite g... |
| gex1 19521 | A group or monoid has expo... |
| ispgp 19522 | A group is a ` P ` -group ... |
| pgpprm 19523 | Reverse closure for the fi... |
| pgpgrp 19524 | Reverse closure for the se... |
| pgpfi1 19525 | A finite group with order ... |
| pgp0 19526 | The identity subgroup is a... |
| subgpgp 19527 | A subgroup of a p-group is... |
| sylow1lem1 19528 | Lemma for ~ sylow1 . The ... |
| sylow1lem2 19529 | Lemma for ~ sylow1 . The ... |
| sylow1lem3 19530 | Lemma for ~ sylow1 . One ... |
| sylow1lem4 19531 | Lemma for ~ sylow1 . The ... |
| sylow1lem5 19532 | Lemma for ~ sylow1 . Usin... |
| sylow1 19533 | Sylow's first theorem. If... |
| odcau 19534 | Cauchy's theorem for the o... |
| pgpfi 19535 | The converse to ~ pgpfi1 .... |
| pgpfi2 19536 | Alternate version of ~ pgp... |
| pgphash 19537 | The order of a p-group. (... |
| isslw 19538 | The property of being a Sy... |
| slwprm 19539 | Reverse closure for the fi... |
| slwsubg 19540 | A Sylow ` P ` -subgroup is... |
| slwispgp 19541 | Defining property of a Syl... |
| slwpss 19542 | A proper superset of a Syl... |
| slwpgp 19543 | A Sylow ` P ` -subgroup is... |
| pgpssslw 19544 | Every ` P ` -subgroup is c... |
| slwn0 19545 | Every finite group contain... |
| subgslw 19546 | A Sylow subgroup that is c... |
| sylow2alem1 19547 | Lemma for ~ sylow2a . An ... |
| sylow2alem2 19548 | Lemma for ~ sylow2a . All... |
| sylow2a 19549 | A named lemma of Sylow's s... |
| sylow2blem1 19550 | Lemma for ~ sylow2b . Eva... |
| sylow2blem2 19551 | Lemma for ~ sylow2b . Lef... |
| sylow2blem3 19552 | Sylow's second theorem. P... |
| sylow2b 19553 | Sylow's second theorem. A... |
| slwhash 19554 | A sylow subgroup has cardi... |
| fislw 19555 | The sylow subgroups of a f... |
| sylow2 19556 | Sylow's second theorem. S... |
| sylow3lem1 19557 | Lemma for ~ sylow3 , first... |
| sylow3lem2 19558 | Lemma for ~ sylow3 , first... |
| sylow3lem3 19559 | Lemma for ~ sylow3 , first... |
| sylow3lem4 19560 | Lemma for ~ sylow3 , first... |
| sylow3lem5 19561 | Lemma for ~ sylow3 , secon... |
| sylow3lem6 19562 | Lemma for ~ sylow3 , secon... |
| sylow3 19563 | Sylow's third theorem. Th... |
| lsmfval 19568 | The subgroup sum function ... |
| lsmvalx 19569 | Subspace sum value (for a ... |
| lsmelvalx 19570 | Subspace sum membership (f... |
| lsmelvalix 19571 | Subspace sum membership (f... |
| oppglsm 19572 | The subspace sum operation... |
| lsmssv 19573 | Subgroup sum is a subset o... |
| lsmless1x 19574 | Subset implies subgroup su... |
| lsmless2x 19575 | Subset implies subgroup su... |
| lsmub1x 19576 | Subgroup sum is an upper b... |
| lsmub2x 19577 | Subgroup sum is an upper b... |
| lsmval 19578 | Subgroup sum value (for a ... |
| lsmelval 19579 | Subgroup sum membership (f... |
| lsmelvali 19580 | Subgroup sum membership (f... |
| lsmelvalm 19581 | Subgroup sum membership an... |
| lsmelvalmi 19582 | Membership of vector subtr... |
| lsmsubm 19583 | The sum of two commuting s... |
| lsmsubg 19584 | The sum of two commuting s... |
| lsmcom2 19585 | Subgroup sum commutes. (C... |
| smndlsmidm 19586 | The direct product is idem... |
| lsmub1 19587 | Subgroup sum is an upper b... |
| lsmub2 19588 | Subgroup sum is an upper b... |
| lsmunss 19589 | Union of subgroups is a su... |
| lsmless1 19590 | Subset implies subgroup su... |
| lsmless2 19591 | Subset implies subgroup su... |
| lsmless12 19592 | Subset implies subgroup su... |
| lsmidm 19593 | Subgroup sum is idempotent... |
| lsmlub 19594 | The least upper bound prop... |
| lsmss1 19595 | Subgroup sum with a subset... |
| lsmss1b 19596 | Subgroup sum with a subset... |
| lsmss2 19597 | Subgroup sum with a subset... |
| lsmss2b 19598 | Subgroup sum with a subset... |
| lsmass 19599 | Subgroup sum is associativ... |
| mndlsmidm 19600 | Subgroup sum is idempotent... |
| lsm01 19601 | Subgroup sum with the zero... |
| lsm02 19602 | Subgroup sum with the zero... |
| subglsm 19603 | The subgroup sum evaluated... |
| lssnle 19604 | Equivalent expressions for... |
| lsmmod 19605 | The modular law holds for ... |
| lsmmod2 19606 | Modular law dual for subgr... |
| lsmpropd 19607 | If two structures have the... |
| cntzrecd 19608 | Commute the "subgroups com... |
| lsmcntz 19609 | The "subgroups commute" pr... |
| lsmcntzr 19610 | The "subgroups commute" pr... |
| lsmdisj 19611 | Disjointness from a subgro... |
| lsmdisj2 19612 | Association of the disjoin... |
| lsmdisj3 19613 | Association of the disjoin... |
| lsmdisjr 19614 | Disjointness from a subgro... |
| lsmdisj2r 19615 | Association of the disjoin... |
| lsmdisj3r 19616 | Association of the disjoin... |
| lsmdisj2a 19617 | Association of the disjoin... |
| lsmdisj2b 19618 | Association of the disjoin... |
| lsmdisj3a 19619 | Association of the disjoin... |
| lsmdisj3b 19620 | Association of the disjoin... |
| subgdisj1 19621 | Vectors belonging to disjo... |
| subgdisj2 19622 | Vectors belonging to disjo... |
| subgdisjb 19623 | Vectors belonging to disjo... |
| pj1fval 19624 | The left projection functi... |
| pj1val 19625 | The left projection functi... |
| pj1eu 19626 | Uniqueness of a left proje... |
| pj1f 19627 | The left projection functi... |
| pj2f 19628 | The right projection funct... |
| pj1id 19629 | Any element of a direct su... |
| pj1eq 19630 | Any element of a direct su... |
| pj1lid 19631 | The left projection functi... |
| pj1rid 19632 | The left projection functi... |
| pj1ghm 19633 | The left projection functi... |
| pj1ghm2 19634 | The left projection functi... |
| lsmhash 19635 | The order of the direct pr... |
| efgmval 19642 | Value of the formal invers... |
| efgmf 19643 | The formal inverse operati... |
| efgmnvl 19644 | The inversion function on ... |
| efgrcl 19645 | Lemma for ~ efgval . (Con... |
| efglem 19646 | Lemma for ~ efgval . (Con... |
| efgval 19647 | Value of the free group co... |
| efger 19648 | Value of the free group co... |
| efgi 19649 | Value of the free group co... |
| efgi0 19650 | Value of the free group co... |
| efgi1 19651 | Value of the free group co... |
| efgtf 19652 | Value of the free group co... |
| efgtval 19653 | Value of the extension fun... |
| efgval2 19654 | Value of the free group co... |
| efgi2 19655 | Value of the free group co... |
| efgtlen 19656 | Value of the free group co... |
| efginvrel2 19657 | The inverse of the reverse... |
| efginvrel1 19658 | The inverse of the reverse... |
| efgsf 19659 | Value of the auxiliary fun... |
| efgsdm 19660 | Elementhood in the domain ... |
| efgsval 19661 | Value of the auxiliary fun... |
| efgsdmi 19662 | Property of the last link ... |
| efgsval2 19663 | Value of the auxiliary fun... |
| efgsrel 19664 | The start and end of any e... |
| efgs1 19665 | A singleton of an irreduci... |
| efgs1b 19666 | Every extension sequence e... |
| efgsp1 19667 | If ` F ` is an extension s... |
| efgsres 19668 | An initial segment of an e... |
| efgsfo 19669 | For any word, there is a s... |
| efgredlema 19670 | The reduced word that form... |
| efgredlemf 19671 | Lemma for ~ efgredleme . ... |
| efgredlemg 19672 | Lemma for ~ efgred . (Con... |
| efgredleme 19673 | Lemma for ~ efgred . (Con... |
| efgredlemd 19674 | The reduced word that form... |
| efgredlemc 19675 | The reduced word that form... |
| efgredlemb 19676 | The reduced word that form... |
| efgredlem 19677 | The reduced word that form... |
| efgred 19678 | The reduced word that form... |
| efgrelexlema 19679 | If two words ` A , B ` are... |
| efgrelexlemb 19680 | If two words ` A , B ` are... |
| efgrelex 19681 | If two words ` A , B ` are... |
| efgredeu 19682 | There is a unique reduced ... |
| efgred2 19683 | Two extension sequences ha... |
| efgcpbllema 19684 | Lemma for ~ efgrelex . De... |
| efgcpbllemb 19685 | Lemma for ~ efgrelex . Sh... |
| efgcpbl 19686 | Two extension sequences ha... |
| efgcpbl2 19687 | Two extension sequences ha... |
| frgpval 19688 | Value of the free group co... |
| frgpcpbl 19689 | Compatibility of the group... |
| frgp0 19690 | The free group is a group.... |
| frgpeccl 19691 | Closure of the quotient ma... |
| frgpgrp 19692 | The free group is a group.... |
| frgpadd 19693 | Addition in the free group... |
| frgpinv 19694 | The inverse of an element ... |
| frgpmhm 19695 | The "natural map" from wor... |
| vrgpfval 19696 | The canonical injection fr... |
| vrgpval 19697 | The value of the generatin... |
| vrgpf 19698 | The mapping from the index... |
| vrgpinv 19699 | The inverse of a generatin... |
| frgpuptf 19700 | Any assignment of the gene... |
| frgpuptinv 19701 | Any assignment of the gene... |
| frgpuplem 19702 | Any assignment of the gene... |
| frgpupf 19703 | Any assignment of the gene... |
| frgpupval 19704 | Any assignment of the gene... |
| frgpup1 19705 | Any assignment of the gene... |
| frgpup2 19706 | The evaluation map has the... |
| frgpup3lem 19707 | The evaluation map has the... |
| frgpup3 19708 | Universal property of the ... |
| 0frgp 19709 | The free group on zero gen... |
| isabl 19714 | The predicate "is an Abeli... |
| ablgrp 19715 | An Abelian group is a grou... |
| ablgrpd 19716 | An Abelian group is a grou... |
| ablcmn 19717 | An Abelian group is a comm... |
| ablcmnd 19718 | An Abelian group is a comm... |
| iscmn 19719 | The predicate "is a commut... |
| isabl2 19720 | The predicate "is an Abeli... |
| cmnpropd 19721 | If two structures have the... |
| ablpropd 19722 | If two structures have the... |
| ablprop 19723 | If two structures have the... |
| iscmnd 19724 | Properties that determine ... |
| isabld 19725 | Properties that determine ... |
| isabli 19726 | Properties that determine ... |
| cmnmnd 19727 | A commutative monoid is a ... |
| cmncom 19728 | A commutative monoid is co... |
| ablcom 19729 | An Abelian group operation... |
| cmn32 19730 | Commutative/associative la... |
| cmn4 19731 | Commutative/associative la... |
| cmn12 19732 | Commutative/associative la... |
| abl32 19733 | Commutative/associative la... |
| cmnmndd 19734 | A commutative monoid is a ... |
| cmnbascntr 19735 | The base set of a commutat... |
| rinvmod 19736 | Uniqueness of a right inve... |
| ablinvadd 19737 | The inverse of an Abelian ... |
| ablsub2inv 19738 | Abelian group subtraction ... |
| ablsubadd 19739 | Relationship between Abeli... |
| ablsub4 19740 | Commutative/associative su... |
| abladdsub4 19741 | Abelian group addition/sub... |
| abladdsub 19742 | Associative-type law for g... |
| ablsubadd23 19743 | Commutative/associative la... |
| ablsubaddsub 19744 | Double subtraction and add... |
| ablpncan2 19745 | Cancellation law for subtr... |
| ablpncan3 19746 | A cancellation law for Abe... |
| ablsubsub 19747 | Law for double subtraction... |
| ablsubsub4 19748 | Law for double subtraction... |
| ablpnpcan 19749 | Cancellation law for mixed... |
| ablnncan 19750 | Cancellation law for group... |
| ablsub32 19751 | Swap the second and third ... |
| ablnnncan 19752 | Cancellation law for group... |
| ablnnncan1 19753 | Cancellation law for group... |
| ablsubsub23 19754 | Swap subtrahend and result... |
| mulgnn0di 19755 | Group multiple of a sum, f... |
| mulgdi 19756 | Group multiple of a sum. ... |
| mulgmhm 19757 | The map from ` x ` to ` n ... |
| mulgghm 19758 | The map from ` x ` to ` n ... |
| mulgsubdi 19759 | Group multiple of a differ... |
| ghmfghm 19760 | The function fulfilling th... |
| ghmcmn 19761 | The image of a commutative... |
| ghmabl 19762 | The image of an abelian gr... |
| invghm 19763 | The inversion map is a gro... |
| eqgabl 19764 | Value of the subgroup cose... |
| qusecsub 19765 | Two subgroup cosets are eq... |
| subgabl 19766 | A subgroup of an abelian g... |
| subcmn 19767 | A submonoid of a commutati... |
| submcmn 19768 | A submonoid of a commutati... |
| submcmn2 19769 | A submonoid is commutative... |
| cntzcmn 19770 | The centralizer of any sub... |
| cntzcmnss 19771 | Any subset in a commutativ... |
| cntrcmnd 19772 | The center of a monoid is ... |
| cntrabl 19773 | The center of a group is a... |
| cntzspan 19774 | If the generators commute,... |
| cntzcmnf 19775 | Discharge the centralizer ... |
| ghmplusg 19776 | The pointwise sum of two l... |
| ablnsg 19777 | Every subgroup of an abeli... |
| odadd1 19778 | The order of a product in ... |
| odadd2 19779 | The order of a product in ... |
| odadd 19780 | The order of a product is ... |
| gex2abl 19781 | A group with exponent 2 (o... |
| gexexlem 19782 | Lemma for ~ gexex . (Cont... |
| gexex 19783 | In an abelian group with f... |
| torsubg 19784 | The set of all elements of... |
| oddvdssubg 19785 | The set of all elements wh... |
| lsmcomx 19786 | Subgroup sum commutes (ext... |
| ablcntzd 19787 | All subgroups in an abelia... |
| lsmcom 19788 | Subgroup sum commutes. (C... |
| lsmsubg2 19789 | The sum of two subgroups i... |
| lsm4 19790 | Commutative/associative la... |
| prdscmnd 19791 | The product of a family of... |
| prdsabld 19792 | The product of a family of... |
| pwscmn 19793 | The structure power on a c... |
| pwsabl 19794 | The structure power on an ... |
| qusabl 19795 | If ` Y ` is a subgroup of ... |
| abl1 19796 | The (smallest) structure r... |
| abln0 19797 | Abelian groups (and theref... |
| cnaddablx 19798 | The complex numbers are an... |
| cnaddabl 19799 | The complex numbers are an... |
| cnaddid 19800 | The group identity element... |
| cnaddinv 19801 | Value of the group inverse... |
| zaddablx 19802 | The integers are an Abelia... |
| frgpnabllem1 19803 | Lemma for ~ frgpnabl . (C... |
| frgpnabllem2 19804 | Lemma for ~ frgpnabl . (C... |
| frgpnabl 19805 | The free group on two or m... |
| imasabl 19806 | The image structure of an ... |
| iscyg 19809 | Definition of a cyclic gro... |
| iscyggen 19810 | The property of being a cy... |
| iscyggen2 19811 | The property of being a cy... |
| iscyg2 19812 | A cyclic group is a group ... |
| cyggeninv 19813 | The inverse of a cyclic ge... |
| cyggenod 19814 | An element is the generato... |
| cyggenod2 19815 | In an infinite cyclic grou... |
| iscyg3 19816 | Definition of a cyclic gro... |
| iscygd 19817 | Definition of a cyclic gro... |
| iscygodd 19818 | Show that a group with an ... |
| cycsubmcmn 19819 | The set of nonnegative int... |
| cyggrp 19820 | A cyclic group is a group.... |
| cygabl 19821 | A cyclic group is abelian.... |
| cygctb 19822 | A cyclic group is countabl... |
| 0cyg 19823 | The trivial group is cycli... |
| prmcyg 19824 | A group with prime order i... |
| lt6abl 19825 | A group with fewer than ` ... |
| ghmcyg 19826 | The image of a cyclic grou... |
| cyggex2 19827 | The exponent of a cyclic g... |
| cyggex 19828 | The exponent of a finite c... |
| cyggexb 19829 | A finite abelian group is ... |
| giccyg 19830 | Cyclicity is a group prope... |
| cycsubgcyg 19831 | The cyclic subgroup genera... |
| cycsubgcyg2 19832 | The cyclic subgroup genera... |
| gsumval3a 19833 | Value of the group sum ope... |
| gsumval3eu 19834 | The group sum as defined i... |
| gsumval3lem1 19835 | Lemma 1 for ~ gsumval3 . ... |
| gsumval3lem2 19836 | Lemma 2 for ~ gsumval3 . ... |
| gsumval3 19837 | Value of the group sum ope... |
| gsumcllem 19838 | Lemma for ~ gsumcl and rel... |
| gsumzres 19839 | Extend a finite group sum ... |
| gsumzcl2 19840 | Closure of a finite group ... |
| gsumzcl 19841 | Closure of a finite group ... |
| gsumzf1o 19842 | Re-index a finite group su... |
| gsumres 19843 | Extend a finite group sum ... |
| gsumcl2 19844 | Closure of a finite group ... |
| gsumcl 19845 | Closure of a finite group ... |
| gsumf1o 19846 | Re-index a finite group su... |
| gsumreidx 19847 | Re-index a finite group su... |
| gsumzsubmcl 19848 | Closure of a group sum in ... |
| gsumsubmcl 19849 | Closure of a group sum in ... |
| gsumsubgcl 19850 | Closure of a group sum in ... |
| gsumzaddlem 19851 | The sum of two group sums.... |
| gsumzadd 19852 | The sum of two group sums.... |
| gsumadd 19853 | The sum of two group sums.... |
| gsummptfsadd 19854 | The sum of two group sums ... |
| gsummptfidmadd 19855 | The sum of two group sums ... |
| gsummptfidmadd2 19856 | The sum of two group sums ... |
| gsumzsplit 19857 | Split a group sum into two... |
| gsumsplit 19858 | Split a group sum into two... |
| gsumsplit2 19859 | Split a group sum into two... |
| gsummptfidmsplit 19860 | Split a group sum expresse... |
| gsummptfidmsplitres 19861 | Split a group sum expresse... |
| gsummptfzsplit 19862 | Split a group sum expresse... |
| gsummptfzsplitl 19863 | Split a group sum expresse... |
| gsumconst 19864 | Sum of a constant series. ... |
| gsumconstf 19865 | Sum of a constant series. ... |
| gsummptshft 19866 | Index shift of a finite gr... |
| gsumzmhm 19867 | Apply a group homomorphism... |
| gsummhm 19868 | Apply a group homomorphism... |
| gsummhm2 19869 | Apply a group homomorphism... |
| gsummptmhm 19870 | Apply a group homomorphism... |
| gsummulglem 19871 | Lemma for ~ gsummulg and ~... |
| gsummulg 19872 | Nonnegative multiple of a ... |
| gsummulgz 19873 | Integer multiple of a grou... |
| gsumzoppg 19874 | The opposite of a group su... |
| gsumzinv 19875 | Inverse of a group sum. (... |
| gsuminv 19876 | Inverse of a group sum. (... |
| gsummptfidminv 19877 | Inverse of a group sum exp... |
| gsumsub 19878 | The difference of two grou... |
| gsummptfssub 19879 | The difference of two grou... |
| gsummptfidmsub 19880 | The difference of two grou... |
| gsumsnfd 19881 | Group sum of a singleton, ... |
| gsumsnd 19882 | Group sum of a singleton, ... |
| gsumsnf 19883 | Group sum of a singleton, ... |
| gsumsn 19884 | Group sum of a singleton. ... |
| gsumpr 19885 | Group sum of a pair. (Con... |
| gsumzunsnd 19886 | Append an element to a fin... |
| gsumunsnfd 19887 | Append an element to a fin... |
| gsumunsnd 19888 | Append an element to a fin... |
| gsumunsnf 19889 | Append an element to a fin... |
| gsumunsn 19890 | Append an element to a fin... |
| gsumdifsnd 19891 | Extract a summand from a f... |
| gsumpt 19892 | Sum of a family that is no... |
| gsummptf1o 19893 | Re-index a finite group su... |
| gsummptun 19894 | Group sum of a disjoint un... |
| gsummpt1n0 19895 | If only one summand in a f... |
| gsummptif1n0 19896 | If only one summand in a f... |
| gsummptcl 19897 | Closure of a finite group ... |
| gsummptfif1o 19898 | Re-index a finite group su... |
| gsummptfzcl 19899 | Closure of a finite group ... |
| gsum2dlem1 19900 | Lemma 1 for ~ gsum2d . (C... |
| gsum2dlem2 19901 | Lemma for ~ gsum2d . (Con... |
| gsum2d 19902 | Write a sum over a two-dim... |
| gsum2d2lem 19903 | Lemma for ~ gsum2d2 : show... |
| gsum2d2 19904 | Write a group sum over a t... |
| gsumcom2 19905 | Two-dimensional commutatio... |
| gsumxp 19906 | Write a group sum over a c... |
| gsumcom 19907 | Commute the arguments of a... |
| gsumcom3 19908 | A commutative law for fini... |
| gsumcom3fi 19909 | A commutative law for fini... |
| gsumxp2 19910 | Write a group sum over a c... |
| prdsgsum 19911 | Finite commutative sums in... |
| pwsgsum 19912 | Finite commutative sums in... |
| fsfnn0gsumfsffz 19913 | Replacing a finitely suppo... |
| nn0gsumfz 19914 | Replacing a finitely suppo... |
| nn0gsumfz0 19915 | Replacing a finitely suppo... |
| gsummptnn0fz 19916 | A final group sum over a f... |
| gsummptnn0fzfv 19917 | A final group sum over a f... |
| telgsumfzslem 19918 | Lemma for ~ telgsumfzs (in... |
| telgsumfzs 19919 | Telescoping group sum rang... |
| telgsumfz 19920 | Telescoping group sum rang... |
| telgsumfz0s 19921 | Telescoping finite group s... |
| telgsumfz0 19922 | Telescoping finite group s... |
| telgsums 19923 | Telescoping finitely suppo... |
| telgsum 19924 | Telescoping finitely suppo... |
| reldmdprd 19929 | The domain of the internal... |
| dmdprd 19930 | The domain of definition o... |
| dmdprdd 19931 | Show that a given family i... |
| dprddomprc 19932 | A family of subgroups inde... |
| dprddomcld 19933 | If a family of subgroups i... |
| dprdval0prc 19934 | The internal direct produc... |
| dprdval 19935 | The value of the internal ... |
| eldprd 19936 | A class ` A ` is an intern... |
| dprdgrp 19937 | Reverse closure for the in... |
| dprdf 19938 | The function ` S ` is a fa... |
| dprdf2 19939 | The function ` S ` is a fa... |
| dprdcntz 19940 | The function ` S ` is a fa... |
| dprddisj 19941 | The function ` S ` is a fa... |
| dprdw 19942 | The property of being a fi... |
| dprdwd 19943 | A mapping being a finitely... |
| dprdff 19944 | A finitely supported funct... |
| dprdfcl 19945 | A finitely supported funct... |
| dprdffsupp 19946 | A finitely supported funct... |
| dprdfcntz 19947 | A function on the elements... |
| dprdssv 19948 | The internal direct produc... |
| dprdfid 19949 | A function mapping all but... |
| eldprdi 19950 | The domain of definition o... |
| dprdfinv 19951 | Take the inverse of a grou... |
| dprdfadd 19952 | Take the sum of group sums... |
| dprdfsub 19953 | Take the difference of gro... |
| dprdfeq0 19954 | The zero function is the o... |
| dprdf11 19955 | Two group sums over a dire... |
| dprdsubg 19956 | The internal direct produc... |
| dprdub 19957 | Each factor is a subset of... |
| dprdlub 19958 | The direct product is smal... |
| dprdspan 19959 | The direct product is the ... |
| dprdres 19960 | Restriction of a direct pr... |
| dprdss 19961 | Create a direct product by... |
| dprdz 19962 | A family consisting entire... |
| dprd0 19963 | The empty family is an int... |
| dprdf1o 19964 | Rearrange the index set of... |
| dprdf1 19965 | Rearrange the index set of... |
| subgdmdprd 19966 | A direct product in a subg... |
| subgdprd 19967 | A direct product in a subg... |
| dprdsn 19968 | A singleton family is an i... |
| dmdprdsplitlem 19969 | Lemma for ~ dmdprdsplit . ... |
| dprdcntz2 19970 | The function ` S ` is a fa... |
| dprddisj2 19971 | The function ` S ` is a fa... |
| dprd2dlem2 19972 | The direct product of a co... |
| dprd2dlem1 19973 | The direct product of a co... |
| dprd2da 19974 | The direct product of a co... |
| dprd2db 19975 | The direct product of a co... |
| dprd2d2 19976 | The direct product of a co... |
| dmdprdsplit2lem 19977 | Lemma for ~ dmdprdsplit . ... |
| dmdprdsplit2 19978 | The direct product splits ... |
| dmdprdsplit 19979 | The direct product splits ... |
| dprdsplit 19980 | The direct product is the ... |
| dmdprdpr 19981 | A singleton family is an i... |
| dprdpr 19982 | A singleton family is an i... |
| dpjlem 19983 | Lemma for theorems about d... |
| dpjcntz 19984 | The two subgroups that app... |
| dpjdisj 19985 | The two subgroups that app... |
| dpjlsm 19986 | The two subgroups that app... |
| dpjfval 19987 | Value of the direct produc... |
| dpjval 19988 | Value of the direct produc... |
| dpjf 19989 | The ` X ` -th index projec... |
| dpjidcl 19990 | The key property of projec... |
| dpjeq 19991 | Decompose a group sum into... |
| dpjid 19992 | The key property of projec... |
| dpjlid 19993 | The ` X ` -th index projec... |
| dpjrid 19994 | The ` Y ` -th index projec... |
| dpjghm 19995 | The direct product is the ... |
| dpjghm2 19996 | The direct product is the ... |
| ablfacrplem 19997 | Lemma for ~ ablfacrp2 . (... |
| ablfacrp 19998 | A finite abelian group who... |
| ablfacrp2 19999 | The factors ` K , L ` of ~... |
| ablfac1lem 20000 | Lemma for ~ ablfac1b . Sa... |
| ablfac1a 20001 | The factors of ~ ablfac1b ... |
| ablfac1b 20002 | Any abelian group is the d... |
| ablfac1c 20003 | The factors of ~ ablfac1b ... |
| ablfac1eulem 20004 | Lemma for ~ ablfac1eu . (... |
| ablfac1eu 20005 | The factorization of ~ abl... |
| pgpfac1lem1 20006 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem2 20007 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3a 20008 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3 20009 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem4 20010 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem5 20011 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1 20012 | Factorization of a finite ... |
| pgpfaclem1 20013 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem2 20014 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem3 20015 | Lemma for ~ pgpfac . (Con... |
| pgpfac 20016 | Full factorization of a fi... |
| ablfaclem1 20017 | Lemma for ~ ablfac . (Con... |
| ablfaclem2 20018 | Lemma for ~ ablfac . (Con... |
| ablfaclem3 20019 | Lemma for ~ ablfac . (Con... |
| ablfac 20020 | The Fundamental Theorem of... |
| ablfac2 20021 | Choose generators for each... |
| issimpg 20024 | The predicate "is a simple... |
| issimpgd 20025 | Deduce a simple group from... |
| simpggrp 20026 | A simple group is a group.... |
| simpggrpd 20027 | A simple group is a group.... |
| simpg2nsg 20028 | A simple group has two nor... |
| trivnsimpgd 20029 | Trivial groups are not sim... |
| simpgntrivd 20030 | Simple groups are nontrivi... |
| simpgnideld 20031 | A simple group contains a ... |
| simpgnsgd 20032 | The only normal subgroups ... |
| simpgnsgeqd 20033 | A normal subgroup of a sim... |
| 2nsgsimpgd 20034 | If any normal subgroup of ... |
| simpgnsgbid 20035 | A nontrivial group is simp... |
| ablsimpnosubgd 20036 | A subgroup of an abelian s... |
| ablsimpg1gend 20037 | An abelian simple group is... |
| ablsimpgcygd 20038 | An abelian simple group is... |
| ablsimpgfindlem1 20039 | Lemma for ~ ablsimpgfind .... |
| ablsimpgfindlem2 20040 | Lemma for ~ ablsimpgfind .... |
| cycsubggenodd 20041 | Relationship between the o... |
| ablsimpgfind 20042 | An abelian simple group is... |
| fincygsubgd 20043 | The subgroup referenced in... |
| fincygsubgodd 20044 | Calculate the order of a s... |
| fincygsubgodexd 20045 | A finite cyclic group has ... |
| prmgrpsimpgd 20046 | A group of prime order is ... |
| ablsimpgprmd 20047 | An abelian simple group ha... |
| ablsimpgd 20048 | An abelian group is simple... |
| fnmgp 20051 | The multiplicative group o... |
| mgpval 20052 | Value of the multiplicatio... |
| mgpplusg 20053 | Value of the group operati... |
| mgpbas 20054 | Base set of the multiplica... |
| mgpsca 20055 | The multiplication monoid ... |
| mgptset 20056 | Topology component of the ... |
| mgptopn 20057 | Topology of the multiplica... |
| mgpds 20058 | Distance function of the m... |
| mgpress 20059 | Subgroup commutes with the... |
| prdsmgp 20060 | The multiplicative monoid ... |
| isrng 20063 | The predicate "is a non-un... |
| rngabl 20064 | A non-unital ring is an (a... |
| rngmgp 20065 | A non-unital ring is a sem... |
| rngmgpf 20066 | Restricted functionality o... |
| rnggrp 20067 | A non-unital ring is a (ad... |
| rngass 20068 | Associative law for the mu... |
| rngdi 20069 | Distributive law for the m... |
| rngdir 20070 | Distributive law for the m... |
| rngacl 20071 | Closure of the addition op... |
| rng0cl 20072 | The zero element of a non-... |
| rngcl 20073 | Closure of the multiplicat... |
| rnglz 20074 | The zero of a non-unital r... |
| rngrz 20075 | The zero of a non-unital r... |
| rngmneg1 20076 | Negation of a product in a... |
| rngmneg2 20077 | Negation of a product in a... |
| rngm2neg 20078 | Double negation of a produ... |
| rngansg 20079 | Every additive subgroup of... |
| rngsubdi 20080 | Ring multiplication distri... |
| rngsubdir 20081 | Ring multiplication distri... |
| isrngd 20082 | Properties that determine ... |
| rngpropd 20083 | If two structures have the... |
| prdsmulrngcl 20084 | Closure of the multiplicat... |
| prdsrngd 20085 | A product of non-unital ri... |
| imasrng 20086 | The image structure of a n... |
| imasrngf1 20087 | The image of a non-unital ... |
| xpsrngd 20088 | A product of two non-unita... |
| qusrng 20089 | The quotient structure of ... |
| ringidval 20092 | The value of the unity ele... |
| dfur2 20093 | The multiplicative identit... |
| ringurd 20094 | Deduce the unity element o... |
| issrg 20097 | The predicate "is a semiri... |
| srgcmn 20098 | A semiring is a commutativ... |
| srgmnd 20099 | A semiring is a monoid. (... |
| srgmgp 20100 | A semiring is a monoid und... |
| srgdilem 20101 | Lemma for ~ srgdi and ~ sr... |
| srgcl 20102 | Closure of the multiplicat... |
| srgass 20103 | Associative law for the mu... |
| srgideu 20104 | The unity element of a sem... |
| srgfcl 20105 | Functionality of the multi... |
| srgdi 20106 | Distributive law for the m... |
| srgdir 20107 | Distributive law for the m... |
| srgidcl 20108 | The unity element of a sem... |
| srg0cl 20109 | The zero element of a semi... |
| srgidmlem 20110 | Lemma for ~ srglidm and ~ ... |
| srglidm 20111 | The unity element of a sem... |
| srgridm 20112 | The unity element of a sem... |
| issrgid 20113 | Properties showing that an... |
| srgacl 20114 | Closure of the addition op... |
| srgcom 20115 | Commutativity of the addit... |
| srgrz 20116 | The zero of a semiring is ... |
| srglz 20117 | The zero of a semiring is ... |
| srgisid 20118 | In a semiring, the only le... |
| o2timesd 20119 | An element of a ring-like ... |
| rglcom4d 20120 | Restricted commutativity o... |
| srgo2times 20121 | A semiring element plus it... |
| srgcom4lem 20122 | Lemma for ~ srgcom4 . Thi... |
| srgcom4 20123 | Restricted commutativity o... |
| srg1zr 20124 | The only semiring with a b... |
| srgen1zr 20125 | The only semiring with one... |
| srgmulgass 20126 | An associative property be... |
| srgpcomp 20127 | If two elements of a semir... |
| srgpcompp 20128 | If two elements of a semir... |
| srgpcomppsc 20129 | If two elements of a semir... |
| srglmhm 20130 | Left-multiplication in a s... |
| srgrmhm 20131 | Right-multiplication in a ... |
| srgsummulcr 20132 | A finite semiring sum mult... |
| sgsummulcl 20133 | A finite semiring sum mult... |
| srg1expzeq1 20134 | The exponentiation (by a n... |
| srgbinomlem1 20135 | Lemma 1 for ~ srgbinomlem ... |
| srgbinomlem2 20136 | Lemma 2 for ~ srgbinomlem ... |
| srgbinomlem3 20137 | Lemma 3 for ~ srgbinomlem ... |
| srgbinomlem4 20138 | Lemma 4 for ~ srgbinomlem ... |
| srgbinomlem 20139 | Lemma for ~ srgbinom . In... |
| srgbinom 20140 | The binomial theorem for c... |
| csrgbinom 20141 | The binomial theorem for c... |
| isring 20146 | The predicate "is a (unita... |
| ringgrp 20147 | A ring is a group. (Contr... |
| ringmgp 20148 | A ring is a monoid under m... |
| iscrng 20149 | A commutative ring is a ri... |
| crngmgp 20150 | A commutative ring's multi... |
| ringgrpd 20151 | A ring is a group. (Contr... |
| ringmnd 20152 | A ring is a monoid under a... |
| ringmgm 20153 | A ring is a magma. (Contr... |
| crngring 20154 | A commutative ring is a ri... |
| crngringd 20155 | A commutative ring is a ri... |
| crnggrpd 20156 | A commutative ring is a gr... |
| mgpf 20157 | Restricted functionality o... |
| ringdilem 20158 | Properties of a unital rin... |
| ringcl 20159 | Closure of the multiplicat... |
| crngcom 20160 | A commutative ring's multi... |
| iscrng2 20161 | A commutative ring is a ri... |
| ringass 20162 | Associative law for multip... |
| ringideu 20163 | The unity element of a rin... |
| crngcomd 20164 | Multiplication is commutat... |
| crngbascntr 20165 | The base set of a commutat... |
| ringassd 20166 | Associative law for multip... |
| crng12d 20167 | Commutative/associative la... |
| crng32d 20168 | Commutative/associative la... |
| ringcld 20169 | Closure of the multiplicat... |
| ringdi 20170 | Distributive law for the m... |
| ringdir 20171 | Distributive law for the m... |
| ringdid 20172 | Distributive law for the m... |
| ringdird 20173 | Distributive law for the m... |
| ringidcl 20174 | The unity element of a rin... |
| ringidcld 20175 | The unity element of a rin... |
| ring0cl 20176 | The zero element of a ring... |
| ringidmlem 20177 | Lemma for ~ ringlidm and ~... |
| ringlidm 20178 | The unity element of a rin... |
| ringridm 20179 | The unity element of a rin... |
| isringid 20180 | Properties showing that an... |
| ringlidmd 20181 | The unity element of a rin... |
| ringridmd 20182 | The unity element of a rin... |
| ringid 20183 | The multiplication operati... |
| ringo2times 20184 | A ring element plus itself... |
| ringadd2 20185 | A ring element plus itself... |
| ringidss 20186 | A subset of the multiplica... |
| ringacl 20187 | Closure of the addition op... |
| ringcomlem 20188 | Lemma for ~ ringcom . Thi... |
| ringcom 20189 | Commutativity of the addit... |
| ringabl 20190 | A ring is an Abelian group... |
| ringcmn 20191 | A ring is a commutative mo... |
| ringabld 20192 | A ring is an Abelian group... |
| ringcmnd 20193 | A ring is a commutative mo... |
| ringrng 20194 | A unital ring is a non-uni... |
| ringssrng 20195 | The unital rings are non-u... |
| isringrng 20196 | The predicate "is a unital... |
| ringpropd 20197 | If two structures have the... |
| crngpropd 20198 | If two structures have the... |
| ringprop 20199 | If two structures have the... |
| isringd 20200 | Properties that determine ... |
| iscrngd 20201 | Properties that determine ... |
| ringlz 20202 | The zero of a unital ring ... |
| ringrz 20203 | The zero of a unital ring ... |
| ringlzd 20204 | The zero of a unital ring ... |
| ringrzd 20205 | The zero of a unital ring ... |
| ringsrg 20206 | Any ring is also a semirin... |
| ring1eq0 20207 | If one and zero are equal,... |
| ring1ne0 20208 | If a ring has at least two... |
| ringinvnz1ne0 20209 | In a unital ring, a left i... |
| ringinvnzdiv 20210 | In a unital ring, a left i... |
| ringnegl 20211 | Negation in a ring is the ... |
| ringnegr 20212 | Negation in a ring is the ... |
| ringmneg1 20213 | Negation of a product in a... |
| ringmneg2 20214 | Negation of a product in a... |
| ringm2neg 20215 | Double negation of a produ... |
| ringsubdi 20216 | Ring multiplication distri... |
| ringsubdir 20217 | Ring multiplication distri... |
| mulgass2 20218 | An associative property be... |
| ring1 20219 | The (smallest) structure r... |
| ringn0 20220 | Rings exist. (Contributed... |
| ringlghm 20221 | Left-multiplication in a r... |
| ringrghm 20222 | Right-multiplication in a ... |
| gsummulc1OLD 20223 | Obsolete version of ~ gsum... |
| gsummulc2OLD 20224 | Obsolete version of ~ gsum... |
| gsummulc1 20225 | A finite ring sum multipli... |
| gsummulc2 20226 | A finite ring sum multipli... |
| gsummgp0 20227 | If one factor in a finite ... |
| gsumdixp 20228 | Distribute a binary produc... |
| prdsmulrcl 20229 | A structure product of rin... |
| prdsringd 20230 | A product of rings is a ri... |
| prdscrngd 20231 | A product of commutative r... |
| prds1 20232 | Value of the ring unity in... |
| pwsring 20233 | A structure power of a rin... |
| pws1 20234 | Value of the ring unity in... |
| pwscrng 20235 | A structure power of a com... |
| pwsmgp 20236 | The multiplicative group o... |
| pwspjmhmmgpd 20237 | The projection given by ~ ... |
| pwsexpg 20238 | Value of a group exponenti... |
| imasring 20239 | The image structure of a r... |
| imasringf1 20240 | The image of a ring under ... |
| xpsringd 20241 | A product of two rings is ... |
| xpsring1d 20242 | The multiplicative identit... |
| qusring2 20243 | The quotient structure of ... |
| crngbinom 20244 | The binomial theorem for c... |
| opprval 20247 | Value of the opposite ring... |
| opprmulfval 20248 | Value of the multiplicatio... |
| opprmul 20249 | Value of the multiplicatio... |
| crngoppr 20250 | In a commutative ring, the... |
| opprlem 20251 | Lemma for ~ opprbas and ~ ... |
| opprbas 20252 | Base set of an opposite ri... |
| oppradd 20253 | Addition operation of an o... |
| opprrng 20254 | An opposite non-unital rin... |
| opprrngb 20255 | A class is a non-unital ri... |
| opprring 20256 | An opposite ring is a ring... |
| opprringb 20257 | Bidirectional form of ~ op... |
| oppr0 20258 | Additive identity of an op... |
| oppr1 20259 | Multiplicative identity of... |
| opprneg 20260 | The negative function in a... |
| opprsubg 20261 | Being a subgroup is a symm... |
| mulgass3 20262 | An associative property be... |
| reldvdsr 20269 | The divides relation is a ... |
| dvdsrval 20270 | Value of the divides relat... |
| dvdsr 20271 | Value of the divides relat... |
| dvdsr2 20272 | Value of the divides relat... |
| dvdsrmul 20273 | A left-multiple of ` X ` i... |
| dvdsrcl 20274 | Closure of a dividing elem... |
| dvdsrcl2 20275 | Closure of a dividing elem... |
| dvdsrid 20276 | An element in a (unital) r... |
| dvdsrtr 20277 | Divisibility is transitive... |
| dvdsrmul1 20278 | The divisibility relation ... |
| dvdsrneg 20279 | An element divides its neg... |
| dvdsr01 20280 | In a ring, zero is divisib... |
| dvdsr02 20281 | Only zero is divisible by ... |
| isunit 20282 | Property of being a unit o... |
| 1unit 20283 | The multiplicative identit... |
| unitcl 20284 | A unit is an element of th... |
| unitss 20285 | The set of units is contai... |
| opprunit 20286 | Being a unit is a symmetri... |
| crngunit 20287 | Property of being a unit i... |
| dvdsunit 20288 | A divisor of a unit is a u... |
| unitmulcl 20289 | The product of units is a ... |
| unitmulclb 20290 | Reversal of ~ unitmulcl in... |
| unitgrpbas 20291 | The base set of the group ... |
| unitgrp 20292 | The group of units is a gr... |
| unitabl 20293 | The group of units of a co... |
| unitgrpid 20294 | The identity of the group ... |
| unitsubm 20295 | The group of units is a su... |
| invrfval 20298 | Multiplicative inverse fun... |
| unitinvcl 20299 | The inverse of a unit exis... |
| unitinvinv 20300 | The inverse of the inverse... |
| ringinvcl 20301 | The inverse of a unit is a... |
| unitlinv 20302 | A unit times its inverse i... |
| unitrinv 20303 | A unit times its inverse i... |
| 1rinv 20304 | The inverse of the ring un... |
| 0unit 20305 | The additive identity is a... |
| unitnegcl 20306 | The negative of a unit is ... |
| ringunitnzdiv 20307 | In a unitary ring, a unit ... |
| ring1nzdiv 20308 | In a unitary ring, the rin... |
| dvrfval 20311 | Division operation in a ri... |
| dvrval 20312 | Division operation in a ri... |
| dvrcl 20313 | Closure of division operat... |
| unitdvcl 20314 | The units are closed under... |
| dvrid 20315 | A ring element divided by ... |
| dvr1 20316 | A ring element divided by ... |
| dvrass 20317 | An associative law for div... |
| dvrcan1 20318 | A cancellation law for div... |
| dvrcan3 20319 | A cancellation law for div... |
| dvreq1 20320 | Equality in terms of ratio... |
| dvrdir 20321 | Distributive law for the d... |
| rdivmuldivd 20322 | Multiplication of two rati... |
| ringinvdv 20323 | Write the inverse function... |
| rngidpropd 20324 | The ring unity depends onl... |
| dvdsrpropd 20325 | The divisibility relation ... |
| unitpropd 20326 | The set of units depends o... |
| invrpropd 20327 | The ring inverse function ... |
| isirred 20328 | An irreducible element of ... |
| isnirred 20329 | The property of being a no... |
| isirred2 20330 | Expand out the class diffe... |
| opprirred 20331 | Irreducibility is symmetri... |
| irredn0 20332 | The additive identity is n... |
| irredcl 20333 | An irreducible element is ... |
| irrednu 20334 | An irreducible element is ... |
| irredn1 20335 | The multiplicative identit... |
| irredrmul 20336 | The product of an irreduci... |
| irredlmul 20337 | The product of a unit and ... |
| irredmul 20338 | If product of two elements... |
| irredneg 20339 | The negative of an irreduc... |
| irrednegb 20340 | An element is irreducible ... |
| rnghmrcl 20347 | Reverse closure of a non-u... |
| rnghmfn 20348 | The mapping of two non-uni... |
| rnghmval 20349 | The set of the non-unital ... |
| isrnghm 20350 | A function is a non-unital... |
| isrnghmmul 20351 | A function is a non-unital... |
| rnghmmgmhm 20352 | A non-unital ring homomorp... |
| rnghmval2 20353 | The non-unital ring homomo... |
| isrngim 20354 | An isomorphism of non-unit... |
| rngimrcl 20355 | Reverse closure for an iso... |
| rnghmghm 20356 | A non-unital ring homomorp... |
| rnghmf 20357 | A ring homomorphism is a f... |
| rnghmmul 20358 | A homomorphism of non-unit... |
| isrnghm2d 20359 | Demonstration of non-unita... |
| isrnghmd 20360 | Demonstration of non-unita... |
| rnghmf1o 20361 | A non-unital ring homomorp... |
| isrngim2 20362 | An isomorphism of non-unit... |
| rngimf1o 20363 | An isomorphism of non-unit... |
| rngimrnghm 20364 | An isomorphism of non-unit... |
| rngimcnv 20365 | The converse of an isomorp... |
| rnghmco 20366 | The composition of non-uni... |
| idrnghm 20367 | The identity homomorphism ... |
| c0mgm 20368 | The constant mapping to ze... |
| c0mhm 20369 | The constant mapping to ze... |
| c0ghm 20370 | The constant mapping to ze... |
| c0snmgmhm 20371 | The constant mapping to ze... |
| c0snmhm 20372 | The constant mapping to ze... |
| c0snghm 20373 | The constant mapping to ze... |
| rngisomfv1 20374 | If there is a non-unital r... |
| rngisom1 20375 | If there is a non-unital r... |
| rngisomring 20376 | If there is a non-unital r... |
| rngisomring1 20377 | If there is a non-unital r... |
| dfrhm2 20383 | The property of a ring hom... |
| rhmrcl1 20385 | Reverse closure of a ring ... |
| rhmrcl2 20386 | Reverse closure of a ring ... |
| isrhm 20387 | A function is a ring homom... |
| rhmmhm 20388 | A ring homomorphism is a h... |
| rhmisrnghm 20389 | Each unital ring homomorph... |
| isrim0OLD 20390 | Obsolete version of ~ isri... |
| rimrcl 20391 | Reverse closure for an iso... |
| isrim0 20392 | A ring isomorphism is a ho... |
| rhmghm 20393 | A ring homomorphism is an ... |
| rhmf 20394 | A ring homomorphism is a f... |
| rhmmul 20395 | A homomorphism of rings pr... |
| isrhm2d 20396 | Demonstration of ring homo... |
| isrhmd 20397 | Demonstration of ring homo... |
| rhm1 20398 | Ring homomorphisms are req... |
| idrhm 20399 | The identity homomorphism ... |
| rhmf1o 20400 | A ring homomorphism is bij... |
| isrim 20401 | An isomorphism of rings is... |
| isrimOLD 20402 | Obsolete version of ~ isri... |
| rimf1o 20403 | An isomorphism of rings is... |
| rimrhmOLD 20404 | Obsolete version of ~ rimr... |
| rimrhm 20405 | A ring isomorphism is a ho... |
| rimgim 20406 | An isomorphism of rings is... |
| rimisrngim 20407 | Each unital ring isomorphi... |
| rhmfn 20408 | The mapping of two rings t... |
| rhmval 20409 | The ring homomorphisms bet... |
| rhmco 20410 | The composition of ring ho... |
| pwsco1rhm 20411 | Right composition with a f... |
| pwsco2rhm 20412 | Left composition with a ri... |
| brric 20413 | The relation "is isomorphi... |
| brrici 20414 | Prove isomorphic by an exp... |
| brric2 20415 | The relation "is isomorphi... |
| ricgic 20416 | If two rings are (ring) is... |
| rhmdvdsr 20417 | A ring homomorphism preser... |
| rhmopp 20418 | A ring homomorphism is als... |
| elrhmunit 20419 | Ring homomorphisms preserv... |
| rhmunitinv 20420 | Ring homomorphisms preserv... |
| isnzr 20423 | Property of a nonzero ring... |
| nzrnz 20424 | One and zero are different... |
| nzrring 20425 | A nonzero ring is a ring. ... |
| nzrringOLD 20426 | Obsolete version of ~ nzrr... |
| isnzr2 20427 | Equivalent characterizatio... |
| isnzr2hash 20428 | Equivalent characterizatio... |
| nzrpropd 20429 | If two structures have the... |
| opprnzrb 20430 | The opposite of a nonzero ... |
| opprnzr 20431 | The opposite of a nonzero ... |
| ringelnzr 20432 | A ring is nonzero if it ha... |
| nzrunit 20433 | A unit is nonzero in any n... |
| 0ringnnzr 20434 | A ring is a zero ring iff ... |
| 0ring 20435 | If a ring has only one ele... |
| 0ringdif 20436 | A zero ring is a ring whic... |
| 0ringbas 20437 | The base set of a zero rin... |
| 0ring01eq 20438 | In a ring with only one el... |
| 01eq0ring 20439 | If the zero and the identi... |
| 01eq0ringOLD 20440 | Obsolete version of ~ 01eq... |
| 0ring01eqbi 20441 | In a unital ring the zero ... |
| 0ring1eq0 20442 | In a zero ring, a ring whi... |
| c0rhm 20443 | The constant mapping to ze... |
| c0rnghm 20444 | The constant mapping to ze... |
| zrrnghm 20445 | The constant mapping to ze... |
| nrhmzr 20446 | There is no ring homomorph... |
| islring 20449 | The predicate "is a local ... |
| lringnzr 20450 | A local ring is a nonzero ... |
| lringring 20451 | A local ring is a ring. (... |
| lringnz 20452 | A local ring is a nonzero ... |
| lringuplu 20453 | If the sum of two elements... |
| issubrng 20456 | The subring of non-unital ... |
| subrngss 20457 | A subring is a subset. (C... |
| subrngid 20458 | Every non-unital ring is a... |
| subrngrng 20459 | A subring is a non-unital ... |
| subrngrcl 20460 | Reverse closure for a subr... |
| subrngsubg 20461 | A subring is a subgroup. ... |
| subrngringnsg 20462 | A subring is a normal subg... |
| subrngbas 20463 | Base set of a subring stru... |
| subrng0 20464 | A subring always has the s... |
| subrngacl 20465 | A subring is closed under ... |
| subrngmcl 20466 | A subring is closed under ... |
| issubrng2 20467 | Characterize the subrings ... |
| opprsubrng 20468 | Being a subring is a symme... |
| subrngint 20469 | The intersection of a none... |
| subrngin 20470 | The intersection of two su... |
| subrngmre 20471 | The subrings of a non-unit... |
| subsubrng 20472 | A subring of a subring is ... |
| subsubrng2 20473 | The set of subrings of a s... |
| rhmimasubrnglem 20474 | Lemma for ~ rhmimasubrng :... |
| rhmimasubrng 20475 | The homomorphic image of a... |
| cntzsubrng 20476 | Centralizers in a non-unit... |
| subrngpropd 20477 | If two structures have the... |
| issubrg 20480 | The subring predicate. (C... |
| subrgss 20481 | A subring is a subset. (C... |
| subrgid 20482 | Every ring is a subring of... |
| subrgring 20483 | A subring is a ring. (Con... |
| subrgcrng 20484 | A subring of a commutative... |
| subrgrcl 20485 | Reverse closure for a subr... |
| subrgsubg 20486 | A subring is a subgroup. ... |
| subrgsubrng 20487 | A subring of a unital ring... |
| subrg0 20488 | A subring always has the s... |
| subrg1cl 20489 | A subring contains the mul... |
| subrgbas 20490 | Base set of a subring stru... |
| subrg1 20491 | A subring always has the s... |
| subrgacl 20492 | A subring is closed under ... |
| subrgmcl 20493 | A subring is closed under ... |
| subrgsubm 20494 | A subring is a submonoid o... |
| subrgdvds 20495 | If an element divides anot... |
| subrguss 20496 | A unit of a subring is a u... |
| subrginv 20497 | A subring always has the s... |
| subrgdv 20498 | A subring always has the s... |
| subrgunit 20499 | An element of a ring is a ... |
| subrgugrp 20500 | The units of a subring for... |
| issubrg2 20501 | Characterize the subrings ... |
| opprsubrg 20502 | Being a subring is a symme... |
| subrgnzr 20503 | A subring of a nonzero rin... |
| subrgint 20504 | The intersection of a none... |
| subrgin 20505 | The intersection of two su... |
| subrgmre 20506 | The subrings of a ring are... |
| subsubrg 20507 | A subring of a subring is ... |
| subsubrg2 20508 | The set of subrings of a s... |
| issubrg3 20509 | A subring is an additive s... |
| resrhm 20510 | Restriction of a ring homo... |
| resrhm2b 20511 | Restriction of the codomai... |
| rhmeql 20512 | The equalizer of two ring ... |
| rhmima 20513 | The homomorphic image of a... |
| rnrhmsubrg 20514 | The range of a ring homomo... |
| cntzsubr 20515 | Centralizers in a ring are... |
| pwsdiagrhm 20516 | Diagonal homomorphism into... |
| subrgpropd 20517 | If two structures have the... |
| rhmpropd 20518 | Ring homomorphism depends ... |
| rgspnval 20521 | Value of the ring-span of ... |
| rgspncl 20522 | The ring-span of a set is ... |
| rgspnssid 20523 | The ring-span of a set con... |
| rgspnmin 20524 | The ring-span is contained... |
| rngcval 20527 | Value of the category of n... |
| rnghmresfn 20528 | The class of non-unital ri... |
| rnghmresel 20529 | An element of the non-unit... |
| rngcbas 20530 | Set of objects of the cate... |
| rngchomfval 20531 | Set of arrows of the categ... |
| rngchom 20532 | Set of arrows of the categ... |
| elrngchom 20533 | A morphism of non-unital r... |
| rngchomfeqhom 20534 | The functionalized Hom-set... |
| rngccofval 20535 | Composition in the categor... |
| rngcco 20536 | Composition in the categor... |
| dfrngc2 20537 | Alternate definition of th... |
| rnghmsscmap2 20538 | The non-unital ring homomo... |
| rnghmsscmap 20539 | The non-unital ring homomo... |
| rnghmsubcsetclem1 20540 | Lemma 1 for ~ rnghmsubcset... |
| rnghmsubcsetclem2 20541 | Lemma 2 for ~ rnghmsubcset... |
| rnghmsubcsetc 20542 | The non-unital ring homomo... |
| rngccat 20543 | The category of non-unital... |
| rngcid 20544 | The identity arrow in the ... |
| rngcsect 20545 | A section in the category ... |
| rngcinv 20546 | An inverse in the category... |
| rngciso 20547 | An isomorphism in the cate... |
| rngcifuestrc 20548 | The "inclusion functor" fr... |
| funcrngcsetc 20549 | The "natural forgetful fun... |
| funcrngcsetcALT 20550 | Alternate proof of ~ funcr... |
| zrinitorngc 20551 | The zero ring is an initia... |
| zrtermorngc 20552 | The zero ring is a termina... |
| zrzeroorngc 20553 | The zero ring is a zero ob... |
| ringcval 20556 | Value of the category of u... |
| rhmresfn 20557 | The class of unital ring h... |
| rhmresel 20558 | An element of the unital r... |
| ringcbas 20559 | Set of objects of the cate... |
| ringchomfval 20560 | Set of arrows of the categ... |
| ringchom 20561 | Set of arrows of the categ... |
| elringchom 20562 | A morphism of unital rings... |
| ringchomfeqhom 20563 | The functionalized Hom-set... |
| ringccofval 20564 | Composition in the categor... |
| ringcco 20565 | Composition in the categor... |
| dfringc2 20566 | Alternate definition of th... |
| rhmsscmap2 20567 | The unital ring homomorphi... |
| rhmsscmap 20568 | The unital ring homomorphi... |
| rhmsubcsetclem1 20569 | Lemma 1 for ~ rhmsubcsetc ... |
| rhmsubcsetclem2 20570 | Lemma 2 for ~ rhmsubcsetc ... |
| rhmsubcsetc 20571 | The unital ring homomorphi... |
| ringccat 20572 | The category of unital rin... |
| ringcid 20573 | The identity arrow in the ... |
| rhmsscrnghm 20574 | The unital ring homomorphi... |
| rhmsubcrngclem1 20575 | Lemma 1 for ~ rhmsubcrngc ... |
| rhmsubcrngclem2 20576 | Lemma 2 for ~ rhmsubcrngc ... |
| rhmsubcrngc 20577 | The unital ring homomorphi... |
| rngcresringcat 20578 | The restriction of the cat... |
| ringcsect 20579 | A section in the category ... |
| ringcinv 20580 | An inverse in the category... |
| ringciso 20581 | An isomorphism in the cate... |
| ringcbasbas 20582 | An element of the base set... |
| funcringcsetc 20583 | The "natural forgetful fun... |
| zrtermoringc 20584 | The zero ring is a termina... |
| zrninitoringc 20585 | The zero ring is not an in... |
| srhmsubclem1 20586 | Lemma 1 for ~ srhmsubc . ... |
| srhmsubclem2 20587 | Lemma 2 for ~ srhmsubc . ... |
| srhmsubclem3 20588 | Lemma 3 for ~ srhmsubc . ... |
| srhmsubc 20589 | According to ~ df-subc , t... |
| sringcat 20590 | The restriction of the cat... |
| crhmsubc 20591 | According to ~ df-subc , t... |
| cringcat 20592 | The restriction of the cat... |
| rngcrescrhm 20593 | The category of non-unital... |
| rhmsubclem1 20594 | Lemma 1 for ~ rhmsubc . (... |
| rhmsubclem2 20595 | Lemma 2 for ~ rhmsubc . (... |
| rhmsubclem3 20596 | Lemma 3 for ~ rhmsubc . (... |
| rhmsubclem4 20597 | Lemma 4 for ~ rhmsubc . (... |
| rhmsubc 20598 | According to ~ df-subc , t... |
| rhmsubccat 20599 | The restriction of the cat... |
| rrgval 20606 | Value of the set or left-r... |
| isrrg 20607 | Membership in the set of l... |
| rrgeq0i 20608 | Property of a left-regular... |
| rrgeq0 20609 | Left-multiplication by a l... |
| rrgsupp 20610 | Left multiplication by a l... |
| rrgss 20611 | Left-regular elements are ... |
| unitrrg 20612 | Units are regular elements... |
| rrgnz 20613 | In a nonzero ring, the zer... |
| isdomn 20614 | Expand definition of a dom... |
| domnnzr 20615 | A domain is a nonzero ring... |
| domnring 20616 | A domain is a ring. (Cont... |
| domneq0 20617 | In a domain, a product is ... |
| domnmuln0 20618 | In a domain, a product of ... |
| isdomn5 20619 | The equivalence between th... |
| isdomn2 20620 | A ring is a domain iff all... |
| isdomn2OLD 20621 | Obsolete version of ~ isdo... |
| domnrrg 20622 | In a domain, a nonzero ele... |
| isdomn6 20623 | A ring is a domain iff the... |
| isdomn3 20624 | Nonzero elements form a mu... |
| isdomn4 20625 | A ring is a domain iff it ... |
| opprdomnb 20626 | A class is a domain if and... |
| opprdomn 20627 | The opposite of a domain i... |
| isdomn4r 20628 | A ring is a domain iff it ... |
| domnlcanb 20629 | Left-cancellation law for ... |
| domnlcan 20630 | Left-cancellation law for ... |
| domnrcanb 20631 | Right-cancellation law for... |
| domnrcan 20632 | Right-cancellation law for... |
| domneq0r 20633 | Right multiplication by a ... |
| isidom 20634 | An integral domain is a co... |
| idomdomd 20635 | An integral domain is a do... |
| idomcringd 20636 | An integral domain is a co... |
| idomringd 20637 | An integral domain is a ri... |
| isdrng 20642 | The predicate "is a divisi... |
| drngunit 20643 | Elementhood in the set of ... |
| drngui 20644 | The set of units of a divi... |
| drngring 20645 | A division ring is a ring.... |
| drngringd 20646 | A division ring is a ring.... |
| drnggrpd 20647 | A division ring is a group... |
| drnggrp 20648 | A division ring is a group... |
| isfld 20649 | A field is a commutative d... |
| flddrngd 20650 | A field is a division ring... |
| fldcrngd 20651 | A field is a commutative r... |
| isdrng2 20652 | A division ring can equiva... |
| drngprop 20653 | If two structures have the... |
| drngmgp 20654 | A division ring contains a... |
| drngid 20655 | A division ring's unity is... |
| drngunz 20656 | A division ring's unity is... |
| drngnzr 20657 | A division ring is a nonze... |
| drngdomn 20658 | A division ring is a domai... |
| drngmcl 20659 | The product of two nonzero... |
| drngmclOLD 20660 | Obsolete version of ~ drng... |
| drngid2 20661 | Properties showing that an... |
| drnginvrcl 20662 | Closure of the multiplicat... |
| drnginvrn0 20663 | The multiplicative inverse... |
| drnginvrcld 20664 | Closure of the multiplicat... |
| drnginvrl 20665 | Property of the multiplica... |
| drnginvrr 20666 | Property of the multiplica... |
| drnginvrld 20667 | Property of the multiplica... |
| drnginvrrd 20668 | Property of the multiplica... |
| drngmul0or 20669 | A product is zero iff one ... |
| drngmul0orOLD 20670 | Obsolete version of ~ drng... |
| drngmulne0 20671 | A product is nonzero iff b... |
| drngmuleq0 20672 | An element is zero iff its... |
| opprdrng 20673 | The opposite of a division... |
| isdrngd 20674 | Properties that characteri... |
| isdrngrd 20675 | Properties that characteri... |
| isdrngdOLD 20676 | Obsolete version of ~ isdr... |
| isdrngrdOLD 20677 | Obsolete version of ~ isdr... |
| drngpropd 20678 | If two structures have the... |
| fldpropd 20679 | If two structures have the... |
| fldidom 20680 | A field is an integral dom... |
| fidomndrnglem 20681 | Lemma for ~ fidomndrng . ... |
| fidomndrng 20682 | A finite domain is a divis... |
| fiidomfld 20683 | A finite integral domain i... |
| rng1nnzr 20684 | The (smallest) structure r... |
| ring1zr 20685 | The only (unital) ring wit... |
| rngen1zr 20686 | The only (unital) ring wit... |
| ringen1zr 20687 | The only unital ring with ... |
| rng1nfld 20688 | The zero ring is not a fie... |
| issubdrg 20689 | Characterize the subfields... |
| drhmsubc 20690 | According to ~ df-subc , t... |
| drngcat 20691 | The restriction of the cat... |
| fldcat 20692 | The restriction of the cat... |
| fldc 20693 | The restriction of the cat... |
| fldhmsubc 20694 | According to ~ df-subc , t... |
| issdrg 20697 | Property of a division sub... |
| sdrgrcl 20698 | Reverse closure for a sub-... |
| sdrgdrng 20699 | A sub-division-ring is a d... |
| sdrgsubrg 20700 | A sub-division-ring is a s... |
| sdrgid 20701 | Every division ring is a d... |
| sdrgss 20702 | A division subring is a su... |
| sdrgbas 20703 | Base set of a sub-division... |
| issdrg2 20704 | Property of a division sub... |
| sdrgunit 20705 | A unit of a sub-division-r... |
| imadrhmcl 20706 | The image of a (nontrivial... |
| fldsdrgfld 20707 | A sub-division-ring of a f... |
| acsfn1p 20708 | Construction of a closure ... |
| subrgacs 20709 | Closure property of subrin... |
| sdrgacs 20710 | Closure property of divisi... |
| cntzsdrg 20711 | Centralizers in division r... |
| subdrgint 20712 | The intersection of a none... |
| sdrgint 20713 | The intersection of a none... |
| primefld 20714 | The smallest sub division ... |
| primefld0cl 20715 | The prime field contains t... |
| primefld1cl 20716 | The prime field contains t... |
| abvfval 20719 | Value of the set of absolu... |
| isabv 20720 | Elementhood in the set of ... |
| isabvd 20721 | Properties that determine ... |
| abvrcl 20722 | Reverse closure for the ab... |
| abvfge0 20723 | An absolute value is a fun... |
| abvf 20724 | An absolute value is a fun... |
| abvcl 20725 | An absolute value is a fun... |
| abvge0 20726 | The absolute value of a nu... |
| abveq0 20727 | The value of an absolute v... |
| abvne0 20728 | The absolute value of a no... |
| abvgt0 20729 | The absolute value of a no... |
| abvmul 20730 | An absolute value distribu... |
| abvtri 20731 | An absolute value satisfie... |
| abv0 20732 | The absolute value of zero... |
| abv1z 20733 | The absolute value of one ... |
| abv1 20734 | The absolute value of one ... |
| abvneg 20735 | The absolute value of a ne... |
| abvsubtri 20736 | An absolute value satisfie... |
| abvrec 20737 | The absolute value distrib... |
| abvdiv 20738 | The absolute value distrib... |
| abvdom 20739 | Any ring with an absolute ... |
| abvres 20740 | The restriction of an abso... |
| abvtrivd 20741 | The trivial absolute value... |
| abvtrivg 20742 | The trivial absolute value... |
| abvtriv 20743 | The trivial absolute value... |
| abvpropd 20744 | If two structures have the... |
| abvn0b 20745 | Another characterization o... |
| staffval 20750 | The functionalization of t... |
| stafval 20751 | The functionalization of t... |
| staffn 20752 | The functionalization is e... |
| issrng 20753 | The predicate "is a star r... |
| srngrhm 20754 | The involution function in... |
| srngring 20755 | A star ring is a ring. (C... |
| srngcnv 20756 | The involution function in... |
| srngf1o 20757 | The involution function in... |
| srngcl 20758 | The involution function in... |
| srngnvl 20759 | The involution function in... |
| srngadd 20760 | The involution function in... |
| srngmul 20761 | The involution function in... |
| srng1 20762 | The conjugate of the ring ... |
| srng0 20763 | The conjugate of the ring ... |
| issrngd 20764 | Properties that determine ... |
| idsrngd 20765 | A commutative ring is a st... |
| islmod 20770 | The predicate "is a left m... |
| lmodlema 20771 | Lemma for properties of a ... |
| islmodd 20772 | Properties that determine ... |
| lmodgrp 20773 | A left module is a group. ... |
| lmodring 20774 | The scalar component of a ... |
| lmodfgrp 20775 | The scalar component of a ... |
| lmodgrpd 20776 | A left module is a group. ... |
| lmodbn0 20777 | The base set of a left mod... |
| lmodacl 20778 | Closure of ring addition f... |
| lmodmcl 20779 | Closure of ring multiplica... |
| lmodsn0 20780 | The set of scalars in a le... |
| lmodvacl 20781 | Closure of vector addition... |
| lmodass 20782 | Left module vector sum is ... |
| lmodlcan 20783 | Left cancellation law for ... |
| lmodvscl 20784 | Closure of scalar product ... |
| lmodvscld 20785 | Closure of scalar product ... |
| scaffval 20786 | The scalar multiplication ... |
| scafval 20787 | The scalar multiplication ... |
| scafeq 20788 | If the scalar multiplicati... |
| scaffn 20789 | The scalar multiplication ... |
| lmodscaf 20790 | The scalar multiplication ... |
| lmodvsdi 20791 | Distributive law for scala... |
| lmodvsdir 20792 | Distributive law for scala... |
| lmodvsass 20793 | Associative law for scalar... |
| lmod0cl 20794 | The ring zero in a left mo... |
| lmod1cl 20795 | The ring unity in a left m... |
| lmodvs1 20796 | Scalar product with the ri... |
| lmod0vcl 20797 | The zero vector is a vecto... |
| lmod0vlid 20798 | Left identity law for the ... |
| lmod0vrid 20799 | Right identity law for the... |
| lmod0vid 20800 | Identity equivalent to the... |
| lmod0vs 20801 | Zero times a vector is the... |
| lmodvs0 20802 | Anything times the zero ve... |
| lmodvsmmulgdi 20803 | Distributive law for a gro... |
| lmodfopnelem1 20804 | Lemma 1 for ~ lmodfopne . ... |
| lmodfopnelem2 20805 | Lemma 2 for ~ lmodfopne . ... |
| lmodfopne 20806 | The (functionalized) opera... |
| lcomf 20807 | A linear-combination sum i... |
| lcomfsupp 20808 | A linear-combination sum i... |
| lmodvnegcl 20809 | Closure of vector negative... |
| lmodvnegid 20810 | Addition of a vector with ... |
| lmodvneg1 20811 | Minus 1 times a vector is ... |
| lmodvsneg 20812 | Multiplication of a vector... |
| lmodvsubcl 20813 | Closure of vector subtract... |
| lmodcom 20814 | Left module vector sum is ... |
| lmodabl 20815 | A left module is an abelia... |
| lmodcmn 20816 | A left module is a commuta... |
| lmodnegadd 20817 | Distribute negation throug... |
| lmod4 20818 | Commutative/associative la... |
| lmodvsubadd 20819 | Relationship between vecto... |
| lmodvaddsub4 20820 | Vector addition/subtractio... |
| lmodvpncan 20821 | Addition/subtraction cance... |
| lmodvnpcan 20822 | Cancellation law for vecto... |
| lmodvsubval2 20823 | Value of vector subtractio... |
| lmodsubvs 20824 | Subtraction of a scalar pr... |
| lmodsubdi 20825 | Scalar multiplication dist... |
| lmodsubdir 20826 | Scalar multiplication dist... |
| lmodsubeq0 20827 | If the difference between ... |
| lmodsubid 20828 | Subtraction of a vector fr... |
| lmodvsghm 20829 | Scalar multiplication of t... |
| lmodprop2d 20830 | If two structures have the... |
| lmodpropd 20831 | If two structures have the... |
| gsumvsmul 20832 | Pull a scalar multiplicati... |
| mptscmfsupp0 20833 | A mapping to a scalar prod... |
| mptscmfsuppd 20834 | A function mapping to a sc... |
| rmodislmodlem 20835 | Lemma for ~ rmodislmod . ... |
| rmodislmod 20836 | The right module ` R ` ind... |
| lssset 20839 | The set of all (not necess... |
| islss 20840 | The predicate "is a subspa... |
| islssd 20841 | Properties that determine ... |
| lssss 20842 | A subspace is a set of vec... |
| lssel 20843 | A subspace member is a vec... |
| lss1 20844 | The set of vectors in a le... |
| lssuni 20845 | The union of all subspaces... |
| lssn0 20846 | A subspace is not empty. ... |
| 00lss 20847 | The empty structure has no... |
| lsscl 20848 | Closure property of a subs... |
| lssvacl 20849 | Closure of vector addition... |
| lssvsubcl 20850 | Closure of vector subtract... |
| lssvancl1 20851 | Non-closure: if one vector... |
| lssvancl2 20852 | Non-closure: if one vector... |
| lss0cl 20853 | The zero vector belongs to... |
| lsssn0 20854 | The singleton of the zero ... |
| lss0ss 20855 | The zero subspace is inclu... |
| lssle0 20856 | No subspace is smaller tha... |
| lssne0 20857 | A nonzero subspace has a n... |
| lssvneln0 20858 | A vector ` X ` which doesn... |
| lssneln0 20859 | A vector ` X ` which doesn... |
| lssssr 20860 | Conclude subspace ordering... |
| lssvscl 20861 | Closure of scalar product ... |
| lssvnegcl 20862 | Closure of negative vector... |
| lsssubg 20863 | All subspaces are subgroup... |
| lsssssubg 20864 | All subspaces are subgroup... |
| islss3 20865 | A linear subspace of a mod... |
| lsslmod 20866 | A submodule is a module. ... |
| lsslss 20867 | The subspaces of a subspac... |
| islss4 20868 | A linear subspace is a sub... |
| lss1d 20869 | One-dimensional subspace (... |
| lssintcl 20870 | The intersection of a none... |
| lssincl 20871 | The intersection of two su... |
| lssmre 20872 | The subspaces of a module ... |
| lssacs 20873 | Submodules are an algebrai... |
| prdsvscacl 20874 | Pointwise scalar multiplic... |
| prdslmodd 20875 | The product of a family of... |
| pwslmod 20876 | A structure power of a lef... |
| lspfval 20879 | The span function for a le... |
| lspf 20880 | The span function on a lef... |
| lspval 20881 | The span of a set of vecto... |
| lspcl 20882 | The span of a set of vecto... |
| lspsncl 20883 | The span of a singleton is... |
| lspprcl 20884 | The span of a pair is a su... |
| lsptpcl 20885 | The span of an unordered t... |
| lspsnsubg 20886 | The span of a singleton is... |
| 00lsp 20887 | ~ fvco4i lemma for linear ... |
| lspid 20888 | The span of a subspace is ... |
| lspssv 20889 | A span is a set of vectors... |
| lspss 20890 | Span preserves subset orde... |
| lspssid 20891 | A set of vectors is a subs... |
| lspidm 20892 | The span of a set of vecto... |
| lspun 20893 | The span of union is the s... |
| lspssp 20894 | If a set of vectors is a s... |
| mrclsp 20895 | Moore closure generalizes ... |
| lspsnss 20896 | The span of the singleton ... |
| ellspsn3 20897 | A member of the span of th... |
| lspprss 20898 | The span of a pair of vect... |
| lspsnid 20899 | A vector belongs to the sp... |
| ellspsn6 20900 | Relationship between a vec... |
| ellspsn5b 20901 | Relationship between a vec... |
| ellspsn5 20902 | Relationship between a vec... |
| lspprid1 20903 | A member of a pair of vect... |
| lspprid2 20904 | A member of a pair of vect... |
| lspprvacl 20905 | The sum of two vectors bel... |
| lssats2 20906 | A way to express atomistic... |
| ellspsni 20907 | A scalar product with a ve... |
| lspsn 20908 | Span of the singleton of a... |
| ellspsn 20909 | Member of span of the sing... |
| lspsnvsi 20910 | Span of a scalar product o... |
| lspsnss2 20911 | Comparable spans of single... |
| lspsnneg 20912 | Negation does not change t... |
| lspsnsub 20913 | Swapping subtraction order... |
| lspsn0 20914 | Span of the singleton of t... |
| lsp0 20915 | Span of the empty set. (C... |
| lspuni0 20916 | Union of the span of the e... |
| lspun0 20917 | The span of a union with t... |
| lspsneq0 20918 | Span of the singleton is t... |
| lspsneq0b 20919 | Equal singleton spans impl... |
| lmodindp1 20920 | Two independent (non-colin... |
| lsslsp 20921 | Spans in submodules corres... |
| lsslspOLD 20922 | Obsolete version of ~ lssl... |
| lss0v 20923 | The zero vector in a submo... |
| lsspropd 20924 | If two structures have the... |
| lsppropd 20925 | If two structures have the... |
| reldmlmhm 20932 | Lemma for module homomorph... |
| lmimfn 20933 | Lemma for module isomorphi... |
| islmhm 20934 | Property of being a homomo... |
| islmhm3 20935 | Property of a module homom... |
| lmhmlem 20936 | Non-quantified consequence... |
| lmhmsca 20937 | A homomorphism of left mod... |
| lmghm 20938 | A homomorphism of left mod... |
| lmhmlmod2 20939 | A homomorphism of left mod... |
| lmhmlmod1 20940 | A homomorphism of left mod... |
| lmhmf 20941 | A homomorphism of left mod... |
| lmhmlin 20942 | A homomorphism of left mod... |
| lmodvsinv 20943 | Multiplication of a vector... |
| lmodvsinv2 20944 | Multiplying a negated vect... |
| islmhm2 20945 | A one-equation proof of li... |
| islmhmd 20946 | Deduction for a module hom... |
| 0lmhm 20947 | The constant zero linear f... |
| idlmhm 20948 | The identity function on a... |
| invlmhm 20949 | The negative function on a... |
| lmhmco 20950 | The composition of two mod... |
| lmhmplusg 20951 | The pointwise sum of two l... |
| lmhmvsca 20952 | The pointwise scalar produ... |
| lmhmf1o 20953 | A bijective module homomor... |
| lmhmima 20954 | The image of a subspace un... |
| lmhmpreima 20955 | The inverse image of a sub... |
| lmhmlsp 20956 | Homomorphisms preserve spa... |
| lmhmrnlss 20957 | The range of a homomorphis... |
| lmhmkerlss 20958 | The kernel of a homomorphi... |
| reslmhm 20959 | Restriction of a homomorph... |
| reslmhm2 20960 | Expansion of the codomain ... |
| reslmhm2b 20961 | Expansion of the codomain ... |
| lmhmeql 20962 | The equalizer of two modul... |
| lspextmo 20963 | A linear function is compl... |
| pwsdiaglmhm 20964 | Diagonal homomorphism into... |
| pwssplit0 20965 | Splitting for structure po... |
| pwssplit1 20966 | Splitting for structure po... |
| pwssplit2 20967 | Splitting for structure po... |
| pwssplit3 20968 | Splitting for structure po... |
| islmim 20969 | An isomorphism of left mod... |
| lmimf1o 20970 | An isomorphism of left mod... |
| lmimlmhm 20971 | An isomorphism of modules ... |
| lmimgim 20972 | An isomorphism of modules ... |
| islmim2 20973 | An isomorphism of left mod... |
| lmimcnv 20974 | The converse of a bijectiv... |
| brlmic 20975 | The relation "is isomorphi... |
| brlmici 20976 | Prove isomorphic by an exp... |
| lmiclcl 20977 | Isomorphism implies the le... |
| lmicrcl 20978 | Isomorphism implies the ri... |
| lmicsym 20979 | Module isomorphism is symm... |
| lmhmpropd 20980 | Module homomorphism depend... |
| islbs 20983 | The predicate " ` B ` is a... |
| lbsss 20984 | A basis is a set of vector... |
| lbsel 20985 | An element of a basis is a... |
| lbssp 20986 | The span of a basis is the... |
| lbsind 20987 | A basis is linearly indepe... |
| lbsind2 20988 | A basis is linearly indepe... |
| lbspss 20989 | No proper subset of a basi... |
| lsmcl 20990 | The sum of two subspaces i... |
| lsmspsn 20991 | Member of subspace sum of ... |
| lsmelval2 20992 | Subspace sum membership in... |
| lsmsp 20993 | Subspace sum in terms of s... |
| lsmsp2 20994 | Subspace sum of spans of s... |
| lsmssspx 20995 | Subspace sum (in its exten... |
| lsmpr 20996 | The span of a pair of vect... |
| lsppreli 20997 | A vector expressed as a su... |
| lsmelpr 20998 | Two ways to say that a vec... |
| lsppr0 20999 | The span of a vector paire... |
| lsppr 21000 | Span of a pair of vectors.... |
| lspprel 21001 | Member of the span of a pa... |
| lspprabs 21002 | Absorption of vector sum i... |
| lspvadd 21003 | The span of a vector sum i... |
| lspsntri 21004 | Triangle-type inequality f... |
| lspsntrim 21005 | Triangle-type inequality f... |
| lbspropd 21006 | If two structures have the... |
| pj1lmhm 21007 | The left projection functi... |
| pj1lmhm2 21008 | The left projection functi... |
| islvec 21011 | The predicate "is a left v... |
| lvecdrng 21012 | The set of scalars of a le... |
| lveclmod 21013 | A left vector space is a l... |
| lveclmodd 21014 | A vector space is a left m... |
| lvecgrpd 21015 | A vector space is a group.... |
| lsslvec 21016 | A vector subspace is a vec... |
| lmhmlvec 21017 | The property for modules t... |
| lvecvs0or 21018 | If a scalar product is zer... |
| lvecvsn0 21019 | A scalar product is nonzer... |
| lssvs0or 21020 | If a scalar product belong... |
| lvecvscan 21021 | Cancellation law for scala... |
| lvecvscan2 21022 | Cancellation law for scala... |
| lvecinv 21023 | Invert coefficient of scal... |
| lspsnvs 21024 | A nonzero scalar product d... |
| lspsneleq 21025 | Membership relation that i... |
| lspsncmp 21026 | Comparable spans of nonzer... |
| lspsnne1 21027 | Two ways to express that v... |
| lspsnne2 21028 | Two ways to express that v... |
| lspsnnecom 21029 | Swap two vectors with diff... |
| lspabs2 21030 | Absorption law for span of... |
| lspabs3 21031 | Absorption law for span of... |
| lspsneq 21032 | Equal spans of singletons ... |
| lspsneu 21033 | Nonzero vectors with equal... |
| ellspsn4 21034 | A member of the span of th... |
| lspdisj 21035 | The span of a vector not i... |
| lspdisjb 21036 | A nonzero vector is not in... |
| lspdisj2 21037 | Unequal spans are disjoint... |
| lspfixed 21038 | Show membership in the spa... |
| lspexch 21039 | Exchange property for span... |
| lspexchn1 21040 | Exchange property for span... |
| lspexchn2 21041 | Exchange property for span... |
| lspindpi 21042 | Partial independence prope... |
| lspindp1 21043 | Alternate way to say 3 vec... |
| lspindp2l 21044 | Alternate way to say 3 vec... |
| lspindp2 21045 | Alternate way to say 3 vec... |
| lspindp3 21046 | Independence of 2 vectors ... |
| lspindp4 21047 | (Partial) independence of ... |
| lvecindp 21048 | Compute the ` X ` coeffici... |
| lvecindp2 21049 | Sums of independent vector... |
| lspsnsubn0 21050 | Unequal singleton spans im... |
| lsmcv 21051 | Subspace sum has the cover... |
| lspsolvlem 21052 | Lemma for ~ lspsolv . (Co... |
| lspsolv 21053 | If ` X ` is in the span of... |
| lssacsex 21054 | In a vector space, subspac... |
| lspsnat 21055 | There is no subspace stric... |
| lspsncv0 21056 | The span of a singleton co... |
| lsppratlem1 21057 | Lemma for ~ lspprat . Let... |
| lsppratlem2 21058 | Lemma for ~ lspprat . Sho... |
| lsppratlem3 21059 | Lemma for ~ lspprat . In ... |
| lsppratlem4 21060 | Lemma for ~ lspprat . In ... |
| lsppratlem5 21061 | Lemma for ~ lspprat . Com... |
| lsppratlem6 21062 | Lemma for ~ lspprat . Neg... |
| lspprat 21063 | A proper subspace of the s... |
| islbs2 21064 | An equivalent formulation ... |
| islbs3 21065 | An equivalent formulation ... |
| lbsacsbs 21066 | Being a basis in a vector ... |
| lvecdim 21067 | The dimension theorem for ... |
| lbsextlem1 21068 | Lemma for ~ lbsext . The ... |
| lbsextlem2 21069 | Lemma for ~ lbsext . Sinc... |
| lbsextlem3 21070 | Lemma for ~ lbsext . A ch... |
| lbsextlem4 21071 | Lemma for ~ lbsext . ~ lbs... |
| lbsextg 21072 | For any linearly independe... |
| lbsext 21073 | For any linearly independe... |
| lbsexg 21074 | Every vector space has a b... |
| lbsex 21075 | Every vector space has a b... |
| lvecprop2d 21076 | If two structures have the... |
| lvecpropd 21077 | If two structures have the... |
| sraval 21082 | Lemma for ~ srabase throug... |
| sralem 21083 | Lemma for ~ srabase and si... |
| srabase 21084 | Base set of a subring alge... |
| sraaddg 21085 | Additive operation of a su... |
| sramulr 21086 | Multiplicative operation o... |
| srasca 21087 | The set of scalars of a su... |
| sravsca 21088 | The scalar product operati... |
| sraip 21089 | The inner product operatio... |
| sratset 21090 | Topology component of a su... |
| sratopn 21091 | Topology component of a su... |
| srads 21092 | Distance function of a sub... |
| sraring 21093 | Condition for a subring al... |
| sralmod 21094 | The subring algebra is a l... |
| sralmod0 21095 | The subring module inherit... |
| issubrgd 21096 | Prove a subring by closure... |
| rlmfn 21097 | ` ringLMod ` is a function... |
| rlmval 21098 | Value of the ring module. ... |
| rlmval2 21099 | Value of the ring module e... |
| rlmbas 21100 | Base set of the ring modul... |
| rlmplusg 21101 | Vector addition in the rin... |
| rlm0 21102 | Zero vector in the ring mo... |
| rlmsub 21103 | Subtraction in the ring mo... |
| rlmmulr 21104 | Ring multiplication in the... |
| rlmsca 21105 | Scalars in the ring module... |
| rlmsca2 21106 | Scalars in the ring module... |
| rlmvsca 21107 | Scalar multiplication in t... |
| rlmtopn 21108 | Topology component of the ... |
| rlmds 21109 | Metric component of the ri... |
| rlmlmod 21110 | The ring module is a modul... |
| rlmlvec 21111 | The ring module over a div... |
| rlmlsm 21112 | Subgroup sum of the ring m... |
| rlmvneg 21113 | Vector negation in the rin... |
| rlmscaf 21114 | Functionalized scalar mult... |
| ixpsnbasval 21115 | The value of an infinite C... |
| lidlval 21120 | Value of the set of ring i... |
| rspval 21121 | Value of the ring span fun... |
| lidlss 21122 | An ideal is a subset of th... |
| lidlssbas 21123 | The base set of the restri... |
| lidlbas 21124 | A (left) ideal of a ring i... |
| islidl 21125 | Predicate of being a (left... |
| rnglidlmcl 21126 | A (left) ideal containing ... |
| rngridlmcl 21127 | A right ideal (which is a ... |
| dflidl2rng 21128 | Alternate (the usual textb... |
| isridlrng 21129 | A right ideal is a left id... |
| lidl0cl 21130 | An ideal contains 0. (Con... |
| lidlacl 21131 | An ideal is closed under a... |
| lidlnegcl 21132 | An ideal contains negative... |
| lidlsubg 21133 | An ideal is a subgroup of ... |
| lidlsubcl 21134 | An ideal is closed under s... |
| lidlmcl 21135 | An ideal is closed under l... |
| lidl1el 21136 | An ideal contains 1 iff it... |
| dflidl2 21137 | Alternate (the usual textb... |
| lidl0ALT 21138 | Alternate proof for ~ lidl... |
| rnglidl0 21139 | Every non-unital ring cont... |
| lidl0 21140 | Every ring contains a zero... |
| lidl1ALT 21141 | Alternate proof for ~ lidl... |
| rnglidl1 21142 | The base set of every non-... |
| lidl1 21143 | Every ring contains a unit... |
| lidlacs 21144 | The ideal system is an alg... |
| rspcl 21145 | The span of a set of ring ... |
| rspssid 21146 | The span of a set of ring ... |
| rsp1 21147 | The span of the identity e... |
| rsp0 21148 | The span of the zero eleme... |
| rspssp 21149 | The ideal span of a set of... |
| elrspsn 21150 | Membership in a principal ... |
| mrcrsp 21151 | Moore closure generalizes ... |
| lidlnz 21152 | A nonzero ideal contains a... |
| drngnidl 21153 | A division ring has only t... |
| lidlrsppropd 21154 | The left ideals and ring s... |
| rnglidlmmgm 21155 | The multiplicative group o... |
| rnglidlmsgrp 21156 | The multiplicative group o... |
| rnglidlrng 21157 | A (left) ideal of a non-un... |
| lidlnsg 21158 | An ideal is a normal subgr... |
| 2idlval 21161 | Definition of a two-sided ... |
| isridl 21162 | A right ideal is a left id... |
| 2idlelb 21163 | Membership in a two-sided ... |
| 2idllidld 21164 | A two-sided ideal is a lef... |
| 2idlridld 21165 | A two-sided ideal is a rig... |
| df2idl2rng 21166 | Alternate (the usual textb... |
| df2idl2 21167 | Alternate (the usual textb... |
| ridl0 21168 | Every ring contains a zero... |
| ridl1 21169 | Every ring contains a unit... |
| 2idl0 21170 | Every ring contains a zero... |
| 2idl1 21171 | Every ring contains a unit... |
| 2idlss 21172 | A two-sided ideal is a sub... |
| 2idlbas 21173 | The base set of a two-side... |
| 2idlelbas 21174 | The base set of a two-side... |
| rng2idlsubrng 21175 | A two-sided ideal of a non... |
| rng2idlnsg 21176 | A two-sided ideal of a non... |
| rng2idl0 21177 | The zero (additive identit... |
| rng2idlsubgsubrng 21178 | A two-sided ideal of a non... |
| rng2idlsubgnsg 21179 | A two-sided ideal of a non... |
| rng2idlsubg0 21180 | The zero (additive identit... |
| 2idlcpblrng 21181 | The coset equivalence rela... |
| 2idlcpbl 21182 | The coset equivalence rela... |
| qus2idrng 21183 | The quotient of a non-unit... |
| qus1 21184 | The multiplicative identit... |
| qusring 21185 | If ` S ` is a two-sided id... |
| qusrhm 21186 | If ` S ` is a two-sided id... |
| rhmpreimaidl 21187 | The preimage of an ideal b... |
| kerlidl 21188 | The kernel of a ring homom... |
| qusmul2idl 21189 | Value of the ring operatio... |
| crngridl 21190 | In a commutative ring, the... |
| crng2idl 21191 | In a commutative ring, a t... |
| qusmulrng 21192 | Value of the multiplicatio... |
| quscrng 21193 | The quotient of a commutat... |
| qusmulcrng 21194 | Value of the ring operatio... |
| rhmqusnsg 21195 | The mapping ` J ` induced ... |
| rngqiprng1elbas 21196 | The ring unity of a two-si... |
| rngqiprngghmlem1 21197 | Lemma 1 for ~ rngqiprngghm... |
| rngqiprngghmlem2 21198 | Lemma 2 for ~ rngqiprngghm... |
| rngqiprngghmlem3 21199 | Lemma 3 for ~ rngqiprngghm... |
| rngqiprngimfolem 21200 | Lemma for ~ rngqiprngimfo ... |
| rngqiprnglinlem1 21201 | Lemma 1 for ~ rngqiprnglin... |
| rngqiprnglinlem2 21202 | Lemma 2 for ~ rngqiprnglin... |
| rngqiprnglinlem3 21203 | Lemma 3 for ~ rngqiprnglin... |
| rngqiprngimf1lem 21204 | Lemma for ~ rngqiprngimf1 ... |
| rngqipbas 21205 | The base set of the produc... |
| rngqiprng 21206 | The product of the quotien... |
| rngqiprngimf 21207 | ` F ` is a function from (... |
| rngqiprngimfv 21208 | The value of the function ... |
| rngqiprngghm 21209 | ` F ` is a homomorphism of... |
| rngqiprngimf1 21210 | ` F ` is a one-to-one func... |
| rngqiprngimfo 21211 | ` F ` is a function from (... |
| rngqiprnglin 21212 | ` F ` is linear with respe... |
| rngqiprngho 21213 | ` F ` is a homomorphism of... |
| rngqiprngim 21214 | ` F ` is an isomorphism of... |
| rng2idl1cntr 21215 | The unity of a two-sided i... |
| rngringbdlem1 21216 | In a unital ring, the quot... |
| rngringbdlem2 21217 | A non-unital ring is unita... |
| rngringbd 21218 | A non-unital ring is unita... |
| ring2idlqus 21219 | For every unital ring ther... |
| ring2idlqusb 21220 | A non-unital ring is unita... |
| rngqiprngfulem1 21221 | Lemma 1 for ~ rngqiprngfu ... |
| rngqiprngfulem2 21222 | Lemma 2 for ~ rngqiprngfu ... |
| rngqiprngfulem3 21223 | Lemma 3 for ~ rngqiprngfu ... |
| rngqiprngfulem4 21224 | Lemma 4 for ~ rngqiprngfu ... |
| rngqiprngfulem5 21225 | Lemma 5 for ~ rngqiprngfu ... |
| rngqipring1 21226 | The ring unity of the prod... |
| rngqiprngfu 21227 | The function value of ` F ... |
| rngqiprngu 21228 | If a non-unital ring has a... |
| ring2idlqus1 21229 | If a non-unital ring has a... |
| lpival 21234 | Value of the set of princi... |
| islpidl 21235 | Property of being a princi... |
| lpi0 21236 | The zero ideal is always p... |
| lpi1 21237 | The unit ideal is always p... |
| islpir 21238 | Principal ideal rings are ... |
| lpiss 21239 | Principal ideals are a sub... |
| islpir2 21240 | Principal ideal rings are ... |
| lpirring 21241 | Principal ideal rings are ... |
| drnglpir 21242 | Division rings are princip... |
| rspsn 21243 | Membership in principal id... |
| lidldvgen 21244 | An element generates an id... |
| lpigen 21245 | An ideal is principal iff ... |
| cnfldstr 21266 | The field of complex numbe... |
| cnfldex 21267 | The field of complex numbe... |
| cnfldbas 21268 | The base set of the field ... |
| mpocnfldadd 21269 | The addition operation of ... |
| cnfldadd 21270 | The addition operation of ... |
| mpocnfldmul 21271 | The multiplication operati... |
| cnfldmul 21272 | The multiplication operati... |
| cnfldcj 21273 | The conjugation operation ... |
| cnfldtset 21274 | The topology component of ... |
| cnfldle 21275 | The ordering of the field ... |
| cnfldds 21276 | The metric of the field of... |
| cnfldunif 21277 | The uniform structure comp... |
| cnfldfun 21278 | The field of complex numbe... |
| cnfldfunALT 21279 | The field of complex numbe... |
| dfcnfldOLD 21280 | Obsolete version of ~ df-c... |
| cnfldstrOLD 21281 | Obsolete version of ~ cnfl... |
| cnfldexOLD 21282 | Obsolete version of ~ cnfl... |
| cnfldbasOLD 21283 | Obsolete version of ~ cnfl... |
| cnfldaddOLD 21284 | Obsolete version of ~ cnfl... |
| cnfldmulOLD 21285 | Obsolete version of ~ cnfl... |
| cnfldcjOLD 21286 | Obsolete version of ~ cnfl... |
| cnfldtsetOLD 21287 | Obsolete version of ~ cnfl... |
| cnfldleOLD 21288 | Obsolete version of ~ cnfl... |
| cnflddsOLD 21289 | Obsolete version of ~ cnfl... |
| cnfldunifOLD 21290 | Obsolete version of ~ cnfl... |
| cnfldfunOLD 21291 | Obsolete version of ~ cnfl... |
| cnfldfunALTOLD 21292 | Obsolete version of ~ cnfl... |
| xrsstr 21293 | The extended real structur... |
| xrsex 21294 | The extended real structur... |
| xrsbas 21295 | The base set of the extend... |
| xrsadd 21296 | The addition operation of ... |
| xrsmul 21297 | The multiplication operati... |
| xrstset 21298 | The topology component of ... |
| xrsle 21299 | The ordering of the extend... |
| cncrng 21300 | The complex numbers form a... |
| cncrngOLD 21301 | Obsolete version of ~ cncr... |
| cnring 21302 | The complex numbers form a... |
| xrsmcmn 21303 | The "multiplicative group"... |
| cnfld0 21304 | Zero is the zero element o... |
| cnfld1 21305 | One is the unity element o... |
| cnfld1OLD 21306 | Obsolete version of ~ cnfl... |
| cnfldneg 21307 | The additive inverse in th... |
| cnfldplusf 21308 | The functionalized additio... |
| cnfldsub 21309 | The subtraction operator i... |
| cndrng 21310 | The complex numbers form a... |
| cndrngOLD 21311 | Obsolete version of ~ cndr... |
| cnflddiv 21312 | The division operation in ... |
| cnflddivOLD 21313 | Obsolete version of ~ cnfl... |
| cnfldinv 21314 | The multiplicative inverse... |
| cnfldmulg 21315 | The group multiple functio... |
| cnfldexp 21316 | The exponentiation operato... |
| cnsrng 21317 | The complex numbers form a... |
| xrsmgm 21318 | The "additive group" of th... |
| xrsnsgrp 21319 | The "additive group" of th... |
| xrsmgmdifsgrp 21320 | The "additive group" of th... |
| xrs1mnd 21321 | The extended real numbers,... |
| xrs10 21322 | The zero of the extended r... |
| xrs1cmn 21323 | The extended real numbers ... |
| xrge0subm 21324 | The nonnegative extended r... |
| xrge0cmn 21325 | The nonnegative extended r... |
| xrsds 21326 | The metric of the extended... |
| xrsdsval 21327 | The metric of the extended... |
| xrsdsreval 21328 | The metric of the extended... |
| xrsdsreclblem 21329 | Lemma for ~ xrsdsreclb . ... |
| xrsdsreclb 21330 | The metric of the extended... |
| cnsubmlem 21331 | Lemma for ~ nn0subm and fr... |
| cnsubglem 21332 | Lemma for ~ resubdrg and f... |
| cnsubrglem 21333 | Lemma for ~ resubdrg and f... |
| cnsubrglemOLD 21334 | Obsolete version of ~ cnsu... |
| cnsubdrglem 21335 | Lemma for ~ resubdrg and f... |
| qsubdrg 21336 | The rational numbers form ... |
| zsubrg 21337 | The integers form a subrin... |
| gzsubrg 21338 | The gaussian integers form... |
| nn0subm 21339 | The nonnegative integers f... |
| rege0subm 21340 | The nonnegative reals form... |
| absabv 21341 | The regular absolute value... |
| zsssubrg 21342 | The integers are a subset ... |
| qsssubdrg 21343 | The rational numbers are a... |
| cnsubrg 21344 | There are no subrings of t... |
| cnmgpabl 21345 | The unit group of the comp... |
| cnmgpid 21346 | The group identity element... |
| cnmsubglem 21347 | Lemma for ~ rpmsubg and fr... |
| rpmsubg 21348 | The positive reals form a ... |
| gzrngunitlem 21349 | Lemma for ~ gzrngunit . (... |
| gzrngunit 21350 | The units on ` ZZ [ _i ] `... |
| gsumfsum 21351 | Relate a group sum on ` CC... |
| regsumfsum 21352 | Relate a group sum on ` ( ... |
| expmhm 21353 | Exponentiation is a monoid... |
| nn0srg 21354 | The nonnegative integers f... |
| rge0srg 21355 | The nonnegative real numbe... |
| zringcrng 21358 | The ring of integers is a ... |
| zringring 21359 | The ring of integers is a ... |
| zringrng 21360 | The ring of integers is a ... |
| zringabl 21361 | The ring of integers is an... |
| zringgrp 21362 | The ring of integers is an... |
| zringbas 21363 | The integers are the base ... |
| zringplusg 21364 | The addition operation of ... |
| zringsub 21365 | The subtraction of element... |
| zringmulg 21366 | The multiplication (group ... |
| zringmulr 21367 | The multiplication operati... |
| zring0 21368 | The zero element of the ri... |
| zring1 21369 | The unity element of the r... |
| zringnzr 21370 | The ring of integers is a ... |
| dvdsrzring 21371 | Ring divisibility in the r... |
| zringlpirlem1 21372 | Lemma for ~ zringlpir . A... |
| zringlpirlem2 21373 | Lemma for ~ zringlpir . A... |
| zringlpirlem3 21374 | Lemma for ~ zringlpir . A... |
| zringinvg 21375 | The additive inverse of an... |
| zringunit 21376 | The units of ` ZZ ` are th... |
| zringlpir 21377 | The integers are a princip... |
| zringndrg 21378 | The integers are not a div... |
| zringcyg 21379 | The integers are a cyclic ... |
| zringsubgval 21380 | Subtraction in the ring of... |
| zringmpg 21381 | The multiplicative group o... |
| prmirredlem 21382 | A positive integer is irre... |
| dfprm2 21383 | The positive irreducible e... |
| prmirred 21384 | The irreducible elements o... |
| expghm 21385 | Exponentiation is a group ... |
| mulgghm2 21386 | The powers of a group elem... |
| mulgrhm 21387 | The powers of the element ... |
| mulgrhm2 21388 | The powers of the element ... |
| irinitoringc 21389 | The ring of integers is an... |
| nzerooringczr 21390 | There is no zero object in... |
| pzriprnglem1 21391 | Lemma 1 for ~ pzriprng : `... |
| pzriprnglem2 21392 | Lemma 2 for ~ pzriprng : ... |
| pzriprnglem3 21393 | Lemma 3 for ~ pzriprng : ... |
| pzriprnglem4 21394 | Lemma 4 for ~ pzriprng : `... |
| pzriprnglem5 21395 | Lemma 5 for ~ pzriprng : `... |
| pzriprnglem6 21396 | Lemma 6 for ~ pzriprng : `... |
| pzriprnglem7 21397 | Lemma 7 for ~ pzriprng : `... |
| pzriprnglem8 21398 | Lemma 8 for ~ pzriprng : `... |
| pzriprnglem9 21399 | Lemma 9 for ~ pzriprng : ... |
| pzriprnglem10 21400 | Lemma 10 for ~ pzriprng : ... |
| pzriprnglem11 21401 | Lemma 11 for ~ pzriprng : ... |
| pzriprnglem12 21402 | Lemma 12 for ~ pzriprng : ... |
| pzriprnglem13 21403 | Lemma 13 for ~ pzriprng : ... |
| pzriprnglem14 21404 | Lemma 14 for ~ pzriprng : ... |
| pzriprngALT 21405 | The non-unital ring ` ( ZZ... |
| pzriprng1ALT 21406 | The ring unity of the ring... |
| pzriprng 21407 | The non-unital ring ` ( ZZ... |
| pzriprng1 21408 | The ring unity of the ring... |
| zrhval 21417 | Define the unique homomorp... |
| zrhval2 21418 | Alternate value of the ` Z... |
| zrhmulg 21419 | Value of the ` ZRHom ` hom... |
| zrhrhmb 21420 | The ` ZRHom ` homomorphism... |
| zrhrhm 21421 | The ` ZRHom ` homomorphism... |
| zrh1 21422 | Interpretation of 1 in a r... |
| zrh0 21423 | Interpretation of 0 in a r... |
| zrhpropd 21424 | The ` ZZ ` ring homomorphi... |
| zlmval 21425 | Augment an abelian group w... |
| zlmlem 21426 | Lemma for ~ zlmbas and ~ z... |
| zlmbas 21427 | Base set of a ` ZZ ` -modu... |
| zlmplusg 21428 | Group operation of a ` ZZ ... |
| zlmmulr 21429 | Ring operation of a ` ZZ `... |
| zlmsca 21430 | Scalar ring of a ` ZZ ` -m... |
| zlmvsca 21431 | Scalar multiplication oper... |
| zlmlmod 21432 | The ` ZZ ` -module operati... |
| chrval 21433 | Definition substitution of... |
| chrcl 21434 | Closure of the characteris... |
| chrid 21435 | The canonical ` ZZ ` ring ... |
| chrdvds 21436 | The ` ZZ ` ring homomorphi... |
| chrcong 21437 | If two integers are congru... |
| dvdschrmulg 21438 | In a ring, any multiple of... |
| fermltlchr 21439 | A generalization of Fermat... |
| chrnzr 21440 | Nonzero rings are precisel... |
| chrrhm 21441 | The characteristic restric... |
| domnchr 21442 | The characteristic of a do... |
| znlidl 21443 | The set ` n ZZ ` is an ide... |
| zncrng2 21444 | Making a commutative ring ... |
| znval 21445 | The value of the ` Z/nZ ` ... |
| znle 21446 | The value of the ` Z/nZ ` ... |
| znval2 21447 | Self-referential expressio... |
| znbaslem 21448 | Lemma for ~ znbas . (Cont... |
| znbas2 21449 | The base set of ` Z/nZ ` i... |
| znadd 21450 | The additive structure of ... |
| znmul 21451 | The multiplicative structu... |
| znzrh 21452 | The ` ZZ ` ring homomorphi... |
| znbas 21453 | The base set of ` Z/nZ ` s... |
| zncrng 21454 | ` Z/nZ ` is a commutative ... |
| znzrh2 21455 | The ` ZZ ` ring homomorphi... |
| znzrhval 21456 | The ` ZZ ` ring homomorphi... |
| znzrhfo 21457 | The ` ZZ ` ring homomorphi... |
| zncyg 21458 | The group ` ZZ / n ZZ ` is... |
| zndvds 21459 | Express equality of equiva... |
| zndvds0 21460 | Special case of ~ zndvds w... |
| znf1o 21461 | The function ` F ` enumera... |
| zzngim 21462 | The ` ZZ ` ring homomorphi... |
| znle2 21463 | The ordering of the ` Z/nZ... |
| znleval 21464 | The ordering of the ` Z/nZ... |
| znleval2 21465 | The ordering of the ` Z/nZ... |
| zntoslem 21466 | Lemma for ~ zntos . (Cont... |
| zntos 21467 | The ` Z/nZ ` structure is ... |
| znhash 21468 | The ` Z/nZ ` structure has... |
| znfi 21469 | The ` Z/nZ ` structure is ... |
| znfld 21470 | The ` Z/nZ ` structure is ... |
| znidomb 21471 | The ` Z/nZ ` structure is ... |
| znchr 21472 | Cyclic rings are defined b... |
| znunit 21473 | The units of ` Z/nZ ` are ... |
| znunithash 21474 | The size of the unit group... |
| znrrg 21475 | The regular elements of ` ... |
| cygznlem1 21476 | Lemma for ~ cygzn . (Cont... |
| cygznlem2a 21477 | Lemma for ~ cygzn . (Cont... |
| cygznlem2 21478 | Lemma for ~ cygzn . (Cont... |
| cygznlem3 21479 | A cyclic group with ` n ` ... |
| cygzn 21480 | A cyclic group with ` n ` ... |
| cygth 21481 | The "fundamental theorem o... |
| cyggic 21482 | Cyclic groups are isomorph... |
| frgpcyg 21483 | A free group is cyclic iff... |
| freshmansdream 21484 | For a prime number ` P ` ,... |
| frobrhm 21485 | In a commutative ring with... |
| cnmsgnsubg 21486 | The signs form a multiplic... |
| cnmsgnbas 21487 | The base set of the sign s... |
| cnmsgngrp 21488 | The group of signs under m... |
| psgnghm 21489 | The sign is a homomorphism... |
| psgnghm2 21490 | The sign is a homomorphism... |
| psgninv 21491 | The sign of a permutation ... |
| psgnco 21492 | Multiplicativity of the pe... |
| zrhpsgnmhm 21493 | Embedding of permutation s... |
| zrhpsgninv 21494 | The embedded sign of a per... |
| evpmss 21495 | Even permutations are perm... |
| psgnevpmb 21496 | A class is an even permuta... |
| psgnodpm 21497 | A permutation which is odd... |
| psgnevpm 21498 | A permutation which is eve... |
| psgnodpmr 21499 | If a permutation has sign ... |
| zrhpsgnevpm 21500 | The sign of an even permut... |
| zrhpsgnodpm 21501 | The sign of an odd permuta... |
| cofipsgn 21502 | Composition of any class `... |
| zrhpsgnelbas 21503 | Embedding of permutation s... |
| zrhcopsgnelbas 21504 | Embedding of permutation s... |
| evpmodpmf1o 21505 | The function for performin... |
| pmtrodpm 21506 | A transposition is an odd ... |
| psgnfix1 21507 | A permutation of a finite ... |
| psgnfix2 21508 | A permutation of a finite ... |
| psgndiflemB 21509 | Lemma 1 for ~ psgndif . (... |
| psgndiflemA 21510 | Lemma 2 for ~ psgndif . (... |
| psgndif 21511 | Embedding of permutation s... |
| copsgndif 21512 | Embedding of permutation s... |
| rebase 21515 | The base of the field of r... |
| remulg 21516 | The multiplication (group ... |
| resubdrg 21517 | The real numbers form a di... |
| resubgval 21518 | Subtraction in the field o... |
| replusg 21519 | The addition operation of ... |
| remulr 21520 | The multiplication operati... |
| re0g 21521 | The zero element of the fi... |
| re1r 21522 | The unity element of the f... |
| rele2 21523 | The ordering relation of t... |
| relt 21524 | The ordering relation of t... |
| reds 21525 | The distance of the field ... |
| redvr 21526 | The division operation of ... |
| retos 21527 | The real numbers are a tot... |
| refld 21528 | The real numbers form a fi... |
| refldcj 21529 | The conjugation operation ... |
| resrng 21530 | The real numbers form a st... |
| regsumsupp 21531 | The group sum over the rea... |
| rzgrp 21532 | The quotient group ` RR / ... |
| isphl 21537 | The predicate "is a genera... |
| phllvec 21538 | A pre-Hilbert space is a l... |
| phllmod 21539 | A pre-Hilbert space is a l... |
| phlsrng 21540 | The scalar ring of a pre-H... |
| phllmhm 21541 | The inner product of a pre... |
| ipcl 21542 | Closure of the inner produ... |
| ipcj 21543 | Conjugate of an inner prod... |
| iporthcom 21544 | Orthogonality (meaning inn... |
| ip0l 21545 | Inner product with a zero ... |
| ip0r 21546 | Inner product with a zero ... |
| ipeq0 21547 | The inner product of a vec... |
| ipdir 21548 | Distributive law for inner... |
| ipdi 21549 | Distributive law for inner... |
| ip2di 21550 | Distributive law for inner... |
| ipsubdir 21551 | Distributive law for inner... |
| ipsubdi 21552 | Distributive law for inner... |
| ip2subdi 21553 | Distributive law for inner... |
| ipass 21554 | Associative law for inner ... |
| ipassr 21555 | "Associative" law for seco... |
| ipassr2 21556 | "Associative" law for inne... |
| ipffval 21557 | The inner product operatio... |
| ipfval 21558 | The inner product operatio... |
| ipfeq 21559 | If the inner product opera... |
| ipffn 21560 | The inner product operatio... |
| phlipf 21561 | The inner product operatio... |
| ip2eq 21562 | Two vectors are equal iff ... |
| isphld 21563 | Properties that determine ... |
| phlpropd 21564 | If two structures have the... |
| ssipeq 21565 | The inner product on a sub... |
| phssipval 21566 | The inner product on a sub... |
| phssip 21567 | The inner product (as a fu... |
| phlssphl 21568 | A subspace of an inner pro... |
| ocvfval 21575 | The orthocomplement operat... |
| ocvval 21576 | Value of the orthocompleme... |
| elocv 21577 | Elementhood in the orthoco... |
| ocvi 21578 | Property of a member of th... |
| ocvss 21579 | The orthocomplement of a s... |
| ocvocv 21580 | A set is contained in its ... |
| ocvlss 21581 | The orthocomplement of a s... |
| ocv2ss 21582 | Orthocomplements reverse s... |
| ocvin 21583 | An orthocomplement has tri... |
| ocvsscon 21584 | Two ways to say that ` S `... |
| ocvlsp 21585 | The orthocomplement of a l... |
| ocv0 21586 | The orthocomplement of the... |
| ocvz 21587 | The orthocomplement of the... |
| ocv1 21588 | The orthocomplement of the... |
| unocv 21589 | The orthocomplement of a u... |
| iunocv 21590 | The orthocomplement of an ... |
| cssval 21591 | The set of closed subspace... |
| iscss 21592 | The predicate "is a closed... |
| cssi 21593 | Property of a closed subsp... |
| cssss 21594 | A closed subspace is a sub... |
| iscss2 21595 | It is sufficient to prove ... |
| ocvcss 21596 | The orthocomplement of any... |
| cssincl 21597 | The zero subspace is a clo... |
| css0 21598 | The zero subspace is a clo... |
| css1 21599 | The whole space is a close... |
| csslss 21600 | A closed subspace of a pre... |
| lsmcss 21601 | A subset of a pre-Hilbert ... |
| cssmre 21602 | The closed subspaces of a ... |
| mrccss 21603 | The Moore closure correspo... |
| thlval 21604 | Value of the Hilbert latti... |
| thlbas 21605 | Base set of the Hilbert la... |
| thlle 21606 | Ordering on the Hilbert la... |
| thlleval 21607 | Ordering on the Hilbert la... |
| thloc 21608 | Orthocomplement on the Hil... |
| pjfval 21615 | The value of the projectio... |
| pjdm 21616 | A subspace is in the domai... |
| pjpm 21617 | The projection map is a pa... |
| pjfval2 21618 | Value of the projection ma... |
| pjval 21619 | Value of the projection ma... |
| pjdm2 21620 | A subspace is in the domai... |
| pjff 21621 | A projection is a linear o... |
| pjf 21622 | A projection is a function... |
| pjf2 21623 | A projection is a function... |
| pjfo 21624 | A projection is a surjecti... |
| pjcss 21625 | A projection subspace is a... |
| ocvpj 21626 | The orthocomplement of a p... |
| ishil 21627 | The predicate "is a Hilber... |
| ishil2 21628 | The predicate "is a Hilber... |
| isobs 21629 | The predicate "is an ortho... |
| obsip 21630 | The inner product of two e... |
| obsipid 21631 | A basis element has length... |
| obsrcl 21632 | Reverse closure for an ort... |
| obsss 21633 | An orthonormal basis is a ... |
| obsne0 21634 | A basis element is nonzero... |
| obsocv 21635 | An orthonormal basis has t... |
| obs2ocv 21636 | The double orthocomplement... |
| obselocv 21637 | A basis element is in the ... |
| obs2ss 21638 | A basis has no proper subs... |
| obslbs 21639 | An orthogonal basis is a l... |
| reldmdsmm 21642 | The direct sum is a well-b... |
| dsmmval 21643 | Value of the module direct... |
| dsmmbase 21644 | Base set of the module dir... |
| dsmmval2 21645 | Self-referential definitio... |
| dsmmbas2 21646 | Base set of the direct sum... |
| dsmmfi 21647 | For finite products, the d... |
| dsmmelbas 21648 | Membership in the finitely... |
| dsmm0cl 21649 | The all-zero vector is con... |
| dsmmacl 21650 | The finite hull is closed ... |
| prdsinvgd2 21651 | Negation of a single coord... |
| dsmmsubg 21652 | The finite hull of a produ... |
| dsmmlss 21653 | The finite hull of a produ... |
| dsmmlmod 21654 | The direct sum of a family... |
| frlmval 21657 | Value of the "free module"... |
| frlmlmod 21658 | The free module is a modul... |
| frlmpws 21659 | The free module as a restr... |
| frlmlss 21660 | The base set of the free m... |
| frlmpwsfi 21661 | The finite free module is ... |
| frlmsca 21662 | The ring of scalars of a f... |
| frlm0 21663 | Zero in a free module (rin... |
| frlmbas 21664 | Base set of the free modul... |
| frlmelbas 21665 | Membership in the base set... |
| frlmrcl 21666 | If a free module is inhabi... |
| frlmbasfsupp 21667 | Elements of the free modul... |
| frlmbasmap 21668 | Elements of the free modul... |
| frlmbasf 21669 | Elements of the free modul... |
| frlmlvec 21670 | The free module over a div... |
| frlmfibas 21671 | The base set of the finite... |
| elfrlmbasn0 21672 | If the dimension of a free... |
| frlmplusgval 21673 | Addition in a free module.... |
| frlmsubgval 21674 | Subtraction in a free modu... |
| frlmvscafval 21675 | Scalar multiplication in a... |
| frlmvplusgvalc 21676 | Coordinates of a sum with ... |
| frlmvscaval 21677 | Coordinates of a scalar mu... |
| frlmplusgvalb 21678 | Addition in a free module ... |
| frlmvscavalb 21679 | Scalar multiplication in a... |
| frlmvplusgscavalb 21680 | Addition combined with sca... |
| frlmgsum 21681 | Finite commutative sums in... |
| frlmsplit2 21682 | Restriction is homomorphic... |
| frlmsslss 21683 | A subset of a free module ... |
| frlmsslss2 21684 | A subset of a free module ... |
| frlmbas3 21685 | An element of the base set... |
| mpofrlmd 21686 | Elements of the free modul... |
| frlmip 21687 | The inner product of a fre... |
| frlmipval 21688 | The inner product of a fre... |
| frlmphllem 21689 | Lemma for ~ frlmphl . (Co... |
| frlmphl 21690 | Conditions for a free modu... |
| uvcfval 21693 | Value of the unit-vector g... |
| uvcval 21694 | Value of a single unit vec... |
| uvcvval 21695 | Value of a unit vector coo... |
| uvcvvcl 21696 | A coordinate of a unit vec... |
| uvcvvcl2 21697 | A unit vector coordinate i... |
| uvcvv1 21698 | The unit vector is one at ... |
| uvcvv0 21699 | The unit vector is zero at... |
| uvcff 21700 | Domain and codomain of the... |
| uvcf1 21701 | In a nonzero ring, each un... |
| uvcresum 21702 | Any element of a free modu... |
| frlmssuvc1 21703 | A scalar multiple of a uni... |
| frlmssuvc2 21704 | A nonzero scalar multiple ... |
| frlmsslsp 21705 | A subset of a free module ... |
| frlmlbs 21706 | The unit vectors comprise ... |
| frlmup1 21707 | Any assignment of unit vec... |
| frlmup2 21708 | The evaluation map has the... |
| frlmup3 21709 | The range of such an evalu... |
| frlmup4 21710 | Universal property of the ... |
| ellspd 21711 | The elements of the span o... |
| elfilspd 21712 | Simplified version of ~ el... |
| rellindf 21717 | The independent-family pre... |
| islinds 21718 | Property of an independent... |
| linds1 21719 | An independent set of vect... |
| linds2 21720 | An independent set of vect... |
| islindf 21721 | Property of an independent... |
| islinds2 21722 | Expanded property of an in... |
| islindf2 21723 | Property of an independent... |
| lindff 21724 | Functional property of a l... |
| lindfind 21725 | A linearly independent fam... |
| lindsind 21726 | A linearly independent set... |
| lindfind2 21727 | In a linearly independent ... |
| lindsind2 21728 | In a linearly independent ... |
| lindff1 21729 | A linearly independent fam... |
| lindfrn 21730 | The range of an independen... |
| f1lindf 21731 | Rearranging and deleting e... |
| lindfres 21732 | Any restriction of an inde... |
| lindsss 21733 | Any subset of an independe... |
| f1linds 21734 | A family constructed from ... |
| islindf3 21735 | In a nonzero ring, indepen... |
| lindfmm 21736 | Linear independence of a f... |
| lindsmm 21737 | Linear independence of a s... |
| lindsmm2 21738 | The monomorphic image of a... |
| lsslindf 21739 | Linear independence is unc... |
| lsslinds 21740 | Linear independence is unc... |
| islbs4 21741 | A basis is an independent ... |
| lbslinds 21742 | A basis is independent. (... |
| islinds3 21743 | A subset is linearly indep... |
| islinds4 21744 | A set is independent in a ... |
| lmimlbs 21745 | The isomorphic image of a ... |
| lmiclbs 21746 | Having a basis is an isomo... |
| islindf4 21747 | A family is independent if... |
| islindf5 21748 | A family is independent if... |
| indlcim 21749 | An independent, spanning f... |
| lbslcic 21750 | A module with a basis is i... |
| lmisfree 21751 | A module has a basis iff i... |
| lvecisfrlm 21752 | Every vector space is isom... |
| lmimco 21753 | The composition of two iso... |
| lmictra 21754 | Module isomorphism is tran... |
| uvcf1o 21755 | In a nonzero ring, the map... |
| uvcendim 21756 | In a nonzero ring, the num... |
| frlmisfrlm 21757 | A free module is isomorphi... |
| frlmiscvec 21758 | Every free module is isomo... |
| isassa 21765 | The properties of an assoc... |
| assalem 21766 | The properties of an assoc... |
| assaass 21767 | Left-associative property ... |
| assaassr 21768 | Right-associative property... |
| assalmod 21769 | An associative algebra is ... |
| assaring 21770 | An associative algebra is ... |
| assasca 21771 | The scalars of an associat... |
| assa2ass 21772 | Left- and right-associativ... |
| assa2ass2 21773 | Left- and right-associativ... |
| isassad 21774 | Sufficient condition for b... |
| issubassa3 21775 | A subring that is also a s... |
| issubassa 21776 | The subalgebras of an asso... |
| sraassab 21777 | A subring algebra is an as... |
| sraassa 21778 | The subring algebra over a... |
| sraassaOLD 21779 | Obsolete version of ~ sraa... |
| rlmassa 21780 | The ring module over a com... |
| assapropd 21781 | If two structures have the... |
| aspval 21782 | Value of the algebraic clo... |
| asplss 21783 | The algebraic span of a se... |
| aspid 21784 | The algebraic span of a su... |
| aspsubrg 21785 | The algebraic span of a se... |
| aspss 21786 | Span preserves subset orde... |
| aspssid 21787 | A set of vectors is a subs... |
| asclfval 21788 | Function value of the alge... |
| asclval 21789 | Value of a mapped algebra ... |
| asclfn 21790 | Unconditional functionalit... |
| asclf 21791 | The algebra scalar lifting... |
| asclghm 21792 | The algebra scalar lifting... |
| ascl0 21793 | The scalar 0 embedded into... |
| ascl1 21794 | The scalar 1 embedded into... |
| asclmul1 21795 | Left multiplication by a l... |
| asclmul2 21796 | Right multiplication by a ... |
| ascldimul 21797 | The algebra scalar lifting... |
| asclinvg 21798 | The group inverse (negatio... |
| asclrhm 21799 | The algebra scalar lifting... |
| rnascl 21800 | The set of lifted scalars ... |
| issubassa2 21801 | A subring of a unital alge... |
| rnasclsubrg 21802 | The scalar multiples of th... |
| rnasclmulcl 21803 | (Vector) multiplication is... |
| rnasclassa 21804 | The scalar multiples of th... |
| ressascl 21805 | The lifting of scalars is ... |
| asclpropd 21806 | If two structures have the... |
| aspval2 21807 | The algebraic closure is t... |
| assamulgscmlem1 21808 | Lemma 1 for ~ assamulgscm ... |
| assamulgscmlem2 21809 | Lemma for ~ assamulgscm (i... |
| assamulgscm 21810 | Exponentiation of a scalar... |
| asclmulg 21811 | Apply group multiplication... |
| zlmassa 21812 | The ` ZZ ` -module operati... |
| reldmpsr 21823 | The multivariate power ser... |
| psrval 21824 | Value of the multivariate ... |
| psrvalstr 21825 | The multivariate power ser... |
| psrbag 21826 | Elementhood in the set of ... |
| psrbagf 21827 | A finite bag is a function... |
| psrbagfsupp 21828 | Finite bags have finite su... |
| snifpsrbag 21829 | A bag containing one eleme... |
| fczpsrbag 21830 | The constant function equa... |
| psrbaglesupp 21831 | The support of a dominated... |
| psrbaglecl 21832 | The set of finite bags is ... |
| psrbagaddcl 21833 | The sum of two finite bags... |
| psrbagcon 21834 | The analogue of the statem... |
| psrbaglefi 21835 | There are finitely many ba... |
| psrbagconcl 21836 | The complement of a bag is... |
| psrbagleadd1 21837 | The analogue of " ` X <_ F... |
| psrbagconf1o 21838 | Bag complementation is a b... |
| gsumbagdiaglem 21839 | Lemma for ~ gsumbagdiag . ... |
| gsumbagdiag 21840 | Two-dimensional commutatio... |
| psrass1lem 21841 | A group sum commutation us... |
| psrbas 21842 | The base set of the multiv... |
| psrelbas 21843 | An element of the set of p... |
| psrelbasfun 21844 | An element of the set of p... |
| psrplusg 21845 | The addition operation of ... |
| psradd 21846 | The addition operation of ... |
| psraddcl 21847 | Closure of the power serie... |
| psraddclOLD 21848 | Obsolete version of ~ psra... |
| rhmpsrlem1 21849 | Lemma for ~ rhmpsr et al. ... |
| rhmpsrlem2 21850 | Lemma for ~ rhmpsr et al. ... |
| psrmulr 21851 | The multiplication operati... |
| psrmulfval 21852 | The multiplication operati... |
| psrmulval 21853 | The multiplication operati... |
| psrmulcllem 21854 | Closure of the power serie... |
| psrmulcl 21855 | Closure of the power serie... |
| psrsca 21856 | The scalar field of the mu... |
| psrvscafval 21857 | The scalar multiplication ... |
| psrvsca 21858 | The scalar multiplication ... |
| psrvscaval 21859 | The scalar multiplication ... |
| psrvscacl 21860 | Closure of the power serie... |
| psr0cl 21861 | The zero element of the ri... |
| psr0lid 21862 | The zero element of the ri... |
| psrnegcl 21863 | The negative function in t... |
| psrlinv 21864 | The negative function in t... |
| psrgrp 21865 | The ring of power series i... |
| psrgrpOLD 21866 | Obsolete version of ~ psrg... |
| psr0 21867 | The zero element of the ri... |
| psrneg 21868 | The negative function of t... |
| psrlmod 21869 | The ring of power series i... |
| psr1cl 21870 | The identity element of th... |
| psrlidm 21871 | The identity element of th... |
| psrridm 21872 | The identity element of th... |
| psrass1 21873 | Associative identity for t... |
| psrdi 21874 | Distributive law for the r... |
| psrdir 21875 | Distributive law for the r... |
| psrass23l 21876 | Associative identity for t... |
| psrcom 21877 | Commutative law for the ri... |
| psrass23 21878 | Associative identities for... |
| psrring 21879 | The ring of power series i... |
| psr1 21880 | The identity element of th... |
| psrcrng 21881 | The ring of power series i... |
| psrassa 21882 | The ring of power series i... |
| resspsrbas 21883 | A restricted power series ... |
| resspsradd 21884 | A restricted power series ... |
| resspsrmul 21885 | A restricted power series ... |
| resspsrvsca 21886 | A restricted power series ... |
| subrgpsr 21887 | A subring of the base ring... |
| psrascl 21888 | Value of the scalar inject... |
| psrasclcl 21889 | A scalar is lifted into a ... |
| mvrfval 21890 | Value of the generating el... |
| mvrval 21891 | Value of the generating el... |
| mvrval2 21892 | Value of the generating el... |
| mvrid 21893 | The ` X i ` -th coefficien... |
| mvrf 21894 | The power series variable ... |
| mvrf1 21895 | The power series variable ... |
| mvrcl2 21896 | A power series variable is... |
| reldmmpl 21897 | The multivariate polynomia... |
| mplval 21898 | Value of the set of multiv... |
| mplbas 21899 | Base set of the set of mul... |
| mplelbas 21900 | Property of being a polyno... |
| mvrcl 21901 | A power series variable is... |
| mvrf2 21902 | The power series/polynomia... |
| mplrcl 21903 | Reverse closure for the po... |
| mplelsfi 21904 | A polynomial treated as a ... |
| mplval2 21905 | Self-referential expressio... |
| mplbasss 21906 | The set of polynomials is ... |
| mplelf 21907 | A polynomial is defined as... |
| mplsubglem 21908 | If ` A ` is an ideal of se... |
| mpllsslem 21909 | If ` A ` is an ideal of su... |
| mplsubglem2 21910 | Lemma for ~ mplsubg and ~ ... |
| mplsubg 21911 | The set of polynomials is ... |
| mpllss 21912 | The set of polynomials is ... |
| mplsubrglem 21913 | Lemma for ~ mplsubrg . (C... |
| mplsubrg 21914 | The set of polynomials is ... |
| mpl0 21915 | The zero polynomial. (Con... |
| mplplusg 21916 | Value of addition in a pol... |
| mplmulr 21917 | Value of multiplication in... |
| mpladd 21918 | The addition operation on ... |
| mplneg 21919 | The negative function on m... |
| mplmul 21920 | The multiplication operati... |
| mpl1 21921 | The identity element of th... |
| mplsca 21922 | The scalar field of a mult... |
| mplvsca2 21923 | The scalar multiplication ... |
| mplvsca 21924 | The scalar multiplication ... |
| mplvscaval 21925 | The scalar multiplication ... |
| mplgrp 21926 | The polynomial ring is a g... |
| mpllmod 21927 | The polynomial ring is a l... |
| mplring 21928 | The polynomial ring is a r... |
| mpllvec 21929 | The polynomial ring is a v... |
| mplcrng 21930 | The polynomial ring is a c... |
| mplassa 21931 | The polynomial ring is an ... |
| mplringd 21932 | The polynomial ring is a r... |
| mpllmodd 21933 | The polynomial ring is a l... |
| ressmplbas2 21934 | The base set of a restrict... |
| ressmplbas 21935 | A restricted polynomial al... |
| ressmpladd 21936 | A restricted polynomial al... |
| ressmplmul 21937 | A restricted polynomial al... |
| ressmplvsca 21938 | A restricted power series ... |
| subrgmpl 21939 | A subring of the base ring... |
| subrgmvr 21940 | The variables in a subring... |
| subrgmvrf 21941 | The variables in a polynom... |
| mplmon 21942 | A monomial is a polynomial... |
| mplmonmul 21943 | The product of two monomia... |
| mplcoe1 21944 | Decompose a polynomial int... |
| mplcoe3 21945 | Decompose a monomial in on... |
| mplcoe5lem 21946 | Lemma for ~ mplcoe4 . (Co... |
| mplcoe5 21947 | Decompose a monomial into ... |
| mplcoe2 21948 | Decompose a monomial into ... |
| mplbas2 21949 | An alternative expression ... |
| ltbval 21950 | Value of the well-order on... |
| ltbwe 21951 | The finite bag order is a ... |
| reldmopsr 21952 | Lemma for ordered power se... |
| opsrval 21953 | The value of the "ordered ... |
| opsrle 21954 | An alternative expression ... |
| opsrval2 21955 | Self-referential expressio... |
| opsrbaslem 21956 | Get a component of the ord... |
| opsrbas 21957 | The base set of the ordere... |
| opsrplusg 21958 | The addition operation of ... |
| opsrmulr 21959 | The multiplication operati... |
| opsrvsca 21960 | The scalar product operati... |
| opsrsca 21961 | The scalar ring of the ord... |
| opsrtoslem1 21962 | Lemma for ~ opsrtos . (Co... |
| opsrtoslem2 21963 | Lemma for ~ opsrtos . (Co... |
| opsrtos 21964 | The ordered power series s... |
| opsrso 21965 | The ordered power series s... |
| opsrcrng 21966 | The ring of ordered power ... |
| opsrassa 21967 | The ring of ordered power ... |
| mplmon2 21968 | Express a scaled monomial.... |
| psrbag0 21969 | The empty bag is a bag. (... |
| psrbagsn 21970 | A singleton bag is a bag. ... |
| mplascl 21971 | Value of the scalar inject... |
| mplasclf 21972 | The scalar injection is a ... |
| subrgascl 21973 | The scalar injection funct... |
| subrgasclcl 21974 | The scalars in a polynomia... |
| mplmon2cl 21975 | A scaled monomial is a pol... |
| mplmon2mul 21976 | Product of scaled monomial... |
| mplind 21977 | Prove a property of polyno... |
| mplcoe4 21978 | Decompose a polynomial int... |
| evlslem4 21983 | The support of a tensor pr... |
| psrbagev1 21984 | A bag of multipliers provi... |
| psrbagev2 21985 | Closure of a sum using a b... |
| evlslem2 21986 | A linear function on the p... |
| evlslem3 21987 | Lemma for ~ evlseu . Poly... |
| evlslem6 21988 | Lemma for ~ evlseu . Fini... |
| evlslem1 21989 | Lemma for ~ evlseu , give ... |
| evlseu 21990 | For a given interpretation... |
| reldmevls 21991 | Well-behaved binary operat... |
| mpfrcl 21992 | Reverse closure for the se... |
| evlsval 21993 | Value of the polynomial ev... |
| evlsval2 21994 | Characterizing properties ... |
| evlsrhm 21995 | Polynomial evaluation is a... |
| evlssca 21996 | Polynomial evaluation maps... |
| evlsvar 21997 | Polynomial evaluation maps... |
| evlsgsumadd 21998 | Polynomial evaluation maps... |
| evlsgsummul 21999 | Polynomial evaluation maps... |
| evlspw 22000 | Polynomial evaluation for ... |
| evlsvarpw 22001 | Polynomial evaluation for ... |
| evlval 22002 | Value of the simple/same r... |
| evlrhm 22003 | The simple evaluation map ... |
| evlsscasrng 22004 | The evaluation of a scalar... |
| evlsca 22005 | Simple polynomial evaluati... |
| evlsvarsrng 22006 | The evaluation of the vari... |
| evlvar 22007 | Simple polynomial evaluati... |
| mpfconst 22008 | Constants are multivariate... |
| mpfproj 22009 | Projections are multivaria... |
| mpfsubrg 22010 | Polynomial functions are a... |
| mpff 22011 | Polynomial functions are f... |
| mpfaddcl 22012 | The sum of multivariate po... |
| mpfmulcl 22013 | The product of multivariat... |
| mpfind 22014 | Prove a property of polyno... |
| selvffval 22020 | Value of the "variable sel... |
| selvfval 22021 | Value of the "variable sel... |
| selvval 22022 | Value of the "variable sel... |
| reldmmhp 22024 | The domain of the homogene... |
| mhpfval 22025 | Value of the "homogeneous ... |
| mhpval 22026 | Value of the "homogeneous ... |
| ismhp 22027 | Property of being a homoge... |
| ismhp2 22028 | Deduce a homogeneous polyn... |
| ismhp3 22029 | A polynomial is homogeneou... |
| mhprcl 22030 | Reverse closure for homoge... |
| mhpmpl 22031 | A homogeneous polynomial i... |
| mhpdeg 22032 | All nonzero terms of a hom... |
| mhp0cl 22033 | The zero polynomial is hom... |
| mhpsclcl 22034 | A scalar (or constant) pol... |
| mhpvarcl 22035 | A power series variable is... |
| mhpmulcl 22036 | A product of homogeneous p... |
| mhppwdeg 22037 | Degree of a homogeneous po... |
| mhpaddcl 22038 | Homogeneous polynomials ar... |
| mhpinvcl 22039 | Homogeneous polynomials ar... |
| mhpsubg 22040 | Homogeneous polynomials fo... |
| mhpvscacl 22041 | Homogeneous polynomials ar... |
| mhplss 22042 | Homogeneous polynomials fo... |
| psdffval 22044 | Value of the power series ... |
| psdfval 22045 | Give a map between power s... |
| psdval 22046 | Evaluate the partial deriv... |
| psdcoef 22047 | Coefficient of a term of t... |
| psdcl 22048 | The derivative of a power ... |
| psdmplcl 22049 | The derivative of a polyno... |
| psdadd 22050 | The derivative of a sum is... |
| psdvsca 22051 | The derivative of a scaled... |
| psdmullem 22052 | Lemma for ~ psdmul . Tran... |
| psdmul 22053 | Product rule for power ser... |
| psd1 22054 | The derivative of one is z... |
| psdascl 22055 | The derivative of a consta... |
| psdmvr 22056 | The partial derivative of ... |
| psdpw 22057 | Power rule for partial der... |
| psr1baslem 22069 | The set of finite bags on ... |
| psr1val 22070 | Value of the ring of univa... |
| psr1crng 22071 | The ring of univariate pow... |
| psr1assa 22072 | The ring of univariate pow... |
| psr1tos 22073 | The ordered power series s... |
| psr1bas2 22074 | The base set of the ring o... |
| psr1bas 22075 | The base set of the ring o... |
| vr1val 22076 | The value of the generator... |
| vr1cl2 22077 | The variable ` X ` is a me... |
| ply1val 22078 | The value of the set of un... |
| ply1bas 22079 | The value of the base set ... |
| ply1basOLD 22080 | Obsolete version of ~ ply1... |
| ply1lss 22081 | Univariate polynomials for... |
| ply1subrg 22082 | Univariate polynomials for... |
| ply1crng 22083 | The ring of univariate pol... |
| ply1assa 22084 | The ring of univariate pol... |
| psr1bascl 22085 | A univariate power series ... |
| psr1basf 22086 | Univariate power series ba... |
| ply1basf 22087 | Univariate polynomial base... |
| ply1bascl 22088 | A univariate polynomial is... |
| ply1bascl2 22089 | A univariate polynomial is... |
| coe1fval 22090 | Value of the univariate po... |
| coe1fv 22091 | Value of an evaluated coef... |
| fvcoe1 22092 | Value of a multivariate co... |
| coe1fval3 22093 | Univariate power series co... |
| coe1f2 22094 | Functionality of univariat... |
| coe1fval2 22095 | Univariate polynomial coef... |
| coe1f 22096 | Functionality of univariat... |
| coe1fvalcl 22097 | A coefficient of a univari... |
| coe1sfi 22098 | Finite support of univaria... |
| coe1fsupp 22099 | The coefficient vector of ... |
| mptcoe1fsupp 22100 | A mapping involving coeffi... |
| coe1ae0 22101 | The coefficient vector of ... |
| vr1cl 22102 | The generator of a univari... |
| opsr0 22103 | Zero in the ordered power ... |
| opsr1 22104 | One in the ordered power s... |
| psr1plusg 22105 | Value of addition in a uni... |
| psr1vsca 22106 | Value of scalar multiplica... |
| psr1mulr 22107 | Value of multiplication in... |
| ply1plusg 22108 | Value of addition in a uni... |
| ply1vsca 22109 | Value of scalar multiplica... |
| ply1mulr 22110 | Value of multiplication in... |
| ply1ass23l 22111 | Associative identity with ... |
| ressply1bas2 22112 | The base set of a restrict... |
| ressply1bas 22113 | A restricted polynomial al... |
| ressply1add 22114 | A restricted polynomial al... |
| ressply1mul 22115 | A restricted polynomial al... |
| ressply1vsca 22116 | A restricted power series ... |
| subrgply1 22117 | A subring of the base ring... |
| gsumply1subr 22118 | Evaluate a group sum in a ... |
| psrbaspropd 22119 | Property deduction for pow... |
| psrplusgpropd 22120 | Property deduction for pow... |
| mplbaspropd 22121 | Property deduction for pol... |
| psropprmul 22122 | Reversing multiplication i... |
| ply1opprmul 22123 | Reversing multiplication i... |
| 00ply1bas 22124 | Lemma for ~ ply1basfvi and... |
| ply1basfvi 22125 | Protection compatibility o... |
| ply1plusgfvi 22126 | Protection compatibility o... |
| ply1baspropd 22127 | Property deduction for uni... |
| ply1plusgpropd 22128 | Property deduction for uni... |
| opsrring 22129 | Ordered power series form ... |
| opsrlmod 22130 | Ordered power series form ... |
| psr1ring 22131 | Univariate power series fo... |
| ply1ring 22132 | Univariate polynomials for... |
| psr1lmod 22133 | Univariate power series fo... |
| psr1sca 22134 | Scalars of a univariate po... |
| psr1sca2 22135 | Scalars of a univariate po... |
| ply1lmod 22136 | Univariate polynomials for... |
| ply1sca 22137 | Scalars of a univariate po... |
| ply1sca2 22138 | Scalars of a univariate po... |
| ply1ascl0 22139 | The zero scalar as a polyn... |
| ply1ascl1 22140 | The multiplicative identit... |
| ply1mpl0 22141 | The univariate polynomial ... |
| ply10s0 22142 | Zero times a univariate po... |
| ply1mpl1 22143 | The univariate polynomial ... |
| ply1ascl 22144 | The univariate polynomial ... |
| subrg1ascl 22145 | The scalar injection funct... |
| subrg1asclcl 22146 | The scalars in a polynomia... |
| subrgvr1 22147 | The variables in a subring... |
| subrgvr1cl 22148 | The variables in a polynom... |
| coe1z 22149 | The coefficient vector of ... |
| coe1add 22150 | The coefficient vector of ... |
| coe1addfv 22151 | A particular coefficient o... |
| coe1subfv 22152 | A particular coefficient o... |
| coe1mul2lem1 22153 | An equivalence for ~ coe1m... |
| coe1mul2lem2 22154 | An equivalence for ~ coe1m... |
| coe1mul2 22155 | The coefficient vector of ... |
| coe1mul 22156 | The coefficient vector of ... |
| ply1moncl 22157 | Closure of the expression ... |
| ply1tmcl 22158 | Closure of the expression ... |
| coe1tm 22159 | Coefficient vector of a po... |
| coe1tmfv1 22160 | Nonzero coefficient of a p... |
| coe1tmfv2 22161 | Zero coefficient of a poly... |
| coe1tmmul2 22162 | Coefficient vector of a po... |
| coe1tmmul 22163 | Coefficient vector of a po... |
| coe1tmmul2fv 22164 | Function value of a right-... |
| coe1pwmul 22165 | Coefficient vector of a po... |
| coe1pwmulfv 22166 | Function value of a right-... |
| ply1scltm 22167 | A scalar is a term with ze... |
| coe1sclmul 22168 | Coefficient vector of a po... |
| coe1sclmulfv 22169 | A single coefficient of a ... |
| coe1sclmul2 22170 | Coefficient vector of a po... |
| ply1sclf 22171 | A scalar polynomial is a p... |
| ply1sclcl 22172 | The value of the algebra s... |
| coe1scl 22173 | Coefficient vector of a sc... |
| ply1sclid 22174 | Recover the base scalar fr... |
| ply1sclf1 22175 | The polynomial scalar func... |
| ply1scl0 22176 | The zero scalar is zero. ... |
| ply1scl0OLD 22177 | Obsolete version of ~ ply1... |
| ply1scln0 22178 | Nonzero scalars create non... |
| ply1scl1 22179 | The one scalar is the unit... |
| ply1scl1OLD 22180 | Obsolete version of ~ ply1... |
| ply1idvr1 22181 | The identity of a polynomi... |
| ply1idvr1OLD 22182 | Obsolete version of ~ ply1... |
| cply1mul 22183 | The product of two constan... |
| ply1coefsupp 22184 | The decomposition of a uni... |
| ply1coe 22185 | Decompose a univariate pol... |
| eqcoe1ply1eq 22186 | Two polynomials over the s... |
| ply1coe1eq 22187 | Two polynomials over the s... |
| cply1coe0 22188 | All but the first coeffici... |
| cply1coe0bi 22189 | A polynomial is constant (... |
| coe1fzgsumdlem 22190 | Lemma for ~ coe1fzgsumd (i... |
| coe1fzgsumd 22191 | Value of an evaluated coef... |
| ply1scleq 22192 | Equality of a constant pol... |
| ply1chr 22193 | The characteristic of a po... |
| gsumsmonply1 22194 | A finite group sum of scal... |
| gsummoncoe1 22195 | A coefficient of the polyn... |
| gsumply1eq 22196 | Two univariate polynomials... |
| lply1binom 22197 | The binomial theorem for l... |
| lply1binomsc 22198 | The binomial theorem for l... |
| ply1fermltlchr 22199 | Fermat's little theorem fo... |
| reldmevls1 22204 | Well-behaved binary operat... |
| ply1frcl 22205 | Reverse closure for the se... |
| evls1fval 22206 | Value of the univariate po... |
| evls1val 22207 | Value of the univariate po... |
| evls1rhmlem 22208 | Lemma for ~ evl1rhm and ~ ... |
| evls1rhm 22209 | Polynomial evaluation is a... |
| evls1sca 22210 | Univariate polynomial eval... |
| evls1gsumadd 22211 | Univariate polynomial eval... |
| evls1gsummul 22212 | Univariate polynomial eval... |
| evls1pw 22213 | Univariate polynomial eval... |
| evls1varpw 22214 | Univariate polynomial eval... |
| evl1fval 22215 | Value of the simple/same r... |
| evl1val 22216 | Value of the simple/same r... |
| evl1fval1lem 22217 | Lemma for ~ evl1fval1 . (... |
| evl1fval1 22218 | Value of the simple/same r... |
| evl1rhm 22219 | Polynomial evaluation is a... |
| fveval1fvcl 22220 | The function value of the ... |
| evl1sca 22221 | Polynomial evaluation maps... |
| evl1scad 22222 | Polynomial evaluation buil... |
| evl1var 22223 | Polynomial evaluation maps... |
| evl1vard 22224 | Polynomial evaluation buil... |
| evls1var 22225 | Univariate polynomial eval... |
| evls1scasrng 22226 | The evaluation of a scalar... |
| evls1varsrng 22227 | The evaluation of the vari... |
| evl1addd 22228 | Polynomial evaluation buil... |
| evl1subd 22229 | Polynomial evaluation buil... |
| evl1muld 22230 | Polynomial evaluation buil... |
| evl1vsd 22231 | Polynomial evaluation buil... |
| evl1expd 22232 | Polynomial evaluation buil... |
| pf1const 22233 | Constants are polynomial f... |
| pf1id 22234 | The identity is a polynomi... |
| pf1subrg 22235 | Polynomial functions are a... |
| pf1rcl 22236 | Reverse closure for the se... |
| pf1f 22237 | Polynomial functions are f... |
| mpfpf1 22238 | Convert a multivariate pol... |
| pf1mpf 22239 | Convert a univariate polyn... |
| pf1addcl 22240 | The sum of multivariate po... |
| pf1mulcl 22241 | The product of multivariat... |
| pf1ind 22242 | Prove a property of polyno... |
| evl1gsumdlem 22243 | Lemma for ~ evl1gsumd (ind... |
| evl1gsumd 22244 | Polynomial evaluation buil... |
| evl1gsumadd 22245 | Univariate polynomial eval... |
| evl1gsumaddval 22246 | Value of a univariate poly... |
| evl1gsummul 22247 | Univariate polynomial eval... |
| evl1varpw 22248 | Univariate polynomial eval... |
| evl1varpwval 22249 | Value of a univariate poly... |
| evl1scvarpw 22250 | Univariate polynomial eval... |
| evl1scvarpwval 22251 | Value of a univariate poly... |
| evl1gsummon 22252 | Value of a univariate poly... |
| evls1scafv 22253 | Value of the univariate po... |
| evls1expd 22254 | Univariate polynomial eval... |
| evls1varpwval 22255 | Univariate polynomial eval... |
| evls1fpws 22256 | Evaluation of a univariate... |
| ressply1evl 22257 | Evaluation of a univariate... |
| evls1addd 22258 | Univariate polynomial eval... |
| evls1muld 22259 | Univariate polynomial eval... |
| evls1vsca 22260 | Univariate polynomial eval... |
| asclply1subcl 22261 | Closure of the algebra sca... |
| evls1fvcl 22262 | Variant of ~ fveval1fvcl f... |
| evls1maprhm 22263 | The function ` F ` mapping... |
| evls1maplmhm 22264 | The function ` F ` mapping... |
| evls1maprnss 22265 | The function ` F ` mapping... |
| evl1maprhm 22266 | The function ` F ` mapping... |
| mhmcompl 22267 | The composition of a monoi... |
| mhmcoaddmpl 22268 | Show that the ring homomor... |
| rhmcomulmpl 22269 | Show that the ring homomor... |
| rhmmpl 22270 | Provide a ring homomorphis... |
| ply1vscl 22271 | Closure of scalar multipli... |
| mhmcoply1 22272 | The composition of a monoi... |
| rhmply1 22273 | Provide a ring homomorphis... |
| rhmply1vr1 22274 | A ring homomorphism betwee... |
| rhmply1vsca 22275 | Apply a ring homomorphism ... |
| rhmply1mon 22276 | Apply a ring homomorphism ... |
| mamufval 22279 | Functional value of the ma... |
| mamuval 22280 | Multiplication of two matr... |
| mamufv 22281 | A cell in the multiplicati... |
| mamudm 22282 | The domain of the matrix m... |
| mamufacex 22283 | Every solution of the equa... |
| mamures 22284 | Rows in a matrix product a... |
| grpvlinv 22285 | Tuple-wise left inverse in... |
| grpvrinv 22286 | Tuple-wise right inverse i... |
| 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... |
| matvsca 22303 | The matrix ring has the sa... |
| mat0 22304 | The matrix ring has the sa... |
| matinvg 22305 | The matrix ring has the sa... |
| mat0op 22306 | Value of a zero matrix as ... |
| matsca2 22307 | The scalars of the matrix ... |
| matbas2 22308 | The base set of the matrix... |
| matbas2i 22309 | A matrix is a function. (... |
| matbas2d 22310 | The base set of the matrix... |
| eqmat 22311 | Two square matrices of the... |
| matecl 22312 | Each entry (according to W... |
| matecld 22313 | Each entry (according to W... |
| matplusg2 22314 | Addition in the matrix rin... |
| matvsca2 22315 | Scalar multiplication in t... |
| matlmod 22316 | The matrix ring is a linea... |
| matgrp 22317 | The matrix ring is a group... |
| matvscl 22318 | Closure of the scalar mult... |
| matsubg 22319 | The matrix ring has the sa... |
| matplusgcell 22320 | Addition in the matrix rin... |
| matsubgcell 22321 | Subtraction in the matrix ... |
| matinvgcell 22322 | Additive inversion in the ... |
| matvscacell 22323 | Scalar multiplication in t... |
| matgsum 22324 | Finite commutative sums in... |
| matmulr 22325 | Multiplication in the matr... |
| mamumat1cl 22326 | The identity matrix (as op... |
| mat1comp 22327 | The components of the iden... |
| mamulid 22328 | The identity matrix (as op... |
| mamurid 22329 | The identity matrix (as op... |
| matring 22330 | Existence of the matrix ri... |
| matassa 22331 | Existence of the matrix al... |
| matmulcell 22332 | Multiplication in the matr... |
| mpomatmul 22333 | Multiplication of two N x ... |
| mat1 22334 | Value of an identity matri... |
| mat1ov 22335 | Entries of an identity mat... |
| mat1bas 22336 | The identity matrix is a m... |
| matsc 22337 | The identity matrix multip... |
| ofco2 22338 | Distribution law for the f... |
| oftpos 22339 | The transposition of the v... |
| mattposcl 22340 | The transpose of a square ... |
| mattpostpos 22341 | The transpose of the trans... |
| mattposvs 22342 | The transposition of a mat... |
| mattpos1 22343 | The transposition of the i... |
| tposmap 22344 | The transposition of an I ... |
| mamutpos 22345 | Behavior of transposes in ... |
| mattposm 22346 | Multiplying two transposed... |
| matgsumcl 22347 | Closure of a group sum ove... |
| madetsumid 22348 | The identity summand in th... |
| matepmcl 22349 | Each entry of a matrix wit... |
| matepm2cl 22350 | Each entry of a matrix wit... |
| madetsmelbas 22351 | A summand of the determina... |
| madetsmelbas2 22352 | A summand of the determina... |
| mat0dimbas0 22353 | The empty set is the one a... |
| mat0dim0 22354 | The zero of the algebra of... |
| mat0dimid 22355 | The identity of the algebr... |
| mat0dimscm 22356 | The scalar multiplication ... |
| mat0dimcrng 22357 | The algebra of matrices wi... |
| mat1dimelbas 22358 | A matrix with dimension 1 ... |
| mat1dimbas 22359 | A matrix with dimension 1 ... |
| mat1dim0 22360 | The zero of the algebra of... |
| mat1dimid 22361 | The identity of the algebr... |
| mat1dimscm 22362 | The scalar multiplication ... |
| mat1dimmul 22363 | The ring multiplication in... |
| mat1dimcrng 22364 | The algebra of matrices wi... |
| mat1f1o 22365 | There is a 1-1 function fr... |
| mat1rhmval 22366 | The value of the ring homo... |
| mat1rhmelval 22367 | The value of the ring homo... |
| mat1rhmcl 22368 | The value of the ring homo... |
| mat1f 22369 | There is a function from a... |
| mat1ghm 22370 | There is a group homomorph... |
| mat1mhm 22371 | There is a monoid homomorp... |
| mat1rhm 22372 | There is a ring homomorphi... |
| mat1rngiso 22373 | There is a ring isomorphis... |
| mat1ric 22374 | A ring is isomorphic to th... |
| dmatval 22379 | The set of ` N ` x ` N ` d... |
| dmatel 22380 | A ` N ` x ` N ` diagonal m... |
| dmatmat 22381 | An ` N ` x ` N ` diagonal ... |
| dmatid 22382 | The identity matrix is a d... |
| dmatelnd 22383 | An extradiagonal entry of ... |
| dmatmul 22384 | The product of two diagona... |
| dmatsubcl 22385 | The difference of two diag... |
| dmatsgrp 22386 | The set of diagonal matric... |
| dmatmulcl 22387 | The product of two diagona... |
| dmatsrng 22388 | The set of diagonal matric... |
| dmatcrng 22389 | The subring of diagonal ma... |
| dmatscmcl 22390 | The multiplication of a di... |
| scmatval 22391 | The set of ` N ` x ` N ` s... |
| scmatel 22392 | An ` N ` x ` N ` scalar ma... |
| scmatscmid 22393 | A scalar matrix can be exp... |
| scmatscmide 22394 | An entry of a scalar matri... |
| scmatscmiddistr 22395 | Distributive law for scala... |
| scmatmat 22396 | An ` N ` x ` N ` scalar ma... |
| scmate 22397 | An entry of an ` N ` x ` N... |
| scmatmats 22398 | The set of an ` N ` x ` N ... |
| scmateALT 22399 | Alternate proof of ~ scmat... |
| scmatscm 22400 | The multiplication of a ma... |
| scmatid 22401 | The identity matrix is a s... |
| scmatdmat 22402 | A scalar matrix is a diago... |
| scmataddcl 22403 | The sum of two scalar matr... |
| scmatsubcl 22404 | The difference of two scal... |
| scmatmulcl 22405 | The product of two scalar ... |
| scmatsgrp 22406 | The set of scalar matrices... |
| scmatsrng 22407 | The set of scalar matrices... |
| scmatcrng 22408 | The subring of scalar matr... |
| scmatsgrp1 22409 | The set of scalar matrices... |
| scmatsrng1 22410 | The set of scalar matrices... |
| smatvscl 22411 | Closure of the scalar mult... |
| scmatlss 22412 | The set of scalar matrices... |
| scmatstrbas 22413 | The set of scalar matrices... |
| scmatrhmval 22414 | The value of the ring homo... |
| scmatrhmcl 22415 | The value of the ring homo... |
| scmatf 22416 | There is a function from a... |
| scmatfo 22417 | There is a function from a... |
| scmatf1 22418 | There is a 1-1 function fr... |
| scmatf1o 22419 | There is a bijection betwe... |
| scmatghm 22420 | There is a group homomorph... |
| scmatmhm 22421 | There is a monoid homomorp... |
| scmatrhm 22422 | There is a ring homomorphi... |
| scmatrngiso 22423 | There is a ring isomorphis... |
| scmatric 22424 | A ring is isomorphic to ev... |
| mat0scmat 22425 | The empty matrix over a ri... |
| mat1scmat 22426 | A 1-dimensional matrix ove... |
| mvmulfval 22429 | Functional value of the ma... |
| mvmulval 22430 | Multiplication of a vector... |
| mvmulfv 22431 | A cell/element in the vect... |
| mavmulval 22432 | Multiplication of a vector... |
| mavmulfv 22433 | A cell/element in the vect... |
| mavmulcl 22434 | Multiplication of an NxN m... |
| 1mavmul 22435 | Multiplication of the iden... |
| mavmulass 22436 | Associativity of the multi... |
| mavmuldm 22437 | The domain of the matrix v... |
| mavmulsolcl 22438 | Every solution of the equa... |
| mavmul0 22439 | Multiplication of a 0-dime... |
| mavmul0g 22440 | The result of the 0-dimens... |
| mvmumamul1 22441 | The multiplication of an M... |
| mavmumamul1 22442 | The multiplication of an N... |
| marrepfval 22447 | First substitution for the... |
| marrepval0 22448 | Second substitution for th... |
| marrepval 22449 | Third substitution for the... |
| marrepeval 22450 | An entry of a matrix with ... |
| marrepcl 22451 | Closure of the row replace... |
| marepvfval 22452 | First substitution for the... |
| marepvval0 22453 | Second substitution for th... |
| marepvval 22454 | Third substitution for the... |
| marepveval 22455 | An entry of a matrix with ... |
| marepvcl 22456 | Closure of the column repl... |
| ma1repvcl 22457 | Closure of the column repl... |
| ma1repveval 22458 | An entry of an identity ma... |
| mulmarep1el 22459 | Element by element multipl... |
| mulmarep1gsum1 22460 | The sum of element by elem... |
| mulmarep1gsum2 22461 | The sum of element by elem... |
| 1marepvmarrepid 22462 | Replacing the ith row by 0... |
| submabas 22465 | Any subset of the index se... |
| submafval 22466 | First substitution for a s... |
| submaval0 22467 | Second substitution for a ... |
| submaval 22468 | Third substitution for a s... |
| submaeval 22469 | An entry of a submatrix of... |
| 1marepvsma1 22470 | The submatrix of the ident... |
| mdetfval 22473 | First substitution for the... |
| mdetleib 22474 | Full substitution of our d... |
| mdetleib2 22475 | Leibniz' formula can also ... |
| nfimdetndef 22476 | The determinant is not def... |
| mdetfval1 22477 | First substitution of an a... |
| mdetleib1 22478 | Full substitution of an al... |
| mdet0pr 22479 | The determinant function f... |
| mdet0f1o 22480 | The determinant function f... |
| mdet0fv0 22481 | The determinant of the emp... |
| mdetf 22482 | Functionality of the deter... |
| mdetcl 22483 | The determinant evaluates ... |
| m1detdiag 22484 | The determinant of a 1-dim... |
| mdetdiaglem 22485 | Lemma for ~ mdetdiag . Pr... |
| mdetdiag 22486 | The determinant of a diago... |
| mdetdiagid 22487 | The determinant of a diago... |
| mdet1 22488 | The determinant of the ide... |
| mdetrlin 22489 | The determinant function i... |
| mdetrsca 22490 | The determinant function i... |
| mdetrsca2 22491 | The determinant function i... |
| mdetr0 22492 | The determinant of a matri... |
| mdet0 22493 | The determinant of the zer... |
| mdetrlin2 22494 | The determinant function i... |
| mdetralt 22495 | The determinant function i... |
| mdetralt2 22496 | The determinant function i... |
| mdetero 22497 | The determinant function i... |
| mdettpos 22498 | Determinant is invariant u... |
| mdetunilem1 22499 | Lemma for ~ mdetuni . (Co... |
| mdetunilem2 22500 | Lemma for ~ mdetuni . (Co... |
| mdetunilem3 22501 | Lemma for ~ mdetuni . (Co... |
| mdetunilem4 22502 | Lemma for ~ mdetuni . (Co... |
| mdetunilem5 22503 | Lemma for ~ mdetuni . (Co... |
| mdetunilem6 22504 | Lemma for ~ mdetuni . (Co... |
| mdetunilem7 22505 | Lemma for ~ mdetuni . (Co... |
| mdetunilem8 22506 | Lemma for ~ mdetuni . (Co... |
| mdetunilem9 22507 | Lemma for ~ mdetuni . (Co... |
| mdetuni0 22508 | Lemma for ~ mdetuni . (Co... |
| mdetuni 22509 | According to the definitio... |
| mdetmul 22510 | Multiplicativity of the de... |
| m2detleiblem1 22511 | Lemma 1 for ~ m2detleib . ... |
| m2detleiblem5 22512 | Lemma 5 for ~ m2detleib . ... |
| m2detleiblem6 22513 | Lemma 6 for ~ m2detleib . ... |
| m2detleiblem7 22514 | Lemma 7 for ~ m2detleib . ... |
| m2detleiblem2 22515 | Lemma 2 for ~ m2detleib . ... |
| m2detleiblem3 22516 | Lemma 3 for ~ m2detleib . ... |
| m2detleiblem4 22517 | Lemma 4 for ~ m2detleib . ... |
| m2detleib 22518 | Leibniz' Formula for 2x2-m... |
| mndifsplit 22523 | Lemma for ~ maducoeval2 . ... |
| madufval 22524 | First substitution for the... |
| maduval 22525 | Second substitution for th... |
| maducoeval 22526 | An entry of the adjunct (c... |
| maducoeval2 22527 | An entry of the adjunct (c... |
| maduf 22528 | Creating the adjunct of ma... |
| madutpos 22529 | The adjuct of a transposed... |
| madugsum 22530 | The determinant of a matri... |
| madurid 22531 | Multiplying a matrix with ... |
| madulid 22532 | Multiplying the adjunct of... |
| minmar1fval 22533 | First substitution for the... |
| minmar1val0 22534 | Second substitution for th... |
| minmar1val 22535 | Third substitution for the... |
| minmar1eval 22536 | An entry of a matrix for a... |
| minmar1marrep 22537 | The minor matrix is a spec... |
| minmar1cl 22538 | Closure of the row replace... |
| maducoevalmin1 22539 | The coefficients of an adj... |
| symgmatr01lem 22540 | Lemma for ~ symgmatr01 . ... |
| symgmatr01 22541 | Applying a permutation tha... |
| gsummatr01lem1 22542 | Lemma A for ~ gsummatr01 .... |
| gsummatr01lem2 22543 | Lemma B for ~ gsummatr01 .... |
| gsummatr01lem3 22544 | Lemma 1 for ~ gsummatr01 .... |
| gsummatr01lem4 22545 | Lemma 2 for ~ gsummatr01 .... |
| gsummatr01 22546 | Lemma 1 for ~ smadiadetlem... |
| marep01ma 22547 | Replacing a row of a squar... |
| smadiadetlem0 22548 | Lemma 0 for ~ smadiadet : ... |
| smadiadetlem1 22549 | Lemma 1 for ~ smadiadet : ... |
| smadiadetlem1a 22550 | Lemma 1a for ~ smadiadet :... |
| smadiadetlem2 22551 | Lemma 2 for ~ smadiadet : ... |
| smadiadetlem3lem0 22552 | Lemma 0 for ~ smadiadetlem... |
| smadiadetlem3lem1 22553 | Lemma 1 for ~ smadiadetlem... |
| smadiadetlem3lem2 22554 | Lemma 2 for ~ smadiadetlem... |
| smadiadetlem3 22555 | Lemma 3 for ~ smadiadet . ... |
| smadiadetlem4 22556 | Lemma 4 for ~ smadiadet . ... |
| smadiadet 22557 | The determinant of a subma... |
| smadiadetglem1 22558 | Lemma 1 for ~ smadiadetg .... |
| smadiadetglem2 22559 | Lemma 2 for ~ smadiadetg .... |
| smadiadetg 22560 | The determinant of a squar... |
| smadiadetg0 22561 | Lemma for ~ smadiadetr : v... |
| smadiadetr 22562 | The determinant of a squar... |
| invrvald 22563 | If a matrix multiplied wit... |
| matinv 22564 | The inverse of a matrix is... |
| matunit 22565 | A matrix is a unit in the ... |
| slesolvec 22566 | Every solution of a system... |
| slesolinv 22567 | The solution of a system o... |
| slesolinvbi 22568 | The solution of a system o... |
| slesolex 22569 | Every system of linear equ... |
| cramerimplem1 22570 | Lemma 1 for ~ cramerimp : ... |
| cramerimplem2 22571 | Lemma 2 for ~ cramerimp : ... |
| cramerimplem3 22572 | Lemma 3 for ~ cramerimp : ... |
| cramerimp 22573 | One direction of Cramer's ... |
| cramerlem1 22574 | Lemma 1 for ~ cramer . (C... |
| cramerlem2 22575 | Lemma 2 for ~ cramer . (C... |
| cramerlem3 22576 | Lemma 3 for ~ cramer . (C... |
| cramer0 22577 | Special case of Cramer's r... |
| cramer 22578 | Cramer's rule. According ... |
| pmatring 22579 | The set of polynomial matr... |
| pmatlmod 22580 | The set of polynomial matr... |
| pmatassa 22581 | The set of polynomial matr... |
| pmat0op 22582 | The zero polynomial matrix... |
| pmat1op 22583 | The identity polynomial ma... |
| pmat1ovd 22584 | Entries of the identity po... |
| pmat0opsc 22585 | The zero polynomial matrix... |
| pmat1opsc 22586 | The identity polynomial ma... |
| pmat1ovscd 22587 | Entries of the identity po... |
| pmatcoe1fsupp 22588 | For a polynomial matrix th... |
| 1pmatscmul 22589 | The scalar product of the ... |
| cpmat 22596 | Value of the constructor o... |
| cpmatpmat 22597 | A constant polynomial matr... |
| cpmatel 22598 | Property of a constant pol... |
| cpmatelimp 22599 | Implication of a set being... |
| cpmatel2 22600 | Another property of a cons... |
| cpmatelimp2 22601 | Another implication of a s... |
| 1elcpmat 22602 | The identity of the ring o... |
| cpmatacl 22603 | The set of all constant po... |
| cpmatinvcl 22604 | The set of all constant po... |
| cpmatmcllem 22605 | Lemma for ~ cpmatmcl . (C... |
| cpmatmcl 22606 | The set of all constant po... |
| cpmatsubgpmat 22607 | The set of all constant po... |
| cpmatsrgpmat 22608 | The set of all constant po... |
| 0elcpmat 22609 | The zero of the ring of al... |
| mat2pmatfval 22610 | Value of the matrix transf... |
| mat2pmatval 22611 | The result of a matrix tra... |
| mat2pmatvalel 22612 | A (matrix) element of the ... |
| mat2pmatbas 22613 | The result of a matrix tra... |
| mat2pmatbas0 22614 | The result of a matrix tra... |
| mat2pmatf 22615 | The matrix transformation ... |
| mat2pmatf1 22616 | The matrix transformation ... |
| mat2pmatghm 22617 | The transformation of matr... |
| mat2pmatmul 22618 | The transformation of matr... |
| mat2pmat1 22619 | The transformation of the ... |
| mat2pmatmhm 22620 | The transformation of matr... |
| mat2pmatrhm 22621 | The transformation of matr... |
| mat2pmatlin 22622 | The transformation of matr... |
| 0mat2pmat 22623 | The transformed zero matri... |
| idmatidpmat 22624 | The transformed identity m... |
| d0mat2pmat 22625 | The transformed empty set ... |
| d1mat2pmat 22626 | The transformation of a ma... |
| mat2pmatscmxcl 22627 | A transformed matrix multi... |
| m2cpm 22628 | The result of a matrix tra... |
| m2cpmf 22629 | The matrix transformation ... |
| m2cpmf1 22630 | The matrix transformation ... |
| m2cpmghm 22631 | The transformation of matr... |
| m2cpmmhm 22632 | The transformation of matr... |
| m2cpmrhm 22633 | The transformation of matr... |
| m2pmfzmap 22634 | The transformed values of ... |
| m2pmfzgsumcl 22635 | Closure of the sum of scal... |
| cpm2mfval 22636 | Value of the inverse matri... |
| cpm2mval 22637 | The result of an inverse m... |
| cpm2mvalel 22638 | A (matrix) element of the ... |
| cpm2mf 22639 | The inverse matrix transfo... |
| m2cpminvid 22640 | The inverse transformation... |
| m2cpminvid2lem 22641 | Lemma for ~ m2cpminvid2 . ... |
| m2cpminvid2 22642 | The transformation applied... |
| m2cpmfo 22643 | The matrix transformation ... |
| m2cpmf1o 22644 | The matrix transformation ... |
| m2cpmrngiso 22645 | The transformation of matr... |
| matcpmric 22646 | The ring of matrices over ... |
| m2cpminv 22647 | The inverse matrix transfo... |
| m2cpminv0 22648 | The inverse matrix transfo... |
| decpmatval0 22651 | The matrix consisting of t... |
| decpmatval 22652 | The matrix consisting of t... |
| decpmate 22653 | An entry of the matrix con... |
| decpmatcl 22654 | Closure of the decompositi... |
| decpmataa0 22655 | The matrix consisting of t... |
| decpmatfsupp 22656 | The mapping to the matrice... |
| decpmatid 22657 | The matrix consisting of t... |
| decpmatmullem 22658 | Lemma for ~ decpmatmul . ... |
| decpmatmul 22659 | The matrix consisting of t... |
| decpmatmulsumfsupp 22660 | Lemma 0 for ~ pm2mpmhm . ... |
| pmatcollpw1lem1 22661 | Lemma 1 for ~ pmatcollpw1 ... |
| pmatcollpw1lem2 22662 | Lemma 2 for ~ pmatcollpw1 ... |
| pmatcollpw1 22663 | Write a polynomial matrix ... |
| pmatcollpw2lem 22664 | Lemma for ~ pmatcollpw2 . ... |
| pmatcollpw2 22665 | Write a polynomial matrix ... |
| monmatcollpw 22666 | The matrix consisting of t... |
| pmatcollpwlem 22667 | Lemma for ~ pmatcollpw . ... |
| pmatcollpw 22668 | Write a polynomial matrix ... |
| pmatcollpwfi 22669 | Write a polynomial matrix ... |
| pmatcollpw3lem 22670 | Lemma for ~ pmatcollpw3 an... |
| pmatcollpw3 22671 | Write a polynomial matrix ... |
| pmatcollpw3fi 22672 | Write a polynomial matrix ... |
| pmatcollpw3fi1lem1 22673 | Lemma 1 for ~ pmatcollpw3f... |
| pmatcollpw3fi1lem2 22674 | Lemma 2 for ~ pmatcollpw3f... |
| pmatcollpw3fi1 22675 | Write a polynomial matrix ... |
| pmatcollpwscmatlem1 22676 | Lemma 1 for ~ pmatcollpwsc... |
| pmatcollpwscmatlem2 22677 | Lemma 2 for ~ pmatcollpwsc... |
| pmatcollpwscmat 22678 | Write a scalar matrix over... |
| pm2mpf1lem 22681 | Lemma for ~ pm2mpf1 . (Co... |
| pm2mpval 22682 | Value of the transformatio... |
| pm2mpfval 22683 | A polynomial matrix transf... |
| pm2mpcl 22684 | The transformation of poly... |
| pm2mpf 22685 | The transformation of poly... |
| pm2mpf1 22686 | The transformation of poly... |
| pm2mpcoe1 22687 | A coefficient of the polyn... |
| idpm2idmp 22688 | The transformation of the ... |
| mptcoe1matfsupp 22689 | The mapping extracting the... |
| mply1topmatcllem 22690 | Lemma for ~ mply1topmatcl ... |
| mply1topmatval 22691 | A polynomial over matrices... |
| mply1topmatcl 22692 | A polynomial over matrices... |
| mp2pm2mplem1 22693 | Lemma 1 for ~ mp2pm2mp . ... |
| mp2pm2mplem2 22694 | Lemma 2 for ~ mp2pm2mp . ... |
| mp2pm2mplem3 22695 | Lemma 3 for ~ mp2pm2mp . ... |
| mp2pm2mplem4 22696 | Lemma 4 for ~ mp2pm2mp . ... |
| mp2pm2mplem5 22697 | Lemma 5 for ~ mp2pm2mp . ... |
| mp2pm2mp 22698 | A polynomial over matrices... |
| pm2mpghmlem2 22699 | Lemma 2 for ~ pm2mpghm . ... |
| pm2mpghmlem1 22700 | Lemma 1 for pm2mpghm . (C... |
| pm2mpfo 22701 | The transformation of poly... |
| pm2mpf1o 22702 | The transformation of poly... |
| pm2mpghm 22703 | The transformation of poly... |
| pm2mpgrpiso 22704 | The transformation of poly... |
| pm2mpmhmlem1 22705 | Lemma 1 for ~ pm2mpmhm . ... |
| pm2mpmhmlem2 22706 | Lemma 2 for ~ pm2mpmhm . ... |
| pm2mpmhm 22707 | The transformation of poly... |
| pm2mprhm 22708 | The transformation of poly... |
| pm2mprngiso 22709 | The transformation of poly... |
| pmmpric 22710 | The ring of polynomial mat... |
| monmat2matmon 22711 | The transformation of a po... |
| pm2mp 22712 | The transformation of a su... |
| chmatcl 22715 | Closure of the characteris... |
| chmatval 22716 | The entries of the charact... |
| chpmatfval 22717 | Value of the characteristi... |
| chpmatval 22718 | The characteristic polynom... |
| chpmatply1 22719 | The characteristic polynom... |
| chpmatval2 22720 | The characteristic polynom... |
| chpmat0d 22721 | The characteristic polynom... |
| chpmat1dlem 22722 | Lemma for ~ chpmat1d . (C... |
| chpmat1d 22723 | The characteristic polynom... |
| chpdmatlem0 22724 | Lemma 0 for ~ chpdmat . (... |
| chpdmatlem1 22725 | Lemma 1 for ~ chpdmat . (... |
| chpdmatlem2 22726 | Lemma 2 for ~ chpdmat . (... |
| chpdmatlem3 22727 | Lemma 3 for ~ chpdmat . (... |
| chpdmat 22728 | The characteristic polynom... |
| chpscmat 22729 | The characteristic polynom... |
| chpscmat0 22730 | The characteristic polynom... |
| chpscmatgsumbin 22731 | The characteristic polynom... |
| chpscmatgsummon 22732 | The characteristic polynom... |
| chp0mat 22733 | The characteristic polynom... |
| chpidmat 22734 | The characteristic polynom... |
| chmaidscmat 22735 | The characteristic polynom... |
| fvmptnn04if 22736 | The function values of a m... |
| fvmptnn04ifa 22737 | The function value of a ma... |
| fvmptnn04ifb 22738 | The function value of a ma... |
| fvmptnn04ifc 22739 | The function value of a ma... |
| fvmptnn04ifd 22740 | The function value of a ma... |
| chfacfisf 22741 | The "characteristic factor... |
| chfacfisfcpmat 22742 | The "characteristic factor... |
| chfacffsupp 22743 | The "characteristic factor... |
| chfacfscmulcl 22744 | Closure of a scaled value ... |
| chfacfscmul0 22745 | A scaled value of the "cha... |
| chfacfscmulfsupp 22746 | A mapping of scaled values... |
| chfacfscmulgsum 22747 | Breaking up a sum of value... |
| chfacfpmmulcl 22748 | Closure of the value of th... |
| chfacfpmmul0 22749 | The value of the "characte... |
| chfacfpmmulfsupp 22750 | A mapping of values of the... |
| chfacfpmmulgsum 22751 | Breaking up a sum of value... |
| chfacfpmmulgsum2 22752 | Breaking up a sum of value... |
| cayhamlem1 22753 | Lemma 1 for ~ cayleyhamilt... |
| cpmadurid 22754 | The right-hand fundamental... |
| cpmidgsum 22755 | Representation of the iden... |
| cpmidgsumm2pm 22756 | Representation of the iden... |
| cpmidpmatlem1 22757 | Lemma 1 for ~ cpmidpmat . ... |
| cpmidpmatlem2 22758 | Lemma 2 for ~ cpmidpmat . ... |
| cpmidpmatlem3 22759 | Lemma 3 for ~ cpmidpmat . ... |
| cpmidpmat 22760 | Representation of the iden... |
| cpmadugsumlemB 22761 | Lemma B for ~ cpmadugsum .... |
| cpmadugsumlemC 22762 | Lemma C for ~ cpmadugsum .... |
| cpmadugsumlemF 22763 | Lemma F for ~ cpmadugsum .... |
| cpmadugsumfi 22764 | The product of the charact... |
| cpmadugsum 22765 | The product of the charact... |
| cpmidgsum2 22766 | Representation of the iden... |
| cpmidg2sum 22767 | Equality of two sums repre... |
| cpmadumatpolylem1 22768 | Lemma 1 for ~ cpmadumatpol... |
| cpmadumatpolylem2 22769 | Lemma 2 for ~ cpmadumatpol... |
| cpmadumatpoly 22770 | The product of the charact... |
| cayhamlem2 22771 | Lemma for ~ cayhamlem3 . ... |
| chcoeffeqlem 22772 | Lemma for ~ chcoeffeq . (... |
| chcoeffeq 22773 | The coefficients of the ch... |
| cayhamlem3 22774 | Lemma for ~ cayhamlem4 . ... |
| cayhamlem4 22775 | Lemma for ~ cayleyhamilton... |
| cayleyhamilton0 22776 | The Cayley-Hamilton theore... |
| cayleyhamilton 22777 | The Cayley-Hamilton theore... |
| cayleyhamiltonALT 22778 | Alternate proof of ~ cayle... |
| cayleyhamilton1 22779 | The Cayley-Hamilton theore... |
| istopg 22782 | Express the predicate " ` ... |
| istop2g 22783 | Express the predicate " ` ... |
| uniopn 22784 | The union of a subset of a... |
| iunopn 22785 | The indexed union of a sub... |
| inopn 22786 | The intersection of two op... |
| fitop 22787 | A topology is closed under... |
| fiinopn 22788 | The intersection of a none... |
| iinopn 22789 | The intersection of a none... |
| unopn 22790 | The union of two open sets... |
| 0opn 22791 | The empty set is an open s... |
| 0ntop 22792 | The empty set is not a top... |
| topopn 22793 | The underlying set of a to... |
| eltopss 22794 | A member of a topology is ... |
| riinopn 22795 | A finite indexed relative ... |
| rintopn 22796 | A finite relative intersec... |
| istopon 22799 | Property of being a topolo... |
| topontop 22800 | A topology on a given base... |
| toponuni 22801 | The base set of a topology... |
| topontopi 22802 | A topology on a given base... |
| toponunii 22803 | The base set of a topology... |
| toptopon 22804 | Alternative definition of ... |
| toptopon2 22805 | A topology is the same thi... |
| topontopon 22806 | A topology on a set is a t... |
| funtopon 22807 | The class ` TopOn ` is a f... |
| toponrestid 22808 | Given a topology on a set,... |
| toponsspwpw 22809 | The set of topologies on a... |
| dmtopon 22810 | The domain of ` TopOn ` is... |
| fntopon 22811 | The class ` TopOn ` is a f... |
| toprntopon 22812 | A topology is the same thi... |
| toponmax 22813 | The base set of a topology... |
| toponss 22814 | A member of a topology is ... |
| toponcom 22815 | If ` K ` is a topology on ... |
| toponcomb 22816 | Biconditional form of ~ to... |
| topgele 22817 | The topologies over the sa... |
| topsn 22818 | The only topology on a sin... |
| istps 22821 | Express the predicate "is ... |
| istps2 22822 | Express the predicate "is ... |
| tpsuni 22823 | The base set of a topologi... |
| tpstop 22824 | The topology extractor on ... |
| tpspropd 22825 | A topological space depend... |
| tpsprop2d 22826 | A topological space depend... |
| topontopn 22827 | Express the predicate "is ... |
| tsettps 22828 | If the topology component ... |
| istpsi 22829 | Properties that determine ... |
| eltpsg 22830 | Properties that determine ... |
| eltpsi 22831 | Properties that determine ... |
| isbasisg 22834 | Express the predicate "the... |
| isbasis2g 22835 | Express the predicate "the... |
| isbasis3g 22836 | Express the predicate "the... |
| basis1 22837 | Property of a basis. (Con... |
| basis2 22838 | Property of a basis. (Con... |
| fiinbas 22839 | If a set is closed under f... |
| basdif0 22840 | A basis is not affected by... |
| baspartn 22841 | A disjoint system of sets ... |
| tgval 22842 | The topology generated by ... |
| tgval2 22843 | Definition of a topology g... |
| eltg 22844 | Membership in a topology g... |
| eltg2 22845 | Membership in a topology g... |
| eltg2b 22846 | Membership in a topology g... |
| eltg4i 22847 | An open set in a topology ... |
| eltg3i 22848 | The union of a set of basi... |
| eltg3 22849 | Membership in a topology g... |
| tgval3 22850 | Alternate expression for t... |
| tg1 22851 | Property of a member of a ... |
| tg2 22852 | Property of a member of a ... |
| bastg 22853 | A member of a basis is a s... |
| unitg 22854 | The topology generated by ... |
| tgss 22855 | Subset relation for genera... |
| tgcl 22856 | Show that a basis generate... |
| tgclb 22857 | The property ~ tgcl can be... |
| tgtopon 22858 | A basis generates a topolo... |
| topbas 22859 | A topology is its own basi... |
| tgtop 22860 | A topology is its own basi... |
| eltop 22861 | Membership in a topology, ... |
| eltop2 22862 | Membership in a topology. ... |
| eltop3 22863 | Membership in a topology. ... |
| fibas 22864 | A collection of finite int... |
| tgdom 22865 | A space has no more open s... |
| tgiun 22866 | The indexed union of a set... |
| tgidm 22867 | The topology generator fun... |
| bastop 22868 | Two ways to express that a... |
| tgtop11 22869 | The topology generation fu... |
| 0top 22870 | The singleton of the empty... |
| en1top 22871 | ` { (/) } ` is the only to... |
| en2top 22872 | If a topology has two elem... |
| tgss3 22873 | A criterion for determinin... |
| tgss2 22874 | A criterion for determinin... |
| basgen 22875 | Given a topology ` J ` , s... |
| basgen2 22876 | Given a topology ` J ` , s... |
| 2basgen 22877 | Conditions that determine ... |
| tgfiss 22878 | If a subbase is included i... |
| tgdif0 22879 | A generated topology is no... |
| bastop1 22880 | A subset of a topology is ... |
| bastop2 22881 | A version of ~ bastop1 tha... |
| distop 22882 | The discrete topology on a... |
| topnex 22883 | The class of all topologie... |
| distopon 22884 | The discrete topology on a... |
| sn0topon 22885 | The singleton of the empty... |
| sn0top 22886 | The singleton of the empty... |
| indislem 22887 | A lemma to eliminate some ... |
| indistopon 22888 | The indiscrete topology on... |
| indistop 22889 | The indiscrete topology on... |
| indisuni 22890 | The base set of the indisc... |
| fctop 22891 | The finite complement topo... |
| fctop2 22892 | The finite complement topo... |
| cctop 22893 | The countable complement t... |
| ppttop 22894 | The particular point topol... |
| pptbas 22895 | The particular point topol... |
| epttop 22896 | The excluded point topolog... |
| indistpsx 22897 | The indiscrete topology on... |
| indistps 22898 | The indiscrete topology on... |
| indistps2 22899 | The indiscrete topology on... |
| indistpsALT 22900 | The indiscrete topology on... |
| indistps2ALT 22901 | The indiscrete topology on... |
| distps 22902 | The discrete topology on a... |
| fncld 22909 | The closed-set generator i... |
| cldval 22910 | The set of closed sets of ... |
| ntrfval 22911 | The interior function on t... |
| clsfval 22912 | The closure function on th... |
| cldrcl 22913 | Reverse closure of the clo... |
| iscld 22914 | The predicate "the class `... |
| iscld2 22915 | A subset of the underlying... |
| cldss 22916 | A closed set is a subset o... |
| cldss2 22917 | The set of closed sets is ... |
| cldopn 22918 | The complement of a closed... |
| isopn2 22919 | A subset of the underlying... |
| opncld 22920 | The complement of an open ... |
| difopn 22921 | The difference of a closed... |
| topcld 22922 | The underlying set of a to... |
| ntrval 22923 | The interior of a subset o... |
| clsval 22924 | The closure of a subset of... |
| 0cld 22925 | The empty set is closed. ... |
| iincld 22926 | The indexed intersection o... |
| intcld 22927 | The intersection of a set ... |
| uncld 22928 | The union of two closed se... |
| cldcls 22929 | A closed subset equals its... |
| incld 22930 | The intersection of two cl... |
| riincld 22931 | An indexed relative inters... |
| iuncld 22932 | A finite indexed union of ... |
| unicld 22933 | A finite union of closed s... |
| clscld 22934 | The closure of a subset of... |
| clsf 22935 | The closure function is a ... |
| ntropn 22936 | The interior of a subset o... |
| clsval2 22937 | Express closure in terms o... |
| ntrval2 22938 | Interior expressed in term... |
| ntrdif 22939 | An interior of a complemen... |
| clsdif 22940 | A closure of a complement ... |
| clsss 22941 | Subset relationship for cl... |
| ntrss 22942 | Subset relationship for in... |
| sscls 22943 | A subset of a topology's u... |
| ntrss2 22944 | A subset includes its inte... |
| ssntr 22945 | An open subset of a set is... |
| clsss3 22946 | The closure of a subset of... |
| ntrss3 22947 | The interior of a subset o... |
| ntrin 22948 | A pairwise intersection of... |
| cmclsopn 22949 | The complement of a closur... |
| cmntrcld 22950 | The complement of an inter... |
| iscld3 22951 | A subset is closed iff it ... |
| iscld4 22952 | A subset is closed iff it ... |
| isopn3 22953 | A subset is open iff it eq... |
| clsidm 22954 | The closure operation is i... |
| ntridm 22955 | The interior operation is ... |
| clstop 22956 | The closure of a topology'... |
| ntrtop 22957 | The interior of a topology... |
| 0ntr 22958 | A subset with an empty int... |
| clsss2 22959 | If a subset is included in... |
| elcls 22960 | Membership in a closure. ... |
| elcls2 22961 | Membership in a closure. ... |
| clsndisj 22962 | Any open set containing a ... |
| ntrcls0 22963 | A subset whose closure has... |
| ntreq0 22964 | Two ways to say that a sub... |
| cldmre 22965 | The closed sets of a topol... |
| mrccls 22966 | Moore closure generalizes ... |
| cls0 22967 | The closure of the empty s... |
| ntr0 22968 | The interior of the empty ... |
| isopn3i 22969 | An open subset equals its ... |
| elcls3 22970 | Membership in a closure in... |
| opncldf1 22971 | A bijection useful for con... |
| opncldf2 22972 | The values of the open-clo... |
| opncldf3 22973 | The values of the converse... |
| isclo 22974 | A set ` A ` is clopen iff ... |
| isclo2 22975 | A set ` A ` is clopen iff ... |
| discld 22976 | The open sets of a discret... |
| sn0cld 22977 | The closed sets of the top... |
| indiscld 22978 | The closed sets of an indi... |
| mretopd 22979 | A Moore collection which i... |
| toponmre 22980 | The topologies over a give... |
| cldmreon 22981 | The closed sets of a topol... |
| iscldtop 22982 | A family is the closed set... |
| mreclatdemoBAD 22983 | The closed subspaces of a ... |
| neifval 22986 | Value of the neighborhood ... |
| neif 22987 | The neighborhood function ... |
| neiss2 22988 | A set with a neighborhood ... |
| neival 22989 | Value of the set of neighb... |
| isnei 22990 | The predicate "the class `... |
| neiint 22991 | An intuitive definition of... |
| isneip 22992 | The predicate "the class `... |
| neii1 22993 | A neighborhood is included... |
| neisspw 22994 | The neighborhoods of any s... |
| neii2 22995 | Property of a neighborhood... |
| neiss 22996 | Any neighborhood of a set ... |
| ssnei 22997 | A set is included in any o... |
| elnei 22998 | A point belongs to any of ... |
| 0nnei 22999 | The empty set is not a nei... |
| neips 23000 | A neighborhood of a set is... |
| opnneissb 23001 | An open set is a neighborh... |
| opnssneib 23002 | Any superset of an open se... |
| ssnei2 23003 | Any subset ` M ` of ` X ` ... |
| neindisj 23004 | Any neighborhood of an ele... |
| opnneiss 23005 | An open set is a neighborh... |
| opnneip 23006 | An open set is a neighborh... |
| opnnei 23007 | A set is open iff it is a ... |
| tpnei 23008 | The underlying set of a to... |
| neiuni 23009 | The union of the neighborh... |
| neindisj2 23010 | A point ` P ` belongs to t... |
| topssnei 23011 | A finer topology has more ... |
| innei 23012 | The intersection of two ne... |
| opnneiid 23013 | Only an open set is a neig... |
| neissex 23014 | For any neighborhood ` N `... |
| 0nei 23015 | The empty set is a neighbo... |
| neipeltop 23016 | Lemma for ~ neiptopreu . ... |
| neiptopuni 23017 | Lemma for ~ neiptopreu . ... |
| neiptoptop 23018 | Lemma for ~ neiptopreu . ... |
| neiptopnei 23019 | Lemma for ~ neiptopreu . ... |
| neiptopreu 23020 | If, to each element ` P ` ... |
| lpfval 23025 | The limit point function o... |
| lpval 23026 | The set of limit points of... |
| islp 23027 | The predicate "the class `... |
| lpsscls 23028 | The limit points of a subs... |
| lpss 23029 | The limit points of a subs... |
| lpdifsn 23030 | ` P ` is a limit point of ... |
| lpss3 23031 | Subset relationship for li... |
| islp2 23032 | The predicate " ` P ` is a... |
| islp3 23033 | The predicate " ` P ` is a... |
| maxlp 23034 | A point is a limit point o... |
| clslp 23035 | The closure of a subset of... |
| islpi 23036 | A point belonging to a set... |
| cldlp 23037 | A subset of a topological ... |
| isperf 23038 | Definition of a perfect sp... |
| isperf2 23039 | Definition of a perfect sp... |
| isperf3 23040 | A perfect space is a topol... |
| perflp 23041 | The limit points of a perf... |
| perfi 23042 | Property of a perfect spac... |
| perftop 23043 | A perfect space is a topol... |
| restrcl 23044 | Reverse closure for the su... |
| restbas 23045 | A subspace topology basis ... |
| tgrest 23046 | A subspace can be generate... |
| resttop 23047 | A subspace topology is a t... |
| resttopon 23048 | A subspace topology is a t... |
| restuni 23049 | The underlying set of a su... |
| stoig 23050 | The topological space buil... |
| restco 23051 | Composition of subspaces. ... |
| restabs 23052 | Equivalence of being a sub... |
| restin 23053 | When the subspace region i... |
| restuni2 23054 | The underlying set of a su... |
| resttopon2 23055 | The underlying set of a su... |
| rest0 23056 | The subspace topology indu... |
| restsn 23057 | The only subspace topology... |
| restsn2 23058 | The subspace topology indu... |
| restcld 23059 | A closed set of a subspace... |
| restcldi 23060 | A closed set is closed in ... |
| restcldr 23061 | A set which is closed in t... |
| restopnb 23062 | If ` B ` is an open subset... |
| ssrest 23063 | If ` K ` is a finer topolo... |
| restopn2 23064 | If ` A ` is open, then ` B... |
| restdis 23065 | A subspace of a discrete t... |
| restfpw 23066 | The restriction of the set... |
| neitr 23067 | The neighborhood of a trac... |
| restcls 23068 | A closure in a subspace to... |
| restntr 23069 | An interior in a subspace ... |
| restlp 23070 | The limit points of a subs... |
| restperf 23071 | Perfection of a subspace. ... |
| perfopn 23072 | An open subset of a perfec... |
| resstopn 23073 | The topology of a restrict... |
| resstps 23074 | A restricted topological s... |
| ordtbaslem 23075 | Lemma for ~ ordtbas . In ... |
| ordtval 23076 | Value of the order topolog... |
| ordtuni 23077 | Value of the order topolog... |
| ordtbas2 23078 | Lemma for ~ ordtbas . (Co... |
| ordtbas 23079 | In a total order, the fini... |
| ordttopon 23080 | Value of the order topolog... |
| ordtopn1 23081 | An upward ray ` ( P , +oo ... |
| ordtopn2 23082 | A downward ray ` ( -oo , P... |
| ordtopn3 23083 | An open interval ` ( A , B... |
| ordtcld1 23084 | A downward ray ` ( -oo , P... |
| ordtcld2 23085 | An upward ray ` [ P , +oo ... |
| ordtcld3 23086 | A closed interval ` [ A , ... |
| ordttop 23087 | The order topology is a to... |
| ordtcnv 23088 | The order dual generates t... |
| ordtrest 23089 | The subspace topology of a... |
| ordtrest2lem 23090 | Lemma for ~ ordtrest2 . (... |
| ordtrest2 23091 | An interval-closed set ` A... |
| letopon 23092 | The topology of the extend... |
| letop 23093 | The topology of the extend... |
| letopuni 23094 | The topology of the extend... |
| xrstopn 23095 | The topology component of ... |
| xrstps 23096 | The extended real number s... |
| leordtvallem1 23097 | Lemma for ~ leordtval . (... |
| leordtvallem2 23098 | Lemma for ~ leordtval . (... |
| leordtval2 23099 | The topology of the extend... |
| leordtval 23100 | The topology of the extend... |
| iccordt 23101 | A closed interval is close... |
| iocpnfordt 23102 | An unbounded above open in... |
| icomnfordt 23103 | An unbounded above open in... |
| iooordt 23104 | An open interval is open i... |
| reordt 23105 | The real numbers are an op... |
| lecldbas 23106 | The set of closed interval... |
| pnfnei 23107 | A neighborhood of ` +oo ` ... |
| mnfnei 23108 | A neighborhood of ` -oo ` ... |
| ordtrestixx 23109 | The restriction of the les... |
| ordtresticc 23110 | The restriction of the les... |
| lmrel 23117 | The topological space conv... |
| lmrcl 23118 | Reverse closure for the co... |
| lmfval 23119 | The relation "sequence ` f... |
| cnfval 23120 | The set of all continuous ... |
| cnpfval 23121 | The function mapping the p... |
| iscn 23122 | The predicate "the class `... |
| cnpval 23123 | The set of all functions f... |
| iscnp 23124 | The predicate "the class `... |
| iscn2 23125 | The predicate "the class `... |
| iscnp2 23126 | The predicate "the class `... |
| cntop1 23127 | Reverse closure for a cont... |
| cntop2 23128 | Reverse closure for a cont... |
| cnptop1 23129 | Reverse closure for a func... |
| cnptop2 23130 | Reverse closure for a func... |
| iscnp3 23131 | The predicate "the class `... |
| cnprcl 23132 | Reverse closure for a func... |
| cnf 23133 | A continuous function is a... |
| cnpf 23134 | A continuous function at p... |
| cnpcl 23135 | The value of a continuous ... |
| cnf2 23136 | A continuous function is a... |
| cnpf2 23137 | A continuous function at p... |
| cnprcl2 23138 | Reverse closure for a func... |
| tgcn 23139 | The continuity predicate w... |
| tgcnp 23140 | The "continuous at a point... |
| subbascn 23141 | The continuity predicate w... |
| ssidcn 23142 | The identity function is a... |
| cnpimaex 23143 | Property of a function con... |
| idcn 23144 | A restricted identity func... |
| lmbr 23145 | Express the binary relatio... |
| lmbr2 23146 | Express the binary relatio... |
| lmbrf 23147 | Express the binary relatio... |
| lmconst 23148 | A constant sequence conver... |
| lmcvg 23149 | Convergence property of a ... |
| iscnp4 23150 | The predicate "the class `... |
| cnpnei 23151 | A condition for continuity... |
| cnima 23152 | An open subset of the codo... |
| cnco 23153 | The composition of two con... |
| cnpco 23154 | The composition of a funct... |
| cnclima 23155 | A closed subset of the cod... |
| iscncl 23156 | A characterization of a co... |
| cncls2i 23157 | Property of the preimage o... |
| cnntri 23158 | Property of the preimage o... |
| cnclsi 23159 | Property of the image of a... |
| cncls2 23160 | Continuity in terms of clo... |
| cncls 23161 | Continuity in terms of clo... |
| cnntr 23162 | Continuity in terms of int... |
| cnss1 23163 | If the topology ` K ` is f... |
| cnss2 23164 | If the topology ` K ` is f... |
| cncnpi 23165 | A continuous function is c... |
| cnsscnp 23166 | The set of continuous func... |
| cncnp 23167 | A continuous function is c... |
| cncnp2 23168 | A continuous function is c... |
| cnnei 23169 | Continuity in terms of nei... |
| cnconst2 23170 | A constant function is con... |
| cnconst 23171 | A constant function is con... |
| cnrest 23172 | Continuity of a restrictio... |
| cnrest2 23173 | Equivalence of continuity ... |
| cnrest2r 23174 | Equivalence of continuity ... |
| cnpresti 23175 | One direction of ~ cnprest... |
| cnprest 23176 | Equivalence of continuity ... |
| cnprest2 23177 | Equivalence of point-conti... |
| cndis 23178 | Every function is continuo... |
| cnindis 23179 | Every function is continuo... |
| cnpdis 23180 | If ` A ` is an isolated po... |
| paste 23181 | Pasting lemma. If ` A ` a... |
| lmfpm 23182 | If ` F ` converges, then `... |
| lmfss 23183 | Inclusion of a function ha... |
| lmcl 23184 | Closure of a limit. (Cont... |
| lmss 23185 | Limit on a subspace. (Con... |
| sslm 23186 | A finer topology has fewer... |
| lmres 23187 | A function converges iff i... |
| lmff 23188 | If ` F ` converges, there ... |
| lmcls 23189 | Any convergent sequence of... |
| lmcld 23190 | Any convergent sequence of... |
| lmcnp 23191 | The image of a convergent ... |
| lmcn 23192 | The image of a convergent ... |
| ist0 23207 | The predicate "is a T_0 sp... |
| ist1 23208 | The predicate "is a T_1 sp... |
| ishaus 23209 | The predicate "is a Hausdo... |
| iscnrm 23210 | The property of being comp... |
| t0sep 23211 | Any two topologically indi... |
| t0dist 23212 | Any two distinct points in... |
| t1sncld 23213 | In a T_1 space, singletons... |
| t1ficld 23214 | In a T_1 space, finite set... |
| hausnei 23215 | Neighborhood property of a... |
| t0top 23216 | A T_0 space is a topologic... |
| t1top 23217 | A T_1 space is a topologic... |
| haustop 23218 | A Hausdorff space is a top... |
| isreg 23219 | The predicate "is a regula... |
| regtop 23220 | A regular space is a topol... |
| regsep 23221 | In a regular space, every ... |
| isnrm 23222 | The predicate "is a normal... |
| nrmtop 23223 | A normal space is a topolo... |
| cnrmtop 23224 | A completely normal space ... |
| iscnrm2 23225 | The property of being comp... |
| ispnrm 23226 | The property of being perf... |
| pnrmnrm 23227 | A perfectly normal space i... |
| pnrmtop 23228 | A perfectly normal space i... |
| pnrmcld 23229 | A closed set in a perfectl... |
| pnrmopn 23230 | An open set in a perfectly... |
| ist0-2 23231 | The predicate "is a T_0 sp... |
| ist0-3 23232 | The predicate "is a T_0 sp... |
| cnt0 23233 | The preimage of a T_0 topo... |
| ist1-2 23234 | An alternate characterizat... |
| t1t0 23235 | A T_1 space is a T_0 space... |
| ist1-3 23236 | A space is T_1 iff every p... |
| cnt1 23237 | The preimage of a T_1 topo... |
| ishaus2 23238 | Express the predicate " ` ... |
| haust1 23239 | A Hausdorff space is a T_1... |
| hausnei2 23240 | The Hausdorff condition st... |
| cnhaus 23241 | The preimage of a Hausdorf... |
| nrmsep3 23242 | In a normal space, given a... |
| nrmsep2 23243 | In a normal space, any two... |
| nrmsep 23244 | In a normal space, disjoin... |
| isnrm2 23245 | An alternate characterizat... |
| isnrm3 23246 | A topological space is nor... |
| cnrmi 23247 | A subspace of a completely... |
| cnrmnrm 23248 | A completely normal space ... |
| restcnrm 23249 | A subspace of a completely... |
| resthauslem 23250 | Lemma for ~ resthaus and s... |
| lpcls 23251 | The limit points of the cl... |
| perfcls 23252 | A subset of a perfect spac... |
| restt0 23253 | A subspace of a T_0 topolo... |
| restt1 23254 | A subspace of a T_1 topolo... |
| resthaus 23255 | A subspace of a Hausdorff ... |
| t1sep2 23256 | Any two points in a T_1 sp... |
| t1sep 23257 | Any two distinct points in... |
| sncld 23258 | A singleton is closed in a... |
| sshauslem 23259 | Lemma for ~ sshaus and sim... |
| sst0 23260 | A topology finer than a T_... |
| sst1 23261 | A topology finer than a T_... |
| sshaus 23262 | A topology finer than a Ha... |
| regsep2 23263 | In a regular space, a clos... |
| isreg2 23264 | A topological space is reg... |
| dnsconst 23265 | If a continuous mapping to... |
| ordtt1 23266 | The order topology is T_1 ... |
| lmmo 23267 | A sequence in a Hausdorff ... |
| lmfun 23268 | The convergence relation i... |
| dishaus 23269 | A discrete topology is Hau... |
| ordthauslem 23270 | Lemma for ~ ordthaus . (C... |
| ordthaus 23271 | The order topology of a to... |
| xrhaus 23272 | The topology of the extend... |
| iscmp 23275 | The predicate "is a compac... |
| cmpcov 23276 | An open cover of a compact... |
| cmpcov2 23277 | Rewrite ~ cmpcov for the c... |
| cmpcovf 23278 | Combine ~ cmpcov with ~ ac... |
| cncmp 23279 | Compactness is respected b... |
| fincmp 23280 | A finite topology is compa... |
| 0cmp 23281 | The singleton of the empty... |
| cmptop 23282 | A compact topology is a to... |
| rncmp 23283 | The image of a compact set... |
| imacmp 23284 | The image of a compact set... |
| discmp 23285 | A discrete topology is com... |
| cmpsublem 23286 | Lemma for ~ cmpsub . (Con... |
| cmpsub 23287 | Two equivalent ways of des... |
| tgcmp 23288 | A topology generated by a ... |
| cmpcld 23289 | A closed subset of a compa... |
| uncmp 23290 | The union of two compact s... |
| fiuncmp 23291 | A finite union of compact ... |
| sscmp 23292 | A subset of a compact topo... |
| hauscmplem 23293 | Lemma for ~ hauscmp . (Co... |
| hauscmp 23294 | A compact subspace of a T2... |
| cmpfi 23295 | If a topology is compact a... |
| cmpfii 23296 | In a compact topology, a s... |
| bwth 23297 | The glorious Bolzano-Weier... |
| isconn 23300 | The predicate ` J ` is a c... |
| isconn2 23301 | The predicate ` J ` is a c... |
| connclo 23302 | The only nonempty clopen s... |
| conndisj 23303 | If a topology is connected... |
| conntop 23304 | A connected topology is a ... |
| indisconn 23305 | The indiscrete topology (o... |
| dfconn2 23306 | An alternate definition of... |
| connsuba 23307 | Connectedness for a subspa... |
| connsub 23308 | Two equivalent ways of say... |
| cnconn 23309 | Connectedness is respected... |
| nconnsubb 23310 | Disconnectedness for a sub... |
| connsubclo 23311 | If a clopen set meets a co... |
| connima 23312 | The image of a connected s... |
| conncn 23313 | A continuous function from... |
| iunconnlem 23314 | Lemma for ~ iunconn . (Co... |
| iunconn 23315 | The indexed union of conne... |
| unconn 23316 | The union of two connected... |
| clsconn 23317 | The closure of a connected... |
| conncompid 23318 | The connected component co... |
| conncompconn 23319 | The connected component co... |
| conncompss 23320 | The connected component co... |
| conncompcld 23321 | The connected component co... |
| conncompclo 23322 | The connected component co... |
| t1connperf 23323 | A connected T_1 space is p... |
| is1stc 23328 | The predicate "is a first-... |
| is1stc2 23329 | An equivalent way of sayin... |
| 1stctop 23330 | A first-countable topology... |
| 1stcclb 23331 | A property of points in a ... |
| 1stcfb 23332 | For any point ` A ` in a f... |
| is2ndc 23333 | The property of being seco... |
| 2ndctop 23334 | A second-countable topolog... |
| 2ndci 23335 | A countable basis generate... |
| 2ndcsb 23336 | Having a countable subbase... |
| 2ndcredom 23337 | A second-countable space h... |
| 2ndc1stc 23338 | A second-countable space i... |
| 1stcrestlem 23339 | Lemma for ~ 1stcrest . (C... |
| 1stcrest 23340 | A subspace of a first-coun... |
| 2ndcrest 23341 | A subspace of a second-cou... |
| 2ndcctbss 23342 | If a topology is second-co... |
| 2ndcdisj 23343 | Any disjoint family of ope... |
| 2ndcdisj2 23344 | Any disjoint collection of... |
| 2ndcomap 23345 | A surjective continuous op... |
| 2ndcsep 23346 | A second-countable topolog... |
| dis2ndc 23347 | A discrete space is second... |
| 1stcelcls 23348 | A point belongs to the clo... |
| 1stccnp 23349 | A mapping is continuous at... |
| 1stccn 23350 | A mapping ` X --> Y ` , wh... |
| islly 23355 | The property of being a lo... |
| isnlly 23356 | The property of being an n... |
| llyeq 23357 | Equality theorem for the `... |
| nllyeq 23358 | Equality theorem for the `... |
| llytop 23359 | A locally ` A ` space is a... |
| nllytop 23360 | A locally ` A ` space is a... |
| llyi 23361 | The property of a locally ... |
| nllyi 23362 | The property of an n-local... |
| nlly2i 23363 | Eliminate the neighborhood... |
| llynlly 23364 | A locally ` A ` space is n... |
| llyssnlly 23365 | A locally ` A ` space is n... |
| llyss 23366 | The "locally" predicate re... |
| nllyss 23367 | The "n-locally" predicate ... |
| subislly 23368 | The property of a subspace... |
| restnlly 23369 | If the property ` A ` pass... |
| restlly 23370 | If the property ` A ` pass... |
| islly2 23371 | An alternative expression ... |
| llyrest 23372 | An open subspace of a loca... |
| nllyrest 23373 | An open subspace of an n-l... |
| loclly 23374 | If ` A ` is a local proper... |
| llyidm 23375 | Idempotence of the "locall... |
| nllyidm 23376 | Idempotence of the "n-loca... |
| toplly 23377 | A topology is locally a to... |
| topnlly 23378 | A topology is n-locally a ... |
| hauslly 23379 | A Hausdorff space is local... |
| hausnlly 23380 | A Hausdorff space is n-loc... |
| hausllycmp 23381 | A compact Hausdorff space ... |
| cldllycmp 23382 | A closed subspace of a loc... |
| lly1stc 23383 | First-countability is a lo... |
| dislly 23384 | The discrete space ` ~P X ... |
| disllycmp 23385 | A discrete space is locall... |
| dis1stc 23386 | A discrete space is first-... |
| hausmapdom 23387 | If ` X ` is a first-counta... |
| hauspwdom 23388 | Simplify the cardinal ` A ... |
| refrel 23395 | Refinement is a relation. ... |
| isref 23396 | The property of being a re... |
| refbas 23397 | A refinement covers the sa... |
| refssex 23398 | Every set in a refinement ... |
| ssref 23399 | A subcover is a refinement... |
| refref 23400 | Reflexivity of refinement.... |
| reftr 23401 | Refinement is transitive. ... |
| refun0 23402 | Adding the empty set prese... |
| isptfin 23403 | The statement "is a point-... |
| islocfin 23404 | The statement "is a locall... |
| finptfin 23405 | A finite cover is a point-... |
| ptfinfin 23406 | A point covered by a point... |
| finlocfin 23407 | A finite cover of a topolo... |
| locfintop 23408 | A locally finite cover cov... |
| locfinbas 23409 | A locally finite cover mus... |
| locfinnei 23410 | A point covered by a local... |
| lfinpfin 23411 | A locally finite cover is ... |
| lfinun 23412 | Adding a finite set preser... |
| locfincmp 23413 | For a compact space, the l... |
| unisngl 23414 | Taking the union of the se... |
| dissnref 23415 | The set of singletons is a... |
| dissnlocfin 23416 | The set of singletons is l... |
| locfindis 23417 | The locally finite covers ... |
| locfincf 23418 | A locally finite cover in ... |
| comppfsc 23419 | A space where every open c... |
| kgenval 23422 | Value of the compact gener... |
| elkgen 23423 | Value of the compact gener... |
| kgeni 23424 | Property of the open sets ... |
| kgentopon 23425 | The compact generator gene... |
| kgenuni 23426 | The base set of the compac... |
| kgenftop 23427 | The compact generator gene... |
| kgenf 23428 | The compact generator is a... |
| kgentop 23429 | A compactly generated spac... |
| kgenss 23430 | The compact generator gene... |
| kgenhaus 23431 | The compact generator gene... |
| kgencmp 23432 | The compact generator topo... |
| kgencmp2 23433 | The compact generator topo... |
| kgenidm 23434 | The compact generator is i... |
| iskgen2 23435 | A space is compactly gener... |
| iskgen3 23436 | Derive the usual definitio... |
| llycmpkgen2 23437 | A locally compact space is... |
| cmpkgen 23438 | A compact space is compact... |
| llycmpkgen 23439 | A locally compact space is... |
| 1stckgenlem 23440 | The one-point compactifica... |
| 1stckgen 23441 | A first-countable space is... |
| kgen2ss 23442 | The compact generator pres... |
| kgencn 23443 | A function from a compactl... |
| kgencn2 23444 | A function ` F : J --> K `... |
| kgencn3 23445 | The set of continuous func... |
| kgen2cn 23446 | A continuous function is a... |
| txval 23451 | Value of the binary topolo... |
| txuni2 23452 | The underlying set of the ... |
| txbasex 23453 | The basis for the product ... |
| txbas 23454 | The set of Cartesian produ... |
| eltx 23455 | A set in a product is open... |
| txtop 23456 | The product of two topolog... |
| ptval 23457 | The value of the product t... |
| ptpjpre1 23458 | The preimage of a projecti... |
| elpt 23459 | Elementhood in the bases o... |
| elptr 23460 | A basic open set in the pr... |
| elptr2 23461 | A basic open set in the pr... |
| ptbasid 23462 | The base set of the produc... |
| ptuni2 23463 | The base set for the produ... |
| ptbasin 23464 | The basis for a product to... |
| ptbasin2 23465 | The basis for a product to... |
| ptbas 23466 | The basis for a product to... |
| ptpjpre2 23467 | The basis for a product to... |
| ptbasfi 23468 | The basis for the product ... |
| pttop 23469 | The product topology is a ... |
| ptopn 23470 | A basic open set in the pr... |
| ptopn2 23471 | A sub-basic open set in th... |
| xkotf 23472 | Functionality of function ... |
| xkobval 23473 | Alternative expression for... |
| xkoval 23474 | Value of the compact-open ... |
| xkotop 23475 | The compact-open topology ... |
| xkoopn 23476 | A basic open set of the co... |
| txtopi 23477 | The product of two topolog... |
| txtopon 23478 | The underlying set of the ... |
| txuni 23479 | The underlying set of the ... |
| txunii 23480 | The underlying set of the ... |
| ptuni 23481 | The base set for the produ... |
| ptunimpt 23482 | Base set of a product topo... |
| pttopon 23483 | The base set for the produ... |
| pttoponconst 23484 | The base set for a product... |
| ptuniconst 23485 | The base set for a product... |
| xkouni 23486 | The base set of the compac... |
| xkotopon 23487 | The base set of the compac... |
| ptval2 23488 | The value of the product t... |
| txopn 23489 | The product of two open se... |
| txcld 23490 | The product of two closed ... |
| txcls 23491 | Closure of a rectangle in ... |
| txss12 23492 | Subset property of the top... |
| txbasval 23493 | It is sufficient to consid... |
| neitx 23494 | The Cartesian product of t... |
| txcnpi 23495 | Continuity of a two-argume... |
| tx1cn 23496 | Continuity of the first pr... |
| tx2cn 23497 | Continuity of the second p... |
| ptpjcn 23498 | Continuity of a projection... |
| ptpjopn 23499 | The projection map is an o... |
| ptcld 23500 | A closed box in the produc... |
| ptcldmpt 23501 | A closed box in the produc... |
| ptclsg 23502 | The closure of a box in th... |
| ptcls 23503 | The closure of a box in th... |
| dfac14lem 23504 | Lemma for ~ dfac14 . By e... |
| dfac14 23505 | Theorem ~ ptcls is an equi... |
| xkoccn 23506 | The "constant function" fu... |
| txcnp 23507 | If two functions are conti... |
| ptcnplem 23508 | Lemma for ~ ptcnp . (Cont... |
| ptcnp 23509 | If every projection of a f... |
| upxp 23510 | Universal property of the ... |
| txcnmpt 23511 | A map into the product of ... |
| uptx 23512 | Universal property of the ... |
| txcn 23513 | A map into the product of ... |
| ptcn 23514 | If every projection of a f... |
| prdstopn 23515 | Topology of a structure pr... |
| prdstps 23516 | A structure product of top... |
| pwstps 23517 | A structure power of a top... |
| txrest 23518 | The subspace of a topologi... |
| txdis 23519 | The topological product of... |
| txindislem 23520 | Lemma for ~ txindis . (Co... |
| txindis 23521 | The topological product of... |
| txdis1cn 23522 | A function is jointly cont... |
| txlly 23523 | If the property ` A ` is p... |
| txnlly 23524 | If the property ` A ` is p... |
| pthaus 23525 | The product of a collectio... |
| ptrescn 23526 | Restriction is a continuou... |
| txtube 23527 | The "tube lemma". If ` X ... |
| txcmplem1 23528 | Lemma for ~ txcmp . (Cont... |
| txcmplem2 23529 | Lemma for ~ txcmp . (Cont... |
| txcmp 23530 | The topological product of... |
| txcmpb 23531 | The topological product of... |
| hausdiag 23532 | A topology is Hausdorff if... |
| hauseqlcld 23533 | In a Hausdorff topology, t... |
| txhaus 23534 | The topological product of... |
| txlm 23535 | Two sequences converge iff... |
| lmcn2 23536 | The image of a convergent ... |
| tx1stc 23537 | The topological product of... |
| tx2ndc 23538 | The topological product of... |
| txkgen 23539 | The topological product of... |
| xkohaus 23540 | If the codomain space is H... |
| xkoptsub 23541 | The compact-open topology ... |
| xkopt 23542 | The compact-open topology ... |
| xkopjcn 23543 | Continuity of a projection... |
| xkoco1cn 23544 | If ` F ` is a continuous f... |
| xkoco2cn 23545 | If ` F ` is a continuous f... |
| xkococnlem 23546 | Continuity of the composit... |
| xkococn 23547 | Continuity of the composit... |
| cnmptid 23548 | The identity function is c... |
| cnmptc 23549 | A constant function is con... |
| cnmpt11 23550 | The composition of continu... |
| cnmpt11f 23551 | The composition of continu... |
| cnmpt1t 23552 | The composition of continu... |
| cnmpt12f 23553 | The composition of continu... |
| cnmpt12 23554 | The composition of continu... |
| cnmpt1st 23555 | The projection onto the fi... |
| cnmpt2nd 23556 | The projection onto the se... |
| cnmpt2c 23557 | A constant function is con... |
| cnmpt21 23558 | The composition of continu... |
| cnmpt21f 23559 | The composition of continu... |
| cnmpt2t 23560 | The composition of continu... |
| cnmpt22 23561 | The composition of continu... |
| cnmpt22f 23562 | The composition of continu... |
| cnmpt1res 23563 | The restriction of a conti... |
| cnmpt2res 23564 | The restriction of a conti... |
| cnmptcom 23565 | The argument converse of a... |
| cnmptkc 23566 | The curried first projecti... |
| cnmptkp 23567 | The evaluation of the inne... |
| cnmptk1 23568 | The composition of a curri... |
| cnmpt1k 23569 | The composition of a one-a... |
| cnmptkk 23570 | The composition of two cur... |
| xkofvcn 23571 | Joint continuity of the fu... |
| cnmptk1p 23572 | The evaluation of a currie... |
| cnmptk2 23573 | The uncurrying of a currie... |
| xkoinjcn 23574 | Continuity of "injection",... |
| cnmpt2k 23575 | The currying of a two-argu... |
| txconn 23576 | The topological product of... |
| imasnopn 23577 | If a relation graph is ope... |
| imasncld 23578 | If a relation graph is clo... |
| imasncls 23579 | If a relation graph is clo... |
| qtopval 23582 | Value of the quotient topo... |
| qtopval2 23583 | Value of the quotient topo... |
| elqtop 23584 | Value of the quotient topo... |
| qtopres 23585 | The quotient topology is u... |
| qtoptop2 23586 | The quotient topology is a... |
| qtoptop 23587 | The quotient topology is a... |
| elqtop2 23588 | Value of the quotient topo... |
| qtopuni 23589 | The base set of the quotie... |
| elqtop3 23590 | Value of the quotient topo... |
| qtoptopon 23591 | The base set of the quotie... |
| qtopid 23592 | A quotient map is a contin... |
| idqtop 23593 | The quotient topology indu... |
| qtopcmplem 23594 | Lemma for ~ qtopcmp and ~ ... |
| qtopcmp 23595 | A quotient of a compact sp... |
| qtopconn 23596 | A quotient of a connected ... |
| qtopkgen 23597 | A quotient of a compactly ... |
| basqtop 23598 | An injection maps bases to... |
| tgqtop 23599 | An injection maps generate... |
| qtopcld 23600 | The property of being a cl... |
| qtopcn 23601 | Universal property of a qu... |
| qtopss 23602 | A surjective continuous fu... |
| qtopeu 23603 | Universal property of the ... |
| qtoprest 23604 | If ` A ` is a saturated op... |
| qtopomap 23605 | If ` F ` is a surjective c... |
| qtopcmap 23606 | If ` F ` is a surjective c... |
| imastopn 23607 | The topology of an image s... |
| imastps 23608 | The image of a topological... |
| qustps 23609 | A quotient structure is a ... |
| kqfval 23610 | Value of the function appe... |
| kqfeq 23611 | Two points in the Kolmogor... |
| kqffn 23612 | The topological indistingu... |
| kqval 23613 | Value of the quotient topo... |
| kqtopon 23614 | The Kolmogorov quotient is... |
| kqid 23615 | The topological indistingu... |
| ist0-4 23616 | The topological indistingu... |
| kqfvima 23617 | When the image set is open... |
| kqsat 23618 | Any open set is saturated ... |
| kqdisj 23619 | A version of ~ imain for t... |
| kqcldsat 23620 | Any closed set is saturate... |
| kqopn 23621 | The topological indistingu... |
| kqcld 23622 | The topological indistingu... |
| kqt0lem 23623 | Lemma for ~ kqt0 . (Contr... |
| isr0 23624 | The property " ` J ` is an... |
| r0cld 23625 | The analogue of the T_1 ax... |
| regr1lem 23626 | Lemma for ~ regr1 . (Cont... |
| regr1lem2 23627 | A Kolmogorov quotient of a... |
| kqreglem1 23628 | A Kolmogorov quotient of a... |
| kqreglem2 23629 | If the Kolmogorov quotient... |
| kqnrmlem1 23630 | A Kolmogorov quotient of a... |
| kqnrmlem2 23631 | If the Kolmogorov quotient... |
| kqtop 23632 | The Kolmogorov quotient is... |
| kqt0 23633 | The Kolmogorov quotient is... |
| kqf 23634 | The Kolmogorov quotient is... |
| r0sep 23635 | The separation property of... |
| nrmr0reg 23636 | A normal R_0 space is also... |
| regr1 23637 | A regular space is R_1, wh... |
| kqreg 23638 | The Kolmogorov quotient of... |
| kqnrm 23639 | The Kolmogorov quotient of... |
| hmeofn 23644 | The set of homeomorphisms ... |
| hmeofval 23645 | The set of all the homeomo... |
| ishmeo 23646 | The predicate F is a homeo... |
| hmeocn 23647 | A homeomorphism is continu... |
| hmeocnvcn 23648 | The converse of a homeomor... |
| hmeocnv 23649 | The converse of a homeomor... |
| hmeof1o2 23650 | A homeomorphism is a 1-1-o... |
| hmeof1o 23651 | A homeomorphism is a 1-1-o... |
| hmeoima 23652 | The image of an open set b... |
| hmeoopn 23653 | Homeomorphisms preserve op... |
| hmeocld 23654 | Homeomorphisms preserve cl... |
| hmeocls 23655 | Homeomorphisms preserve cl... |
| hmeontr 23656 | Homeomorphisms preserve in... |
| hmeoimaf1o 23657 | The function mapping open ... |
| hmeores 23658 | The restriction of a homeo... |
| hmeoco 23659 | The composite of two homeo... |
| idhmeo 23660 | The identity function is a... |
| hmeocnvb 23661 | The converse of a homeomor... |
| hmeoqtop 23662 | A homeomorphism is a quoti... |
| hmph 23663 | Express the predicate ` J ... |
| hmphi 23664 | If there is a homeomorphis... |
| hmphtop 23665 | Reverse closure for the ho... |
| hmphtop1 23666 | The relation "being homeom... |
| hmphtop2 23667 | The relation "being homeom... |
| hmphref 23668 | "Is homeomorphic to" is re... |
| hmphsym 23669 | "Is homeomorphic to" is sy... |
| hmphtr 23670 | "Is homeomorphic to" is tr... |
| hmpher 23671 | "Is homeomorphic to" is an... |
| hmphen 23672 | Homeomorphisms preserve th... |
| hmphsymb 23673 | "Is homeomorphic to" is sy... |
| haushmphlem 23674 | Lemma for ~ haushmph and s... |
| cmphmph 23675 | Compactness is a topologic... |
| connhmph 23676 | Connectedness is a topolog... |
| t0hmph 23677 | T_0 is a topological prope... |
| t1hmph 23678 | T_1 is a topological prope... |
| haushmph 23679 | Hausdorff-ness is a topolo... |
| reghmph 23680 | Regularity is a topologica... |
| nrmhmph 23681 | Normality is a topological... |
| hmph0 23682 | A topology homeomorphic to... |
| hmphdis 23683 | Homeomorphisms preserve to... |
| hmphindis 23684 | Homeomorphisms preserve to... |
| indishmph 23685 | Equinumerous sets equipped... |
| hmphen2 23686 | Homeomorphisms preserve th... |
| cmphaushmeo 23687 | A continuous bijection fro... |
| ordthmeolem 23688 | Lemma for ~ ordthmeo . (C... |
| ordthmeo 23689 | An order isomorphism is a ... |
| txhmeo 23690 | Lift a pair of homeomorphi... |
| txswaphmeolem 23691 | Show inverse for the "swap... |
| txswaphmeo 23692 | There is a homeomorphism f... |
| pt1hmeo 23693 | The canonical homeomorphis... |
| ptuncnv 23694 | Exhibit the converse funct... |
| ptunhmeo 23695 | Define a homeomorphism fro... |
| xpstopnlem1 23696 | The function ` F ` used in... |
| xpstps 23697 | A binary product of topolo... |
| xpstopnlem2 23698 | Lemma for ~ xpstopn . (Co... |
| xpstopn 23699 | The topology on a binary p... |
| ptcmpfi 23700 | A topological product of f... |
| xkocnv 23701 | The inverse of the "curryi... |
| xkohmeo 23702 | The Exponential Law for to... |
| qtopf1 23703 | If a quotient map is injec... |
| qtophmeo 23704 | If two functions on a base... |
| t0kq 23705 | A topological space is T_0... |
| kqhmph 23706 | A topological space is T_0... |
| ist1-5lem 23707 | Lemma for ~ ist1-5 and sim... |
| t1r0 23708 | A T_1 space is R_0. That ... |
| ist1-5 23709 | A topological space is T_1... |
| ishaus3 23710 | A topological space is Hau... |
| nrmreg 23711 | A normal T_1 space is regu... |
| reghaus 23712 | A regular T_0 space is Hau... |
| nrmhaus 23713 | A T_1 normal space is Haus... |
| elmptrab 23714 | Membership in a one-parame... |
| elmptrab2 23715 | Membership in a one-parame... |
| isfbas 23716 | The predicate " ` F ` is a... |
| fbasne0 23717 | There are no empty filter ... |
| 0nelfb 23718 | No filter base contains th... |
| fbsspw 23719 | A filter base on a set is ... |
| fbelss 23720 | An element of the filter b... |
| fbdmn0 23721 | The domain of a filter bas... |
| isfbas2 23722 | The predicate " ` F ` is a... |
| fbasssin 23723 | A filter base contains sub... |
| fbssfi 23724 | A filter base contains sub... |
| fbssint 23725 | A filter base contains sub... |
| fbncp 23726 | A filter base does not con... |
| fbun 23727 | A necessary and sufficient... |
| fbfinnfr 23728 | No filter base containing ... |
| opnfbas 23729 | The collection of open sup... |
| trfbas2 23730 | Conditions for the trace o... |
| trfbas 23731 | Conditions for the trace o... |
| isfil 23734 | The predicate "is a filter... |
| filfbas 23735 | A filter is a filter base.... |
| 0nelfil 23736 | The empty set doesn't belo... |
| fileln0 23737 | An element of a filter is ... |
| filsspw 23738 | A filter is a subset of th... |
| filelss 23739 | An element of a filter is ... |
| filss 23740 | A filter is closed under t... |
| filin 23741 | A filter is closed under t... |
| filtop 23742 | The underlying set belongs... |
| isfil2 23743 | Derive the standard axioms... |
| isfildlem 23744 | Lemma for ~ isfild . (Con... |
| isfild 23745 | Sufficient condition for a... |
| filfi 23746 | A filter is closed under t... |
| filinn0 23747 | The intersection of two el... |
| filintn0 23748 | A filter has the finite in... |
| filn0 23749 | The empty set is not a fil... |
| infil 23750 | The intersection of two fi... |
| snfil 23751 | A singleton is a filter. ... |
| fbasweak 23752 | A filter base on any set i... |
| snfbas 23753 | Condition for a singleton ... |
| fsubbas 23754 | A condition for a set to g... |
| fbasfip 23755 | A filter base has the fini... |
| fbunfip 23756 | A helpful lemma for showin... |
| fgval 23757 | The filter generating clas... |
| elfg 23758 | A condition for elements o... |
| ssfg 23759 | A filter base is a subset ... |
| fgss 23760 | A bigger base generates a ... |
| fgss2 23761 | A condition for a filter t... |
| fgfil 23762 | A filter generates itself.... |
| elfilss 23763 | An element belongs to a fi... |
| filfinnfr 23764 | No filter containing a fin... |
| fgcl 23765 | A generated filter is a fi... |
| fgabs 23766 | Absorption law for filter ... |
| neifil 23767 | The neighborhoods of a non... |
| filunibas 23768 | Recover the base set from ... |
| filunirn 23769 | Two ways to express a filt... |
| filconn 23770 | A filter gives rise to a c... |
| fbasrn 23771 | Given a filter on a domain... |
| filuni 23772 | The union of a nonempty se... |
| trfil1 23773 | Conditions for the trace o... |
| trfil2 23774 | Conditions for the trace o... |
| trfil3 23775 | Conditions for the trace o... |
| trfilss 23776 | If ` A ` is a member of th... |
| fgtr 23777 | If ` A ` is a member of th... |
| trfg 23778 | The trace operation and th... |
| trnei 23779 | The trace, over a set ` A ... |
| cfinfil 23780 | Relative complements of th... |
| csdfil 23781 | The set of all elements wh... |
| supfil 23782 | The supersets of a nonempt... |
| zfbas 23783 | The set of upper sets of i... |
| uzrest 23784 | The restriction of the set... |
| uzfbas 23785 | The set of upper sets of i... |
| isufil 23790 | The property of being an u... |
| ufilfil 23791 | An ultrafilter is a filter... |
| ufilss 23792 | For any subset of the base... |
| ufilb 23793 | The complement is in an ul... |
| ufilmax 23794 | Any filter finer than an u... |
| isufil2 23795 | The maximal property of an... |
| ufprim 23796 | An ultrafilter is a prime ... |
| trufil 23797 | Conditions for the trace o... |
| filssufilg 23798 | A filter is contained in s... |
| filssufil 23799 | A filter is contained in s... |
| isufl 23800 | Define the (strong) ultraf... |
| ufli 23801 | Property of a set that sat... |
| numufl 23802 | Consequence of ~ filssufil... |
| fiufl 23803 | A finite set satisfies the... |
| acufl 23804 | The axiom of choice implie... |
| ssufl 23805 | If ` Y ` is a subset of ` ... |
| ufileu 23806 | If the ultrafilter contain... |
| filufint 23807 | A filter is equal to the i... |
| uffix 23808 | Lemma for ~ fixufil and ~ ... |
| fixufil 23809 | The condition describing a... |
| uffixfr 23810 | An ultrafilter is either f... |
| uffix2 23811 | A classification of fixed ... |
| uffixsn 23812 | The singleton of the gener... |
| ufildom1 23813 | An ultrafilter is generate... |
| uffinfix 23814 | An ultrafilter containing ... |
| cfinufil 23815 | An ultrafilter is free iff... |
| ufinffr 23816 | An infinite subset is cont... |
| ufilen 23817 | Any infinite set has an ul... |
| ufildr 23818 | An ultrafilter gives rise ... |
| fin1aufil 23819 | There are no definable fre... |
| fmval 23830 | Introduce a function that ... |
| fmfil 23831 | A mapping filter is a filt... |
| fmf 23832 | Pushing-forward via a func... |
| fmss 23833 | A finer filter produces a ... |
| elfm 23834 | An element of a mapping fi... |
| elfm2 23835 | An element of a mapping fi... |
| fmfg 23836 | The image filter of a filt... |
| elfm3 23837 | An alternate formulation o... |
| imaelfm 23838 | An image of a filter eleme... |
| rnelfmlem 23839 | Lemma for ~ rnelfm . (Con... |
| rnelfm 23840 | A condition for a filter t... |
| fmfnfmlem1 23841 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem2 23842 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem3 23843 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem4 23844 | Lemma for ~ fmfnfm . (Con... |
| fmfnfm 23845 | A filter finer than an ima... |
| fmufil 23846 | An image filter of an ultr... |
| fmid 23847 | The filter map applied to ... |
| fmco 23848 | Composition of image filte... |
| ufldom 23849 | The ultrafilter lemma prop... |
| flimval 23850 | The set of limit points of... |
| elflim2 23851 | The predicate "is a limit ... |
| flimtop 23852 | Reverse closure for the li... |
| flimneiss 23853 | A filter contains the neig... |
| flimnei 23854 | A filter contains all of t... |
| flimelbas 23855 | A limit point of a filter ... |
| flimfil 23856 | Reverse closure for the li... |
| flimtopon 23857 | Reverse closure for the li... |
| elflim 23858 | The predicate "is a limit ... |
| flimss2 23859 | A limit point of a filter ... |
| flimss1 23860 | A limit point of a filter ... |
| neiflim 23861 | A point is a limit point o... |
| flimopn 23862 | The condition for being a ... |
| fbflim 23863 | A condition for a filter t... |
| fbflim2 23864 | A condition for a filter b... |
| flimclsi 23865 | The convergent points of a... |
| hausflimlem 23866 | If ` A ` and ` B ` are bot... |
| hausflimi 23867 | One direction of ~ hausfli... |
| hausflim 23868 | A condition for a topology... |
| flimcf 23869 | Fineness is properly chara... |
| flimrest 23870 | The set of limit points in... |
| flimclslem 23871 | Lemma for ~ flimcls . (Co... |
| flimcls 23872 | Closure in terms of filter... |
| flimsncls 23873 | If ` A ` is a limit point ... |
| hauspwpwf1 23874 | Lemma for ~ hauspwpwdom . ... |
| hauspwpwdom 23875 | If ` X ` is a Hausdorff sp... |
| flffval 23876 | Given a topology and a fil... |
| flfval 23877 | Given a function from a fi... |
| flfnei 23878 | The property of being a li... |
| flfneii 23879 | A neighborhood of a limit ... |
| isflf 23880 | The property of being a li... |
| flfelbas 23881 | A limit point of a functio... |
| flffbas 23882 | Limit points of a function... |
| flftg 23883 | Limit points of a function... |
| hausflf 23884 | If a function has its valu... |
| hausflf2 23885 | If a convergent function h... |
| cnpflfi 23886 | Forward direction of ~ cnp... |
| cnpflf2 23887 | ` F ` is continuous at poi... |
| cnpflf 23888 | Continuity of a function a... |
| cnflf 23889 | A function is continuous i... |
| cnflf2 23890 | A function is continuous i... |
| flfcnp 23891 | A continuous function pres... |
| lmflf 23892 | The topological limit rela... |
| txflf 23893 | Two sequences converge in ... |
| flfcnp2 23894 | The image of a convergent ... |
| fclsval 23895 | The set of all cluster poi... |
| isfcls 23896 | A cluster point of a filte... |
| fclsfil 23897 | Reverse closure for the cl... |
| fclstop 23898 | Reverse closure for the cl... |
| fclstopon 23899 | Reverse closure for the cl... |
| isfcls2 23900 | A cluster point of a filte... |
| fclsopn 23901 | Write the cluster point co... |
| fclsopni 23902 | An open neighborhood of a ... |
| fclselbas 23903 | A cluster point is in the ... |
| fclsneii 23904 | A neighborhood of a cluste... |
| fclssscls 23905 | The set of cluster points ... |
| fclsnei 23906 | Cluster points in terms of... |
| supnfcls 23907 | The filter of supersets of... |
| fclsbas 23908 | Cluster points in terms of... |
| fclsss1 23909 | A finer topology has fewer... |
| fclsss2 23910 | A finer filter has fewer c... |
| fclsrest 23911 | The set of cluster points ... |
| fclscf 23912 | Characterization of finene... |
| flimfcls 23913 | A limit point is a cluster... |
| fclsfnflim 23914 | A filter clusters at a poi... |
| flimfnfcls 23915 | A filter converges to a po... |
| fclscmpi 23916 | Forward direction of ~ fcl... |
| fclscmp 23917 | A space is compact iff eve... |
| uffclsflim 23918 | The cluster points of an u... |
| ufilcmp 23919 | A space is compact iff eve... |
| fcfval 23920 | The set of cluster points ... |
| isfcf 23921 | The property of being a cl... |
| fcfnei 23922 | The property of being a cl... |
| fcfelbas 23923 | A cluster point of a funct... |
| fcfneii 23924 | A neighborhood of a cluste... |
| flfssfcf 23925 | A limit point of a functio... |
| uffcfflf 23926 | If the domain filter is an... |
| cnpfcfi 23927 | Lemma for ~ cnpfcf . If a... |
| cnpfcf 23928 | A function ` F ` is contin... |
| cnfcf 23929 | Continuity of a function i... |
| flfcntr 23930 | A continuous function's va... |
| alexsublem 23931 | Lemma for ~ alexsub . (Co... |
| alexsub 23932 | The Alexander Subbase Theo... |
| alexsubb 23933 | Biconditional form of the ... |
| alexsubALTlem1 23934 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem2 23935 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem3 23936 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem4 23937 | Lemma for ~ alexsubALT . ... |
| alexsubALT 23938 | The Alexander Subbase Theo... |
| ptcmplem1 23939 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem2 23940 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem3 23941 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem4 23942 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem5 23943 | Lemma for ~ ptcmp . (Cont... |
| ptcmpg 23944 | Tychonoff's theorem: The ... |
| ptcmp 23945 | Tychonoff's theorem: The ... |
| cnextval 23948 | The function applying cont... |
| cnextfval 23949 | The continuous extension o... |
| cnextrel 23950 | In the general case, a con... |
| cnextfun 23951 | If the target space is Hau... |
| cnextfvval 23952 | The value of the continuou... |
| cnextf 23953 | Extension by continuity. ... |
| cnextcn 23954 | Extension by continuity. ... |
| cnextfres1 23955 | ` F ` and its extension by... |
| cnextfres 23956 | ` F ` and its extension by... |
| istmd 23961 | The predicate "is a topolo... |
| tmdmnd 23962 | A topological monoid is a ... |
| tmdtps 23963 | A topological monoid is a ... |
| istgp 23964 | The predicate "is a topolo... |
| tgpgrp 23965 | A topological group is a g... |
| tgptmd 23966 | A topological group is a t... |
| tgptps 23967 | A topological group is a t... |
| tmdtopon 23968 | The topology of a topologi... |
| tgptopon 23969 | The topology of a topologi... |
| tmdcn 23970 | In a topological monoid, t... |
| tgpcn 23971 | In a topological group, th... |
| tgpinv 23972 | In a topological group, th... |
| grpinvhmeo 23973 | The inverse function in a ... |
| cnmpt1plusg 23974 | Continuity of the group su... |
| cnmpt2plusg 23975 | Continuity of the group su... |
| tmdcn2 23976 | Write out the definition o... |
| tgpsubcn 23977 | In a topological group, th... |
| istgp2 23978 | A group with a topology is... |
| tmdmulg 23979 | In a topological monoid, t... |
| tgpmulg 23980 | In a topological group, th... |
| tgpmulg2 23981 | In a topological monoid, t... |
| tmdgsum 23982 | In a topological monoid, t... |
| tmdgsum2 23983 | For any neighborhood ` U `... |
| oppgtmd 23984 | The opposite of a topologi... |
| oppgtgp 23985 | The opposite of a topologi... |
| distgp 23986 | Any group equipped with th... |
| indistgp 23987 | Any group equipped with th... |
| efmndtmd 23988 | The monoid of endofunction... |
| tmdlactcn 23989 | The left group action of e... |
| tgplacthmeo 23990 | The left group action of e... |
| submtmd 23991 | A submonoid of a topologic... |
| subgtgp 23992 | A subgroup of a topologica... |
| symgtgp 23993 | The symmetric group is a t... |
| subgntr 23994 | A subgroup of a topologica... |
| opnsubg 23995 | An open subgroup of a topo... |
| clssubg 23996 | The closure of a subgroup ... |
| clsnsg 23997 | The closure of a normal su... |
| cldsubg 23998 | A subgroup of finite index... |
| tgpconncompeqg 23999 | The connected component co... |
| tgpconncomp 24000 | The identity component, th... |
| tgpconncompss 24001 | The identity component is ... |
| ghmcnp 24002 | A group homomorphism on to... |
| snclseqg 24003 | The coset of the closure o... |
| tgphaus 24004 | A topological group is Hau... |
| tgpt1 24005 | Hausdorff and T1 are equiv... |
| tgpt0 24006 | Hausdorff and T0 are equiv... |
| qustgpopn 24007 | A quotient map in a topolo... |
| qustgplem 24008 | Lemma for ~ qustgp . (Con... |
| qustgp 24009 | The quotient of a topologi... |
| qustgphaus 24010 | The quotient of a topologi... |
| prdstmdd 24011 | The product of a family of... |
| prdstgpd 24012 | The product of a family of... |
| tsmsfbas 24015 | The collection of all sets... |
| tsmslem1 24016 | The finite partial sums of... |
| tsmsval2 24017 | Definition of the topologi... |
| tsmsval 24018 | Definition of the topologi... |
| tsmspropd 24019 | The group sum depends only... |
| eltsms 24020 | The property of being a su... |
| tsmsi 24021 | The property of being a su... |
| tsmscl 24022 | A sum in a topological gro... |
| haustsms 24023 | In a Hausdorff topological... |
| haustsms2 24024 | In a Hausdorff topological... |
| tsmscls 24025 | One half of ~ tgptsmscls ,... |
| tsmsgsum 24026 | The convergent points of a... |
| tsmsid 24027 | If a sum is finite, the us... |
| haustsmsid 24028 | In a Hausdorff topological... |
| tsms0 24029 | The sum of zero is zero. ... |
| tsmssubm 24030 | Evaluate an infinite group... |
| tsmsres 24031 | Extend an infinite group s... |
| tsmsf1o 24032 | Re-index an infinite group... |
| tsmsmhm 24033 | Apply a continuous group h... |
| tsmsadd 24034 | The sum of two infinite gr... |
| tsmsinv 24035 | Inverse of an infinite gro... |
| tsmssub 24036 | The difference of two infi... |
| tgptsmscls 24037 | A sum in a topological gro... |
| tgptsmscld 24038 | The set of limit points to... |
| tsmssplit 24039 | Split a topological group ... |
| tsmsxplem1 24040 | Lemma for ~ tsmsxp . (Con... |
| tsmsxplem2 24041 | Lemma for ~ tsmsxp . (Con... |
| tsmsxp 24042 | Write a sum over a two-dim... |
| istrg 24051 | Express the predicate " ` ... |
| trgtmd 24052 | The multiplicative monoid ... |
| istdrg 24053 | Express the predicate " ` ... |
| tdrgunit 24054 | The unit group of a topolo... |
| trgtgp 24055 | A topological ring is a to... |
| trgtmd2 24056 | A topological ring is a to... |
| trgtps 24057 | A topological ring is a to... |
| trgring 24058 | A topological ring is a ri... |
| trggrp 24059 | A topological ring is a gr... |
| tdrgtrg 24060 | A topological division rin... |
| tdrgdrng 24061 | A topological division rin... |
| tdrgring 24062 | A topological division rin... |
| tdrgtmd 24063 | A topological division rin... |
| tdrgtps 24064 | A topological division rin... |
| istdrg2 24065 | A topological-ring divisio... |
| mulrcn 24066 | The functionalization of t... |
| invrcn2 24067 | The multiplicative inverse... |
| invrcn 24068 | The multiplicative inverse... |
| cnmpt1mulr 24069 | Continuity of ring multipl... |
| cnmpt2mulr 24070 | Continuity of ring multipl... |
| dvrcn 24071 | The division function is c... |
| istlm 24072 | The predicate " ` W ` is a... |
| vscacn 24073 | The scalar multiplication ... |
| tlmtmd 24074 | A topological module is a ... |
| tlmtps 24075 | A topological module is a ... |
| tlmlmod 24076 | A topological module is a ... |
| tlmtrg 24077 | The scalar ring of a topol... |
| tlmscatps 24078 | The scalar ring of a topol... |
| istvc 24079 | A topological vector space... |
| tvctdrg 24080 | The scalar field of a topo... |
| cnmpt1vsca 24081 | Continuity of scalar multi... |
| cnmpt2vsca 24082 | Continuity of scalar multi... |
| tlmtgp 24083 | A topological vector space... |
| tvctlm 24084 | A topological vector space... |
| tvclmod 24085 | A topological vector space... |
| tvclvec 24086 | A topological vector space... |
| ustfn 24089 | The defined uniform struct... |
| ustval 24090 | The class of all uniform s... |
| isust 24091 | The predicate " ` U ` is a... |
| ustssxp 24092 | Entourages are subsets of ... |
| ustssel 24093 | A uniform structure is upw... |
| ustbasel 24094 | The full set is always an ... |
| ustincl 24095 | A uniform structure is clo... |
| ustdiag 24096 | The diagonal set is includ... |
| ustinvel 24097 | If ` V ` is an entourage, ... |
| ustexhalf 24098 | For each entourage ` V ` t... |
| ustrel 24099 | The elements of uniform st... |
| ustfilxp 24100 | A uniform structure on a n... |
| ustne0 24101 | A uniform structure cannot... |
| ustssco 24102 | In an uniform structure, a... |
| ustexsym 24103 | In an uniform structure, f... |
| ustex2sym 24104 | In an uniform structure, f... |
| ustex3sym 24105 | In an uniform structure, f... |
| ustref 24106 | Any element of the base se... |
| ust0 24107 | The unique uniform structu... |
| ustn0 24108 | The empty set is not an un... |
| ustund 24109 | If two intersecting sets `... |
| ustelimasn 24110 | Any point ` A ` is near en... |
| ustneism 24111 | For a point ` A ` in ` X `... |
| elrnustOLD 24112 | Obsolete version of ~ elfv... |
| ustbas2 24113 | Second direction for ~ ust... |
| ustuni 24114 | The set union of a uniform... |
| ustbas 24115 | Recover the base of an uni... |
| ustimasn 24116 | Lemma for ~ ustuqtop . (C... |
| trust 24117 | The trace of a uniform str... |
| utopval 24120 | The topology induced by a ... |
| elutop 24121 | Open sets in the topology ... |
| utoptop 24122 | The topology induced by a ... |
| utopbas 24123 | The base of the topology i... |
| utoptopon 24124 | Topology induced by a unif... |
| restutop 24125 | Restriction of a topology ... |
| restutopopn 24126 | The restriction of the top... |
| ustuqtoplem 24127 | Lemma for ~ ustuqtop . (C... |
| ustuqtop0 24128 | Lemma for ~ ustuqtop . (C... |
| ustuqtop1 24129 | Lemma for ~ ustuqtop , sim... |
| ustuqtop2 24130 | Lemma for ~ ustuqtop . (C... |
| ustuqtop3 24131 | Lemma for ~ ustuqtop , sim... |
| ustuqtop4 24132 | Lemma for ~ ustuqtop . (C... |
| ustuqtop5 24133 | Lemma for ~ ustuqtop . (C... |
| ustuqtop 24134 | For a given uniform struct... |
| utopsnneiplem 24135 | The neighborhoods of a poi... |
| utopsnneip 24136 | The neighborhoods of a poi... |
| utopsnnei 24137 | Images of singletons by en... |
| utop2nei 24138 | For any symmetrical entour... |
| utop3cls 24139 | Relation between a topolog... |
| utopreg 24140 | All Hausdorff uniform spac... |
| ussval 24147 | The uniform structure on u... |
| ussid 24148 | In case the base of the ` ... |
| isusp 24149 | The predicate ` W ` is a u... |
| ressuss 24150 | Value of the uniform struc... |
| ressust 24151 | The uniform structure of a... |
| ressusp 24152 | The restriction of a unifo... |
| tusval 24153 | The value of the uniform s... |
| tuslem 24154 | Lemma for ~ tusbas , ~ tus... |
| tusbas 24155 | The base set of a construc... |
| tusunif 24156 | The uniform structure of a... |
| tususs 24157 | The uniform structure of a... |
| tustopn 24158 | The topology induced by a ... |
| tususp 24159 | A constructed uniform spac... |
| tustps 24160 | A constructed uniform spac... |
| uspreg 24161 | If a uniform space is Haus... |
| ucnval 24164 | The set of all uniformly c... |
| isucn 24165 | The predicate " ` F ` is a... |
| isucn2 24166 | The predicate " ` F ` is a... |
| ucnimalem 24167 | Reformulate the ` G ` func... |
| ucnima 24168 | An equivalent statement of... |
| ucnprima 24169 | The preimage by a uniforml... |
| iducn 24170 | The identity is uniformly ... |
| cstucnd 24171 | A constant function is uni... |
| ucncn 24172 | Uniform continuity implies... |
| iscfilu 24175 | The predicate " ` F ` is a... |
| cfilufbas 24176 | A Cauchy filter base is a ... |
| cfiluexsm 24177 | For a Cauchy filter base a... |
| fmucndlem 24178 | Lemma for ~ fmucnd . (Con... |
| fmucnd 24179 | The image of a Cauchy filt... |
| cfilufg 24180 | The filter generated by a ... |
| trcfilu 24181 | Condition for the trace of... |
| cfiluweak 24182 | A Cauchy filter base is al... |
| neipcfilu 24183 | In an uniform space, a nei... |
| iscusp 24186 | The predicate " ` W ` is a... |
| cuspusp 24187 | A complete uniform space i... |
| cuspcvg 24188 | In a complete uniform spac... |
| iscusp2 24189 | The predicate " ` W ` is a... |
| cnextucn 24190 | Extension by continuity. ... |
| ucnextcn 24191 | Extension by continuity. ... |
| ispsmet 24192 | Express the predicate " ` ... |
| psmetdmdm 24193 | Recover the base set from ... |
| psmetf 24194 | The distance function of a... |
| psmetcl 24195 | Closure of the distance fu... |
| psmet0 24196 | The distance function of a... |
| psmettri2 24197 | Triangle inequality for th... |
| psmetsym 24198 | The distance function of a... |
| psmettri 24199 | Triangle inequality for th... |
| psmetge0 24200 | The distance function of a... |
| psmetxrge0 24201 | The distance function of a... |
| psmetres2 24202 | Restriction of a pseudomet... |
| psmetlecl 24203 | Real closure of an extende... |
| distspace 24204 | A set ` X ` together with ... |
| ismet 24211 | Express the predicate " ` ... |
| isxmet 24212 | Express the predicate " ` ... |
| ismeti 24213 | Properties that determine ... |
| isxmetd 24214 | Properties that determine ... |
| isxmet2d 24215 | It is safe to only require... |
| metflem 24216 | Lemma for ~ metf and other... |
| xmetf 24217 | Mapping of the distance fu... |
| metf 24218 | Mapping of the distance fu... |
| xmetcl 24219 | Closure of the distance fu... |
| metcl 24220 | Closure of the distance fu... |
| ismet2 24221 | An extended metric is a me... |
| metxmet 24222 | A metric is an extended me... |
| xmetdmdm 24223 | Recover the base set from ... |
| metdmdm 24224 | Recover the base set from ... |
| xmetunirn 24225 | Two ways to express an ext... |
| xmeteq0 24226 | The value of an extended m... |
| meteq0 24227 | The value of a metric is z... |
| xmettri2 24228 | Triangle inequality for th... |
| mettri2 24229 | Triangle inequality for th... |
| xmet0 24230 | The distance function of a... |
| met0 24231 | The distance function of a... |
| xmetge0 24232 | The distance function of a... |
| metge0 24233 | The distance function of a... |
| xmetlecl 24234 | Real closure of an extende... |
| xmetsym 24235 | The distance function of a... |
| xmetpsmet 24236 | An extended metric is a ps... |
| xmettpos 24237 | The distance function of a... |
| metsym 24238 | The distance function of a... |
| xmettri 24239 | Triangle inequality for th... |
| mettri 24240 | Triangle inequality for th... |
| xmettri3 24241 | Triangle inequality for th... |
| mettri3 24242 | Triangle inequality for th... |
| xmetrtri 24243 | One half of the reverse tr... |
| xmetrtri2 24244 | The reverse triangle inequ... |
| metrtri 24245 | Reverse triangle inequalit... |
| xmetgt0 24246 | The distance function of a... |
| metgt0 24247 | The distance function of a... |
| metn0 24248 | A metric space is nonempty... |
| xmetres2 24249 | Restriction of an extended... |
| metreslem 24250 | Lemma for ~ metres . (Con... |
| metres2 24251 | Lemma for ~ metres . (Con... |
| xmetres 24252 | A restriction of an extend... |
| metres 24253 | A restriction of a metric ... |
| 0met 24254 | The empty metric. (Contri... |
| prdsdsf 24255 | The product metric is a fu... |
| prdsxmetlem 24256 | The product metric is an e... |
| prdsxmet 24257 | The product metric is an e... |
| prdsmet 24258 | The product metric is a me... |
| ressprdsds 24259 | Restriction of a product m... |
| resspwsds 24260 | Restriction of a power met... |
| imasdsf1olem 24261 | Lemma for ~ imasdsf1o . (... |
| imasdsf1o 24262 | The distance function is t... |
| imasf1oxmet 24263 | The image of an extended m... |
| imasf1omet 24264 | The image of a metric is a... |
| xpsdsfn 24265 | Closure of the metric in a... |
| xpsdsfn2 24266 | Closure of the metric in a... |
| xpsxmetlem 24267 | Lemma for ~ xpsxmet . (Co... |
| xpsxmet 24268 | A product metric of extend... |
| xpsdsval 24269 | Value of the metric in a b... |
| xpsmet 24270 | The direct product of two ... |
| blfvalps 24271 | The value of the ball func... |
| blfval 24272 | The value of the ball func... |
| blvalps 24273 | The ball around a point ` ... |
| blval 24274 | The ball around a point ` ... |
| elblps 24275 | Membership in a ball. (Co... |
| elbl 24276 | Membership in a ball. (Co... |
| elbl2ps 24277 | Membership in a ball. (Co... |
| elbl2 24278 | Membership in a ball. (Co... |
| elbl3ps 24279 | Membership in a ball, with... |
| elbl3 24280 | Membership in a ball, with... |
| blcomps 24281 | Commute the arguments to t... |
| blcom 24282 | Commute the arguments to t... |
| xblpnfps 24283 | The infinity ball in an ex... |
| xblpnf 24284 | The infinity ball in an ex... |
| blpnf 24285 | The infinity ball in a sta... |
| bldisj 24286 | Two balls are disjoint if ... |
| blgt0 24287 | A nonempty ball implies th... |
| bl2in 24288 | Two balls are disjoint if ... |
| xblss2ps 24289 | One ball is contained in a... |
| xblss2 24290 | One ball is contained in a... |
| blss2ps 24291 | One ball is contained in a... |
| blss2 24292 | One ball is contained in a... |
| blhalf 24293 | A ball of radius ` R / 2 `... |
| blfps 24294 | Mapping of a ball. (Contr... |
| blf 24295 | Mapping of a ball. (Contr... |
| blrnps 24296 | Membership in the range of... |
| blrn 24297 | Membership in the range of... |
| xblcntrps 24298 | A ball contains its center... |
| xblcntr 24299 | A ball contains its center... |
| blcntrps 24300 | A ball contains its center... |
| blcntr 24301 | A ball contains its center... |
| xbln0 24302 | A ball is nonempty iff the... |
| bln0 24303 | A ball is not empty. (Con... |
| blelrnps 24304 | A ball belongs to the set ... |
| blelrn 24305 | A ball belongs to the set ... |
| blssm 24306 | A ball is a subset of the ... |
| unirnblps 24307 | The union of the set of ba... |
| unirnbl 24308 | The union of the set of ba... |
| blin 24309 | The intersection of two ba... |
| ssblps 24310 | The size of a ball increas... |
| ssbl 24311 | The size of a ball increas... |
| blssps 24312 | Any point ` P ` in a ball ... |
| blss 24313 | Any point ` P ` in a ball ... |
| blssexps 24314 | Two ways to express the ex... |
| blssex 24315 | Two ways to express the ex... |
| ssblex 24316 | A nested ball exists whose... |
| blin2 24317 | Given any two balls and a ... |
| blbas 24318 | The balls of a metric spac... |
| blres 24319 | A ball in a restricted met... |
| xmeterval 24320 | Value of the "finitely sep... |
| xmeter 24321 | The "finitely separated" r... |
| xmetec 24322 | The equivalence classes un... |
| blssec 24323 | A ball centered at ` P ` i... |
| blpnfctr 24324 | The infinity ball in an ex... |
| xmetresbl 24325 | An extended metric restric... |
| mopnval 24326 | An open set is a subset of... |
| mopntopon 24327 | The set of open sets of a ... |
| mopntop 24328 | The set of open sets of a ... |
| mopnuni 24329 | The union of all open sets... |
| elmopn 24330 | The defining property of a... |
| mopnfss 24331 | The family of open sets of... |
| mopnm 24332 | The base set of a metric s... |
| elmopn2 24333 | A defining property of an ... |
| mopnss 24334 | An open set of a metric sp... |
| isxms 24335 | Express the predicate " ` ... |
| isxms2 24336 | Express the predicate " ` ... |
| isms 24337 | Express the predicate " ` ... |
| isms2 24338 | Express the predicate " ` ... |
| xmstopn 24339 | The topology component of ... |
| mstopn 24340 | The topology component of ... |
| xmstps 24341 | An extended metric space i... |
| msxms 24342 | A metric space is an exten... |
| mstps 24343 | A metric space is a topolo... |
| xmsxmet 24344 | The distance function, sui... |
| msmet 24345 | The distance function, sui... |
| msf 24346 | The distance function of a... |
| xmsxmet2 24347 | The distance function, sui... |
| msmet2 24348 | The distance function, sui... |
| mscl 24349 | Closure of the distance fu... |
| xmscl 24350 | Closure of the distance fu... |
| xmsge0 24351 | The distance function in a... |
| xmseq0 24352 | The distance between two p... |
| xmssym 24353 | The distance function in a... |
| xmstri2 24354 | Triangle inequality for th... |
| mstri2 24355 | Triangle inequality for th... |
| xmstri 24356 | Triangle inequality for th... |
| mstri 24357 | Triangle inequality for th... |
| xmstri3 24358 | Triangle inequality for th... |
| mstri3 24359 | Triangle inequality for th... |
| msrtri 24360 | Reverse triangle inequalit... |
| xmspropd 24361 | Property deduction for an ... |
| mspropd 24362 | Property deduction for a m... |
| setsmsbas 24363 | The base set of a construc... |
| setsmsds 24364 | The distance function of a... |
| setsmstset 24365 | The topology of a construc... |
| setsmstopn 24366 | The topology of a construc... |
| setsxms 24367 | The constructed metric spa... |
| setsms 24368 | The constructed metric spa... |
| tmsval 24369 | For any metric there is an... |
| tmslem 24370 | Lemma for ~ tmsbas , ~ tms... |
| tmsbas 24371 | The base set of a construc... |
| tmsds 24372 | The metric of a constructe... |
| tmstopn 24373 | The topology of a construc... |
| tmsxms 24374 | The constructed metric spa... |
| tmsms 24375 | The constructed metric spa... |
| imasf1obl 24376 | The image of a metric spac... |
| imasf1oxms 24377 | The image of a metric spac... |
| imasf1oms 24378 | The image of a metric spac... |
| prdsbl 24379 | A ball in the product metr... |
| mopni 24380 | An open set of a metric sp... |
| mopni2 24381 | An open set of a metric sp... |
| mopni3 24382 | An open set of a metric sp... |
| blssopn 24383 | The balls of a metric spac... |
| unimopn 24384 | The union of a collection ... |
| mopnin 24385 | The intersection of two op... |
| mopn0 24386 | The empty set is an open s... |
| rnblopn 24387 | A ball of a metric space i... |
| blopn 24388 | A ball of a metric space i... |
| neibl 24389 | The neighborhoods around a... |
| blnei 24390 | A ball around a point is a... |
| lpbl 24391 | Every ball around a limit ... |
| blsscls2 24392 | A smaller closed ball is c... |
| blcld 24393 | A "closed ball" in a metri... |
| blcls 24394 | The closure of an open bal... |
| blsscls 24395 | If two concentric balls ha... |
| metss 24396 | Two ways of saying that me... |
| metequiv 24397 | Two ways of saying that tw... |
| metequiv2 24398 | If there is a sequence of ... |
| metss2lem 24399 | Lemma for ~ metss2 . (Con... |
| metss2 24400 | If the metric ` D ` is "st... |
| comet 24401 | The composition of an exte... |
| stdbdmetval 24402 | Value of the standard boun... |
| stdbdxmet 24403 | The standard bounded metri... |
| stdbdmet 24404 | The standard bounded metri... |
| stdbdbl 24405 | The standard bounded metri... |
| stdbdmopn 24406 | The standard bounded metri... |
| mopnex 24407 | The topology generated by ... |
| methaus 24408 | The topology generated by ... |
| met1stc 24409 | The topology generated by ... |
| met2ndci 24410 | A separable metric space (... |
| met2ndc 24411 | A metric space is second-c... |
| metrest 24412 | Two alternate formulations... |
| ressxms 24413 | The restriction of a metri... |
| ressms 24414 | The restriction of a metri... |
| prdsmslem1 24415 | Lemma for ~ prdsms . The ... |
| prdsxmslem1 24416 | Lemma for ~ prdsms . The ... |
| prdsxmslem2 24417 | Lemma for ~ prdsxms . The... |
| prdsxms 24418 | The indexed product struct... |
| prdsms 24419 | The indexed product struct... |
| pwsxms 24420 | A power of an extended met... |
| pwsms 24421 | A power of a metric space ... |
| xpsxms 24422 | A binary product of metric... |
| xpsms 24423 | A binary product of metric... |
| tmsxps 24424 | Express the product of two... |
| tmsxpsmopn 24425 | Express the product of two... |
| tmsxpsval 24426 | Value of the product of tw... |
| tmsxpsval2 24427 | Value of the product of tw... |
| metcnp3 24428 | Two ways to express that `... |
| metcnp 24429 | Two ways to say a mapping ... |
| metcnp2 24430 | Two ways to say a mapping ... |
| metcn 24431 | Two ways to say a mapping ... |
| metcnpi 24432 | Epsilon-delta property of ... |
| metcnpi2 24433 | Epsilon-delta property of ... |
| metcnpi3 24434 | Epsilon-delta property of ... |
| txmetcnp 24435 | Continuity of a binary ope... |
| txmetcn 24436 | Continuity of a binary ope... |
| metuval 24437 | Value of the uniform struc... |
| metustel 24438 | Define a filter base ` F `... |
| metustss 24439 | Range of the elements of t... |
| metustrel 24440 | Elements of the filter bas... |
| metustto 24441 | Any two elements of the fi... |
| metustid 24442 | The identity diagonal is i... |
| metustsym 24443 | Elements of the filter bas... |
| metustexhalf 24444 | For any element ` A ` of t... |
| metustfbas 24445 | The filter base generated ... |
| metust 24446 | The uniform structure gene... |
| cfilucfil 24447 | Given a metric ` D ` and a... |
| metuust 24448 | The uniform structure gene... |
| cfilucfil2 24449 | Given a metric ` D ` and a... |
| blval2 24450 | The ball around a point ` ... |
| elbl4 24451 | Membership in a ball, alte... |
| metuel 24452 | Elementhood in the uniform... |
| metuel2 24453 | Elementhood in the uniform... |
| metustbl 24454 | The "section" image of an ... |
| psmetutop 24455 | The topology induced by a ... |
| xmetutop 24456 | The topology induced by a ... |
| xmsusp 24457 | If the uniform set of a me... |
| restmetu 24458 | The uniform structure gene... |
| metucn 24459 | Uniform continuity in metr... |
| dscmet 24460 | The discrete metric on any... |
| dscopn 24461 | The discrete metric genera... |
| nrmmetd 24462 | Show that a group norm gen... |
| abvmet 24463 | An absolute value ` F ` ge... |
| nmfval 24476 | The value of the norm func... |
| nmval 24477 | The value of the norm as t... |
| nmfval0 24478 | The value of the norm func... |
| nmfval2 24479 | The value of the norm func... |
| nmval2 24480 | The value of the norm on a... |
| nmf2 24481 | The norm on a metric group... |
| nmpropd 24482 | Weak property deduction fo... |
| nmpropd2 24483 | Strong property deduction ... |
| isngp 24484 | The property of being a no... |
| isngp2 24485 | The property of being a no... |
| isngp3 24486 | The property of being a no... |
| ngpgrp 24487 | A normed group is a group.... |
| ngpms 24488 | A normed group is a metric... |
| ngpxms 24489 | A normed group is an exten... |
| ngptps 24490 | A normed group is a topolo... |
| ngpmet 24491 | The (induced) metric of a ... |
| ngpds 24492 | Value of the distance func... |
| ngpdsr 24493 | Value of the distance func... |
| ngpds2 24494 | Write the distance between... |
| ngpds2r 24495 | Write the distance between... |
| ngpds3 24496 | Write the distance between... |
| ngpds3r 24497 | Write the distance between... |
| ngprcan 24498 | Cancel right addition insi... |
| ngplcan 24499 | Cancel left addition insid... |
| isngp4 24500 | Express the property of be... |
| ngpinvds 24501 | Two elements are the same ... |
| ngpsubcan 24502 | Cancel right subtraction i... |
| nmf 24503 | The norm on a normed group... |
| nmcl 24504 | The norm of a normed group... |
| nmge0 24505 | The norm of a normed group... |
| nmeq0 24506 | The identity is the only e... |
| nmne0 24507 | The norm of a nonzero elem... |
| nmrpcl 24508 | The norm of a nonzero elem... |
| nminv 24509 | The norm of a negated elem... |
| nmmtri 24510 | The triangle inequality fo... |
| nmsub 24511 | The norm of the difference... |
| nmrtri 24512 | Reverse triangle inequalit... |
| nm2dif 24513 | Inequality for the differe... |
| nmtri 24514 | The triangle inequality fo... |
| nmtri2 24515 | Triangle inequality for th... |
| ngpi 24516 | The properties of a normed... |
| nm0 24517 | Norm of the identity eleme... |
| nmgt0 24518 | The norm of a nonzero elem... |
| sgrim 24519 | The induced metric on a su... |
| sgrimval 24520 | The induced metric on a su... |
| subgnm 24521 | The norm in a subgroup. (... |
| subgnm2 24522 | A substructure assigns the... |
| subgngp 24523 | A normed group restricted ... |
| ngptgp 24524 | A normed abelian group is ... |
| ngppropd 24525 | Property deduction for a n... |
| reldmtng 24526 | The function ` toNrmGrp ` ... |
| tngval 24527 | Value of the function whic... |
| tnglem 24528 | Lemma for ~ tngbas and sim... |
| tngbas 24529 | The base set of a structur... |
| tngplusg 24530 | The group addition of a st... |
| tng0 24531 | The group identity of a st... |
| tngmulr 24532 | The ring multiplication of... |
| tngsca 24533 | The scalar ring of a struc... |
| tngvsca 24534 | The scalar multiplication ... |
| tngip 24535 | The inner product operatio... |
| tngds 24536 | The metric function of a s... |
| tngtset 24537 | The topology generated by ... |
| tngtopn 24538 | The topology generated by ... |
| tngnm 24539 | The topology generated by ... |
| tngngp2 24540 | A norm turns a group into ... |
| tngngpd 24541 | Derive the axioms for a no... |
| tngngp 24542 | Derive the axioms for a no... |
| tnggrpr 24543 | If a structure equipped wi... |
| tngngp3 24544 | Alternate definition of a ... |
| nrmtngdist 24545 | The augmentation of a norm... |
| nrmtngnrm 24546 | The augmentation of a norm... |
| tngngpim 24547 | The induced metric of a no... |
| isnrg 24548 | A normed ring is a ring wi... |
| nrgabv 24549 | The norm of a normed ring ... |
| nrgngp 24550 | A normed ring is a normed ... |
| nrgring 24551 | A normed ring is a ring. ... |
| nmmul 24552 | The norm of a product in a... |
| nrgdsdi 24553 | Distribute a distance calc... |
| nrgdsdir 24554 | Distribute a distance calc... |
| nm1 24555 | The norm of one in a nonze... |
| unitnmn0 24556 | The norm of a unit is nonz... |
| nminvr 24557 | The norm of an inverse in ... |
| nmdvr 24558 | The norm of a division in ... |
| nrgdomn 24559 | A nonzero normed ring is a... |
| nrgtgp 24560 | A normed ring is a topolog... |
| subrgnrg 24561 | A normed ring restricted t... |
| tngnrg 24562 | Given any absolute value o... |
| isnlm 24563 | A normed (left) module is ... |
| nmvs 24564 | Defining property of a nor... |
| nlmngp 24565 | A normed module is a norme... |
| nlmlmod 24566 | A normed module is a left ... |
| nlmnrg 24567 | The scalar component of a ... |
| nlmngp2 24568 | The scalar component of a ... |
| nlmdsdi 24569 | Distribute a distance calc... |
| nlmdsdir 24570 | Distribute a distance calc... |
| nlmmul0or 24571 | If a scalar product is zer... |
| sranlm 24572 | The subring algebra over a... |
| nlmvscnlem2 24573 | Lemma for ~ nlmvscn . Com... |
| nlmvscnlem1 24574 | Lemma for ~ nlmvscn . (Co... |
| nlmvscn 24575 | The scalar multiplication ... |
| rlmnlm 24576 | The ring module over a nor... |
| rlmnm 24577 | The norm function in the r... |
| nrgtrg 24578 | A normed ring is a topolog... |
| nrginvrcnlem 24579 | Lemma for ~ nrginvrcn . C... |
| nrginvrcn 24580 | The ring inverse function ... |
| nrgtdrg 24581 | A normed division ring is ... |
| nlmtlm 24582 | A normed module is a topol... |
| isnvc 24583 | A normed vector space is j... |
| nvcnlm 24584 | A normed vector space is a... |
| nvclvec 24585 | A normed vector space is a... |
| nvclmod 24586 | A normed vector space is a... |
| isnvc2 24587 | A normed vector space is j... |
| nvctvc 24588 | A normed vector space is a... |
| lssnlm 24589 | A subspace of a normed mod... |
| lssnvc 24590 | A subspace of a normed vec... |
| rlmnvc 24591 | The ring module over a nor... |
| ngpocelbl 24592 | Membership of an off-cente... |
| nmoffn 24599 | The function producing ope... |
| reldmnghm 24600 | Lemma for normed group hom... |
| reldmnmhm 24601 | Lemma for module homomorph... |
| nmofval 24602 | Value of the operator norm... |
| nmoval 24603 | Value of the operator norm... |
| nmogelb 24604 | Property of the operator n... |
| nmolb 24605 | Any upper bound on the val... |
| nmolb2d 24606 | Any upper bound on the val... |
| nmof 24607 | The operator norm is a fun... |
| nmocl 24608 | The operator norm of an op... |
| nmoge0 24609 | The operator norm of an op... |
| nghmfval 24610 | A normed group homomorphis... |
| isnghm 24611 | A normed group homomorphis... |
| isnghm2 24612 | A normed group homomorphis... |
| isnghm3 24613 | A normed group homomorphis... |
| bddnghm 24614 | A bounded group homomorphi... |
| nghmcl 24615 | A normed group homomorphis... |
| nmoi 24616 | The operator norm achieves... |
| nmoix 24617 | The operator norm is a bou... |
| nmoi2 24618 | The operator norm is a bou... |
| nmoleub 24619 | The operator norm, defined... |
| nghmrcl1 24620 | Reverse closure for a norm... |
| nghmrcl2 24621 | Reverse closure for a norm... |
| nghmghm 24622 | A normed group homomorphis... |
| nmo0 24623 | The operator norm of the z... |
| nmoeq0 24624 | The operator norm is zero ... |
| nmoco 24625 | An upper bound on the oper... |
| nghmco 24626 | The composition of normed ... |
| nmotri 24627 | Triangle inequality for th... |
| nghmplusg 24628 | The sum of two bounded lin... |
| 0nghm 24629 | The zero operator is a nor... |
| nmoid 24630 | The operator norm of the i... |
| idnghm 24631 | The identity operator is a... |
| nmods 24632 | Upper bound for the distan... |
| nghmcn 24633 | A normed group homomorphis... |
| isnmhm 24634 | A normed module homomorphi... |
| nmhmrcl1 24635 | Reverse closure for a norm... |
| nmhmrcl2 24636 | Reverse closure for a norm... |
| nmhmlmhm 24637 | A normed module homomorphi... |
| nmhmnghm 24638 | A normed module homomorphi... |
| nmhmghm 24639 | A normed module homomorphi... |
| isnmhm2 24640 | A normed module homomorphi... |
| nmhmcl 24641 | A normed module homomorphi... |
| idnmhm 24642 | The identity operator is a... |
| 0nmhm 24643 | The zero operator is a bou... |
| nmhmco 24644 | The composition of bounded... |
| nmhmplusg 24645 | The sum of two bounded lin... |
| qtopbaslem 24646 | The set of open intervals ... |
| qtopbas 24647 | The set of open intervals ... |
| retopbas 24648 | A basis for the standard t... |
| retop 24649 | The standard topology on t... |
| uniretop 24650 | The underlying set of the ... |
| retopon 24651 | The standard topology on t... |
| retps 24652 | The standard topological s... |
| iooretop 24653 | Open intervals are open se... |
| icccld 24654 | Closed intervals are close... |
| icopnfcld 24655 | Right-unbounded closed int... |
| iocmnfcld 24656 | Left-unbounded closed inte... |
| qdensere 24657 | ` QQ ` is dense in the sta... |
| cnmetdval 24658 | Value of the distance func... |
| cnmet 24659 | The absolute value metric ... |
| cnxmet 24660 | The absolute value metric ... |
| cnbl0 24661 | Two ways to write the open... |
| cnblcld 24662 | Two ways to write the clos... |
| cnfldms 24663 | The complex number field i... |
| cnfldxms 24664 | The complex number field i... |
| cnfldtps 24665 | The complex number field i... |
| cnfldnm 24666 | The norm of the field of c... |
| cnngp 24667 | The complex numbers form a... |
| cnnrg 24668 | The complex numbers form a... |
| cnfldtopn 24669 | The topology of the comple... |
| cnfldtopon 24670 | The topology of the comple... |
| cnfldtop 24671 | The topology of the comple... |
| cnfldhaus 24672 | The topology of the comple... |
| unicntop 24673 | The underlying set of the ... |
| cnopn 24674 | The set of complex numbers... |
| cnn0opn 24675 | The set of nonzero complex... |
| zringnrg 24676 | The ring of integers is a ... |
| remetdval 24677 | Value of the distance func... |
| remet 24678 | The absolute value metric ... |
| rexmet 24679 | The absolute value metric ... |
| bl2ioo 24680 | A ball in terms of an open... |
| ioo2bl 24681 | An open interval of reals ... |
| ioo2blex 24682 | An open interval of reals ... |
| blssioo 24683 | The balls of the standard ... |
| tgioo 24684 | The topology generated by ... |
| qdensere2 24685 | ` QQ ` is dense in ` RR ` ... |
| blcvx 24686 | An open ball in the comple... |
| rehaus 24687 | The standard topology on t... |
| tgqioo 24688 | The topology generated by ... |
| re2ndc 24689 | The standard topology on t... |
| resubmet 24690 | The subspace topology indu... |
| tgioo2 24691 | The standard topology on t... |
| rerest 24692 | The subspace topology indu... |
| tgioo4 24693 | The standard topology on t... |
| tgioo3 24694 | The standard topology on t... |
| xrtgioo 24695 | The topology on the extend... |
| xrrest 24696 | The subspace topology indu... |
| xrrest2 24697 | The subspace topology indu... |
| xrsxmet 24698 | The metric on the extended... |
| xrsdsre 24699 | The metric on the extended... |
| xrsblre 24700 | Any ball of the metric of ... |
| xrsmopn 24701 | The metric on the extended... |
| zcld 24702 | The integers are a closed ... |
| recld2 24703 | The real numbers are a clo... |
| zcld2 24704 | The integers are a closed ... |
| zdis 24705 | The integers are a discret... |
| sszcld 24706 | Every subset of the intege... |
| reperflem 24707 | A subset of the real numbe... |
| reperf 24708 | The real numbers are a per... |
| cnperf 24709 | The complex numbers are a ... |
| iccntr 24710 | The interior of a closed i... |
| icccmplem1 24711 | Lemma for ~ icccmp . (Con... |
| icccmplem2 24712 | Lemma for ~ icccmp . (Con... |
| icccmplem3 24713 | Lemma for ~ icccmp . (Con... |
| icccmp 24714 | A closed interval in ` RR ... |
| reconnlem1 24715 | Lemma for ~ reconn . Conn... |
| reconnlem2 24716 | Lemma for ~ reconn . (Con... |
| reconn 24717 | A subset of the reals is c... |
| retopconn 24718 | Corollary of ~ reconn . T... |
| iccconn 24719 | A closed interval is conne... |
| opnreen 24720 | Every nonempty open set is... |
| rectbntr0 24721 | A countable subset of the ... |
| xrge0gsumle 24722 | A finite sum in the nonneg... |
| xrge0tsms 24723 | Any finite or infinite sum... |
| xrge0tsms2 24724 | Any finite or infinite sum... |
| metdcnlem 24725 | The metric function of a m... |
| xmetdcn2 24726 | The metric function of an ... |
| xmetdcn 24727 | The metric function of an ... |
| metdcn2 24728 | The metric function of a m... |
| metdcn 24729 | The metric function of a m... |
| msdcn 24730 | The metric function of a m... |
| cnmpt1ds 24731 | Continuity of the metric f... |
| cnmpt2ds 24732 | Continuity of the metric f... |
| nmcn 24733 | The norm of a normed group... |
| ngnmcncn 24734 | The norm of a normed group... |
| abscn 24735 | The absolute value functio... |
| metdsval 24736 | Value of the "distance to ... |
| metdsf 24737 | The distance from a point ... |
| metdsge 24738 | The distance from the poin... |
| metds0 24739 | If a point is in a set, it... |
| metdstri 24740 | A generalization of the tr... |
| metdsle 24741 | The distance from a point ... |
| metdsre 24742 | The distance from a point ... |
| metdseq0 24743 | The distance from a point ... |
| metdscnlem 24744 | Lemma for ~ metdscn . (Co... |
| metdscn 24745 | The function ` F ` which g... |
| metdscn2 24746 | The function ` F ` which g... |
| metnrmlem1a 24747 | Lemma for ~ metnrm . (Con... |
| metnrmlem1 24748 | Lemma for ~ metnrm . (Con... |
| metnrmlem2 24749 | Lemma for ~ metnrm . (Con... |
| metnrmlem3 24750 | Lemma for ~ metnrm . (Con... |
| metnrm 24751 | A metric space is normal. ... |
| metreg 24752 | A metric space is regular.... |
| addcnlem 24753 | Lemma for ~ addcn , ~ subc... |
| addcn 24754 | Complex number addition is... |
| subcn 24755 | Complex number subtraction... |
| mulcn 24756 | Complex number multiplicat... |
| divcnOLD 24757 | Obsolete version of ~ divc... |
| mpomulcn 24758 | Complex number multiplicat... |
| divcn 24759 | Complex number division is... |
| cnfldtgp 24760 | The complex numbers form a... |
| fsumcn 24761 | A finite sum of functions ... |
| fsum2cn 24762 | Version of ~ fsumcn for tw... |
| expcn 24763 | The power function on comp... |
| divccn 24764 | Division by a nonzero cons... |
| expcnOLD 24765 | Obsolete version of ~ expc... |
| divccnOLD 24766 | Obsolete version of ~ divc... |
| sqcn 24767 | The square function on com... |
| iitopon 24772 | The unit interval is a top... |
| iitop 24773 | The unit interval is a top... |
| iiuni 24774 | The base set of the unit i... |
| dfii2 24775 | Alternate definition of th... |
| dfii3 24776 | Alternate definition of th... |
| dfii4 24777 | Alternate definition of th... |
| dfii5 24778 | The unit interval expresse... |
| iicmp 24779 | The unit interval is compa... |
| iiconn 24780 | The unit interval is conne... |
| cncfval 24781 | The value of the continuou... |
| elcncf 24782 | Membership in the set of c... |
| elcncf2 24783 | Version of ~ elcncf with a... |
| cncfrss 24784 | Reverse closure of the con... |
| cncfrss2 24785 | Reverse closure of the con... |
| cncff 24786 | A continuous complex funct... |
| cncfi 24787 | Defining property of a con... |
| elcncf1di 24788 | Membership in the set of c... |
| elcncf1ii 24789 | Membership in the set of c... |
| rescncf 24790 | A continuous complex funct... |
| cncfcdm 24791 | Change the codomain of a c... |
| cncfss 24792 | The set of continuous func... |
| climcncf 24793 | Image of a limit under a c... |
| abscncf 24794 | Absolute value is continuo... |
| recncf 24795 | Real part is continuous. ... |
| imcncf 24796 | Imaginary part is continuo... |
| cjcncf 24797 | Complex conjugate is conti... |
| mulc1cncf 24798 | Multiplication by a consta... |
| divccncf 24799 | Division by a constant is ... |
| cncfco 24800 | The composition of two con... |
| cncfcompt2 24801 | Composition of continuous ... |
| cncfmet 24802 | Relate complex function co... |
| cncfcn 24803 | Relate complex function co... |
| cncfcn1 24804 | Relate complex function co... |
| cncfmptc 24805 | A constant function is a c... |
| cncfmptid 24806 | The identity function is a... |
| cncfmpt1f 24807 | Composition of continuous ... |
| cncfmpt2f 24808 | Composition of continuous ... |
| cncfmpt2ss 24809 | Composition of continuous ... |
| addccncf 24810 | Adding a constant is a con... |
| idcncf 24811 | The identity function is a... |
| sub1cncf 24812 | Subtracting a constant is ... |
| sub2cncf 24813 | Subtraction from a constan... |
| cdivcncf 24814 | Division with a constant n... |
| negcncf 24815 | The negative function is c... |
| negcncfOLD 24816 | Obsolete version of ~ negc... |
| negfcncf 24817 | The negative of a continuo... |
| abscncfALT 24818 | Absolute value is continuo... |
| cncfcnvcn 24819 | Rewrite ~ cmphaushmeo for ... |
| expcncf 24820 | The power function on comp... |
| cnmptre 24821 | Lemma for ~ iirevcn and re... |
| cnmpopc 24822 | Piecewise definition of a ... |
| iirev 24823 | Reverse the unit interval.... |
| iirevcn 24824 | The reversion function is ... |
| iihalf1 24825 | Map the first half of ` II... |
| iihalf1cn 24826 | The first half function is... |
| iihalf1cnOLD 24827 | Obsolete version of ~ iiha... |
| iihalf2 24828 | Map the second half of ` I... |
| iihalf2cn 24829 | The second half function i... |
| iihalf2cnOLD 24830 | Obsolete version of ~ iiha... |
| elii1 24831 | Divide the unit interval i... |
| elii2 24832 | Divide the unit interval i... |
| iimulcl 24833 | The unit interval is close... |
| iimulcn 24834 | Multiplication is a contin... |
| iimulcnOLD 24835 | Obsolete version of ~ iimu... |
| icoopnst 24836 | A half-open interval start... |
| iocopnst 24837 | A half-open interval endin... |
| icchmeo 24838 | The natural bijection from... |
| icchmeoOLD 24839 | Obsolete version of ~ icch... |
| icopnfcnv 24840 | Define a bijection from ` ... |
| icopnfhmeo 24841 | The defined bijection from... |
| iccpnfcnv 24842 | Define a bijection from ` ... |
| iccpnfhmeo 24843 | The defined bijection from... |
| xrhmeo 24844 | The bijection from ` [ -u ... |
| xrhmph 24845 | The extended reals are hom... |
| xrcmp 24846 | The topology of the extend... |
| xrconn 24847 | The topology of the extend... |
| icccvx 24848 | A linear combination of tw... |
| oprpiece1res1 24849 | Restriction to the first p... |
| oprpiece1res2 24850 | Restriction to the second ... |
| cnrehmeo 24851 | The canonical bijection fr... |
| cnrehmeoOLD 24852 | Obsolete version of ~ cnre... |
| cnheiborlem 24853 | Lemma for ~ cnheibor . (C... |
| cnheibor 24854 | Heine-Borel theorem for co... |
| cnllycmp 24855 | The topology on the comple... |
| rellycmp 24856 | The topology on the reals ... |
| bndth 24857 | The Boundedness Theorem. ... |
| evth 24858 | The Extreme Value Theorem.... |
| evth2 24859 | The Extreme Value Theorem,... |
| lebnumlem1 24860 | Lemma for ~ lebnum . The ... |
| lebnumlem2 24861 | Lemma for ~ lebnum . As a... |
| lebnumlem3 24862 | Lemma for ~ lebnum . By t... |
| lebnum 24863 | The Lebesgue number lemma,... |
| xlebnum 24864 | Generalize ~ lebnum to ext... |
| lebnumii 24865 | Specialize the Lebesgue nu... |
| ishtpy 24871 | Membership in the class of... |
| htpycn 24872 | A homotopy is a continuous... |
| htpyi 24873 | A homotopy evaluated at it... |
| ishtpyd 24874 | Deduction for membership i... |
| htpycom 24875 | Given a homotopy from ` F ... |
| htpyid 24876 | A homotopy from a function... |
| htpyco1 24877 | Compose a homotopy with a ... |
| htpyco2 24878 | Compose a homotopy with a ... |
| htpycc 24879 | Concatenate two homotopies... |
| isphtpy 24880 | Membership in the class of... |
| phtpyhtpy 24881 | A path homotopy is a homot... |
| phtpycn 24882 | A path homotopy is a conti... |
| phtpyi 24883 | Membership in the class of... |
| phtpy01 24884 | Two path-homotopic paths h... |
| isphtpyd 24885 | Deduction for membership i... |
| isphtpy2d 24886 | Deduction for membership i... |
| phtpycom 24887 | Given a homotopy from ` F ... |
| phtpyid 24888 | A homotopy from a path to ... |
| phtpyco2 24889 | Compose a path homotopy wi... |
| phtpycc 24890 | Concatenate two path homot... |
| phtpcrel 24892 | The path homotopy relation... |
| isphtpc 24893 | The relation "is path homo... |
| phtpcer 24894 | Path homotopy is an equiva... |
| phtpc01 24895 | Path homotopic paths have ... |
| reparphti 24896 | Lemma for ~ reparpht . (C... |
| reparphtiOLD 24897 | Obsolete version of ~ repa... |
| reparpht 24898 | Reparametrization lemma. ... |
| phtpcco2 24899 | Compose a path homotopy wi... |
| pcofval 24910 | The value of the path conc... |
| pcoval 24911 | The concatenation of two p... |
| pcovalg 24912 | Evaluate the concatenation... |
| pcoval1 24913 | Evaluate the concatenation... |
| pco0 24914 | The starting point of a pa... |
| pco1 24915 | The ending point of a path... |
| pcoval2 24916 | Evaluate the concatenation... |
| pcocn 24917 | The concatenation of two p... |
| copco 24918 | The composition of a conca... |
| pcohtpylem 24919 | Lemma for ~ pcohtpy . (Co... |
| pcohtpy 24920 | Homotopy invariance of pat... |
| pcoptcl 24921 | A constant function is a p... |
| pcopt 24922 | Concatenation with a point... |
| pcopt2 24923 | Concatenation with a point... |
| pcoass 24924 | Order of concatenation doe... |
| pcorevcl 24925 | Closure for a reversed pat... |
| pcorevlem 24926 | Lemma for ~ pcorev . Prov... |
| pcorev 24927 | Concatenation with the rev... |
| pcorev2 24928 | Concatenation with the rev... |
| pcophtb 24929 | The path homotopy equivale... |
| om1val 24930 | The definition of the loop... |
| om1bas 24931 | The base set of the loop s... |
| om1elbas 24932 | Elementhood in the base se... |
| om1addcl 24933 | Closure of the group opera... |
| om1plusg 24934 | The group operation (which... |
| om1tset 24935 | The topology of the loop s... |
| om1opn 24936 | The topology of the loop s... |
| pi1val 24937 | The definition of the fund... |
| pi1bas 24938 | The base set of the fundam... |
| pi1blem 24939 | Lemma for ~ pi1buni . (Co... |
| pi1buni 24940 | Another way to write the l... |
| pi1bas2 24941 | The base set of the fundam... |
| pi1eluni 24942 | Elementhood in the base se... |
| pi1bas3 24943 | The base set of the fundam... |
| pi1cpbl 24944 | The group operation, loop ... |
| elpi1 24945 | The elements of the fundam... |
| elpi1i 24946 | The elements of the fundam... |
| pi1addf 24947 | The group operation of ` p... |
| pi1addval 24948 | The concatenation of two p... |
| pi1grplem 24949 | Lemma for ~ pi1grp . (Con... |
| pi1grp 24950 | The fundamental group is a... |
| pi1id 24951 | The identity element of th... |
| pi1inv 24952 | An inverse in the fundamen... |
| pi1xfrf 24953 | Functionality of the loop ... |
| pi1xfrval 24954 | The value of the loop tran... |
| pi1xfr 24955 | Given a path ` F ` and its... |
| pi1xfrcnvlem 24956 | Given a path ` F ` between... |
| pi1xfrcnv 24957 | Given a path ` F ` between... |
| pi1xfrgim 24958 | The mapping ` G ` between ... |
| pi1cof 24959 | Functionality of the loop ... |
| pi1coval 24960 | The value of the loop tran... |
| pi1coghm 24961 | The mapping ` G ` between ... |
| isclm 24964 | A subcomplex module is a l... |
| clmsca 24965 | The ring of scalars ` F ` ... |
| clmsubrg 24966 | The base set of the ring o... |
| clmlmod 24967 | A subcomplex module is a l... |
| clmgrp 24968 | A subcomplex module is an ... |
| clmabl 24969 | A subcomplex module is an ... |
| clmring 24970 | The scalar ring of a subco... |
| clmfgrp 24971 | The scalar ring of a subco... |
| clm0 24972 | The zero of the scalar rin... |
| clm1 24973 | The identity of the scalar... |
| clmadd 24974 | The addition of the scalar... |
| clmmul 24975 | The multiplication of the ... |
| clmcj 24976 | The conjugation of the sca... |
| isclmi 24977 | Reverse direction of ~ isc... |
| clmzss 24978 | The scalar ring of a subco... |
| clmsscn 24979 | The scalar ring of a subco... |
| clmsub 24980 | Subtraction in the scalar ... |
| clmneg 24981 | Negation in the scalar rin... |
| clmneg1 24982 | Minus one is in the scalar... |
| clmabs 24983 | Norm in the scalar ring of... |
| clmacl 24984 | Closure of ring addition f... |
| clmmcl 24985 | Closure of ring multiplica... |
| clmsubcl 24986 | Closure of ring subtractio... |
| lmhmclm 24987 | The domain of a linear ope... |
| clmvscl 24988 | Closure of scalar product ... |
| clmvsass 24989 | Associative law for scalar... |
| clmvscom 24990 | Commutative law for the sc... |
| clmvsdir 24991 | Distributive law for scala... |
| clmvsdi 24992 | Distributive law for scala... |
| clmvs1 24993 | Scalar product with ring u... |
| clmvs2 24994 | A vector plus itself is tw... |
| clm0vs 24995 | Zero times a vector is the... |
| clmopfne 24996 | The (functionalized) opera... |
| isclmp 24997 | The predicate "is a subcom... |
| isclmi0 24998 | Properties that determine ... |
| clmvneg1 24999 | Minus 1 times a vector is ... |
| clmvsneg 25000 | Multiplication of a vector... |
| clmmulg 25001 | The group multiple functio... |
| clmsubdir 25002 | Scalar multiplication dist... |
| clmpm1dir 25003 | Subtractive distributive l... |
| clmnegneg 25004 | Double negative of a vecto... |
| clmnegsubdi2 25005 | Distribution of negative o... |
| clmsub4 25006 | Rearrangement of 4 terms i... |
| clmvsrinv 25007 | A vector minus itself. (C... |
| clmvslinv 25008 | Minus a vector plus itself... |
| clmvsubval 25009 | Value of vector subtractio... |
| clmvsubval2 25010 | Value of vector subtractio... |
| clmvz 25011 | Two ways to express the ne... |
| zlmclm 25012 | The ` ZZ ` -module operati... |
| clmzlmvsca 25013 | The scalar product of a su... |
| nmoleub2lem 25014 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem3 25015 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem2 25016 | Lemma for ~ nmoleub2a and ... |
| nmoleub2a 25017 | The operator norm is the s... |
| nmoleub2b 25018 | The operator norm is the s... |
| nmoleub3 25019 | The operator norm is the s... |
| nmhmcn 25020 | A linear operator over a n... |
| cmodscexp 25021 | The powers of ` _i ` belon... |
| cmodscmulexp 25022 | The scalar product of a ve... |
| cvslvec 25025 | A subcomplex vector space ... |
| cvsclm 25026 | A subcomplex vector space ... |
| iscvs 25027 | A subcomplex vector space ... |
| iscvsp 25028 | The predicate "is a subcom... |
| iscvsi 25029 | Properties that determine ... |
| cvsi 25030 | The properties of a subcom... |
| cvsunit 25031 | Unit group of the scalar r... |
| cvsdiv 25032 | Division of the scalar rin... |
| cvsdivcl 25033 | The scalar field of a subc... |
| cvsmuleqdivd 25034 | An equality involving rati... |
| cvsdiveqd 25035 | An equality involving rati... |
| cnlmodlem1 25036 | Lemma 1 for ~ cnlmod . (C... |
| cnlmodlem2 25037 | Lemma 2 for ~ cnlmod . (C... |
| cnlmodlem3 25038 | Lemma 3 for ~ cnlmod . (C... |
| cnlmod4 25039 | Lemma 4 for ~ cnlmod . (C... |
| cnlmod 25040 | The set of complex numbers... |
| cnstrcvs 25041 | The set of complex numbers... |
| cnrbas 25042 | The set of complex numbers... |
| cnrlmod 25043 | The complex left module of... |
| cnrlvec 25044 | The complex left module of... |
| cncvs 25045 | The complex left module of... |
| recvs 25046 | The field of the real numb... |
| qcvs 25047 | The field of rational numb... |
| zclmncvs 25048 | The ring of integers as le... |
| isncvsngp 25049 | A normed subcomplex vector... |
| isncvsngpd 25050 | Properties that determine ... |
| ncvsi 25051 | The properties of a normed... |
| ncvsprp 25052 | Proportionality property o... |
| ncvsge0 25053 | The norm of a scalar produ... |
| ncvsm1 25054 | The norm of the opposite o... |
| ncvsdif 25055 | The norm of the difference... |
| ncvspi 25056 | The norm of a vector plus ... |
| ncvs1 25057 | From any nonzero vector of... |
| cnrnvc 25058 | The module of complex numb... |
| cnncvs 25059 | The module of complex numb... |
| cnnm 25060 | The norm of the normed sub... |
| ncvspds 25061 | Value of the distance func... |
| cnindmet 25062 | The metric induced on the ... |
| cnncvsaddassdemo 25063 | Derive the associative law... |
| cnncvsmulassdemo 25064 | Derive the associative law... |
| cnncvsabsnegdemo 25065 | Derive the absolute value ... |
| iscph 25070 | A subcomplex pre-Hilbert s... |
| cphphl 25071 | A subcomplex pre-Hilbert s... |
| cphnlm 25072 | A subcomplex pre-Hilbert s... |
| cphngp 25073 | A subcomplex pre-Hilbert s... |
| cphlmod 25074 | A subcomplex pre-Hilbert s... |
| cphlvec 25075 | A subcomplex pre-Hilbert s... |
| cphnvc 25076 | A subcomplex pre-Hilbert s... |
| cphsubrglem 25077 | Lemma for ~ cphsubrg . (C... |
| cphreccllem 25078 | Lemma for ~ cphreccl . (C... |
| cphsca 25079 | A subcomplex pre-Hilbert s... |
| cphsubrg 25080 | The scalar field of a subc... |
| cphreccl 25081 | The scalar field of a subc... |
| cphdivcl 25082 | The scalar field of a subc... |
| cphcjcl 25083 | The scalar field of a subc... |
| cphsqrtcl 25084 | The scalar field of a subc... |
| cphabscl 25085 | The scalar field of a subc... |
| cphsqrtcl2 25086 | The scalar field of a subc... |
| cphsqrtcl3 25087 | If the scalar field of a s... |
| cphqss 25088 | The scalar field of a subc... |
| cphclm 25089 | A subcomplex pre-Hilbert s... |
| cphnmvs 25090 | Norm of a scalar product. ... |
| cphipcl 25091 | An inner product is a memb... |
| cphnmfval 25092 | The value of the norm in a... |
| cphnm 25093 | The square of the norm is ... |
| nmsq 25094 | The square of the norm is ... |
| cphnmf 25095 | The norm of a vector is a ... |
| cphnmcl 25096 | The norm of a vector is a ... |
| reipcl 25097 | An inner product of an ele... |
| ipge0 25098 | The inner product in a sub... |
| cphipcj 25099 | Conjugate of an inner prod... |
| cphipipcj 25100 | An inner product times its... |
| cphorthcom 25101 | Orthogonality (meaning inn... |
| cphip0l 25102 | Inner product with a zero ... |
| cphip0r 25103 | Inner product with a zero ... |
| cphipeq0 25104 | The inner product of a vec... |
| cphdir 25105 | Distributive law for inner... |
| cphdi 25106 | Distributive law for inner... |
| cph2di 25107 | Distributive law for inner... |
| cphsubdir 25108 | Distributive law for inner... |
| cphsubdi 25109 | Distributive law for inner... |
| cph2subdi 25110 | Distributive law for inner... |
| cphass 25111 | Associative law for inner ... |
| cphassr 25112 | "Associative" law for seco... |
| cph2ass 25113 | Move scalar multiplication... |
| cphassi 25114 | Associative law for the fi... |
| cphassir 25115 | "Associative" law for the ... |
| cphpyth 25116 | The pythagorean theorem fo... |
| tcphex 25117 | Lemma for ~ tcphbas and si... |
| tcphval 25118 | Define a function to augme... |
| tcphbas 25119 | The base set of a subcompl... |
| tchplusg 25120 | The addition operation of ... |
| tcphsub 25121 | The subtraction operation ... |
| tcphmulr 25122 | The ring operation of a su... |
| tcphsca 25123 | The scalar field of a subc... |
| tcphvsca 25124 | The scalar multiplication ... |
| tcphip 25125 | The inner product of a sub... |
| tcphtopn 25126 | The topology of a subcompl... |
| tcphphl 25127 | Augmentation of a subcompl... |
| tchnmfval 25128 | The norm of a subcomplex p... |
| tcphnmval 25129 | The norm of a subcomplex p... |
| cphtcphnm 25130 | The norm of a norm-augment... |
| tcphds 25131 | The distance of a pre-Hilb... |
| phclm 25132 | A pre-Hilbert space whose ... |
| tcphcphlem3 25133 | Lemma for ~ tcphcph : real... |
| ipcau2 25134 | The Cauchy-Schwarz inequal... |
| tcphcphlem1 25135 | Lemma for ~ tcphcph : the ... |
| tcphcphlem2 25136 | Lemma for ~ tcphcph : homo... |
| tcphcph 25137 | The standard definition of... |
| ipcau 25138 | The Cauchy-Schwarz inequal... |
| nmparlem 25139 | Lemma for ~ nmpar . (Cont... |
| nmpar 25140 | A subcomplex pre-Hilbert s... |
| cphipval2 25141 | Value of the inner product... |
| 4cphipval2 25142 | Four times the inner produ... |
| cphipval 25143 | Value of the inner product... |
| ipcnlem2 25144 | The inner product operatio... |
| ipcnlem1 25145 | The inner product operatio... |
| ipcn 25146 | The inner product operatio... |
| cnmpt1ip 25147 | Continuity of inner produc... |
| cnmpt2ip 25148 | Continuity of inner produc... |
| csscld 25149 | A "closed subspace" in a s... |
| clsocv 25150 | The orthogonal complement ... |
| cphsscph 25151 | A subspace of a subcomplex... |
| lmmbr 25158 | Express the binary relatio... |
| lmmbr2 25159 | Express the binary relatio... |
| lmmbr3 25160 | Express the binary relatio... |
| lmmcvg 25161 | Convergence property of a ... |
| lmmbrf 25162 | Express the binary relatio... |
| lmnn 25163 | A condition that implies c... |
| cfilfval 25164 | The set of Cauchy filters ... |
| iscfil 25165 | The property of being a Ca... |
| iscfil2 25166 | The property of being a Ca... |
| cfilfil 25167 | A Cauchy filter is a filte... |
| cfili 25168 | Property of a Cauchy filte... |
| cfil3i 25169 | A Cauchy filter contains b... |
| cfilss 25170 | A filter finer than a Cauc... |
| fgcfil 25171 | The Cauchy filter conditio... |
| fmcfil 25172 | The Cauchy filter conditio... |
| iscfil3 25173 | A filter is Cauchy iff it ... |
| cfilfcls 25174 | Similar to ultrafilters ( ... |
| caufval 25175 | The set of Cauchy sequence... |
| iscau 25176 | Express the property " ` F... |
| iscau2 25177 | Express the property " ` F... |
| iscau3 25178 | Express the Cauchy sequenc... |
| iscau4 25179 | Express the property " ` F... |
| iscauf 25180 | Express the property " ` F... |
| caun0 25181 | A metric with a Cauchy seq... |
| caufpm 25182 | Inclusion of a Cauchy sequ... |
| caucfil 25183 | A Cauchy sequence predicat... |
| iscmet 25184 | The property " ` D ` is a ... |
| cmetcvg 25185 | The convergence of a Cauch... |
| cmetmet 25186 | A complete metric space is... |
| cmetmeti 25187 | A complete metric space is... |
| cmetcaulem 25188 | Lemma for ~ cmetcau . (Co... |
| cmetcau 25189 | The convergence of a Cauch... |
| iscmet3lem3 25190 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem1 25191 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem2 25192 | Lemma for ~ iscmet3 . (Co... |
| iscmet3 25193 | The property " ` D ` is a ... |
| iscmet2 25194 | A metric ` D ` is complete... |
| cfilresi 25195 | A Cauchy filter on a metri... |
| cfilres 25196 | Cauchy filter on a metric ... |
| caussi 25197 | Cauchy sequence on a metri... |
| causs 25198 | Cauchy sequence on a metri... |
| equivcfil 25199 | If the metric ` D ` is "st... |
| equivcau 25200 | If the metric ` D ` is "st... |
| lmle 25201 | If the distance from each ... |
| nglmle 25202 | If the norm of each member... |
| lmclim 25203 | Relate a limit on the metr... |
| lmclimf 25204 | Relate a limit on the metr... |
| metelcls 25205 | A point belongs to the clo... |
| metcld 25206 | A subset of a metric space... |
| metcld2 25207 | A subset of a metric space... |
| caubl 25208 | Sufficient condition to en... |
| caublcls 25209 | The convergent point of a ... |
| metcnp4 25210 | Two ways to say a mapping ... |
| metcn4 25211 | Two ways to say a mapping ... |
| iscmet3i 25212 | Properties that determine ... |
| lmcau 25213 | Every convergent sequence ... |
| flimcfil 25214 | Every convergent filter in... |
| metsscmetcld 25215 | A complete subspace of a m... |
| cmetss 25216 | A subspace of a complete m... |
| equivcmet 25217 | If two metrics are strongl... |
| relcmpcmet 25218 | If ` D ` is a metric space... |
| cmpcmet 25219 | A compact metric space is ... |
| cfilucfil3 25220 | Given a metric ` D ` and a... |
| cfilucfil4 25221 | Given a metric ` D ` and a... |
| cncmet 25222 | The set of complex numbers... |
| recmet 25223 | The real numbers are a com... |
| bcthlem1 25224 | Lemma for ~ bcth . Substi... |
| bcthlem2 25225 | Lemma for ~ bcth . The ba... |
| bcthlem3 25226 | Lemma for ~ bcth . The li... |
| bcthlem4 25227 | Lemma for ~ bcth . Given ... |
| bcthlem5 25228 | Lemma for ~ bcth . The pr... |
| bcth 25229 | Baire's Category Theorem. ... |
| bcth2 25230 | Baire's Category Theorem, ... |
| bcth3 25231 | Baire's Category Theorem, ... |
| isbn 25238 | A Banach space is a normed... |
| bnsca 25239 | The scalar field of a Bana... |
| bnnvc 25240 | A Banach space is a normed... |
| bnnlm 25241 | A Banach space is a normed... |
| bnngp 25242 | A Banach space is a normed... |
| bnlmod 25243 | A Banach space is a left m... |
| bncms 25244 | A Banach space is a comple... |
| iscms 25245 | A complete metric space is... |
| cmscmet 25246 | The induced metric on a co... |
| bncmet 25247 | The induced metric on Bana... |
| cmsms 25248 | A complete metric space is... |
| cmspropd 25249 | Property deduction for a c... |
| cmssmscld 25250 | The restriction of a metri... |
| cmsss 25251 | The restriction of a compl... |
| lssbn 25252 | A subspace of a Banach spa... |
| cmetcusp1 25253 | If the uniform set of a co... |
| cmetcusp 25254 | The uniform space generate... |
| cncms 25255 | The field of complex numbe... |
| cnflduss 25256 | The uniform structure of t... |
| cnfldcusp 25257 | The field of complex numbe... |
| resscdrg 25258 | The real numbers are a sub... |
| cncdrg 25259 | The only complete subfield... |
| srabn 25260 | The subring algebra over a... |
| rlmbn 25261 | The ring module over a com... |
| ishl 25262 | The predicate "is a subcom... |
| hlbn 25263 | Every subcomplex Hilbert s... |
| hlcph 25264 | Every subcomplex Hilbert s... |
| hlphl 25265 | Every subcomplex Hilbert s... |
| hlcms 25266 | Every subcomplex Hilbert s... |
| hlprlem 25267 | Lemma for ~ hlpr . (Contr... |
| hlress 25268 | The scalar field of a subc... |
| hlpr 25269 | The scalar field of a subc... |
| ishl2 25270 | A Hilbert space is a compl... |
| cphssphl 25271 | A Banach subspace of a sub... |
| cmslssbn 25272 | A complete linear subspace... |
| cmscsscms 25273 | A closed subspace of a com... |
| bncssbn 25274 | A closed subspace of a Ban... |
| cssbn 25275 | A complete subspace of a n... |
| csschl 25276 | A complete subspace of a c... |
| cmslsschl 25277 | A complete linear subspace... |
| chlcsschl 25278 | A closed subspace of a sub... |
| retopn 25279 | The topology of the real n... |
| recms 25280 | The real numbers form a co... |
| reust 25281 | The Uniform structure of t... |
| recusp 25282 | The real numbers form a co... |
| rrxval 25287 | Value of the generalized E... |
| rrxbase 25288 | The base of the generalize... |
| rrxprds 25289 | Expand the definition of t... |
| rrxip 25290 | The inner product of the g... |
| rrxnm 25291 | The norm of the generalize... |
| rrxcph 25292 | Generalized Euclidean real... |
| rrxds 25293 | The distance over generali... |
| rrxvsca 25294 | The scalar product over ge... |
| rrxplusgvscavalb 25295 | The result of the addition... |
| rrxsca 25296 | The field of real numbers ... |
| rrx0 25297 | The zero ("origin") in a g... |
| rrx0el 25298 | The zero ("origin") in a g... |
| csbren 25299 | Cauchy-Schwarz-Bunjakovsky... |
| trirn 25300 | Triangle inequality in R^n... |
| rrxf 25301 | Euclidean vectors as funct... |
| rrxfsupp 25302 | Euclidean vectors are of f... |
| rrxsuppss 25303 | Support of Euclidean vecto... |
| rrxmvallem 25304 | Support of the function us... |
| rrxmval 25305 | The value of the Euclidean... |
| rrxmfval 25306 | The value of the Euclidean... |
| rrxmetlem 25307 | Lemma for ~ rrxmet . (Con... |
| rrxmet 25308 | Euclidean space is a metri... |
| rrxdstprj1 25309 | The distance between two p... |
| rrxbasefi 25310 | The base of the generalize... |
| rrxdsfi 25311 | The distance over generali... |
| rrxmetfi 25312 | Euclidean space is a metri... |
| rrxdsfival 25313 | The value of the Euclidean... |
| ehlval 25314 | Value of the Euclidean spa... |
| ehlbase 25315 | The base of the Euclidean ... |
| ehl0base 25316 | The base of the Euclidean ... |
| ehl0 25317 | The Euclidean space of dim... |
| ehleudis 25318 | The Euclidean distance fun... |
| ehleudisval 25319 | The value of the Euclidean... |
| ehl1eudis 25320 | The Euclidean distance fun... |
| ehl1eudisval 25321 | The value of the Euclidean... |
| ehl2eudis 25322 | The Euclidean distance fun... |
| ehl2eudisval 25323 | The value of the Euclidean... |
| minveclem1 25324 | Lemma for ~ minvec . The ... |
| minveclem4c 25325 | Lemma for ~ minvec . The ... |
| minveclem2 25326 | Lemma for ~ minvec . Any ... |
| minveclem3a 25327 | Lemma for ~ minvec . ` D `... |
| minveclem3b 25328 | Lemma for ~ minvec . The ... |
| minveclem3 25329 | Lemma for ~ minvec . The ... |
| minveclem4a 25330 | Lemma for ~ minvec . ` F `... |
| minveclem4b 25331 | Lemma for ~ minvec . The ... |
| minveclem4 25332 | Lemma for ~ minvec . The ... |
| minveclem5 25333 | Lemma for ~ minvec . Disc... |
| minveclem6 25334 | Lemma for ~ minvec . Any ... |
| minveclem7 25335 | Lemma for ~ minvec . Sinc... |
| minvec 25336 | Minimizing vector theorem,... |
| pjthlem1 25337 | Lemma for ~ pjth . (Contr... |
| pjthlem2 25338 | Lemma for ~ pjth . (Contr... |
| pjth 25339 | Projection Theorem: Any H... |
| pjth2 25340 | Projection Theorem with ab... |
| cldcss 25341 | Corollary of the Projectio... |
| cldcss2 25342 | Corollary of the Projectio... |
| hlhil 25343 | Corollary of the Projectio... |
| addcncf 25344 | The addition of two contin... |
| subcncf 25345 | The subtraction of two con... |
| mulcncf 25346 | The multiplication of two ... |
| mulcncfOLD 25347 | Obsolete version of ~ mulc... |
| divcncf 25348 | The quotient of two contin... |
| pmltpclem1 25349 | Lemma for ~ pmltpc . (Con... |
| pmltpclem2 25350 | Lemma for ~ pmltpc . (Con... |
| pmltpc 25351 | Any function on the reals ... |
| ivthlem1 25352 | Lemma for ~ ivth . The se... |
| ivthlem2 25353 | Lemma for ~ ivth . Show t... |
| ivthlem3 25354 | Lemma for ~ ivth , the int... |
| ivth 25355 | The intermediate value the... |
| ivth2 25356 | The intermediate value the... |
| ivthle 25357 | The intermediate value the... |
| ivthle2 25358 | The intermediate value the... |
| ivthicc 25359 | The interval between any t... |
| evthicc 25360 | Specialization of the Extr... |
| evthicc2 25361 | Combine ~ ivthicc with ~ e... |
| cniccbdd 25362 | A continuous function on a... |
| ovolfcl 25367 | Closure for the interval e... |
| ovolfioo 25368 | Unpack the interval coveri... |
| ovolficc 25369 | Unpack the interval coveri... |
| ovolficcss 25370 | Any (closed) interval cove... |
| ovolfsval 25371 | The value of the interval ... |
| ovolfsf 25372 | Closure for the interval l... |
| ovolsf 25373 | Closure for the partial su... |
| ovolval 25374 | The value of the outer mea... |
| elovolmlem 25375 | Lemma for ~ elovolm and re... |
| elovolm 25376 | Elementhood in the set ` M... |
| elovolmr 25377 | Sufficient condition for e... |
| ovolmge0 25378 | The set ` M ` is composed ... |
| ovolcl 25379 | The volume of a set is an ... |
| ovollb 25380 | The outer volume is a lowe... |
| ovolgelb 25381 | The outer volume is the gr... |
| ovolge0 25382 | The volume of a set is alw... |
| ovolf 25383 | The domain and codomain of... |
| ovollecl 25384 | If an outer volume is boun... |
| ovolsslem 25385 | Lemma for ~ ovolss . (Con... |
| ovolss 25386 | The volume of a set is mon... |
| ovolsscl 25387 | If a set is contained in a... |
| ovolssnul 25388 | A subset of a nullset is n... |
| ovollb2lem 25389 | Lemma for ~ ovollb2 . (Co... |
| ovollb2 25390 | It is often more convenien... |
| ovolctb 25391 | The volume of a denumerabl... |
| ovolq 25392 | The rational numbers have ... |
| ovolctb2 25393 | The volume of a countable ... |
| ovol0 25394 | The empty set has 0 outer ... |
| ovolfi 25395 | A finite set has 0 outer L... |
| ovolsn 25396 | A singleton has 0 outer Le... |
| ovolunlem1a 25397 | Lemma for ~ ovolun . (Con... |
| ovolunlem1 25398 | Lemma for ~ ovolun . (Con... |
| ovolunlem2 25399 | Lemma for ~ ovolun . (Con... |
| ovolun 25400 | The Lebesgue outer measure... |
| ovolunnul 25401 | Adding a nullset does not ... |
| ovolfiniun 25402 | The Lebesgue outer measure... |
| ovoliunlem1 25403 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem2 25404 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem3 25405 | Lemma for ~ ovoliun . (Co... |
| ovoliun 25406 | The Lebesgue outer measure... |
| ovoliun2 25407 | The Lebesgue outer measure... |
| ovoliunnul 25408 | A countable union of nulls... |
| shft2rab 25409 | If ` B ` is a shift of ` A... |
| ovolshftlem1 25410 | Lemma for ~ ovolshft . (C... |
| ovolshftlem2 25411 | Lemma for ~ ovolshft . (C... |
| ovolshft 25412 | The Lebesgue outer measure... |
| sca2rab 25413 | If ` B ` is a scale of ` A... |
| ovolscalem1 25414 | Lemma for ~ ovolsca . (Co... |
| ovolscalem2 25415 | Lemma for ~ ovolshft . (C... |
| ovolsca 25416 | The Lebesgue outer measure... |
| ovolicc1 25417 | The measure of a closed in... |
| ovolicc2lem1 25418 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem2 25419 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem3 25420 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem4 25421 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem5 25422 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2 25423 | The measure of a closed in... |
| ovolicc 25424 | The measure of a closed in... |
| ovolicopnf 25425 | The measure of a right-unb... |
| ovolre 25426 | The measure of the real nu... |
| ismbl 25427 | The predicate " ` A ` is L... |
| ismbl2 25428 | From ~ ovolun , it suffice... |
| volres 25429 | A self-referencing abbrevi... |
| volf 25430 | The domain and codomain of... |
| mblvol 25431 | The volume of a measurable... |
| mblss 25432 | A measurable set is a subs... |
| mblsplit 25433 | The defining property of m... |
| volss 25434 | The Lebesgue measure is mo... |
| cmmbl 25435 | The complement of a measur... |
| nulmbl 25436 | A nullset is measurable. ... |
| nulmbl2 25437 | A set of outer measure zer... |
| unmbl 25438 | A union of measurable sets... |
| shftmbl 25439 | A shift of a measurable se... |
| 0mbl 25440 | The empty set is measurabl... |
| rembl 25441 | The set of all real number... |
| unidmvol 25442 | The union of the Lebesgue ... |
| inmbl 25443 | An intersection of measura... |
| difmbl 25444 | A difference of measurable... |
| finiunmbl 25445 | A finite union of measurab... |
| volun 25446 | The Lebesgue measure funct... |
| volinun 25447 | Addition of non-disjoint s... |
| volfiniun 25448 | The volume of a disjoint f... |
| iundisj 25449 | Rewrite a countable union ... |
| iundisj2 25450 | A disjoint union is disjoi... |
| voliunlem1 25451 | Lemma for ~ voliun . (Con... |
| voliunlem2 25452 | Lemma for ~ voliun . (Con... |
| voliunlem3 25453 | Lemma for ~ voliun . (Con... |
| iunmbl 25454 | The measurable sets are cl... |
| voliun 25455 | The Lebesgue measure funct... |
| volsuplem 25456 | Lemma for ~ volsup . (Con... |
| volsup 25457 | The volume of the limit of... |
| iunmbl2 25458 | The measurable sets are cl... |
| ioombl1lem1 25459 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem2 25460 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem3 25461 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem4 25462 | Lemma for ~ ioombl1 . (Co... |
| ioombl1 25463 | An open right-unbounded in... |
| icombl1 25464 | A closed unbounded-above i... |
| icombl 25465 | A closed-below, open-above... |
| ioombl 25466 | An open real interval is m... |
| iccmbl 25467 | A closed real interval is ... |
| iccvolcl 25468 | A closed real interval has... |
| ovolioo 25469 | The measure of an open int... |
| volioo 25470 | The measure of an open int... |
| ioovolcl 25471 | An open real interval has ... |
| ovolfs2 25472 | Alternative expression for... |
| ioorcl2 25473 | An open interval with fini... |
| ioorf 25474 | Define a function from ope... |
| ioorval 25475 | Define a function from ope... |
| ioorinv2 25476 | The function ` F ` is an "... |
| ioorinv 25477 | The function ` F ` is an "... |
| ioorcl 25478 | The function ` F ` does no... |
| uniiccdif 25479 | A union of closed interval... |
| uniioovol 25480 | A disjoint union of open i... |
| uniiccvol 25481 | An almost-disjoint union o... |
| uniioombllem1 25482 | Lemma for ~ uniioombl . (... |
| uniioombllem2a 25483 | Lemma for ~ uniioombl . (... |
| uniioombllem2 25484 | Lemma for ~ uniioombl . (... |
| uniioombllem3a 25485 | Lemma for ~ uniioombl . (... |
| uniioombllem3 25486 | Lemma for ~ uniioombl . (... |
| uniioombllem4 25487 | Lemma for ~ uniioombl . (... |
| uniioombllem5 25488 | Lemma for ~ uniioombl . (... |
| uniioombllem6 25489 | Lemma for ~ uniioombl . (... |
| uniioombl 25490 | A disjoint union of open i... |
| uniiccmbl 25491 | An almost-disjoint union o... |
| dyadf 25492 | The function ` F ` returns... |
| dyadval 25493 | Value of the dyadic ration... |
| dyadovol 25494 | Volume of a dyadic rationa... |
| dyadss 25495 | Two closed dyadic rational... |
| dyaddisjlem 25496 | Lemma for ~ dyaddisj . (C... |
| dyaddisj 25497 | Two closed dyadic rational... |
| dyadmaxlem 25498 | Lemma for ~ dyadmax . (Co... |
| dyadmax 25499 | Any nonempty set of dyadic... |
| dyadmbllem 25500 | Lemma for ~ dyadmbl . (Co... |
| dyadmbl 25501 | Any union of dyadic ration... |
| opnmbllem 25502 | Lemma for ~ opnmbl . (Con... |
| opnmbl 25503 | All open sets are measurab... |
| opnmblALT 25504 | All open sets are measurab... |
| subopnmbl 25505 | Sets which are open in a m... |
| volsup2 25506 | The volume of ` A ` is the... |
| volcn 25507 | The function formed by res... |
| volivth 25508 | The Intermediate Value The... |
| vitalilem1 25509 | Lemma for ~ vitali . (Con... |
| vitalilem2 25510 | Lemma for ~ vitali . (Con... |
| vitalilem3 25511 | Lemma for ~ vitali . (Con... |
| vitalilem4 25512 | Lemma for ~ vitali . (Con... |
| vitalilem5 25513 | Lemma for ~ vitali . (Con... |
| vitali 25514 | If the reals can be well-o... |
| ismbf1 25525 | The predicate " ` F ` is a... |
| mbff 25526 | A measurable function is a... |
| mbfdm 25527 | The domain of a measurable... |
| mbfconstlem 25528 | Lemma for ~ mbfconst and r... |
| ismbf 25529 | The predicate " ` F ` is a... |
| ismbfcn 25530 | A complex function is meas... |
| mbfima 25531 | Definitional property of a... |
| mbfimaicc 25532 | The preimage of any closed... |
| mbfimasn 25533 | The preimage of a point un... |
| mbfconst 25534 | A constant function is mea... |
| mbf0 25535 | The empty function is meas... |
| mbfid 25536 | The identity function is m... |
| mbfmptcl 25537 | Lemma for the ` MblFn ` pr... |
| mbfdm2 25538 | The domain of a measurable... |
| ismbfcn2 25539 | A complex function is meas... |
| ismbfd 25540 | Deduction to prove measura... |
| ismbf2d 25541 | Deduction to prove measura... |
| mbfeqalem1 25542 | Lemma for ~ mbfeqalem2 . ... |
| mbfeqalem2 25543 | Lemma for ~ mbfeqa . (Con... |
| mbfeqa 25544 | If two functions are equal... |
| mbfres 25545 | The restriction of a measu... |
| mbfres2 25546 | Measurability of a piecewi... |
| mbfss 25547 | Change the domain of a mea... |
| mbfmulc2lem 25548 | Multiplication by a consta... |
| mbfmulc2re 25549 | Multiplication by a consta... |
| mbfmax 25550 | The maximum of two functio... |
| mbfneg 25551 | The negative of a measurab... |
| mbfpos 25552 | The positive part of a mea... |
| mbfposr 25553 | Converse to ~ mbfpos . (C... |
| mbfposb 25554 | A function is measurable i... |
| ismbf3d 25555 | Simplified form of ~ ismbf... |
| mbfimaopnlem 25556 | Lemma for ~ mbfimaopn . (... |
| mbfimaopn 25557 | The preimage of any open s... |
| mbfimaopn2 25558 | The preimage of any set op... |
| cncombf 25559 | The composition of a conti... |
| cnmbf 25560 | A continuous function is m... |
| mbfaddlem 25561 | The sum of two measurable ... |
| mbfadd 25562 | The sum of two measurable ... |
| mbfsub 25563 | The difference of two meas... |
| mbfmulc2 25564 | A complex constant times a... |
| mbfsup 25565 | The supremum of a sequence... |
| mbfinf 25566 | The infimum of a sequence ... |
| mbflimsup 25567 | The limit supremum of a se... |
| mbflimlem 25568 | The pointwise limit of a s... |
| mbflim 25569 | The pointwise limit of a s... |
| 0pval 25572 | The zero function evaluate... |
| 0plef 25573 | Two ways to say that the f... |
| 0pledm 25574 | Adjust the domain of the l... |
| isi1f 25575 | The predicate " ` F ` is a... |
| i1fmbf 25576 | Simple functions are measu... |
| i1ff 25577 | A simple function is a fun... |
| i1frn 25578 | A simple function has fini... |
| i1fima 25579 | Any preimage of a simple f... |
| i1fima2 25580 | Any preimage of a simple f... |
| i1fima2sn 25581 | Preimage of a singleton. ... |
| i1fd 25582 | A simplified set of assump... |
| i1f0rn 25583 | Any simple function takes ... |
| itg1val 25584 | The value of the integral ... |
| itg1val2 25585 | The value of the integral ... |
| itg1cl 25586 | Closure of the integral on... |
| itg1ge0 25587 | Closure of the integral on... |
| i1f0 25588 | The zero function is simpl... |
| itg10 25589 | The zero function has zero... |
| i1f1lem 25590 | Lemma for ~ i1f1 and ~ itg... |
| i1f1 25591 | Base case simple functions... |
| itg11 25592 | The integral of an indicat... |
| itg1addlem1 25593 | Decompose a preimage, whic... |
| i1faddlem 25594 | Decompose the preimage of ... |
| i1fmullem 25595 | Decompose the preimage of ... |
| i1fadd 25596 | The sum of two simple func... |
| i1fmul 25597 | The pointwise product of t... |
| itg1addlem2 25598 | Lemma for ~ itg1add . The... |
| itg1addlem3 25599 | Lemma for ~ itg1add . (Co... |
| itg1addlem4 25600 | Lemma for ~ itg1add . (Co... |
| itg1addlem5 25601 | Lemma for ~ itg1add . (Co... |
| itg1add 25602 | The integral of a sum of s... |
| i1fmulclem 25603 | Decompose the preimage of ... |
| i1fmulc 25604 | A nonnegative constant tim... |
| itg1mulc 25605 | The integral of a constant... |
| i1fres 25606 | The "restriction" of a sim... |
| i1fpos 25607 | The positive part of a sim... |
| i1fposd 25608 | Deduction form of ~ i1fpos... |
| i1fsub 25609 | The difference of two simp... |
| itg1sub 25610 | The integral of a differen... |
| itg10a 25611 | The integral of a simple f... |
| itg1ge0a 25612 | The integral of an almost ... |
| itg1lea 25613 | Approximate version of ~ i... |
| itg1le 25614 | If one simple function dom... |
| itg1climres 25615 | Restricting the simple fun... |
| mbfi1fseqlem1 25616 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem2 25617 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem3 25618 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem4 25619 | Lemma for ~ mbfi1fseq . T... |
| mbfi1fseqlem5 25620 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseqlem6 25621 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseq 25622 | A characterization of meas... |
| mbfi1flimlem 25623 | Lemma for ~ mbfi1flim . (... |
| mbfi1flim 25624 | Any real measurable functi... |
| mbfmullem2 25625 | Lemma for ~ mbfmul . (Con... |
| mbfmullem 25626 | Lemma for ~ mbfmul . (Con... |
| mbfmul 25627 | The product of two measura... |
| itg2lcl 25628 | The set of lower sums is a... |
| itg2val 25629 | Value of the integral on n... |
| itg2l 25630 | Elementhood in the set ` L... |
| itg2lr 25631 | Sufficient condition for e... |
| xrge0f 25632 | A real function is a nonne... |
| itg2cl 25633 | The integral of a nonnegat... |
| itg2ub 25634 | The integral of a nonnegat... |
| itg2leub 25635 | Any upper bound on the int... |
| itg2ge0 25636 | The integral of a nonnegat... |
| itg2itg1 25637 | The integral of a nonnegat... |
| itg20 25638 | The integral of the zero f... |
| itg2lecl 25639 | If an ` S.2 ` integral is ... |
| itg2le 25640 | If one function dominates ... |
| itg2const 25641 | Integral of a constant fun... |
| itg2const2 25642 | When the base set of a con... |
| itg2seq 25643 | Definitional property of t... |
| itg2uba 25644 | Approximate version of ~ i... |
| itg2lea 25645 | Approximate version of ~ i... |
| itg2eqa 25646 | Approximate equality of in... |
| itg2mulclem 25647 | Lemma for ~ itg2mulc . (C... |
| itg2mulc 25648 | The integral of a nonnegat... |
| itg2splitlem 25649 | Lemma for ~ itg2split . (... |
| itg2split 25650 | The ` S.2 ` integral split... |
| itg2monolem1 25651 | Lemma for ~ itg2mono . We... |
| itg2monolem2 25652 | Lemma for ~ itg2mono . (C... |
| itg2monolem3 25653 | Lemma for ~ itg2mono . (C... |
| itg2mono 25654 | The Monotone Convergence T... |
| itg2i1fseqle 25655 | Subject to the conditions ... |
| itg2i1fseq 25656 | Subject to the conditions ... |
| itg2i1fseq2 25657 | In an extension to the res... |
| itg2i1fseq3 25658 | Special case of ~ itg2i1fs... |
| itg2addlem 25659 | Lemma for ~ itg2add . (Co... |
| itg2add 25660 | The ` S.2 ` integral is li... |
| itg2gt0 25661 | If the function ` F ` is s... |
| itg2cnlem1 25662 | Lemma for ~ itgcn . (Cont... |
| itg2cnlem2 25663 | Lemma for ~ itgcn . (Cont... |
| itg2cn 25664 | A sort of absolute continu... |
| ibllem 25665 | Conditioned equality theor... |
| isibl 25666 | The predicate " ` F ` is i... |
| isibl2 25667 | The predicate " ` F ` is i... |
| iblmbf 25668 | An integrable function is ... |
| iblitg 25669 | If a function is integrabl... |
| dfitg 25670 | Evaluate the class substit... |
| itgex 25671 | An integral is a set. (Co... |
| itgeq1f 25672 | Equality theorem for an in... |
| itgeq1fOLD 25673 | Obsolete version of ~ itge... |
| itgeq1 25674 | Equality theorem for an in... |
| nfitg1 25675 | Bound-variable hypothesis ... |
| nfitg 25676 | Bound-variable hypothesis ... |
| cbvitg 25677 | Change bound variable in a... |
| cbvitgv 25678 | Change bound variable in a... |
| itgeq2 25679 | Equality theorem for an in... |
| itgresr 25680 | The domain of an integral ... |
| itg0 25681 | The integral of anything o... |
| itgz 25682 | The integral of zero on an... |
| itgeq2dv 25683 | Equality theorem for an in... |
| itgmpt 25684 | Change bound variable in a... |
| itgcl 25685 | The integral of an integra... |
| itgvallem 25686 | Substitution lemma. (Cont... |
| itgvallem3 25687 | Lemma for ~ itgposval and ... |
| ibl0 25688 | The zero function is integ... |
| iblcnlem1 25689 | Lemma for ~ iblcnlem . (C... |
| iblcnlem 25690 | Expand out the universal q... |
| itgcnlem 25691 | Expand out the sum in ~ df... |
| iblrelem 25692 | Integrability of a real fu... |
| iblposlem 25693 | Lemma for ~ iblpos . (Con... |
| iblpos 25694 | Integrability of a nonnega... |
| iblre 25695 | Integrability of a real fu... |
| itgrevallem1 25696 | Lemma for ~ itgposval and ... |
| itgposval 25697 | The integral of a nonnegat... |
| itgreval 25698 | Decompose the integral of ... |
| itgrecl 25699 | Real closure of an integra... |
| iblcn 25700 | Integrability of a complex... |
| itgcnval 25701 | Decompose the integral of ... |
| itgre 25702 | Real part of an integral. ... |
| itgim 25703 | Imaginary part of an integ... |
| iblneg 25704 | The negative of an integra... |
| itgneg 25705 | Negation of an integral. ... |
| iblss 25706 | A subset of an integrable ... |
| iblss2 25707 | Change the domain of an in... |
| itgitg2 25708 | Transfer an integral using... |
| i1fibl 25709 | A simple function is integ... |
| itgitg1 25710 | Transfer an integral using... |
| itgle 25711 | Monotonicity of an integra... |
| itgge0 25712 | The integral of a positive... |
| itgss 25713 | Expand the set of an integ... |
| itgss2 25714 | Expand the set of an integ... |
| itgeqa 25715 | Approximate equality of in... |
| itgss3 25716 | Expand the set of an integ... |
| itgioo 25717 | Equality of integrals on o... |
| itgless 25718 | Expand the integral of a n... |
| iblconst 25719 | A constant function is int... |
| itgconst 25720 | Integral of a constant fun... |
| ibladdlem 25721 | Lemma for ~ ibladd . (Con... |
| ibladd 25722 | Add two integrals over the... |
| iblsub 25723 | Subtract two integrals ove... |
| itgaddlem1 25724 | Lemma for ~ itgadd . (Con... |
| itgaddlem2 25725 | Lemma for ~ itgadd . (Con... |
| itgadd 25726 | Add two integrals over the... |
| itgsub 25727 | Subtract two integrals ove... |
| itgfsum 25728 | Take a finite sum of integ... |
| iblabslem 25729 | Lemma for ~ iblabs . (Con... |
| iblabs 25730 | The absolute value of an i... |
| iblabsr 25731 | A measurable function is i... |
| iblmulc2 25732 | Multiply an integral by a ... |
| itgmulc2lem1 25733 | Lemma for ~ itgmulc2 : pos... |
| itgmulc2lem2 25734 | Lemma for ~ itgmulc2 : rea... |
| itgmulc2 25735 | Multiply an integral by a ... |
| itgabs 25736 | The triangle inequality fo... |
| itgsplit 25737 | The ` S. ` integral splits... |
| itgspliticc 25738 | The ` S. ` integral splits... |
| itgsplitioo 25739 | The ` S. ` integral splits... |
| bddmulibl 25740 | A bounded function times a... |
| bddibl 25741 | A bounded function is inte... |
| cniccibl 25742 | A continuous function on a... |
| bddiblnc 25743 | Choice-free proof of ~ bdd... |
| cnicciblnc 25744 | Choice-free proof of ~ cni... |
| itggt0 25745 | The integral of a strictly... |
| itgcn 25746 | Transfer ~ itg2cn to the f... |
| ditgeq1 25749 | Equality theorem for the d... |
| ditgeq2 25750 | Equality theorem for the d... |
| ditgeq3 25751 | Equality theorem for the d... |
| ditgeq3dv 25752 | Equality theorem for the d... |
| ditgex 25753 | A directed integral is a s... |
| ditg0 25754 | Value of the directed inte... |
| cbvditg 25755 | Change bound variable in a... |
| cbvditgv 25756 | Change bound variable in a... |
| ditgpos 25757 | Value of the directed inte... |
| ditgneg 25758 | Value of the directed inte... |
| ditgcl 25759 | Closure of a directed inte... |
| ditgswap 25760 | Reverse a directed integra... |
| ditgsplitlem 25761 | Lemma for ~ ditgsplit . (... |
| ditgsplit 25762 | This theorem is the raison... |
| reldv 25771 | The derivative function is... |
| limcvallem 25772 | Lemma for ~ ellimc . (Con... |
| limcfval 25773 | Value and set bounds on th... |
| ellimc 25774 | Value of the limit predica... |
| limcrcl 25775 | Reverse closure for the li... |
| limccl 25776 | Closure of the limit opera... |
| limcdif 25777 | It suffices to consider fu... |
| ellimc2 25778 | Write the definition of a ... |
| limcnlp 25779 | If ` B ` is not a limit po... |
| ellimc3 25780 | Write the epsilon-delta de... |
| limcflflem 25781 | Lemma for ~ limcflf . (Co... |
| limcflf 25782 | The limit operator can be ... |
| limcmo 25783 | If ` B ` is a limit point ... |
| limcmpt 25784 | Express the limit operator... |
| limcmpt2 25785 | Express the limit operator... |
| limcresi 25786 | Any limit of ` F ` is also... |
| limcres 25787 | If ` B ` is an interior po... |
| cnplimc 25788 | A function is continuous a... |
| cnlimc 25789 | ` F ` is a continuous func... |
| cnlimci 25790 | If ` F ` is a continuous f... |
| cnmptlimc 25791 | If ` F ` is a continuous f... |
| limccnp 25792 | If the limit of ` F ` at `... |
| limccnp2 25793 | The image of a convergent ... |
| limcco 25794 | Composition of two limits.... |
| limciun 25795 | A point is a limit of ` F ... |
| limcun 25796 | A point is a limit of ` F ... |
| dvlem 25797 | Closure for a difference q... |
| dvfval 25798 | Value and set bounds on th... |
| eldv 25799 | The differentiable predica... |
| dvcl 25800 | The derivative function ta... |
| dvbssntr 25801 | The set of differentiable ... |
| dvbss 25802 | The set of differentiable ... |
| dvbsss 25803 | The set of differentiable ... |
| perfdvf 25804 | The derivative is a functi... |
| recnprss 25805 | Both ` RR ` and ` CC ` are... |
| recnperf 25806 | Both ` RR ` and ` CC ` are... |
| dvfg 25807 | Explicitly write out the f... |
| dvf 25808 | The derivative is a functi... |
| dvfcn 25809 | The derivative is a functi... |
| dvreslem 25810 | Lemma for ~ dvres . (Cont... |
| dvres2lem 25811 | Lemma for ~ dvres2 . (Con... |
| dvres 25812 | Restriction of a derivativ... |
| dvres2 25813 | Restriction of the base se... |
| dvres3 25814 | Restriction of a complex d... |
| dvres3a 25815 | Restriction of a complex d... |
| dvidlem 25816 | Lemma for ~ dvid and ~ dvc... |
| dvmptresicc 25817 | Derivative of a function r... |
| dvconst 25818 | Derivative of a constant f... |
| dvid 25819 | Derivative of the identity... |
| dvcnp 25820 | The difference quotient is... |
| dvcnp2 25821 | A function is continuous a... |
| dvcnp2OLD 25822 | Obsolete version of ~ dvcn... |
| dvcn 25823 | A differentiable function ... |
| dvnfval 25824 | Value of the iterated deri... |
| dvnff 25825 | The iterated derivative is... |
| dvn0 25826 | Zero times iterated deriva... |
| dvnp1 25827 | Successor iterated derivat... |
| dvn1 25828 | One times iterated derivat... |
| dvnf 25829 | The N-times derivative is ... |
| dvnbss 25830 | The set of N-times differe... |
| dvnadd 25831 | The ` N ` -th derivative o... |
| dvn2bss 25832 | An N-times differentiable ... |
| dvnres 25833 | Multiple derivative versio... |
| cpnfval 25834 | Condition for n-times cont... |
| fncpn 25835 | The ` C^n ` object is a fu... |
| elcpn 25836 | Condition for n-times cont... |
| cpnord 25837 | ` C^n ` conditions are ord... |
| cpncn 25838 | A ` C^n ` function is cont... |
| cpnres 25839 | The restriction of a ` C^n... |
| dvaddbr 25840 | The sum rule for derivativ... |
| dvmulbr 25841 | The product rule for deriv... |
| dvmulbrOLD 25842 | Obsolete version of ~ dvmu... |
| dvadd 25843 | The sum rule for derivativ... |
| dvmul 25844 | The product rule for deriv... |
| dvaddf 25845 | The sum rule for everywher... |
| dvmulf 25846 | The product rule for every... |
| dvcmul 25847 | The product rule when one ... |
| dvcmulf 25848 | The product rule when one ... |
| dvcobr 25849 | The chain rule for derivat... |
| dvcobrOLD 25850 | Obsolete version of ~ dvco... |
| dvco 25851 | The chain rule for derivat... |
| dvcof 25852 | The chain rule for everywh... |
| dvcjbr 25853 | The derivative of the conj... |
| dvcj 25854 | The derivative of the conj... |
| dvfre 25855 | The derivative of a real f... |
| dvnfre 25856 | The ` N ` -th derivative o... |
| dvexp 25857 | Derivative of a power func... |
| dvexp2 25858 | Derivative of an exponenti... |
| dvrec 25859 | Derivative of the reciproc... |
| dvmptres3 25860 | Function-builder for deriv... |
| dvmptid 25861 | Function-builder for deriv... |
| dvmptc 25862 | Function-builder for deriv... |
| dvmptcl 25863 | Closure lemma for ~ dvmptc... |
| dvmptadd 25864 | Function-builder for deriv... |
| dvmptmul 25865 | Function-builder for deriv... |
| dvmptres2 25866 | Function-builder for deriv... |
| dvmptres 25867 | Function-builder for deriv... |
| dvmptcmul 25868 | Function-builder for deriv... |
| dvmptdivc 25869 | Function-builder for deriv... |
| dvmptneg 25870 | Function-builder for deriv... |
| dvmptsub 25871 | Function-builder for deriv... |
| dvmptcj 25872 | Function-builder for deriv... |
| dvmptre 25873 | Function-builder for deriv... |
| dvmptim 25874 | Function-builder for deriv... |
| dvmptntr 25875 | Function-builder for deriv... |
| dvmptco 25876 | Function-builder for deriv... |
| dvrecg 25877 | Derivative of the reciproc... |
| dvmptdiv 25878 | Function-builder for deriv... |
| dvmptfsum 25879 | Function-builder for deriv... |
| dvcnvlem 25880 | Lemma for ~ dvcnvre . (Co... |
| dvcnv 25881 | A weak version of ~ dvcnvr... |
| dvexp3 25882 | Derivative of an exponenti... |
| dveflem 25883 | Derivative of the exponent... |
| dvef 25884 | Derivative of the exponent... |
| dvsincos 25885 | Derivative of the sine and... |
| dvsin 25886 | Derivative of the sine fun... |
| dvcos 25887 | Derivative of the cosine f... |
| dvferm1lem 25888 | Lemma for ~ dvferm . (Con... |
| dvferm1 25889 | One-sided version of ~ dvf... |
| dvferm2lem 25890 | Lemma for ~ dvferm . (Con... |
| dvferm2 25891 | One-sided version of ~ dvf... |
| dvferm 25892 | Fermat's theorem on statio... |
| rollelem 25893 | Lemma for ~ rolle . (Cont... |
| rolle 25894 | Rolle's theorem. If ` F `... |
| cmvth 25895 | Cauchy's Mean Value Theore... |
| cmvthOLD 25896 | Obsolete version of ~ cmvt... |
| mvth 25897 | The Mean Value Theorem. I... |
| dvlip 25898 | A function with derivative... |
| dvlipcn 25899 | A complex function with de... |
| dvlip2 25900 | Combine the results of ~ d... |
| c1liplem1 25901 | Lemma for ~ c1lip1 . (Con... |
| c1lip1 25902 | C^1 functions are Lipschit... |
| c1lip2 25903 | C^1 functions are Lipschit... |
| c1lip3 25904 | C^1 functions are Lipschit... |
| dveq0 25905 | If a continuous function h... |
| dv11cn 25906 | Two functions defined on a... |
| dvgt0lem1 25907 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0lem2 25908 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0 25909 | A function on a closed int... |
| dvlt0 25910 | A function on a closed int... |
| dvge0 25911 | A function on a closed int... |
| dvle 25912 | If ` A ( x ) , C ( x ) ` a... |
| dvivthlem1 25913 | Lemma for ~ dvivth . (Con... |
| dvivthlem2 25914 | Lemma for ~ dvivth . (Con... |
| dvivth 25915 | Darboux' theorem, or the i... |
| dvne0 25916 | A function on a closed int... |
| dvne0f1 25917 | A function on a closed int... |
| lhop1lem 25918 | Lemma for ~ lhop1 . (Cont... |
| lhop1 25919 | L'Hôpital's Rule for... |
| lhop2 25920 | L'Hôpital's Rule for... |
| lhop 25921 | L'Hôpital's Rule. I... |
| dvcnvrelem1 25922 | Lemma for ~ dvcnvre . (Co... |
| dvcnvrelem2 25923 | Lemma for ~ dvcnvre . (Co... |
| dvcnvre 25924 | The derivative rule for in... |
| dvcvx 25925 | A real function with stric... |
| dvfsumle 25926 | Compare a finite sum to an... |
| dvfsumleOLD 25927 | Obsolete version of ~ dvfs... |
| dvfsumge 25928 | Compare a finite sum to an... |
| dvfsumabs 25929 | Compare a finite sum to an... |
| dvmptrecl 25930 | Real closure of a derivati... |
| dvfsumrlimf 25931 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem1 25932 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2 25933 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2OLD 25934 | Obsolete version of ~ dvfs... |
| dvfsumlem3 25935 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem4 25936 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlimge0 25937 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlim 25938 | Compare a finite sum to an... |
| dvfsumrlim2 25939 | Compare a finite sum to an... |
| dvfsumrlim3 25940 | Conjoin the statements of ... |
| dvfsum2 25941 | The reverse of ~ dvfsumrli... |
| ftc1lem1 25942 | Lemma for ~ ftc1a and ~ ft... |
| ftc1lem2 25943 | Lemma for ~ ftc1 . (Contr... |
| ftc1a 25944 | The Fundamental Theorem of... |
| ftc1lem3 25945 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem4 25946 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem5 25947 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem6 25948 | Lemma for ~ ftc1 . (Contr... |
| ftc1 25949 | The Fundamental Theorem of... |
| ftc1cn 25950 | Strengthen the assumptions... |
| ftc2 25951 | The Fundamental Theorem of... |
| ftc2ditglem 25952 | Lemma for ~ ftc2ditg . (C... |
| ftc2ditg 25953 | Directed integral analogue... |
| itgparts 25954 | Integration by parts. If ... |
| itgsubstlem 25955 | Lemma for ~ itgsubst . (C... |
| itgsubst 25956 | Integration by ` u ` -subs... |
| itgpowd 25957 | The integral of a monomial... |
| reldmmdeg 25962 | Multivariate degree is a b... |
| tdeglem1 25963 | Functionality of the total... |
| tdeglem3 25964 | Additivity of the total de... |
| tdeglem4 25965 | There is only one multi-in... |
| tdeglem2 25966 | Simplification of total de... |
| mdegfval 25967 | Value of the multivariate ... |
| mdegval 25968 | Value of the multivariate ... |
| mdegleb 25969 | Property of being of limit... |
| mdeglt 25970 | If there is an upper limit... |
| mdegldg 25971 | A nonzero polynomial has s... |
| mdegxrcl 25972 | Closure of polynomial degr... |
| mdegxrf 25973 | Functionality of polynomia... |
| mdegcl 25974 | Sharp closure for multivar... |
| mdeg0 25975 | Degree of the zero polynom... |
| mdegnn0cl 25976 | Degree of a nonzero polyno... |
| degltlem1 25977 | Theorem on arithmetic of e... |
| degltp1le 25978 | Theorem on arithmetic of e... |
| mdegaddle 25979 | The degree of a sum is at ... |
| mdegvscale 25980 | The degree of a scalar mul... |
| mdegvsca 25981 | The degree of a scalar mul... |
| mdegle0 25982 | A polynomial has nonpositi... |
| mdegmullem 25983 | Lemma for ~ mdegmulle2 . ... |
| mdegmulle2 25984 | The multivariate degree of... |
| deg1fval 25985 | Relate univariate polynomi... |
| deg1xrf 25986 | Functionality of univariat... |
| deg1xrcl 25987 | Closure of univariate poly... |
| deg1cl 25988 | Sharp closure of univariat... |
| mdegpropd 25989 | Property deduction for pol... |
| deg1fvi 25990 | Univariate polynomial degr... |
| deg1propd 25991 | Property deduction for pol... |
| deg1z 25992 | Degree of the zero univari... |
| deg1nn0cl 25993 | Degree of a nonzero univar... |
| deg1n0ima 25994 | Degree image of a set of p... |
| deg1nn0clb 25995 | A polynomial is nonzero if... |
| deg1lt0 25996 | A polynomial is zero iff i... |
| deg1ldg 25997 | A nonzero univariate polyn... |
| deg1ldgn 25998 | An index at which a polyno... |
| deg1ldgdomn 25999 | A nonzero univariate polyn... |
| deg1leb 26000 | Property of being of limit... |
| deg1val 26001 | Value of the univariate de... |
| deg1lt 26002 | If the degree of a univari... |
| deg1ge 26003 | Conversely, a nonzero coef... |
| coe1mul3 26004 | The coefficient vector of ... |
| coe1mul4 26005 | Value of the "leading" coe... |
| deg1addle 26006 | The degree of a sum is at ... |
| deg1addle2 26007 | If both factors have degre... |
| deg1add 26008 | Exact degree of a sum of t... |
| deg1vscale 26009 | The degree of a scalar tim... |
| deg1vsca 26010 | The degree of a scalar tim... |
| deg1invg 26011 | The degree of the negated ... |
| deg1suble 26012 | The degree of a difference... |
| deg1sub 26013 | Exact degree of a differen... |
| deg1mulle2 26014 | Produce a bound on the pro... |
| deg1sublt 26015 | Subtraction of two polynom... |
| deg1le0 26016 | A polynomial has nonpositi... |
| deg1sclle 26017 | A scalar polynomial has no... |
| deg1scl 26018 | A nonzero scalar polynomia... |
| deg1mul2 26019 | Degree of multiplication o... |
| deg1mul 26020 | Degree of multiplication o... |
| deg1mul3 26021 | Degree of multiplication o... |
| deg1mul3le 26022 | Degree of multiplication o... |
| deg1tmle 26023 | Limiting degree of a polyn... |
| deg1tm 26024 | Exact degree of a polynomi... |
| deg1pwle 26025 | Limiting degree of a varia... |
| deg1pw 26026 | Exact degree of a variable... |
| ply1nz 26027 | Univariate polynomials ove... |
| ply1nzb 26028 | Univariate polynomials are... |
| ply1domn 26029 | Corollary of ~ deg1mul2 : ... |
| ply1idom 26030 | The ring of univariate pol... |
| ply1divmo 26041 | Uniqueness of a quotient i... |
| ply1divex 26042 | Lemma for ~ ply1divalg : e... |
| ply1divalg 26043 | The division algorithm for... |
| ply1divalg2 26044 | Reverse the order of multi... |
| uc1pval 26045 | Value of the set of unitic... |
| isuc1p 26046 | Being a unitic polynomial.... |
| mon1pval 26047 | Value of the set of monic ... |
| ismon1p 26048 | Being a monic polynomial. ... |
| uc1pcl 26049 | Unitic polynomials are pol... |
| mon1pcl 26050 | Monic polynomials are poly... |
| uc1pn0 26051 | Unitic polynomials are not... |
| mon1pn0 26052 | Monic polynomials are not ... |
| uc1pdeg 26053 | Unitic polynomials have no... |
| uc1pldg 26054 | Unitic polynomials have un... |
| mon1pldg 26055 | Unitic polynomials have on... |
| mon1puc1p 26056 | Monic polynomials are unit... |
| uc1pmon1p 26057 | Make a unitic polynomial m... |
| deg1submon1p 26058 | The difference of two moni... |
| mon1pid 26059 | Monicity and degree of the... |
| q1pval 26060 | Value of the univariate po... |
| q1peqb 26061 | Characterizing property of... |
| q1pcl 26062 | Closure of the quotient by... |
| r1pval 26063 | Value of the polynomial re... |
| r1pcl 26064 | Closure of remainder follo... |
| r1pdeglt 26065 | The remainder has a degree... |
| r1pid 26066 | Express the original polyn... |
| r1pid2 26067 | Identity law for polynomia... |
| dvdsq1p 26068 | Divisibility in a polynomi... |
| dvdsr1p 26069 | Divisibility in a polynomi... |
| ply1remlem 26070 | A term of the form ` x - N... |
| ply1rem 26071 | The polynomial remainder t... |
| facth1 26072 | The factor theorem and its... |
| fta1glem1 26073 | Lemma for ~ fta1g . (Cont... |
| fta1glem2 26074 | Lemma for ~ fta1g . (Cont... |
| fta1g 26075 | The one-sided fundamental ... |
| fta1blem 26076 | Lemma for ~ fta1b . (Cont... |
| fta1b 26077 | The assumption that ` R ` ... |
| idomrootle 26078 | No element of an integral ... |
| drnguc1p 26079 | Over a division ring, all ... |
| ig1peu 26080 | There is a unique monic po... |
| ig1pval 26081 | Substitutions for the poly... |
| ig1pval2 26082 | Generator of the zero idea... |
| ig1pval3 26083 | Characterizing properties ... |
| ig1pcl 26084 | The monic generator of an ... |
| ig1pdvds 26085 | The monic generator of an ... |
| ig1prsp 26086 | Any ideal of polynomials o... |
| ply1lpir 26087 | The ring of polynomials ov... |
| ply1pid 26088 | The polynomials over a fie... |
| plyco0 26097 | Two ways to say that a fun... |
| plyval 26098 | Value of the polynomial se... |
| plybss 26099 | Reverse closure of the par... |
| elply 26100 | Definition of a polynomial... |
| elply2 26101 | The coefficient function c... |
| plyun0 26102 | The set of polynomials is ... |
| plyf 26103 | A polynomial is a function... |
| plyss 26104 | The polynomial set functio... |
| plyssc 26105 | Every polynomial ring is c... |
| elplyr 26106 | Sufficient condition for e... |
| elplyd 26107 | Sufficient condition for e... |
| ply1termlem 26108 | Lemma for ~ ply1term . (C... |
| ply1term 26109 | A one-term polynomial. (C... |
| plypow 26110 | A power is a polynomial. ... |
| plyconst 26111 | A constant function is a p... |
| ne0p 26112 | A test to show that a poly... |
| ply0 26113 | The zero function is a pol... |
| plyid 26114 | The identity function is a... |
| plyeq0lem 26115 | Lemma for ~ plyeq0 . If `... |
| plyeq0 26116 | If a polynomial is zero at... |
| plypf1 26117 | Write the set of complex p... |
| plyaddlem1 26118 | Derive the coefficient fun... |
| plymullem1 26119 | Derive the coefficient fun... |
| plyaddlem 26120 | Lemma for ~ plyadd . (Con... |
| plymullem 26121 | Lemma for ~ plymul . (Con... |
| plyadd 26122 | The sum of two polynomials... |
| plymul 26123 | The product of two polynom... |
| plysub 26124 | The difference of two poly... |
| plyaddcl 26125 | The sum of two polynomials... |
| plymulcl 26126 | The product of two polynom... |
| plysubcl 26127 | The difference of two poly... |
| coeval 26128 | Value of the coefficient f... |
| coeeulem 26129 | Lemma for ~ coeeu . (Cont... |
| coeeu 26130 | Uniqueness of the coeffici... |
| coelem 26131 | Lemma for properties of th... |
| coeeq 26132 | If ` A ` satisfies the pro... |
| dgrval 26133 | Value of the degree functi... |
| dgrlem 26134 | Lemma for ~ dgrcl and simi... |
| coef 26135 | The domain and codomain of... |
| coef2 26136 | The domain and codomain of... |
| coef3 26137 | The domain and codomain of... |
| dgrcl 26138 | The degree of any polynomi... |
| dgrub 26139 | If the ` M ` -th coefficie... |
| dgrub2 26140 | All the coefficients above... |
| dgrlb 26141 | If all the coefficients ab... |
| coeidlem 26142 | Lemma for ~ coeid . (Cont... |
| coeid 26143 | Reconstruct a polynomial a... |
| coeid2 26144 | Reconstruct a polynomial a... |
| coeid3 26145 | Reconstruct a polynomial a... |
| plyco 26146 | The composition of two pol... |
| coeeq2 26147 | Compute the coefficient fu... |
| dgrle 26148 | Given an explicit expressi... |
| dgreq 26149 | If the highest term in a p... |
| 0dgr 26150 | A constant function has de... |
| 0dgrb 26151 | A function has degree zero... |
| dgrnznn 26152 | A nonzero polynomial with ... |
| coefv0 26153 | The result of evaluating a... |
| coeaddlem 26154 | Lemma for ~ coeadd and ~ d... |
| coemullem 26155 | Lemma for ~ coemul and ~ d... |
| coeadd 26156 | The coefficient function o... |
| coemul 26157 | A coefficient of a product... |
| coe11 26158 | The coefficient function i... |
| coemulhi 26159 | The leading coefficient of... |
| coemulc 26160 | The coefficient function i... |
| coe0 26161 | The coefficients of the ze... |
| coesub 26162 | The coefficient function o... |
| coe1termlem 26163 | The coefficient function o... |
| coe1term 26164 | The coefficient function o... |
| dgr1term 26165 | The degree of a monomial. ... |
| plycn 26166 | A polynomial is a continuo... |
| plycnOLD 26167 | Obsolete version of ~ plyc... |
| dgr0 26168 | The degree of the zero pol... |
| coeidp 26169 | The coefficients of the id... |
| dgrid 26170 | The degree of the identity... |
| dgreq0 26171 | The leading coefficient of... |
| dgrlt 26172 | Two ways to say that the d... |
| dgradd 26173 | The degree of a sum of pol... |
| dgradd2 26174 | The degree of a sum of pol... |
| dgrmul2 26175 | The degree of a product of... |
| dgrmul 26176 | The degree of a product of... |
| dgrmulc 26177 | Scalar multiplication by a... |
| dgrsub 26178 | The degree of a difference... |
| dgrcolem1 26179 | The degree of a compositio... |
| dgrcolem2 26180 | Lemma for ~ dgrco . (Cont... |
| dgrco 26181 | The degree of a compositio... |
| plycjlem 26182 | Lemma for ~ plycj and ~ co... |
| plycj 26183 | The double conjugation of ... |
| coecj 26184 | Double conjugation of a po... |
| plycjOLD 26185 | Obsolete version of ~ plyc... |
| coecjOLD 26186 | Obsolete version of ~ coec... |
| plyrecj 26187 | A polynomial with real coe... |
| plymul0or 26188 | Polynomial multiplication ... |
| ofmulrt 26189 | The set of roots of a prod... |
| plyreres 26190 | Real-coefficient polynomia... |
| dvply1 26191 | Derivative of a polynomial... |
| dvply2g 26192 | The derivative of a polyno... |
| dvply2gOLD 26193 | Obsolete version of ~ dvpl... |
| dvply2 26194 | The derivative of a polyno... |
| dvnply2 26195 | Polynomials have polynomia... |
| dvnply 26196 | Polynomials have polynomia... |
| plycpn 26197 | Polynomials are smooth. (... |
| quotval 26200 | Value of the quotient func... |
| plydivlem1 26201 | Lemma for ~ plydivalg . (... |
| plydivlem2 26202 | Lemma for ~ plydivalg . (... |
| plydivlem3 26203 | Lemma for ~ plydivex . Ba... |
| plydivlem4 26204 | Lemma for ~ plydivex . In... |
| plydivex 26205 | Lemma for ~ plydivalg . (... |
| plydiveu 26206 | Lemma for ~ plydivalg . (... |
| plydivalg 26207 | The division algorithm on ... |
| quotlem 26208 | Lemma for properties of th... |
| quotcl 26209 | The quotient of two polyno... |
| quotcl2 26210 | Closure of the quotient fu... |
| quotdgr 26211 | Remainder property of the ... |
| plyremlem 26212 | Closure of a linear factor... |
| plyrem 26213 | The polynomial remainder t... |
| facth 26214 | The factor theorem. If a ... |
| fta1lem 26215 | Lemma for ~ fta1 . (Contr... |
| fta1 26216 | The easy direction of the ... |
| quotcan 26217 | Exact division with a mult... |
| vieta1lem1 26218 | Lemma for ~ vieta1 . (Con... |
| vieta1lem2 26219 | Lemma for ~ vieta1 : induc... |
| vieta1 26220 | The first-order Vieta's fo... |
| plyexmo 26221 | An infinite set of values ... |
| elaa 26224 | Elementhood in the set of ... |
| aacn 26225 | An algebraic number is a c... |
| aasscn 26226 | The algebraic numbers are ... |
| elqaalem1 26227 | Lemma for ~ elqaa . The f... |
| elqaalem2 26228 | Lemma for ~ elqaa . (Cont... |
| elqaalem3 26229 | Lemma for ~ elqaa . (Cont... |
| elqaa 26230 | The set of numbers generat... |
| qaa 26231 | Every rational number is a... |
| qssaa 26232 | The rational numbers are c... |
| iaa 26233 | The imaginary unit is alge... |
| aareccl 26234 | The reciprocal of an algeb... |
| aacjcl 26235 | The conjugate of an algebr... |
| aannenlem1 26236 | Lemma for ~ aannen . (Con... |
| aannenlem2 26237 | Lemma for ~ aannen . (Con... |
| aannenlem3 26238 | The algebraic numbers are ... |
| aannen 26239 | The algebraic numbers are ... |
| aalioulem1 26240 | Lemma for ~ aaliou . An i... |
| aalioulem2 26241 | Lemma for ~ aaliou . (Con... |
| aalioulem3 26242 | Lemma for ~ aaliou . (Con... |
| aalioulem4 26243 | Lemma for ~ aaliou . (Con... |
| aalioulem5 26244 | Lemma for ~ aaliou . (Con... |
| aalioulem6 26245 | Lemma for ~ aaliou . (Con... |
| aaliou 26246 | Liouville's theorem on dio... |
| geolim3 26247 | Geometric series convergen... |
| aaliou2 26248 | Liouville's approximation ... |
| aaliou2b 26249 | Liouville's approximation ... |
| aaliou3lem1 26250 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem2 26251 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem3 26252 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem8 26253 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem4 26254 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem5 26255 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem6 26256 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem7 26257 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem9 26258 | Example of a "Liouville nu... |
| aaliou3 26259 | Example of a "Liouville nu... |
| taylfvallem1 26264 | Lemma for ~ taylfval . (C... |
| taylfvallem 26265 | Lemma for ~ taylfval . (C... |
| taylfval 26266 | Define the Taylor polynomi... |
| eltayl 26267 | Value of the Taylor series... |
| taylf 26268 | The Taylor series defines ... |
| tayl0 26269 | The Taylor series is alway... |
| taylplem1 26270 | Lemma for ~ taylpfval and ... |
| taylplem2 26271 | Lemma for ~ taylpfval and ... |
| taylpfval 26272 | Define the Taylor polynomi... |
| taylpf 26273 | The Taylor polynomial is a... |
| taylpval 26274 | Value of the Taylor polyno... |
| taylply2 26275 | The Taylor polynomial is a... |
| taylply2OLD 26276 | Obsolete version of ~ tayl... |
| taylply 26277 | The Taylor polynomial is a... |
| dvtaylp 26278 | The derivative of the Tayl... |
| dvntaylp 26279 | The ` M ` -th derivative o... |
| dvntaylp0 26280 | The first ` N ` derivative... |
| taylthlem1 26281 | Lemma for ~ taylth . This... |
| taylthlem2 26282 | Lemma for ~ taylth . (Con... |
| taylthlem2OLD 26283 | Obsolete version of ~ tayl... |
| taylth 26284 | Taylor's theorem. The Tay... |
| ulmrel 26287 | The uniform limit relation... |
| ulmscl 26288 | Closure of the base set in... |
| ulmval 26289 | Express the predicate: Th... |
| ulmcl 26290 | Closure of a uniform limit... |
| ulmf 26291 | Closure of a uniform limit... |
| ulmpm 26292 | Closure of a uniform limit... |
| ulmf2 26293 | Closure of a uniform limit... |
| ulm2 26294 | Simplify ~ ulmval when ` F... |
| ulmi 26295 | The uniform limit property... |
| ulmclm 26296 | A uniform limit of functio... |
| ulmres 26297 | A sequence of functions co... |
| ulmshftlem 26298 | Lemma for ~ ulmshft . (Co... |
| ulmshft 26299 | A sequence of functions co... |
| ulm0 26300 | Every function converges u... |
| ulmuni 26301 | A sequence of functions un... |
| ulmdm 26302 | Two ways to express that a... |
| ulmcaulem 26303 | Lemma for ~ ulmcau and ~ u... |
| ulmcau 26304 | A sequence of functions co... |
| ulmcau2 26305 | A sequence of functions co... |
| ulmss 26306 | A uniform limit of functio... |
| ulmbdd 26307 | A uniform limit of bounded... |
| ulmcn 26308 | A uniform limit of continu... |
| ulmdvlem1 26309 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem2 26310 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem3 26311 | Lemma for ~ ulmdv . (Cont... |
| ulmdv 26312 | If ` F ` is a sequence of ... |
| mtest 26313 | The Weierstrass M-test. I... |
| mtestbdd 26314 | Given the hypotheses of th... |
| mbfulm 26315 | A uniform limit of measura... |
| iblulm 26316 | A uniform limit of integra... |
| itgulm 26317 | A uniform limit of integra... |
| itgulm2 26318 | A uniform limit of integra... |
| pserval 26319 | Value of the function ` G ... |
| pserval2 26320 | Value of the function ` G ... |
| psergf 26321 | The sequence of terms in t... |
| radcnvlem1 26322 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem2 26323 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem3 26324 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnv0 26325 | Zero is always a convergen... |
| radcnvcl 26326 | The radius of convergence ... |
| radcnvlt1 26327 | If ` X ` is within the ope... |
| radcnvlt2 26328 | If ` X ` is within the ope... |
| radcnvle 26329 | If ` X ` is a convergent p... |
| dvradcnv 26330 | The radius of convergence ... |
| pserulm 26331 | If ` S ` is a region conta... |
| psercn2 26332 | Since by ~ pserulm the ser... |
| psercn2OLD 26333 | Obsolete version of ~ pser... |
| psercnlem2 26334 | Lemma for ~ psercn . (Con... |
| psercnlem1 26335 | Lemma for ~ psercn . (Con... |
| psercn 26336 | An infinite series converg... |
| pserdvlem1 26337 | Lemma for ~ pserdv . (Con... |
| pserdvlem2 26338 | Lemma for ~ pserdv . (Con... |
| pserdv 26339 | The derivative of a power ... |
| pserdv2 26340 | The derivative of a power ... |
| abelthlem1 26341 | Lemma for ~ abelth . (Con... |
| abelthlem2 26342 | Lemma for ~ abelth . The ... |
| abelthlem3 26343 | Lemma for ~ abelth . (Con... |
| abelthlem4 26344 | Lemma for ~ abelth . (Con... |
| abelthlem5 26345 | Lemma for ~ abelth . (Con... |
| abelthlem6 26346 | Lemma for ~ abelth . (Con... |
| abelthlem7a 26347 | Lemma for ~ abelth . (Con... |
| abelthlem7 26348 | Lemma for ~ abelth . (Con... |
| abelthlem8 26349 | Lemma for ~ abelth . (Con... |
| abelthlem9 26350 | Lemma for ~ abelth . By a... |
| abelth 26351 | Abel's theorem. If the po... |
| abelth2 26352 | Abel's theorem, restricted... |
| efcn 26353 | The exponential function i... |
| sincn 26354 | Sine is continuous. (Cont... |
| coscn 26355 | Cosine is continuous. (Co... |
| reeff1olem 26356 | Lemma for ~ reeff1o . (Co... |
| reeff1o 26357 | The real exponential funct... |
| reefiso 26358 | The exponential function o... |
| efcvx 26359 | The exponential function o... |
| reefgim 26360 | The exponential function i... |
| pilem1 26361 | Lemma for ~ pire , ~ pigt2... |
| pilem2 26362 | Lemma for ~ pire , ~ pigt2... |
| pilem3 26363 | Lemma for ~ pire , ~ pigt2... |
| pigt2lt4 26364 | ` _pi ` is between 2 and 4... |
| sinpi 26365 | The sine of ` _pi ` is 0. ... |
| pire 26366 | ` _pi ` is a real number. ... |
| picn 26367 | ` _pi ` is a complex numbe... |
| pipos 26368 | ` _pi ` is positive. (Con... |
| pine0 26369 | ` _pi ` is nonzero. (Cont... |
| pirp 26370 | ` _pi ` is a positive real... |
| negpicn 26371 | ` -u _pi ` is a real numbe... |
| sinhalfpilem 26372 | Lemma for ~ sinhalfpi and ... |
| halfpire 26373 | ` _pi / 2 ` is real. (Con... |
| neghalfpire 26374 | ` -u _pi / 2 ` is real. (... |
| neghalfpirx 26375 | ` -u _pi / 2 ` is an exten... |
| pidiv2halves 26376 | Adding ` _pi / 2 ` to itse... |
| sinhalfpi 26377 | The sine of ` _pi / 2 ` is... |
| coshalfpi 26378 | The cosine of ` _pi / 2 ` ... |
| cosneghalfpi 26379 | The cosine of ` -u _pi / 2... |
| efhalfpi 26380 | The exponential of ` _i _p... |
| cospi 26381 | The cosine of ` _pi ` is `... |
| efipi 26382 | The exponential of ` _i x.... |
| eulerid 26383 | Euler's identity. (Contri... |
| sin2pi 26384 | The sine of ` 2 _pi ` is 0... |
| cos2pi 26385 | The cosine of ` 2 _pi ` is... |
| ef2pi 26386 | The exponential of ` 2 _pi... |
| ef2kpi 26387 | If ` K ` is an integer, th... |
| efper 26388 | The exponential function i... |
| sinperlem 26389 | Lemma for ~ sinper and ~ c... |
| sinper 26390 | The sine function is perio... |
| cosper 26391 | The cosine function is per... |
| sin2kpi 26392 | If ` K ` is an integer, th... |
| cos2kpi 26393 | If ` K ` is an integer, th... |
| sin2pim 26394 | Sine of a number subtracte... |
| cos2pim 26395 | Cosine of a number subtrac... |
| sinmpi 26396 | Sine of a number less ` _p... |
| cosmpi 26397 | Cosine of a number less ` ... |
| sinppi 26398 | Sine of a number plus ` _p... |
| cosppi 26399 | Cosine of a number plus ` ... |
| efimpi 26400 | The exponential function a... |
| sinhalfpip 26401 | The sine of ` _pi / 2 ` pl... |
| sinhalfpim 26402 | The sine of ` _pi / 2 ` mi... |
| coshalfpip 26403 | The cosine of ` _pi / 2 ` ... |
| coshalfpim 26404 | The cosine of ` _pi / 2 ` ... |
| ptolemy 26405 | Ptolemy's Theorem. This t... |
| sincosq1lem 26406 | Lemma for ~ sincosq1sgn . ... |
| sincosq1sgn 26407 | The signs of the sine and ... |
| sincosq2sgn 26408 | The signs of the sine and ... |
| sincosq3sgn 26409 | The signs of the sine and ... |
| sincosq4sgn 26410 | The signs of the sine and ... |
| coseq00topi 26411 | Location of the zeroes of ... |
| coseq0negpitopi 26412 | Location of the zeroes of ... |
| tanrpcl 26413 | Positive real closure of t... |
| tangtx 26414 | The tangent function is gr... |
| tanabsge 26415 | The tangent function is gr... |
| sinq12gt0 26416 | The sine of a number stric... |
| sinq12ge0 26417 | The sine of a number betwe... |
| sinq34lt0t 26418 | The sine of a number stric... |
| cosq14gt0 26419 | The cosine of a number str... |
| cosq14ge0 26420 | The cosine of a number bet... |
| sincosq1eq 26421 | Complementarity of the sin... |
| sincos4thpi 26422 | The sine and cosine of ` _... |
| tan4thpi 26423 | The tangent of ` _pi / 4 `... |
| tan4thpiOLD 26424 | Obsolete version of ~ tan4... |
| sincos6thpi 26425 | The sine and cosine of ` _... |
| sincos3rdpi 26426 | The sine and cosine of ` _... |
| pigt3 26427 | ` _pi ` is greater than 3.... |
| pige3 26428 | ` _pi ` is greater than or... |
| pige3ALT 26429 | Alternate proof of ~ pige3... |
| abssinper 26430 | The absolute value of sine... |
| sinkpi 26431 | The sine of an integer mul... |
| coskpi 26432 | The absolute value of the ... |
| sineq0 26433 | A complex number whose sin... |
| coseq1 26434 | A complex number whose cos... |
| cos02pilt1 26435 | Cosine is less than one be... |
| cosq34lt1 26436 | Cosine is less than one in... |
| efeq1 26437 | A complex number whose exp... |
| cosne0 26438 | The cosine function has no... |
| cosordlem 26439 | Lemma for ~ cosord . (Con... |
| cosord 26440 | Cosine is decreasing over ... |
| cos0pilt1 26441 | Cosine is between minus on... |
| cos11 26442 | Cosine is one-to-one over ... |
| sinord 26443 | Sine is increasing over th... |
| recosf1o 26444 | The cosine function is a b... |
| resinf1o 26445 | The sine function is a bij... |
| tanord1 26446 | The tangent function is st... |
| tanord 26447 | The tangent function is st... |
| tanregt0 26448 | The real part of the tange... |
| negpitopissre 26449 | The interval ` ( -u _pi (,... |
| efgh 26450 | The exponential function o... |
| efif1olem1 26451 | Lemma for ~ efif1o . (Con... |
| efif1olem2 26452 | Lemma for ~ efif1o . (Con... |
| efif1olem3 26453 | Lemma for ~ efif1o . (Con... |
| efif1olem4 26454 | The exponential function o... |
| efif1o 26455 | The exponential function o... |
| efifo 26456 | The exponential function o... |
| eff1olem 26457 | The exponential function m... |
| eff1o 26458 | The exponential function m... |
| efabl 26459 | The image of a subgroup of... |
| efsubm 26460 | The image of a subgroup of... |
| circgrp 26461 | The circle group ` T ` is ... |
| circsubm 26462 | The circle group ` T ` is ... |
| logrn 26467 | The range of the natural l... |
| ellogrn 26468 | Write out the property ` A... |
| dflog2 26469 | The natural logarithm func... |
| relogrn 26470 | The range of the natural l... |
| logrncn 26471 | The range of the natural l... |
| eff1o2 26472 | The exponential function r... |
| logf1o 26473 | The natural logarithm func... |
| dfrelog 26474 | The natural logarithm func... |
| relogf1o 26475 | The natural logarithm func... |
| logrncl 26476 | Closure of the natural log... |
| logcl 26477 | Closure of the natural log... |
| logimcl 26478 | Closure of the imaginary p... |
| logcld 26479 | The logarithm of a nonzero... |
| logimcld 26480 | The imaginary part of the ... |
| logimclad 26481 | The imaginary part of the ... |
| abslogimle 26482 | The imaginary part of the ... |
| logrnaddcl 26483 | The range of the natural l... |
| relogcl 26484 | Closure of the natural log... |
| eflog 26485 | Relationship between the n... |
| logeq0im1 26486 | If the logarithm of a numb... |
| logccne0 26487 | The logarithm isn't 0 if i... |
| logne0 26488 | Logarithm of a non-1 posit... |
| reeflog 26489 | Relationship between the n... |
| logef 26490 | Relationship between the n... |
| relogef 26491 | Relationship between the n... |
| logeftb 26492 | Relationship between the n... |
| relogeftb 26493 | Relationship between the n... |
| log1 26494 | The natural logarithm of `... |
| loge 26495 | The natural logarithm of `... |
| logi 26496 | The natural logarithm of `... |
| logneg 26497 | The natural logarithm of a... |
| logm1 26498 | The natural logarithm of n... |
| lognegb 26499 | If a number has imaginary ... |
| relogoprlem 26500 | Lemma for ~ relogmul and ~... |
| relogmul 26501 | The natural logarithm of t... |
| relogdiv 26502 | The natural logarithm of t... |
| explog 26503 | Exponentiation of a nonzer... |
| reexplog 26504 | Exponentiation of a positi... |
| relogexp 26505 | The natural logarithm of p... |
| relog 26506 | Real part of a logarithm. ... |
| relogiso 26507 | The natural logarithm func... |
| reloggim 26508 | The natural logarithm is a... |
| logltb 26509 | The natural logarithm func... |
| logfac 26510 | The logarithm of a factori... |
| eflogeq 26511 | Solve an equation involvin... |
| logleb 26512 | Natural logarithm preserve... |
| rplogcl 26513 | Closure of the logarithm f... |
| logge0 26514 | The logarithm of a number ... |
| logcj 26515 | The natural logarithm dist... |
| efiarg 26516 | The exponential of the "ar... |
| cosargd 26517 | The cosine of the argument... |
| cosarg0d 26518 | The cosine of the argument... |
| argregt0 26519 | Closure of the argument of... |
| argrege0 26520 | Closure of the argument of... |
| argimgt0 26521 | Closure of the argument of... |
| argimlt0 26522 | Closure of the argument of... |
| logimul 26523 | Multiplying a number by ` ... |
| logneg2 26524 | The logarithm of the negat... |
| logmul2 26525 | Generalization of ~ relogm... |
| logdiv2 26526 | Generalization of ~ relogd... |
| abslogle 26527 | Bound on the magnitude of ... |
| tanarg 26528 | The basic relation between... |
| logdivlti 26529 | The ` log x / x ` function... |
| logdivlt 26530 | The ` log x / x ` function... |
| logdivle 26531 | The ` log x / x ` function... |
| relogcld 26532 | Closure of the natural log... |
| reeflogd 26533 | Relationship between the n... |
| relogmuld 26534 | The natural logarithm of t... |
| relogdivd 26535 | The natural logarithm of t... |
| logled 26536 | Natural logarithm preserve... |
| relogefd 26537 | Relationship between the n... |
| rplogcld 26538 | Closure of the logarithm f... |
| logge0d 26539 | The logarithm of a number ... |
| logge0b 26540 | The logarithm of a number ... |
| loggt0b 26541 | The logarithm of a number ... |
| logle1b 26542 | The logarithm of a number ... |
| loglt1b 26543 | The logarithm of a number ... |
| divlogrlim 26544 | The inverse logarithm func... |
| logno1 26545 | The logarithm function is ... |
| dvrelog 26546 | The derivative of the real... |
| relogcn 26547 | The real logarithm functio... |
| ellogdm 26548 | Elementhood in the "contin... |
| logdmn0 26549 | A number in the continuous... |
| logdmnrp 26550 | A number in the continuous... |
| logdmss 26551 | The continuity domain of `... |
| logcnlem2 26552 | Lemma for ~ logcn . (Cont... |
| logcnlem3 26553 | Lemma for ~ logcn . (Cont... |
| logcnlem4 26554 | Lemma for ~ logcn . (Cont... |
| logcnlem5 26555 | Lemma for ~ logcn . (Cont... |
| logcn 26556 | The logarithm function is ... |
| dvloglem 26557 | Lemma for ~ dvlog . (Cont... |
| logdmopn 26558 | The "continuous domain" of... |
| logf1o2 26559 | The logarithm maps its con... |
| dvlog 26560 | The derivative of the comp... |
| dvlog2lem 26561 | Lemma for ~ dvlog2 . (Con... |
| dvlog2 26562 | The derivative of the comp... |
| advlog 26563 | The antiderivative of the ... |
| advlogexp 26564 | The antiderivative of a po... |
| efopnlem1 26565 | Lemma for ~ efopn . (Cont... |
| efopnlem2 26566 | Lemma for ~ efopn . (Cont... |
| efopn 26567 | The exponential map is an ... |
| logtayllem 26568 | Lemma for ~ logtayl . (Co... |
| logtayl 26569 | The Taylor series for ` -u... |
| logtaylsum 26570 | The Taylor series for ` -u... |
| logtayl2 26571 | Power series expression fo... |
| logccv 26572 | The natural logarithm func... |
| cxpval 26573 | Value of the complex power... |
| cxpef 26574 | Value of the complex power... |
| 0cxp 26575 | Value of the complex power... |
| cxpexpz 26576 | Relate the complex power f... |
| cxpexp 26577 | Relate the complex power f... |
| logcxp 26578 | Logarithm of a complex pow... |
| cxp0 26579 | Value of the complex power... |
| cxp1 26580 | Value of the complex power... |
| 1cxp 26581 | Value of the complex power... |
| ecxp 26582 | Write the exponential func... |
| cxpcl 26583 | Closure of the complex pow... |
| recxpcl 26584 | Real closure of the comple... |
| rpcxpcl 26585 | Positive real closure of t... |
| cxpne0 26586 | Complex exponentiation is ... |
| cxpeq0 26587 | Complex exponentiation is ... |
| cxpadd 26588 | Sum of exponents law for c... |
| cxpp1 26589 | Value of a nonzero complex... |
| cxpneg 26590 | Value of a complex number ... |
| cxpsub 26591 | Exponent subtraction law f... |
| cxpge0 26592 | Nonnegative exponentiation... |
| mulcxplem 26593 | Lemma for ~ mulcxp . (Con... |
| mulcxp 26594 | Complex exponentiation of ... |
| cxprec 26595 | Complex exponentiation of ... |
| divcxp 26596 | Complex exponentiation of ... |
| cxpmul 26597 | Product of exponents law f... |
| cxpmul2 26598 | Product of exponents law f... |
| cxproot 26599 | The complex power function... |
| cxpmul2z 26600 | Generalize ~ cxpmul2 to ne... |
| abscxp 26601 | Absolute value of a power,... |
| abscxp2 26602 | Absolute value of a power,... |
| cxplt 26603 | Ordering property for comp... |
| cxple 26604 | Ordering property for comp... |
| cxplea 26605 | Ordering property for comp... |
| cxple2 26606 | Ordering property for comp... |
| cxplt2 26607 | Ordering property for comp... |
| cxple2a 26608 | Ordering property for comp... |
| cxplt3 26609 | Ordering property for comp... |
| cxple3 26610 | Ordering property for comp... |
| cxpsqrtlem 26611 | Lemma for ~ cxpsqrt . (Co... |
| cxpsqrt 26612 | The complex exponential fu... |
| logsqrt 26613 | Logarithm of a square root... |
| cxp0d 26614 | Value of the complex power... |
| cxp1d 26615 | Value of the complex power... |
| 1cxpd 26616 | Value of the complex power... |
| cxpcld 26617 | Closure of the complex pow... |
| cxpmul2d 26618 | Product of exponents law f... |
| 0cxpd 26619 | Value of the complex power... |
| cxpexpzd 26620 | Relate the complex power f... |
| cxpefd 26621 | Value of the complex power... |
| cxpne0d 26622 | Complex exponentiation is ... |
| cxpp1d 26623 | Value of a nonzero complex... |
| cxpnegd 26624 | Value of a complex number ... |
| cxpmul2zd 26625 | Generalize ~ cxpmul2 to ne... |
| cxpaddd 26626 | Sum of exponents law for c... |
| cxpsubd 26627 | Exponent subtraction law f... |
| cxpltd 26628 | Ordering property for comp... |
| cxpled 26629 | Ordering property for comp... |
| cxplead 26630 | Ordering property for comp... |
| divcxpd 26631 | Complex exponentiation of ... |
| recxpcld 26632 | Positive real closure of t... |
| cxpge0d 26633 | Nonnegative exponentiation... |
| cxple2ad 26634 | Ordering property for comp... |
| cxplt2d 26635 | Ordering property for comp... |
| cxple2d 26636 | Ordering property for comp... |
| mulcxpd 26637 | Complex exponentiation of ... |
| recxpf1lem 26638 | Complex exponentiation on ... |
| cxpsqrtth 26639 | Square root theorem over t... |
| 2irrexpq 26640 | There exist irrational num... |
| cxprecd 26641 | Complex exponentiation of ... |
| rpcxpcld 26642 | Positive real closure of t... |
| logcxpd 26643 | Logarithm of a complex pow... |
| cxplt3d 26644 | Ordering property for comp... |
| cxple3d 26645 | Ordering property for comp... |
| cxpmuld 26646 | Product of exponents law f... |
| cxpgt0d 26647 | A positive real raised to ... |
| cxpcom 26648 | Commutative law for real e... |
| dvcxp1 26649 | The derivative of a comple... |
| dvcxp2 26650 | The derivative of a comple... |
| dvsqrt 26651 | The derivative of the real... |
| dvcncxp1 26652 | Derivative of complex powe... |
| dvcnsqrt 26653 | Derivative of square root ... |
| cxpcn 26654 | Domain of continuity of th... |
| cxpcnOLD 26655 | Obsolete version of ~ cxpc... |
| cxpcn2 26656 | Continuity of the complex ... |
| cxpcn3lem 26657 | Lemma for ~ cxpcn3 . (Con... |
| cxpcn3 26658 | Extend continuity of the c... |
| resqrtcn 26659 | Continuity of the real squ... |
| sqrtcn 26660 | Continuity of the square r... |
| cxpaddlelem 26661 | Lemma for ~ cxpaddle . (C... |
| cxpaddle 26662 | Ordering property for comp... |
| abscxpbnd 26663 | Bound on the absolute valu... |
| root1id 26664 | Property of an ` N ` -th r... |
| root1eq1 26665 | The only powers of an ` N ... |
| root1cj 26666 | Within the ` N ` -th roots... |
| cxpeq 26667 | Solve an equation involvin... |
| zrtelqelz 26668 | If the ` N ` -th root of a... |
| zrtdvds 26669 | A positive integer root di... |
| rtprmirr 26670 | The root of a prime number... |
| loglesqrt 26671 | An upper bound on the loga... |
| logreclem 26672 | Symmetry of the natural lo... |
| logrec 26673 | Logarithm of a reciprocal ... |
| logbval 26676 | Define the value of the ` ... |
| logbcl 26677 | General logarithm closure.... |
| logbid1 26678 | General logarithm is 1 whe... |
| logb1 26679 | The logarithm of ` 1 ` to ... |
| elogb 26680 | The general logarithm of a... |
| logbchbase 26681 | Change of base for logarit... |
| relogbval 26682 | Value of the general logar... |
| relogbcl 26683 | Closure of the general log... |
| relogbzcl 26684 | Closure of the general log... |
| relogbreexp 26685 | Power law for the general ... |
| relogbzexp 26686 | Power law for the general ... |
| relogbmul 26687 | The logarithm of the produ... |
| relogbmulexp 26688 | The logarithm of the produ... |
| relogbdiv 26689 | The logarithm of the quoti... |
| relogbexp 26690 | Identity law for general l... |
| nnlogbexp 26691 | Identity law for general l... |
| logbrec 26692 | Logarithm of a reciprocal ... |
| logbleb 26693 | The general logarithm func... |
| logblt 26694 | The general logarithm func... |
| relogbcxp 26695 | Identity law for the gener... |
| cxplogb 26696 | Identity law for the gener... |
| relogbcxpb 26697 | The logarithm is the inver... |
| logbmpt 26698 | The general logarithm to a... |
| logbf 26699 | The general logarithm to a... |
| logbfval 26700 | The general logarithm of a... |
| relogbf 26701 | The general logarithm to a... |
| logblog 26702 | The general logarithm to t... |
| logbgt0b 26703 | The logarithm of a positiv... |
| logbgcd1irr 26704 | The logarithm of an intege... |
| 2logb9irr 26705 | Example for ~ logbgcd1irr ... |
| logbprmirr 26706 | The logarithm of a prime t... |
| 2logb3irr 26707 | Example for ~ logbprmirr .... |
| 2logb9irrALT 26708 | Alternate proof of ~ 2logb... |
| sqrt2cxp2logb9e3 26709 | The square root of two to ... |
| 2irrexpqALT 26710 | Alternate proof of ~ 2irre... |
| angval 26711 | Define the angle function,... |
| angcan 26712 | Cancel a constant multipli... |
| angneg 26713 | Cancel a negative sign in ... |
| angvald 26714 | The (signed) angle between... |
| angcld 26715 | The (signed) angle between... |
| angrteqvd 26716 | Two vectors are at a right... |
| cosangneg2d 26717 | The cosine of the angle be... |
| angrtmuld 26718 | Perpendicularity of two ve... |
| ang180lem1 26719 | Lemma for ~ ang180 . Show... |
| ang180lem2 26720 | Lemma for ~ ang180 . Show... |
| ang180lem3 26721 | Lemma for ~ ang180 . Sinc... |
| ang180lem4 26722 | Lemma for ~ ang180 . Redu... |
| ang180lem5 26723 | Lemma for ~ ang180 : Redu... |
| ang180 26724 | The sum of angles ` m A B ... |
| lawcoslem1 26725 | Lemma for ~ lawcos . Here... |
| lawcos 26726 | Law of cosines (also known... |
| pythag 26727 | Pythagorean theorem. Give... |
| isosctrlem1 26728 | Lemma for ~ isosctr . (Co... |
| isosctrlem2 26729 | Lemma for ~ isosctr . Cor... |
| isosctrlem3 26730 | Lemma for ~ isosctr . Cor... |
| isosctr 26731 | Isosceles triangle theorem... |
| ssscongptld 26732 | If two triangles have equa... |
| affineequiv 26733 | Equivalence between two wa... |
| affineequiv2 26734 | Equivalence between two wa... |
| affineequiv3 26735 | Equivalence between two wa... |
| affineequiv4 26736 | Equivalence between two wa... |
| affineequivne 26737 | Equivalence between two wa... |
| angpieqvdlem 26738 | Equivalence used in the pr... |
| angpieqvdlem2 26739 | Equivalence used in ~ angp... |
| angpined 26740 | If the angle at ABC is ` _... |
| angpieqvd 26741 | The angle ABC is ` _pi ` i... |
| chordthmlem 26742 | If ` M ` is the midpoint o... |
| chordthmlem2 26743 | If M is the midpoint of AB... |
| chordthmlem3 26744 | If M is the midpoint of AB... |
| chordthmlem4 26745 | If P is on the segment AB ... |
| chordthmlem5 26746 | If P is on the segment AB ... |
| chordthm 26747 | The intersecting chords th... |
| heron 26748 | Heron's formula gives the ... |
| quad2 26749 | The quadratic equation, wi... |
| quad 26750 | The quadratic equation. (... |
| 1cubrlem 26751 | The cube roots of unity. ... |
| 1cubr 26752 | The cube roots of unity. ... |
| dcubic1lem 26753 | Lemma for ~ dcubic1 and ~ ... |
| dcubic2 26754 | Reverse direction of ~ dcu... |
| dcubic1 26755 | Forward direction of ~ dcu... |
| dcubic 26756 | Solutions to the depressed... |
| mcubic 26757 | Solutions to a monic cubic... |
| cubic2 26758 | The solution to the genera... |
| cubic 26759 | The cubic equation, which ... |
| binom4 26760 | Work out a quartic binomia... |
| dquartlem1 26761 | Lemma for ~ dquart . (Con... |
| dquartlem2 26762 | Lemma for ~ dquart . (Con... |
| dquart 26763 | Solve a depressed quartic ... |
| quart1cl 26764 | Closure lemmas for ~ quart... |
| quart1lem 26765 | Lemma for ~ quart1 . (Con... |
| quart1 26766 | Depress a quartic equation... |
| quartlem1 26767 | Lemma for ~ quart . (Cont... |
| quartlem2 26768 | Closure lemmas for ~ quart... |
| quartlem3 26769 | Closure lemmas for ~ quart... |
| quartlem4 26770 | Closure lemmas for ~ quart... |
| quart 26771 | The quartic equation, writ... |
| asinlem 26778 | The argument to the logari... |
| asinlem2 26779 | The argument to the logari... |
| asinlem3a 26780 | Lemma for ~ asinlem3 . (C... |
| asinlem3 26781 | The argument to the logari... |
| asinf 26782 | Domain and codomain of the... |
| asincl 26783 | Closure for the arcsin fun... |
| acosf 26784 | Domain and codoamin of the... |
| acoscl 26785 | Closure for the arccos fun... |
| atandm 26786 | Since the property is a li... |
| atandm2 26787 | This form of ~ atandm is a... |
| atandm3 26788 | A compact form of ~ atandm... |
| atandm4 26789 | A compact form of ~ atandm... |
| atanf 26790 | Domain and codoamin of the... |
| atancl 26791 | Closure for the arctan fun... |
| asinval 26792 | Value of the arcsin functi... |
| acosval 26793 | Value of the arccos functi... |
| atanval 26794 | Value of the arctan functi... |
| atanre 26795 | A real number is in the do... |
| asinneg 26796 | The arcsine function is od... |
| acosneg 26797 | The negative symmetry rela... |
| efiasin 26798 | The exponential of the arc... |
| sinasin 26799 | The arcsine function is an... |
| cosacos 26800 | The arccosine function is ... |
| asinsinlem 26801 | Lemma for ~ asinsin . (Co... |
| asinsin 26802 | The arcsine function compo... |
| acoscos 26803 | The arccosine function is ... |
| asin1 26804 | The arcsine of ` 1 ` is ` ... |
| acos1 26805 | The arccosine of ` 1 ` is ... |
| reasinsin 26806 | The arcsine function compo... |
| asinsinb 26807 | Relationship between sine ... |
| acoscosb 26808 | Relationship between cosin... |
| asinbnd 26809 | The arcsine function has r... |
| acosbnd 26810 | The arccosine function has... |
| asinrebnd 26811 | Bounds on the arcsine func... |
| asinrecl 26812 | The arcsine function is re... |
| acosrecl 26813 | The arccosine function is ... |
| cosasin 26814 | The cosine of the arcsine ... |
| sinacos 26815 | The sine of the arccosine ... |
| atandmneg 26816 | The domain of the arctange... |
| atanneg 26817 | The arctangent function is... |
| atan0 26818 | The arctangent of zero is ... |
| atandmcj 26819 | The arctangent function di... |
| atancj 26820 | The arctangent function di... |
| atanrecl 26821 | The arctangent function is... |
| efiatan 26822 | Value of the exponential o... |
| atanlogaddlem 26823 | Lemma for ~ atanlogadd . ... |
| atanlogadd 26824 | The rule ` sqrt ( z w ) = ... |
| atanlogsublem 26825 | Lemma for ~ atanlogsub . ... |
| atanlogsub 26826 | A variation on ~ atanlogad... |
| efiatan2 26827 | Value of the exponential o... |
| 2efiatan 26828 | Value of the exponential o... |
| tanatan 26829 | The arctangent function is... |
| atandmtan 26830 | The tangent function has r... |
| cosatan 26831 | The cosine of an arctangen... |
| cosatanne0 26832 | The arctangent function ha... |
| atantan 26833 | The arctangent function is... |
| atantanb 26834 | Relationship between tange... |
| atanbndlem 26835 | Lemma for ~ atanbnd . (Co... |
| atanbnd 26836 | The arctangent function is... |
| atanord 26837 | The arctangent function is... |
| atan1 26838 | The arctangent of ` 1 ` is... |
| bndatandm 26839 | A point in the open unit d... |
| atans 26840 | The "domain of continuity"... |
| atans2 26841 | It suffices to show that `... |
| atansopn 26842 | The domain of continuity o... |
| atansssdm 26843 | The domain of continuity o... |
| ressatans 26844 | The real number line is a ... |
| dvatan 26845 | The derivative of the arct... |
| atancn 26846 | The arctangent is a contin... |
| atantayl 26847 | The Taylor series for ` ar... |
| atantayl2 26848 | The Taylor series for ` ar... |
| atantayl3 26849 | The Taylor series for ` ar... |
| leibpilem1 26850 | Lemma for ~ leibpi . (Con... |
| leibpilem2 26851 | The Leibniz formula for ` ... |
| leibpi 26852 | The Leibniz formula for ` ... |
| leibpisum 26853 | The Leibniz formula for ` ... |
| log2cnv 26854 | Using the Taylor series fo... |
| log2tlbnd 26855 | Bound the error term in th... |
| log2ublem1 26856 | Lemma for ~ log2ub . The ... |
| log2ublem2 26857 | Lemma for ~ log2ub . (Con... |
| log2ublem3 26858 | Lemma for ~ log2ub . In d... |
| log2ub 26859 | ` log 2 ` is less than ` 2... |
| log2le1 26860 | ` log 2 ` is less than ` 1... |
| birthdaylem1 26861 | Lemma for ~ birthday . (C... |
| birthdaylem2 26862 | For general ` N ` and ` K ... |
| birthdaylem3 26863 | For general ` N ` and ` K ... |
| birthday 26864 | The Birthday Problem. The... |
| dmarea 26867 | The domain of the area fun... |
| areambl 26868 | The fibers of a measurable... |
| areass 26869 | A measurable region is a s... |
| dfarea 26870 | Rewrite ~ df-area self-ref... |
| areaf 26871 | Area measurement is a func... |
| areacl 26872 | The area of a measurable r... |
| areage0 26873 | The area of a measurable r... |
| areaval 26874 | The area of a measurable r... |
| rlimcnp 26875 | Relate a limit of a real-v... |
| rlimcnp2 26876 | Relate a limit of a real-v... |
| rlimcnp3 26877 | Relate a limit of a real-v... |
| xrlimcnp 26878 | Relate a limit of a real-v... |
| efrlim 26879 | The limit of the sequence ... |
| efrlimOLD 26880 | Obsolete version of ~ efrl... |
| dfef2 26881 | The limit of the sequence ... |
| cxplim 26882 | A power to a negative expo... |
| sqrtlim 26883 | The inverse square root fu... |
| rlimcxp 26884 | Any power to a positive ex... |
| o1cxp 26885 | An eventually bounded func... |
| cxp2limlem 26886 | A linear factor grows slow... |
| cxp2lim 26887 | Any power grows slower tha... |
| cxploglim 26888 | The logarithm grows slower... |
| cxploglim2 26889 | Every power of the logarit... |
| divsqrtsumlem 26890 | Lemma for ~ divsqrsum and ... |
| divsqrsumf 26891 | The function ` F ` used in... |
| divsqrsum 26892 | The sum ` sum_ n <_ x ( 1 ... |
| divsqrtsum2 26893 | A bound on the distance of... |
| divsqrtsumo1 26894 | The sum ` sum_ n <_ x ( 1 ... |
| cvxcl 26895 | Closure of a 0-1 linear co... |
| scvxcvx 26896 | A strictly convex function... |
| jensenlem1 26897 | Lemma for ~ jensen . (Con... |
| jensenlem2 26898 | Lemma for ~ jensen . (Con... |
| jensen 26899 | Jensen's inequality, a fin... |
| amgmlem 26900 | Lemma for ~ amgm . (Contr... |
| amgm 26901 | Inequality of arithmetic a... |
| logdifbnd 26904 | Bound on the difference of... |
| logdiflbnd 26905 | Lower bound on the differe... |
| emcllem1 26906 | Lemma for ~ emcl . The se... |
| emcllem2 26907 | Lemma for ~ emcl . ` F ` i... |
| emcllem3 26908 | Lemma for ~ emcl . The fu... |
| emcllem4 26909 | Lemma for ~ emcl . The di... |
| emcllem5 26910 | Lemma for ~ emcl . The pa... |
| emcllem6 26911 | Lemma for ~ emcl . By the... |
| emcllem7 26912 | Lemma for ~ emcl and ~ har... |
| emcl 26913 | Closure and bounds for the... |
| harmonicbnd 26914 | A bound on the harmonic se... |
| harmonicbnd2 26915 | A bound on the harmonic se... |
| emre 26916 | The Euler-Mascheroni const... |
| emgt0 26917 | The Euler-Mascheroni const... |
| harmonicbnd3 26918 | A bound on the harmonic se... |
| harmoniclbnd 26919 | A bound on the harmonic se... |
| harmonicubnd 26920 | A bound on the harmonic se... |
| harmonicbnd4 26921 | The asymptotic behavior of... |
| fsumharmonic 26922 | Bound a finite sum based o... |
| zetacvg 26925 | The zeta series is converg... |
| eldmgm 26932 | Elementhood in the set of ... |
| dmgmaddn0 26933 | If ` A ` is not a nonposit... |
| dmlogdmgm 26934 | If ` A ` is in the continu... |
| rpdmgm 26935 | A positive real number is ... |
| dmgmn0 26936 | If ` A ` is not a nonposit... |
| dmgmaddnn0 26937 | If ` A ` is not a nonposit... |
| dmgmdivn0 26938 | Lemma for ~ lgamf . (Cont... |
| lgamgulmlem1 26939 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem2 26940 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem3 26941 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem4 26942 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem5 26943 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem6 26944 | The series ` G ` is unifor... |
| lgamgulm 26945 | The series ` G ` is unifor... |
| lgamgulm2 26946 | Rewrite the limit of the s... |
| lgambdd 26947 | The log-Gamma function is ... |
| lgamucov 26948 | The ` U ` regions used in ... |
| lgamucov2 26949 | The ` U ` regions used in ... |
| lgamcvglem 26950 | Lemma for ~ lgamf and ~ lg... |
| lgamcl 26951 | The log-Gamma function is ... |
| lgamf 26952 | The log-Gamma function is ... |
| gamf 26953 | The Gamma function is a co... |
| gamcl 26954 | The exponential of the log... |
| eflgam 26955 | The exponential of the log... |
| gamne0 26956 | The Gamma function is neve... |
| igamval 26957 | Value of the inverse Gamma... |
| igamz 26958 | Value of the inverse Gamma... |
| igamgam 26959 | Value of the inverse Gamma... |
| igamlgam 26960 | Value of the inverse Gamma... |
| igamf 26961 | Closure of the inverse Gam... |
| igamcl 26962 | Closure of the inverse Gam... |
| gamigam 26963 | The Gamma function is the ... |
| lgamcvg 26964 | The series ` G ` converges... |
| lgamcvg2 26965 | The series ` G ` converges... |
| gamcvg 26966 | The pointwise exponential ... |
| lgamp1 26967 | The functional equation of... |
| gamp1 26968 | The functional equation of... |
| gamcvg2lem 26969 | Lemma for ~ gamcvg2 . (Co... |
| gamcvg2 26970 | An infinite product expres... |
| regamcl 26971 | The Gamma function is real... |
| relgamcl 26972 | The log-Gamma function is ... |
| rpgamcl 26973 | The log-Gamma function is ... |
| lgam1 26974 | The log-Gamma function at ... |
| gam1 26975 | The log-Gamma function at ... |
| facgam 26976 | The Gamma function general... |
| gamfac 26977 | The Gamma function general... |
| wilthlem1 26978 | The only elements that are... |
| wilthlem2 26979 | Lemma for ~ wilth : induct... |
| wilthlem3 26980 | Lemma for ~ wilth . Here ... |
| wilth 26981 | Wilson's theorem. A numbe... |
| wilthimp 26982 | The forward implication of... |
| ftalem1 26983 | Lemma for ~ fta : "growth... |
| ftalem2 26984 | Lemma for ~ fta . There e... |
| ftalem3 26985 | Lemma for ~ fta . There e... |
| ftalem4 26986 | Lemma for ~ fta : Closure... |
| ftalem5 26987 | Lemma for ~ fta : Main pr... |
| ftalem6 26988 | Lemma for ~ fta : Dischar... |
| ftalem7 26989 | Lemma for ~ fta . Shift t... |
| fta 26990 | The Fundamental Theorem of... |
| basellem1 26991 | Lemma for ~ basel . Closu... |
| basellem2 26992 | Lemma for ~ basel . Show ... |
| basellem3 26993 | Lemma for ~ basel . Using... |
| basellem4 26994 | Lemma for ~ basel . By ~ ... |
| basellem5 26995 | Lemma for ~ basel . Using... |
| basellem6 26996 | Lemma for ~ basel . The f... |
| basellem7 26997 | Lemma for ~ basel . The f... |
| basellem8 26998 | Lemma for ~ basel . The f... |
| basellem9 26999 | Lemma for ~ basel . Since... |
| basel 27000 | The sum of the inverse squ... |
| efnnfsumcl 27013 | Finite sum closure in the ... |
| ppisval 27014 | The set of primes less tha... |
| ppisval2 27015 | The set of primes less tha... |
| ppifi 27016 | The set of primes less tha... |
| prmdvdsfi 27017 | The set of prime divisors ... |
| chtf 27018 | Domain and codoamin of the... |
| chtcl 27019 | Real closure of the Chebys... |
| chtval 27020 | Value of the Chebyshev fun... |
| efchtcl 27021 | The Chebyshev function is ... |
| chtge0 27022 | The Chebyshev function is ... |
| vmaval 27023 | Value of the von Mangoldt ... |
| isppw 27024 | Two ways to say that ` A `... |
| isppw2 27025 | Two ways to say that ` A `... |
| vmappw 27026 | Value of the von Mangoldt ... |
| vmaprm 27027 | Value of the von Mangoldt ... |
| vmacl 27028 | Closure for the von Mangol... |
| vmaf 27029 | Functionality of the von M... |
| efvmacl 27030 | The von Mangoldt is closed... |
| vmage0 27031 | The von Mangoldt function ... |
| chpval 27032 | Value of the second Chebys... |
| chpf 27033 | Functionality of the secon... |
| chpcl 27034 | Closure for the second Che... |
| efchpcl 27035 | The second Chebyshev funct... |
| chpge0 27036 | The second Chebyshev funct... |
| ppival 27037 | Value of the prime-countin... |
| ppival2 27038 | Value of the prime-countin... |
| ppival2g 27039 | Value of the prime-countin... |
| ppif 27040 | Domain and codomain of the... |
| ppicl 27041 | Real closure of the prime-... |
| muval 27042 | The value of the Möbi... |
| muval1 27043 | The value of the Möbi... |
| muval2 27044 | The value of the Möbi... |
| isnsqf 27045 | Two ways to say that a num... |
| issqf 27046 | Two ways to say that a num... |
| sqfpc 27047 | The prime count of a squar... |
| dvdssqf 27048 | A divisor of a squarefree ... |
| sqf11 27049 | A squarefree number is com... |
| muf 27050 | The Möbius function i... |
| mucl 27051 | Closure of the Möbius... |
| sgmval 27052 | The value of the divisor f... |
| sgmval2 27053 | The value of the divisor f... |
| 0sgm 27054 | The value of the sum-of-di... |
| sgmf 27055 | The divisor function is a ... |
| sgmcl 27056 | Closure of the divisor fun... |
| sgmnncl 27057 | Closure of the divisor fun... |
| mule1 27058 | The Möbius function t... |
| chtfl 27059 | The Chebyshev function doe... |
| chpfl 27060 | The second Chebyshev funct... |
| ppiprm 27061 | The prime-counting functio... |
| ppinprm 27062 | The prime-counting functio... |
| chtprm 27063 | The Chebyshev function at ... |
| chtnprm 27064 | The Chebyshev function at ... |
| chpp1 27065 | The second Chebyshev funct... |
| chtwordi 27066 | The Chebyshev function is ... |
| chpwordi 27067 | The second Chebyshev funct... |
| chtdif 27068 | The difference of the Cheb... |
| efchtdvds 27069 | The exponentiated Chebyshe... |
| ppifl 27070 | The prime-counting functio... |
| ppip1le 27071 | The prime-counting functio... |
| ppiwordi 27072 | The prime-counting functio... |
| ppidif 27073 | The difference of the prim... |
| ppi1 27074 | The prime-counting functio... |
| cht1 27075 | The Chebyshev function at ... |
| vma1 27076 | The von Mangoldt function ... |
| chp1 27077 | The second Chebyshev funct... |
| ppi1i 27078 | Inference form of ~ ppiprm... |
| ppi2i 27079 | Inference form of ~ ppinpr... |
| ppi2 27080 | The prime-counting functio... |
| ppi3 27081 | The prime-counting functio... |
| cht2 27082 | The Chebyshev function at ... |
| cht3 27083 | The Chebyshev function at ... |
| ppinncl 27084 | Closure of the prime-count... |
| chtrpcl 27085 | Closure of the Chebyshev f... |
| ppieq0 27086 | The prime-counting functio... |
| ppiltx 27087 | The prime-counting functio... |
| prmorcht 27088 | Relate the primorial (prod... |
| mumullem1 27089 | Lemma for ~ mumul . A mul... |
| mumullem2 27090 | Lemma for ~ mumul . The p... |
| mumul 27091 | The Möbius function i... |
| sqff1o 27092 | There is a bijection from ... |
| fsumdvdsdiaglem 27093 | A "diagonal commutation" o... |
| fsumdvdsdiag 27094 | A "diagonal commutation" o... |
| fsumdvdscom 27095 | A double commutation of di... |
| dvdsppwf1o 27096 | A bijection between the di... |
| dvdsflf1o 27097 | A bijection from the numbe... |
| dvdsflsumcom 27098 | A sum commutation from ` s... |
| fsumfldivdiaglem 27099 | Lemma for ~ fsumfldivdiag ... |
| fsumfldivdiag 27100 | The right-hand side of ~ d... |
| musum 27101 | The sum of the Möbius... |
| musumsum 27102 | Evaluate a collapsing sum ... |
| muinv 27103 | The Möbius inversion ... |
| mpodvdsmulf1o 27104 | If ` M ` and ` N ` are two... |
| fsumdvdsmul 27105 | Product of two divisor sum... |
| dvdsmulf1o 27106 | If ` M ` and ` N ` are two... |
| fsumdvdsmulOLD 27107 | Obsolete version of ~ fsum... |
| sgmppw 27108 | The value of the divisor f... |
| 0sgmppw 27109 | A prime power ` P ^ K ` ha... |
| 1sgmprm 27110 | The sum of divisors for a ... |
| 1sgm2ppw 27111 | The sum of the divisors of... |
| sgmmul 27112 | The divisor function for f... |
| ppiublem1 27113 | Lemma for ~ ppiub . (Cont... |
| ppiublem2 27114 | A prime greater than ` 3 `... |
| ppiub 27115 | An upper bound on the prim... |
| vmalelog 27116 | The von Mangoldt function ... |
| chtlepsi 27117 | The first Chebyshev functi... |
| chprpcl 27118 | Closure of the second Cheb... |
| chpeq0 27119 | The second Chebyshev funct... |
| chteq0 27120 | The first Chebyshev functi... |
| chtleppi 27121 | Upper bound on the ` theta... |
| chtublem 27122 | Lemma for ~ chtub . (Cont... |
| chtub 27123 | An upper bound on the Cheb... |
| fsumvma 27124 | Rewrite a sum over the von... |
| fsumvma2 27125 | Apply ~ fsumvma for the co... |
| pclogsum 27126 | The logarithmic analogue o... |
| vmasum 27127 | The sum of the von Mangold... |
| logfac2 27128 | Another expression for the... |
| chpval2 27129 | Express the second Chebysh... |
| chpchtsum 27130 | The second Chebyshev funct... |
| chpub 27131 | An upper bound on the seco... |
| logfacubnd 27132 | A simple upper bound on th... |
| logfaclbnd 27133 | A lower bound on the logar... |
| logfacbnd3 27134 | Show the stronger statemen... |
| logfacrlim 27135 | Combine the estimates ~ lo... |
| logexprlim 27136 | The sum ` sum_ n <_ x , lo... |
| logfacrlim2 27137 | Write out ~ logfacrlim as ... |
| mersenne 27138 | A Mersenne prime is a prim... |
| perfect1 27139 | Euclid's contribution to t... |
| perfectlem1 27140 | Lemma for ~ perfect . (Co... |
| perfectlem2 27141 | Lemma for ~ perfect . (Co... |
| perfect 27142 | The Euclid-Euler theorem, ... |
| dchrval 27145 | Value of the group of Diri... |
| dchrbas 27146 | Base set of the group of D... |
| dchrelbas 27147 | A Dirichlet character is a... |
| dchrelbas2 27148 | A Dirichlet character is a... |
| dchrelbas3 27149 | A Dirichlet character is a... |
| dchrelbasd 27150 | A Dirichlet character is a... |
| dchrrcl 27151 | Reverse closure for a Diri... |
| dchrmhm 27152 | A Dirichlet character is a... |
| dchrf 27153 | A Dirichlet character is a... |
| dchrelbas4 27154 | A Dirichlet character is a... |
| dchrzrh1 27155 | Value of a Dirichlet chara... |
| dchrzrhcl 27156 | A Dirichlet character take... |
| dchrzrhmul 27157 | A Dirichlet character is c... |
| dchrplusg 27158 | Group operation on the gro... |
| dchrmul 27159 | Group operation on the gro... |
| dchrmulcl 27160 | Closure of the group opera... |
| dchrn0 27161 | A Dirichlet character is n... |
| dchr1cl 27162 | Closure of the principal D... |
| dchrmullid 27163 | Left identity for the prin... |
| dchrinvcl 27164 | Closure of the group inver... |
| dchrabl 27165 | The set of Dirichlet chara... |
| dchrfi 27166 | The group of Dirichlet cha... |
| dchrghm 27167 | A Dirichlet character rest... |
| dchr1 27168 | Value of the principal Dir... |
| dchreq 27169 | A Dirichlet character is d... |
| dchrresb 27170 | A Dirichlet character is d... |
| dchrabs 27171 | A Dirichlet character take... |
| dchrinv 27172 | The inverse of a Dirichlet... |
| dchrabs2 27173 | A Dirichlet character take... |
| dchr1re 27174 | The principal Dirichlet ch... |
| dchrptlem1 27175 | Lemma for ~ dchrpt . (Con... |
| dchrptlem2 27176 | Lemma for ~ dchrpt . (Con... |
| dchrptlem3 27177 | Lemma for ~ dchrpt . (Con... |
| dchrpt 27178 | For any element other than... |
| dchrsum2 27179 | An orthogonality relation ... |
| dchrsum 27180 | An orthogonality relation ... |
| sumdchr2 27181 | Lemma for ~ sumdchr . (Co... |
| dchrhash 27182 | There are exactly ` phi ( ... |
| sumdchr 27183 | An orthogonality relation ... |
| dchr2sum 27184 | An orthogonality relation ... |
| sum2dchr 27185 | An orthogonality relation ... |
| bcctr 27186 | Value of the central binom... |
| pcbcctr 27187 | Prime count of a central b... |
| bcmono 27188 | The binomial coefficient i... |
| bcmax 27189 | The binomial coefficient t... |
| bcp1ctr 27190 | Ratio of two central binom... |
| bclbnd 27191 | A bound on the binomial co... |
| efexple 27192 | Convert a bound on a power... |
| bpos1lem 27193 | Lemma for ~ bpos1 . (Cont... |
| bpos1 27194 | Bertrand's postulate, chec... |
| bposlem1 27195 | An upper bound on the prim... |
| bposlem2 27196 | There are no odd primes in... |
| bposlem3 27197 | Lemma for ~ bpos . Since ... |
| bposlem4 27198 | Lemma for ~ bpos . (Contr... |
| bposlem5 27199 | Lemma for ~ bpos . Bound ... |
| bposlem6 27200 | Lemma for ~ bpos . By usi... |
| bposlem7 27201 | Lemma for ~ bpos . The fu... |
| bposlem8 27202 | Lemma for ~ bpos . Evalua... |
| bposlem9 27203 | Lemma for ~ bpos . Derive... |
| bpos 27204 | Bertrand's postulate: ther... |
| zabsle1 27207 | ` { -u 1 , 0 , 1 } ` is th... |
| lgslem1 27208 | When ` a ` is coprime to t... |
| lgslem2 27209 | The set ` Z ` of all integ... |
| lgslem3 27210 | The set ` Z ` of all integ... |
| lgslem4 27211 | Lemma for ~ lgsfcl2 . (Co... |
| lgsval 27212 | Value of the Legendre symb... |
| lgsfval 27213 | Value of the function ` F ... |
| lgsfcl2 27214 | The function ` F ` is clos... |
| lgscllem 27215 | The Legendre symbol is an ... |
| lgsfcl 27216 | Closure of the function ` ... |
| lgsfle1 27217 | The function ` F ` has mag... |
| lgsval2lem 27218 | Lemma for ~ lgsval2 . (Co... |
| lgsval4lem 27219 | Lemma for ~ lgsval4 . (Co... |
| lgscl2 27220 | The Legendre symbol is an ... |
| lgs0 27221 | The Legendre symbol when t... |
| lgscl 27222 | The Legendre symbol is an ... |
| lgsle1 27223 | The Legendre symbol has ab... |
| lgsval2 27224 | The Legendre symbol at a p... |
| lgs2 27225 | The Legendre symbol at ` 2... |
| lgsval3 27226 | The Legendre symbol at an ... |
| lgsvalmod 27227 | The Legendre symbol is equ... |
| lgsval4 27228 | Restate ~ lgsval for nonze... |
| lgsfcl3 27229 | Closure of the function ` ... |
| lgsval4a 27230 | Same as ~ lgsval4 for posi... |
| lgscl1 27231 | The value of the Legendre ... |
| lgsneg 27232 | The Legendre symbol is eit... |
| lgsneg1 27233 | The Legendre symbol for no... |
| lgsmod 27234 | The Legendre (Jacobi) symb... |
| lgsdilem 27235 | Lemma for ~ lgsdi and ~ lg... |
| lgsdir2lem1 27236 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem2 27237 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem3 27238 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem4 27239 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem5 27240 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2 27241 | The Legendre symbol is com... |
| lgsdirprm 27242 | The Legendre symbol is com... |
| lgsdir 27243 | The Legendre symbol is com... |
| lgsdilem2 27244 | Lemma for ~ lgsdi . (Cont... |
| lgsdi 27245 | The Legendre symbol is com... |
| lgsne0 27246 | The Legendre symbol is non... |
| lgsabs1 27247 | The Legendre symbol is non... |
| lgssq 27248 | The Legendre symbol at a s... |
| lgssq2 27249 | The Legendre symbol at a s... |
| lgsprme0 27250 | The Legendre symbol at any... |
| 1lgs 27251 | The Legendre symbol at ` 1... |
| lgs1 27252 | The Legendre symbol at ` 1... |
| lgsmodeq 27253 | The Legendre (Jacobi) symb... |
| lgsmulsqcoprm 27254 | The Legendre (Jacobi) symb... |
| lgsdirnn0 27255 | Variation on ~ lgsdir vali... |
| lgsdinn0 27256 | Variation on ~ lgsdi valid... |
| lgsqrlem1 27257 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem2 27258 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem3 27259 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem4 27260 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem5 27261 | Lemma for ~ lgsqr . (Cont... |
| lgsqr 27262 | The Legendre symbol for od... |
| lgsqrmod 27263 | If the Legendre symbol of ... |
| lgsqrmodndvds 27264 | If the Legendre symbol of ... |
| lgsdchrval 27265 | The Legendre symbol functi... |
| lgsdchr 27266 | The Legendre symbol functi... |
| gausslemma2dlem0a 27267 | Auxiliary lemma 1 for ~ ga... |
| gausslemma2dlem0b 27268 | Auxiliary lemma 2 for ~ ga... |
| gausslemma2dlem0c 27269 | Auxiliary lemma 3 for ~ ga... |
| gausslemma2dlem0d 27270 | Auxiliary lemma 4 for ~ ga... |
| gausslemma2dlem0e 27271 | Auxiliary lemma 5 for ~ ga... |
| gausslemma2dlem0f 27272 | Auxiliary lemma 6 for ~ ga... |
| gausslemma2dlem0g 27273 | Auxiliary lemma 7 for ~ ga... |
| gausslemma2dlem0h 27274 | Auxiliary lemma 8 for ~ ga... |
| gausslemma2dlem0i 27275 | Auxiliary lemma 9 for ~ ga... |
| gausslemma2dlem1a 27276 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem1 27277 | Lemma 1 for ~ gausslemma2d... |
| gausslemma2dlem2 27278 | Lemma 2 for ~ gausslemma2d... |
| gausslemma2dlem3 27279 | Lemma 3 for ~ gausslemma2d... |
| gausslemma2dlem4 27280 | Lemma 4 for ~ gausslemma2d... |
| gausslemma2dlem5a 27281 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem5 27282 | Lemma 5 for ~ gausslemma2d... |
| gausslemma2dlem6 27283 | Lemma 6 for ~ gausslemma2d... |
| gausslemma2dlem7 27284 | Lemma 7 for ~ gausslemma2d... |
| gausslemma2d 27285 | Gauss' Lemma (see also the... |
| lgseisenlem1 27286 | Lemma for ~ lgseisen . If... |
| lgseisenlem2 27287 | Lemma for ~ lgseisen . Th... |
| lgseisenlem3 27288 | Lemma for ~ lgseisen . (C... |
| lgseisenlem4 27289 | Lemma for ~ lgseisen . (C... |
| lgseisen 27290 | Eisenstein's lemma, an exp... |
| lgsquadlem1 27291 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem2 27292 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem3 27293 | Lemma for ~ lgsquad . (Co... |
| lgsquad 27294 | The Law of Quadratic Recip... |
| lgsquad2lem1 27295 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2lem2 27296 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2 27297 | Extend ~ lgsquad to coprim... |
| lgsquad3 27298 | Extend ~ lgsquad2 to integ... |
| m1lgs 27299 | The first supplement to th... |
| 2lgslem1a1 27300 | Lemma 1 for ~ 2lgslem1a . ... |
| 2lgslem1a2 27301 | Lemma 2 for ~ 2lgslem1a . ... |
| 2lgslem1a 27302 | Lemma 1 for ~ 2lgslem1 . ... |
| 2lgslem1b 27303 | Lemma 2 for ~ 2lgslem1 . ... |
| 2lgslem1c 27304 | Lemma 3 for ~ 2lgslem1 . ... |
| 2lgslem1 27305 | Lemma 1 for ~ 2lgs . (Con... |
| 2lgslem2 27306 | Lemma 2 for ~ 2lgs . (Con... |
| 2lgslem3a 27307 | Lemma for ~ 2lgslem3a1 . ... |
| 2lgslem3b 27308 | Lemma for ~ 2lgslem3b1 . ... |
| 2lgslem3c 27309 | Lemma for ~ 2lgslem3c1 . ... |
| 2lgslem3d 27310 | Lemma for ~ 2lgslem3d1 . ... |
| 2lgslem3a1 27311 | Lemma 1 for ~ 2lgslem3 . ... |
| 2lgslem3b1 27312 | Lemma 2 for ~ 2lgslem3 . ... |
| 2lgslem3c1 27313 | Lemma 3 for ~ 2lgslem3 . ... |
| 2lgslem3d1 27314 | Lemma 4 for ~ 2lgslem3 . ... |
| 2lgslem3 27315 | Lemma 3 for ~ 2lgs . (Con... |
| 2lgs2 27316 | The Legendre symbol for ` ... |
| 2lgslem4 27317 | Lemma 4 for ~ 2lgs : speci... |
| 2lgs 27318 | The second supplement to t... |
| 2lgsoddprmlem1 27319 | Lemma 1 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem2 27320 | Lemma 2 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem3a 27321 | Lemma 1 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3b 27322 | Lemma 2 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3c 27323 | Lemma 3 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3d 27324 | Lemma 4 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3 27325 | Lemma 3 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem4 27326 | Lemma 4 for ~ 2lgsoddprm .... |
| 2lgsoddprm 27327 | The second supplement to t... |
| 2sqlem1 27328 | Lemma for ~ 2sq . (Contri... |
| 2sqlem2 27329 | Lemma for ~ 2sq . (Contri... |
| mul2sq 27330 | Fibonacci's identity (actu... |
| 2sqlem3 27331 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem4 27332 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem5 27333 | Lemma for ~ 2sq . If a nu... |
| 2sqlem6 27334 | Lemma for ~ 2sq . If a nu... |
| 2sqlem7 27335 | Lemma for ~ 2sq . (Contri... |
| 2sqlem8a 27336 | Lemma for ~ 2sqlem8 . (Co... |
| 2sqlem8 27337 | Lemma for ~ 2sq . (Contri... |
| 2sqlem9 27338 | Lemma for ~ 2sq . (Contri... |
| 2sqlem10 27339 | Lemma for ~ 2sq . Every f... |
| 2sqlem11 27340 | Lemma for ~ 2sq . (Contri... |
| 2sq 27341 | All primes of the form ` 4... |
| 2sqblem 27342 | Lemma for ~ 2sqb . (Contr... |
| 2sqb 27343 | The converse to ~ 2sq . (... |
| 2sq2 27344 | ` 2 ` is the sum of square... |
| 2sqn0 27345 | If the sum of two squares ... |
| 2sqcoprm 27346 | If the sum of two squares ... |
| 2sqmod 27347 | Given two decompositions o... |
| 2sqmo 27348 | There exists at most one d... |
| 2sqnn0 27349 | All primes of the form ` 4... |
| 2sqnn 27350 | All primes of the form ` 4... |
| addsq2reu 27351 | For each complex number ` ... |
| addsqn2reu 27352 | For each complex number ` ... |
| addsqrexnreu 27353 | For each complex number, t... |
| addsqnreup 27354 | There is no unique decompo... |
| addsq2nreurex 27355 | For each complex number ` ... |
| addsqn2reurex2 27356 | For each complex number ` ... |
| 2sqreulem1 27357 | Lemma 1 for ~ 2sqreu . (C... |
| 2sqreultlem 27358 | Lemma for ~ 2sqreult . (C... |
| 2sqreultblem 27359 | Lemma for ~ 2sqreultb . (... |
| 2sqreunnlem1 27360 | Lemma 1 for ~ 2sqreunn . ... |
| 2sqreunnltlem 27361 | Lemma for ~ 2sqreunnlt . ... |
| 2sqreunnltblem 27362 | Lemma for ~ 2sqreunnltb . ... |
| 2sqreulem2 27363 | Lemma 2 for ~ 2sqreu etc. ... |
| 2sqreulem3 27364 | Lemma 3 for ~ 2sqreu etc. ... |
| 2sqreulem4 27365 | Lemma 4 for ~ 2sqreu et. ... |
| 2sqreunnlem2 27366 | Lemma 2 for ~ 2sqreunn . ... |
| 2sqreu 27367 | There exists a unique deco... |
| 2sqreunn 27368 | There exists a unique deco... |
| 2sqreult 27369 | There exists a unique deco... |
| 2sqreultb 27370 | There exists a unique deco... |
| 2sqreunnlt 27371 | There exists a unique deco... |
| 2sqreunnltb 27372 | There exists a unique deco... |
| 2sqreuop 27373 | There exists a unique deco... |
| 2sqreuopnn 27374 | There exists a unique deco... |
| 2sqreuoplt 27375 | There exists a unique deco... |
| 2sqreuopltb 27376 | There exists a unique deco... |
| 2sqreuopnnlt 27377 | There exists a unique deco... |
| 2sqreuopnnltb 27378 | There exists a unique deco... |
| 2sqreuopb 27379 | There exists a unique deco... |
| chebbnd1lem1 27380 | Lemma for ~ chebbnd1 : sho... |
| chebbnd1lem2 27381 | Lemma for ~ chebbnd1 : Sh... |
| chebbnd1lem3 27382 | Lemma for ~ chebbnd1 : get... |
| chebbnd1 27383 | The Chebyshev bound: The ... |
| chtppilimlem1 27384 | Lemma for ~ chtppilim . (... |
| chtppilimlem2 27385 | Lemma for ~ chtppilim . (... |
| chtppilim 27386 | The ` theta ` function is ... |
| chto1ub 27387 | The ` theta ` function is ... |
| chebbnd2 27388 | The Chebyshev bound, part ... |
| chto1lb 27389 | The ` theta ` function is ... |
| chpchtlim 27390 | The ` psi ` and ` theta ` ... |
| chpo1ub 27391 | The ` psi ` function is up... |
| chpo1ubb 27392 | The ` psi ` function is up... |
| vmadivsum 27393 | The sum of the von Mangold... |
| vmadivsumb 27394 | Give a total bound on the ... |
| rplogsumlem1 27395 | Lemma for ~ rplogsum . (C... |
| rplogsumlem2 27396 | Lemma for ~ rplogsum . Eq... |
| dchrisum0lem1a 27397 | Lemma for ~ dchrisum0lem1 ... |
| rpvmasumlem 27398 | Lemma for ~ rpvmasum . Ca... |
| dchrisumlema 27399 | Lemma for ~ dchrisum . Le... |
| dchrisumlem1 27400 | Lemma for ~ dchrisum . Le... |
| dchrisumlem2 27401 | Lemma for ~ dchrisum . Le... |
| dchrisumlem3 27402 | Lemma for ~ dchrisum . Le... |
| dchrisum 27403 | If ` n e. [ M , +oo ) |-> ... |
| dchrmusumlema 27404 | Lemma for ~ dchrmusum and ... |
| dchrmusum2 27405 | The sum of the Möbius... |
| dchrvmasumlem1 27406 | An alternative expression ... |
| dchrvmasum2lem 27407 | Give an expression for ` l... |
| dchrvmasum2if 27408 | Combine the results of ~ d... |
| dchrvmasumlem2 27409 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlem3 27410 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlema 27411 | Lemma for ~ dchrvmasum and... |
| dchrvmasumiflem1 27412 | Lemma for ~ dchrvmasumif .... |
| dchrvmasumiflem2 27413 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumif 27414 | An asymptotic approximatio... |
| dchrvmaeq0 27415 | The set ` W ` is the colle... |
| dchrisum0fval 27416 | Value of the function ` F ... |
| dchrisum0fmul 27417 | The function ` F ` , the d... |
| dchrisum0ff 27418 | The function ` F ` is a re... |
| dchrisum0flblem1 27419 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flblem2 27420 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flb 27421 | The divisor sum of a real ... |
| dchrisum0fno1 27422 | The sum ` sum_ k <_ x , F ... |
| rpvmasum2 27423 | A partial result along the... |
| dchrisum0re 27424 | Suppose ` X ` is a non-pri... |
| dchrisum0lema 27425 | Lemma for ~ dchrisum0 . A... |
| dchrisum0lem1b 27426 | Lemma for ~ dchrisum0lem1 ... |
| dchrisum0lem1 27427 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2a 27428 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2 27429 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem3 27430 | Lemma for ~ dchrisum0 . (... |
| dchrisum0 27431 | The sum ` sum_ n e. NN , X... |
| dchrisumn0 27432 | The sum ` sum_ n e. NN , X... |
| dchrmusumlem 27433 | The sum of the Möbius... |
| dchrvmasumlem 27434 | The sum of the Möbius... |
| dchrmusum 27435 | The sum of the Möbius... |
| dchrvmasum 27436 | The sum of the von Mangold... |
| rpvmasum 27437 | The sum of the von Mangold... |
| rplogsum 27438 | The sum of ` log p / p ` o... |
| dirith2 27439 | Dirichlet's theorem: there... |
| dirith 27440 | Dirichlet's theorem: there... |
| mudivsum 27441 | Asymptotic formula for ` s... |
| mulogsumlem 27442 | Lemma for ~ mulogsum . (C... |
| mulogsum 27443 | Asymptotic formula for ... |
| logdivsum 27444 | Asymptotic analysis of ... |
| mulog2sumlem1 27445 | Asymptotic formula for ... |
| mulog2sumlem2 27446 | Lemma for ~ mulog2sum . (... |
| mulog2sumlem3 27447 | Lemma for ~ mulog2sum . (... |
| mulog2sum 27448 | Asymptotic formula for ... |
| vmalogdivsum2 27449 | The sum ` sum_ n <_ x , La... |
| vmalogdivsum 27450 | The sum ` sum_ n <_ x , La... |
| 2vmadivsumlem 27451 | Lemma for ~ 2vmadivsum . ... |
| 2vmadivsum 27452 | The sum ` sum_ m n <_ x , ... |
| logsqvma 27453 | A formula for ` log ^ 2 ( ... |
| logsqvma2 27454 | The Möbius inverse of... |
| log2sumbnd 27455 | Bound on the difference be... |
| selberglem1 27456 | Lemma for ~ selberg . Est... |
| selberglem2 27457 | Lemma for ~ selberg . (Co... |
| selberglem3 27458 | Lemma for ~ selberg . Est... |
| selberg 27459 | Selberg's symmetry formula... |
| selbergb 27460 | Convert eventual boundedne... |
| selberg2lem 27461 | Lemma for ~ selberg2 . Eq... |
| selberg2 27462 | Selberg's symmetry formula... |
| selberg2b 27463 | Convert eventual boundedne... |
| chpdifbndlem1 27464 | Lemma for ~ chpdifbnd . (... |
| chpdifbndlem2 27465 | Lemma for ~ chpdifbnd . (... |
| chpdifbnd 27466 | A bound on the difference ... |
| logdivbnd 27467 | A bound on a sum of logs, ... |
| selberg3lem1 27468 | Introduce a log weighting ... |
| selberg3lem2 27469 | Lemma for ~ selberg3 . Eq... |
| selberg3 27470 | Introduce a log weighting ... |
| selberg4lem1 27471 | Lemma for ~ selberg4 . Eq... |
| selberg4 27472 | The Selberg symmetry formu... |
| pntrval 27473 | Define the residual of the... |
| pntrf 27474 | Functionality of the resid... |
| pntrmax 27475 | There is a bound on the re... |
| pntrsumo1 27476 | A bound on a sum over ` R ... |
| pntrsumbnd 27477 | A bound on a sum over ` R ... |
| pntrsumbnd2 27478 | A bound on a sum over ` R ... |
| selbergr 27479 | Selberg's symmetry formula... |
| selberg3r 27480 | Selberg's symmetry formula... |
| selberg4r 27481 | Selberg's symmetry formula... |
| selberg34r 27482 | The sum of ~ selberg3r and... |
| pntsval 27483 | Define the "Selberg functi... |
| pntsf 27484 | Functionality of the Selbe... |
| selbergs 27485 | Selberg's symmetry formula... |
| selbergsb 27486 | Selberg's symmetry formula... |
| pntsval2 27487 | The Selberg function can b... |
| pntrlog2bndlem1 27488 | The sum of ~ selberg3r and... |
| pntrlog2bndlem2 27489 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem3 27490 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem4 27491 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem5 27492 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem6a 27493 | Lemma for ~ pntrlog2bndlem... |
| pntrlog2bndlem6 27494 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bnd 27495 | A bound on ` R ( x ) log ^... |
| pntpbnd1a 27496 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd1 27497 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd2 27498 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd 27499 | Lemma for ~ pnt . Establi... |
| pntibndlem1 27500 | Lemma for ~ pntibnd . (Co... |
| pntibndlem2a 27501 | Lemma for ~ pntibndlem2 . ... |
| pntibndlem2 27502 | Lemma for ~ pntibnd . The... |
| pntibndlem3 27503 | Lemma for ~ pntibnd . Pac... |
| pntibnd 27504 | Lemma for ~ pnt . Establi... |
| pntlemd 27505 | Lemma for ~ pnt . Closure... |
| pntlemc 27506 | Lemma for ~ pnt . Closure... |
| pntlema 27507 | Lemma for ~ pnt . Closure... |
| pntlemb 27508 | Lemma for ~ pnt . Unpack ... |
| pntlemg 27509 | Lemma for ~ pnt . Closure... |
| pntlemh 27510 | Lemma for ~ pnt . Bounds ... |
| pntlemn 27511 | Lemma for ~ pnt . The "na... |
| pntlemq 27512 | Lemma for ~ pntlemj . (Co... |
| pntlemr 27513 | Lemma for ~ pntlemj . (Co... |
| pntlemj 27514 | Lemma for ~ pnt . The ind... |
| pntlemi 27515 | Lemma for ~ pnt . Elimina... |
| pntlemf 27516 | Lemma for ~ pnt . Add up ... |
| pntlemk 27517 | Lemma for ~ pnt . Evaluat... |
| pntlemo 27518 | Lemma for ~ pnt . Combine... |
| pntleme 27519 | Lemma for ~ pnt . Package... |
| pntlem3 27520 | Lemma for ~ pnt . Equatio... |
| pntlemp 27521 | Lemma for ~ pnt . Wrappin... |
| pntleml 27522 | Lemma for ~ pnt . Equatio... |
| pnt3 27523 | The Prime Number Theorem, ... |
| pnt2 27524 | The Prime Number Theorem, ... |
| pnt 27525 | The Prime Number Theorem: ... |
| abvcxp 27526 | Raising an absolute value ... |
| padicfval 27527 | Value of the p-adic absolu... |
| padicval 27528 | Value of the p-adic absolu... |
| ostth2lem1 27529 | Lemma for ~ ostth2 , altho... |
| qrngbas 27530 | The base set of the field ... |
| qdrng 27531 | The rationals form a divis... |
| qrng0 27532 | The zero element of the fi... |
| qrng1 27533 | The unity element of the f... |
| qrngneg 27534 | The additive inverse in th... |
| qrngdiv 27535 | The division operation in ... |
| qabvle 27536 | By using induction on ` N ... |
| qabvexp 27537 | Induct the product rule ~ ... |
| ostthlem1 27538 | Lemma for ~ ostth . If tw... |
| ostthlem2 27539 | Lemma for ~ ostth . Refin... |
| qabsabv 27540 | The regular absolute value... |
| padicabv 27541 | The p-adic absolute value ... |
| padicabvf 27542 | The p-adic absolute value ... |
| padicabvcxp 27543 | All positive powers of the... |
| ostth1 27544 | - Lemma for ~ ostth : triv... |
| ostth2lem2 27545 | Lemma for ~ ostth2 . (Con... |
| ostth2lem3 27546 | Lemma for ~ ostth2 . (Con... |
| ostth2lem4 27547 | Lemma for ~ ostth2 . (Con... |
| ostth2 27548 | - Lemma for ~ ostth : regu... |
| ostth3 27549 | - Lemma for ~ ostth : p-ad... |
| ostth 27550 | Ostrowski's theorem, which... |
| elno 27557 | Membership in the surreals... |
| elnoOLD 27558 | Obsolete version of ~ elno... |
| sltval 27559 | The value of the surreal l... |
| bdayval 27560 | The value of the birthday ... |
| nofun 27561 | A surreal is a function. ... |
| nodmon 27562 | The domain of a surreal is... |
| norn 27563 | The range of a surreal is ... |
| nofnbday 27564 | A surreal is a function ov... |
| nodmord 27565 | The domain of a surreal ha... |
| elno2 27566 | An alternative condition f... |
| elno3 27567 | Another condition for memb... |
| sltval2 27568 | Alternate expression for s... |
| nofv 27569 | The function value of a su... |
| nosgnn0 27570 | ` (/) ` is not a surreal s... |
| nosgnn0i 27571 | If ` X ` is a surreal sign... |
| noreson 27572 | The restriction of a surre... |
| sltintdifex 27573 |
If ` A |
| sltres 27574 | If the restrictions of two... |
| noxp1o 27575 | The Cartesian product of a... |
| noseponlem 27576 | Lemma for ~ nosepon . Con... |
| nosepon 27577 | Given two unequal surreals... |
| noextend 27578 | Extending a surreal by one... |
| noextendseq 27579 | Extend a surreal by a sequ... |
| noextenddif 27580 | Calculate the place where ... |
| noextendlt 27581 | Extending a surreal with a... |
| noextendgt 27582 | Extending a surreal with a... |
| nolesgn2o 27583 | Given ` A ` less-than or e... |
| nolesgn2ores 27584 | Given ` A ` less-than or e... |
| nogesgn1o 27585 | Given ` A ` greater than o... |
| nogesgn1ores 27586 | Given ` A ` greater than o... |
| sltsolem1 27587 | Lemma for ~ sltso . The "... |
| sltso 27588 | Less-than totally orders t... |
| bdayfo 27589 | The birthday function maps... |
| fvnobday 27590 | The value of a surreal at ... |
| nosepnelem 27591 | Lemma for ~ nosepne . (Co... |
| nosepne 27592 | The value of two non-equal... |
| nosep1o 27593 | If the value of a surreal ... |
| nosep2o 27594 | If the value of a surreal ... |
| nosepdmlem 27595 | Lemma for ~ nosepdm . (Co... |
| nosepdm 27596 | The first place two surrea... |
| nosepeq 27597 | The values of two surreals... |
| nosepssdm 27598 | Given two non-equal surrea... |
| nodenselem4 27599 | Lemma for ~ nodense . Sho... |
| nodenselem5 27600 | Lemma for ~ nodense . If ... |
| nodenselem6 27601 | The restriction of a surre... |
| nodenselem7 27602 | Lemma for ~ nodense . ` A ... |
| nodenselem8 27603 | Lemma for ~ nodense . Giv... |
| nodense 27604 | Given two distinct surreal... |
| bdayimaon 27605 | Lemma for full-eta propert... |
| nolt02olem 27606 | Lemma for ~ nolt02o . If ... |
| nolt02o 27607 | Given ` A ` less-than ` B ... |
| nogt01o 27608 | Given ` A ` greater than `... |
| noresle 27609 | Restriction law for surrea... |
| nomaxmo 27610 | A class of surreals has at... |
| nominmo 27611 | A class of surreals has at... |
| nosupprefixmo 27612 | In any class of surreals, ... |
| noinfprefixmo 27613 | In any class of surreals, ... |
| nosupcbv 27614 | Lemma to change bound vari... |
| nosupno 27615 | The next several theorems ... |
| nosupdm 27616 | The domain of the surreal ... |
| nosupbday 27617 | Birthday bounding law for ... |
| nosupfv 27618 | The value of surreal supre... |
| nosupres 27619 | A restriction law for surr... |
| nosupbnd1lem1 27620 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1lem2 27621 | Lemma for ~ nosupbnd1 . W... |
| nosupbnd1lem3 27622 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem4 27623 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem5 27624 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem6 27625 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1 27626 | Bounding law from below fo... |
| nosupbnd2lem1 27627 | Bounding law from above wh... |
| nosupbnd2 27628 | Bounding law from above fo... |
| noinfcbv 27629 | Change bound variables for... |
| noinfno 27630 | The next several theorems ... |
| noinfdm 27631 | Next, we calculate the dom... |
| noinfbday 27632 | Birthday bounding law for ... |
| noinffv 27633 | The value of surreal infim... |
| noinfres 27634 | The restriction of surreal... |
| noinfbnd1lem1 27635 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1lem2 27636 | Lemma for ~ noinfbnd1 . W... |
| noinfbnd1lem3 27637 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem4 27638 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem5 27639 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem6 27640 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1 27641 | Bounding law from above fo... |
| noinfbnd2lem1 27642 | Bounding law from below wh... |
| noinfbnd2 27643 | Bounding law from below fo... |
| nosupinfsep 27644 | Given two sets of surreals... |
| noetasuplem1 27645 | Lemma for ~ noeta . Estab... |
| noetasuplem2 27646 | Lemma for ~ noeta . The r... |
| noetasuplem3 27647 | Lemma for ~ noeta . ` Z ` ... |
| noetasuplem4 27648 | Lemma for ~ noeta . When ... |
| noetainflem1 27649 | Lemma for ~ noeta . Estab... |
| noetainflem2 27650 | Lemma for ~ noeta . The r... |
| noetainflem3 27651 | Lemma for ~ noeta . ` W ` ... |
| noetainflem4 27652 | Lemma for ~ noeta . If ` ... |
| noetalem1 27653 | Lemma for ~ noeta . Eithe... |
| noetalem2 27654 | Lemma for ~ noeta . The f... |
| noeta 27655 | The full-eta axiom for the... |
| sltirr 27658 | Surreal less-than is irref... |
| slttr 27659 | Surreal less-than is trans... |
| sltasym 27660 | Surreal less-than is asymm... |
| sltlin 27661 | Surreal less-than obeys tr... |
| slttrieq2 27662 | Trichotomy law for surreal... |
| slttrine 27663 | Trichotomy law for surreal... |
| slenlt 27664 | Surreal less-than or equal... |
| sltnle 27665 | Surreal less-than in terms... |
| sleloe 27666 | Surreal less-than or equal... |
| sletri3 27667 | Trichotomy law for surreal... |
| sltletr 27668 | Surreal transitive law. (... |
| slelttr 27669 | Surreal transitive law. (... |
| sletr 27670 | Surreal transitive law. (... |
| slttrd 27671 | Surreal less-than is trans... |
| sltletrd 27672 | Surreal less-than is trans... |
| slelttrd 27673 | Surreal less-than is trans... |
| sletrd 27674 | Surreal less-than or equal... |
| slerflex 27675 | Surreal less-than or equal... |
| sletric 27676 | Surreal trichotomy law. (... |
| maxs1 27677 | A surreal is less than or ... |
| maxs2 27678 | A surreal is less than or ... |
| mins1 27679 | The minimum of two surreal... |
| mins2 27680 | The minimum of two surreal... |
| sltled 27681 | Surreal less-than implies ... |
| sltne 27682 | Surreal less-than implies ... |
| sltlend 27683 | Surreal less-than in terms... |
| bdayfun 27684 | The birthday function is a... |
| bdayfn 27685 | The birthday function is a... |
| bdaydm 27686 | The birthday function's do... |
| bdayrn 27687 | The birthday function's ra... |
| bdayelon 27688 | The value of the birthday ... |
| nocvxminlem 27689 | Lemma for ~ nocvxmin . Gi... |
| nocvxmin 27690 | Given a nonempty convex cl... |
| noprc 27691 | The surreal numbers are a ... |
| noeta2 27696 | A version of ~ noeta with ... |
| brsslt 27697 | Binary relation form of th... |
| ssltex1 27698 | The first argument of surr... |
| ssltex2 27699 | The second argument of sur... |
| ssltss1 27700 | The first argument of surr... |
| ssltss2 27701 | The second argument of sur... |
| ssltsep 27702 | The separation property of... |
| ssltd 27703 | Deduce surreal set less-th... |
| ssltsn 27704 | Surreal set less-than of t... |
| ssltsepc 27705 | Two elements of separated ... |
| ssltsepcd 27706 | Two elements of separated ... |
| sssslt1 27707 | Relation between surreal s... |
| sssslt2 27708 | Relation between surreal s... |
| nulsslt 27709 | The empty set is less-than... |
| nulssgt 27710 | The empty set is greater t... |
| conway 27711 | Conway's Simplicity Theore... |
| scutval 27712 | The value of the surreal c... |
| scutcut 27713 | Cut properties of the surr... |
| scutcl 27714 | Closure law for surreal cu... |
| scutcld 27715 | Closure law for surreal cu... |
| scutbday 27716 | The birthday of the surrea... |
| eqscut 27717 | Condition for equality to ... |
| eqscut2 27718 | Condition for equality to ... |
| sslttr 27719 | Transitive law for surreal... |
| ssltun1 27720 | Union law for surreal set ... |
| ssltun2 27721 | Union law for surreal set ... |
| scutun12 27722 | Union law for surreal cuts... |
| dmscut 27723 | The domain of the surreal ... |
| scutf 27724 | Functionality statement fo... |
| etasslt 27725 | A restatement of ~ noeta u... |
| etasslt2 27726 | A version of ~ etasslt wit... |
| scutbdaybnd 27727 | An upper bound on the birt... |
| scutbdaybnd2 27728 | An upper bound on the birt... |
| scutbdaybnd2lim 27729 | An upper bound on the birt... |
| scutbdaylt 27730 | If a surreal lies in a gap... |
| slerec 27731 | A comparison law for surre... |
| sltrec 27732 | A comparison law for surre... |
| ssltdisj 27733 | If ` A ` preceeds ` B ` , ... |
| 0sno 27738 | Surreal zero is a surreal.... |
| 1sno 27739 | Surreal one is a surreal. ... |
| bday0s 27740 | Calculate the birthday of ... |
| 0slt1s 27741 | Surreal zero is less than ... |
| bday0b 27742 | The only surreal with birt... |
| bday1s 27743 | The birthday of surreal on... |
| cuteq0 27744 | Condition for a surreal cu... |
| cutneg 27745 | The simplest number greate... |
| cuteq1 27746 | Condition for a surreal cu... |
| sgt0ne0 27747 | A positive surreal is not ... |
| sgt0ne0d 27748 | A positive surreal is not ... |
| 1sne0s 27749 | Surreal zero does not equa... |
| madeval 27760 | The value of the made by f... |
| madeval2 27761 | Alternative characterizati... |
| oldval 27762 | The value of the old optio... |
| newval 27763 | The value of the new optio... |
| madef 27764 | The made function is a fun... |
| oldf 27765 | The older function is a fu... |
| newf 27766 | The new function is a func... |
| old0 27767 | No surreal is older than `... |
| madessno 27768 | Made sets are surreals. (... |
| oldssno 27769 | Old sets are surreals. (C... |
| newssno 27770 | New sets are surreals. (C... |
| leftval 27771 | The value of the left opti... |
| rightval 27772 | The value of the right opt... |
| elleft 27773 | Membership in the left set... |
| elright 27774 | Membership in the right se... |
| leftlt 27775 | A member of a surreal's le... |
| rightgt 27776 | A member of a surreal's ri... |
| leftf 27777 | The functionality of the l... |
| rightf 27778 | The functionality of the r... |
| elmade 27779 | Membership in the made fun... |
| elmade2 27780 | Membership in the made fun... |
| elold 27781 | Membership in an old set. ... |
| ssltleft 27782 | A surreal is greater than ... |
| ssltright 27783 | A surreal is less than its... |
| lltropt 27784 | The left options of a surr... |
| made0 27785 | The only surreal made on d... |
| new0 27786 | The only surreal new on da... |
| old1 27787 | The only surreal older tha... |
| madess 27788 | If ` A ` is less than or e... |
| oldssmade 27789 | The older-than set is a su... |
| leftssold 27790 | The left options are a sub... |
| rightssold 27791 | The right options are a su... |
| leftssno 27792 | The left set of a surreal ... |
| rightssno 27793 | The right set of a surreal... |
| madecut 27794 | Given a section that is a ... |
| madeun 27795 | The made set is the union ... |
| madeoldsuc 27796 | The made set is the old se... |
| oldsuc 27797 | The value of the old set a... |
| oldlim 27798 | The value of the old set a... |
| madebdayim 27799 | If a surreal is a member o... |
| oldbdayim 27800 | If ` X ` is in the old set... |
| oldirr 27801 | No surreal is a member of ... |
| leftirr 27802 | No surreal is a member of ... |
| rightirr 27803 | No surreal is a member of ... |
| left0s 27804 | The left set of ` 0s ` is ... |
| right0s 27805 | The right set of ` 0s ` is... |
| left1s 27806 | The left set of ` 1s ` is ... |
| right1s 27807 | The right set of ` 1s ` is... |
| lrold 27808 | The union of the left and ... |
| madebdaylemold 27809 | Lemma for ~ madebday . If... |
| madebdaylemlrcut 27810 | Lemma for ~ madebday . If... |
| madebday 27811 | A surreal is part of the s... |
| oldbday 27812 | A surreal is part of the s... |
| newbday 27813 | A surreal is an element of... |
| newbdayim 27814 | One direction of the bicon... |
| lrcut 27815 | A surreal is equal to the ... |
| scutfo 27816 | The surreal cut function i... |
| sltn0 27817 | If ` X ` is less than ` Y ... |
| lruneq 27818 | If two surreals share a bi... |
| sltlpss 27819 | If two surreals share a bi... |
| slelss 27820 | If two surreals ` A ` and ... |
| 0elold 27821 | Zero is in the old set of ... |
| 0elleft 27822 | Zero is in the left set of... |
| 0elright 27823 | Zero is in the right set o... |
| madefi 27824 | The made set of an ordinal... |
| oldfi 27825 | The old set of an ordinal ... |
| cofsslt 27826 | If every element of ` A ` ... |
| coinitsslt 27827 | If ` B ` is coinitial with... |
| cofcut1 27828 | If ` C ` is cofinal with `... |
| cofcut1d 27829 | If ` C ` is cofinal with `... |
| cofcut2 27830 | If ` A ` and ` C ` are mut... |
| cofcut2d 27831 | If ` A ` and ` C ` are mut... |
| cofcutr 27832 | If ` X ` is the cut of ` A... |
| cofcutr1d 27833 | If ` X ` is the cut of ` A... |
| cofcutr2d 27834 | If ` X ` is the cut of ` A... |
| cofcutrtime 27835 | If ` X ` is the cut of ` A... |
| cofcutrtime1d 27836 | If ` X ` is a timely cut o... |
| cofcutrtime2d 27837 | If ` X ` is a timely cut o... |
| cofss 27838 | Cofinality for a subset. ... |
| coiniss 27839 | Coinitiality for a subset.... |
| cutlt 27840 | Eliminating all elements b... |
| cutpos 27841 | Reduce the elements of a c... |
| cutmax 27842 | If ` A ` has a maximum, th... |
| cutmin 27843 | If ` B ` has a minimum, th... |
| lrrecval 27846 | The next step in the devel... |
| lrrecval2 27847 | Next, we establish an alte... |
| lrrecpo 27848 | Now, we establish that ` R... |
| lrrecse 27849 | Next, we show that ` R ` i... |
| lrrecfr 27850 | Now we show that ` R ` is ... |
| lrrecpred 27851 | Finally, we calculate the ... |
| noinds 27852 | Induction principle for a ... |
| norecfn 27853 | Surreal recursion over one... |
| norecov 27854 | Calculate the value of the... |
| noxpordpo 27857 | To get through most of the... |
| noxpordfr 27858 | Next we establish the foun... |
| noxpordse 27859 | Next we establish the set-... |
| noxpordpred 27860 | Next we calculate the pred... |
| no2indslem 27861 | Double induction on surrea... |
| no2inds 27862 | Double induction on surrea... |
| norec2fn 27863 | The double-recursion opera... |
| norec2ov 27864 | The value of the double-re... |
| no3inds 27865 | Triple induction over surr... |
| addsfn 27868 | Surreal addition is a func... |
| addsval 27869 | The value of surreal addit... |
| addsval2 27870 | The value of surreal addit... |
| addsrid 27871 | Surreal addition to zero i... |
| addsridd 27872 | Surreal addition to zero i... |
| addscom 27873 | Surreal addition commutes.... |
| addscomd 27874 | Surreal addition commutes.... |
| addslid 27875 | Surreal addition to zero i... |
| addsproplem1 27876 | Lemma for surreal addition... |
| addsproplem2 27877 | Lemma for surreal addition... |
| addsproplem3 27878 | Lemma for surreal addition... |
| addsproplem4 27879 | Lemma for surreal addition... |
| addsproplem5 27880 | Lemma for surreal addition... |
| addsproplem6 27881 | Lemma for surreal addition... |
| addsproplem7 27882 | Lemma for surreal addition... |
| addsprop 27883 | Inductively show that surr... |
| addscutlem 27884 | Lemma for ~ addscut . Sho... |
| addscut 27885 | Demonstrate the cut proper... |
| addscut2 27886 | Show that the cut involved... |
| addscld 27887 | Surreal numbers are closed... |
| addscl 27888 | Surreal numbers are closed... |
| addsf 27889 | Function statement for sur... |
| addsfo 27890 | Surreal addition is onto. ... |
| peano2no 27891 | A theorem for surreals tha... |
| sltadd1im 27892 | Surreal less-than is prese... |
| sltadd2im 27893 | Surreal less-than is prese... |
| sleadd1im 27894 | Surreal less-than or equal... |
| sleadd2im 27895 | Surreal less-than or equal... |
| sleadd1 27896 | Addition to both sides of ... |
| sleadd2 27897 | Addition to both sides of ... |
| sltadd2 27898 | Addition to both sides of ... |
| sltadd1 27899 | Addition to both sides of ... |
| addscan2 27900 | Cancellation law for surre... |
| addscan1 27901 | Cancellation law for surre... |
| sleadd1d 27902 | Addition to both sides of ... |
| sleadd2d 27903 | Addition to both sides of ... |
| sltadd2d 27904 | Addition to both sides of ... |
| sltadd1d 27905 | Addition to both sides of ... |
| addscan2d 27906 | Cancellation law for surre... |
| addscan1d 27907 | Cancellation law for surre... |
| addsuniflem 27908 | Lemma for ~ addsunif . St... |
| addsunif 27909 | Uniformity theorem for sur... |
| addsasslem1 27910 | Lemma for addition associa... |
| addsasslem2 27911 | Lemma for addition associa... |
| addsass 27912 | Surreal addition is associ... |
| addsassd 27913 | Surreal addition is associ... |
| adds32d 27914 | Commutative/associative la... |
| adds12d 27915 | Commutative/associative la... |
| adds4d 27916 | Rearrangement of four term... |
| adds42d 27917 | Rearrangement of four term... |
| sltaddpos1d 27918 | Addition of a positive num... |
| sltaddpos2d 27919 | Addition of a positive num... |
| slt2addd 27920 | Adding both sides of two s... |
| addsgt0d 27921 | The sum of two positive su... |
| sltp1d 27922 | A surreal is less than its... |
| addsbdaylem 27923 | Lemma for ~ addsbday . (C... |
| addsbday 27924 | The birthday of the sum of... |
| negsfn 27929 | Surreal negation is a func... |
| subsfn 27930 | Surreal subtraction is a f... |
| negsval 27931 | The value of the surreal n... |
| negs0s 27932 | Negative surreal zero is s... |
| negs1s 27933 | An expression for negative... |
| negsproplem1 27934 | Lemma for surreal negation... |
| negsproplem2 27935 | Lemma for surreal negation... |
| negsproplem3 27936 | Lemma for surreal negation... |
| negsproplem4 27937 | Lemma for surreal negation... |
| negsproplem5 27938 | Lemma for surreal negation... |
| negsproplem6 27939 | Lemma for surreal negation... |
| negsproplem7 27940 | Lemma for surreal negation... |
| negsprop 27941 | Show closure and ordering ... |
| negscl 27942 | The surreals are closed un... |
| negscld 27943 | The surreals are closed un... |
| sltnegim 27944 | The forward direction of t... |
| negscut 27945 | The cut properties of surr... |
| negscut2 27946 | The cut that defines surre... |
| negsid 27947 | Surreal addition of a numb... |
| negsidd 27948 | Surreal addition of a numb... |
| negsex 27949 | Every surreal has a negati... |
| negnegs 27950 | A surreal is equal to the ... |
| sltneg 27951 | Negative of both sides of ... |
| sleneg 27952 | Negative of both sides of ... |
| sltnegd 27953 | Negative of both sides of ... |
| slenegd 27954 | Negative of both sides of ... |
| negs11 27955 | Surreal negation is one-to... |
| negsdi 27956 | Distribution of surreal ne... |
| slt0neg2d 27957 | Comparison of a surreal an... |
| negsf 27958 | Function statement for sur... |
| negsfo 27959 | Function statement for sur... |
| negsf1o 27960 | Surreal negation is a bije... |
| negsunif 27961 | Uniformity property for su... |
| negsbdaylem 27962 | Lemma for ~ negsbday . Bo... |
| negsbday 27963 | Negation of a surreal numb... |
| subsval 27964 | The value of surreal subtr... |
| subsvald 27965 | The value of surreal subtr... |
| subscl 27966 | Closure law for surreal su... |
| subscld 27967 | Closure law for surreal su... |
| subsf 27968 | Function statement for sur... |
| subsfo 27969 | Surreal subtraction is an ... |
| negsval2 27970 | Surreal negation in terms ... |
| negsval2d 27971 | Surreal negation in terms ... |
| subsid1 27972 | Identity law for subtracti... |
| subsid 27973 | Subtraction of a surreal f... |
| subadds 27974 | Relationship between addit... |
| subaddsd 27975 | Relationship between addit... |
| pncans 27976 | Cancellation law for surre... |
| pncan3s 27977 | Subtraction and addition o... |
| pncan2s 27978 | Cancellation law for surre... |
| npcans 27979 | Cancellation law for surre... |
| sltsub1 27980 | Subtraction from both side... |
| sltsub2 27981 | Subtraction from both side... |
| sltsub1d 27982 | Subtraction from both side... |
| sltsub2d 27983 | Subtraction from both side... |
| negsubsdi2d 27984 | Distribution of negative o... |
| addsubsassd 27985 | Associative-type law for s... |
| addsubsd 27986 | Law for surreal addition a... |
| sltsubsubbd 27987 | Equivalence for the surrea... |
| sltsubsub2bd 27988 | Equivalence for the surrea... |
| sltsubsub3bd 27989 | Equivalence for the surrea... |
| slesubsubbd 27990 | Equivalence for the surrea... |
| slesubsub2bd 27991 | Equivalence for the surrea... |
| slesubsub3bd 27992 | Equivalence for the surrea... |
| sltsubaddd 27993 | Surreal less-than relation... |
| sltsubadd2d 27994 | Surreal less-than relation... |
| sltaddsubd 27995 | Surreal less-than relation... |
| sltaddsub2d 27996 | Surreal less-than relation... |
| slesubaddd 27997 | Surreal less-than or equal... |
| subsubs4d 27998 | Law for double surreal sub... |
| subsubs2d 27999 | Law for double surreal sub... |
| nncansd 28000 | Cancellation law for surre... |
| posdifsd 28001 | Comparison of two surreals... |
| sltsubposd 28002 | Subtraction of a positive ... |
| subsge0d 28003 | Non-negative subtraction. ... |
| addsubs4d 28004 | Rearrangement of four term... |
| sltm1d 28005 | A surreal is greater than ... |
| subscan1d 28006 | Cancellation law for surre... |
| subscan2d 28007 | Cancellation law for surre... |
| subseq0d 28008 | The difference between two... |
| mulsfn 28011 | Surreal multiplication is ... |
| mulsval 28012 | The value of surreal multi... |
| mulsval2lem 28013 | Lemma for ~ mulsval2 . Ch... |
| mulsval2 28014 | The value of surreal multi... |
| muls01 28015 | Surreal multiplication by ... |
| mulsrid 28016 | Surreal one is a right ide... |
| mulsridd 28017 | Surreal one is a right ide... |
| mulsproplemcbv 28018 | Lemma for surreal multipli... |
| mulsproplem1 28019 | Lemma for surreal multipli... |
| mulsproplem2 28020 | Lemma for surreal multipli... |
| mulsproplem3 28021 | Lemma for surreal multipli... |
| mulsproplem4 28022 | Lemma for surreal multipli... |
| mulsproplem5 28023 | Lemma for surreal multipli... |
| mulsproplem6 28024 | Lemma for surreal multipli... |
| mulsproplem7 28025 | Lemma for surreal multipli... |
| mulsproplem8 28026 | Lemma for surreal multipli... |
| mulsproplem9 28027 | Lemma for surreal multipli... |
| mulsproplem10 28028 | Lemma for surreal multipli... |
| mulsproplem11 28029 | Lemma for surreal multipli... |
| mulsproplem12 28030 | Lemma for surreal multipli... |
| mulsproplem13 28031 | Lemma for surreal multipli... |
| mulsproplem14 28032 | Lemma for surreal multipli... |
| mulsprop 28033 | Surreals are closed under ... |
| mulscutlem 28034 | Lemma for ~ mulscut . Sta... |
| mulscut 28035 | Show the cut properties of... |
| mulscut2 28036 | Show that the cut involved... |
| mulscl 28037 | The surreals are closed un... |
| mulscld 28038 | The surreals are closed un... |
| sltmul 28039 | An ordering relationship f... |
| sltmuld 28040 | An ordering relationship f... |
| slemuld 28041 | An ordering relationship f... |
| mulscom 28042 | Surreal multiplication com... |
| mulscomd 28043 | Surreal multiplication com... |
| muls02 28044 | Surreal multiplication by ... |
| mulslid 28045 | Surreal one is a left iden... |
| mulslidd 28046 | Surreal one is a left iden... |
| mulsgt0 28047 | The product of two positiv... |
| mulsgt0d 28048 | The product of two positiv... |
| mulsge0d 28049 | The product of two non-neg... |
| ssltmul1 28050 | One surreal set less-than ... |
| ssltmul2 28051 | One surreal set less-than ... |
| mulsuniflem 28052 | Lemma for ~ mulsunif . St... |
| mulsunif 28053 | Surreal multiplication has... |
| addsdilem1 28054 | Lemma for surreal distribu... |
| addsdilem2 28055 | Lemma for surreal distribu... |
| addsdilem3 28056 | Lemma for ~ addsdi . Show... |
| addsdilem4 28057 | Lemma for ~ addsdi . Show... |
| addsdi 28058 | Distributive law for surre... |
| addsdid 28059 | Distributive law for surre... |
| addsdird 28060 | Distributive law for surre... |
| subsdid 28061 | Distribution of surreal mu... |
| subsdird 28062 | Distribution of surreal mu... |
| mulnegs1d 28063 | Product with negative is n... |
| mulnegs2d 28064 | Product with negative is n... |
| mul2negsd 28065 | Surreal product of two neg... |
| mulsasslem1 28066 | Lemma for ~ mulsass . Exp... |
| mulsasslem2 28067 | Lemma for ~ mulsass . Exp... |
| mulsasslem3 28068 | Lemma for ~ mulsass . Dem... |
| mulsass 28069 | Associative law for surrea... |
| mulsassd 28070 | Associative law for surrea... |
| muls4d 28071 | Rearrangement of four surr... |
| mulsunif2lem 28072 | Lemma for ~ mulsunif2 . S... |
| mulsunif2 28073 | Alternate expression for s... |
| sltmul2 28074 | Multiplication of both sid... |
| sltmul2d 28075 | Multiplication of both sid... |
| sltmul1d 28076 | Multiplication of both sid... |
| slemul2d 28077 | Multiplication of both sid... |
| slemul1d 28078 | Multiplication of both sid... |
| sltmulneg1d 28079 | Multiplication of both sid... |
| sltmulneg2d 28080 | Multiplication of both sid... |
| mulscan2dlem 28081 | Lemma for ~ mulscan2d . C... |
| mulscan2d 28082 | Cancellation of surreal mu... |
| mulscan1d 28083 | Cancellation of surreal mu... |
| muls12d 28084 | Commutative/associative la... |
| slemul1ad 28085 | Multiplication of both sid... |
| sltmul12ad 28086 | Comparison of the product ... |
| divsmo 28087 | Uniqueness of surreal inve... |
| muls0ord 28088 | If a surreal product is ze... |
| mulsne0bd 28089 | The product of two non-zer... |
| divsval 28092 | The value of surreal divis... |
| norecdiv 28093 | If a surreal has a recipro... |
| noreceuw 28094 | If a surreal has a recipro... |
| recsne0 28095 | If a surreal has a recipro... |
| divsmulw 28096 | Relationship between surre... |
| divsmulwd 28097 | Relationship between surre... |
| divsclw 28098 | Weak division closure law.... |
| divsclwd 28099 | Weak division closure law.... |
| divscan2wd 28100 | A weak cancellation law fo... |
| divscan1wd 28101 | A weak cancellation law fo... |
| sltdivmulwd 28102 | Surreal less-than relation... |
| sltdivmul2wd 28103 | Surreal less-than relation... |
| sltmuldivwd 28104 | Surreal less-than relation... |
| sltmuldiv2wd 28105 | Surreal less-than relation... |
| divsasswd 28106 | An associative law for sur... |
| divs1 28107 | A surreal divided by one i... |
| precsexlemcbv 28108 | Lemma for surreal reciproc... |
| precsexlem1 28109 | Lemma for surreal reciproc... |
| precsexlem2 28110 | Lemma for surreal reciproc... |
| precsexlem3 28111 | Lemma for surreal reciproc... |
| precsexlem4 28112 | Lemma for surreal reciproc... |
| precsexlem5 28113 | Lemma for surreal reciproc... |
| precsexlem6 28114 | Lemma for surreal reciproc... |
| precsexlem7 28115 | Lemma for surreal reciproc... |
| precsexlem8 28116 | Lemma for surreal reciproc... |
| precsexlem9 28117 | Lemma for surreal reciproc... |
| precsexlem10 28118 | Lemma for surreal reciproc... |
| precsexlem11 28119 | Lemma for surreal reciproc... |
| precsex 28120 | Every positive surreal has... |
| recsex 28121 | A non-zero surreal has a r... |
| recsexd 28122 | A non-zero surreal has a r... |
| divsmul 28123 | Relationship between surre... |
| divsmuld 28124 | Relationship between surre... |
| divscl 28125 | Surreal division closure l... |
| divscld 28126 | Surreal division closure l... |
| divscan2d 28127 | A cancellation law for sur... |
| divscan1d 28128 | A cancellation law for sur... |
| sltdivmuld 28129 | Surreal less-than relation... |
| sltdivmul2d 28130 | Surreal less-than relation... |
| sltmuldivd 28131 | Surreal less-than relation... |
| sltmuldiv2d 28132 | Surreal less-than relation... |
| divsassd 28133 | An associative law for sur... |
| divmuldivsd 28134 | Multiplication of two surr... |
| divdivs1d 28135 | Surreal division into a fr... |
| divsrecd 28136 | Relationship between surre... |
| divsdird 28137 | Distribution of surreal di... |
| divscan3d 28138 | A cancellation law for sur... |
| abssval 28141 | The value of surreal absol... |
| absscl 28142 | Closure law for surreal ab... |
| abssid 28143 | The absolute value of a no... |
| abs0s 28144 | The absolute value of surr... |
| abssnid 28145 | For a negative surreal, it... |
| absmuls 28146 | Surreal absolute value dis... |
| abssge0 28147 | The absolute value of a su... |
| abssor 28148 | The absolute value of a su... |
| abssneg 28149 | Surreal absolute value of ... |
| sleabs 28150 | A surreal is less than or ... |
| absslt 28151 | Surreal absolute value and... |
| elons 28154 | Membership in the class of... |
| onssno 28155 | The surreal ordinals are a... |
| onsno 28156 | A surreal ordinal is a sur... |
| 0ons 28157 | Surreal zero is a surreal ... |
| 1ons 28158 | Surreal one is a surreal o... |
| elons2 28159 | A surreal is ordinal iff i... |
| elons2d 28160 | The cut of any set of surr... |
| onsleft 28161 | The left set of a surreal ... |
| sltonold 28162 | The class of ordinals less... |
| sltonex 28163 | The class of ordinals less... |
| onscutleft 28164 | A surreal ordinal is equal... |
| onscutlt 28165 | A surreal ordinal is the s... |
| bday11on 28166 | The birthday function is o... |
| onnolt 28167 | If a surreal ordinal is le... |
| onslt 28168 | Less-than is the same as b... |
| onsiso 28169 | The birthday function rest... |
| onswe 28170 | Surreal less-than well-ord... |
| onsse 28171 | Surreal less-than is set-l... |
| onsis 28172 | Transfinite induction sche... |
| bdayon 28173 | The birthday of a surreal ... |
| onaddscl 28174 | The surreal ordinals are c... |
| onmulscl 28175 | The surreal ordinals are c... |
| peano2ons 28176 | The successor of a surreal... |
| seqsex 28179 | Existence of the surreal s... |
| seqseq123d 28180 | Equality deduction for the... |
| nfseqs 28181 | Hypothesis builder for the... |
| seqsval 28182 | The value of the surreal s... |
| noseqex 28183 | The next several theorems ... |
| noseq0 28184 | The surreal ` A ` is a mem... |
| noseqp1 28185 | One plus an element of ` Z... |
| noseqind 28186 | Peano's inductive postulat... |
| noseqinds 28187 | Induction schema for surre... |
| noseqssno 28188 | A surreal sequence is a su... |
| noseqno 28189 | An element of a surreal se... |
| om2noseq0 28190 | The mapping ` G ` is a one... |
| om2noseqsuc 28191 | The value of ` G ` at a su... |
| om2noseqfo 28192 | Function statement for ` G... |
| om2noseqlt 28193 | Surreal less-than relation... |
| om2noseqlt2 28194 | The mapping ` G ` preserve... |
| om2noseqf1o 28195 | ` G ` is a bijection. (Co... |
| om2noseqiso 28196 | ` G ` is an isomorphism fr... |
| om2noseqoi 28197 | An alternative definition ... |
| om2noseqrdg 28198 | A helper lemma for the val... |
| noseqrdglem 28199 | A helper lemma for the val... |
| noseqrdgfn 28200 | The recursive definition g... |
| noseqrdg0 28201 | Initial value of a recursi... |
| noseqrdgsuc 28202 | Successor value of a recur... |
| seqsfn 28203 | The surreal sequence build... |
| seqs1 28204 | The value of the surreal s... |
| seqsp1 28205 | The value of the surreal s... |
| n0sex 28210 | The set of all non-negativ... |
| nnsex 28211 | The set of all positive su... |
| peano5n0s 28212 | Peano's inductive postulat... |
| n0ssno 28213 | The non-negative surreal i... |
| nnssn0s 28214 | The positive surreal integ... |
| nnssno 28215 | The positive surreal integ... |
| n0sno 28216 | A non-negative surreal int... |
| nnsno 28217 | A positive surreal integer... |
| n0snod 28218 | A non-negative surreal int... |
| nnsnod 28219 | A positive surreal integer... |
| nnn0s 28220 | A positive surreal integer... |
| nnn0sd 28221 | A positive surreal integer... |
| 0n0s 28222 | Peano postulate: ` 0s ` is... |
| peano2n0s 28223 | Peano postulate: the succe... |
| dfn0s2 28224 | Alternate definition of th... |
| n0sind 28225 | Principle of Mathematical ... |
| n0scut 28226 | A cut form for non-negativ... |
| n0scut2 28227 | A cut form for the success... |
| n0ons 28228 | A surreal natural is a sur... |
| nnne0s 28229 | A surreal positive integer... |
| n0sge0 28230 | A non-negative integer is ... |
| nnsgt0 28231 | A positive integer is grea... |
| elnns 28232 | Membership in the positive... |
| elnns2 28233 | A positive surreal integer... |
| n0s0suc 28234 | A non-negative surreal int... |
| nnsge1 28235 | A positive surreal integer... |
| n0addscl 28236 | The non-negative surreal i... |
| n0mulscl 28237 | The non-negative surreal i... |
| nnaddscl 28238 | The positive surreal integ... |
| nnmulscl 28239 | The positive surreal integ... |
| 1n0s 28240 | Surreal one is a non-negat... |
| 1nns 28241 | Surreal one is a positive ... |
| peano2nns 28242 | Peano postulate for positi... |
| nnsrecgt0d 28243 | The reciprocal of a positi... |
| n0sbday 28244 | A non-negative surreal int... |
| n0ssold 28245 | The non-negative surreal i... |
| n0sfincut 28246 | The simplest number greate... |
| onsfi 28247 | A surreal ordinal with a f... |
| onltn0s 28248 | A surreal ordinal that is ... |
| n0cutlt 28249 | A non-negative surreal int... |
| seqn0sfn 28250 | The surreal sequence build... |
| eln0s 28251 | A non-negative surreal int... |
| n0s0m1 28252 | Every non-negative surreal... |
| n0subs 28253 | Subtraction of non-negativ... |
| n0subs2 28254 | Subtraction of non-negativ... |
| n0sltp1le 28255 | Non-negative surreal order... |
| n0sleltp1 28256 | Non-negative surreal order... |
| n0slem1lt 28257 | Non-negative surreal order... |
| bdayn0p1 28258 | The birthday of ` A +s 1s ... |
| bdayn0sf1o 28259 | The birthday function rest... |
| n0p1nns 28260 | One plus a non-negative su... |
| dfnns2 28261 | Alternate definition of th... |
| nnsind 28262 | Principle of Mathematical ... |
| nn1m1nns 28263 | Every positive surreal int... |
| nnm1n0s 28264 | A positive surreal integer... |
| eucliddivs 28265 | Euclid's division lemma fo... |
| zsex 28268 | The surreal integers form ... |
| zssno 28269 | The surreal integers are a... |
| zno 28270 | A surreal integer is a sur... |
| znod 28271 | A surreal integer is a sur... |
| elzs 28272 | Membership in the set of s... |
| nnzsubs 28273 | The difference of two surr... |
| nnzs 28274 | A positive surreal integer... |
| nnzsd 28275 | A positive surreal integer... |
| 0zs 28276 | Zero is a surreal integer.... |
| n0zs 28277 | A non-negative surreal int... |
| n0zsd 28278 | A non-negative surreal int... |
| 1zs 28279 | One is a surreal integer. ... |
| znegscl 28280 | The surreal integers are c... |
| znegscld 28281 | The surreal integers are c... |
| zaddscl 28282 | The surreal integers are c... |
| zaddscld 28283 | The surreal integers are c... |
| zsubscld 28284 | The surreal integers are c... |
| zmulscld 28285 | The surreal integers are c... |
| elzn0s 28286 | A surreal integer is a sur... |
| elzs2 28287 | A surreal integer is eithe... |
| eln0zs 28288 | Non-negative surreal integ... |
| elnnzs 28289 | Positive surreal integer p... |
| elznns 28290 | Surreal integer property e... |
| zn0subs 28291 | The non-negative differenc... |
| peano5uzs 28292 | Peano's inductive postulat... |
| uzsind 28293 | Induction on the upper sur... |
| zsbday 28294 | A surreal integer has a fi... |
| zscut 28295 | A cut expression for surre... |
| 1p1e2s 28302 | One plus one is two. Surr... |
| no2times 28303 | Version of ~ 2times for su... |
| 2nns 28304 | Surreal two is a surreal n... |
| 2sno 28305 | Surreal two is a surreal n... |
| 2ne0s 28306 | Surreal two is non-zero. ... |
| n0seo 28307 | A non-negative surreal int... |
| zseo 28308 | A surreal integer is eithe... |
| twocut 28309 | Two times the cut of zero ... |
| nohalf 28310 | An explicit expression for... |
| expsval 28311 | The value of surreal expon... |
| expsnnval 28312 | Value of surreal exponenti... |
| exps0 28313 | Surreal exponentiation to ... |
| exps1 28314 | Surreal exponentiation to ... |
| expsp1 28315 | Value of a surreal number ... |
| expscllem 28316 | Lemma for proving non-nega... |
| expscl 28317 | Closure law for surreal ex... |
| n0expscl 28318 | Closure law for non-negati... |
| nnexpscl 28319 | Closure law for positive s... |
| expadds 28320 | Sum of exponents law for s... |
| expsne0 28321 | A non-negative surreal int... |
| expsgt0 28322 | A non-negative surreal int... |
| pw2recs 28323 | Any power of two has a mul... |
| pw2divscld 28324 | Division closure for power... |
| pw2divsmuld 28325 | Relationship between surre... |
| pw2divscan3d 28326 | Cancellation law for surre... |
| pw2divscan2d 28327 | A cancellation law for sur... |
| pw2gt0divsd 28328 | Division of a positive sur... |
| pw2ge0divsd 28329 | Divison of a non-negative ... |
| pw2divsrecd 28330 | Relationship between surre... |
| pw2divsdird 28331 | Distribution of surreal di... |
| pw2divsnegd 28332 | Move negative sign inside ... |
| halfcut 28333 | Relate the cut of twice of... |
| addhalfcut 28334 | The cut of a surreal non-n... |
| pw2cut 28335 | Extend ~ halfcut to arbitr... |
| pw2cutp1 28336 | Simplify ~ pw2cut in the c... |
| elzs12 28337 | Membership in the dyadic f... |
| zs12ex 28338 | The class of dyadic fracti... |
| zzs12 28339 | A surreal integer is a dya... |
| zs12negscl 28340 | The dyadics are closed und... |
| zs12negsclb 28341 | A surreal is a dyadic frac... |
| zs12ge0 28342 | An expression for non-nega... |
| zs12bday 28343 | A dyadic fraction has a fi... |
| elreno 28346 | Membership in the set of s... |
| recut 28347 | The cut involved in defini... |
| 0reno 28348 | Surreal zero is a surreal ... |
| renegscl 28349 | The surreal reals are clos... |
| readdscl 28350 | The surreal reals are clos... |
| remulscllem1 28351 | Lemma for ~ remulscl . Sp... |
| remulscllem2 28352 | Lemma for ~ remulscl . Bo... |
| remulscl 28353 | The surreal reals are clos... |
| itvndx 28364 | Index value of the Interva... |
| lngndx 28365 | Index value of the "line" ... |
| itvid 28366 | Utility theorem: index-ind... |
| lngid 28367 | Utility theorem: index-ind... |
| slotsinbpsd 28368 | The slots ` Base ` , ` +g ... |
| slotslnbpsd 28369 | The slots ` Base ` , ` +g ... |
| lngndxnitvndx 28370 | The slot for the line is n... |
| trkgstr 28371 | Functionality of a Tarski ... |
| trkgbas 28372 | The base set of a Tarski g... |
| trkgdist 28373 | The measure of a distance ... |
| trkgitv 28374 | The congruence relation in... |
| istrkgc 28381 | Property of being a Tarski... |
| istrkgb 28382 | Property of being a Tarski... |
| istrkgcb 28383 | Property of being a Tarski... |
| istrkge 28384 | Property of fulfilling Euc... |
| istrkgl 28385 | Building lines from the se... |
| istrkgld 28386 | Property of fulfilling the... |
| istrkg2ld 28387 | Property of fulfilling the... |
| istrkg3ld 28388 | Property of fulfilling the... |
| axtgcgrrflx 28389 | Axiom of reflexivity of co... |
| axtgcgrid 28390 | Axiom of identity of congr... |
| axtgsegcon 28391 | Axiom of segment construct... |
| axtg5seg 28392 | Five segments axiom, Axiom... |
| axtgbtwnid 28393 | Identity of Betweenness. ... |
| axtgpasch 28394 | Axiom of (Inner) Pasch, Ax... |
| axtgcont1 28395 | Axiom of Continuity. Axio... |
| axtgcont 28396 | Axiom of Continuity. Axio... |
| axtglowdim2 28397 | Lower dimension axiom for ... |
| axtgupdim2 28398 | Upper dimension axiom for ... |
| axtgeucl 28399 | Euclid's Axiom. Axiom A10... |
| tgjustf 28400 | Given any function ` F ` ,... |
| tgjustr 28401 | Given any equivalence rela... |
| tgjustc1 28402 | A justification for using ... |
| tgjustc2 28403 | A justification for using ... |
| tgcgrcomimp 28404 | Congruence commutes on the... |
| tgcgrcomr 28405 | Congruence commutes on the... |
| tgcgrcoml 28406 | Congruence commutes on the... |
| tgcgrcomlr 28407 | Congruence commutes on bot... |
| tgcgreqb 28408 | Congruence and equality. ... |
| tgcgreq 28409 | Congruence and equality. ... |
| tgcgrneq 28410 | Congruence and equality. ... |
| tgcgrtriv 28411 | Degenerate segments are co... |
| tgcgrextend 28412 | Link congruence over a pai... |
| tgsegconeq 28413 | Two points that satisfy th... |
| tgbtwntriv2 28414 | Betweenness always holds f... |
| tgbtwncom 28415 | Betweenness commutes. The... |
| tgbtwncomb 28416 | Betweenness commutes, bico... |
| tgbtwnne 28417 | Betweenness and inequality... |
| tgbtwntriv1 28418 | Betweenness always holds f... |
| tgbtwnswapid 28419 | If you can swap the first ... |
| tgbtwnintr 28420 | Inner transitivity law for... |
| tgbtwnexch3 28421 | Exchange the first endpoin... |
| tgbtwnouttr2 28422 | Outer transitivity law for... |
| tgbtwnexch2 28423 | Exchange the outer point o... |
| tgbtwnouttr 28424 | Outer transitivity law for... |
| tgbtwnexch 28425 | Outer transitivity law for... |
| tgtrisegint 28426 | A line segment between two... |
| tglowdim1 28427 | Lower dimension axiom for ... |
| tglowdim1i 28428 | Lower dimension axiom for ... |
| tgldimor 28429 | Excluded-middle like state... |
| tgldim0eq 28430 | In dimension zero, any two... |
| tgldim0itv 28431 | In dimension zero, any two... |
| tgldim0cgr 28432 | In dimension zero, any two... |
| tgbtwndiff 28433 | There is always a ` c ` di... |
| tgdim01 28434 | In geometries of dimension... |
| tgifscgr 28435 | Inner five segment congrue... |
| tgcgrsub 28436 | Removing identical parts f... |
| iscgrg 28439 | The congruence property fo... |
| iscgrgd 28440 | The property for two seque... |
| iscgrglt 28441 | The property for two seque... |
| trgcgrg 28442 | The property for two trian... |
| trgcgr 28443 | Triangle congruence. (Con... |
| ercgrg 28444 | The shape congruence relat... |
| tgcgrxfr 28445 | A line segment can be divi... |
| cgr3id 28446 | Reflexivity law for three-... |
| cgr3simp1 28447 | Deduce segment congruence ... |
| cgr3simp2 28448 | Deduce segment congruence ... |
| cgr3simp3 28449 | Deduce segment congruence ... |
| cgr3swap12 28450 | Permutation law for three-... |
| cgr3swap23 28451 | Permutation law for three-... |
| cgr3swap13 28452 | Permutation law for three-... |
| cgr3rotr 28453 | Permutation law for three-... |
| cgr3rotl 28454 | Permutation law for three-... |
| trgcgrcom 28455 | Commutative law for three-... |
| cgr3tr 28456 | Transitivity law for three... |
| tgbtwnxfr 28457 | A condition for extending ... |
| tgcgr4 28458 | Two quadrilaterals to be c... |
| isismt 28461 | Property of being an isome... |
| ismot 28462 | Property of being an isome... |
| motcgr 28463 | Property of a motion: dist... |
| idmot 28464 | The identity is a motion. ... |
| motf1o 28465 | Motions are bijections. (... |
| motcl 28466 | Closure of motions. (Cont... |
| motco 28467 | The composition of two mot... |
| cnvmot 28468 | The converse of a motion i... |
| motplusg 28469 | The operation for motions ... |
| motgrp 28470 | The motions of a geometry ... |
| motcgrg 28471 | Property of a motion: dist... |
| motcgr3 28472 | Property of a motion: dist... |
| tglng 28473 | Lines of a Tarski Geometry... |
| tglnfn 28474 | Lines as functions. (Cont... |
| tglnunirn 28475 | Lines are sets of points. ... |
| tglnpt 28476 | Lines are sets of points. ... |
| tglngne 28477 | It takes two different poi... |
| tglngval 28478 | The line going through poi... |
| tglnssp 28479 | Lines are subset of the ge... |
| tgellng 28480 | Property of lying on the l... |
| tgcolg 28481 | We choose the notation ` (... |
| btwncolg1 28482 | Betweenness implies coline... |
| btwncolg2 28483 | Betweenness implies coline... |
| btwncolg3 28484 | Betweenness implies coline... |
| colcom 28485 | Swapping the points defini... |
| colrot1 28486 | Rotating the points defini... |
| colrot2 28487 | Rotating the points defini... |
| ncolcom 28488 | Swapping non-colinear poin... |
| ncolrot1 28489 | Rotating non-colinear poin... |
| ncolrot2 28490 | Rotating non-colinear poin... |
| tgdim01ln 28491 | In geometries of dimension... |
| ncoltgdim2 28492 | If there are three non-col... |
| lnxfr 28493 | Transfer law for colineari... |
| lnext 28494 | Extend a line with a missi... |
| tgfscgr 28495 | Congruence law for the gen... |
| lncgr 28496 | Congruence rule for lines.... |
| lnid 28497 | Identity law for points on... |
| tgidinside 28498 | Law for finding a point in... |
| tgbtwnconn1lem1 28499 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem2 28500 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem3 28501 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1 28502 | Connectivity law for betwe... |
| tgbtwnconn2 28503 | Another connectivity law f... |
| tgbtwnconn3 28504 | Inner connectivity law for... |
| tgbtwnconnln3 28505 | Derive colinearity from be... |
| tgbtwnconn22 28506 | Double connectivity law fo... |
| tgbtwnconnln1 28507 | Derive colinearity from be... |
| tgbtwnconnln2 28508 | Derive colinearity from be... |
| legval 28511 | Value of the less-than rel... |
| legov 28512 | Value of the less-than rel... |
| legov2 28513 | An equivalent definition o... |
| legid 28514 | Reflexivity of the less-th... |
| btwnleg 28515 | Betweenness implies less-t... |
| legtrd 28516 | Transitivity of the less-t... |
| legtri3 28517 | Equality from the less-tha... |
| legtrid 28518 | Trichotomy law for the les... |
| leg0 28519 | Degenerated (zero-length) ... |
| legeq 28520 | Deduce equality from "less... |
| legbtwn 28521 | Deduce betweenness from "l... |
| tgcgrsub2 28522 | Removing identical parts f... |
| ltgseg 28523 | The set ` E ` denotes the ... |
| ltgov 28524 | Strict "shorter than" geom... |
| legov3 28525 | An equivalent definition o... |
| legso 28526 | The "shorter than" relatio... |
| ishlg 28529 | Rays : Definition 6.1 of ... |
| hlcomb 28530 | The half-line relation com... |
| hlcomd 28531 | The half-line relation com... |
| hlne1 28532 | The half-line relation imp... |
| hlne2 28533 | The half-line relation imp... |
| hlln 28534 | The half-line relation imp... |
| hleqnid 28535 | The endpoint does not belo... |
| hlid 28536 | The half-line relation is ... |
| hltr 28537 | The half-line relation is ... |
| hlbtwn 28538 | Betweenness is a sufficien... |
| btwnhl1 28539 | Deduce half-line from betw... |
| btwnhl2 28540 | Deduce half-line from betw... |
| btwnhl 28541 | Swap betweenness for a hal... |
| lnhl 28542 | Either a point ` C ` on th... |
| hlcgrex 28543 | Construct a point on a hal... |
| hlcgreulem 28544 | Lemma for ~ hlcgreu . (Co... |
| hlcgreu 28545 | The point constructed in ~... |
| btwnlng1 28546 | Betweenness implies coline... |
| btwnlng2 28547 | Betweenness implies coline... |
| btwnlng3 28548 | Betweenness implies coline... |
| lncom 28549 | Swapping the points defini... |
| lnrot1 28550 | Rotating the points defini... |
| lnrot2 28551 | Rotating the points defini... |
| ncolne1 28552 | Non-colinear points are di... |
| ncolne2 28553 | Non-colinear points are di... |
| tgisline 28554 | The property of being a pr... |
| tglnne 28555 | It takes two different poi... |
| tglndim0 28556 | There are no lines in dime... |
| tgelrnln 28557 | The property of being a pr... |
| tglineeltr 28558 | Transitivity law for lines... |
| tglineelsb2 28559 | If ` S ` lies on PQ , then... |
| tglinerflx1 28560 | Reflexivity law for line m... |
| tglinerflx2 28561 | Reflexivity law for line m... |
| tglinecom 28562 | Commutativity law for line... |
| tglinethru 28563 | If ` A ` is a line contain... |
| tghilberti1 28564 | There is a line through an... |
| tghilberti2 28565 | There is at most one line ... |
| tglinethrueu 28566 | There is a unique line goi... |
| tglnne0 28567 | A line ` A ` has at least ... |
| tglnpt2 28568 | Find a second point on a l... |
| tglineintmo 28569 | Two distinct lines interse... |
| tglineineq 28570 | Two distinct lines interse... |
| tglineneq 28571 | Given three non-colinear p... |
| tglineinteq 28572 | Two distinct lines interse... |
| ncolncol 28573 | Deduce non-colinearity fro... |
| coltr 28574 | A transitivity law for col... |
| coltr3 28575 | A transitivity law for col... |
| colline 28576 | Three points are colinear ... |
| tglowdim2l 28577 | Reformulation of the lower... |
| tglowdim2ln 28578 | There is always one point ... |
| mirreu3 28581 | Existential uniqueness of ... |
| mirval 28582 | Value of the point inversi... |
| mirfv 28583 | Value of the point inversi... |
| mircgr 28584 | Property of the image by t... |
| mirbtwn 28585 | Property of the image by t... |
| ismir 28586 | Property of the image by t... |
| mirf 28587 | Point inversion as functio... |
| mircl 28588 | Closure of the point inver... |
| mirmir 28589 | The point inversion functi... |
| mircom 28590 | Variation on ~ mirmir . (... |
| mirreu 28591 | Any point has a unique ant... |
| mireq 28592 | Equality deduction for poi... |
| mirinv 28593 | The only invariant point o... |
| mirne 28594 | Mirror of non-center point... |
| mircinv 28595 | The center point is invari... |
| mirf1o 28596 | The point inversion functi... |
| miriso 28597 | The point inversion functi... |
| mirbtwni 28598 | Point inversion preserves ... |
| mirbtwnb 28599 | Point inversion preserves ... |
| mircgrs 28600 | Point inversion preserves ... |
| mirmir2 28601 | Point inversion of a point... |
| mirmot 28602 | Point investion is a motio... |
| mirln 28603 | If two points are on the s... |
| mirln2 28604 | If a point and its mirror ... |
| mirconn 28605 | Point inversion of connect... |
| mirhl 28606 | If two points ` X ` and ` ... |
| mirbtwnhl 28607 | If the center of the point... |
| mirhl2 28608 | Deduce half-line relation ... |
| mircgrextend 28609 | Link congruence over a pai... |
| mirtrcgr 28610 | Point inversion of one poi... |
| mirauto 28611 | Point inversion preserves ... |
| miduniq 28612 | Uniqueness of the middle p... |
| miduniq1 28613 | Uniqueness of the middle p... |
| miduniq2 28614 | If two point inversions co... |
| colmid 28615 | Colinearity and equidistan... |
| symquadlem 28616 | Lemma of the symetrial qua... |
| krippenlem 28617 | Lemma for ~ krippen . We ... |
| krippen 28618 | Krippenlemma (German for c... |
| midexlem 28619 | Lemma for the existence of... |
| israg 28624 | Property for 3 points A, B... |
| ragcom 28625 | Commutative rule for right... |
| ragcol 28626 | The right angle property i... |
| ragmir 28627 | Right angle property is pr... |
| mirrag 28628 | Right angle is conserved b... |
| ragtrivb 28629 | Trivial right angle. Theo... |
| ragflat2 28630 | Deduce equality from two r... |
| ragflat 28631 | Deduce equality from two r... |
| ragtriva 28632 | Trivial right angle. Theo... |
| ragflat3 28633 | Right angle and colinearit... |
| ragcgr 28634 | Right angle and colinearit... |
| motrag 28635 | Right angles are preserved... |
| ragncol 28636 | Right angle implies non-co... |
| perpln1 28637 | Derive a line from perpend... |
| perpln2 28638 | Derive a line from perpend... |
| isperp 28639 | Property for 2 lines A, B ... |
| perpcom 28640 | The "perpendicular" relati... |
| perpneq 28641 | Two perpendicular lines ar... |
| isperp2 28642 | Property for 2 lines A, B,... |
| isperp2d 28643 | One direction of ~ isperp2... |
| ragperp 28644 | Deduce that two lines are ... |
| footexALT 28645 | Alternative version of ~ f... |
| footexlem1 28646 | Lemma for ~ footex . (Con... |
| footexlem2 28647 | Lemma for ~ footex . (Con... |
| footex 28648 | From a point ` C ` outside... |
| foot 28649 | From a point ` C ` outside... |
| footne 28650 | Uniqueness of the foot poi... |
| footeq 28651 | Uniqueness of the foot poi... |
| hlperpnel 28652 | A point on a half-line whi... |
| perprag 28653 | Deduce a right angle from ... |
| perpdragALT 28654 | Deduce a right angle from ... |
| perpdrag 28655 | Deduce a right angle from ... |
| colperp 28656 | Deduce a perpendicularity ... |
| colperpexlem1 28657 | Lemma for ~ colperp . Fir... |
| colperpexlem2 28658 | Lemma for ~ colperpex . S... |
| colperpexlem3 28659 | Lemma for ~ colperpex . C... |
| colperpex 28660 | In dimension 2 and above, ... |
| mideulem2 28661 | Lemma for ~ opphllem , whi... |
| opphllem 28662 | Lemma 8.24 of [Schwabhause... |
| mideulem 28663 | Lemma for ~ mideu . We ca... |
| midex 28664 | Existence of the midpoint,... |
| mideu 28665 | Existence and uniqueness o... |
| islnopp 28666 | The property for two point... |
| islnoppd 28667 | Deduce that ` A ` and ` B ... |
| oppne1 28668 | Points lying on opposite s... |
| oppne2 28669 | Points lying on opposite s... |
| oppne3 28670 | Points lying on opposite s... |
| oppcom 28671 | Commutativity rule for "op... |
| opptgdim2 28672 | If two points opposite to ... |
| oppnid 28673 | The "opposite to a line" r... |
| opphllem1 28674 | Lemma for ~ opphl . (Cont... |
| opphllem2 28675 | Lemma for ~ opphl . Lemma... |
| opphllem3 28676 | Lemma for ~ opphl : We as... |
| opphllem4 28677 | Lemma for ~ opphl . (Cont... |
| opphllem5 28678 | Second part of Lemma 9.4 o... |
| opphllem6 28679 | First part of Lemma 9.4 of... |
| oppperpex 28680 | Restating ~ colperpex usin... |
| opphl 28681 | If two points ` A ` and ` ... |
| outpasch 28682 | Axiom of Pasch, outer form... |
| hlpasch 28683 | An application of the axio... |
| ishpg 28686 | Value of the half-plane re... |
| hpgbr 28687 | Half-planes : property for... |
| hpgne1 28688 | Points on the open half pl... |
| hpgne2 28689 | Points on the open half pl... |
| lnopp2hpgb 28690 | Theorem 9.8 of [Schwabhaus... |
| lnoppnhpg 28691 | If two points lie on the o... |
| hpgerlem 28692 | Lemma for the proof that t... |
| hpgid 28693 | The half-plane relation is... |
| hpgcom 28694 | The half-plane relation co... |
| hpgtr 28695 | The half-plane relation is... |
| colopp 28696 | Opposite sides of a line f... |
| colhp 28697 | Half-plane relation for co... |
| hphl 28698 | If two points are on the s... |
| midf 28703 | Midpoint as a function. (... |
| midcl 28704 | Closure of the midpoint. ... |
| ismidb 28705 | Property of the midpoint. ... |
| midbtwn 28706 | Betweenness of midpoint. ... |
| midcgr 28707 | Congruence of midpoint. (... |
| midid 28708 | Midpoint of a null segment... |
| midcom 28709 | Commutativity rule for the... |
| mirmid 28710 | Point inversion preserves ... |
| lmieu 28711 | Uniqueness of the line mir... |
| lmif 28712 | Line mirror as a function.... |
| lmicl 28713 | Closure of the line mirror... |
| islmib 28714 | Property of the line mirro... |
| lmicom 28715 | The line mirroring functio... |
| lmilmi 28716 | Line mirroring is an invol... |
| lmireu 28717 | Any point has a unique ant... |
| lmieq 28718 | Equality deduction for lin... |
| lmiinv 28719 | The invariants of the line... |
| lmicinv 28720 | The mirroring line is an i... |
| lmimid 28721 | If we have a right angle, ... |
| lmif1o 28722 | The line mirroring functio... |
| lmiisolem 28723 | Lemma for ~ lmiiso . (Con... |
| lmiiso 28724 | The line mirroring functio... |
| lmimot 28725 | Line mirroring is a motion... |
| hypcgrlem1 28726 | Lemma for ~ hypcgr , case ... |
| hypcgrlem2 28727 | Lemma for ~ hypcgr , case ... |
| hypcgr 28728 | If the catheti of two righ... |
| lmiopp 28729 | Line mirroring produces po... |
| lnperpex 28730 | Existence of a perpendicul... |
| trgcopy 28731 | Triangle construction: a c... |
| trgcopyeulem 28732 | Lemma for ~ trgcopyeu . (... |
| trgcopyeu 28733 | Triangle construction: a c... |
| iscgra 28736 | Property for two angles AB... |
| iscgra1 28737 | A special version of ~ isc... |
| iscgrad 28738 | Sufficient conditions for ... |
| cgrane1 28739 | Angles imply inequality. ... |
| cgrane2 28740 | Angles imply inequality. ... |
| cgrane3 28741 | Angles imply inequality. ... |
| cgrane4 28742 | Angles imply inequality. ... |
| cgrahl1 28743 | Angle congruence is indepe... |
| cgrahl2 28744 | Angle congruence is indepe... |
| cgracgr 28745 | First direction of proposi... |
| cgraid 28746 | Angle congruence is reflex... |
| cgraswap 28747 | Swap rays in a congruence ... |
| cgrcgra 28748 | Triangle congruence implie... |
| cgracom 28749 | Angle congruence commutes.... |
| cgratr 28750 | Angle congruence is transi... |
| flatcgra 28751 | Flat angles are congruent.... |
| cgraswaplr 28752 | Swap both side of angle co... |
| cgrabtwn 28753 | Angle congruence preserves... |
| cgrahl 28754 | Angle congruence preserves... |
| cgracol 28755 | Angle congruence preserves... |
| cgrancol 28756 | Angle congruence preserves... |
| dfcgra2 28757 | This is the full statement... |
| sacgr 28758 | Supplementary angles of co... |
| oacgr 28759 | Vertical angle theorem. V... |
| acopy 28760 | Angle construction. Theor... |
| acopyeu 28761 | Angle construction. Theor... |
| isinag 28765 | Property for point ` X ` t... |
| isinagd 28766 | Sufficient conditions for ... |
| inagflat 28767 | Any point lies in a flat a... |
| inagswap 28768 | Swap the order of the half... |
| inagne1 28769 | Deduce inequality from the... |
| inagne2 28770 | Deduce inequality from the... |
| inagne3 28771 | Deduce inequality from the... |
| inaghl 28772 | The "point lie in angle" r... |
| isleag 28774 | Geometrical "less than" pr... |
| isleagd 28775 | Sufficient condition for "... |
| leagne1 28776 | Deduce inequality from the... |
| leagne2 28777 | Deduce inequality from the... |
| leagne3 28778 | Deduce inequality from the... |
| leagne4 28779 | Deduce inequality from the... |
| cgrg3col4 28780 | Lemma 11.28 of [Schwabhaus... |
| tgsas1 28781 | First congruence theorem: ... |
| tgsas 28782 | First congruence theorem: ... |
| tgsas2 28783 | First congruence theorem: ... |
| tgsas3 28784 | First congruence theorem: ... |
| tgasa1 28785 | Second congruence theorem:... |
| tgasa 28786 | Second congruence theorem:... |
| tgsss1 28787 | Third congruence theorem: ... |
| tgsss2 28788 | Third congruence theorem: ... |
| tgsss3 28789 | Third congruence theorem: ... |
| dfcgrg2 28790 | Congruence for two triangl... |
| isoas 28791 | Congruence theorem for iso... |
| iseqlg 28794 | Property of a triangle bei... |
| iseqlgd 28795 | Condition for a triangle t... |
| f1otrgds 28796 | Convenient lemma for ~ f1o... |
| f1otrgitv 28797 | Convenient lemma for ~ f1o... |
| f1otrg 28798 | A bijection between bases ... |
| f1otrge 28799 | A bijection between bases ... |
| ttgval 28802 | Define a function to augme... |
| ttglem 28803 | Lemma for ~ ttgbas , ~ ttg... |
| ttgbas 28804 | The base set of a subcompl... |
| ttgplusg 28805 | The addition operation of ... |
| ttgsub 28806 | The subtraction operation ... |
| ttgvsca 28807 | The scalar product of a su... |
| ttgds 28808 | The metric of a subcomplex... |
| ttgitvval 28809 | Betweenness for a subcompl... |
| ttgelitv 28810 | Betweenness for a subcompl... |
| ttgbtwnid 28811 | Any subcomplex module equi... |
| ttgcontlem1 28812 | Lemma for % ttgcont . (Co... |
| xmstrkgc 28813 | Any metric space fulfills ... |
| cchhllem 28814 | Lemma for chlbas and chlvs... |
| elee 28821 | Membership in a Euclidean ... |
| mptelee 28822 | A condition for a mapping ... |
| eleenn 28823 | If ` A ` is in ` ( EE `` N... |
| eleei 28824 | The forward direction of ~... |
| eedimeq 28825 | A point belongs to at most... |
| brbtwn 28826 | The binary relation form o... |
| brcgr 28827 | The binary relation form o... |
| fveere 28828 | The function value of a po... |
| fveecn 28829 | The function value of a po... |
| eqeefv 28830 | Two points are equal iff t... |
| eqeelen 28831 | Two points are equal iff t... |
| brbtwn2 28832 | Alternate characterization... |
| colinearalglem1 28833 | Lemma for ~ colinearalg . ... |
| colinearalglem2 28834 | Lemma for ~ colinearalg . ... |
| colinearalglem3 28835 | Lemma for ~ colinearalg . ... |
| colinearalglem4 28836 | Lemma for ~ colinearalg . ... |
| colinearalg 28837 | An algebraic characterizat... |
| eleesub 28838 | Membership of a subtractio... |
| eleesubd 28839 | Membership of a subtractio... |
| axdimuniq 28840 | The unique dimension axiom... |
| axcgrrflx 28841 | ` A ` is as far from ` B `... |
| axcgrtr 28842 | Congruence is transitive. ... |
| axcgrid 28843 | If there is no distance be... |
| axsegconlem1 28844 | Lemma for ~ axsegcon . Ha... |
| axsegconlem2 28845 | Lemma for ~ axsegcon . Sh... |
| axsegconlem3 28846 | Lemma for ~ axsegcon . Sh... |
| axsegconlem4 28847 | Lemma for ~ axsegcon . Sh... |
| axsegconlem5 28848 | Lemma for ~ axsegcon . Sh... |
| axsegconlem6 28849 | Lemma for ~ axsegcon . Sh... |
| axsegconlem7 28850 | Lemma for ~ axsegcon . Sh... |
| axsegconlem8 28851 | Lemma for ~ axsegcon . Sh... |
| axsegconlem9 28852 | Lemma for ~ axsegcon . Sh... |
| axsegconlem10 28853 | Lemma for ~ axsegcon . Sh... |
| axsegcon 28854 | Any segment ` A B ` can be... |
| ax5seglem1 28855 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem2 28856 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem3a 28857 | Lemma for ~ ax5seg . (Con... |
| ax5seglem3 28858 | Lemma for ~ ax5seg . Comb... |
| ax5seglem4 28859 | Lemma for ~ ax5seg . Give... |
| ax5seglem5 28860 | Lemma for ~ ax5seg . If `... |
| ax5seglem6 28861 | Lemma for ~ ax5seg . Give... |
| ax5seglem7 28862 | Lemma for ~ ax5seg . An a... |
| ax5seglem8 28863 | Lemma for ~ ax5seg . Use ... |
| ax5seglem9 28864 | Lemma for ~ ax5seg . Take... |
| ax5seg 28865 | The five segment axiom. T... |
| axbtwnid 28866 | Points are indivisible. T... |
| axpaschlem 28867 | Lemma for ~ axpasch . Set... |
| axpasch 28868 | The inner Pasch axiom. Ta... |
| axlowdimlem1 28869 | Lemma for ~ axlowdim . Es... |
| axlowdimlem2 28870 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem3 28871 | Lemma for ~ axlowdim . Se... |
| axlowdimlem4 28872 | Lemma for ~ axlowdim . Se... |
| axlowdimlem5 28873 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem6 28874 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem7 28875 | Lemma for ~ axlowdim . Se... |
| axlowdimlem8 28876 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem9 28877 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem10 28878 | Lemma for ~ axlowdim . Se... |
| axlowdimlem11 28879 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem12 28880 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem13 28881 | Lemma for ~ axlowdim . Es... |
| axlowdimlem14 28882 | Lemma for ~ axlowdim . Ta... |
| axlowdimlem15 28883 | Lemma for ~ axlowdim . Se... |
| axlowdimlem16 28884 | Lemma for ~ axlowdim . Se... |
| axlowdimlem17 28885 | Lemma for ~ axlowdim . Es... |
| axlowdim1 28886 | The lower dimension axiom ... |
| axlowdim2 28887 | The lower two-dimensional ... |
| axlowdim 28888 | The general lower dimensio... |
| axeuclidlem 28889 | Lemma for ~ axeuclid . Ha... |
| axeuclid 28890 | Euclid's axiom. Take an a... |
| axcontlem1 28891 | Lemma for ~ axcont . Chan... |
| axcontlem2 28892 | Lemma for ~ axcont . The ... |
| axcontlem3 28893 | Lemma for ~ axcont . Give... |
| axcontlem4 28894 | Lemma for ~ axcont . Give... |
| axcontlem5 28895 | Lemma for ~ axcont . Comp... |
| axcontlem6 28896 | Lemma for ~ axcont . Stat... |
| axcontlem7 28897 | Lemma for ~ axcont . Give... |
| axcontlem8 28898 | Lemma for ~ axcont . A po... |
| axcontlem9 28899 | Lemma for ~ axcont . Give... |
| axcontlem10 28900 | Lemma for ~ axcont . Give... |
| axcontlem11 28901 | Lemma for ~ axcont . Elim... |
| axcontlem12 28902 | Lemma for ~ axcont . Elim... |
| axcont 28903 | The axiom of continuity. ... |
| eengv 28906 | The value of the Euclidean... |
| eengstr 28907 | The Euclidean geometry as ... |
| eengbas 28908 | The Base of the Euclidean ... |
| ebtwntg 28909 | The betweenness relation u... |
| ecgrtg 28910 | The congruence relation us... |
| elntg 28911 | The line definition in the... |
| elntg2 28912 | The line definition in the... |
| eengtrkg 28913 | The geometry structure for... |
| eengtrkge 28914 | The geometry structure for... |
| edgfid 28917 | Utility theorem: index-ind... |
| edgfndx 28918 | Index value of the ~ df-ed... |
| edgfndxnn 28919 | The index value of the edg... |
| edgfndxid 28920 | The value of the edge func... |
| basendxltedgfndx 28921 | The index value of the ` B... |
| basendxnedgfndx 28922 | The slots ` Base ` and ` .... |
| vtxval 28927 | The set of vertices of a g... |
| iedgval 28928 | The set of indexed edges o... |
| 1vgrex 28929 | A graph with at least one ... |
| opvtxval 28930 | The set of vertices of a g... |
| opvtxfv 28931 | The set of vertices of a g... |
| opvtxov 28932 | The set of vertices of a g... |
| opiedgval 28933 | The set of indexed edges o... |
| opiedgfv 28934 | The set of indexed edges o... |
| opiedgov 28935 | The set of indexed edges o... |
| opvtxfvi 28936 | The set of vertices of a g... |
| opiedgfvi 28937 | The set of indexed edges o... |
| funvtxdmge2val 28938 | The set of vertices of an ... |
| funiedgdmge2val 28939 | The set of indexed edges o... |
| funvtxdm2val 28940 | The set of vertices of an ... |
| funiedgdm2val 28941 | The set of indexed edges o... |
| funvtxval0 28942 | The set of vertices of an ... |
| basvtxval 28943 | The set of vertices of a g... |
| edgfiedgval 28944 | The set of indexed edges o... |
| funvtxval 28945 | The set of vertices of a g... |
| funiedgval 28946 | The set of indexed edges o... |
| structvtxvallem 28947 | Lemma for ~ structvtxval a... |
| structvtxval 28948 | The set of vertices of an ... |
| structiedg0val 28949 | The set of indexed edges o... |
| structgrssvtxlem 28950 | Lemma for ~ structgrssvtx ... |
| structgrssvtx 28951 | The set of vertices of a g... |
| structgrssiedg 28952 | The set of indexed edges o... |
| struct2grstr 28953 | A graph represented as an ... |
| struct2grvtx 28954 | The set of vertices of a g... |
| struct2griedg 28955 | The set of indexed edges o... |
| graop 28956 | Any representation of a gr... |
| grastruct 28957 | Any representation of a gr... |
| gropd 28958 | If any representation of a... |
| grstructd 28959 | If any representation of a... |
| gropeld 28960 | If any representation of a... |
| grstructeld 28961 | If any representation of a... |
| setsvtx 28962 | The vertices of a structur... |
| setsiedg 28963 | The (indexed) edges of a s... |
| snstrvtxval 28964 | The set of vertices of a g... |
| snstriedgval 28965 | The set of indexed edges o... |
| vtxval0 28966 | Degenerated case 1 for ver... |
| iedgval0 28967 | Degenerated case 1 for edg... |
| vtxvalsnop 28968 | Degenerated case 2 for ver... |
| iedgvalsnop 28969 | Degenerated case 2 for edg... |
| vtxval3sn 28970 | Degenerated case 3 for ver... |
| iedgval3sn 28971 | Degenerated case 3 for edg... |
| vtxvalprc 28972 | Degenerated case 4 for ver... |
| iedgvalprc 28973 | Degenerated case 4 for edg... |
| edgval 28976 | The edges of a graph. (Co... |
| iedgedg 28977 | An indexed edge is an edge... |
| edgopval 28978 | The edges of a graph repre... |
| edgov 28979 | The edges of a graph repre... |
| edgstruct 28980 | The edges of a graph repre... |
| edgiedgb 28981 | A set is an edge iff it is... |
| edg0iedg0 28982 | There is no edge in a grap... |
| isuhgr 28987 | The predicate "is an undir... |
| isushgr 28988 | The predicate "is an undir... |
| uhgrf 28989 | The edge function of an un... |
| ushgrf 28990 | The edge function of an un... |
| uhgrss 28991 | An edge is a subset of ver... |
| uhgreq12g 28992 | If two sets have the same ... |
| uhgrfun 28993 | The edge function of an un... |
| uhgrn0 28994 | An edge is a nonempty subs... |
| lpvtx 28995 | The endpoints of a loop (w... |
| ushgruhgr 28996 | An undirected simple hyper... |
| isuhgrop 28997 | The property of being an u... |
| uhgr0e 28998 | The empty graph, with vert... |
| uhgr0vb 28999 | The null graph, with no ve... |
| uhgr0 29000 | The null graph represented... |
| uhgrun 29001 | The union ` U ` of two (un... |
| uhgrunop 29002 | The union of two (undirect... |
| ushgrun 29003 | The union ` U ` of two (un... |
| ushgrunop 29004 | The union of two (undirect... |
| uhgrstrrepe 29005 | Replacing (or adding) the ... |
| incistruhgr 29006 | An _incidence structure_ `... |
| isupgr 29011 | The property of being an u... |
| wrdupgr 29012 | The property of being an u... |
| upgrf 29013 | The edge function of an un... |
| upgrfn 29014 | The edge function of an un... |
| upgrss 29015 | An edge is a subset of ver... |
| upgrn0 29016 | An edge is a nonempty subs... |
| upgrle 29017 | An edge of an undirected p... |
| upgrfi 29018 | An edge is a finite subset... |
| upgrex 29019 | An edge is an unordered pa... |
| upgrbi 29020 | Show that an unordered pai... |
| upgrop 29021 | A pseudograph represented ... |
| isumgr 29022 | The property of being an u... |
| isumgrs 29023 | The simplified property of... |
| wrdumgr 29024 | The property of being an u... |
| umgrf 29025 | The edge function of an un... |
| umgrfn 29026 | The edge function of an un... |
| umgredg2 29027 | An edge of a multigraph ha... |
| umgrbi 29028 | Show that an unordered pai... |
| upgruhgr 29029 | An undirected pseudograph ... |
| umgrupgr 29030 | An undirected multigraph i... |
| umgruhgr 29031 | An undirected multigraph i... |
| upgrle2 29032 | An edge of an undirected p... |
| umgrnloopv 29033 | In a multigraph, there is ... |
| umgredgprv 29034 | In a multigraph, an edge i... |
| umgrnloop 29035 | In a multigraph, there is ... |
| umgrnloop0 29036 | A multigraph has no loops.... |
| umgr0e 29037 | The empty graph, with vert... |
| upgr0e 29038 | The empty graph, with vert... |
| upgr1elem 29039 | Lemma for ~ upgr1e and ~ u... |
| upgr1e 29040 | A pseudograph with one edg... |
| upgr0eop 29041 | The empty graph, with vert... |
| upgr1eop 29042 | A pseudograph with one edg... |
| upgr0eopALT 29043 | Alternate proof of ~ upgr0... |
| upgr1eopALT 29044 | Alternate proof of ~ upgr1... |
| upgrun 29045 | The union ` U ` of two pse... |
| upgrunop 29046 | The union of two pseudogra... |
| umgrun 29047 | The union ` U ` of two mul... |
| umgrunop 29048 | The union of two multigrap... |
| umgrislfupgrlem 29049 | Lemma for ~ umgrislfupgr a... |
| umgrislfupgr 29050 | A multigraph is a loop-fre... |
| lfgredgge2 29051 | An edge of a loop-free gra... |
| lfgrnloop 29052 | A loop-free graph has no l... |
| uhgredgiedgb 29053 | In a hypergraph, a set is ... |
| uhgriedg0edg0 29054 | A hypergraph has no edges ... |
| uhgredgn0 29055 | An edge of a hypergraph is... |
| edguhgr 29056 | An edge of a hypergraph is... |
| uhgredgrnv 29057 | An edge of a hypergraph co... |
| uhgredgss 29058 | The set of edges of a hype... |
| upgredgss 29059 | The set of edges of a pseu... |
| umgredgss 29060 | The set of edges of a mult... |
| edgupgr 29061 | Properties of an edge of a... |
| edgumgr 29062 | Properties of an edge of a... |
| uhgrvtxedgiedgb 29063 | In a hypergraph, a vertex ... |
| upgredg 29064 | For each edge in a pseudog... |
| umgredg 29065 | For each edge in a multigr... |
| upgrpredgv 29066 | An edge of a pseudograph a... |
| umgrpredgv 29067 | An edge of a multigraph al... |
| upgredg2vtx 29068 | For a vertex incident to a... |
| upgredgpr 29069 | If a proper pair (of verti... |
| edglnl 29070 | The edges incident with a ... |
| numedglnl 29071 | The number of edges incide... |
| umgredgne 29072 | An edge of a multigraph al... |
| umgrnloop2 29073 | A multigraph has no loops.... |
| umgredgnlp 29074 | An edge of a multigraph is... |
| isuspgr 29079 | The property of being a si... |
| isusgr 29080 | The property of being a si... |
| uspgrf 29081 | The edge function of a sim... |
| usgrf 29082 | The edge function of a sim... |
| isusgrs 29083 | The property of being a si... |
| usgrfs 29084 | The edge function of a sim... |
| usgrfun 29085 | The edge function of a sim... |
| usgredgss 29086 | The set of edges of a simp... |
| edgusgr 29087 | An edge of a simple graph ... |
| isuspgrop 29088 | The property of being an u... |
| isusgrop 29089 | The property of being an u... |
| usgrop 29090 | A simple graph represented... |
| isausgr 29091 | The property of an unorder... |
| ausgrusgrb 29092 | The equivalence of the def... |
| usgrausgri 29093 | A simple graph represented... |
| ausgrumgri 29094 | If an alternatively define... |
| ausgrusgri 29095 | The equivalence of the def... |
| usgrausgrb 29096 | The equivalence of the def... |
| usgredgop 29097 | An edge of a simple graph ... |
| usgrf1o 29098 | The edge function of a sim... |
| usgrf1 29099 | The edge function of a sim... |
| uspgrf1oedg 29100 | The edge function of a sim... |
| usgrss 29101 | An edge is a subset of ver... |
| uspgredgiedg 29102 | In a simple pseudograph, f... |
| uspgriedgedg 29103 | In a simple pseudograph, f... |
| uspgrushgr 29104 | A simple pseudograph is an... |
| uspgrupgr 29105 | A simple pseudograph is an... |
| uspgrupgrushgr 29106 | A graph is a simple pseudo... |
| usgruspgr 29107 | A simple graph is a simple... |
| usgrumgr 29108 | A simple graph is an undir... |
| usgrumgruspgr 29109 | A graph is a simple graph ... |
| usgruspgrb 29110 | A class is a simple graph ... |
| uspgruhgr 29111 | An undirected simple pseud... |
| usgrupgr 29112 | A simple graph is an undir... |
| usgruhgr 29113 | A simple graph is an undir... |
| usgrislfuspgr 29114 | A simple graph is a loop-f... |
| uspgrun 29115 | The union ` U ` of two sim... |
| uspgrunop 29116 | The union of two simple ps... |
| usgrun 29117 | The union ` U ` of two sim... |
| usgrunop 29118 | The union of two simple gr... |
| usgredg2 29119 | The value of the "edge fun... |
| usgredg2ALT 29120 | Alternate proof of ~ usgre... |
| usgredgprv 29121 | In a simple graph, an edge... |
| usgredgprvALT 29122 | Alternate proof of ~ usgre... |
| usgredgppr 29123 | An edge of a simple graph ... |
| usgrpredgv 29124 | An edge of a simple graph ... |
| edgssv2 29125 | An edge of a simple graph ... |
| usgredg 29126 | For each edge in a simple ... |
| usgrnloopv 29127 | In a simple graph, there i... |
| usgrnloopvALT 29128 | Alternate proof of ~ usgrn... |
| usgrnloop 29129 | In a simple graph, there i... |
| usgrnloopALT 29130 | Alternate proof of ~ usgrn... |
| usgrnloop0 29131 | A simple graph has no loop... |
| usgrnloop0ALT 29132 | Alternate proof of ~ usgrn... |
| usgredgne 29133 | An edge of a simple graph ... |
| usgrf1oedg 29134 | The edge function of a sim... |
| uhgr2edg 29135 | If a vertex is adjacent to... |
| umgr2edg 29136 | If a vertex is adjacent to... |
| usgr2edg 29137 | If a vertex is adjacent to... |
| umgr2edg1 29138 | If a vertex is adjacent to... |
| usgr2edg1 29139 | If a vertex is adjacent to... |
| umgrvad2edg 29140 | If a vertex is adjacent to... |
| umgr2edgneu 29141 | If a vertex is adjacent to... |
| usgrsizedg 29142 | In a simple graph, the siz... |
| usgredg3 29143 | The value of the "edge fun... |
| usgredg4 29144 | For a vertex incident to a... |
| usgredgreu 29145 | For a vertex incident to a... |
| usgredg2vtx 29146 | For a vertex incident to a... |
| uspgredg2vtxeu 29147 | For a vertex incident to a... |
| usgredg2vtxeu 29148 | For a vertex incident to a... |
| usgredg2vtxeuALT 29149 | Alternate proof of ~ usgre... |
| uspgredg2vlem 29150 | Lemma for ~ uspgredg2v . ... |
| uspgredg2v 29151 | In a simple pseudograph, t... |
| usgredg2vlem1 29152 | Lemma 1 for ~ usgredg2v . ... |
| usgredg2vlem2 29153 | Lemma 2 for ~ usgredg2v . ... |
| usgredg2v 29154 | In a simple graph, the map... |
| usgriedgleord 29155 | Alternate version of ~ usg... |
| ushgredgedg 29156 | In a simple hypergraph the... |
| usgredgedg 29157 | In a simple graph there is... |
| ushgredgedgloop 29158 | In a simple hypergraph the... |
| uspgredgleord 29159 | In a simple pseudograph th... |
| usgredgleord 29160 | In a simple graph the numb... |
| usgredgleordALT 29161 | Alternate proof for ~ usgr... |
| usgrstrrepe 29162 | Replacing (or adding) the ... |
| usgr0e 29163 | The empty graph, with vert... |
| usgr0vb 29164 | The null graph, with no ve... |
| uhgr0v0e 29165 | The null graph, with no ve... |
| uhgr0vsize0 29166 | The size of a hypergraph w... |
| uhgr0edgfi 29167 | A graph of order 0 (i.e. w... |
| usgr0v 29168 | The null graph, with no ve... |
| uhgr0vusgr 29169 | The null graph, with no ve... |
| usgr0 29170 | The null graph represented... |
| uspgr1e 29171 | A simple pseudograph with ... |
| usgr1e 29172 | A simple graph with one ed... |
| usgr0eop 29173 | The empty graph, with vert... |
| uspgr1eop 29174 | A simple pseudograph with ... |
| uspgr1ewop 29175 | A simple pseudograph with ... |
| uspgr1v1eop 29176 | A simple pseudograph with ... |
| usgr1eop 29177 | A simple graph with (at le... |
| uspgr2v1e2w 29178 | A simple pseudograph with ... |
| usgr2v1e2w 29179 | A simple graph with two ve... |
| edg0usgr 29180 | A class without edges is a... |
| lfuhgr1v0e 29181 | A loop-free hypergraph wit... |
| usgr1vr 29182 | A simple graph with one ve... |
| usgr1v 29183 | A class with one (or no) v... |
| usgr1v0edg 29184 | A class with one (or no) v... |
| usgrexmpldifpr 29185 | Lemma for ~ usgrexmpledg :... |
| usgrexmplef 29186 | Lemma for ~ usgrexmpl . (... |
| usgrexmpllem 29187 | Lemma for ~ usgrexmpl . (... |
| usgrexmplvtx 29188 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpledg 29189 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl 29190 | ` G ` is a simple graph of... |
| griedg0prc 29191 | The class of empty graphs ... |
| griedg0ssusgr 29192 | The class of all simple gr... |
| usgrprc 29193 | The class of simple graphs... |
| relsubgr 29196 | The class of the subgraph ... |
| subgrv 29197 | If a class is a subgraph o... |
| issubgr 29198 | The property of a set to b... |
| issubgr2 29199 | The property of a set to b... |
| subgrprop 29200 | The properties of a subgra... |
| subgrprop2 29201 | The properties of a subgra... |
| uhgrissubgr 29202 | The property of a hypergra... |
| subgrprop3 29203 | The properties of a subgra... |
| egrsubgr 29204 | An empty graph consisting ... |
| 0grsubgr 29205 | The null graph (represente... |
| 0uhgrsubgr 29206 | The null graph (as hypergr... |
| uhgrsubgrself 29207 | A hypergraph is a subgraph... |
| subgrfun 29208 | The edge function of a sub... |
| subgruhgrfun 29209 | The edge function of a sub... |
| subgreldmiedg 29210 | An element of the domain o... |
| subgruhgredgd 29211 | An edge of a subgraph of a... |
| subumgredg2 29212 | An edge of a subgraph of a... |
| subuhgr 29213 | A subgraph of a hypergraph... |
| subupgr 29214 | A subgraph of a pseudograp... |
| subumgr 29215 | A subgraph of a multigraph... |
| subusgr 29216 | A subgraph of a simple gra... |
| uhgrspansubgrlem 29217 | Lemma for ~ uhgrspansubgr ... |
| uhgrspansubgr 29218 | A spanning subgraph ` S ` ... |
| uhgrspan 29219 | A spanning subgraph ` S ` ... |
| upgrspan 29220 | A spanning subgraph ` S ` ... |
| umgrspan 29221 | A spanning subgraph ` S ` ... |
| usgrspan 29222 | A spanning subgraph ` S ` ... |
| uhgrspanop 29223 | A spanning subgraph of a h... |
| upgrspanop 29224 | A spanning subgraph of a p... |
| umgrspanop 29225 | A spanning subgraph of a m... |
| usgrspanop 29226 | A spanning subgraph of a s... |
| uhgrspan1lem1 29227 | Lemma 1 for ~ uhgrspan1 . ... |
| uhgrspan1lem2 29228 | Lemma 2 for ~ uhgrspan1 . ... |
| uhgrspan1lem3 29229 | Lemma 3 for ~ uhgrspan1 . ... |
| uhgrspan1 29230 | The induced subgraph ` S `... |
| upgrreslem 29231 | Lemma for ~ upgrres . (Co... |
| umgrreslem 29232 | Lemma for ~ umgrres and ~ ... |
| upgrres 29233 | A subgraph obtained by rem... |
| umgrres 29234 | A subgraph obtained by rem... |
| usgrres 29235 | A subgraph obtained by rem... |
| upgrres1lem1 29236 | Lemma 1 for ~ upgrres1 . ... |
| umgrres1lem 29237 | Lemma for ~ umgrres1 . (C... |
| upgrres1lem2 29238 | Lemma 2 for ~ upgrres1 . ... |
| upgrres1lem3 29239 | Lemma 3 for ~ upgrres1 . ... |
| upgrres1 29240 | A pseudograph obtained by ... |
| umgrres1 29241 | A multigraph obtained by r... |
| usgrres1 29242 | Restricting a simple graph... |
| isfusgr 29245 | The property of being a fi... |
| fusgrvtxfi 29246 | A finite simple graph has ... |
| isfusgrf1 29247 | The property of being a fi... |
| isfusgrcl 29248 | The property of being a fi... |
| fusgrusgr 29249 | A finite simple graph is a... |
| opfusgr 29250 | A finite simple graph repr... |
| usgredgffibi 29251 | The number of edges in a s... |
| fusgredgfi 29252 | In a finite simple graph t... |
| usgr1v0e 29253 | The size of a (finite) sim... |
| usgrfilem 29254 | In a finite simple graph, ... |
| fusgrfisbase 29255 | Induction base for ~ fusgr... |
| fusgrfisstep 29256 | Induction step in ~ fusgrf... |
| fusgrfis 29257 | A finite simple graph is o... |
| fusgrfupgrfs 29258 | A finite simple graph is a... |
| nbgrprc0 29261 | The set of neighbors is em... |
| nbgrcl 29262 | If a class ` X ` has at le... |
| nbgrval 29263 | The set of neighbors of a ... |
| dfnbgr2 29264 | Alternate definition of th... |
| dfnbgr3 29265 | Alternate definition of th... |
| nbgrnvtx0 29266 | If a class ` X ` is not a ... |
| nbgrel 29267 | Characterization of a neig... |
| nbgrisvtx 29268 | Every neighbor ` N ` of a ... |
| nbgrssvtx 29269 | The neighbors of a vertex ... |
| nbuhgr 29270 | The set of neighbors of a ... |
| nbupgr 29271 | The set of neighbors of a ... |
| nbupgrel 29272 | A neighbor of a vertex in ... |
| nbumgrvtx 29273 | The set of neighbors of a ... |
| nbumgr 29274 | The set of neighbors of an... |
| nbusgrvtx 29275 | The set of neighbors of a ... |
| nbusgr 29276 | The set of neighbors of an... |
| nbgr2vtx1edg 29277 | If a graph has two vertice... |
| nbuhgr2vtx1edgblem 29278 | Lemma for ~ nbuhgr2vtx1edg... |
| nbuhgr2vtx1edgb 29279 | If a hypergraph has two ve... |
| nbusgreledg 29280 | A class/vertex is a neighb... |
| uhgrnbgr0nb 29281 | A vertex which is not endp... |
| nbgr0vtx 29282 | In a null graph (with no v... |
| nbgr0edglem 29283 | Lemma for ~ nbgr0edg and ~... |
| nbgr0edg 29284 | In an empty graph (with no... |
| nbgr1vtx 29285 | In a graph with one vertex... |
| nbgrnself 29286 | A vertex in a graph is not... |
| nbgrnself2 29287 | A class ` X ` is not a nei... |
| nbgrssovtx 29288 | The neighbors of a vertex ... |
| nbgrssvwo2 29289 | The neighbors of a vertex ... |
| nbgrsym 29290 | In a graph, the neighborho... |
| nbupgrres 29291 | The neighborhood of a vert... |
| usgrnbcnvfv 29292 | Applying the edge function... |
| nbusgredgeu 29293 | For each neighbor of a ver... |
| edgnbusgreu 29294 | For each edge incident to ... |
| nbusgredgeu0 29295 | For each neighbor of a ver... |
| nbusgrf1o0 29296 | The mapping of neighbors o... |
| nbusgrf1o1 29297 | The set of neighbors of a ... |
| nbusgrf1o 29298 | The set of neighbors of a ... |
| nbedgusgr 29299 | The number of neighbors of... |
| edgusgrnbfin 29300 | The number of neighbors of... |
| nbusgrfi 29301 | The class of neighbors of ... |
| nbfiusgrfi 29302 | The class of neighbors of ... |
| hashnbusgrnn0 29303 | The number of neighbors of... |
| nbfusgrlevtxm1 29304 | The number of neighbors of... |
| nbfusgrlevtxm2 29305 | If there is a vertex which... |
| nbusgrvtxm1 29306 | If the number of neighbors... |
| nb3grprlem1 29307 | Lemma 1 for ~ nb3grpr . (... |
| nb3grprlem2 29308 | Lemma 2 for ~ nb3grpr . (... |
| nb3grpr 29309 | The neighbors of a vertex ... |
| nb3grpr2 29310 | The neighbors of a vertex ... |
| nb3gr2nb 29311 | If the neighbors of two ve... |
| uvtxval 29314 | The set of all universal v... |
| uvtxel 29315 | A universal vertex, i.e. a... |
| uvtxisvtx 29316 | A universal vertex is a ve... |
| uvtxssvtx 29317 | The set of the universal v... |
| vtxnbuvtx 29318 | A universal vertex has all... |
| uvtxnbgrss 29319 | A universal vertex has all... |
| uvtxnbgrvtx 29320 | A universal vertex is neig... |
| uvtx0 29321 | There is no universal vert... |
| isuvtx 29322 | The set of all universal v... |
| uvtxel1 29323 | Characterization of a univ... |
| uvtx01vtx 29324 | If a graph/class has no ed... |
| uvtx2vtx1edg 29325 | If a graph has two vertice... |
| uvtx2vtx1edgb 29326 | If a hypergraph has two ve... |
| uvtxnbgr 29327 | A universal vertex has all... |
| uvtxnbgrb 29328 | A vertex is universal iff ... |
| uvtxusgr 29329 | The set of all universal v... |
| uvtxusgrel 29330 | A universal vertex, i.e. a... |
| uvtxnm1nbgr 29331 | A universal vertex has ` n... |
| nbusgrvtxm1uvtx 29332 | If the number of neighbors... |
| uvtxnbvtxm1 29333 | A universal vertex has ` n... |
| nbupgruvtxres 29334 | The neighborhood of a univ... |
| uvtxupgrres 29335 | A universal vertex is univ... |
| cplgruvtxb 29340 | A graph ` G ` is complete ... |
| prcliscplgr 29341 | A proper class (representi... |
| iscplgr 29342 | The property of being a co... |
| iscplgrnb 29343 | A graph is complete iff al... |
| iscplgredg 29344 | A graph ` G ` is complete ... |
| iscusgr 29345 | The property of being a co... |
| cusgrusgr 29346 | A complete simple graph is... |
| cusgrcplgr 29347 | A complete simple graph is... |
| iscusgrvtx 29348 | A simple graph is complete... |
| cusgruvtxb 29349 | A simple graph is complete... |
| iscusgredg 29350 | A simple graph is complete... |
| cusgredg 29351 | In a complete simple graph... |
| cplgr0 29352 | The null graph (with no ve... |
| cusgr0 29353 | The null graph (with no ve... |
| cplgr0v 29354 | A null graph (with no vert... |
| cusgr0v 29355 | A graph with no vertices a... |
| cplgr1vlem 29356 | Lemma for ~ cplgr1v and ~ ... |
| cplgr1v 29357 | A graph with one vertex is... |
| cusgr1v 29358 | A graph with one vertex an... |
| cplgr2v 29359 | An undirected hypergraph w... |
| cplgr2vpr 29360 | An undirected hypergraph w... |
| nbcplgr 29361 | In a complete graph, each ... |
| cplgr3v 29362 | A pseudograph with three (... |
| cusgr3vnbpr 29363 | The neighbors of a vertex ... |
| cplgrop 29364 | A complete graph represent... |
| cusgrop 29365 | A complete simple graph re... |
| cusgrexilem1 29366 | Lemma 1 for ~ cusgrexi . ... |
| usgrexilem 29367 | Lemma for ~ usgrexi . (Co... |
| usgrexi 29368 | An arbitrary set regarded ... |
| cusgrexilem2 29369 | Lemma 2 for ~ cusgrexi . ... |
| cusgrexi 29370 | An arbitrary set ` V ` reg... |
| cusgrexg 29371 | For each set there is a se... |
| structtousgr 29372 | Any (extensible) structure... |
| structtocusgr 29373 | Any (extensible) structure... |
| cffldtocusgr 29374 | The field of complex numbe... |
| cffldtocusgrOLD 29375 | Obsolete version of ~ cffl... |
| cusgrres 29376 | Restricting a complete sim... |
| cusgrsizeindb0 29377 | Base case of the induction... |
| cusgrsizeindb1 29378 | Base case of the induction... |
| cusgrsizeindslem 29379 | Lemma for ~ cusgrsizeinds ... |
| cusgrsizeinds 29380 | Part 1 of induction step i... |
| cusgrsize2inds 29381 | Induction step in ~ cusgrs... |
| cusgrsize 29382 | The size of a finite compl... |
| cusgrfilem1 29383 | Lemma 1 for ~ cusgrfi . (... |
| cusgrfilem2 29384 | Lemma 2 for ~ cusgrfi . (... |
| cusgrfilem3 29385 | Lemma 3 for ~ cusgrfi . (... |
| cusgrfi 29386 | If the size of a complete ... |
| usgredgsscusgredg 29387 | A simple graph is a subgra... |
| usgrsscusgr 29388 | A simple graph is a subgra... |
| sizusglecusglem1 29389 | Lemma 1 for ~ sizusglecusg... |
| sizusglecusglem2 29390 | Lemma 2 for ~ sizusglecusg... |
| sizusglecusg 29391 | The size of a simple graph... |
| fusgrmaxsize 29392 | The maximum size of a fini... |
| vtxdgfval 29395 | The value of the vertex de... |
| vtxdgval 29396 | The degree of a vertex. (... |
| vtxdgfival 29397 | The degree of a vertex for... |
| vtxdgop 29398 | The vertex degree expresse... |
| vtxdgf 29399 | The vertex degree function... |
| vtxdgelxnn0 29400 | The degree of a vertex is ... |
| vtxdg0v 29401 | The degree of a vertex in ... |
| vtxdg0e 29402 | The degree of a vertex in ... |
| vtxdgfisnn0 29403 | The degree of a vertex in ... |
| vtxdgfisf 29404 | The vertex degree function... |
| vtxdeqd 29405 | Equality theorem for the v... |
| vtxduhgr0e 29406 | The degree of a vertex in ... |
| vtxdlfuhgr1v 29407 | The degree of the vertex i... |
| vdumgr0 29408 | A vertex in a multigraph h... |
| vtxdun 29409 | The degree of a vertex in ... |
| vtxdfiun 29410 | The degree of a vertex in ... |
| vtxduhgrun 29411 | The degree of a vertex in ... |
| vtxduhgrfiun 29412 | The degree of a vertex in ... |
| vtxdlfgrval 29413 | The value of the vertex de... |
| vtxdumgrval 29414 | The value of the vertex de... |
| vtxdusgrval 29415 | The value of the vertex de... |
| vtxd0nedgb 29416 | A vertex has degree 0 iff ... |
| vtxdushgrfvedglem 29417 | Lemma for ~ vtxdushgrfvedg... |
| vtxdushgrfvedg 29418 | The value of the vertex de... |
| vtxdusgrfvedg 29419 | The value of the vertex de... |
| vtxduhgr0nedg 29420 | If a vertex in a hypergrap... |
| vtxdumgr0nedg 29421 | If a vertex in a multigrap... |
| vtxduhgr0edgnel 29422 | A vertex in a hypergraph h... |
| vtxdusgr0edgnel 29423 | A vertex in a simple graph... |
| vtxdusgr0edgnelALT 29424 | Alternate proof of ~ vtxdu... |
| vtxdgfusgrf 29425 | The vertex degree function... |
| vtxdgfusgr 29426 | In a finite simple graph, ... |
| fusgrn0degnn0 29427 | In a nonempty, finite grap... |
| 1loopgruspgr 29428 | A graph with one edge whic... |
| 1loopgredg 29429 | The set of edges in a grap... |
| 1loopgrnb0 29430 | In a graph (simple pseudog... |
| 1loopgrvd2 29431 | The vertex degree of a one... |
| 1loopgrvd0 29432 | The vertex degree of a one... |
| 1hevtxdg0 29433 | The vertex degree of verte... |
| 1hevtxdg1 29434 | The vertex degree of verte... |
| 1hegrvtxdg1 29435 | The vertex degree of a gra... |
| 1hegrvtxdg1r 29436 | The vertex degree of a gra... |
| 1egrvtxdg1 29437 | The vertex degree of a one... |
| 1egrvtxdg1r 29438 | The vertex degree of a one... |
| 1egrvtxdg0 29439 | The vertex degree of a one... |
| p1evtxdeqlem 29440 | Lemma for ~ p1evtxdeq and ... |
| p1evtxdeq 29441 | If an edge ` E ` which doe... |
| p1evtxdp1 29442 | If an edge ` E ` (not bein... |
| uspgrloopvtx 29443 | The set of vertices in a g... |
| uspgrloopvtxel 29444 | A vertex in a graph (simpl... |
| uspgrloopiedg 29445 | The set of edges in a grap... |
| uspgrloopedg 29446 | The set of edges in a grap... |
| uspgrloopnb0 29447 | In a graph (simple pseudog... |
| uspgrloopvd2 29448 | The vertex degree of a one... |
| umgr2v2evtx 29449 | The set of vertices in a m... |
| umgr2v2evtxel 29450 | A vertex in a multigraph w... |
| umgr2v2eiedg 29451 | The edge function in a mul... |
| umgr2v2eedg 29452 | The set of edges in a mult... |
| umgr2v2e 29453 | A multigraph with two edge... |
| umgr2v2enb1 29454 | In a multigraph with two e... |
| umgr2v2evd2 29455 | In a multigraph with two e... |
| hashnbusgrvd 29456 | In a simple graph, the num... |
| usgruvtxvdb 29457 | In a finite simple graph w... |
| vdiscusgrb 29458 | A finite simple graph with... |
| vdiscusgr 29459 | In a finite complete simpl... |
| vtxdusgradjvtx 29460 | The degree of a vertex in ... |
| usgrvd0nedg 29461 | If a vertex in a simple gr... |
| uhgrvd00 29462 | If every vertex in a hyper... |
| usgrvd00 29463 | If every vertex in a simpl... |
| vdegp1ai 29464 | The induction step for a v... |
| vdegp1bi 29465 | The induction step for a v... |
| vdegp1ci 29466 | The induction step for a v... |
| vtxdginducedm1lem1 29467 | Lemma 1 for ~ vtxdginduced... |
| vtxdginducedm1lem2 29468 | Lemma 2 for ~ vtxdginduced... |
| vtxdginducedm1lem3 29469 | Lemma 3 for ~ vtxdginduced... |
| vtxdginducedm1lem4 29470 | Lemma 4 for ~ vtxdginduced... |
| vtxdginducedm1 29471 | The degree of a vertex ` v... |
| vtxdginducedm1fi 29472 | The degree of a vertex ` v... |
| finsumvtxdg2ssteplem1 29473 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem2 29474 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem3 29475 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem4 29476 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2sstep 29477 | Induction step of ~ finsum... |
| finsumvtxdg2size 29478 | The sum of the degrees of ... |
| fusgr1th 29479 | The sum of the degrees of ... |
| finsumvtxdgeven 29480 | The sum of the degrees of ... |
| vtxdgoddnumeven 29481 | The number of vertices of ... |
| fusgrvtxdgonume 29482 | The number of vertices of ... |
| isrgr 29487 | The property of a class be... |
| rgrprop 29488 | The properties of a k-regu... |
| isrusgr 29489 | The property of being a k-... |
| rusgrprop 29490 | The properties of a k-regu... |
| rusgrrgr 29491 | A k-regular simple graph i... |
| rusgrusgr 29492 | A k-regular simple graph i... |
| finrusgrfusgr 29493 | A finite regular simple gr... |
| isrusgr0 29494 | The property of being a k-... |
| rusgrprop0 29495 | The properties of a k-regu... |
| usgreqdrusgr 29496 | If all vertices in a simpl... |
| fusgrregdegfi 29497 | In a nonempty finite simpl... |
| fusgrn0eqdrusgr 29498 | If all vertices in a nonem... |
| frusgrnn0 29499 | In a nonempty finite k-reg... |
| 0edg0rgr 29500 | A graph is 0-regular if it... |
| uhgr0edg0rgr 29501 | A hypergraph is 0-regular ... |
| uhgr0edg0rgrb 29502 | A hypergraph is 0-regular ... |
| usgr0edg0rusgr 29503 | A simple graph is 0-regula... |
| 0vtxrgr 29504 | A null graph (with no vert... |
| 0vtxrusgr 29505 | A graph with no vertices a... |
| 0uhgrrusgr 29506 | The null graph as hypergra... |
| 0grrusgr 29507 | The null graph represented... |
| 0grrgr 29508 | The null graph represented... |
| cusgrrusgr 29509 | A complete simple graph wi... |
| cusgrm1rusgr 29510 | A finite simple graph with... |
| rusgrpropnb 29511 | The properties of a k-regu... |
| rusgrpropedg 29512 | The properties of a k-regu... |
| rusgrpropadjvtx 29513 | The properties of a k-regu... |
| rusgrnumwrdl2 29514 | In a k-regular simple grap... |
| rusgr1vtxlem 29515 | Lemma for ~ rusgr1vtx . (... |
| rusgr1vtx 29516 | If a k-regular simple grap... |
| rgrusgrprc 29517 | The class of 0-regular sim... |
| rusgrprc 29518 | The class of 0-regular sim... |
| rgrprc 29519 | The class of 0-regular gra... |
| rgrprcx 29520 | The class of 0-regular gra... |
| rgrx0ndm 29521 | 0 is not in the domain of ... |
| rgrx0nd 29522 | The potentially alternativ... |
| ewlksfval 29529 | The set of s-walks of edge... |
| isewlk 29530 | Conditions for a function ... |
| ewlkprop 29531 | Properties of an s-walk of... |
| ewlkinedg 29532 | The intersection (common v... |
| ewlkle 29533 | An s-walk of edges is also... |
| upgrewlkle2 29534 | In a pseudograph, there is... |
| wkslem1 29535 | Lemma 1 for walks to subst... |
| wkslem2 29536 | Lemma 2 for walks to subst... |
| wksfval 29537 | The set of walks (in an un... |
| iswlk 29538 | Properties of a pair of fu... |
| wlkprop 29539 | Properties of a walk. (Co... |
| wlkv 29540 | The classes involved in a ... |
| iswlkg 29541 | Generalization of ~ iswlk ... |
| wlkf 29542 | The mapping enumerating th... |
| wlkcl 29543 | A walk has length ` # ( F ... |
| wlkp 29544 | The mapping enumerating th... |
| wlkpwrd 29545 | The sequence of vertices o... |
| wlklenvp1 29546 | The number of vertices of ... |
| wksv 29547 | The class of walks is a se... |
| wksvOLD 29548 | Obsolete version of ~ wksv... |
| wlkn0 29549 | The sequence of vertices o... |
| wlklenvm1 29550 | The number of edges of a w... |
| ifpsnprss 29551 | Lemma for ~ wlkvtxeledg : ... |
| wlkvtxeledg 29552 | Each pair of adjacent vert... |
| wlkvtxiedg 29553 | The vertices of a walk are... |
| relwlk 29554 | The set ` ( Walks `` G ) `... |
| wlkvv 29555 | If there is at least one w... |
| wlkop 29556 | A walk is an ordered pair.... |
| wlkcpr 29557 | A walk as class with two c... |
| wlk2f 29558 | If there is a walk ` W ` t... |
| wlkcomp 29559 | A walk expressed by proper... |
| wlkcompim 29560 | Implications for the prope... |
| wlkelwrd 29561 | The components of a walk a... |
| wlkeq 29562 | Conditions for two walks (... |
| edginwlk 29563 | The value of the edge func... |
| upgredginwlk 29564 | The value of the edge func... |
| iedginwlk 29565 | The value of the edge func... |
| wlkl1loop 29566 | A walk of length 1 from a ... |
| wlk1walk 29567 | A walk is a 1-walk "on the... |
| wlk1ewlk 29568 | A walk is an s-walk "on th... |
| upgriswlk 29569 | Properties of a pair of fu... |
| upgrwlkedg 29570 | The edges of a walk in a p... |
| upgrwlkcompim 29571 | Implications for the prope... |
| wlkvtxedg 29572 | The vertices of a walk are... |
| upgrwlkvtxedg 29573 | The pairs of connected ver... |
| uspgr2wlkeq 29574 | Conditions for two walks w... |
| uspgr2wlkeq2 29575 | Conditions for two walks w... |
| uspgr2wlkeqi 29576 | Conditions for two walks w... |
| umgrwlknloop 29577 | In a multigraph, each walk... |
| wlkResOLD 29578 | Obsolete version of ~ opab... |
| wlkv0 29579 | If there is a walk in the ... |
| g0wlk0 29580 | There is no walk in a null... |
| 0wlk0 29581 | There is no walk for the e... |
| wlk0prc 29582 | There is no walk in a null... |
| wlklenvclwlk 29583 | The number of vertices in ... |
| wlkson 29584 | The set of walks between t... |
| iswlkon 29585 | Properties of a pair of fu... |
| wlkonprop 29586 | Properties of a walk betwe... |
| wlkpvtx 29587 | A walk connects vertices. ... |
| wlkepvtx 29588 | The endpoints of a walk ar... |
| wlkoniswlk 29589 | A walk between two vertice... |
| wlkonwlk 29590 | A walk is a walk between i... |
| wlkonwlk1l 29591 | A walk is a walk from its ... |
| wlksoneq1eq2 29592 | Two walks with identical s... |
| wlkonl1iedg 29593 | If there is a walk between... |
| wlkon2n0 29594 | The length of a walk betwe... |
| 2wlklem 29595 | Lemma for theorems for wal... |
| upgr2wlk 29596 | Properties of a pair of fu... |
| wlkreslem 29597 | Lemma for ~ wlkres . (Con... |
| wlkres 29598 | The restriction ` <. H , Q... |
| redwlklem 29599 | Lemma for ~ redwlk . (Con... |
| redwlk 29600 | A walk ending at the last ... |
| wlkp1lem1 29601 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem2 29602 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem3 29603 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem4 29604 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem5 29605 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem6 29606 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem7 29607 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem8 29608 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1 29609 | Append one path segment (e... |
| wlkdlem1 29610 | Lemma 1 for ~ wlkd . (Con... |
| wlkdlem2 29611 | Lemma 2 for ~ wlkd . (Con... |
| wlkdlem3 29612 | Lemma 3 for ~ wlkd . (Con... |
| wlkdlem4 29613 | Lemma 4 for ~ wlkd . (Con... |
| wlkd 29614 | Two words representing a w... |
| lfgrwlkprop 29615 | Two adjacent vertices in a... |
| lfgriswlk 29616 | Conditions for a pair of f... |
| lfgrwlknloop 29617 | In a loop-free graph, each... |
| reltrls 29622 | The set ` ( Trails `` G ) ... |
| trlsfval 29623 | The set of trails (in an u... |
| istrl 29624 | Conditions for a pair of c... |
| trliswlk 29625 | A trail is a walk. (Contr... |
| trlf1 29626 | The enumeration ` F ` of a... |
| trlreslem 29627 | Lemma for ~ trlres . Form... |
| trlres 29628 | The restriction ` <. H , Q... |
| upgrtrls 29629 | The set of trails in a pse... |
| upgristrl 29630 | Properties of a pair of fu... |
| upgrf1istrl 29631 | Properties of a pair of a ... |
| wksonproplem 29632 | Lemma for theorems for pro... |
| wksonproplemOLD 29633 | Obsolete version of ~ wkso... |
| trlsonfval 29634 | The set of trails between ... |
| istrlson 29635 | Properties of a pair of fu... |
| trlsonprop 29636 | Properties of a trail betw... |
| trlsonistrl 29637 | A trail between two vertic... |
| trlsonwlkon 29638 | A trail between two vertic... |
| trlontrl 29639 | A trail is a trail between... |
| relpths 29648 | The set ` ( Paths `` G ) `... |
| pthsfval 29649 | The set of paths (in an un... |
| spthsfval 29650 | The set of simple paths (i... |
| ispth 29651 | Conditions for a pair of c... |
| isspth 29652 | Conditions for a pair of c... |
| pthistrl 29653 | A path is a trail (in an u... |
| spthispth 29654 | A simple path is a path (i... |
| pthiswlk 29655 | A path is a walk (in an un... |
| spthiswlk 29656 | A simple path is a walk (i... |
| pthdivtx 29657 | The inner vertices of a pa... |
| pthdadjvtx 29658 | The adjacent vertices of a... |
| dfpth2 29659 | Alternate definition for a... |
| pthdifv 29660 | The vertices of a path are... |
| 2pthnloop 29661 | A path of length at least ... |
| upgr2pthnlp 29662 | A path of length at least ... |
| spthdifv 29663 | The vertices of a simple p... |
| spthdep 29664 | A simple path (at least of... |
| pthdepisspth 29665 | A path with different star... |
| upgrwlkdvdelem 29666 | Lemma for ~ upgrwlkdvde . ... |
| upgrwlkdvde 29667 | In a pseudograph, all edge... |
| upgrspthswlk 29668 | The set of simple paths in... |
| upgrwlkdvspth 29669 | A walk consisting of diffe... |
| pthsonfval 29670 | The set of paths between t... |
| spthson 29671 | The set of simple paths be... |
| ispthson 29672 | Properties of a pair of fu... |
| isspthson 29673 | Properties of a pair of fu... |
| pthsonprop 29674 | Properties of a path betwe... |
| spthonprop 29675 | Properties of a simple pat... |
| pthonispth 29676 | A path between two vertice... |
| pthontrlon 29677 | A path between two vertice... |
| pthonpth 29678 | A path is a path between i... |
| isspthonpth 29679 | A pair of functions is a s... |
| spthonisspth 29680 | A simple path between to v... |
| spthonpthon 29681 | A simple path between two ... |
| spthonepeq 29682 | The endpoints of a simple ... |
| uhgrwkspthlem1 29683 | Lemma 1 for ~ uhgrwkspth .... |
| uhgrwkspthlem2 29684 | Lemma 2 for ~ uhgrwkspth .... |
| uhgrwkspth 29685 | Any walk of length 1 betwe... |
| usgr2wlkneq 29686 | The vertices and edges are... |
| usgr2wlkspthlem1 29687 | Lemma 1 for ~ usgr2wlkspth... |
| usgr2wlkspthlem2 29688 | Lemma 2 for ~ usgr2wlkspth... |
| usgr2wlkspth 29689 | In a simple graph, any wal... |
| usgr2trlncl 29690 | In a simple graph, any tra... |
| usgr2trlspth 29691 | In a simple graph, any tra... |
| usgr2pthspth 29692 | In a simple graph, any pat... |
| usgr2pthlem 29693 | Lemma for ~ usgr2pth . (C... |
| usgr2pth 29694 | In a simple graph, there i... |
| usgr2pth0 29695 | In a simply graph, there i... |
| pthdlem1 29696 | Lemma 1 for ~ pthd . (Con... |
| pthdlem2lem 29697 | Lemma for ~ pthdlem2 . (C... |
| pthdlem2 29698 | Lemma 2 for ~ pthd . (Con... |
| pthd 29699 | Two words representing a t... |
| clwlks 29702 | The set of closed walks (i... |
| isclwlk 29703 | A pair of functions repres... |
| clwlkiswlk 29704 | A closed walk is a walk (i... |
| clwlkwlk 29705 | Closed walks are walks (in... |
| clwlkswks 29706 | Closed walks are walks (in... |
| isclwlke 29707 | Properties of a pair of fu... |
| isclwlkupgr 29708 | Properties of a pair of fu... |
| clwlkcomp 29709 | A closed walk expressed by... |
| clwlkcompim 29710 | Implications for the prope... |
| upgrclwlkcompim 29711 | Implications for the prope... |
| clwlkcompbp 29712 | Basic properties of the co... |
| clwlkl1loop 29713 | A closed walk of length 1 ... |
| crcts 29718 | The set of circuits (in an... |
| cycls 29719 | The set of cycles (in an u... |
| iscrct 29720 | Sufficient and necessary c... |
| iscycl 29721 | Sufficient and necessary c... |
| crctprop 29722 | The properties of a circui... |
| cyclprop 29723 | The properties of a cycle:... |
| crctisclwlk 29724 | A circuit is a closed walk... |
| crctistrl 29725 | A circuit is a trail. (Co... |
| crctiswlk 29726 | A circuit is a walk. (Con... |
| cyclispth 29727 | A cycle is a path. (Contr... |
| cycliswlk 29728 | A cycle is a walk. (Contr... |
| cycliscrct 29729 | A cycle is a circuit. (Co... |
| cyclnumvtx 29730 | The number of vertices of ... |
| cyclnspth 29731 | A (non-trivial) cycle is n... |
| pthisspthorcycl 29732 | A path is either a simple ... |
| pthspthcyc 29733 | A pair ` <. F , P >. ` rep... |
| cyclispthon 29734 | A cycle is a path starting... |
| lfgrn1cycl 29735 | In a loop-free graph there... |
| usgr2trlncrct 29736 | In a simple graph, any tra... |
| umgrn1cycl 29737 | In a multigraph graph (wit... |
| uspgrn2crct 29738 | In a simple pseudograph th... |
| usgrn2cycl 29739 | In a simple graph there ar... |
| crctcshwlkn0lem1 29740 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem2 29741 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem3 29742 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem4 29743 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem5 29744 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem6 29745 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem7 29746 | Lemma for ~ crctcshwlkn0 .... |
| crctcshlem1 29747 | Lemma for ~ crctcsh . (Co... |
| crctcshlem2 29748 | Lemma for ~ crctcsh . (Co... |
| crctcshlem3 29749 | Lemma for ~ crctcsh . (Co... |
| crctcshlem4 29750 | Lemma for ~ crctcsh . (Co... |
| crctcshwlkn0 29751 | Cyclically shifting the in... |
| crctcshwlk 29752 | Cyclically shifting the in... |
| crctcshtrl 29753 | Cyclically shifting the in... |
| crctcsh 29754 | Cyclically shifting the in... |
| wwlks 29765 | The set of walks (in an un... |
| iswwlks 29766 | A word over the set of ver... |
| wwlksn 29767 | The set of walks (in an un... |
| iswwlksn 29768 | A word over the set of ver... |
| wwlksnprcl 29769 | Derivation of the length o... |
| iswwlksnx 29770 | Properties of a word to re... |
| wwlkbp 29771 | Basic properties of a walk... |
| wwlknbp 29772 | Basic properties of a walk... |
| wwlknp 29773 | Properties of a set being ... |
| wwlknbp1 29774 | Other basic properties of ... |
| wwlknvtx 29775 | The symbols of a word ` W ... |
| wwlknllvtx 29776 | If a word ` W ` represents... |
| wwlknlsw 29777 | If a word represents a wal... |
| wspthsn 29778 | The set of simple paths of... |
| iswspthn 29779 | An element of the set of s... |
| wspthnp 29780 | Properties of a set being ... |
| wwlksnon 29781 | The set of walks of a fixe... |
| wspthsnon 29782 | The set of simple paths of... |
| iswwlksnon 29783 | The set of walks of a fixe... |
| wwlksnon0 29784 | Sufficient conditions for ... |
| wwlksonvtx 29785 | If a word ` W ` represents... |
| iswspthsnon 29786 | The set of simple paths of... |
| wwlknon 29787 | An element of the set of w... |
| wspthnon 29788 | An element of the set of s... |
| wspthnonp 29789 | Properties of a set being ... |
| wspthneq1eq2 29790 | Two simple paths with iden... |
| wwlksn0s 29791 | The set of all walks as wo... |
| wwlkssswrd 29792 | Walks (represented by word... |
| wwlksn0 29793 | A walk of length 0 is repr... |
| 0enwwlksnge1 29794 | In graphs without edges, t... |
| wwlkswwlksn 29795 | A walk of a fixed length a... |
| wwlkssswwlksn 29796 | The walks of a fixed lengt... |
| wlkiswwlks1 29797 | The sequence of vertices i... |
| wlklnwwlkln1 29798 | The sequence of vertices i... |
| wlkiswwlks2lem1 29799 | Lemma 1 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem2 29800 | Lemma 2 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem3 29801 | Lemma 3 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem4 29802 | Lemma 4 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem5 29803 | Lemma 5 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem6 29804 | Lemma 6 for ~ wlkiswwlks2 ... |
| wlkiswwlks2 29805 | A walk as word corresponds... |
| wlkiswwlks 29806 | A walk as word corresponds... |
| wlkiswwlksupgr2 29807 | A walk as word corresponds... |
| wlkiswwlkupgr 29808 | A walk as word corresponds... |
| wlkswwlksf1o 29809 | The mapping of (ordinary) ... |
| wlkswwlksen 29810 | The set of walks as words ... |
| wwlksm1edg 29811 | Removing the trailing edge... |
| wlklnwwlkln2lem 29812 | Lemma for ~ wlklnwwlkln2 a... |
| wlklnwwlkln2 29813 | A walk of length ` N ` as ... |
| wlklnwwlkn 29814 | A walk of length ` N ` as ... |
| wlklnwwlklnupgr2 29815 | A walk of length ` N ` as ... |
| wlklnwwlknupgr 29816 | A walk of length ` N ` as ... |
| wlknewwlksn 29817 | If a walk in a pseudograph... |
| wlknwwlksnbij 29818 | The mapping ` ( t e. T |->... |
| wlknwwlksnen 29819 | In a simple pseudograph, t... |
| wlknwwlksneqs 29820 | The set of walks of a fixe... |
| wwlkseq 29821 | Equality of two walks (as ... |
| wwlksnred 29822 | Reduction of a walk (as wo... |
| wwlksnext 29823 | Extension of a walk (as wo... |
| wwlksnextbi 29824 | Extension of a walk (as wo... |
| wwlksnredwwlkn 29825 | For each walk (as word) of... |
| wwlksnredwwlkn0 29826 | For each walk (as word) of... |
| wwlksnextwrd 29827 | Lemma for ~ wwlksnextbij .... |
| wwlksnextfun 29828 | Lemma for ~ wwlksnextbij .... |
| wwlksnextinj 29829 | Lemma for ~ wwlksnextbij .... |
| wwlksnextsurj 29830 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij0 29831 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij 29832 | There is a bijection betwe... |
| wwlksnexthasheq 29833 | The number of the extensio... |
| disjxwwlksn 29834 | Sets of walks (as words) e... |
| wwlksnndef 29835 | Conditions for ` WWalksN `... |
| wwlksnfi 29836 | The number of walks repres... |
| wlksnfi 29837 | The number of walks of fix... |
| wlksnwwlknvbij 29838 | There is a bijection betwe... |
| wwlksnextproplem1 29839 | Lemma 1 for ~ wwlksnextpro... |
| wwlksnextproplem2 29840 | Lemma 2 for ~ wwlksnextpro... |
| wwlksnextproplem3 29841 | Lemma 3 for ~ wwlksnextpro... |
| wwlksnextprop 29842 | Adding additional properti... |
| disjxwwlkn 29843 | Sets of walks (as words) e... |
| hashwwlksnext 29844 | Number of walks (as words)... |
| wwlksnwwlksnon 29845 | A walk of fixed length is ... |
| wspthsnwspthsnon 29846 | A simple path of fixed len... |
| wspthsnonn0vne 29847 | If the set of simple paths... |
| wspthsswwlkn 29848 | The set of simple paths of... |
| wspthnfi 29849 | In a finite graph, the set... |
| wwlksnonfi 29850 | In a finite graph, the set... |
| wspthsswwlknon 29851 | The set of simple paths of... |
| wspthnonfi 29852 | In a finite graph, the set... |
| wspniunwspnon 29853 | The set of nonempty simple... |
| wspn0 29854 | If there are no vertices, ... |
| 2wlkdlem1 29855 | Lemma 1 for ~ 2wlkd . (Co... |
| 2wlkdlem2 29856 | Lemma 2 for ~ 2wlkd . (Co... |
| 2wlkdlem3 29857 | Lemma 3 for ~ 2wlkd . (Co... |
| 2wlkdlem4 29858 | Lemma 4 for ~ 2wlkd . (Co... |
| 2wlkdlem5 29859 | Lemma 5 for ~ 2wlkd . (Co... |
| 2pthdlem1 29860 | Lemma 1 for ~ 2pthd . (Co... |
| 2wlkdlem6 29861 | Lemma 6 for ~ 2wlkd . (Co... |
| 2wlkdlem7 29862 | Lemma 7 for ~ 2wlkd . (Co... |
| 2wlkdlem8 29863 | Lemma 8 for ~ 2wlkd . (Co... |
| 2wlkdlem9 29864 | Lemma 9 for ~ 2wlkd . (Co... |
| 2wlkdlem10 29865 | Lemma 10 for ~ 3wlkd . (C... |
| 2wlkd 29866 | Construction of a walk fro... |
| 2wlkond 29867 | A walk of length 2 from on... |
| 2trld 29868 | Construction of a trail fr... |
| 2trlond 29869 | A trail of length 2 from o... |
| 2pthd 29870 | A path of length 2 from on... |
| 2spthd 29871 | A simple path of length 2 ... |
| 2pthond 29872 | A simple path of length 2 ... |
| 2pthon3v 29873 | For a vertex adjacent to t... |
| umgr2adedgwlklem 29874 | Lemma for ~ umgr2adedgwlk ... |
| umgr2adedgwlk 29875 | In a multigraph, two adjac... |
| umgr2adedgwlkon 29876 | In a multigraph, two adjac... |
| umgr2adedgwlkonALT 29877 | Alternate proof for ~ umgr... |
| umgr2adedgspth 29878 | In a multigraph, two adjac... |
| umgr2wlk 29879 | In a multigraph, there is ... |
| umgr2wlkon 29880 | For each pair of adjacent ... |
| elwwlks2s3 29881 | A walk of length 2 as word... |
| midwwlks2s3 29882 | There is a vertex between ... |
| wwlks2onv 29883 | If a length 3 string repre... |
| elwwlks2ons3im 29884 | A walk as word of length 2... |
| elwwlks2ons3 29885 | For each walk of length 2 ... |
| s3wwlks2on 29886 | A length 3 string which re... |
| umgrwwlks2on 29887 | A walk of length 2 between... |
| wwlks2onsym 29888 | There is a walk of length ... |
| elwwlks2on 29889 | A walk of length 2 between... |
| elwspths2on 29890 | A simple path of length 2 ... |
| wpthswwlks2on 29891 | For two different vertices... |
| 2wspdisj 29892 | All simple paths of length... |
| 2wspiundisj 29893 | All simple paths of length... |
| usgr2wspthons3 29894 | A simple path of length 2 ... |
| usgr2wspthon 29895 | A simple path of length 2 ... |
| elwwlks2 29896 | A walk of length 2 between... |
| elwspths2spth 29897 | A simple path of length 2 ... |
| rusgrnumwwlkl1 29898 | In a k-regular graph, ther... |
| rusgrnumwwlkslem 29899 | Lemma for ~ rusgrnumwwlks ... |
| rusgrnumwwlklem 29900 | Lemma for ~ rusgrnumwwlk e... |
| rusgrnumwwlkb0 29901 | Induction base 0 for ~ rus... |
| rusgrnumwwlkb1 29902 | Induction base 1 for ~ rus... |
| rusgr0edg 29903 | Special case for graphs wi... |
| rusgrnumwwlks 29904 | Induction step for ~ rusgr... |
| rusgrnumwwlk 29905 | In a ` K `-regular graph, ... |
| rusgrnumwwlkg 29906 | In a ` K `-regular graph, ... |
| rusgrnumwlkg 29907 | In a k-regular graph, the ... |
| clwwlknclwwlkdif 29908 | The set ` A ` of walks of ... |
| clwwlknclwwlkdifnum 29909 | In a ` K `-regular graph, ... |
| clwwlk 29912 | The set of closed walks (i... |
| isclwwlk 29913 | Properties of a word to re... |
| clwwlkbp 29914 | Basic properties of a clos... |
| clwwlkgt0 29915 | There is no empty closed w... |
| clwwlksswrd 29916 | Closed walks (represented ... |
| clwwlk1loop 29917 | A closed walk of length 1 ... |
| clwwlkccatlem 29918 | Lemma for ~ clwwlkccat : i... |
| clwwlkccat 29919 | The concatenation of two w... |
| umgrclwwlkge2 29920 | A closed walk in a multigr... |
| clwlkclwwlklem2a1 29921 | Lemma 1 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a2 29922 | Lemma 2 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a3 29923 | Lemma 3 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2fv1 29924 | Lemma 4a for ~ clwlkclwwlk... |
| clwlkclwwlklem2fv2 29925 | Lemma 4b for ~ clwlkclwwlk... |
| clwlkclwwlklem2a4 29926 | Lemma 4 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a 29927 | Lemma for ~ clwlkclwwlklem... |
| clwlkclwwlklem1 29928 | Lemma 1 for ~ clwlkclwwlk ... |
| clwlkclwwlklem2 29929 | Lemma 2 for ~ clwlkclwwlk ... |
| clwlkclwwlklem3 29930 | Lemma 3 for ~ clwlkclwwlk ... |
| clwlkclwwlk 29931 | A closed walk as word of l... |
| clwlkclwwlk2 29932 | A closed walk corresponds ... |
| clwlkclwwlkflem 29933 | Lemma for ~ clwlkclwwlkf .... |
| clwlkclwwlkf1lem2 29934 | Lemma 2 for ~ clwlkclwwlkf... |
| clwlkclwwlkf1lem3 29935 | Lemma 3 for ~ clwlkclwwlkf... |
| clwlkclwwlkfolem 29936 | Lemma for ~ clwlkclwwlkfo ... |
| clwlkclwwlkf 29937 | ` F ` is a function from t... |
| clwlkclwwlkfo 29938 | ` F ` is a function from t... |
| clwlkclwwlkf1 29939 | ` F ` is a one-to-one func... |
| clwlkclwwlkf1o 29940 | ` F ` is a bijection betwe... |
| clwlkclwwlken 29941 | The set of the nonempty cl... |
| clwwisshclwwslemlem 29942 | Lemma for ~ clwwisshclwwsl... |
| clwwisshclwwslem 29943 | Lemma for ~ clwwisshclwws ... |
| clwwisshclwws 29944 | Cyclically shifting a clos... |
| clwwisshclwwsn 29945 | Cyclically shifting a clos... |
| erclwwlkrel 29946 | ` .~ ` is a relation. (Co... |
| erclwwlkeq 29947 | Two classes are equivalent... |
| erclwwlkeqlen 29948 | If two classes are equival... |
| erclwwlkref 29949 | ` .~ ` is a reflexive rela... |
| erclwwlksym 29950 | ` .~ ` is a symmetric rela... |
| erclwwlktr 29951 | ` .~ ` is a transitive rel... |
| erclwwlk 29952 | ` .~ ` is an equivalence r... |
| clwwlkn 29955 | The set of closed walks of... |
| isclwwlkn 29956 | A word over the set of ver... |
| clwwlkn0 29957 | There is no closed walk of... |
| clwwlkneq0 29958 | Sufficient conditions for ... |
| clwwlkclwwlkn 29959 | A closed walk of a fixed l... |
| clwwlksclwwlkn 29960 | The closed walks of a fixe... |
| clwwlknlen 29961 | The length of a word repre... |
| clwwlknnn 29962 | The length of a closed wal... |
| clwwlknwrd 29963 | A closed walk of a fixed l... |
| clwwlknbp 29964 | Basic properties of a clos... |
| isclwwlknx 29965 | Characterization of a word... |
| clwwlknp 29966 | Properties of a set being ... |
| clwwlknwwlksn 29967 | A word representing a clos... |
| clwwlknlbonbgr1 29968 | The last but one vertex in... |
| clwwlkinwwlk 29969 | If the initial vertex of a... |
| clwwlkn1 29970 | A closed walk of length 1 ... |
| loopclwwlkn1b 29971 | The singleton word consist... |
| clwwlkn1loopb 29972 | A word represents a closed... |
| clwwlkn2 29973 | A closed walk of length 2 ... |
| clwwlknfi 29974 | If there is only a finite ... |
| clwwlkel 29975 | Obtaining a closed walk (a... |
| clwwlkf 29976 | Lemma 1 for ~ clwwlkf1o : ... |
| clwwlkfv 29977 | Lemma 2 for ~ clwwlkf1o : ... |
| clwwlkf1 29978 | Lemma 3 for ~ clwwlkf1o : ... |
| clwwlkfo 29979 | Lemma 4 for ~ clwwlkf1o : ... |
| clwwlkf1o 29980 | F is a 1-1 onto function, ... |
| clwwlken 29981 | The set of closed walks of... |
| clwwlknwwlkncl 29982 | Obtaining a closed walk (a... |
| clwwlkwwlksb 29983 | A nonempty word over verti... |
| clwwlknwwlksnb 29984 | A word over vertices repre... |
| clwwlkext2edg 29985 | If a word concatenated wit... |
| wwlksext2clwwlk 29986 | If a word represents a wal... |
| wwlksubclwwlk 29987 | Any prefix of a word repre... |
| clwwnisshclwwsn 29988 | Cyclically shifting a clos... |
| eleclclwwlknlem1 29989 | Lemma 1 for ~ eleclclwwlkn... |
| eleclclwwlknlem2 29990 | Lemma 2 for ~ eleclclwwlkn... |
| clwwlknscsh 29991 | The set of cyclical shifts... |
| clwwlknccat 29992 | The concatenation of two w... |
| umgr2cwwk2dif 29993 | If a word represents a clo... |
| umgr2cwwkdifex 29994 | If a word represents a clo... |
| erclwwlknrel 29995 | ` .~ ` is a relation. (Co... |
| erclwwlkneq 29996 | Two classes are equivalent... |
| erclwwlkneqlen 29997 | If two classes are equival... |
| erclwwlknref 29998 | ` .~ ` is a reflexive rela... |
| erclwwlknsym 29999 | ` .~ ` is a symmetric rela... |
| erclwwlkntr 30000 | ` .~ ` is a transitive rel... |
| erclwwlkn 30001 | ` .~ ` is an equivalence r... |
| qerclwwlknfi 30002 | The quotient set of the se... |
| hashclwwlkn0 30003 | The number of closed walks... |
| eclclwwlkn1 30004 | An equivalence class accor... |
| eleclclwwlkn 30005 | A member of an equivalence... |
| hashecclwwlkn1 30006 | The size of every equivale... |
| umgrhashecclwwlk 30007 | The size of every equivale... |
| fusgrhashclwwlkn 30008 | The size of the set of clo... |
| clwwlkndivn 30009 | The size of the set of clo... |
| clwlknf1oclwwlknlem1 30010 | Lemma 1 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem2 30011 | Lemma 2 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem3 30012 | Lemma 3 for ~ clwlknf1oclw... |
| clwlknf1oclwwlkn 30013 | There is a one-to-one onto... |
| clwlkssizeeq 30014 | The size of the set of clo... |
| clwlksndivn 30015 | The size of the set of clo... |
| clwwlknonmpo 30018 | ` ( ClWWalksNOn `` G ) ` i... |
| clwwlknon 30019 | The set of closed walks on... |
| isclwwlknon 30020 | A word over the set of ver... |
| clwwlk0on0 30021 | There is no word over the ... |
| clwwlknon0 30022 | Sufficient conditions for ... |
| clwwlknonfin 30023 | In a finite graph ` G ` , ... |
| clwwlknonel 30024 | Characterization of a word... |
| clwwlknonccat 30025 | The concatenation of two w... |
| clwwlknon1 30026 | The set of closed walks on... |
| clwwlknon1loop 30027 | If there is a loop at vert... |
| clwwlknon1nloop 30028 | If there is no loop at ver... |
| clwwlknon1sn 30029 | The set of (closed) walks ... |
| clwwlknon1le1 30030 | There is at most one (clos... |
| clwwlknon2 30031 | The set of closed walks on... |
| clwwlknon2x 30032 | The set of closed walks on... |
| s2elclwwlknon2 30033 | Sufficient conditions of a... |
| clwwlknon2num 30034 | In a ` K `-regular graph `... |
| clwwlknonwwlknonb 30035 | A word over vertices repre... |
| clwwlknonex2lem1 30036 | Lemma 1 for ~ clwwlknonex2... |
| clwwlknonex2lem2 30037 | Lemma 2 for ~ clwwlknonex2... |
| clwwlknonex2 30038 | Extending a closed walk ` ... |
| clwwlknonex2e 30039 | Extending a closed walk ` ... |
| clwwlknondisj 30040 | The sets of closed walks o... |
| clwwlknun 30041 | The set of closed walks of... |
| clwwlkvbij 30042 | There is a bijection betwe... |
| 0ewlk 30043 | The empty set (empty seque... |
| 1ewlk 30044 | A sequence of 1 edge is an... |
| 0wlk 30045 | A pair of an empty set (of... |
| is0wlk 30046 | A pair of an empty set (of... |
| 0wlkonlem1 30047 | Lemma 1 for ~ 0wlkon and ~... |
| 0wlkonlem2 30048 | Lemma 2 for ~ 0wlkon and ~... |
| 0wlkon 30049 | A walk of length 0 from a ... |
| 0wlkons1 30050 | A walk of length 0 from a ... |
| 0trl 30051 | A pair of an empty set (of... |
| is0trl 30052 | A pair of an empty set (of... |
| 0trlon 30053 | A trail of length 0 from a... |
| 0pth 30054 | A pair of an empty set (of... |
| 0spth 30055 | A pair of an empty set (of... |
| 0pthon 30056 | A path of length 0 from a ... |
| 0pthon1 30057 | A path of length 0 from a ... |
| 0pthonv 30058 | For each vertex there is a... |
| 0clwlk 30059 | A pair of an empty set (of... |
| 0clwlkv 30060 | Any vertex (more precisely... |
| 0clwlk0 30061 | There is no closed walk in... |
| 0crct 30062 | A pair of an empty set (of... |
| 0cycl 30063 | A pair of an empty set (of... |
| 1pthdlem1 30064 | Lemma 1 for ~ 1pthd . (Co... |
| 1pthdlem2 30065 | Lemma 2 for ~ 1pthd . (Co... |
| 1wlkdlem1 30066 | Lemma 1 for ~ 1wlkd . (Co... |
| 1wlkdlem2 30067 | Lemma 2 for ~ 1wlkd . (Co... |
| 1wlkdlem3 30068 | Lemma 3 for ~ 1wlkd . (Co... |
| 1wlkdlem4 30069 | Lemma 4 for ~ 1wlkd . (Co... |
| 1wlkd 30070 | In a graph with two vertic... |
| 1trld 30071 | In a graph with two vertic... |
| 1pthd 30072 | In a graph with two vertic... |
| 1pthond 30073 | In a graph with two vertic... |
| upgr1wlkdlem1 30074 | Lemma 1 for ~ upgr1wlkd . ... |
| upgr1wlkdlem2 30075 | Lemma 2 for ~ upgr1wlkd . ... |
| upgr1wlkd 30076 | In a pseudograph with two ... |
| upgr1trld 30077 | In a pseudograph with two ... |
| upgr1pthd 30078 | In a pseudograph with two ... |
| upgr1pthond 30079 | In a pseudograph with two ... |
| lppthon 30080 | A loop (which is an edge a... |
| lp1cycl 30081 | A loop (which is an edge a... |
| 1pthon2v 30082 | For each pair of adjacent ... |
| 1pthon2ve 30083 | For each pair of adjacent ... |
| wlk2v2elem1 30084 | Lemma 1 for ~ wlk2v2e : ` ... |
| wlk2v2elem2 30085 | Lemma 2 for ~ wlk2v2e : T... |
| wlk2v2e 30086 | In a graph with two vertic... |
| ntrl2v2e 30087 | A walk which is not a trai... |
| 3wlkdlem1 30088 | Lemma 1 for ~ 3wlkd . (Co... |
| 3wlkdlem2 30089 | Lemma 2 for ~ 3wlkd . (Co... |
| 3wlkdlem3 30090 | Lemma 3 for ~ 3wlkd . (Co... |
| 3wlkdlem4 30091 | Lemma 4 for ~ 3wlkd . (Co... |
| 3wlkdlem5 30092 | Lemma 5 for ~ 3wlkd . (Co... |
| 3pthdlem1 30093 | Lemma 1 for ~ 3pthd . (Co... |
| 3wlkdlem6 30094 | Lemma 6 for ~ 3wlkd . (Co... |
| 3wlkdlem7 30095 | Lemma 7 for ~ 3wlkd . (Co... |
| 3wlkdlem8 30096 | Lemma 8 for ~ 3wlkd . (Co... |
| 3wlkdlem9 30097 | Lemma 9 for ~ 3wlkd . (Co... |
| 3wlkdlem10 30098 | Lemma 10 for ~ 3wlkd . (C... |
| 3wlkd 30099 | Construction of a walk fro... |
| 3wlkond 30100 | A walk of length 3 from on... |
| 3trld 30101 | Construction of a trail fr... |
| 3trlond 30102 | A trail of length 3 from o... |
| 3pthd 30103 | A path of length 3 from on... |
| 3pthond 30104 | A path of length 3 from on... |
| 3spthd 30105 | A simple path of length 3 ... |
| 3spthond 30106 | A simple path of length 3 ... |
| 3cycld 30107 | Construction of a 3-cycle ... |
| 3cyclpd 30108 | Construction of a 3-cycle ... |
| upgr3v3e3cycl 30109 | If there is a cycle of len... |
| uhgr3cyclexlem 30110 | Lemma for ~ uhgr3cyclex . ... |
| uhgr3cyclex 30111 | If there are three differe... |
| umgr3cyclex 30112 | If there are three (differ... |
| umgr3v3e3cycl 30113 | If and only if there is a ... |
| upgr4cycl4dv4e 30114 | If there is a cycle of len... |
| dfconngr1 30117 | Alternative definition of ... |
| isconngr 30118 | The property of being a co... |
| isconngr1 30119 | The property of being a co... |
| cusconngr 30120 | A complete hypergraph is c... |
| 0conngr 30121 | A graph without vertices i... |
| 0vconngr 30122 | A graph without vertices i... |
| 1conngr 30123 | A graph with (at most) one... |
| conngrv2edg 30124 | A vertex in a connected gr... |
| vdn0conngrumgrv2 30125 | A vertex in a connected mu... |
| releupth 30128 | The set ` ( EulerPaths `` ... |
| eupths 30129 | The Eulerian paths on the ... |
| iseupth 30130 | The property " ` <. F , P ... |
| iseupthf1o 30131 | The property " ` <. F , P ... |
| eupthi 30132 | Properties of an Eulerian ... |
| eupthf1o 30133 | The ` F ` function in an E... |
| eupthfi 30134 | Any graph with an Eulerian... |
| eupthseg 30135 | The ` N ` -th edge in an e... |
| upgriseupth 30136 | The property " ` <. F , P ... |
| upgreupthi 30137 | Properties of an Eulerian ... |
| upgreupthseg 30138 | The ` N ` -th edge in an e... |
| eupthcl 30139 | An Eulerian path has lengt... |
| eupthistrl 30140 | An Eulerian path is a trai... |
| eupthiswlk 30141 | An Eulerian path is a walk... |
| eupthpf 30142 | The ` P ` function in an E... |
| eupth0 30143 | There is an Eulerian path ... |
| eupthres 30144 | The restriction ` <. H , Q... |
| eupthp1 30145 | Append one path segment to... |
| eupth2eucrct 30146 | Append one path segment to... |
| eupth2lem1 30147 | Lemma for ~ eupth2 . (Con... |
| eupth2lem2 30148 | Lemma for ~ eupth2 . (Con... |
| trlsegvdeglem1 30149 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem2 30150 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem3 30151 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem4 30152 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem5 30153 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem6 30154 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem7 30155 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeg 30156 | Formerly part of proof of ... |
| eupth2lem3lem1 30157 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem2 30158 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem3 30159 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem4 30160 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem5 30161 | Lemma for ~ eupth2 . (Con... |
| eupth2lem3lem6 30162 | Formerly part of proof of ... |
| eupth2lem3lem7 30163 | Lemma for ~ eupth2lem3 : ... |
| eupthvdres 30164 | Formerly part of proof of ... |
| eupth2lem3 30165 | Lemma for ~ eupth2 . (Con... |
| eupth2lemb 30166 | Lemma for ~ eupth2 (induct... |
| eupth2lems 30167 | Lemma for ~ eupth2 (induct... |
| eupth2 30168 | The only vertices of odd d... |
| eulerpathpr 30169 | A graph with an Eulerian p... |
| eulerpath 30170 | A pseudograph with an Eule... |
| eulercrct 30171 | A pseudograph with an Eule... |
| eucrctshift 30172 | Cyclically shifting the in... |
| eucrct2eupth1 30173 | Removing one edge ` ( I ``... |
| eucrct2eupth 30174 | Removing one edge ` ( I ``... |
| konigsbergvtx 30175 | The set of vertices of the... |
| konigsbergiedg 30176 | The indexed edges of the K... |
| konigsbergiedgw 30177 | The indexed edges of the K... |
| konigsbergssiedgwpr 30178 | Each subset of the indexed... |
| konigsbergssiedgw 30179 | Each subset of the indexed... |
| konigsbergumgr 30180 | The Königsberg graph ... |
| konigsberglem1 30181 | Lemma 1 for ~ konigsberg :... |
| konigsberglem2 30182 | Lemma 2 for ~ konigsberg :... |
| konigsberglem3 30183 | Lemma 3 for ~ konigsberg :... |
| konigsberglem4 30184 | Lemma 4 for ~ konigsberg :... |
| konigsberglem5 30185 | Lemma 5 for ~ konigsberg :... |
| konigsberg 30186 | The Königsberg Bridge... |
| isfrgr 30189 | The property of being a fr... |
| frgrusgr 30190 | A friendship graph is a si... |
| frgr0v 30191 | Any null graph (set with n... |
| frgr0vb 30192 | Any null graph (without ve... |
| frgruhgr0v 30193 | Any null graph (without ve... |
| frgr0 30194 | The null graph (graph with... |
| frcond1 30195 | The friendship condition: ... |
| frcond2 30196 | The friendship condition: ... |
| frgreu 30197 | Variant of ~ frcond2 : An... |
| frcond3 30198 | The friendship condition, ... |
| frcond4 30199 | The friendship condition, ... |
| frgr1v 30200 | Any graph with (at most) o... |
| nfrgr2v 30201 | Any graph with two (differ... |
| frgr3vlem1 30202 | Lemma 1 for ~ frgr3v . (C... |
| frgr3vlem2 30203 | Lemma 2 for ~ frgr3v . (C... |
| frgr3v 30204 | Any graph with three verti... |
| 1vwmgr 30205 | Every graph with one verte... |
| 3vfriswmgrlem 30206 | Lemma for ~ 3vfriswmgr . ... |
| 3vfriswmgr 30207 | Every friendship graph wit... |
| 1to2vfriswmgr 30208 | Every friendship graph wit... |
| 1to3vfriswmgr 30209 | Every friendship graph wit... |
| 1to3vfriendship 30210 | The friendship theorem for... |
| 2pthfrgrrn 30211 | Between any two (different... |
| 2pthfrgrrn2 30212 | Between any two (different... |
| 2pthfrgr 30213 | Between any two (different... |
| 3cyclfrgrrn1 30214 | Every vertex in a friendsh... |
| 3cyclfrgrrn 30215 | Every vertex in a friendsh... |
| 3cyclfrgrrn2 30216 | Every vertex in a friendsh... |
| 3cyclfrgr 30217 | Every vertex in a friendsh... |
| 4cycl2v2nb 30218 | In a (maybe degenerate) 4-... |
| 4cycl2vnunb 30219 | In a 4-cycle, two distinct... |
| n4cyclfrgr 30220 | There is no 4-cycle in a f... |
| 4cyclusnfrgr 30221 | A graph with a 4-cycle is ... |
| frgrnbnb 30222 | If two neighbors ` U ` and... |
| frgrconngr 30223 | A friendship graph is conn... |
| vdgn0frgrv2 30224 | A vertex in a friendship g... |
| vdgn1frgrv2 30225 | Any vertex in a friendship... |
| vdgn1frgrv3 30226 | Any vertex in a friendship... |
| vdgfrgrgt2 30227 | Any vertex in a friendship... |
| frgrncvvdeqlem1 30228 | Lemma 1 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem2 30229 | Lemma 2 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem3 30230 | Lemma 3 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem4 30231 | Lemma 4 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem5 30232 | Lemma 5 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem6 30233 | Lemma 6 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem7 30234 | Lemma 7 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem8 30235 | Lemma 8 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem9 30236 | Lemma 9 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem10 30237 | Lemma 10 for ~ frgrncvvdeq... |
| frgrncvvdeq 30238 | In a friendship graph, two... |
| frgrwopreglem4a 30239 | In a friendship graph any ... |
| frgrwopreglem5a 30240 | If a friendship graph has ... |
| frgrwopreglem1 30241 | Lemma 1 for ~ frgrwopreg :... |
| frgrwopreglem2 30242 | Lemma 2 for ~ frgrwopreg .... |
| frgrwopreglem3 30243 | Lemma 3 for ~ frgrwopreg .... |
| frgrwopreglem4 30244 | Lemma 4 for ~ frgrwopreg .... |
| frgrwopregasn 30245 | According to statement 5 i... |
| frgrwopregbsn 30246 | According to statement 5 i... |
| frgrwopreg1 30247 | According to statement 5 i... |
| frgrwopreg2 30248 | According to statement 5 i... |
| frgrwopreglem5lem 30249 | Lemma for ~ frgrwopreglem5... |
| frgrwopreglem5 30250 | Lemma 5 for ~ frgrwopreg .... |
| frgrwopreglem5ALT 30251 | Alternate direct proof of ... |
| frgrwopreg 30252 | In a friendship graph ther... |
| frgrregorufr0 30253 | In a friendship graph ther... |
| frgrregorufr 30254 | If there is a vertex havin... |
| frgrregorufrg 30255 | If there is a vertex havin... |
| frgr2wwlkeu 30256 | For two different vertices... |
| frgr2wwlkn0 30257 | In a friendship graph, the... |
| frgr2wwlk1 30258 | In a friendship graph, the... |
| frgr2wsp1 30259 | In a friendship graph, the... |
| frgr2wwlkeqm 30260 | If there is a (simple) pat... |
| frgrhash2wsp 30261 | The number of simple paths... |
| fusgreg2wsplem 30262 | Lemma for ~ fusgreg2wsp an... |
| fusgr2wsp2nb 30263 | The set of paths of length... |
| fusgreghash2wspv 30264 | According to statement 7 i... |
| fusgreg2wsp 30265 | In a finite simple graph, ... |
| 2wspmdisj 30266 | The sets of paths of lengt... |
| fusgreghash2wsp 30267 | In a finite k-regular grap... |
| frrusgrord0lem 30268 | Lemma for ~ frrusgrord0 . ... |
| frrusgrord0 30269 | If a nonempty finite frien... |
| frrusgrord 30270 | If a nonempty finite frien... |
| numclwwlk2lem1lem 30271 | Lemma for ~ numclwwlk2lem1... |
| 2clwwlklem 30272 | Lemma for ~ clwwnonrepclww... |
| clwwnrepclwwn 30273 | If the initial vertex of a... |
| clwwnonrepclwwnon 30274 | If the initial vertex of a... |
| 2clwwlk2clwwlklem 30275 | Lemma for ~ 2clwwlk2clwwlk... |
| 2clwwlk 30276 | Value of operation ` C ` ,... |
| 2clwwlk2 30277 | The set ` ( X C 2 ) ` of d... |
| 2clwwlkel 30278 | Characterization of an ele... |
| 2clwwlk2clwwlk 30279 | An element of the value of... |
| numclwwlk1lem2foalem 30280 | Lemma for ~ numclwwlk1lem2... |
| extwwlkfab 30281 | The set ` ( X C N ) ` of d... |
| extwwlkfabel 30282 | Characterization of an ele... |
| numclwwlk1lem2foa 30283 | Going forth and back from ... |
| numclwwlk1lem2f 30284 | ` T ` is a function, mappi... |
| numclwwlk1lem2fv 30285 | Value of the function ` T ... |
| numclwwlk1lem2f1 30286 | ` T ` is a 1-1 function. ... |
| numclwwlk1lem2fo 30287 | ` T ` is an onto function.... |
| numclwwlk1lem2f1o 30288 | ` T ` is a 1-1 onto functi... |
| numclwwlk1lem2 30289 | The set of double loops of... |
| numclwwlk1 30290 | Statement 9 in [Huneke] p.... |
| clwwlknonclwlknonf1o 30291 | ` F ` is a bijection betwe... |
| clwwlknonclwlknonen 30292 | The sets of the two repres... |
| dlwwlknondlwlknonf1olem1 30293 | Lemma 1 for ~ dlwwlknondlw... |
| dlwwlknondlwlknonf1o 30294 | ` F ` is a bijection betwe... |
| dlwwlknondlwlknonen 30295 | The sets of the two repres... |
| wlkl0 30296 | There is exactly one walk ... |
| clwlknon2num 30297 | There are k walks of lengt... |
| numclwlk1lem1 30298 | Lemma 1 for ~ numclwlk1 (S... |
| numclwlk1lem2 30299 | Lemma 2 for ~ numclwlk1 (S... |
| numclwlk1 30300 | Statement 9 in [Huneke] p.... |
| numclwwlkovh0 30301 | Value of operation ` H ` ,... |
| numclwwlkovh 30302 | Value of operation ` H ` ,... |
| numclwwlkovq 30303 | Value of operation ` Q ` ,... |
| numclwwlkqhash 30304 | In a ` K `-regular graph, ... |
| numclwwlk2lem1 30305 | In a friendship graph, for... |
| numclwlk2lem2f 30306 | ` R ` is a function mappin... |
| numclwlk2lem2fv 30307 | Value of the function ` R ... |
| numclwlk2lem2f1o 30308 | ` R ` is a 1-1 onto functi... |
| numclwwlk2lem3 30309 | In a friendship graph, the... |
| numclwwlk2 30310 | Statement 10 in [Huneke] p... |
| numclwwlk3lem1 30311 | Lemma 2 for ~ numclwwlk3 .... |
| numclwwlk3lem2lem 30312 | Lemma for ~ numclwwlk3lem2... |
| numclwwlk3lem2 30313 | Lemma 1 for ~ numclwwlk3 :... |
| numclwwlk3 30314 | Statement 12 in [Huneke] p... |
| numclwwlk4 30315 | The total number of closed... |
| numclwwlk5lem 30316 | Lemma for ~ numclwwlk5 . ... |
| numclwwlk5 30317 | Statement 13 in [Huneke] p... |
| numclwwlk7lem 30318 | Lemma for ~ numclwwlk7 , ~... |
| numclwwlk6 30319 | For a prime divisor ` P ` ... |
| numclwwlk7 30320 | Statement 14 in [Huneke] p... |
| numclwwlk8 30321 | The size of the set of clo... |
| frgrreggt1 30322 | If a finite nonempty frien... |
| frgrreg 30323 | If a finite nonempty frien... |
| frgrregord013 30324 | If a finite friendship gra... |
| frgrregord13 30325 | If a nonempty finite frien... |
| frgrogt3nreg 30326 | If a finite friendship gra... |
| friendshipgt3 30327 | The friendship theorem for... |
| friendship 30328 | The friendship theorem: I... |
| conventions 30329 |
H... |
| conventions-labels 30330 |
... |
| conventions-comments 30331 |
... |
| natded 30332 | Here are typical n... |
| ex-natded5.2 30333 | Theorem 5.2 of [Clemente] ... |
| ex-natded5.2-2 30334 | A more efficient proof of ... |
| ex-natded5.2i 30335 | The same as ~ ex-natded5.2... |
| ex-natded5.3 30336 | Theorem 5.3 of [Clemente] ... |
| ex-natded5.3-2 30337 | A more efficient proof of ... |
| ex-natded5.3i 30338 | The same as ~ ex-natded5.3... |
| ex-natded5.5 30339 | Theorem 5.5 of [Clemente] ... |
| ex-natded5.7 30340 | Theorem 5.7 of [Clemente] ... |
| ex-natded5.7-2 30341 | A more efficient proof of ... |
| ex-natded5.8 30342 | Theorem 5.8 of [Clemente] ... |
| ex-natded5.8-2 30343 | A more efficient proof of ... |
| ex-natded5.13 30344 | Theorem 5.13 of [Clemente]... |
| ex-natded5.13-2 30345 | A more efficient proof of ... |
| ex-natded9.20 30346 | Theorem 9.20 of [Clemente]... |
| ex-natded9.20-2 30347 | A more efficient proof of ... |
| ex-natded9.26 30348 | Theorem 9.26 of [Clemente]... |
| ex-natded9.26-2 30349 | A more efficient proof of ... |
| ex-or 30350 | Example for ~ df-or . Exa... |
| ex-an 30351 | Example for ~ df-an . Exa... |
| ex-dif 30352 | Example for ~ df-dif . Ex... |
| ex-un 30353 | Example for ~ df-un . Exa... |
| ex-in 30354 | Example for ~ df-in . Exa... |
| ex-uni 30355 | Example for ~ df-uni . Ex... |
| ex-ss 30356 | Example for ~ df-ss . Exa... |
| ex-pss 30357 | Example for ~ df-pss . Ex... |
| ex-pw 30358 | Example for ~ df-pw . Exa... |
| ex-pr 30359 | Example for ~ df-pr . (Co... |
| ex-br 30360 | Example for ~ df-br . Exa... |
| ex-opab 30361 | Example for ~ df-opab . E... |
| ex-eprel 30362 | Example for ~ df-eprel . ... |
| ex-id 30363 | Example for ~ df-id . Exa... |
| ex-po 30364 | Example for ~ df-po . Exa... |
| ex-xp 30365 | Example for ~ df-xp . Exa... |
| ex-cnv 30366 | Example for ~ df-cnv . Ex... |
| ex-co 30367 | Example for ~ df-co . Exa... |
| ex-dm 30368 | Example for ~ df-dm . Exa... |
| ex-rn 30369 | Example for ~ df-rn . Exa... |
| ex-res 30370 | Example for ~ df-res . Ex... |
| ex-ima 30371 | Example for ~ df-ima . Ex... |
| ex-fv 30372 | Example for ~ df-fv . Exa... |
| ex-1st 30373 | Example for ~ df-1st . Ex... |
| ex-2nd 30374 | Example for ~ df-2nd . Ex... |
| 1kp2ke3k 30375 | Example for ~ df-dec , 100... |
| ex-fl 30376 | Example for ~ df-fl . Exa... |
| ex-ceil 30377 | Example for ~ df-ceil . (... |
| ex-mod 30378 | Example for ~ df-mod . (C... |
| ex-exp 30379 | Example for ~ df-exp . (C... |
| ex-fac 30380 | Example for ~ df-fac . (C... |
| ex-bc 30381 | Example for ~ df-bc . (Co... |
| ex-hash 30382 | Example for ~ df-hash . (... |
| ex-sqrt 30383 | Example for ~ df-sqrt . (... |
| ex-abs 30384 | Example for ~ df-abs . (C... |
| ex-dvds 30385 | Example for ~ df-dvds : 3 ... |
| ex-gcd 30386 | Example for ~ df-gcd . (C... |
| ex-lcm 30387 | Example for ~ df-lcm . (C... |
| ex-prmo 30388 | Example for ~ df-prmo : ` ... |
| aevdemo 30389 | Proof illustrating the com... |
| ex-ind-dvds 30390 | Example of a proof by indu... |
| ex-fpar 30391 | Formalized example provide... |
| avril1 30392 | Poisson d'Avril's Theorem.... |
| 2bornot2b 30393 | The law of excluded middle... |
| helloworld 30394 | The classic "Hello world" ... |
| 1p1e2apr1 30395 | One plus one equals two. ... |
| eqid1 30396 | Law of identity (reflexivi... |
| 1div0apr 30397 | Division by zero is forbid... |
| topnfbey 30398 | Nothing seems to be imposs... |
| 9p10ne21 30399 | 9 + 10 is not equal to 21.... |
| 9p10ne21fool 30400 | 9 + 10 equals 21. This as... |
| nrt2irr 30402 | The ` N ` -th root of 2 is... |
| isplig 30405 | The predicate "is a planar... |
| ispligb 30406 | The predicate "is a planar... |
| tncp 30407 | In any planar incidence ge... |
| l2p 30408 | For any line in a planar i... |
| lpni 30409 | For any line in a planar i... |
| nsnlplig 30410 | There is no "one-point lin... |
| nsnlpligALT 30411 | Alternate version of ~ nsn... |
| n0lplig 30412 | There is no "empty line" i... |
| n0lpligALT 30413 | Alternate version of ~ n0l... |
| eulplig 30414 | Through two distinct point... |
| pliguhgr 30415 | Any planar incidence geome... |
| dummylink 30416 | Alias for ~ a1ii that may ... |
| id1 30417 | Alias for ~ idALT that may... |
| isgrpo 30426 | The predicate "is a group ... |
| isgrpoi 30427 | Properties that determine ... |
| grpofo 30428 | A group operation maps ont... |
| grpocl 30429 | Closure law for a group op... |
| grpolidinv 30430 | A group has a left identit... |
| grpon0 30431 | The base set of a group is... |
| grpoass 30432 | A group operation is assoc... |
| grpoidinvlem1 30433 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem2 30434 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem3 30435 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem4 30436 | Lemma for ~ grpoidinv . (... |
| grpoidinv 30437 | A group has a left and rig... |
| grpoideu 30438 | The left identity element ... |
| grporndm 30439 | A group's range in terms o... |
| 0ngrp 30440 | The empty set is not a gro... |
| gidval 30441 | The value of the identity ... |
| grpoidval 30442 | Lemma for ~ grpoidcl and o... |
| grpoidcl 30443 | The identity element of a ... |
| grpoidinv2 30444 | A group's properties using... |
| grpolid 30445 | The identity element of a ... |
| grporid 30446 | The identity element of a ... |
| grporcan 30447 | Right cancellation law for... |
| grpoinveu 30448 | The left inverse element o... |
| grpoid 30449 | Two ways of saying that an... |
| grporn 30450 | The range of a group opera... |
| grpoinvfval 30451 | The inverse function of a ... |
| grpoinvval 30452 | The inverse of a group ele... |
| grpoinvcl 30453 | A group element's inverse ... |
| grpoinv 30454 | The properties of a group ... |
| grpolinv 30455 | The left inverse of a grou... |
| grporinv 30456 | The right inverse of a gro... |
| grpoinvid1 30457 | The inverse of a group ele... |
| grpoinvid2 30458 | The inverse of a group ele... |
| grpolcan 30459 | Left cancellation law for ... |
| grpo2inv 30460 | Double inverse law for gro... |
| grpoinvf 30461 | Mapping of the inverse fun... |
| grpoinvop 30462 | The inverse of the group o... |
| grpodivfval 30463 | Group division (or subtrac... |
| grpodivval 30464 | Group division (or subtrac... |
| grpodivinv 30465 | Group division by an inver... |
| grpoinvdiv 30466 | Inverse of a group divisio... |
| grpodivf 30467 | Mapping for group division... |
| grpodivcl 30468 | Closure of group division ... |
| grpodivdiv 30469 | Double group division. (C... |
| grpomuldivass 30470 | Associative-type law for m... |
| grpodivid 30471 | Division of a group member... |
| grponpcan 30472 | Cancellation law for group... |
| isablo 30475 | The predicate "is an Abeli... |
| ablogrpo 30476 | An Abelian group operation... |
| ablocom 30477 | An Abelian group operation... |
| ablo32 30478 | Commutative/associative la... |
| ablo4 30479 | Commutative/associative la... |
| isabloi 30480 | Properties that determine ... |
| ablomuldiv 30481 | Law for group multiplicati... |
| ablodivdiv 30482 | Law for double group divis... |
| ablodivdiv4 30483 | Law for double group divis... |
| ablodiv32 30484 | Swap the second and third ... |
| ablonncan 30485 | Cancellation law for group... |
| ablonnncan1 30486 | Cancellation law for group... |
| vcrel 30489 | The class of all complex v... |
| vciOLD 30490 | Obsolete version of ~ cvsi... |
| vcsm 30491 | Functionality of th scalar... |
| vccl 30492 | Closure of the scalar prod... |
| vcidOLD 30493 | Identity element for the s... |
| vcdi 30494 | Distributive law for the s... |
| vcdir 30495 | Distributive law for the s... |
| vcass 30496 | Associative law for the sc... |
| vc2OLD 30497 | A vector plus itself is tw... |
| vcablo 30498 | Vector addition is an Abel... |
| vcgrp 30499 | Vector addition is a group... |
| vclcan 30500 | Left cancellation law for ... |
| vczcl 30501 | The zero vector is a vecto... |
| vc0rid 30502 | The zero vector is a right... |
| vc0 30503 | Zero times a vector is the... |
| vcz 30504 | Anything times the zero ve... |
| vcm 30505 | Minus 1 times a vector is ... |
| isvclem 30506 | Lemma for ~ isvcOLD . (Co... |
| vcex 30507 | The components of a comple... |
| isvcOLD 30508 | The predicate "is a comple... |
| isvciOLD 30509 | Properties that determine ... |
| cnaddabloOLD 30510 | Obsolete version of ~ cnad... |
| cnidOLD 30511 | Obsolete version of ~ cnad... |
| cncvcOLD 30512 | Obsolete version of ~ cncv... |
| nvss 30522 | Structure of the class of ... |
| nvvcop 30523 | A normed complex vector sp... |
| nvrel 30531 | The class of all normed co... |
| vafval 30532 | Value of the function for ... |
| bafval 30533 | Value of the function for ... |
| smfval 30534 | Value of the function for ... |
| 0vfval 30535 | Value of the function for ... |
| nmcvfval 30536 | Value of the norm function... |
| nvop2 30537 | A normed complex vector sp... |
| nvvop 30538 | The vector space component... |
| isnvlem 30539 | Lemma for ~ isnv . (Contr... |
| nvex 30540 | The components of a normed... |
| isnv 30541 | The predicate "is a normed... |
| isnvi 30542 | Properties that determine ... |
| nvi 30543 | The properties of a normed... |
| nvvc 30544 | The vector space component... |
| nvablo 30545 | The vector addition operat... |
| nvgrp 30546 | The vector addition operat... |
| nvgf 30547 | Mapping for the vector add... |
| nvsf 30548 | Mapping for the scalar mul... |
| nvgcl 30549 | Closure law for the vector... |
| nvcom 30550 | The vector addition (group... |
| nvass 30551 | The vector addition (group... |
| nvadd32 30552 | Commutative/associative la... |
| nvrcan 30553 | Right cancellation law for... |
| nvadd4 30554 | Rearrangement of 4 terms i... |
| nvscl 30555 | Closure law for the scalar... |
| nvsid 30556 | Identity element for the s... |
| nvsass 30557 | Associative law for the sc... |
| nvscom 30558 | Commutative law for the sc... |
| nvdi 30559 | Distributive law for the s... |
| nvdir 30560 | Distributive law for the s... |
| nv2 30561 | A vector plus itself is tw... |
| vsfval 30562 | Value of the function for ... |
| nvzcl 30563 | Closure law for the zero v... |
| nv0rid 30564 | The zero vector is a right... |
| nv0lid 30565 | The zero vector is a left ... |
| nv0 30566 | Zero times a vector is the... |
| nvsz 30567 | Anything times the zero ve... |
| nvinv 30568 | Minus 1 times a vector is ... |
| nvinvfval 30569 | Function for the negative ... |
| nvm 30570 | Vector subtraction in term... |
| nvmval 30571 | Value of vector subtractio... |
| nvmval2 30572 | Value of vector subtractio... |
| nvmfval 30573 | Value of the function for ... |
| nvmf 30574 | Mapping for the vector sub... |
| nvmcl 30575 | Closure law for the vector... |
| nvnnncan1 30576 | Cancellation law for vecto... |
| nvmdi 30577 | Distributive law for scala... |
| nvnegneg 30578 | Double negative of a vecto... |
| nvmul0or 30579 | If a scalar product is zer... |
| nvrinv 30580 | A vector minus itself. (C... |
| nvlinv 30581 | Minus a vector plus itself... |
| nvpncan2 30582 | Cancellation law for vecto... |
| nvpncan 30583 | Cancellation law for vecto... |
| nvaddsub 30584 | Commutative/associative la... |
| nvnpcan 30585 | Cancellation law for a nor... |
| nvaddsub4 30586 | Rearrangement of 4 terms i... |
| nvmeq0 30587 | The difference between two... |
| nvmid 30588 | A vector minus itself is t... |
| nvf 30589 | Mapping for the norm funct... |
| nvcl 30590 | The norm of a normed compl... |
| nvcli 30591 | The norm of a normed compl... |
| nvs 30592 | Proportionality property o... |
| nvsge0 30593 | The norm of a scalar produ... |
| nvm1 30594 | The norm of the negative o... |
| nvdif 30595 | The norm of the difference... |
| nvpi 30596 | The norm of a vector plus ... |
| nvz0 30597 | The norm of a zero vector ... |
| nvz 30598 | The norm of a vector is ze... |
| nvtri 30599 | Triangle inequality for th... |
| nvmtri 30600 | Triangle inequality for th... |
| nvabs 30601 | Norm difference property o... |
| nvge0 30602 | The norm of a normed compl... |
| nvgt0 30603 | A nonzero norm is positive... |
| nv1 30604 | From any nonzero vector, c... |
| nvop 30605 | A complex inner product sp... |
| cnnv 30606 | The set of complex numbers... |
| cnnvg 30607 | The vector addition (group... |
| cnnvba 30608 | The base set of the normed... |
| cnnvs 30609 | The scalar product operati... |
| cnnvnm 30610 | The norm operation of the ... |
| cnnvm 30611 | The vector subtraction ope... |
| elimnv 30612 | Hypothesis elimination lem... |
| elimnvu 30613 | Hypothesis elimination lem... |
| imsval 30614 | Value of the induced metri... |
| imsdval 30615 | Value of the induced metri... |
| imsdval2 30616 | Value of the distance func... |
| nvnd 30617 | The norm of a normed compl... |
| imsdf 30618 | Mapping for the induced me... |
| imsmetlem 30619 | Lemma for ~ imsmet . (Con... |
| imsmet 30620 | The induced metric of a no... |
| imsxmet 30621 | The induced metric of a no... |
| cnims 30622 | The metric induced on the ... |
| vacn 30623 | Vector addition is jointly... |
| nmcvcn 30624 | The norm of a normed compl... |
| nmcnc 30625 | The norm of a normed compl... |
| smcnlem 30626 | Lemma for ~ smcn . (Contr... |
| smcn 30627 | Scalar multiplication is j... |
| vmcn 30628 | Vector subtraction is join... |
| dipfval 30631 | The inner product function... |
| ipval 30632 | Value of the inner product... |
| ipval2lem2 30633 | Lemma for ~ ipval3 . (Con... |
| ipval2lem3 30634 | Lemma for ~ ipval3 . (Con... |
| ipval2lem4 30635 | Lemma for ~ ipval3 . (Con... |
| ipval2 30636 | Expansion of the inner pro... |
| 4ipval2 30637 | Four times the inner produ... |
| ipval3 30638 | Expansion of the inner pro... |
| ipidsq 30639 | The inner product of a vec... |
| ipnm 30640 | Norm expressed in terms of... |
| dipcl 30641 | An inner product is a comp... |
| ipf 30642 | Mapping for the inner prod... |
| dipcj 30643 | The complex conjugate of a... |
| ipipcj 30644 | An inner product times its... |
| diporthcom 30645 | Orthogonality (meaning inn... |
| dip0r 30646 | Inner product with a zero ... |
| dip0l 30647 | Inner product with a zero ... |
| ipz 30648 | The inner product of a vec... |
| dipcn 30649 | Inner product is jointly c... |
| sspval 30652 | The set of all subspaces o... |
| isssp 30653 | The predicate "is a subspa... |
| sspid 30654 | A normed complex vector sp... |
| sspnv 30655 | A subspace is a normed com... |
| sspba 30656 | The base set of a subspace... |
| sspg 30657 | Vector addition on a subsp... |
| sspgval 30658 | Vector addition on a subsp... |
| ssps 30659 | Scalar multiplication on a... |
| sspsval 30660 | Scalar multiplication on a... |
| sspmlem 30661 | Lemma for ~ sspm and other... |
| sspmval 30662 | Vector addition on a subsp... |
| sspm 30663 | Vector subtraction on a su... |
| sspz 30664 | The zero vector of a subsp... |
| sspn 30665 | The norm on a subspace is ... |
| sspnval 30666 | The norm on a subspace in ... |
| sspimsval 30667 | The induced metric on a su... |
| sspims 30668 | The induced metric on a su... |
| lnoval 30681 | The set of linear operator... |
| islno 30682 | The predicate "is a linear... |
| lnolin 30683 | Basic linearity property o... |
| lnof 30684 | A linear operator is a map... |
| lno0 30685 | The value of a linear oper... |
| lnocoi 30686 | The composition of two lin... |
| lnoadd 30687 | Addition property of a lin... |
| lnosub 30688 | Subtraction property of a ... |
| lnomul 30689 | Scalar multiplication prop... |
| nvo00 30690 | Two ways to express a zero... |
| nmoofval 30691 | The operator norm function... |
| nmooval 30692 | The operator norm function... |
| nmosetre 30693 | The set in the supremum of... |
| nmosetn0 30694 | The set in the supremum of... |
| nmoxr 30695 | The norm of an operator is... |
| nmooge0 30696 | The norm of an operator is... |
| nmorepnf 30697 | The norm of an operator is... |
| nmoreltpnf 30698 | The norm of any operator i... |
| nmogtmnf 30699 | The norm of an operator is... |
| nmoolb 30700 | A lower bound for an opera... |
| nmoubi 30701 | An upper bound for an oper... |
| nmoub3i 30702 | An upper bound for an oper... |
| nmoub2i 30703 | An upper bound for an oper... |
| nmobndi 30704 | Two ways to express that a... |
| nmounbi 30705 | Two ways two express that ... |
| nmounbseqi 30706 | An unbounded operator dete... |
| nmounbseqiALT 30707 | Alternate shorter proof of... |
| nmobndseqi 30708 | A bounded sequence determi... |
| nmobndseqiALT 30709 | Alternate shorter proof of... |
| bloval 30710 | The class of bounded linea... |
| isblo 30711 | The predicate "is a bounde... |
| isblo2 30712 | The predicate "is a bounde... |
| bloln 30713 | A bounded operator is a li... |
| blof 30714 | A bounded operator is an o... |
| nmblore 30715 | The norm of a bounded oper... |
| 0ofval 30716 | The zero operator between ... |
| 0oval 30717 | Value of the zero operator... |
| 0oo 30718 | The zero operator is an op... |
| 0lno 30719 | The zero operator is linea... |
| nmoo0 30720 | The operator norm of the z... |
| 0blo 30721 | The zero operator is a bou... |
| nmlno0lem 30722 | Lemma for ~ nmlno0i . (Co... |
| nmlno0i 30723 | The norm of a linear opera... |
| nmlno0 30724 | The norm of a linear opera... |
| nmlnoubi 30725 | An upper bound for the ope... |
| nmlnogt0 30726 | The norm of a nonzero line... |
| lnon0 30727 | The domain of a nonzero li... |
| nmblolbii 30728 | A lower bound for the norm... |
| nmblolbi 30729 | A lower bound for the norm... |
| isblo3i 30730 | The predicate "is a bounde... |
| blo3i 30731 | Properties that determine ... |
| blometi 30732 | Upper bound for the distan... |
| blocnilem 30733 | Lemma for ~ blocni and ~ l... |
| blocni 30734 | A linear operator is conti... |
| lnocni 30735 | If a linear operator is co... |
| blocn 30736 | A linear operator is conti... |
| blocn2 30737 | A bounded linear operator ... |
| ajfval 30738 | The adjoint function. (Co... |
| hmoval 30739 | The set of Hermitian (self... |
| ishmo 30740 | The predicate "is a hermit... |
| phnv 30743 | Every complex inner produc... |
| phrel 30744 | The class of all complex i... |
| phnvi 30745 | Every complex inner produc... |
| isphg 30746 | The predicate "is a comple... |
| phop 30747 | A complex inner product sp... |
| cncph 30748 | The set of complex numbers... |
| elimph 30749 | Hypothesis elimination lem... |
| elimphu 30750 | Hypothesis elimination lem... |
| isph 30751 | The predicate "is an inner... |
| phpar2 30752 | The parallelogram law for ... |
| phpar 30753 | The parallelogram law for ... |
| ip0i 30754 | A slight variant of Equati... |
| ip1ilem 30755 | Lemma for ~ ip1i . (Contr... |
| ip1i 30756 | Equation 6.47 of [Ponnusam... |
| ip2i 30757 | Equation 6.48 of [Ponnusam... |
| ipdirilem 30758 | Lemma for ~ ipdiri . (Con... |
| ipdiri 30759 | Distributive law for inner... |
| ipasslem1 30760 | Lemma for ~ ipassi . Show... |
| ipasslem2 30761 | Lemma for ~ ipassi . Show... |
| ipasslem3 30762 | Lemma for ~ ipassi . Show... |
| ipasslem4 30763 | Lemma for ~ ipassi . Show... |
| ipasslem5 30764 | Lemma for ~ ipassi . Show... |
| ipasslem7 30765 | Lemma for ~ ipassi . Show... |
| ipasslem8 30766 | Lemma for ~ ipassi . By ~... |
| ipasslem9 30767 | Lemma for ~ ipassi . Conc... |
| ipasslem10 30768 | Lemma for ~ ipassi . Show... |
| ipasslem11 30769 | Lemma for ~ ipassi . Show... |
| ipassi 30770 | Associative law for inner ... |
| dipdir 30771 | Distributive law for inner... |
| dipdi 30772 | Distributive law for inner... |
| ip2dii 30773 | Inner product of two sums.... |
| dipass 30774 | Associative law for inner ... |
| dipassr 30775 | "Associative" law for seco... |
| dipassr2 30776 | "Associative" law for inne... |
| dipsubdir 30777 | Distributive law for inner... |
| dipsubdi 30778 | Distributive law for inner... |
| pythi 30779 | The Pythagorean theorem fo... |
| siilem1 30780 | Lemma for ~ sii . (Contri... |
| siilem2 30781 | Lemma for ~ sii . (Contri... |
| siii 30782 | Inference from ~ sii . (C... |
| sii 30783 | Obsolete version of ~ ipca... |
| ipblnfi 30784 | A function ` F ` generated... |
| ip2eqi 30785 | Two vectors are equal iff ... |
| phoeqi 30786 | A condition implying that ... |
| ajmoi 30787 | Every operator has at most... |
| ajfuni 30788 | The adjoint function is a ... |
| ajfun 30789 | The adjoint function is a ... |
| ajval 30790 | Value of the adjoint funct... |
| iscbn 30793 | A complex Banach space is ... |
| cbncms 30794 | The induced metric on comp... |
| bnnv 30795 | Every complex Banach space... |
| bnrel 30796 | The class of all complex B... |
| bnsscmcl 30797 | A subspace of a Banach spa... |
| cnbn 30798 | The set of complex numbers... |
| ubthlem1 30799 | Lemma for ~ ubth . The fu... |
| ubthlem2 30800 | Lemma for ~ ubth . Given ... |
| ubthlem3 30801 | Lemma for ~ ubth . Prove ... |
| ubth 30802 | Uniform Boundedness Theore... |
| minvecolem1 30803 | Lemma for ~ minveco . The... |
| minvecolem2 30804 | Lemma for ~ minveco . Any... |
| minvecolem3 30805 | Lemma for ~ minveco . The... |
| minvecolem4a 30806 | Lemma for ~ minveco . ` F ... |
| minvecolem4b 30807 | Lemma for ~ minveco . The... |
| minvecolem4c 30808 | Lemma for ~ minveco . The... |
| minvecolem4 30809 | Lemma for ~ minveco . The... |
| minvecolem5 30810 | Lemma for ~ minveco . Dis... |
| minvecolem6 30811 | Lemma for ~ minveco . Any... |
| minvecolem7 30812 | Lemma for ~ minveco . Sin... |
| minveco 30813 | Minimizing vector theorem,... |
| ishlo 30816 | The predicate "is a comple... |
| hlobn 30817 | Every complex Hilbert spac... |
| hlph 30818 | Every complex Hilbert spac... |
| hlrel 30819 | The class of all complex H... |
| hlnv 30820 | Every complex Hilbert spac... |
| hlnvi 30821 | Every complex Hilbert spac... |
| hlvc 30822 | Every complex Hilbert spac... |
| hlcmet 30823 | The induced metric on a co... |
| hlmet 30824 | The induced metric on a co... |
| hlpar2 30825 | The parallelogram law sati... |
| hlpar 30826 | The parallelogram law sati... |
| hlex 30827 | The base set of a Hilbert ... |
| hladdf 30828 | Mapping for Hilbert space ... |
| hlcom 30829 | Hilbert space vector addit... |
| hlass 30830 | Hilbert space vector addit... |
| hl0cl 30831 | The Hilbert space zero vec... |
| hladdid 30832 | Hilbert space addition wit... |
| hlmulf 30833 | Mapping for Hilbert space ... |
| hlmulid 30834 | Hilbert space scalar multi... |
| hlmulass 30835 | Hilbert space scalar multi... |
| hldi 30836 | Hilbert space scalar multi... |
| hldir 30837 | Hilbert space scalar multi... |
| hlmul0 30838 | Hilbert space scalar multi... |
| hlipf 30839 | Mapping for Hilbert space ... |
| hlipcj 30840 | Conjugate law for Hilbert ... |
| hlipdir 30841 | Distributive law for Hilbe... |
| hlipass 30842 | Associative law for Hilber... |
| hlipgt0 30843 | The inner product of a Hil... |
| hlcompl 30844 | Completeness of a Hilbert ... |
| cnchl 30845 | The set of complex numbers... |
| htthlem 30846 | Lemma for ~ htth . The co... |
| htth 30847 | Hellinger-Toeplitz Theorem... |
| The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here | |
| h2hva 30903 | The group (addition) opera... |
| h2hsm 30904 | The scalar product operati... |
| h2hnm 30905 | The norm function of Hilbe... |
| h2hvs 30906 | The vector subtraction ope... |
| h2hmetdval 30907 | Value of the distance func... |
| h2hcau 30908 | The Cauchy sequences of Hi... |
| h2hlm 30909 | The limit sequences of Hil... |
| axhilex-zf 30910 | Derive Axiom ~ ax-hilex fr... |
| axhfvadd-zf 30911 | Derive Axiom ~ ax-hfvadd f... |
| axhvcom-zf 30912 | Derive Axiom ~ ax-hvcom fr... |
| axhvass-zf 30913 | Derive Axiom ~ ax-hvass fr... |
| axhv0cl-zf 30914 | Derive Axiom ~ ax-hv0cl fr... |
| axhvaddid-zf 30915 | Derive Axiom ~ ax-hvaddid ... |
| axhfvmul-zf 30916 | Derive Axiom ~ ax-hfvmul f... |
| axhvmulid-zf 30917 | Derive Axiom ~ ax-hvmulid ... |
| axhvmulass-zf 30918 | Derive Axiom ~ ax-hvmulass... |
| axhvdistr1-zf 30919 | Derive Axiom ~ ax-hvdistr1... |
| axhvdistr2-zf 30920 | Derive Axiom ~ ax-hvdistr2... |
| axhvmul0-zf 30921 | Derive Axiom ~ ax-hvmul0 f... |
| axhfi-zf 30922 | Derive Axiom ~ ax-hfi from... |
| axhis1-zf 30923 | Derive Axiom ~ ax-his1 fro... |
| axhis2-zf 30924 | Derive Axiom ~ ax-his2 fro... |
| axhis3-zf 30925 | Derive Axiom ~ ax-his3 fro... |
| axhis4-zf 30926 | Derive Axiom ~ ax-his4 fro... |
| axhcompl-zf 30927 | Derive Axiom ~ ax-hcompl f... |
| hvmulex 30940 | The Hilbert space scalar p... |
| hvaddcl 30941 | Closure of vector addition... |
| hvmulcl 30942 | Closure of scalar multipli... |
| hvmulcli 30943 | Closure inference for scal... |
| hvsubf 30944 | Mapping domain and codomai... |
| hvsubval 30945 | Value of vector subtractio... |
| hvsubcl 30946 | Closure of vector subtract... |
| hvaddcli 30947 | Closure of vector addition... |
| hvcomi 30948 | Commutation of vector addi... |
| hvsubvali 30949 | Value of vector subtractio... |
| hvsubcli 30950 | Closure of vector subtract... |
| ifhvhv0 30951 | Prove ` if ( A e. ~H , A ,... |
| hvaddlid 30952 | Addition with the zero vec... |
| hvmul0 30953 | Scalar multiplication with... |
| hvmul0or 30954 | If a scalar product is zer... |
| hvsubid 30955 | Subtraction of a vector fr... |
| hvnegid 30956 | Addition of negative of a ... |
| hv2neg 30957 | Two ways to express the ne... |
| hvaddlidi 30958 | Addition with the zero vec... |
| hvnegidi 30959 | Addition of negative of a ... |
| hv2negi 30960 | Two ways to express the ne... |
| hvm1neg 30961 | Convert minus one times a ... |
| hvaddsubval 30962 | Value of vector addition i... |
| hvadd32 30963 | Commutative/associative la... |
| hvadd12 30964 | Commutative/associative la... |
| hvadd4 30965 | Hilbert vector space addit... |
| hvsub4 30966 | Hilbert vector space addit... |
| hvaddsub12 30967 | Commutative/associative la... |
| hvpncan 30968 | Addition/subtraction cance... |
| hvpncan2 30969 | Addition/subtraction cance... |
| hvaddsubass 30970 | Associativity of sum and d... |
| hvpncan3 30971 | Subtraction and addition o... |
| hvmulcom 30972 | Scalar multiplication comm... |
| hvsubass 30973 | Hilbert vector space assoc... |
| hvsub32 30974 | Hilbert vector space commu... |
| hvmulassi 30975 | Scalar multiplication asso... |
| hvmulcomi 30976 | Scalar multiplication comm... |
| hvmul2negi 30977 | Double negative in scalar ... |
| hvsubdistr1 30978 | Scalar multiplication dist... |
| hvsubdistr2 30979 | Scalar multiplication dist... |
| hvdistr1i 30980 | Scalar multiplication dist... |
| hvsubdistr1i 30981 | Scalar multiplication dist... |
| hvassi 30982 | Hilbert vector space assoc... |
| hvadd32i 30983 | Hilbert vector space commu... |
| hvsubassi 30984 | Hilbert vector space assoc... |
| hvsub32i 30985 | Hilbert vector space commu... |
| hvadd12i 30986 | Hilbert vector space commu... |
| hvadd4i 30987 | Hilbert vector space addit... |
| hvsubsub4i 30988 | Hilbert vector space addit... |
| hvsubsub4 30989 | Hilbert vector space addit... |
| hv2times 30990 | Two times a vector. (Cont... |
| hvnegdii 30991 | Distribution of negative o... |
| hvsubeq0i 30992 | If the difference between ... |
| hvsubcan2i 30993 | Vector cancellation law. ... |
| hvaddcani 30994 | Cancellation law for vecto... |
| hvsubaddi 30995 | Relationship between vecto... |
| hvnegdi 30996 | Distribution of negative o... |
| hvsubeq0 30997 | If the difference between ... |
| hvaddeq0 30998 | If the sum of two vectors ... |
| hvaddcan 30999 | Cancellation law for vecto... |
| hvaddcan2 31000 | Cancellation law for vecto... |
| hvmulcan 31001 | Cancellation law for scala... |
| hvmulcan2 31002 | Cancellation law for scala... |
| hvsubcan 31003 | Cancellation law for vecto... |
| hvsubcan2 31004 | Cancellation law for vecto... |
| hvsub0 31005 | Subtraction of a zero vect... |
| hvsubadd 31006 | Relationship between vecto... |
| hvaddsub4 31007 | Hilbert vector space addit... |
| hicl 31009 | Closure of inner product. ... |
| hicli 31010 | Closure inference for inne... |
| his5 31015 | Associative law for inner ... |
| his52 31016 | Associative law for inner ... |
| his35 31017 | Move scalar multiplication... |
| his35i 31018 | Move scalar multiplication... |
| his7 31019 | Distributive law for inner... |
| hiassdi 31020 | Distributive/associative l... |
| his2sub 31021 | Distributive law for inner... |
| his2sub2 31022 | Distributive law for inner... |
| hire 31023 | A necessary and sufficient... |
| hiidrcl 31024 | Real closure of inner prod... |
| hi01 31025 | Inner product with the 0 v... |
| hi02 31026 | Inner product with the 0 v... |
| hiidge0 31027 | Inner product with self is... |
| his6 31028 | Zero inner product with se... |
| his1i 31029 | Conjugate law for inner pr... |
| abshicom 31030 | Commuted inner products ha... |
| hial0 31031 | A vector whose inner produ... |
| hial02 31032 | A vector whose inner produ... |
| hisubcomi 31033 | Two vector subtractions si... |
| hi2eq 31034 | Lemma used to prove equali... |
| hial2eq 31035 | Two vectors whose inner pr... |
| hial2eq2 31036 | Two vectors whose inner pr... |
| orthcom 31037 | Orthogonality commutes. (... |
| normlem0 31038 | Lemma used to derive prope... |
| normlem1 31039 | Lemma used to derive prope... |
| normlem2 31040 | Lemma used to derive prope... |
| normlem3 31041 | Lemma used to derive prope... |
| normlem4 31042 | Lemma used to derive prope... |
| normlem5 31043 | Lemma used to derive prope... |
| normlem6 31044 | Lemma used to derive prope... |
| normlem7 31045 | Lemma used to derive prope... |
| normlem8 31046 | Lemma used to derive prope... |
| normlem9 31047 | Lemma used to derive prope... |
| normlem7tALT 31048 | Lemma used to derive prope... |
| bcseqi 31049 | Equality case of Bunjakova... |
| normlem9at 31050 | Lemma used to derive prope... |
| dfhnorm2 31051 | Alternate definition of th... |
| normf 31052 | The norm function maps fro... |
| normval 31053 | The value of the norm of a... |
| normcl 31054 | Real closure of the norm o... |
| normge0 31055 | The norm of a vector is no... |
| normgt0 31056 | The norm of nonzero vector... |
| norm0 31057 | The norm of a zero vector.... |
| norm-i 31058 | Theorem 3.3(i) of [Beran] ... |
| normne0 31059 | A norm is nonzero iff its ... |
| normcli 31060 | Real closure of the norm o... |
| normsqi 31061 | The square of a norm. (Co... |
| norm-i-i 31062 | Theorem 3.3(i) of [Beran] ... |
| normsq 31063 | The square of a norm. (Co... |
| normsub0i 31064 | Two vectors are equal iff ... |
| normsub0 31065 | Two vectors are equal iff ... |
| norm-ii-i 31066 | Triangle inequality for no... |
| norm-ii 31067 | Triangle inequality for no... |
| norm-iii-i 31068 | Theorem 3.3(iii) of [Beran... |
| norm-iii 31069 | Theorem 3.3(iii) of [Beran... |
| normsubi 31070 | Negative doesn't change th... |
| normpythi 31071 | Analogy to Pythagorean the... |
| normsub 31072 | Swapping order of subtract... |
| normneg 31073 | The norm of a vector equal... |
| normpyth 31074 | Analogy to Pythagorean the... |
| normpyc 31075 | Corollary to Pythagorean t... |
| norm3difi 31076 | Norm of differences around... |
| norm3adifii 31077 | Norm of differences around... |
| norm3lem 31078 | Lemma involving norm of di... |
| norm3dif 31079 | Norm of differences around... |
| norm3dif2 31080 | Norm of differences around... |
| norm3lemt 31081 | Lemma involving norm of di... |
| norm3adifi 31082 | Norm of differences around... |
| normpari 31083 | Parallelogram law for norm... |
| normpar 31084 | Parallelogram law for norm... |
| normpar2i 31085 | Corollary of parallelogram... |
| polid2i 31086 | Generalized polarization i... |
| polidi 31087 | Polarization identity. Re... |
| polid 31088 | Polarization identity. Re... |
| hilablo 31089 | Hilbert space vector addit... |
| hilid 31090 | The group identity element... |
| hilvc 31091 | Hilbert space is a complex... |
| hilnormi 31092 | Hilbert space norm in term... |
| hilhhi 31093 | Deduce the structure of Hi... |
| hhnv 31094 | Hilbert space is a normed ... |
| hhva 31095 | The group (addition) opera... |
| hhba 31096 | The base set of Hilbert sp... |
| hh0v 31097 | The zero vector of Hilbert... |
| hhsm 31098 | The scalar product operati... |
| hhvs 31099 | The vector subtraction ope... |
| hhnm 31100 | The norm function of Hilbe... |
| hhims 31101 | The induced metric of Hilb... |
| hhims2 31102 | Hilbert space distance met... |
| hhmet 31103 | The induced metric of Hilb... |
| hhxmet 31104 | The induced metric of Hilb... |
| hhmetdval 31105 | Value of the distance func... |
| hhip 31106 | The inner product operatio... |
| hhph 31107 | The Hilbert space of the H... |
| bcsiALT 31108 | Bunjakovaskij-Cauchy-Schwa... |
| bcsiHIL 31109 | Bunjakovaskij-Cauchy-Schwa... |
| bcs 31110 | Bunjakovaskij-Cauchy-Schwa... |
| bcs2 31111 | Corollary of the Bunjakova... |
| bcs3 31112 | Corollary of the Bunjakova... |
| hcau 31113 | Member of the set of Cauch... |
| hcauseq 31114 | A Cauchy sequences on a Hi... |
| hcaucvg 31115 | A Cauchy sequence on a Hil... |
| seq1hcau 31116 | A sequence on a Hilbert sp... |
| hlimi 31117 | Express the predicate: Th... |
| hlimseqi 31118 | A sequence with a limit on... |
| hlimveci 31119 | Closure of the limit of a ... |
| hlimconvi 31120 | Convergence of a sequence ... |
| hlim2 31121 | The limit of a sequence on... |
| hlimadd 31122 | Limit of the sum of two se... |
| hilmet 31123 | The Hilbert space norm det... |
| hilxmet 31124 | The Hilbert space norm det... |
| hilmetdval 31125 | Value of the distance func... |
| hilims 31126 | Hilbert space distance met... |
| hhcau 31127 | The Cauchy sequences of Hi... |
| hhlm 31128 | The limit sequences of Hil... |
| hhcmpl 31129 | Lemma used for derivation ... |
| hilcompl 31130 | Lemma used for derivation ... |
| hhcms 31132 | The Hilbert space induced ... |
| hhhl 31133 | The Hilbert space structur... |
| hilcms 31134 | The Hilbert space norm det... |
| hilhl 31135 | The Hilbert space of the H... |
| issh 31137 | Subspace ` H ` of a Hilber... |
| issh2 31138 | Subspace ` H ` of a Hilber... |
| shss 31139 | A subspace is a subset of ... |
| shel 31140 | A member of a subspace of ... |
| shex 31141 | The set of subspaces of a ... |
| shssii 31142 | A closed subspace of a Hil... |
| sheli 31143 | A member of a subspace of ... |
| shelii 31144 | A member of a subspace of ... |
| sh0 31145 | The zero vector belongs to... |
| shaddcl 31146 | Closure of vector addition... |
| shmulcl 31147 | Closure of vector scalar m... |
| issh3 31148 | Subspace ` H ` of a Hilber... |
| shsubcl 31149 | Closure of vector subtract... |
| isch 31151 | Closed subspace ` H ` of a... |
| isch2 31152 | Closed subspace ` H ` of a... |
| chsh 31153 | A closed subspace is a sub... |
| chsssh 31154 | Closed subspaces are subsp... |
| chex 31155 | The set of closed subspace... |
| chshii 31156 | A closed subspace is a sub... |
| ch0 31157 | The zero vector belongs to... |
| chss 31158 | A closed subspace of a Hil... |
| chel 31159 | A member of a closed subsp... |
| chssii 31160 | A closed subspace of a Hil... |
| cheli 31161 | A member of a closed subsp... |
| chelii 31162 | A member of a closed subsp... |
| chlimi 31163 | The limit property of a cl... |
| hlim0 31164 | The zero sequence in Hilbe... |
| hlimcaui 31165 | If a sequence in Hilbert s... |
| hlimf 31166 | Function-like behavior of ... |
| hlimuni 31167 | A Hilbert space sequence c... |
| hlimreui 31168 | The limit of a Hilbert spa... |
| hlimeui 31169 | The limit of a Hilbert spa... |
| isch3 31170 | A Hilbert subspace is clos... |
| chcompl 31171 | Completeness of a closed s... |
| helch 31172 | The Hilbert lattice one (w... |
| ifchhv 31173 | Prove ` if ( A e. CH , A ,... |
| helsh 31174 | Hilbert space is a subspac... |
| shsspwh 31175 | Subspaces are subsets of H... |
| chsspwh 31176 | Closed subspaces are subse... |
| hsn0elch 31177 | The zero subspace belongs ... |
| norm1 31178 | From any nonzero Hilbert s... |
| norm1exi 31179 | A normalized vector exists... |
| norm1hex 31180 | A normalized vector can ex... |
| elch0 31183 | Membership in zero for clo... |
| h0elch 31184 | The zero subspace is a clo... |
| h0elsh 31185 | The zero subspace is a sub... |
| hhssva 31186 | The vector addition operat... |
| hhsssm 31187 | The scalar multiplication ... |
| hhssnm 31188 | The norm operation on a su... |
| issubgoilem 31189 | Lemma for ~ hhssabloilem .... |
| hhssabloilem 31190 | Lemma for ~ hhssabloi . F... |
| hhssabloi 31191 | Abelian group property of ... |
| hhssablo 31192 | Abelian group property of ... |
| hhssnv 31193 | Normed complex vector spac... |
| hhssnvt 31194 | Normed complex vector spac... |
| hhsst 31195 | A member of ` SH ` is a su... |
| hhshsslem1 31196 | Lemma for ~ hhsssh . (Con... |
| hhshsslem2 31197 | Lemma for ~ hhsssh . (Con... |
| hhsssh 31198 | The predicate " ` H ` is a... |
| hhsssh2 31199 | The predicate " ` H ` is a... |
| hhssba 31200 | The base set of a subspace... |
| hhssvs 31201 | The vector subtraction ope... |
| hhssvsf 31202 | Mapping of the vector subt... |
| hhssims 31203 | Induced metric of a subspa... |
| hhssims2 31204 | Induced metric of a subspa... |
| hhssmet 31205 | Induced metric of a subspa... |
| hhssmetdval 31206 | Value of the distance func... |
| hhsscms 31207 | The induced metric of a cl... |
| hhssbnOLD 31208 | Obsolete version of ~ cssb... |
| ocval 31209 | Value of orthogonal comple... |
| ocel 31210 | Membership in orthogonal c... |
| shocel 31211 | Membership in orthogonal c... |
| ocsh 31212 | The orthogonal complement ... |
| shocsh 31213 | The orthogonal complement ... |
| ocss 31214 | An orthogonal complement i... |
| shocss 31215 | An orthogonal complement i... |
| occon 31216 | Contraposition law for ort... |
| occon2 31217 | Double contraposition for ... |
| occon2i 31218 | Double contraposition for ... |
| oc0 31219 | The zero vector belongs to... |
| ocorth 31220 | Members of a subset and it... |
| shocorth 31221 | Members of a subspace and ... |
| ococss 31222 | Inclusion in complement of... |
| shococss 31223 | Inclusion in complement of... |
| shorth 31224 | Members of orthogonal subs... |
| ocin 31225 | Intersection of a Hilbert ... |
| occon3 31226 | Hilbert lattice contraposi... |
| ocnel 31227 | A nonzero vector in the co... |
| chocvali 31228 | Value of the orthogonal co... |
| shuni 31229 | Two subspaces with trivial... |
| chocunii 31230 | Lemma for uniqueness part ... |
| pjhthmo 31231 | Projection Theorem, unique... |
| occllem 31232 | Lemma for ~ occl . (Contr... |
| occl 31233 | Closure of complement of H... |
| shoccl 31234 | Closure of complement of H... |
| choccl 31235 | Closure of complement of H... |
| choccli 31236 | Closure of ` CH ` orthocom... |
| shsval 31241 | Value of subspace sum of t... |
| shsss 31242 | The subspace sum is a subs... |
| shsel 31243 | Membership in the subspace... |
| shsel3 31244 | Membership in the subspace... |
| shseli 31245 | Membership in subspace sum... |
| shscli 31246 | Closure of subspace sum. ... |
| shscl 31247 | Closure of subspace sum. ... |
| shscom 31248 | Commutative law for subspa... |
| shsva 31249 | Vector sum belongs to subs... |
| shsel1 31250 | A subspace sum contains a ... |
| shsel2 31251 | A subspace sum contains a ... |
| shsvs 31252 | Vector subtraction belongs... |
| shsub1 31253 | Subspace sum is an upper b... |
| shsub2 31254 | Subspace sum is an upper b... |
| choc0 31255 | The orthocomplement of the... |
| choc1 31256 | The orthocomplement of the... |
| chocnul 31257 | Orthogonal complement of t... |
| shintcli 31258 | Closure of intersection of... |
| shintcl 31259 | The intersection of a none... |
| chintcli 31260 | The intersection of a none... |
| chintcl 31261 | The intersection (infimum)... |
| spanval 31262 | Value of the linear span o... |
| hsupval 31263 | Value of supremum of set o... |
| chsupval 31264 | The value of the supremum ... |
| spancl 31265 | The span of a subset of Hi... |
| elspancl 31266 | A member of a span is a ve... |
| shsupcl 31267 | Closure of the subspace su... |
| hsupcl 31268 | Closure of supremum of set... |
| chsupcl 31269 | Closure of supremum of sub... |
| hsupss 31270 | Subset relation for suprem... |
| chsupss 31271 | Subset relation for suprem... |
| hsupunss 31272 | The union of a set of Hilb... |
| chsupunss 31273 | The union of a set of clos... |
| spanss2 31274 | A subset of Hilbert space ... |
| shsupunss 31275 | The union of a set of subs... |
| spanid 31276 | A subspace of Hilbert spac... |
| spanss 31277 | Ordering relationship for ... |
| spanssoc 31278 | The span of a subset of Hi... |
| sshjval 31279 | Value of join for subsets ... |
| shjval 31280 | Value of join in ` SH ` . ... |
| chjval 31281 | Value of join in ` CH ` . ... |
| chjvali 31282 | Value of join in ` CH ` . ... |
| sshjval3 31283 | Value of join for subsets ... |
| sshjcl 31284 | Closure of join for subset... |
| shjcl 31285 | Closure of join in ` SH ` ... |
| chjcl 31286 | Closure of join in ` CH ` ... |
| shjcom 31287 | Commutative law for Hilber... |
| shless 31288 | Subset implies subset of s... |
| shlej1 31289 | Add disjunct to both sides... |
| shlej2 31290 | Add disjunct to both sides... |
| shincli 31291 | Closure of intersection of... |
| shscomi 31292 | Commutative law for subspa... |
| shsvai 31293 | Vector sum belongs to subs... |
| shsel1i 31294 | A subspace sum contains a ... |
| shsel2i 31295 | A subspace sum contains a ... |
| shsvsi 31296 | Vector subtraction belongs... |
| shunssi 31297 | Union is smaller than subs... |
| shunssji 31298 | Union is smaller than Hilb... |
| shsleji 31299 | Subspace sum is smaller th... |
| shjcomi 31300 | Commutative law for join i... |
| shsub1i 31301 | Subspace sum is an upper b... |
| shsub2i 31302 | Subspace sum is an upper b... |
| shub1i 31303 | Hilbert lattice join is an... |
| shjcli 31304 | Closure of ` CH ` join. (... |
| shjshcli 31305 | ` SH ` closure of join. (... |
| shlessi 31306 | Subset implies subset of s... |
| shlej1i 31307 | Add disjunct to both sides... |
| shlej2i 31308 | Add disjunct to both sides... |
| shslej 31309 | Subspace sum is smaller th... |
| shincl 31310 | Closure of intersection of... |
| shub1 31311 | Hilbert lattice join is an... |
| shub2 31312 | A subspace is a subset of ... |
| shsidmi 31313 | Idempotent law for Hilbert... |
| shslubi 31314 | The least upper bound law ... |
| shlesb1i 31315 | Hilbert lattice ordering i... |
| shsval2i 31316 | An alternate way to expres... |
| shsval3i 31317 | An alternate way to expres... |
| shmodsi 31318 | The modular law holds for ... |
| shmodi 31319 | The modular law is implied... |
| pjhthlem1 31320 | Lemma for ~ pjhth . (Cont... |
| pjhthlem2 31321 | Lemma for ~ pjhth . (Cont... |
| pjhth 31322 | Projection Theorem: Any H... |
| pjhtheu 31323 | Projection Theorem: Any H... |
| pjhfval 31325 | The value of the projectio... |
| pjhval 31326 | Value of a projection. (C... |
| pjpreeq 31327 | Equality with a projection... |
| pjeq 31328 | Equality with a projection... |
| axpjcl 31329 | Closure of a projection in... |
| pjhcl 31330 | Closure of a projection in... |
| omlsilem 31331 | Lemma for orthomodular law... |
| omlsii 31332 | Subspace inference form of... |
| omlsi 31333 | Subspace form of orthomodu... |
| ococi 31334 | Complement of complement o... |
| ococ 31335 | Complement of complement o... |
| dfch2 31336 | Alternate definition of th... |
| ococin 31337 | The double complement is t... |
| hsupval2 31338 | Alternate definition of su... |
| chsupval2 31339 | The value of the supremum ... |
| sshjval2 31340 | Value of join in the set o... |
| chsupid 31341 | A subspace is the supremum... |
| chsupsn 31342 | Value of supremum of subse... |
| shlub 31343 | Hilbert lattice join is th... |
| shlubi 31344 | Hilbert lattice join is th... |
| pjhtheu2 31345 | Uniqueness of ` y ` for th... |
| pjcli 31346 | Closure of a projection in... |
| pjhcli 31347 | Closure of a projection in... |
| pjpjpre 31348 | Decomposition of a vector ... |
| axpjpj 31349 | Decomposition of a vector ... |
| pjclii 31350 | Closure of a projection in... |
| pjhclii 31351 | Closure of a projection in... |
| pjpj0i 31352 | Decomposition of a vector ... |
| pjpji 31353 | Decomposition of a vector ... |
| pjpjhth 31354 | Projection Theorem: Any H... |
| pjpjhthi 31355 | Projection Theorem: Any H... |
| pjop 31356 | Orthocomplement projection... |
| pjpo 31357 | Projection in terms of ort... |
| pjopi 31358 | Orthocomplement projection... |
| pjpoi 31359 | Projection in terms of ort... |
| pjoc1i 31360 | Projection of a vector in ... |
| pjchi 31361 | Projection of a vector in ... |
| pjoccl 31362 | The part of a vector that ... |
| pjoc1 31363 | Projection of a vector in ... |
| pjomli 31364 | Subspace form of orthomodu... |
| pjoml 31365 | Subspace form of orthomodu... |
| pjococi 31366 | Proof of orthocomplement t... |
| pjoc2i 31367 | Projection of a vector in ... |
| pjoc2 31368 | Projection of a vector in ... |
| sh0le 31369 | The zero subspace is the s... |
| ch0le 31370 | The zero subspace is the s... |
| shle0 31371 | No subspace is smaller tha... |
| chle0 31372 | No Hilbert lattice element... |
| chnlen0 31373 | A Hilbert lattice element ... |
| ch0pss 31374 | The zero subspace is a pro... |
| orthin 31375 | The intersection of orthog... |
| ssjo 31376 | The lattice join of a subs... |
| shne0i 31377 | A nonzero subspace has a n... |
| shs0i 31378 | Hilbert subspace sum with ... |
| shs00i 31379 | Two subspaces are zero iff... |
| ch0lei 31380 | The closed subspace zero i... |
| chle0i 31381 | No Hilbert closed subspace... |
| chne0i 31382 | A nonzero closed subspace ... |
| chocini 31383 | Intersection of a closed s... |
| chj0i 31384 | Join with lattice zero in ... |
| chm1i 31385 | Meet with lattice one in `... |
| chjcli 31386 | Closure of ` CH ` join. (... |
| chsleji 31387 | Subspace sum is smaller th... |
| chseli 31388 | Membership in subspace sum... |
| chincli 31389 | Closure of Hilbert lattice... |
| chsscon3i 31390 | Hilbert lattice contraposi... |
| chsscon1i 31391 | Hilbert lattice contraposi... |
| chsscon2i 31392 | Hilbert lattice contraposi... |
| chcon2i 31393 | Hilbert lattice contraposi... |
| chcon1i 31394 | Hilbert lattice contraposi... |
| chcon3i 31395 | Hilbert lattice contraposi... |
| chunssji 31396 | Union is smaller than ` CH... |
| chjcomi 31397 | Commutative law for join i... |
| chub1i 31398 | ` CH ` join is an upper bo... |
| chub2i 31399 | ` CH ` join is an upper bo... |
| chlubi 31400 | Hilbert lattice join is th... |
| chlubii 31401 | Hilbert lattice join is th... |
| chlej1i 31402 | Add join to both sides of ... |
| chlej2i 31403 | Add join to both sides of ... |
| chlej12i 31404 | Add join to both sides of ... |
| chlejb1i 31405 | Hilbert lattice ordering i... |
| chdmm1i 31406 | De Morgan's law for meet i... |
| chdmm2i 31407 | De Morgan's law for meet i... |
| chdmm3i 31408 | De Morgan's law for meet i... |
| chdmm4i 31409 | De Morgan's law for meet i... |
| chdmj1i 31410 | De Morgan's law for join i... |
| chdmj2i 31411 | De Morgan's law for join i... |
| chdmj3i 31412 | De Morgan's law for join i... |
| chdmj4i 31413 | De Morgan's law for join i... |
| chnlei 31414 | Equivalent expressions for... |
| chjassi 31415 | Associative law for Hilber... |
| chj00i 31416 | Two Hilbert lattice elemen... |
| chjoi 31417 | The join of a closed subsp... |
| chj1i 31418 | Join with Hilbert lattice ... |
| chm0i 31419 | Meet with Hilbert lattice ... |
| chm0 31420 | Meet with Hilbert lattice ... |
| shjshsi 31421 | Hilbert lattice join equal... |
| shjshseli 31422 | A closed subspace sum equa... |
| chne0 31423 | A nonzero closed subspace ... |
| chocin 31424 | Intersection of a closed s... |
| chssoc 31425 | A closed subspace less tha... |
| chj0 31426 | Join with Hilbert lattice ... |
| chslej 31427 | Subspace sum is smaller th... |
| chincl 31428 | Closure of Hilbert lattice... |
| chsscon3 31429 | Hilbert lattice contraposi... |
| chsscon1 31430 | Hilbert lattice contraposi... |
| chsscon2 31431 | Hilbert lattice contraposi... |
| chpsscon3 31432 | Hilbert lattice contraposi... |
| chpsscon1 31433 | Hilbert lattice contraposi... |
| chpsscon2 31434 | Hilbert lattice contraposi... |
| chjcom 31435 | Commutative law for Hilber... |
| chub1 31436 | Hilbert lattice join is gr... |
| chub2 31437 | Hilbert lattice join is gr... |
| chlub 31438 | Hilbert lattice join is th... |
| chlej1 31439 | Add join to both sides of ... |
| chlej2 31440 | Add join to both sides of ... |
| chlejb1 31441 | Hilbert lattice ordering i... |
| chlejb2 31442 | Hilbert lattice ordering i... |
| chnle 31443 | Equivalent expressions for... |
| chjo 31444 | The join of a closed subsp... |
| chabs1 31445 | Hilbert lattice absorption... |
| chabs2 31446 | Hilbert lattice absorption... |
| chabs1i 31447 | Hilbert lattice absorption... |
| chabs2i 31448 | Hilbert lattice absorption... |
| chjidm 31449 | Idempotent law for Hilbert... |
| chjidmi 31450 | Idempotent law for Hilbert... |
| chj12i 31451 | A rearrangement of Hilbert... |
| chj4i 31452 | Rearrangement of the join ... |
| chjjdiri 31453 | Hilbert lattice join distr... |
| chdmm1 31454 | De Morgan's law for meet i... |
| chdmm2 31455 | De Morgan's law for meet i... |
| chdmm3 31456 | De Morgan's law for meet i... |
| chdmm4 31457 | De Morgan's law for meet i... |
| chdmj1 31458 | De Morgan's law for join i... |
| chdmj2 31459 | De Morgan's law for join i... |
| chdmj3 31460 | De Morgan's law for join i... |
| chdmj4 31461 | De Morgan's law for join i... |
| chjass 31462 | Associative law for Hilber... |
| chj12 31463 | A rearrangement of Hilbert... |
| chj4 31464 | Rearrangement of the join ... |
| ledii 31465 | An ortholattice is distrib... |
| lediri 31466 | An ortholattice is distrib... |
| lejdii 31467 | An ortholattice is distrib... |
| lejdiri 31468 | An ortholattice is distrib... |
| ledi 31469 | An ortholattice is distrib... |
| spansn0 31470 | The span of the singleton ... |
| span0 31471 | The span of the empty set ... |
| elspani 31472 | Membership in the span of ... |
| spanuni 31473 | The span of a union is the... |
| spanun 31474 | The span of a union is the... |
| sshhococi 31475 | The join of two Hilbert sp... |
| hne0 31476 | Hilbert space has a nonzer... |
| chsup0 31477 | The supremum of the empty ... |
| h1deoi 31478 | Membership in orthocomplem... |
| h1dei 31479 | Membership in 1-dimensiona... |
| h1did 31480 | A generating vector belong... |
| h1dn0 31481 | A nonzero vector generates... |
| h1de2i 31482 | Membership in 1-dimensiona... |
| h1de2bi 31483 | Membership in 1-dimensiona... |
| h1de2ctlem 31484 | Lemma for ~ h1de2ci . (Co... |
| h1de2ci 31485 | Membership in 1-dimensiona... |
| spansni 31486 | The span of a singleton in... |
| elspansni 31487 | Membership in the span of ... |
| spansn 31488 | The span of a singleton in... |
| spansnch 31489 | The span of a Hilbert spac... |
| spansnsh 31490 | The span of a Hilbert spac... |
| spansnchi 31491 | The span of a singleton in... |
| spansnid 31492 | A vector belongs to the sp... |
| spansnmul 31493 | A scalar product with a ve... |
| elspansncl 31494 | A member of a span of a si... |
| elspansn 31495 | Membership in the span of ... |
| elspansn2 31496 | Membership in the span of ... |
| spansncol 31497 | The singletons of collinea... |
| spansneleqi 31498 | Membership relation implie... |
| spansneleq 31499 | Membership relation that i... |
| spansnss 31500 | The span of the singleton ... |
| elspansn3 31501 | A member of the span of th... |
| elspansn4 31502 | A span membership conditio... |
| elspansn5 31503 | A vector belonging to both... |
| spansnss2 31504 | The span of the singleton ... |
| normcan 31505 | Cancellation-type law that... |
| pjspansn 31506 | A projection on the span o... |
| spansnpji 31507 | A subset of Hilbert space ... |
| spanunsni 31508 | The span of the union of a... |
| spanpr 31509 | The span of a pair of vect... |
| h1datomi 31510 | A 1-dimensional subspace i... |
| h1datom 31511 | A 1-dimensional subspace i... |
| cmbr 31513 | Binary relation expressing... |
| pjoml2i 31514 | Variation of orthomodular ... |
| pjoml3i 31515 | Variation of orthomodular ... |
| pjoml4i 31516 | Variation of orthomodular ... |
| pjoml5i 31517 | The orthomodular law. Rem... |
| pjoml6i 31518 | An equivalent of the ortho... |
| cmbri 31519 | Binary relation expressing... |
| cmcmlem 31520 | Commutation is symmetric. ... |
| cmcmi 31521 | Commutation is symmetric. ... |
| cmcm2i 31522 | Commutation with orthocomp... |
| cmcm3i 31523 | Commutation with orthocomp... |
| cmcm4i 31524 | Commutation with orthocomp... |
| cmbr2i 31525 | Alternate definition of th... |
| cmcmii 31526 | Commutation is symmetric. ... |
| cmcm2ii 31527 | Commutation with orthocomp... |
| cmcm3ii 31528 | Commutation with orthocomp... |
| cmbr3i 31529 | Alternate definition for t... |
| cmbr4i 31530 | Alternate definition for t... |
| lecmi 31531 | Comparable Hilbert lattice... |
| lecmii 31532 | Comparable Hilbert lattice... |
| cmj1i 31533 | A Hilbert lattice element ... |
| cmj2i 31534 | A Hilbert lattice element ... |
| cmm1i 31535 | A Hilbert lattice element ... |
| cmm2i 31536 | A Hilbert lattice element ... |
| cmbr3 31537 | Alternate definition for t... |
| cm0 31538 | The zero Hilbert lattice e... |
| cmidi 31539 | The commutes relation is r... |
| pjoml2 31540 | Variation of orthomodular ... |
| pjoml3 31541 | Variation of orthomodular ... |
| pjoml5 31542 | The orthomodular law. Rem... |
| cmcm 31543 | Commutation is symmetric. ... |
| cmcm3 31544 | Commutation with orthocomp... |
| cmcm2 31545 | Commutation with orthocomp... |
| lecm 31546 | Comparable Hilbert lattice... |
| fh1 31547 | Foulis-Holland Theorem. I... |
| fh2 31548 | Foulis-Holland Theorem. I... |
| cm2j 31549 | A lattice element that com... |
| fh1i 31550 | Foulis-Holland Theorem. I... |
| fh2i 31551 | Foulis-Holland Theorem. I... |
| fh3i 31552 | Variation of the Foulis-Ho... |
| fh4i 31553 | Variation of the Foulis-Ho... |
| cm2ji 31554 | A lattice element that com... |
| cm2mi 31555 | A lattice element that com... |
| qlax1i 31556 | One of the equations showi... |
| qlax2i 31557 | One of the equations showi... |
| qlax3i 31558 | One of the equations showi... |
| qlax4i 31559 | One of the equations showi... |
| qlax5i 31560 | One of the equations showi... |
| qlaxr1i 31561 | One of the conditions show... |
| qlaxr2i 31562 | One of the conditions show... |
| qlaxr4i 31563 | One of the conditions show... |
| qlaxr5i 31564 | One of the conditions show... |
| qlaxr3i 31565 | A variation of the orthomo... |
| chscllem1 31566 | Lemma for ~ chscl . (Cont... |
| chscllem2 31567 | Lemma for ~ chscl . (Cont... |
| chscllem3 31568 | Lemma for ~ chscl . (Cont... |
| chscllem4 31569 | Lemma for ~ chscl . (Cont... |
| chscl 31570 | The subspace sum of two cl... |
| osumi 31571 | If two closed subspaces of... |
| osumcori 31572 | Corollary of ~ osumi . (C... |
| osumcor2i 31573 | Corollary of ~ osumi , sho... |
| osum 31574 | If two closed subspaces of... |
| spansnji 31575 | The subspace sum of a clos... |
| spansnj 31576 | The subspace sum of a clos... |
| spansnscl 31577 | The subspace sum of a clos... |
| sumspansn 31578 | The sum of two vectors bel... |
| spansnm0i 31579 | The meet of different one-... |
| nonbooli 31580 | A Hilbert lattice with two... |
| spansncvi 31581 | Hilbert space has the cove... |
| spansncv 31582 | Hilbert space has the cove... |
| 5oalem1 31583 | Lemma for orthoarguesian l... |
| 5oalem2 31584 | Lemma for orthoarguesian l... |
| 5oalem3 31585 | Lemma for orthoarguesian l... |
| 5oalem4 31586 | Lemma for orthoarguesian l... |
| 5oalem5 31587 | Lemma for orthoarguesian l... |
| 5oalem6 31588 | Lemma for orthoarguesian l... |
| 5oalem7 31589 | Lemma for orthoarguesian l... |
| 5oai 31590 | Orthoarguesian law 5OA. Th... |
| 3oalem1 31591 | Lemma for 3OA (weak) ortho... |
| 3oalem2 31592 | Lemma for 3OA (weak) ortho... |
| 3oalem3 31593 | Lemma for 3OA (weak) ortho... |
| 3oalem4 31594 | Lemma for 3OA (weak) ortho... |
| 3oalem5 31595 | Lemma for 3OA (weak) ortho... |
| 3oalem6 31596 | Lemma for 3OA (weak) ortho... |
| 3oai 31597 | 3OA (weak) orthoarguesian ... |
| pjorthi 31598 | Projection components on o... |
| pjch1 31599 | Property of identity proje... |
| pjo 31600 | The orthogonal projection.... |
| pjcompi 31601 | Component of a projection.... |
| pjidmi 31602 | A projection is idempotent... |
| pjadjii 31603 | A projection is self-adjoi... |
| pjaddii 31604 | Projection of vector sum i... |
| pjinormii 31605 | The inner product of a pro... |
| pjmulii 31606 | Projection of (scalar) pro... |
| pjsubii 31607 | Projection of vector diffe... |
| pjsslem 31608 | Lemma for subset relations... |
| pjss2i 31609 | Subset relationship for pr... |
| pjssmii 31610 | Projection meet property. ... |
| pjssge0ii 31611 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormii 31612 | Theorem 4.5(v)<->(vi) of [... |
| pjcji 31613 | The projection on a subspa... |
| pjadji 31614 | A projection is self-adjoi... |
| pjaddi 31615 | Projection of vector sum i... |
| pjinormi 31616 | The inner product of a pro... |
| pjsubi 31617 | Projection of vector diffe... |
| pjmuli 31618 | Projection of scalar produ... |
| pjige0i 31619 | The inner product of a pro... |
| pjige0 31620 | The inner product of a pro... |
| pjcjt2 31621 | The projection on a subspa... |
| pj0i 31622 | The projection of the zero... |
| pjch 31623 | Projection of a vector in ... |
| pjid 31624 | The projection of a vector... |
| pjvec 31625 | The set of vectors belongi... |
| pjocvec 31626 | The set of vectors belongi... |
| pjocini 31627 | Membership of projection i... |
| pjini 31628 | Membership of projection i... |
| pjjsi 31629 | A sufficient condition for... |
| pjfni 31630 | Functionality of a project... |
| pjrni 31631 | The range of a projection.... |
| pjfoi 31632 | A projection maps onto its... |
| pjfi 31633 | The mapping of a projectio... |
| pjvi 31634 | The value of a projection ... |
| pjhfo 31635 | A projection maps onto its... |
| pjrn 31636 | The range of a projection.... |
| pjhf 31637 | The mapping of a projectio... |
| pjfn 31638 | Functionality of a project... |
| pjsumi 31639 | The projection on a subspa... |
| pj11i 31640 | One-to-one correspondence ... |
| pjdsi 31641 | Vector decomposition into ... |
| pjds3i 31642 | Vector decomposition into ... |
| pj11 31643 | One-to-one correspondence ... |
| pjmfn 31644 | Functionality of the proje... |
| pjmf1 31645 | The projector function map... |
| pjoi0 31646 | The inner product of proje... |
| pjoi0i 31647 | The inner product of proje... |
| pjopythi 31648 | Pythagorean theorem for pr... |
| pjopyth 31649 | Pythagorean theorem for pr... |
| pjnormi 31650 | The norm of the projection... |
| pjpythi 31651 | Pythagorean theorem for pr... |
| pjneli 31652 | If a vector does not belon... |
| pjnorm 31653 | The norm of the projection... |
| pjpyth 31654 | Pythagorean theorem for pr... |
| pjnel 31655 | If a vector does not belon... |
| pjnorm2 31656 | A vector belongs to the su... |
| mayete3i 31657 | Mayet's equation E_3. Par... |
| mayetes3i 31658 | Mayet's equation E^*_3, de... |
| hosmval 31664 | Value of the sum of two Hi... |
| hommval 31665 | Value of the scalar produc... |
| hodmval 31666 | Value of the difference of... |
| hfsmval 31667 | Value of the sum of two Hi... |
| hfmmval 31668 | Value of the scalar produc... |
| hosval 31669 | Value of the sum of two Hi... |
| homval 31670 | Value of the scalar produc... |
| hodval 31671 | Value of the difference of... |
| hfsval 31672 | Value of the sum of two Hi... |
| hfmval 31673 | Value of the scalar produc... |
| hoscl 31674 | Closure of the sum of two ... |
| homcl 31675 | Closure of the scalar prod... |
| hodcl 31676 | Closure of the difference ... |
| ho0val 31679 | Value of the zero Hilbert ... |
| ho0f 31680 | Functionality of the zero ... |
| df0op2 31681 | Alternate definition of Hi... |
| dfiop2 31682 | Alternate definition of Hi... |
| hoif 31683 | Functionality of the Hilbe... |
| hoival 31684 | The value of the Hilbert s... |
| hoico1 31685 | Composition with the Hilbe... |
| hoico2 31686 | Composition with the Hilbe... |
| hoaddcl 31687 | The sum of Hilbert space o... |
| homulcl 31688 | The scalar product of a Hi... |
| hoeq 31689 | Equality of Hilbert space ... |
| hoeqi 31690 | Equality of Hilbert space ... |
| hoscli 31691 | Closure of Hilbert space o... |
| hodcli 31692 | Closure of Hilbert space o... |
| hocoi 31693 | Composition of Hilbert spa... |
| hococli 31694 | Closure of composition of ... |
| hocofi 31695 | Mapping of composition of ... |
| hocofni 31696 | Functionality of compositi... |
| hoaddcli 31697 | Mapping of sum of Hilbert ... |
| hosubcli 31698 | Mapping of difference of H... |
| hoaddfni 31699 | Functionality of sum of Hi... |
| hosubfni 31700 | Functionality of differenc... |
| hoaddcomi 31701 | Commutativity of sum of Hi... |
| hosubcl 31702 | Mapping of difference of H... |
| hoaddcom 31703 | Commutativity of sum of Hi... |
| hodsi 31704 | Relationship between Hilbe... |
| hoaddassi 31705 | Associativity of sum of Hi... |
| hoadd12i 31706 | Commutative/associative la... |
| hoadd32i 31707 | Commutative/associative la... |
| hocadddiri 31708 | Distributive law for Hilbe... |
| hocsubdiri 31709 | Distributive law for Hilbe... |
| ho2coi 31710 | Double composition of Hilb... |
| hoaddass 31711 | Associativity of sum of Hi... |
| hoadd32 31712 | Commutative/associative la... |
| hoadd4 31713 | Rearrangement of 4 terms i... |
| hocsubdir 31714 | Distributive law for Hilbe... |
| hoaddridi 31715 | Sum of a Hilbert space ope... |
| hodidi 31716 | Difference of a Hilbert sp... |
| ho0coi 31717 | Composition of the zero op... |
| hoid1i 31718 | Composition of Hilbert spa... |
| hoid1ri 31719 | Composition of Hilbert spa... |
| hoaddrid 31720 | Sum of a Hilbert space ope... |
| hodid 31721 | Difference of a Hilbert sp... |
| hon0 31722 | A Hilbert space operator i... |
| hodseqi 31723 | Subtraction and addition o... |
| ho0subi 31724 | Subtraction of Hilbert spa... |
| honegsubi 31725 | Relationship between Hilbe... |
| ho0sub 31726 | Subtraction of Hilbert spa... |
| hosubid1 31727 | The zero operator subtract... |
| honegsub 31728 | Relationship between Hilbe... |
| homullid 31729 | An operator equals its sca... |
| homco1 31730 | Associative law for scalar... |
| homulass 31731 | Scalar product associative... |
| hoadddi 31732 | Scalar product distributiv... |
| hoadddir 31733 | Scalar product reverse dis... |
| homul12 31734 | Swap first and second fact... |
| honegneg 31735 | Double negative of a Hilbe... |
| hosubneg 31736 | Relationship between opera... |
| hosubdi 31737 | Scalar product distributiv... |
| honegdi 31738 | Distribution of negative o... |
| honegsubdi 31739 | Distribution of negative o... |
| honegsubdi2 31740 | Distribution of negative o... |
| hosubsub2 31741 | Law for double subtraction... |
| hosub4 31742 | Rearrangement of 4 terms i... |
| hosubadd4 31743 | Rearrangement of 4 terms i... |
| hoaddsubass 31744 | Associative-type law for a... |
| hoaddsub 31745 | Law for operator addition ... |
| hosubsub 31746 | Law for double subtraction... |
| hosubsub4 31747 | Law for double subtraction... |
| ho2times 31748 | Two times a Hilbert space ... |
| hoaddsubassi 31749 | Associativity of sum and d... |
| hoaddsubi 31750 | Law for sum and difference... |
| hosd1i 31751 | Hilbert space operator sum... |
| hosd2i 31752 | Hilbert space operator sum... |
| hopncani 31753 | Hilbert space operator can... |
| honpcani 31754 | Hilbert space operator can... |
| hosubeq0i 31755 | If the difference between ... |
| honpncani 31756 | Hilbert space operator can... |
| ho01i 31757 | A condition implying that ... |
| ho02i 31758 | A condition implying that ... |
| hoeq1 31759 | A condition implying that ... |
| hoeq2 31760 | A condition implying that ... |
| adjmo 31761 | Every Hilbert space operat... |
| adjsym 31762 | Symmetry property of an ad... |
| eigrei 31763 | A necessary and sufficient... |
| eigre 31764 | A necessary and sufficient... |
| eigposi 31765 | A sufficient condition (fi... |
| eigorthi 31766 | A necessary and sufficient... |
| eigorth 31767 | A necessary and sufficient... |
| nmopval 31785 | Value of the norm of a Hil... |
| elcnop 31786 | Property defining a contin... |
| ellnop 31787 | Property defining a linear... |
| lnopf 31788 | A linear Hilbert space ope... |
| elbdop 31789 | Property defining a bounde... |
| bdopln 31790 | A bounded linear Hilbert s... |
| bdopf 31791 | A bounded linear Hilbert s... |
| nmopsetretALT 31792 | The set in the supremum of... |
| nmopsetretHIL 31793 | The set in the supremum of... |
| nmopsetn0 31794 | The set in the supremum of... |
| nmopxr 31795 | The norm of a Hilbert spac... |
| nmoprepnf 31796 | The norm of a Hilbert spac... |
| nmopgtmnf 31797 | The norm of a Hilbert spac... |
| nmopreltpnf 31798 | The norm of a Hilbert spac... |
| nmopre 31799 | The norm of a bounded oper... |
| elbdop2 31800 | Property defining a bounde... |
| elunop 31801 | Property defining a unitar... |
| elhmop 31802 | Property defining a Hermit... |
| hmopf 31803 | A Hermitian operator is a ... |
| hmopex 31804 | The class of Hermitian ope... |
| nmfnval 31805 | Value of the norm of a Hil... |
| nmfnsetre 31806 | The set in the supremum of... |
| nmfnsetn0 31807 | The set in the supremum of... |
| nmfnxr 31808 | The norm of any Hilbert sp... |
| nmfnrepnf 31809 | The norm of a Hilbert spac... |
| nlfnval 31810 | Value of the null space of... |
| elcnfn 31811 | Property defining a contin... |
| ellnfn 31812 | Property defining a linear... |
| lnfnf 31813 | A linear Hilbert space fun... |
| dfadj2 31814 | Alternate definition of th... |
| funadj 31815 | Functionality of the adjoi... |
| dmadjss 31816 | The domain of the adjoint ... |
| dmadjop 31817 | A member of the domain of ... |
| adjeu 31818 | Elementhood in the domain ... |
| adjval 31819 | Value of the adjoint funct... |
| adjval2 31820 | Value of the adjoint funct... |
| cnvadj 31821 | The adjoint function equal... |
| funcnvadj 31822 | The converse of the adjoin... |
| adj1o 31823 | The adjoint function maps ... |
| dmadjrn 31824 | The adjoint of an operator... |
| eigvecval 31825 | The set of eigenvectors of... |
| eigvalfval 31826 | The eigenvalues of eigenve... |
| specval 31827 | The value of the spectrum ... |
| speccl 31828 | The spectrum of an operato... |
| hhlnoi 31829 | The linear operators of Hi... |
| hhnmoi 31830 | The norm of an operator in... |
| hhbloi 31831 | A bounded linear operator ... |
| hh0oi 31832 | The zero operator in Hilbe... |
| hhcno 31833 | The continuous operators o... |
| hhcnf 31834 | The continuous functionals... |
| dmadjrnb 31835 | The adjoint of an operator... |
| nmoplb 31836 | A lower bound for an opera... |
| nmopub 31837 | An upper bound for an oper... |
| nmopub2tALT 31838 | An upper bound for an oper... |
| nmopub2tHIL 31839 | An upper bound for an oper... |
| nmopge0 31840 | The norm of any Hilbert sp... |
| nmopgt0 31841 | A linear Hilbert space ope... |
| cnopc 31842 | Basic continuity property ... |
| lnopl 31843 | Basic linearity property o... |
| unop 31844 | Basic inner product proper... |
| unopf1o 31845 | A unitary operator in Hilb... |
| unopnorm 31846 | A unitary operator is idem... |
| cnvunop 31847 | The inverse (converse) of ... |
| unopadj 31848 | The inverse (converse) of ... |
| unoplin 31849 | A unitary operator is line... |
| counop 31850 | The composition of two uni... |
| hmop 31851 | Basic inner product proper... |
| hmopre 31852 | The inner product of the v... |
| nmfnlb 31853 | A lower bound for a functi... |
| nmfnleub 31854 | An upper bound for the nor... |
| nmfnleub2 31855 | An upper bound for the nor... |
| nmfnge0 31856 | The norm of any Hilbert sp... |
| elnlfn 31857 | Membership in the null spa... |
| elnlfn2 31858 | Membership in the null spa... |
| cnfnc 31859 | Basic continuity property ... |
| lnfnl 31860 | Basic linearity property o... |
| adjcl 31861 | Closure of the adjoint of ... |
| adj1 31862 | Property of an adjoint Hil... |
| adj2 31863 | Property of an adjoint Hil... |
| adjeq 31864 | A property that determines... |
| adjadj 31865 | Double adjoint. Theorem 3... |
| adjvalval 31866 | Value of the value of the ... |
| unopadj2 31867 | The adjoint of a unitary o... |
| hmopadj 31868 | A Hermitian operator is se... |
| hmdmadj 31869 | Every Hermitian operator h... |
| hmopadj2 31870 | An operator is Hermitian i... |
| hmoplin 31871 | A Hermitian operator is li... |
| brafval 31872 | The bra of a vector, expre... |
| braval 31873 | A bra-ket juxtaposition, e... |
| braadd 31874 | Linearity property of bra ... |
| bramul 31875 | Linearity property of bra ... |
| brafn 31876 | The bra function is a func... |
| bralnfn 31877 | The Dirac bra function is ... |
| bracl 31878 | Closure of the bra functio... |
| bra0 31879 | The Dirac bra of the zero ... |
| brafnmul 31880 | Anti-linearity property of... |
| kbfval 31881 | The outer product of two v... |
| kbop 31882 | The outer product of two v... |
| kbval 31883 | The value of the operator ... |
| kbmul 31884 | Multiplication property of... |
| kbpj 31885 | If a vector ` A ` has norm... |
| eleigvec 31886 | Membership in the set of e... |
| eleigvec2 31887 | Membership in the set of e... |
| eleigveccl 31888 | Closure of an eigenvector ... |
| eigvalval 31889 | The eigenvalue of an eigen... |
| eigvalcl 31890 | An eigenvalue is a complex... |
| eigvec1 31891 | Property of an eigenvector... |
| eighmre 31892 | The eigenvalues of a Hermi... |
| eighmorth 31893 | Eigenvectors of a Hermitia... |
| nmopnegi 31894 | Value of the norm of the n... |
| lnop0 31895 | The value of a linear Hilb... |
| lnopmul 31896 | Multiplicative property of... |
| lnopli 31897 | Basic scalar product prope... |
| lnopfi 31898 | A linear Hilbert space ope... |
| lnop0i 31899 | The value of a linear Hilb... |
| lnopaddi 31900 | Additive property of a lin... |
| lnopmuli 31901 | Multiplicative property of... |
| lnopaddmuli 31902 | Sum/product property of a ... |
| lnopsubi 31903 | Subtraction property for a... |
| lnopsubmuli 31904 | Subtraction/product proper... |
| lnopmulsubi 31905 | Product/subtraction proper... |
| homco2 31906 | Move a scalar product out ... |
| idunop 31907 | The identity function (res... |
| 0cnop 31908 | The identically zero funct... |
| 0cnfn 31909 | The identically zero funct... |
| idcnop 31910 | The identity function (res... |
| idhmop 31911 | The Hilbert space identity... |
| 0hmop 31912 | The identically zero funct... |
| 0lnop 31913 | The identically zero funct... |
| 0lnfn 31914 | The identically zero funct... |
| nmop0 31915 | The norm of the zero opera... |
| nmfn0 31916 | The norm of the identicall... |
| hmopbdoptHIL 31917 | A Hermitian operator is a ... |
| hoddii 31918 | Distributive law for Hilbe... |
| hoddi 31919 | Distributive law for Hilbe... |
| nmop0h 31920 | The norm of any operator o... |
| idlnop 31921 | The identity function (res... |
| 0bdop 31922 | The identically zero opera... |
| adj0 31923 | Adjoint of the zero operat... |
| nmlnop0iALT 31924 | A linear operator with a z... |
| nmlnop0iHIL 31925 | A linear operator with a z... |
| nmlnopgt0i 31926 | A linear Hilbert space ope... |
| nmlnop0 31927 | A linear operator with a z... |
| nmlnopne0 31928 | A linear operator with a n... |
| lnopmi 31929 | The scalar product of a li... |
| lnophsi 31930 | The sum of two linear oper... |
| lnophdi 31931 | The difference of two line... |
| lnopcoi 31932 | The composition of two lin... |
| lnopco0i 31933 | The composition of a linea... |
| lnopeq0lem1 31934 | Lemma for ~ lnopeq0i . Ap... |
| lnopeq0lem2 31935 | Lemma for ~ lnopeq0i . (C... |
| lnopeq0i 31936 | A condition implying that ... |
| lnopeqi 31937 | Two linear Hilbert space o... |
| lnopeq 31938 | Two linear Hilbert space o... |
| lnopunilem1 31939 | Lemma for ~ lnopunii . (C... |
| lnopunilem2 31940 | Lemma for ~ lnopunii . (C... |
| lnopunii 31941 | If a linear operator (whos... |
| elunop2 31942 | An operator is unitary iff... |
| nmopun 31943 | Norm of a unitary Hilbert ... |
| unopbd 31944 | A unitary operator is a bo... |
| lnophmlem1 31945 | Lemma for ~ lnophmi . (Co... |
| lnophmlem2 31946 | Lemma for ~ lnophmi . (Co... |
| lnophmi 31947 | A linear operator is Hermi... |
| lnophm 31948 | A linear operator is Hermi... |
| hmops 31949 | The sum of two Hermitian o... |
| hmopm 31950 | The scalar product of a He... |
| hmopd 31951 | The difference of two Herm... |
| hmopco 31952 | The composition of two com... |
| nmbdoplbi 31953 | A lower bound for the norm... |
| nmbdoplb 31954 | A lower bound for the norm... |
| nmcexi 31955 | Lemma for ~ nmcopexi and ~... |
| nmcopexi 31956 | The norm of a continuous l... |
| nmcoplbi 31957 | A lower bound for the norm... |
| nmcopex 31958 | The norm of a continuous l... |
| nmcoplb 31959 | A lower bound for the norm... |
| nmophmi 31960 | The norm of the scalar pro... |
| bdophmi 31961 | The scalar product of a bo... |
| lnconi 31962 | Lemma for ~ lnopconi and ~... |
| lnopconi 31963 | A condition equivalent to ... |
| lnopcon 31964 | A condition equivalent to ... |
| lnopcnbd 31965 | A linear operator is conti... |
| lncnopbd 31966 | A continuous linear operat... |
| lncnbd 31967 | A continuous linear operat... |
| lnopcnre 31968 | A linear operator is conti... |
| lnfnli 31969 | Basic property of a linear... |
| lnfnfi 31970 | A linear Hilbert space fun... |
| lnfn0i 31971 | The value of a linear Hilb... |
| lnfnaddi 31972 | Additive property of a lin... |
| lnfnmuli 31973 | Multiplicative property of... |
| lnfnaddmuli 31974 | Sum/product property of a ... |
| lnfnsubi 31975 | Subtraction property for a... |
| lnfn0 31976 | The value of a linear Hilb... |
| lnfnmul 31977 | Multiplicative property of... |
| nmbdfnlbi 31978 | A lower bound for the norm... |
| nmbdfnlb 31979 | A lower bound for the norm... |
| nmcfnexi 31980 | The norm of a continuous l... |
| nmcfnlbi 31981 | A lower bound for the norm... |
| nmcfnex 31982 | The norm of a continuous l... |
| nmcfnlb 31983 | A lower bound of the norm ... |
| lnfnconi 31984 | A condition equivalent to ... |
| lnfncon 31985 | A condition equivalent to ... |
| lnfncnbd 31986 | A linear functional is con... |
| imaelshi 31987 | The image of a subspace un... |
| rnelshi 31988 | The range of a linear oper... |
| nlelshi 31989 | The null space of a linear... |
| nlelchi 31990 | The null space of a contin... |
| riesz3i 31991 | A continuous linear functi... |
| riesz4i 31992 | A continuous linear functi... |
| riesz4 31993 | A continuous linear functi... |
| riesz1 31994 | Part 1 of the Riesz repres... |
| riesz2 31995 | Part 2 of the Riesz repres... |
| cnlnadjlem1 31996 | Lemma for ~ cnlnadji (Theo... |
| cnlnadjlem2 31997 | Lemma for ~ cnlnadji . ` G... |
| cnlnadjlem3 31998 | Lemma for ~ cnlnadji . By... |
| cnlnadjlem4 31999 | Lemma for ~ cnlnadji . Th... |
| cnlnadjlem5 32000 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem6 32001 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem7 32002 | Lemma for ~ cnlnadji . He... |
| cnlnadjlem8 32003 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem9 32004 | Lemma for ~ cnlnadji . ` F... |
| cnlnadji 32005 | Every continuous linear op... |
| cnlnadjeui 32006 | Every continuous linear op... |
| cnlnadjeu 32007 | Every continuous linear op... |
| cnlnadj 32008 | Every continuous linear op... |
| cnlnssadj 32009 | Every continuous linear Hi... |
| bdopssadj 32010 | Every bounded linear Hilbe... |
| bdopadj 32011 | Every bounded linear Hilbe... |
| adjbdln 32012 | The adjoint of a bounded l... |
| adjbdlnb 32013 | An operator is bounded and... |
| adjbd1o 32014 | The mapping of adjoints of... |
| adjlnop 32015 | The adjoint of an operator... |
| adjsslnop 32016 | Every operator with an adj... |
| nmopadjlei 32017 | Property of the norm of an... |
| nmopadjlem 32018 | Lemma for ~ nmopadji . (C... |
| nmopadji 32019 | Property of the norm of an... |
| adjeq0 32020 | An operator is zero iff it... |
| adjmul 32021 | The adjoint of the scalar ... |
| adjadd 32022 | The adjoint of the sum of ... |
| nmoptrii 32023 | Triangle inequality for th... |
| nmopcoi 32024 | Upper bound for the norm o... |
| bdophsi 32025 | The sum of two bounded lin... |
| bdophdi 32026 | The difference between two... |
| bdopcoi 32027 | The composition of two bou... |
| nmoptri2i 32028 | Triangle-type inequality f... |
| adjcoi 32029 | The adjoint of a compositi... |
| nmopcoadji 32030 | The norm of an operator co... |
| nmopcoadj2i 32031 | The norm of an operator co... |
| nmopcoadj0i 32032 | An operator composed with ... |
| unierri 32033 | If we approximate a chain ... |
| branmfn 32034 | The norm of the bra functi... |
| brabn 32035 | The bra of a vector is a b... |
| rnbra 32036 | The set of bras equals the... |
| bra11 32037 | The bra function maps vect... |
| bracnln 32038 | A bra is a continuous line... |
| cnvbraval 32039 | Value of the converse of t... |
| cnvbracl 32040 | Closure of the converse of... |
| cnvbrabra 32041 | The converse bra of the br... |
| bracnvbra 32042 | The bra of the converse br... |
| bracnlnval 32043 | The vector that a continuo... |
| cnvbramul 32044 | Multiplication property of... |
| kbass1 32045 | Dirac bra-ket associative ... |
| kbass2 32046 | Dirac bra-ket associative ... |
| kbass3 32047 | Dirac bra-ket associative ... |
| kbass4 32048 | Dirac bra-ket associative ... |
| kbass5 32049 | Dirac bra-ket associative ... |
| kbass6 32050 | Dirac bra-ket associative ... |
| leopg 32051 | Ordering relation for posi... |
| leop 32052 | Ordering relation for oper... |
| leop2 32053 | Ordering relation for oper... |
| leop3 32054 | Operator ordering in terms... |
| leoppos 32055 | Binary relation defining a... |
| leoprf2 32056 | The ordering relation for ... |
| leoprf 32057 | The ordering relation for ... |
| leopsq 32058 | The square of a Hermitian ... |
| 0leop 32059 | The zero operator is a pos... |
| idleop 32060 | The identity operator is a... |
| leopadd 32061 | The sum of two positive op... |
| leopmuli 32062 | The scalar product of a no... |
| leopmul 32063 | The scalar product of a po... |
| leopmul2i 32064 | Scalar product applied to ... |
| leoptri 32065 | The positive operator orde... |
| leoptr 32066 | The positive operator orde... |
| leopnmid 32067 | A bounded Hermitian operat... |
| nmopleid 32068 | A nonzero, bounded Hermiti... |
| opsqrlem1 32069 | Lemma for opsqri . (Contr... |
| opsqrlem2 32070 | Lemma for opsqri . ` F `` ... |
| opsqrlem3 32071 | Lemma for opsqri . (Contr... |
| opsqrlem4 32072 | Lemma for opsqri . (Contr... |
| opsqrlem5 32073 | Lemma for opsqri . (Contr... |
| opsqrlem6 32074 | Lemma for opsqri . (Contr... |
| pjhmopi 32075 | A projector is a Hermitian... |
| pjlnopi 32076 | A projector is a linear op... |
| pjnmopi 32077 | The operator norm of a pro... |
| pjbdlni 32078 | A projector is a bounded l... |
| pjhmop 32079 | A projection is a Hermitia... |
| hmopidmchi 32080 | An idempotent Hermitian op... |
| hmopidmpji 32081 | An idempotent Hermitian op... |
| hmopidmch 32082 | An idempotent Hermitian op... |
| hmopidmpj 32083 | An idempotent Hermitian op... |
| pjsdii 32084 | Distributive law for Hilbe... |
| pjddii 32085 | Distributive law for Hilbe... |
| pjsdi2i 32086 | Chained distributive law f... |
| pjcoi 32087 | Composition of projections... |
| pjcocli 32088 | Closure of composition of ... |
| pjcohcli 32089 | Closure of composition of ... |
| pjadjcoi 32090 | Adjoint of composition of ... |
| pjcofni 32091 | Functionality of compositi... |
| pjss1coi 32092 | Subset relationship for pr... |
| pjss2coi 32093 | Subset relationship for pr... |
| pjssmi 32094 | Projection meet property. ... |
| pjssge0i 32095 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormi 32096 | Theorem 4.5(v)<->(vi) of [... |
| pjnormssi 32097 | Theorem 4.5(i)<->(vi) of [... |
| pjorthcoi 32098 | Composition of projections... |
| pjscji 32099 | The projection of orthogon... |
| pjssumi 32100 | The projection on a subspa... |
| pjssposi 32101 | Projector ordering can be ... |
| pjordi 32102 | The definition of projecto... |
| pjssdif2i 32103 | The projection subspace of... |
| pjssdif1i 32104 | A necessary and sufficient... |
| pjimai 32105 | The image of a projection.... |
| pjidmcoi 32106 | A projection is idempotent... |
| pjoccoi 32107 | Composition of projections... |
| pjtoi 32108 | Subspace sum of projection... |
| pjoci 32109 | Projection of orthocomplem... |
| pjidmco 32110 | A projection operator is i... |
| dfpjop 32111 | Definition of projection o... |
| pjhmopidm 32112 | Two ways to express the se... |
| elpjidm 32113 | A projection operator is i... |
| elpjhmop 32114 | A projection operator is H... |
| 0leopj 32115 | A projector is a positive ... |
| pjadj2 32116 | A projector is self-adjoin... |
| pjadj3 32117 | A projector is self-adjoin... |
| elpjch 32118 | Reconstruction of the subs... |
| elpjrn 32119 | Reconstruction of the subs... |
| pjinvari 32120 | A closed subspace ` H ` wi... |
| pjin1i 32121 | Lemma for Theorem 1.22 of ... |
| pjin2i 32122 | Lemma for Theorem 1.22 of ... |
| pjin3i 32123 | Lemma for Theorem 1.22 of ... |
| pjclem1 32124 | Lemma for projection commu... |
| pjclem2 32125 | Lemma for projection commu... |
| pjclem3 32126 | Lemma for projection commu... |
| pjclem4a 32127 | Lemma for projection commu... |
| pjclem4 32128 | Lemma for projection commu... |
| pjci 32129 | Two subspaces commute iff ... |
| pjcmul1i 32130 | A necessary and sufficient... |
| pjcmul2i 32131 | The projection subspace of... |
| pjcohocli 32132 | Closure of composition of ... |
| pjadj2coi 32133 | Adjoint of double composit... |
| pj2cocli 32134 | Closure of double composit... |
| pj3lem1 32135 | Lemma for projection tripl... |
| pj3si 32136 | Stronger projection triple... |
| pj3i 32137 | Projection triplet theorem... |
| pj3cor1i 32138 | Projection triplet corolla... |
| pjs14i 32139 | Theorem S-14 of Watanabe, ... |
| isst 32142 | Property of a state. (Con... |
| ishst 32143 | Property of a complex Hilb... |
| sticl 32144 | ` [ 0 , 1 ] ` closure of t... |
| stcl 32145 | Real closure of the value ... |
| hstcl 32146 | Closure of the value of a ... |
| hst1a 32147 | Unit value of a Hilbert-sp... |
| hstel2 32148 | Properties of a Hilbert-sp... |
| hstorth 32149 | Orthogonality property of ... |
| hstosum 32150 | Orthogonal sum property of... |
| hstoc 32151 | Sum of a Hilbert-space-val... |
| hstnmoc 32152 | Sum of norms of a Hilbert-... |
| stge0 32153 | The value of a state is no... |
| stle1 32154 | The value of a state is le... |
| hstle1 32155 | The norm of the value of a... |
| hst1h 32156 | The norm of a Hilbert-spac... |
| hst0h 32157 | The norm of a Hilbert-spac... |
| hstpyth 32158 | Pythagorean property of a ... |
| hstle 32159 | Ordering property of a Hil... |
| hstles 32160 | Ordering property of a Hil... |
| hstoh 32161 | A Hilbert-space-valued sta... |
| hst0 32162 | A Hilbert-space-valued sta... |
| sthil 32163 | The value of a state at th... |
| stj 32164 | The value of a state on a ... |
| sto1i 32165 | The state of a subspace pl... |
| sto2i 32166 | The state of the orthocomp... |
| stge1i 32167 | If a state is greater than... |
| stle0i 32168 | If a state is less than or... |
| stlei 32169 | Ordering law for states. ... |
| stlesi 32170 | Ordering law for states. ... |
| stji1i 32171 | Join of components of Sasa... |
| stm1i 32172 | State of component of unit... |
| stm1ri 32173 | State of component of unit... |
| stm1addi 32174 | Sum of states whose meet i... |
| staddi 32175 | If the sum of 2 states is ... |
| stm1add3i 32176 | Sum of states whose meet i... |
| stadd3i 32177 | If the sum of 3 states is ... |
| st0 32178 | The state of the zero subs... |
| strlem1 32179 | Lemma for strong state the... |
| strlem2 32180 | Lemma for strong state the... |
| strlem3a 32181 | Lemma for strong state the... |
| strlem3 32182 | Lemma for strong state the... |
| strlem4 32183 | Lemma for strong state the... |
| strlem5 32184 | Lemma for strong state the... |
| strlem6 32185 | Lemma for strong state the... |
| stri 32186 | Strong state theorem. The... |
| strb 32187 | Strong state theorem (bidi... |
| hstrlem2 32188 | Lemma for strong set of CH... |
| hstrlem3a 32189 | Lemma for strong set of CH... |
| hstrlem3 32190 | Lemma for strong set of CH... |
| hstrlem4 32191 | Lemma for strong set of CH... |
| hstrlem5 32192 | Lemma for strong set of CH... |
| hstrlem6 32193 | Lemma for strong set of CH... |
| hstri 32194 | Hilbert space admits a str... |
| hstrbi 32195 | Strong CH-state theorem (b... |
| largei 32196 | A Hilbert lattice admits a... |
| jplem1 32197 | Lemma for Jauch-Piron theo... |
| jplem2 32198 | Lemma for Jauch-Piron theo... |
| jpi 32199 | The function ` S ` , that ... |
| golem1 32200 | Lemma for Godowski's equat... |
| golem2 32201 | Lemma for Godowski's equat... |
| goeqi 32202 | Godowski's equation, shown... |
| stcltr1i 32203 | Property of a strong class... |
| stcltr2i 32204 | Property of a strong class... |
| stcltrlem1 32205 | Lemma for strong classical... |
| stcltrlem2 32206 | Lemma for strong classical... |
| stcltrthi 32207 | Theorem for classically st... |
| cvbr 32211 | Binary relation expressing... |
| cvbr2 32212 | Binary relation expressing... |
| cvcon3 32213 | Contraposition law for the... |
| cvpss 32214 | The covers relation implie... |
| cvnbtwn 32215 | The covers relation implie... |
| cvnbtwn2 32216 | The covers relation implie... |
| cvnbtwn3 32217 | The covers relation implie... |
| cvnbtwn4 32218 | The covers relation implie... |
| cvnsym 32219 | The covers relation is not... |
| cvnref 32220 | The covers relation is not... |
| cvntr 32221 | The covers relation is not... |
| spansncv2 32222 | Hilbert space has the cove... |
| mdbr 32223 | Binary relation expressing... |
| mdi 32224 | Consequence of the modular... |
| mdbr2 32225 | Binary relation expressing... |
| mdbr3 32226 | Binary relation expressing... |
| mdbr4 32227 | Binary relation expressing... |
| dmdbr 32228 | Binary relation expressing... |
| dmdmd 32229 | The dual modular pair prop... |
| mddmd 32230 | The modular pair property ... |
| dmdi 32231 | Consequence of the dual mo... |
| dmdbr2 32232 | Binary relation expressing... |
| dmdi2 32233 | Consequence of the dual mo... |
| dmdbr3 32234 | Binary relation expressing... |
| dmdbr4 32235 | Binary relation expressing... |
| dmdi4 32236 | Consequence of the dual mo... |
| dmdbr5 32237 | Binary relation expressing... |
| mddmd2 32238 | Relationship between modul... |
| mdsl0 32239 | A sublattice condition tha... |
| ssmd1 32240 | Ordering implies the modul... |
| ssmd2 32241 | Ordering implies the modul... |
| ssdmd1 32242 | Ordering implies the dual ... |
| ssdmd2 32243 | Ordering implies the dual ... |
| dmdsl3 32244 | Sublattice mapping for a d... |
| mdsl3 32245 | Sublattice mapping for a m... |
| mdslle1i 32246 | Order preservation of the ... |
| mdslle2i 32247 | Order preservation of the ... |
| mdslj1i 32248 | Join preservation of the o... |
| mdslj2i 32249 | Meet preservation of the r... |
| mdsl1i 32250 | If the modular pair proper... |
| mdsl2i 32251 | If the modular pair proper... |
| mdsl2bi 32252 | If the modular pair proper... |
| cvmdi 32253 | The covering property impl... |
| mdslmd1lem1 32254 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem2 32255 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem3 32256 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem4 32257 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1i 32258 | Preservation of the modula... |
| mdslmd2i 32259 | Preservation of the modula... |
| mdsldmd1i 32260 | Preservation of the dual m... |
| mdslmd3i 32261 | Modular pair conditions th... |
| mdslmd4i 32262 | Modular pair condition tha... |
| csmdsymi 32263 | Cross-symmetry implies M-s... |
| mdexchi 32264 | An exchange lemma for modu... |
| cvmd 32265 | The covering property impl... |
| cvdmd 32266 | The covering property impl... |
| ela 32268 | Atoms in a Hilbert lattice... |
| elat2 32269 | Expanded membership relati... |
| elatcv0 32270 | A Hilbert lattice element ... |
| atcv0 32271 | An atom covers the zero su... |
| atssch 32272 | Atoms are a subset of the ... |
| atelch 32273 | An atom is a Hilbert latti... |
| atne0 32274 | An atom is not the Hilbert... |
| atss 32275 | A lattice element smaller ... |
| atsseq 32276 | Two atoms in a subset rela... |
| atcveq0 32277 | A Hilbert lattice element ... |
| h1da 32278 | A 1-dimensional subspace i... |
| spansna 32279 | The span of the singleton ... |
| sh1dle 32280 | A 1-dimensional subspace i... |
| ch1dle 32281 | A 1-dimensional subspace i... |
| atom1d 32282 | The 1-dimensional subspace... |
| superpos 32283 | Superposition Principle. ... |
| chcv1 32284 | The Hilbert lattice has th... |
| chcv2 32285 | The Hilbert lattice has th... |
| chjatom 32286 | The join of a closed subsp... |
| shatomici 32287 | The lattice of Hilbert sub... |
| hatomici 32288 | The Hilbert lattice is ato... |
| hatomic 32289 | A Hilbert lattice is atomi... |
| shatomistici 32290 | The lattice of Hilbert sub... |
| hatomistici 32291 | ` CH ` is atomistic, i.e. ... |
| chpssati 32292 | Two Hilbert lattice elemen... |
| chrelati 32293 | The Hilbert lattice is rel... |
| chrelat2i 32294 | A consequence of relative ... |
| cvati 32295 | If a Hilbert lattice eleme... |
| cvbr4i 32296 | An alternate way to expres... |
| cvexchlem 32297 | Lemma for ~ cvexchi . (Co... |
| cvexchi 32298 | The Hilbert lattice satisf... |
| chrelat2 32299 | A consequence of relative ... |
| chrelat3 32300 | A consequence of relative ... |
| chrelat3i 32301 | A consequence of the relat... |
| chrelat4i 32302 | A consequence of relative ... |
| cvexch 32303 | The Hilbert lattice satisf... |
| cvp 32304 | The Hilbert lattice satisf... |
| atnssm0 32305 | The meet of a Hilbert latt... |
| atnemeq0 32306 | The meet of distinct atoms... |
| atssma 32307 | The meet with an atom's su... |
| atcv0eq 32308 | Two atoms covering the zer... |
| atcv1 32309 | Two atoms covering the zer... |
| atexch 32310 | The Hilbert lattice satisf... |
| atomli 32311 | An assertion holding in at... |
| atoml2i 32312 | An assertion holding in at... |
| atordi 32313 | An ordering law for a Hilb... |
| atcvatlem 32314 | Lemma for ~ atcvati . (Co... |
| atcvati 32315 | A nonzero Hilbert lattice ... |
| atcvat2i 32316 | A Hilbert lattice element ... |
| atord 32317 | An ordering law for a Hilb... |
| atcvat2 32318 | A Hilbert lattice element ... |
| chirredlem1 32319 | Lemma for ~ chirredi . (C... |
| chirredlem2 32320 | Lemma for ~ chirredi . (C... |
| chirredlem3 32321 | Lemma for ~ chirredi . (C... |
| chirredlem4 32322 | Lemma for ~ chirredi . (C... |
| chirredi 32323 | The Hilbert lattice is irr... |
| chirred 32324 | The Hilbert lattice is irr... |
| atcvat3i 32325 | A condition implying that ... |
| atcvat4i 32326 | A condition implying exist... |
| atdmd 32327 | Two Hilbert lattice elemen... |
| atmd 32328 | Two Hilbert lattice elemen... |
| atmd2 32329 | Two Hilbert lattice elemen... |
| atabsi 32330 | Absorption of an incompara... |
| atabs2i 32331 | Absorption of an incompara... |
| mdsymlem1 32332 | Lemma for ~ mdsymi . (Con... |
| mdsymlem2 32333 | Lemma for ~ mdsymi . (Con... |
| mdsymlem3 32334 | Lemma for ~ mdsymi . (Con... |
| mdsymlem4 32335 | Lemma for ~ mdsymi . This... |
| mdsymlem5 32336 | Lemma for ~ mdsymi . (Con... |
| mdsymlem6 32337 | Lemma for ~ mdsymi . This... |
| mdsymlem7 32338 | Lemma for ~ mdsymi . Lemm... |
| mdsymlem8 32339 | Lemma for ~ mdsymi . Lemm... |
| mdsymi 32340 | M-symmetry of the Hilbert ... |
| mdsym 32341 | M-symmetry of the Hilbert ... |
| dmdsym 32342 | Dual M-symmetry of the Hil... |
| atdmd2 32343 | Two Hilbert lattice elemen... |
| sumdmdii 32344 | If the subspace sum of two... |
| cmmdi 32345 | Commuting subspaces form a... |
| cmdmdi 32346 | Commuting subspaces form a... |
| sumdmdlem 32347 | Lemma for ~ sumdmdi . The... |
| sumdmdlem2 32348 | Lemma for ~ sumdmdi . (Co... |
| sumdmdi 32349 | The subspace sum of two Hi... |
| dmdbr4ati 32350 | Dual modular pair property... |
| dmdbr5ati 32351 | Dual modular pair property... |
| dmdbr6ati 32352 | Dual modular pair property... |
| dmdbr7ati 32353 | Dual modular pair property... |
| mdoc1i 32354 | Orthocomplements form a mo... |
| mdoc2i 32355 | Orthocomplements form a mo... |
| dmdoc1i 32356 | Orthocomplements form a du... |
| dmdoc2i 32357 | Orthocomplements form a du... |
| mdcompli 32358 | A condition equivalent to ... |
| dmdcompli 32359 | A condition equivalent to ... |
| mddmdin0i 32360 | If dual modular implies mo... |
| cdjreui 32361 | A member of the sum of dis... |
| cdj1i 32362 | Two ways to express " ` A ... |
| cdj3lem1 32363 | A property of " ` A ` and ... |
| cdj3lem2 32364 | Lemma for ~ cdj3i . Value... |
| cdj3lem2a 32365 | Lemma for ~ cdj3i . Closu... |
| cdj3lem2b 32366 | Lemma for ~ cdj3i . The f... |
| cdj3lem3 32367 | Lemma for ~ cdj3i . Value... |
| cdj3lem3a 32368 | Lemma for ~ cdj3i . Closu... |
| cdj3lem3b 32369 | Lemma for ~ cdj3i . The s... |
| cdj3i 32370 | Two ways to express " ` A ... |
| The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here | |
| mathbox 32371 | (_This theorem is a dummy ... |
| sa-abvi 32372 | A theorem about the univer... |
| xfree 32373 | A partial converse to ~ 19... |
| xfree2 32374 | A partial converse to ~ 19... |
| addltmulALT 32375 | A proof readability experi... |
| ad11antr 32376 | Deduction adding 11 conjun... |
| simp-12l 32377 | Simplification of a conjun... |
| simp-12r 32378 | Simplification of a conjun... |
| an42ds 32379 | Inference exchanging the l... |
| an52ds 32380 | Inference exchanging the l... |
| an62ds 32381 | Inference exchanging the l... |
| an72ds 32382 | Inference exchanging the l... |
| an82ds 32383 | Inference exchanging the l... |
| syl22anbrc 32384 | Syllogism inference. (Con... |
| bian1d 32385 | Adding a superfluous conju... |
| bian1dOLD 32386 | Obsolete version of ~ bian... |
| orim12da 32387 | Deduce a disjunction from ... |
| or3di 32388 | Distributive law for disju... |
| or3dir 32389 | Distributive law for disju... |
| 3o1cs 32390 | Deduction eliminating disj... |
| 3o2cs 32391 | Deduction eliminating disj... |
| 3o3cs 32392 | Deduction eliminating disj... |
| 13an22anass 32393 | Associative law for four c... |
| sbc2iedf 32394 | Conversion of implicit sub... |
| rspc2daf 32395 | Double restricted speciali... |
| ralcom4f 32396 | Commutation of restricted ... |
| rexcom4f 32397 | Commutation of restricted ... |
| 19.9d2rf 32398 | A deduction version of one... |
| 19.9d2r 32399 | A deduction version of one... |
| r19.29ffa 32400 | A commonly used pattern ba... |
| n0limd 32401 | Deduction rule for nonempt... |
| reu6dv 32402 | A condition which implies ... |
| eqtrb 32403 | A transposition of equalit... |
| eqelbid 32404 | A variable elimination law... |
| opsbc2ie 32405 | Conversion of implicit sub... |
| opreu2reuALT 32406 | Correspondence between uni... |
| 2reucom 32409 | Double restricted existent... |
| 2reu2rex1 32410 | Double restricted existent... |
| 2reureurex 32411 | Double restricted existent... |
| 2reu2reu2 32412 | Double restricted existent... |
| opreu2reu1 32413 | Equivalent definition of t... |
| sq2reunnltb 32414 | There exists a unique deco... |
| addsqnot2reu 32415 | For each complex number ` ... |
| sbceqbidf 32416 | Equality theorem for class... |
| sbcies 32417 | A special version of class... |
| mo5f 32418 | Alternate definition of "a... |
| nmo 32419 | Negation of "at most one".... |
| reuxfrdf 32420 | Transfer existential uniqu... |
| rexunirn 32421 | Restricted existential qua... |
| rmoxfrd 32422 | Transfer "at most one" res... |
| rmoun 32423 | "At most one" restricted e... |
| rmounid 32424 | A case where an "at most o... |
| riotaeqbidva 32425 | Equivalent wff's yield equ... |
| dmrab 32426 | Domain of a restricted cla... |
| difrab2 32427 | Difference of two restrict... |
| rabexgfGS 32428 | Separation Scheme in terms... |
| rabsnel 32429 | Truth implied by equality ... |
| rabsspr 32430 | Conditions for a restricte... |
| rabsstp 32431 | Conditions for a restricte... |
| 3unrab 32432 | Union of three restricted ... |
| foresf1o 32433 | From a surjective function... |
| rabfodom 32434 | Domination relation for re... |
| rabrexfi 32435 | Conditions for a class abs... |
| abrexdomjm 32436 | An indexed set is dominate... |
| abrexdom2jm 32437 | An indexed set is dominate... |
| abrexexd 32438 | Existence of a class abstr... |
| elabreximd 32439 | Class substitution in an i... |
| elabreximdv 32440 | Class substitution in an i... |
| abrexss 32441 | A necessary condition for ... |
| nelun 32442 | Negated membership for a u... |
| snsssng 32443 | If a singleton is a subset... |
| n0nsnel 32444 | If a class with one elemen... |
| inin 32445 | Intersection with an inter... |
| difininv 32446 | Condition for the intersec... |
| difeq 32447 | Rewriting an equation with... |
| eqdif 32448 | If both set differences of... |
| indifbi 32449 | Two ways to express equali... |
| diffib 32450 | Case where ~ diffi is a bi... |
| difxp1ss 32451 | Difference law for Cartesi... |
| difxp2ss 32452 | Difference law for Cartesi... |
| indifundif 32453 | A remarkable equation with... |
| elpwincl1 32454 | Closure of intersection wi... |
| elpwdifcl 32455 | Closure of class differenc... |
| elpwiuncl 32456 | Closure of indexed union w... |
| elpreq 32457 | Equality wihin a pair. (C... |
| prssad 32458 | If a pair is a subset of a... |
| prssbd 32459 | If a pair is a subset of a... |
| nelpr 32460 | A set ` A ` not in a pair ... |
| inpr0 32461 | Rewrite an empty intersect... |
| neldifpr1 32462 | The first element of a pai... |
| neldifpr2 32463 | The second element of a pa... |
| unidifsnel 32464 | The other element of a pai... |
| unidifsnne 32465 | The other element of a pai... |
| tpssg 32466 | An unordered triple of ele... |
| tpssd 32467 | Deduction version of tpssi... |
| tpssad 32468 | If an ordered triple is a ... |
| tpssbd 32469 | If an ordered triple is a ... |
| tpsscd 32470 | If an ordered triple is a ... |
| ifeqeqx 32471 | An equality theorem tailor... |
| elimifd 32472 | Elimination of a condition... |
| elim2if 32473 | Elimination of two conditi... |
| elim2ifim 32474 | Elimination of two conditi... |
| ifeq3da 32475 | Given an expression ` C ` ... |
| ifnetrue 32476 | Deduce truth from a condit... |
| ifnefals 32477 | Deduce falsehood from a co... |
| ifnebib 32478 | The converse of ~ ifbi hol... |
| uniinn0 32479 | Sufficient and necessary c... |
| uniin1 32480 | Union of intersection. Ge... |
| uniin2 32481 | Union of intersection. Ge... |
| difuncomp 32482 | Express a class difference... |
| elpwunicl 32483 | Closure of a set union wit... |
| cbviunf 32484 | Rule used to change the bo... |
| iuneq12daf 32485 | Equality deduction for ind... |
| iunin1f 32486 | Indexed union of intersect... |
| ssiun3 32487 | Subset equivalence for an ... |
| ssiun2sf 32488 | Subset relationship for an... |
| iuninc 32489 | The union of an increasing... |
| iundifdifd 32490 | The intersection of a set ... |
| iundifdif 32491 | The intersection of a set ... |
| iunrdx 32492 | Re-index an indexed union.... |
| iunpreima 32493 | Preimage of an indexed uni... |
| iunrnmptss 32494 | A subset relation for an i... |
| iunxunsn 32495 | Appending a set to an inde... |
| iunxunpr 32496 | Appending two sets to an i... |
| iunxpssiun1 32497 | Provide an upper bound for... |
| iinabrex 32498 | Rewriting an indexed inter... |
| disjnf 32499 | In case ` x ` is not free ... |
| cbvdisjf 32500 | Change bound variables in ... |
| disjss1f 32501 | A subset of a disjoint col... |
| disjeq1f 32502 | Equality theorem for disjo... |
| disjxun0 32503 | Simplify a disjoint union.... |
| disjdifprg 32504 | A trivial partition into a... |
| disjdifprg2 32505 | A trivial partition of a s... |
| disji2f 32506 | Property of a disjoint col... |
| disjif 32507 | Property of a disjoint col... |
| disjorf 32508 | Two ways to say that a col... |
| disjorsf 32509 | Two ways to say that a col... |
| disjif2 32510 | Property of a disjoint col... |
| disjabrex 32511 | Rewriting a disjoint colle... |
| disjabrexf 32512 | Rewriting a disjoint colle... |
| disjpreima 32513 | A preimage of a disjoint s... |
| disjrnmpt 32514 | Rewriting a disjoint colle... |
| disjin 32515 | If a collection is disjoin... |
| disjin2 32516 | If a collection is disjoin... |
| disjxpin 32517 | Derive a disjunction over ... |
| iundisjf 32518 | Rewrite a countable union ... |
| iundisj2f 32519 | A disjoint union is disjoi... |
| disjrdx 32520 | Re-index a disjunct collec... |
| disjex 32521 | Two ways to say that two c... |
| disjexc 32522 | A variant of ~ disjex , ap... |
| disjunsn 32523 | Append an element to a dis... |
| disjun0 32524 | Adding the empty element p... |
| disjiunel 32525 | A set of elements B of a d... |
| disjuniel 32526 | A set of elements B of a d... |
| xpdisjres 32527 | Restriction of a constant ... |
| opeldifid 32528 | Ordered pair elementhood o... |
| difres 32529 | Case when class difference... |
| imadifxp 32530 | Image of the difference wi... |
| relfi 32531 | A relation (set) is finite... |
| 0res 32532 | Restriction of the empty f... |
| fcoinver 32533 | Build an equivalence relat... |
| fcoinvbr 32534 | Binary relation for the eq... |
| brab2d 32535 | Expressing that two sets a... |
| brabgaf 32536 | The law of concretion for ... |
| brelg 32537 | Two things in a binary rel... |
| br8d 32538 | Substitution for an eight-... |
| opabdm 32539 | Domain of an ordered-pair ... |
| opabrn 32540 | Range of an ordered-pair c... |
| opabssi 32541 | Sufficient condition for a... |
| opabid2ss 32542 | One direction of ~ opabid2... |
| ssrelf 32543 | A subclass relationship de... |
| eqrelrd2 32544 | A version of ~ eqrelrdv2 w... |
| erbr3b 32545 | Biconditional for equivale... |
| iunsnima 32546 | Image of a singleton by an... |
| iunsnima2 32547 | Version of ~ iunsnima with... |
| ac6sf2 32548 | Alternate version of ~ ac6... |
| ac6mapd 32549 | Axiom of choice equivalent... |
| fnresin 32550 | Restriction of a function ... |
| f1o3d 32551 | Describe an implicit one-t... |
| eldmne0 32552 | A function of nonempty dom... |
| f1rnen 32553 | Equinumerosity of the rang... |
| rinvf1o 32554 | Sufficient conditions for ... |
| fresf1o 32555 | Conditions for a restricti... |
| nfpconfp 32556 | The set of fixed points of... |
| fmptco1f1o 32557 | The action of composing (t... |
| cofmpt2 32558 | Express composition of a m... |
| f1mptrn 32559 | Express injection for a ma... |
| dfimafnf 32560 | Alternate definition of th... |
| funimass4f 32561 | Membership relation for th... |
| suppss2f 32562 | Show that the support of a... |
| ofrn 32563 | The range of the function ... |
| ofrn2 32564 | The range of the function ... |
| off2 32565 | The function operation pro... |
| ofresid 32566 | Applying an operation rest... |
| unipreima 32567 | Preimage of a class union.... |
| opfv 32568 | Value of a function produc... |
| xppreima 32569 | The preimage of a Cartesia... |
| 2ndimaxp 32570 | Image of a cartesian produ... |
| dmdju 32571 | Domain of a disjoint union... |
| djussxp2 32572 | Stronger version of ~ djus... |
| 2ndresdju 32573 | The ` 2nd ` function restr... |
| 2ndresdjuf1o 32574 | The ` 2nd ` function restr... |
| xppreima2 32575 | The preimage of a Cartesia... |
| abfmpunirn 32576 | Membership in a union of a... |
| rabfmpunirn 32577 | Membership in a union of a... |
| abfmpeld 32578 | Membership in an element o... |
| abfmpel 32579 | Membership in an element o... |
| fmptdF 32580 | Domain and codomain of the... |
| fmptcof2 32581 | Composition of two functio... |
| fcomptf 32582 | Express composition of two... |
| acunirnmpt 32583 | Axiom of choice for the un... |
| acunirnmpt2 32584 | Axiom of choice for the un... |
| acunirnmpt2f 32585 | Axiom of choice for the un... |
| aciunf1lem 32586 | Choice in an index union. ... |
| aciunf1 32587 | Choice in an index union. ... |
| ofoprabco 32588 | Function operation as a co... |
| ofpreima 32589 | Express the preimage of a ... |
| ofpreima2 32590 | Express the preimage of a ... |
| funcnvmpt 32591 | Condition for a function i... |
| funcnv5mpt 32592 | Two ways to say that a fun... |
| funcnv4mpt 32593 | Two ways to say that a fun... |
| preimane 32594 | Different elements have di... |
| fnpreimac 32595 | Choose a set ` x ` contain... |
| fgreu 32596 | Exactly one point of a fun... |
| fcnvgreu 32597 | If the converse of a relat... |
| rnmposs 32598 | The range of an operation ... |
| mptssALT 32599 | Deduce subset relation of ... |
| dfcnv2 32600 | Alternative definition of ... |
| mpomptxf 32601 | Express a two-argument fun... |
| of0r 32602 | Function operation with th... |
| elmaprd 32603 | Deduction associated with ... |
| suppovss 32604 | A bound for the support of... |
| elsuppfnd 32605 | Deduce membership in the s... |
| fisuppov1 32606 | Formula building theorem f... |
| suppun2 32607 | The support of a union is ... |
| fdifsupp 32608 | Express the support of a f... |
| suppiniseg 32609 | Relation between the suppo... |
| fsuppinisegfi 32610 | The initial segment ` ( ``... |
| fressupp 32611 | The restriction of a funct... |
| fdifsuppconst 32612 | A function is a zero const... |
| ressupprn 32613 | The range of a function re... |
| supppreima 32614 | Express the support of a f... |
| fsupprnfi 32615 | Finite support implies fin... |
| mptiffisupp 32616 | Conditions for a mapping f... |
| cosnopne 32617 | Composition of two ordered... |
| cosnop 32618 | Composition of two ordered... |
| cnvprop 32619 | Converse of a pair of orde... |
| brprop 32620 | Binary relation for a pair... |
| mptprop 32621 | Rewrite pairs of ordered p... |
| coprprop 32622 | Composition of two pairs o... |
| fmptunsnop 32623 | Two ways to express a func... |
| gtiso 32624 | Two ways to write a strict... |
| isoun 32625 | Infer an isomorphism from ... |
| disjdsct 32626 | A disjoint collection is d... |
| df1stres 32627 | Definition for a restricti... |
| df2ndres 32628 | Definition for a restricti... |
| 1stpreimas 32629 | The preimage of a singleto... |
| 1stpreima 32630 | The preimage by ` 1st ` is... |
| 2ndpreima 32631 | The preimage by ` 2nd ` is... |
| curry2ima 32632 | The image of a curried fun... |
| preiman0 32633 | The preimage of a nonempty... |
| intimafv 32634 | The intersection of an ima... |
| imafi2 32635 | The image by a finite set ... |
| unifi3 32636 | If a union is finite, then... |
| snct 32637 | A singleton is countable. ... |
| prct 32638 | An unordered pair is count... |
| mpocti 32639 | An operation is countable ... |
| abrexct 32640 | An image set of a countabl... |
| mptctf 32641 | A countable mapping set is... |
| abrexctf 32642 | An image set of a countabl... |
| padct 32643 | Index a countable set with... |
| f1od2 32644 | Sufficient condition for a... |
| fcobij 32645 | Composing functions with a... |
| fcobijfs 32646 | Composing finitely support... |
| suppss3 32647 | Deduce a function's suppor... |
| fsuppcurry1 32648 | Finite support of a currie... |
| fsuppcurry2 32649 | Finite support of a currie... |
| offinsupp1 32650 | Finite support for a funct... |
| ffs2 32651 | Rewrite a function's suppo... |
| ffsrn 32652 | The range of a finitely su... |
| resf1o 32653 | Restriction of functions t... |
| maprnin 32654 | Restricting the range of t... |
| fpwrelmapffslem 32655 | Lemma for ~ fpwrelmapffs .... |
| fpwrelmap 32656 | Define a canonical mapping... |
| fpwrelmapffs 32657 | Define a canonical mapping... |
| sgnval2 32658 | Value of the signum of a r... |
| creq0 32659 | The real representation of... |
| 1nei 32660 | The imaginary unit ` _i ` ... |
| 1neg1t1neg1 32661 | An integer unit times itse... |
| nnmulge 32662 | Multiplying by a positive ... |
| submuladdd 32663 | The product of a differenc... |
| muldivdid 32664 | Distribution of division o... |
| binom2subadd 32665 | The difference of the squa... |
| cjsubd 32666 | Complex conjugate distribu... |
| re0cj 32667 | The conjugate of a pure im... |
| receqid 32668 | Real numbers equal to thei... |
| pythagreim 32669 | A simplified version of th... |
| efiargd 32670 | The exponential of the "ar... |
| arginv 32671 | The argument of the invers... |
| argcj 32672 | The argument of the conjug... |
| quad3d 32673 | Variant of quadratic equat... |
| lt2addrd 32674 | If the right-hand side of ... |
| xrlelttric 32675 | Trichotomy law for extende... |
| xaddeq0 32676 | Two extended reals which a... |
| rexmul2 32677 | If the result ` A ` of an ... |
| xrinfm 32678 | The extended real numbers ... |
| le2halvesd 32679 | A sum is less than the who... |
| xraddge02 32680 | A number is less than or e... |
| xrge0addge 32681 | A number is less than or e... |
| xlt2addrd 32682 | If the right-hand side of ... |
| xrge0infss 32683 | Any subset of nonnegative ... |
| xrge0infssd 32684 | Inequality deduction for i... |
| xrge0addcld 32685 | Nonnegative extended reals... |
| xrge0subcld 32686 | Condition for closure of n... |
| infxrge0lb 32687 | A member of a set of nonne... |
| infxrge0glb 32688 | The infimum of a set of no... |
| infxrge0gelb 32689 | The infimum of a set of no... |
| xrofsup 32690 | The supremum is preserved ... |
| supxrnemnf 32691 | The supremum of a nonempty... |
| xnn0gt0 32692 | Nonzero extended nonnegati... |
| xnn01gt 32693 | An extended nonnegative in... |
| nn0xmulclb 32694 | Finite multiplication in t... |
| xnn0nn0d 32695 | Conditions for an extended... |
| xnn0nnd 32696 | Conditions for an extended... |
| joiniooico 32697 | Disjoint joining an open i... |
| ubico 32698 | A right-open interval does... |
| xeqlelt 32699 | Equality in terms of 'less... |
| eliccelico 32700 | Relate elementhood to a cl... |
| elicoelioo 32701 | Relate elementhood to a cl... |
| iocinioc2 32702 | Intersection between two o... |
| xrdifh 32703 | Class difference of a half... |
| iocinif 32704 | Relate intersection of two... |
| difioo 32705 | The difference between two... |
| difico 32706 | The difference between two... |
| uzssico 32707 | Upper integer sets are a s... |
| fz2ssnn0 32708 | A finite set of sequential... |
| nndiffz1 32709 | Upper set of the positive ... |
| ssnnssfz 32710 | For any finite subset of `... |
| fzm1ne1 32711 | Elementhood of an integer ... |
| fzspl 32712 | Split the last element of ... |
| fzdif2 32713 | Split the last element of ... |
| fzodif2 32714 | Split the last element of ... |
| fzodif1 32715 | Set difference of two half... |
| fzsplit3 32716 | Split a finite interval of... |
| elfzodif0 32717 | If an integer ` M ` is in ... |
| bcm1n 32718 | The proportion of one bino... |
| iundisjfi 32719 | Rewrite a countable union ... |
| iundisj2fi 32720 | A disjoint union is disjoi... |
| iundisjcnt 32721 | Rewrite a countable union ... |
| iundisj2cnt 32722 | A countable disjoint union... |
| fzone1 32723 | Elementhood in a half-open... |
| fzom1ne1 32724 | Elementhood in a half-open... |
| f1ocnt 32725 | Given a countable set ` A ... |
| fz1nnct 32726 | NN and integer ranges star... |
| fz1nntr 32727 | NN and integer ranges star... |
| fzo0opth 32728 | Equality for a half open i... |
| nn0difffzod 32729 | A nonnegative integer that... |
| suppssnn0 32730 | Show that the support of a... |
| hashunif 32731 | The cardinality of a disjo... |
| hashxpe 32732 | The size of the Cartesian ... |
| hashgt1 32733 | Restate "set contains at l... |
| hashpss 32734 | The size of a proper subse... |
| hashne0 32735 | Deduce that the size of a ... |
| elq2 32736 | Elementhood in the rationa... |
| znumd 32737 | Numerator of an integer. ... |
| zdend 32738 | Denominator of an integer.... |
| numdenneg 32739 | Numerator and denominator ... |
| divnumden2 32740 | Calculate the reduced form... |
| expgt0b 32741 | A real number ` A ` raised... |
| nn0split01 32742 | Split 0 and 1 from the non... |
| nn0disj01 32743 | The pair ` { 0 , 1 } ` doe... |
| nnindf 32744 | Principle of Mathematical ... |
| nn0min 32745 | Extracting the minimum pos... |
| subne0nn 32746 | A nonnegative difference i... |
| ltesubnnd 32747 | Subtracting an integer num... |
| fprodeq02 32748 | If one of the factors is z... |
| pr01ssre 32749 | The range of the indicator... |
| fprodex01 32750 | A product of factors equal... |
| prodpr 32751 | A product over a pair is t... |
| prodtp 32752 | A product over a triple is... |
| fsumub 32753 | An upper bound for a term ... |
| fsumiunle 32754 | Upper bound for a sum of n... |
| dfdec100 32755 | Split the hundreds from a ... |
| sgncl 32756 | Closure of the signum. (C... |
| sgnclre 32757 | Closure of the signum. (C... |
| sgnneg 32758 | Negation of the signum. (... |
| sgn3da 32759 | A conditional containing a... |
| sgnmul 32760 | Signum of a product. (Con... |
| sgnmulrp2 32761 | Multiplication by a positi... |
| sgnsub 32762 | Subtraction of a number of... |
| sgnnbi 32763 | Negative signum. (Contrib... |
| sgnpbi 32764 | Positive signum. (Contrib... |
| sgn0bi 32765 | Zero signum. (Contributed... |
| sgnsgn 32766 | Signum is idempotent. (Co... |
| sgnmulsgn 32767 | If two real numbers are of... |
| sgnmulsgp 32768 | If two real numbers are of... |
| nexple 32769 | A lower bound for an expon... |
| 2exple2exp 32770 | If a nonnegative integer `... |
| expevenpos 32771 | Even powers are positive. ... |
| oexpled 32772 | Odd power monomials are mo... |
| indv 32775 | Value of the indicator fun... |
| indval 32776 | Value of the indicator fun... |
| indval2 32777 | Alternate value of the ind... |
| indf 32778 | An indicator function as a... |
| indfval 32779 | Value of the indicator fun... |
| ind1 32780 | Value of the indicator fun... |
| ind0 32781 | Value of the indicator fun... |
| ind1a 32782 | Value of the indicator fun... |
| indpi1 32783 | Preimage of the singleton ... |
| indsum 32784 | Finite sum of a product wi... |
| indsumin 32785 | Finite sum of a product wi... |
| prodindf 32786 | The product of indicators ... |
| indf1o 32787 | The bijection between a po... |
| indpreima 32788 | A function with range ` { ... |
| indf1ofs 32789 | The bijection between fini... |
| indsupp 32790 | The support of the indicat... |
| dp2eq1 32793 | Equality theorem for the d... |
| dp2eq2 32794 | Equality theorem for the d... |
| dp2eq1i 32795 | Equality theorem for the d... |
| dp2eq2i 32796 | Equality theorem for the d... |
| dp2eq12i 32797 | Equality theorem for the d... |
| dp20u 32798 | Add a zero in the tenths (... |
| dp20h 32799 | Add a zero in the unit pla... |
| dp2cl 32800 | Closure for the decimal fr... |
| dp2clq 32801 | Closure for a decimal frac... |
| rpdp2cl 32802 | Closure for a decimal frac... |
| rpdp2cl2 32803 | Closure for a decimal frac... |
| dp2lt10 32804 | Decimal fraction builds re... |
| dp2lt 32805 | Comparing two decimal frac... |
| dp2ltsuc 32806 | Comparing a decimal fracti... |
| dp2ltc 32807 | Comparing two decimal expa... |
| dpval 32810 | Define the value of the de... |
| dpcl 32811 | Prove that the closure of ... |
| dpfrac1 32812 | Prove a simple equivalence... |
| dpval2 32813 | Value of the decimal point... |
| dpval3 32814 | Value of the decimal point... |
| dpmul10 32815 | Multiply by 10 a decimal e... |
| decdiv10 32816 | Divide a decimal number by... |
| dpmul100 32817 | Multiply by 100 a decimal ... |
| dp3mul10 32818 | Multiply by 10 a decimal e... |
| dpmul1000 32819 | Multiply by 1000 a decimal... |
| dpval3rp 32820 | Value of the decimal point... |
| dp0u 32821 | Add a zero in the tenths p... |
| dp0h 32822 | Remove a zero in the units... |
| rpdpcl 32823 | Closure of the decimal poi... |
| dplt 32824 | Comparing two decimal expa... |
| dplti 32825 | Comparing a decimal expans... |
| dpgti 32826 | Comparing a decimal expans... |
| dpltc 32827 | Comparing two decimal inte... |
| dpexpp1 32828 | Add one zero to the mantis... |
| 0dp2dp 32829 | Multiply by 10 a decimal e... |
| dpadd2 32830 | Addition with one decimal,... |
| dpadd 32831 | Addition with one decimal.... |
| dpadd3 32832 | Addition with two decimals... |
| dpmul 32833 | Multiplication with one de... |
| dpmul4 32834 | An upper bound to multipli... |
| threehalves 32835 | Example theorem demonstrat... |
| 1mhdrd 32836 | Example theorem demonstrat... |
| xdivval 32839 | Value of division: the (un... |
| xrecex 32840 | Existence of reciprocal of... |
| xmulcand 32841 | Cancellation law for exten... |
| xreceu 32842 | Existential uniqueness of ... |
| xdivcld 32843 | Closure law for the extend... |
| xdivcl 32844 | Closure law for the extend... |
| xdivmul 32845 | Relationship between divis... |
| rexdiv 32846 | The extended real division... |
| xdivrec 32847 | Relationship between divis... |
| xdivid 32848 | A number divided by itself... |
| xdiv0 32849 | Division into zero is zero... |
| xdiv0rp 32850 | Division into zero is zero... |
| eliccioo 32851 | Membership in a closed int... |
| elxrge02 32852 | Elementhood in the set of ... |
| xdivpnfrp 32853 | Plus infinity divided by a... |
| rpxdivcld 32854 | Closure law for extended d... |
| xrpxdivcld 32855 | Closure law for extended d... |
| wrdres 32856 | Condition for the restrict... |
| wrdsplex 32857 | Existence of a split of a ... |
| wrdfsupp 32858 | A word has finite support.... |
| wrdpmcl 32859 | Closure of a word with per... |
| pfx1s2 32860 | The prefix of length 1 of ... |
| pfxrn2 32861 | The range of a prefix of a... |
| pfxrn3 32862 | Express the range of a pre... |
| pfxf1 32863 | Condition for a prefix to ... |
| s1f1 32864 | Conditions for a length 1 ... |
| s2rnOLD 32865 | Obsolete version of ~ s2rn... |
| s2f1 32866 | Conditions for a length 2 ... |
| s3rnOLD 32867 | Obsolete version of ~ s2rn... |
| s3f1 32868 | Conditions for a length 3 ... |
| s3clhash 32869 | Closure of the words of le... |
| ccatf1 32870 | Conditions for a concatena... |
| ccatdmss 32871 | The domain of a concatenat... |
| pfxlsw2ccat 32872 | Reconstruct a word from it... |
| ccatws1f1o 32873 | Conditions for the concate... |
| ccatws1f1olast 32874 | Two ways to reorder symbol... |
| wrdt2ind 32875 | Perform an induction over ... |
| swrdrn2 32876 | The range of a subword is ... |
| swrdrn3 32877 | Express the range of a sub... |
| swrdf1 32878 | Condition for a subword to... |
| swrdrndisj 32879 | Condition for the range of... |
| splfv3 32880 | Symbols to the right of a ... |
| 1cshid 32881 | Cyclically shifting a sing... |
| cshw1s2 32882 | Cyclically shifting a leng... |
| cshwrnid 32883 | Cyclically shifting a word... |
| cshf1o 32884 | Condition for the cyclic s... |
| ressplusf 32885 | The group operation functi... |
| ressnm 32886 | The norm in a restricted s... |
| abvpropd2 32887 | Weaker version of ~ abvpro... |
| oppgle 32888 | less-than relation of an o... |
| oppglt 32889 | less-than relation of an o... |
| ressprs 32890 | The restriction of a prose... |
| posrasymb 32891 | A poset ordering is asymet... |
| resspos 32892 | The restriction of a Poset... |
| resstos 32893 | The restriction of a Toset... |
| odutos 32894 | Being a toset is a self-du... |
| tlt2 32895 | In a Toset, two elements m... |
| tlt3 32896 | In a Toset, two elements m... |
| trleile 32897 | In a Toset, two elements m... |
| toslublem 32898 | Lemma for ~ toslub and ~ x... |
| toslub 32899 | In a toset, the lowest upp... |
| tosglblem 32900 | Lemma for ~ tosglb and ~ x... |
| tosglb 32901 | Same theorem as ~ toslub ,... |
| clatp0cl 32902 | The poset zero of a comple... |
| clatp1cl 32903 | The poset one of a complet... |
| mntoval 32908 | Operation value of the mon... |
| ismnt 32909 | Express the statement " ` ... |
| ismntd 32910 | Property of being a monoto... |
| mntf 32911 | A monotone function is a f... |
| mgcoval 32912 | Operation value of the mon... |
| mgcval 32913 | Monotone Galois connection... |
| mgcf1 32914 | The lower adjoint ` F ` of... |
| mgcf2 32915 | The upper adjoint ` G ` of... |
| mgccole1 32916 | An inequality for the kern... |
| mgccole2 32917 | Inequality for the closure... |
| mgcmnt1 32918 | The lower adjoint ` F ` of... |
| mgcmnt2 32919 | The upper adjoint ` G ` of... |
| mgcmntco 32920 | A Galois connection like s... |
| dfmgc2lem 32921 | Lemma for dfmgc2, backward... |
| dfmgc2 32922 | Alternate definition of th... |
| mgcmnt1d 32923 | Galois connection implies ... |
| mgcmnt2d 32924 | Galois connection implies ... |
| mgccnv 32925 | The inverse Galois connect... |
| pwrssmgc 32926 | Given a function ` F ` , e... |
| mgcf1olem1 32927 | Property of a Galois conne... |
| mgcf1olem2 32928 | Property of a Galois conne... |
| mgcf1o 32929 | Given a Galois connection,... |
| ischn 32932 | Property of being a chain.... |
| chnwrd 32933 | A chain is an ordered sequ... |
| chnltm1 32934 | Basic property of a chain.... |
| pfxchn 32935 | A prefix of a chain is sti... |
| s1chn 32936 | A singleton word is always... |
| chnind 32937 | Induction over a chain. S... |
| chnub 32938 | In a chain, the last eleme... |
| chnlt 32939 | Compare any two elements i... |
| chnso 32940 | A chain induces a total or... |
| chnccats1 32941 | Extend a chain with a sing... |
| xrs0 32944 | The zero of the extended r... |
| xrslt 32945 | The "strictly less than" r... |
| xrsinvgval 32946 | The inversion operation in... |
| xrsmulgzz 32947 | The "multiple" function in... |
| xrstos 32948 | The extended real numbers ... |
| xrsclat 32949 | The extended real numbers ... |
| xrsp0 32950 | The poset 0 of the extende... |
| xrsp1 32951 | The poset 1 of the extende... |
| xrge0base 32952 | The base of the extended n... |
| xrge00 32953 | The zero of the extended n... |
| xrge0plusg 32954 | The additive law of the ex... |
| xrge0le 32955 | The "less than or equal to... |
| xrge0mulgnn0 32956 | The group multiple functio... |
| xrge0addass 32957 | Associativity of extended ... |
| xrge0addgt0 32958 | The sum of nonnegative and... |
| xrge0adddir 32959 | Right-distributivity of ex... |
| xrge0adddi 32960 | Left-distributivity of ext... |
| xrge0npcan 32961 | Extended nonnegative real ... |
| fsumrp0cl 32962 | Closure of a finite sum of... |
| mndcld 32963 | Closure of the operation o... |
| mndassd 32964 | A monoid operation is asso... |
| mndlrinv 32965 | In a monoid, if an element... |
| mndlrinvb 32966 | In a monoid, if an element... |
| mndlactf1 32967 | If an element ` X ` of a m... |
| mndlactfo 32968 | An element ` X ` of a mono... |
| mndractf1 32969 | If an element ` X ` of a m... |
| mndractfo 32970 | An element ` X ` of a mono... |
| mndlactf1o 32971 | An element ` X ` of a mono... |
| mndractf1o 32972 | An element ` X ` of a mono... |
| cmn4d 32973 | Commutative/associative la... |
| cmn246135 32974 | Rearrange terms in a commu... |
| cmn145236 32975 | Rearrange terms in a commu... |
| submcld 32976 | Submonoids are closed unde... |
| abliso 32977 | The image of an Abelian gr... |
| lmhmghmd 32978 | A module homomorphism is a... |
| mhmimasplusg 32979 | Value of the operation of ... |
| lmhmimasvsca 32980 | Value of the scalar produc... |
| grpsubcld 32981 | Closure of group subtracti... |
| subgcld 32982 | A subgroup is closed under... |
| subgsubcld 32983 | A subgroup is closed under... |
| subgmulgcld 32984 | Closure of the group multi... |
| ressmulgnn0d 32985 | Values for the group multi... |
| gsumsubg 32986 | The group sum in a subgrou... |
| gsumsra 32987 | The group sum in a subring... |
| gsummpt2co 32988 | Split a finite sum into a ... |
| gsummpt2d 32989 | Express a finite sum over ... |
| lmodvslmhm 32990 | Scalar multiplication in a... |
| gsumvsmul1 32991 | Pull a scalar multiplicati... |
| gsummptres 32992 | Extend a finite group sum ... |
| gsummptres2 32993 | Extend a finite group sum ... |
| gsummptfsf1o 32994 | Re-index a finite group su... |
| gsumfs2d 32995 | Express a finite sum over ... |
| gsumzresunsn 32996 | Append an element to a fin... |
| gsumpart 32997 | Express a group sum as a d... |
| gsumtp 32998 | Group sum of an unordered ... |
| gsumzrsum 32999 | Relate a group sum on ` ZZ... |
| gsummulgc2 33000 | A finite group sum multipl... |
| gsumhashmul 33001 | Express a group sum by gro... |
| xrge0tsmsd 33002 | Any finite or infinite sum... |
| xrge0tsmsbi 33003 | Any limit of a finite or i... |
| xrge0tsmseq 33004 | Any limit of a finite or i... |
| gsumwun 33005 | In a commutative ring, a g... |
| gsumwrd2dccatlem 33006 | Lemma for ~ gsumwrd2dccat ... |
| gsumwrd2dccat 33007 | Rewrite a sum ranging over... |
| cntzun 33008 | The centralizer of a union... |
| cntzsnid 33009 | The centralizer of the ide... |
| cntrcrng 33010 | The center of a ring is a ... |
| isomnd 33015 | A (left) ordered monoid is... |
| isogrp 33016 | A (left-)ordered group is ... |
| ogrpgrp 33017 | A left-ordered group is a ... |
| omndmnd 33018 | A left-ordered monoid is a... |
| omndtos 33019 | A left-ordered monoid is a... |
| omndadd 33020 | In an ordered monoid, the ... |
| omndaddr 33021 | In a right ordered monoid,... |
| omndadd2d 33022 | In a commutative left orde... |
| omndadd2rd 33023 | In a left- and right- orde... |
| submomnd 33024 | A submonoid of an ordered ... |
| xrge0omnd 33025 | The nonnegative extended r... |
| omndmul2 33026 | In an ordered monoid, the ... |
| omndmul3 33027 | In an ordered monoid, the ... |
| omndmul 33028 | In a commutative ordered m... |
| ogrpinv0le 33029 | In an ordered group, the o... |
| ogrpsub 33030 | In an ordered group, the o... |
| ogrpaddlt 33031 | In an ordered group, stric... |
| ogrpaddltbi 33032 | In a right ordered group, ... |
| ogrpaddltrd 33033 | In a right ordered group, ... |
| ogrpaddltrbid 33034 | In a right ordered group, ... |
| ogrpsublt 33035 | In an ordered group, stric... |
| ogrpinv0lt 33036 | In an ordered group, the o... |
| ogrpinvlt 33037 | In an ordered group, the o... |
| gsumle 33038 | A finite sum in an ordered... |
| symgfcoeu 33039 | Uniqueness property of per... |
| symgcom 33040 | Two permutations ` X ` and... |
| symgcom2 33041 | Two permutations ` X ` and... |
| symgcntz 33042 | All elements of a (finite)... |
| odpmco 33043 | The composition of two odd... |
| symgsubg 33044 | The value of the group sub... |
| pmtrprfv2 33045 | In a transposition of two ... |
| pmtrcnel 33046 | Composing a permutation ` ... |
| pmtrcnel2 33047 | Variation on ~ pmtrcnel . ... |
| pmtrcnelor 33048 | Composing a permutation ` ... |
| fzo0pmtrlast 33049 | Reorder a half-open intege... |
| wrdpmtrlast 33050 | Reorder a word, so that th... |
| pmtridf1o 33051 | Transpositions of ` X ` an... |
| pmtridfv1 33052 | Value at X of the transpos... |
| pmtridfv2 33053 | Value at Y of the transpos... |
| psgnid 33054 | Permutation sign of the id... |
| psgndmfi 33055 | For a finite base set, the... |
| pmtrto1cl 33056 | Useful lemma for the follo... |
| psgnfzto1stlem 33057 | Lemma for ~ psgnfzto1st . ... |
| fzto1stfv1 33058 | Value of our permutation `... |
| fzto1st1 33059 | Special case where the per... |
| fzto1st 33060 | The function moving one el... |
| fzto1stinvn 33061 | Value of the inverse of ou... |
| psgnfzto1st 33062 | The permutation sign for m... |
| tocycval 33065 | Value of the cycle builder... |
| tocycfv 33066 | Function value of a permut... |
| tocycfvres1 33067 | A cyclic permutation is a ... |
| tocycfvres2 33068 | A cyclic permutation is th... |
| cycpmfvlem 33069 | Lemma for ~ cycpmfv1 and ~... |
| cycpmfv1 33070 | Value of a cycle function ... |
| cycpmfv2 33071 | Value of a cycle function ... |
| cycpmfv3 33072 | Values outside of the orbi... |
| cycpmcl 33073 | Cyclic permutations are pe... |
| tocycf 33074 | The permutation cycle buil... |
| tocyc01 33075 | Permutation cycles built f... |
| cycpm2tr 33076 | A cyclic permutation of 2 ... |
| cycpm2cl 33077 | Closure for the 2-cycles. ... |
| cyc2fv1 33078 | Function value of a 2-cycl... |
| cyc2fv2 33079 | Function value of a 2-cycl... |
| trsp2cyc 33080 | Exhibit the word a transpo... |
| cycpmco2f1 33081 | The word U used in ~ cycpm... |
| cycpmco2rn 33082 | The orbit of the compositi... |
| cycpmco2lem1 33083 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem2 33084 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem3 33085 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem4 33086 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem5 33087 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem6 33088 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem7 33089 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2 33090 | The composition of a cycli... |
| cyc2fvx 33091 | Function value of a 2-cycl... |
| cycpm3cl 33092 | Closure of the 3-cycles in... |
| cycpm3cl2 33093 | Closure of the 3-cycles in... |
| cyc3fv1 33094 | Function value of a 3-cycl... |
| cyc3fv2 33095 | Function value of a 3-cycl... |
| cyc3fv3 33096 | Function value of a 3-cycl... |
| cyc3co2 33097 | Represent a 3-cycle as a c... |
| cycpmconjvlem 33098 | Lemma for ~ cycpmconjv . ... |
| cycpmconjv 33099 | A formula for computing co... |
| cycpmrn 33100 | The range of the word used... |
| tocyccntz 33101 | All elements of a (finite)... |
| evpmval 33102 | Value of the set of even p... |
| cnmsgn0g 33103 | The neutral element of the... |
| evpmsubg 33104 | The alternating group is a... |
| evpmid 33105 | The identity is an even pe... |
| altgnsg 33106 | The alternating group ` ( ... |
| cyc3evpm 33107 | 3-Cycles are even permutat... |
| cyc3genpmlem 33108 | Lemma for ~ cyc3genpm . (... |
| cyc3genpm 33109 | The alternating group ` A ... |
| cycpmgcl 33110 | Cyclic permutations are pe... |
| cycpmconjslem1 33111 | Lemma for ~ cycpmconjs . ... |
| cycpmconjslem2 33112 | Lemma for ~ cycpmconjs . ... |
| cycpmconjs 33113 | All cycles of the same len... |
| cyc3conja 33114 | All 3-cycles are conjugate... |
| sgnsv 33117 | The sign mapping. (Contri... |
| sgnsval 33118 | The sign value. (Contribu... |
| sgnsf 33119 | The sign function. (Contr... |
| fxpval 33122 | Value of the set of fixed ... |
| fxpss 33123 | The set of fixed points is... |
| fxpgaval 33124 | Value of the set of fixed ... |
| isfxp 33125 | Property of being a fixed ... |
| fxpgaeq 33126 | A fixed point ` X ` is inv... |
| conjga 33127 | Group conjugation induces ... |
| cntrval2 33128 | Express the center ` Z ` o... |
| fxpsubm 33129 | Provided the group action ... |
| inftmrel 33134 | The infinitesimal relation... |
| isinftm 33135 | Express ` x ` is infinites... |
| isarchi 33136 | Express the predicate " ` ... |
| pnfinf 33137 | Plus infinity is an infini... |
| xrnarchi 33138 | The completed real line is... |
| isarchi2 33139 | Alternative way to express... |
| submarchi 33140 | A submonoid is archimedean... |
| isarchi3 33141 | This is the usual definiti... |
| archirng 33142 | Property of Archimedean or... |
| archirngz 33143 | Property of Archimedean le... |
| archiexdiv 33144 | In an Archimedean group, g... |
| archiabllem1a 33145 | Lemma for ~ archiabl : In... |
| archiabllem1b 33146 | Lemma for ~ archiabl . (C... |
| archiabllem1 33147 | Archimedean ordered groups... |
| archiabllem2a 33148 | Lemma for ~ archiabl , whi... |
| archiabllem2c 33149 | Lemma for ~ archiabl . (C... |
| archiabllem2b 33150 | Lemma for ~ archiabl . (C... |
| archiabllem2 33151 | Archimedean ordered groups... |
| archiabl 33152 | Archimedean left- and righ... |
| isslmd 33155 | The predicate "is a semimo... |
| slmdlema 33156 | Lemma for properties of a ... |
| lmodslmd 33157 | Left semimodules generaliz... |
| slmdcmn 33158 | A semimodule is a commutat... |
| slmdmnd 33159 | A semimodule is a monoid. ... |
| slmdsrg 33160 | The scalar component of a ... |
| slmdbn0 33161 | The base set of a semimodu... |
| slmdacl 33162 | Closure of ring addition f... |
| slmdmcl 33163 | Closure of ring multiplica... |
| slmdsn0 33164 | The set of scalars in a se... |
| slmdvacl 33165 | Closure of vector addition... |
| slmdass 33166 | Semiring left module vecto... |
| slmdvscl 33167 | Closure of scalar product ... |
| slmdvsdi 33168 | Distributive law for scala... |
| slmdvsdir 33169 | Distributive law for scala... |
| slmdvsass 33170 | Associative law for scalar... |
| slmd0cl 33171 | The ring zero in a semimod... |
| slmd1cl 33172 | The ring unity in a semiri... |
| slmdvs1 33173 | Scalar product with ring u... |
| slmd0vcl 33174 | The zero vector is a vecto... |
| slmd0vlid 33175 | Left identity law for the ... |
| slmd0vrid 33176 | Right identity law for the... |
| slmd0vs 33177 | Zero times a vector is the... |
| slmdvs0 33178 | Anything times the zero ve... |
| gsumvsca1 33179 | Scalar product of a finite... |
| gsumvsca2 33180 | Scalar product of a finite... |
| prmsimpcyc 33181 | A group of prime order is ... |
| ringdi22 33182 | Expand the product of two ... |
| urpropd 33183 | Sufficient condition for r... |
| subrgmcld 33184 | A subring is closed under ... |
| ress1r 33185 | ` 1r ` is unaffected by re... |
| ringinvval 33186 | The ring inverse expressed... |
| dvrcan5 33187 | Cancellation law for commo... |
| subrgchr 33188 | If ` A ` is a subring of `... |
| rmfsupp2 33189 | A mapping of a multiplicat... |
| unitnz 33190 | In a nonzero ring, a unit ... |
| isunit2 33191 | Alternate definition of be... |
| isunit3 33192 | Alternate definition of be... |
| elrgspnlem1 33193 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem2 33194 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem3 33195 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem4 33196 | Lemma for ~ elrgspn . (Co... |
| elrgspn 33197 | Membership in the subring ... |
| elrgspnsubrunlem1 33198 | Lemma for ~ elrgspnsubrun ... |
| elrgspnsubrunlem2 33199 | Lemma for ~ elrgspnsubrun ... |
| elrgspnsubrun 33200 | Membership in the ring spa... |
| irrednzr 33201 | A ring with an irreducible... |
| 0ringsubrg 33202 | A subring of a zero ring i... |
| 0ringcring 33203 | The zero ring is commutati... |
| reldmrloc 33208 | Ring localization is a pro... |
| erlval 33209 | Value of the ring localiza... |
| rlocval 33210 | Expand the value of the ri... |
| erlcl1 33211 | Closure for the ring local... |
| erlcl2 33212 | Closure for the ring local... |
| erldi 33213 | Main property of the ring ... |
| erlbrd 33214 | Deduce the ring localizati... |
| erlbr2d 33215 | Deduce the ring localizati... |
| erler 33216 | The relation used to build... |
| elrlocbasi 33217 | Membership in the basis of... |
| rlocbas 33218 | The base set of a ring loc... |
| rlocaddval 33219 | Value of the addition in t... |
| rlocmulval 33220 | Value of the addition in t... |
| rloccring 33221 | The ring localization ` L ... |
| rloc0g 33222 | The zero of a ring localiz... |
| rloc1r 33223 | The multiplicative identit... |
| rlocf1 33224 | The embedding ` F ` of a r... |
| domnmuln0rd 33225 | In a domain, factors of a ... |
| domnprodn0 33226 | In a domain, a finite prod... |
| domnpropd 33227 | If two structures have the... |
| idompropd 33228 | If two structures have the... |
| idomrcan 33229 | Right-cancellation law for... |
| domnlcanOLD 33230 | Obsolete version of ~ domn... |
| domnlcanbOLD 33231 | Obsolete version of ~ domn... |
| idomrcanOLD 33232 | Obsolete version of ~ idom... |
| 1rrg 33233 | The multiplicative identit... |
| rrgsubm 33234 | The left regular elements ... |
| subrdom 33235 | A subring of a domain is a... |
| subridom 33236 | A subring of an integral d... |
| subrfld 33237 | A subring of a field is an... |
| eufndx 33240 | Index value of the Euclide... |
| eufid 33241 | Utility theorem: index-ind... |
| ringinveu 33244 | If a ring unit element ` X... |
| isdrng4 33245 | A division ring is a ring ... |
| rndrhmcl 33246 | The image of a division ri... |
| qfld 33247 | The field of rational numb... |
| subsdrg 33248 | A subring of a sub-divisio... |
| sdrgdvcl 33249 | A sub-division-ring is clo... |
| sdrginvcl 33250 | A sub-division-ring is clo... |
| primefldchr 33251 | The characteristic of a pr... |
| fracval 33254 | Value of the field of frac... |
| fracbas 33255 | The base of the field of f... |
| fracerl 33256 | Rewrite the ring localizat... |
| fracf1 33257 | The embedding of a commuta... |
| fracfld 33258 | The field of fractions of ... |
| idomsubr 33259 | Every integral domain is i... |
| fldgenval 33262 | Value of the field generat... |
| fldgenssid 33263 | The field generated by a s... |
| fldgensdrg 33264 | A generated subfield is a ... |
| fldgenssv 33265 | A generated subfield is a ... |
| fldgenss 33266 | Generated subfields preser... |
| fldgenidfld 33267 | The subfield generated by ... |
| fldgenssp 33268 | The field generated by a s... |
| fldgenid 33269 | The subfield of a field ` ... |
| fldgenfld 33270 | A generated subfield is a ... |
| primefldgen1 33271 | The prime field of a divis... |
| 1fldgenq 33272 | The field of rational numb... |
| isorng 33277 | An ordered ring is a ring ... |
| orngring 33278 | An ordered ring is a ring.... |
| orngogrp 33279 | An ordered ring is an orde... |
| isofld 33280 | An ordered field is a fiel... |
| orngmul 33281 | In an ordered ring, the or... |
| orngsqr 33282 | In an ordered ring, all sq... |
| ornglmulle 33283 | In an ordered ring, multip... |
| orngrmulle 33284 | In an ordered ring, multip... |
| ornglmullt 33285 | In an ordered ring, multip... |
| orngrmullt 33286 | In an ordered ring, multip... |
| orngmullt 33287 | In an ordered ring, the st... |
| ofldfld 33288 | An ordered field is a fiel... |
| ofldtos 33289 | An ordered field is a tota... |
| orng0le1 33290 | In an ordered ring, the ri... |
| ofldlt1 33291 | In an ordered field, the r... |
| ofldchr 33292 | The characteristic of an o... |
| suborng 33293 | Every subring of an ordere... |
| subofld 33294 | Every subfield of an order... |
| isarchiofld 33295 | Axiom of Archimedes : a ch... |
| rhmdvd 33296 | A ring homomorphism preser... |
| kerunit 33297 | If a unit element lies in ... |
| reldmresv 33300 | The scalar restriction is ... |
| resvval 33301 | Value of structure restric... |
| resvid2 33302 | General behavior of trivia... |
| resvval2 33303 | Value of nontrivial struct... |
| resvsca 33304 | Base set of a structure re... |
| resvlem 33305 | Other elements of a scalar... |
| resvbas 33306 | ` Base ` is unaffected by ... |
| resvplusg 33307 | ` +g ` is unaffected by sc... |
| resvvsca 33308 | ` .s ` is unaffected by sc... |
| resvmulr 33309 | ` .r ` is unaffected by sc... |
| resv0g 33310 | ` 0g ` is unaffected by sc... |
| resv1r 33311 | ` 1r ` is unaffected by sc... |
| resvcmn 33312 | Scalar restriction preserv... |
| gzcrng 33313 | The gaussian integers form... |
| cnfldfld 33314 | The complex numbers form a... |
| reofld 33315 | The real numbers form an o... |
| nn0omnd 33316 | The nonnegative integers f... |
| rearchi 33317 | The field of the real numb... |
| nn0archi 33318 | The monoid of the nonnegat... |
| xrge0slmod 33319 | The extended nonnegative r... |
| qusker 33320 | The kernel of a quotient m... |
| eqgvscpbl 33321 | The left coset equivalence... |
| qusvscpbl 33322 | The quotient map distribut... |
| qusvsval 33323 | Value of the scalar multip... |
| imaslmod 33324 | The image structure of a l... |
| imasmhm 33325 | Given a function ` F ` wit... |
| imasghm 33326 | Given a function ` F ` wit... |
| imasrhm 33327 | Given a function ` F ` wit... |
| imaslmhm 33328 | Given a function ` F ` wit... |
| quslmod 33329 | If ` G ` is a submodule in... |
| quslmhm 33330 | If ` G ` is a submodule of... |
| quslvec 33331 | If ` S ` is a vector subsp... |
| ecxpid 33332 | The equivalence class of a... |
| qsxpid 33333 | The quotient set of a cart... |
| qusxpid 33334 | The Group quotient equival... |
| qustriv 33335 | The quotient of a group ` ... |
| qustrivr 33336 | Converse of ~ qustriv . (... |
| znfermltl 33337 | Fermat's little theorem in... |
| islinds5 33338 | A set is linearly independ... |
| ellspds 33339 | Variation on ~ ellspd . (... |
| 0ellsp 33340 | Zero is in all spans. (Co... |
| 0nellinds 33341 | The group identity cannot ... |
| rspsnid 33342 | A principal ideal contains... |
| elrsp 33343 | Write the elements of a ri... |
| ellpi 33344 | Elementhood in a left prin... |
| lpirlidllpi 33345 | In a principal ideal ring,... |
| rspidlid 33346 | The ideal span of an ideal... |
| pidlnz 33347 | A principal ideal generate... |
| lbslsp 33348 | Any element of a left modu... |
| lindssn 33349 | Any singleton of a nonzero... |
| lindflbs 33350 | Conditions for an independ... |
| islbs5 33351 | An equivalent formulation ... |
| linds2eq 33352 | Deduce equality of element... |
| lindfpropd 33353 | Property deduction for lin... |
| lindspropd 33354 | Property deduction for lin... |
| dvdsruassoi 33355 | If two elements ` X ` and ... |
| dvdsruasso 33356 | Two elements ` X ` and ` Y... |
| dvdsruasso2 33357 | A reformulation of ~ dvdsr... |
| dvdsrspss 33358 | In a ring, an element ` X ... |
| rspsnasso 33359 | Two elements ` X ` and ` Y... |
| unitprodclb 33360 | A finite product is a unit... |
| elgrplsmsn 33361 | Membership in a sumset wit... |
| lsmsnorb 33362 | The sumset of a group with... |
| lsmsnorb2 33363 | The sumset of a single ele... |
| elringlsm 33364 | Membership in a product of... |
| elringlsmd 33365 | Membership in a product of... |
| ringlsmss 33366 | Closure of the product of ... |
| ringlsmss1 33367 | The product of an ideal ` ... |
| ringlsmss2 33368 | The product with an ideal ... |
| lsmsnpridl 33369 | The product of the ring wi... |
| lsmsnidl 33370 | The product of the ring wi... |
| lsmidllsp 33371 | The sum of two ideals is t... |
| lsmidl 33372 | The sum of two ideals is a... |
| lsmssass 33373 | Group sum is associative, ... |
| grplsm0l 33374 | Sumset with the identity s... |
| grplsmid 33375 | The direct sum of an eleme... |
| quslsm 33376 | Express the image by the q... |
| qusbas2 33377 | Alternate definition of th... |
| qus0g 33378 | The identity element of a ... |
| qusima 33379 | The image of a subgroup by... |
| qusrn 33380 | The natural map from eleme... |
| nsgqus0 33381 | A normal subgroup ` N ` is... |
| nsgmgclem 33382 | Lemma for ~ nsgmgc . (Con... |
| nsgmgc 33383 | There is a monotone Galois... |
| nsgqusf1olem1 33384 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem2 33385 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem3 33386 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1o 33387 | The canonical projection h... |
| lmhmqusker 33388 | A surjective module homomo... |
| lmicqusker 33389 | The image ` H ` of a modul... |
| lidlmcld 33390 | An ideal is closed under l... |
| intlidl 33391 | The intersection of a none... |
| 0ringidl 33392 | The zero ideal is the only... |
| pidlnzb 33393 | A principal ideal is nonze... |
| lidlunitel 33394 | If an ideal ` I ` contains... |
| unitpidl1 33395 | The ideal ` I ` generated ... |
| rhmquskerlem 33396 | The mapping ` J ` induced ... |
| rhmqusker 33397 | A surjective ring homomorp... |
| ricqusker 33398 | The image ` H ` of a ring ... |
| elrspunidl 33399 | Elementhood in the span of... |
| elrspunsn 33400 | Membership to the span of ... |
| lidlincl 33401 | Ideals are closed under in... |
| idlinsubrg 33402 | The intersection between a... |
| rhmimaidl 33403 | The image of an ideal ` I ... |
| drngidl 33404 | A nonzero ring is a divisi... |
| drngidlhash 33405 | A ring is a division ring ... |
| prmidlval 33408 | The class of prime ideals ... |
| isprmidl 33409 | The predicate "is a prime ... |
| prmidlnr 33410 | A prime ideal is a proper ... |
| prmidl 33411 | The main property of a pri... |
| prmidl2 33412 | A condition that shows an ... |
| idlmulssprm 33413 | Let ` P ` be a prime ideal... |
| pridln1 33414 | A proper ideal cannot cont... |
| prmidlidl 33415 | A prime ideal is an ideal.... |
| prmidlssidl 33416 | Prime ideals as a subset o... |
| cringm4 33417 | Commutative/associative la... |
| isprmidlc 33418 | The predicate "is prime id... |
| prmidlc 33419 | Property of a prime ideal ... |
| 0ringprmidl 33420 | The trivial ring does not ... |
| prmidl0 33421 | The zero ideal of a commut... |
| rhmpreimaprmidl 33422 | The preimage of a prime id... |
| qsidomlem1 33423 | If the quotient ring of a ... |
| qsidomlem2 33424 | A quotient by a prime idea... |
| qsidom 33425 | An ideal ` I ` in the comm... |
| qsnzr 33426 | A quotient of a non-zero r... |
| ssdifidllem 33427 | Lemma for ~ ssdifidl : Th... |
| ssdifidl 33428 | Let ` R ` be a ring, and l... |
| ssdifidlprm 33429 | If the set ` S ` of ~ ssdi... |
| mxidlval 33432 | The set of maximal ideals ... |
| ismxidl 33433 | The predicate "is a maxima... |
| mxidlidl 33434 | A maximal ideal is an idea... |
| mxidlnr 33435 | A maximal ideal is proper.... |
| mxidlmax 33436 | A maximal ideal is a maxim... |
| mxidln1 33437 | One is not contained in an... |
| mxidlnzr 33438 | A ring with a maximal idea... |
| mxidlmaxv 33439 | An ideal ` I ` strictly co... |
| crngmxidl 33440 | In a commutative ring, max... |
| mxidlprm 33441 | Every maximal ideal is pri... |
| mxidlirredi 33442 | In an integral domain, the... |
| mxidlirred 33443 | In a principal ideal domai... |
| ssmxidllem 33444 | The set ` P ` used in the ... |
| ssmxidl 33445 | Let ` R ` be a ring, and l... |
| drnglidl1ne0 33446 | In a nonzero ring, the zer... |
| drng0mxidl 33447 | In a division ring, the ze... |
| drngmxidl 33448 | The zero ideal is the only... |
| drngmxidlr 33449 | If a ring's only maximal i... |
| krull 33450 | Krull's theorem: Any nonz... |
| mxidlnzrb 33451 | A ring is nonzero if and o... |
| krullndrng 33452 | Krull's theorem for non-di... |
| opprabs 33453 | The opposite ring of the o... |
| oppreqg 33454 | Group coset equivalence re... |
| opprnsg 33455 | Normal subgroups of the op... |
| opprlidlabs 33456 | The ideals of the opposite... |
| oppr2idl 33457 | Two sided ideal of the opp... |
| opprmxidlabs 33458 | The maximal ideal of the o... |
| opprqusbas 33459 | The base of the quotient o... |
| opprqusplusg 33460 | The group operation of the... |
| opprqus0g 33461 | The group identity element... |
| opprqusmulr 33462 | The multiplication operati... |
| opprqus1r 33463 | The ring unity of the quot... |
| opprqusdrng 33464 | The quotient of the opposi... |
| qsdrngilem 33465 | Lemma for ~ qsdrngi . (Co... |
| qsdrngi 33466 | A quotient by a maximal le... |
| qsdrnglem2 33467 | Lemma for ~ qsdrng . (Con... |
| qsdrng 33468 | An ideal ` M ` is both lef... |
| qsfld 33469 | An ideal ` M ` in the comm... |
| mxidlprmALT 33470 | Every maximal ideal is pri... |
| idlsrgstr 33473 | A constructed semiring of ... |
| idlsrgval 33474 | Lemma for ~ idlsrgbas thro... |
| idlsrgbas 33475 | Base of the ideals of a ri... |
| idlsrgplusg 33476 | Additive operation of the ... |
| idlsrg0g 33477 | The zero ideal is the addi... |
| idlsrgmulr 33478 | Multiplicative operation o... |
| idlsrgtset 33479 | Topology component of the ... |
| idlsrgmulrval 33480 | Value of the ring multipli... |
| idlsrgmulrcl 33481 | Ideals of a ring ` R ` are... |
| idlsrgmulrss1 33482 | In a commutative ring, the... |
| idlsrgmulrss2 33483 | The product of two ideals ... |
| idlsrgmulrssin 33484 | In a commutative ring, the... |
| idlsrgmnd 33485 | The ideals of a ring form ... |
| idlsrgcmnd 33486 | The ideals of a ring form ... |
| rprmval 33487 | The prime elements of a ri... |
| isrprm 33488 | Property for ` P ` to be a... |
| rprmcl 33489 | A ring prime is an element... |
| rprmdvds 33490 | If a ring prime ` Q ` divi... |
| rprmnz 33491 | A ring prime is nonzero. ... |
| rprmnunit 33492 | A ring prime is not a unit... |
| rsprprmprmidl 33493 | In a commutative ring, ide... |
| rsprprmprmidlb 33494 | In an integral domain, an ... |
| rprmndvdsr1 33495 | A ring prime element does ... |
| rprmasso 33496 | In an integral domain, the... |
| rprmasso2 33497 | In an integral domain, if ... |
| rprmasso3 33498 | In an integral domain, if ... |
| unitmulrprm 33499 | A ring unit multiplied by ... |
| rprmndvdsru 33500 | A ring prime element does ... |
| rprmirredlem 33501 | Lemma for ~ rprmirred . (... |
| rprmirred 33502 | In an integral domain, rin... |
| rprmirredb 33503 | In a principal ideal domai... |
| rprmdvdspow 33504 | If a prime element divides... |
| rprmdvdsprod 33505 | If a prime element ` Q ` d... |
| 1arithidomlem1 33506 | Lemma for ~ 1arithidom . ... |
| 1arithidomlem2 33507 | Lemma for ~ 1arithidom : i... |
| 1arithidom 33508 | Uniqueness of prime factor... |
| isufd 33511 | The property of being a Un... |
| ufdprmidl 33512 | In a unique factorization ... |
| ufdidom 33513 | A nonzero unique factoriza... |
| pidufd 33514 | Every principal ideal doma... |
| 1arithufdlem1 33515 | Lemma for ~ 1arithufd . T... |
| 1arithufdlem2 33516 | Lemma for ~ 1arithufd . T... |
| 1arithufdlem3 33517 | Lemma for ~ 1arithufd . I... |
| 1arithufdlem4 33518 | Lemma for ~ 1arithufd . N... |
| 1arithufd 33519 | Existence of a factorizati... |
| dfufd2lem 33520 | Lemma for ~ dfufd2 . (Con... |
| dfufd2 33521 | Alternative definition of ... |
| zringidom 33522 | The ring of integers is an... |
| zringpid 33523 | The ring of integers is a ... |
| dfprm3 33524 | The (positive) prime eleme... |
| zringfrac 33525 | The field of fractions of ... |
| 0ringmon1p 33526 | There are no monic polynom... |
| fply1 33527 | Conditions for a function ... |
| ply1lvec 33528 | In a division ring, the un... |
| evls1fn 33529 | Functionality of the subri... |
| evls1dm 33530 | The domain of the subring ... |
| evls1fvf 33531 | The subring evaluation fun... |
| evl1fvf 33532 | The univariate polynomial ... |
| evl1fpws 33533 | Evaluation of a univariate... |
| ressply1evls1 33534 | Subring evaluation of a un... |
| ressdeg1 33535 | The degree of a univariate... |
| ressply10g 33536 | A restricted polynomial al... |
| ressply1mon1p 33537 | The monic polynomials of a... |
| ressply1invg 33538 | An element of a restricted... |
| ressply1sub 33539 | A restricted polynomial al... |
| ressasclcl 33540 | Closure of the univariate ... |
| evls1subd 33541 | Univariate polynomial eval... |
| deg1le0eq0 33542 | A polynomial with nonposit... |
| ply1asclunit 33543 | A non-zero scalar polynomi... |
| ply1unit 33544 | In a field ` F ` , a polyn... |
| evl1deg1 33545 | Evaluation of a univariate... |
| evl1deg2 33546 | Evaluation of a univariate... |
| evl1deg3 33547 | Evaluation of a univariate... |
| ply1dg1rt 33548 | Express the root ` - B / A... |
| ply1dg1rtn0 33549 | Polynomials of degree 1 ov... |
| ply1mulrtss 33550 | The roots of a factor ` F ... |
| ply1dg3rt0irred 33551 | If a cubic polynomial over... |
| m1pmeq 33552 | If two monic polynomials `... |
| ply1fermltl 33553 | Fermat's little theorem fo... |
| coe1mon 33554 | Coefficient vector of a mo... |
| ply1moneq 33555 | Two monomials are equal if... |
| coe1zfv 33556 | The coefficients of the ze... |
| coe1vr1 33557 | Polynomial coefficient of ... |
| deg1vr 33558 | The degree of the variable... |
| vr1nz 33559 | A univariate polynomial va... |
| ply1degltel 33560 | Characterize elementhood i... |
| ply1degleel 33561 | Characterize elementhood i... |
| ply1degltlss 33562 | The space ` S ` of the uni... |
| gsummoncoe1fzo 33563 | A coefficient of the polyn... |
| ply1gsumz 33564 | If a polynomial given as a... |
| deg1addlt 33565 | If both factors have degre... |
| ig1pnunit 33566 | The polynomial ideal gener... |
| ig1pmindeg 33567 | The polynomial ideal gener... |
| q1pdir 33568 | Distribution of univariate... |
| q1pvsca 33569 | Scalar multiplication prop... |
| r1pvsca 33570 | Scalar multiplication prop... |
| r1p0 33571 | Polynomial remainder opera... |
| r1pcyc 33572 | The polynomial remainder o... |
| r1padd1 33573 | Addition property of the p... |
| r1pid2OLD 33574 | Obsolete version of ~ r1pi... |
| r1plmhm 33575 | The univariate polynomial ... |
| r1pquslmic 33576 | The univariate polynomial ... |
| sra1r 33577 | The unity element of a sub... |
| sradrng 33578 | Condition for a subring al... |
| sraidom 33579 | Condition for a subring al... |
| srasubrg 33580 | A subring of the original ... |
| sralvec 33581 | Given a sub division ring ... |
| srafldlvec 33582 | Given a subfield ` F ` of ... |
| resssra 33583 | The subring algebra of a r... |
| lsssra 33584 | A subring is a subspace of... |
| drgext0g 33585 | The additive neutral eleme... |
| drgextvsca 33586 | The scalar multiplication ... |
| drgext0gsca 33587 | The additive neutral eleme... |
| drgextsubrg 33588 | The scalar field is a subr... |
| drgextlsp 33589 | The scalar field is a subs... |
| drgextgsum 33590 | Group sum in a division ri... |
| lvecdimfi 33591 | Finite version of ~ lvecdi... |
| exsslsb 33592 | Any finite generating set ... |
| lbslelsp 33593 | The size of a basis ` X ` ... |
| dimval 33596 | The dimension of a vector ... |
| dimvalfi 33597 | The dimension of a vector ... |
| dimcl 33598 | Closure of the vector spac... |
| lmimdim 33599 | Module isomorphisms preser... |
| lmicdim 33600 | Module isomorphisms preser... |
| lvecdim0i 33601 | A vector space of dimensio... |
| lvecdim0 33602 | A vector space of dimensio... |
| lssdimle 33603 | The dimension of a linear ... |
| dimpropd 33604 | If two structures have the... |
| rlmdim 33605 | The left vector space indu... |
| rgmoddimOLD 33606 | Obsolete version of ~ rlmd... |
| frlmdim 33607 | Dimension of a free left m... |
| tnglvec 33608 | Augmenting a structure wit... |
| tngdim 33609 | Dimension of a left vector... |
| rrxdim 33610 | Dimension of the generaliz... |
| matdim 33611 | Dimension of the space of ... |
| lbslsat 33612 | A nonzero vector ` X ` is ... |
| lsatdim 33613 | A line, spanned by a nonze... |
| drngdimgt0 33614 | The dimension of a vector ... |
| lmhmlvec2 33615 | A homomorphism of left vec... |
| kerlmhm 33616 | The kernel of a vector spa... |
| imlmhm 33617 | The image of a vector spac... |
| ply1degltdimlem 33618 | Lemma for ~ ply1degltdim .... |
| ply1degltdim 33619 | The space ` S ` of the uni... |
| lindsunlem 33620 | Lemma for ~ lindsun . (Co... |
| lindsun 33621 | Condition for the union of... |
| lbsdiflsp0 33622 | The linear spans of two di... |
| dimkerim 33623 | Given a linear map ` F ` b... |
| qusdimsum 33624 | Let ` W ` be a vector spac... |
| fedgmullem1 33625 | Lemma for ~ fedgmul . (Co... |
| fedgmullem2 33626 | Lemma for ~ fedgmul . (Co... |
| fedgmul 33627 | The multiplicativity formu... |
| dimlssid 33628 | If the dimension of a line... |
| lvecendof1f1o 33629 | If an endomorphism ` U ` o... |
| lactlmhm 33630 | In an associative algebra ... |
| assalactf1o 33631 | In an associative algebra ... |
| assarrginv 33632 | If an element ` X ` of an ... |
| assafld 33633 | If an algebra ` A ` of fin... |
| relfldext 33640 | The field extension is a r... |
| brfldext 33641 | The field extension relati... |
| ccfldextrr 33642 | The field of the complex n... |
| fldextfld1 33643 | A field extension is only ... |
| fldextfld2 33644 | A field extension is only ... |
| fldextsubrg 33645 | Field extension implies a ... |
| sdrgfldext 33646 | A field ` E ` and any sub-... |
| fldextress 33647 | Field extension implies a ... |
| brfinext 33648 | The finite field extension... |
| extdgval 33649 | Value of the field extensi... |
| fldextsdrg 33650 | Deduce sub-division-ring f... |
| fldextsralvec 33651 | The subring algebra associ... |
| extdgcl 33652 | Closure of the field exten... |
| extdggt0 33653 | Degrees of field extension... |
| fldexttr 33654 | Field extension is a trans... |
| fldextid 33655 | The field extension relati... |
| extdgid 33656 | A trivial field extension ... |
| fldsdrgfldext 33657 | A sub-division-ring of a f... |
| fldsdrgfldext2 33658 | A sub-sub-division-ring of... |
| extdgmul 33659 | The multiplicativity formu... |
| finexttrb 33660 | The extension ` E ` of ` K... |
| extdg1id 33661 | If the degree of the exten... |
| extdg1b 33662 | The degree of the extensio... |
| fldgenfldext 33663 | A subfield ` F ` extended ... |
| fldextchr 33664 | The characteristic of a su... |
| evls1fldgencl 33665 | Closure of the subring pol... |
| ccfldsrarelvec 33666 | The subring algebra of the... |
| ccfldextdgrr 33667 | The degree of the field ex... |
| fldextrspunlsplem 33668 | Lemma for ~ fldextrspunlsp... |
| fldextrspunlsp 33669 | Lemma for ~ fldextrspunfld... |
| fldextrspunlem1 33670 | Lemma for ~ fldextrspunfld... |
| fldextrspunfld 33671 | The ring generated by the ... |
| fldextrspunlem2 33672 | Part of the proof of Propo... |
| fldextrspundgle 33673 | Inequality involving the d... |
| fldextrspundglemul 33674 | Given two field extensions... |
| fldextrspundgdvdslem 33675 | Lemma for ~ fldextrspundgd... |
| fldextrspundgdvds 33676 | Given two finite extension... |
| fldext2rspun 33677 | Given two field extensions... |
| irngval 33680 | The elements of a field ` ... |
| elirng 33681 | Property for an element ` ... |
| irngss 33682 | All elements of a subring ... |
| irngssv 33683 | An integral element is an ... |
| 0ringirng 33684 | A zero ring ` R ` has no i... |
| irngnzply1lem 33685 | In the case of a field ` E... |
| irngnzply1 33686 | In the case of a field ` E... |
| ply1annidllem 33691 | Write the set ` Q ` of pol... |
| ply1annidl 33692 | The set ` Q ` of polynomia... |
| ply1annnr 33693 | The set ` Q ` of polynomia... |
| ply1annig1p 33694 | The ideal ` Q ` of polynom... |
| minplyval 33695 | Expand the value of the mi... |
| minplycl 33696 | The minimal polynomial is ... |
| ply1annprmidl 33697 | The set ` Q ` of polynomia... |
| minplymindeg 33698 | The minimal polynomial of ... |
| minplyann 33699 | The minimal polynomial for... |
| minplyirredlem 33700 | Lemma for ~ minplyirred . ... |
| minplyirred 33701 | A nonzero minimal polynomi... |
| irngnminplynz 33702 | Integral elements have non... |
| minplym1p 33703 | A minimal polynomial is mo... |
| minplynzm1p 33704 | If a minimal polynomial is... |
| minplyelirng 33705 | If the minimial polynomial... |
| irredminply 33706 | An irreducible, monic, ann... |
| algextdeglem1 33707 | Lemma for ~ algextdeg . (... |
| algextdeglem2 33708 | Lemma for ~ algextdeg . B... |
| algextdeglem3 33709 | Lemma for ~ algextdeg . T... |
| algextdeglem4 33710 | Lemma for ~ algextdeg . B... |
| algextdeglem5 33711 | Lemma for ~ algextdeg . T... |
| algextdeglem6 33712 | Lemma for ~ algextdeg . B... |
| algextdeglem7 33713 | Lemma for ~ algextdeg . T... |
| algextdeglem8 33714 | Lemma for ~ algextdeg . T... |
| algextdeg 33715 | The degree of an algebraic... |
| rtelextdg2lem 33716 | Lemma for ~ rtelextdg2 : ... |
| rtelextdg2 33717 | If an element ` X ` is a s... |
| fldext2chn 33718 | In a non-empty chain ` T `... |
| constrrtll 33721 | In the construction of con... |
| constrrtlc1 33722 | In the construction of con... |
| constrrtlc2 33723 | In the construction of con... |
| constrrtcclem 33724 | In the construction of con... |
| constrrtcc 33725 | In the construction of con... |
| isconstr 33726 | Property of being a constr... |
| constr0 33727 | The first step of the cons... |
| constrsuc 33728 | Membership in the successo... |
| constrlim 33729 | Limit step of the construc... |
| constrsscn 33730 | Closure of the constructib... |
| constrsslem 33731 | Lemma for ~ constrss . Th... |
| constr01 33732 | ` 0 ` and ` 1 ` are in all... |
| constrss 33733 | Constructed points are in ... |
| constrmon 33734 | The construction of constr... |
| constrconj 33735 | If a point ` X ` of the co... |
| constrfin 33736 | Each step of the construct... |
| constrelextdg2 33737 | If the ` N ` -th step ` ( ... |
| constrextdg2lem 33738 | Lemma for ~ constrextdg2 (... |
| constrextdg2 33739 | Any step ` ( C `` N ) ` of... |
| constrext2chnlem 33740 | Lemma for ~ constrext2chn ... |
| constrfiss 33741 | For any finite set ` A ` o... |
| constrllcllem 33742 | Constructible numbers are ... |
| constrlccllem 33743 | Constructible numbers are ... |
| constrcccllem 33744 | Constructible numbers are ... |
| constrcbvlem 33745 | Technical lemma for elimin... |
| constrllcl 33746 | Constructible numbers are ... |
| constrlccl 33747 | Constructible numbers are ... |
| constrcccl 33748 | Constructible numbers are ... |
| constrext2chn 33749 | If a constructible number ... |
| constrcn 33750 | Constructible numbers are ... |
| nn0constr 33751 | Nonnegative integers are c... |
| constraddcl 33752 | Constructive numbers are c... |
| constrnegcl 33753 | Constructible numbers are ... |
| zconstr 33754 | Integers are constructible... |
| constrdircl 33755 | Constructible numbers are ... |
| iconstr 33756 | The imaginary unit ` _i ` ... |
| constrremulcl 33757 | If two real numbers ` X ` ... |
| constrcjcl 33758 | Constructible numbers are ... |
| constrrecl 33759 | Constructible numbers are ... |
| constrimcl 33760 | Constructible numbers are ... |
| constrmulcl 33761 | Constructible numbers are ... |
| constrreinvcl 33762 | If a real number ` X ` is ... |
| constrinvcl 33763 | Constructible numbers are ... |
| constrcon 33764 | Contradiction of construct... |
| constrsdrg 33765 | Constructible numbers form... |
| constrfld 33766 | The constructible numbers ... |
| constrresqrtcl 33767 | If a positive real number ... |
| constrabscl 33768 | Constructible numbers are ... |
| constrsqrtcl 33769 | Constructible numbers are ... |
| 2sqr3minply 33770 | The polynomial ` ( ( X ^ 3... |
| 2sqr3nconstr 33771 | Doubling the cube is an im... |
| cos9thpiminplylem1 33772 | The polynomial ` ( ( X ^ 3... |
| cos9thpiminplylem2 33773 | The polynomial ` ( ( X ^ 3... |
| cos9thpiminplylem3 33774 | Lemma for ~ cos9thpiminply... |
| cos9thpiminplylem4 33775 | Lemma for ~ cos9thpiminply... |
| cos9thpiminplylem5 33776 | The constructed complex nu... |
| cos9thpiminplylem6 33777 | Evaluation of the polynomi... |
| cos9thpiminply 33778 | The polynomial ` ( ( X ^ 3... |
| cos9thpinconstrlem1 33779 | The complex number ` O ` ,... |
| cos9thpinconstrlem2 33780 | The complex number ` A ` i... |
| cos9thpinconstr 33781 | Trisecting an angle is an ... |
| trisecnconstr 33782 | Not all angles can be tris... |
| smatfval 33785 | Value of the submatrix. (... |
| smatrcl 33786 | Closure of the rectangular... |
| smatlem 33787 | Lemma for the next theorem... |
| smattl 33788 | Entries of a submatrix, to... |
| smattr 33789 | Entries of a submatrix, to... |
| smatbl 33790 | Entries of a submatrix, bo... |
| smatbr 33791 | Entries of a submatrix, bo... |
| smatcl 33792 | Closure of the square subm... |
| matmpo 33793 | Write a square matrix as a... |
| 1smat1 33794 | The submatrix of the ident... |
| submat1n 33795 | One case where the submatr... |
| submatres 33796 | Special case where the sub... |
| submateqlem1 33797 | Lemma for ~ submateq . (C... |
| submateqlem2 33798 | Lemma for ~ submateq . (C... |
| submateq 33799 | Sufficient condition for t... |
| submatminr1 33800 | If we take a submatrix by ... |
| lmatval 33803 | Value of the literal matri... |
| lmatfval 33804 | Entries of a literal matri... |
| lmatfvlem 33805 | Useful lemma to extract li... |
| lmatcl 33806 | Closure of the literal mat... |
| lmat22lem 33807 | Lemma for ~ lmat22e11 and ... |
| lmat22e11 33808 | Entry of a 2x2 literal mat... |
| lmat22e12 33809 | Entry of a 2x2 literal mat... |
| lmat22e21 33810 | Entry of a 2x2 literal mat... |
| lmat22e22 33811 | Entry of a 2x2 literal mat... |
| lmat22det 33812 | The determinant of a liter... |
| mdetpmtr1 33813 | The determinant of a matri... |
| mdetpmtr2 33814 | The determinant of a matri... |
| mdetpmtr12 33815 | The determinant of a matri... |
| mdetlap1 33816 | A Laplace expansion of the... |
| madjusmdetlem1 33817 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem2 33818 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem3 33819 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem4 33820 | Lemma for ~ madjusmdet . ... |
| madjusmdet 33821 | Express the cofactor of th... |
| mdetlap 33822 | Laplace expansion of the d... |
| ist0cld 33823 | The predicate "is a T_0 sp... |
| txomap 33824 | Given two open maps ` F ` ... |
| qtopt1 33825 | If every equivalence class... |
| qtophaus 33826 | If an open map's graph in ... |
| circtopn 33827 | The topology of the unit c... |
| circcn 33828 | The function gluing the re... |
| reff 33829 | For any cover refinement, ... |
| locfinreflem 33830 | A locally finite refinemen... |
| locfinref 33831 | A locally finite refinemen... |
| iscref 33834 | The property that every op... |
| crefeq 33835 | Equality theorem for the "... |
| creftop 33836 | A space where every open c... |
| crefi 33837 | The property that every op... |
| crefdf 33838 | A formulation of ~ crefi e... |
| crefss 33839 | The "every open cover has ... |
| cmpcref 33840 | Equivalent definition of c... |
| cmpfiref 33841 | Every open cover of a Comp... |
| ldlfcntref 33844 | Every open cover of a Lind... |
| ispcmp 33847 | The predicate "is a paraco... |
| cmppcmp 33848 | Every compact space is par... |
| dispcmp 33849 | Every discrete space is pa... |
| pcmplfin 33850 | Given a paracompact topolo... |
| pcmplfinf 33851 | Given a paracompact topolo... |
| rspecval 33854 | Value of the spectrum of t... |
| rspecbas 33855 | The prime ideals form the ... |
| rspectset 33856 | Topology component of the ... |
| rspectopn 33857 | The topology component of ... |
| zarcls0 33858 | The closure of the identit... |
| zarcls1 33859 | The unit ideal ` B ` is th... |
| zarclsun 33860 | The union of two closed se... |
| zarclsiin 33861 | In a Zariski topology, the... |
| zarclsint 33862 | The intersection of a fami... |
| zarclssn 33863 | The closed points of Zaris... |
| zarcls 33864 | The open sets of the Zaris... |
| zartopn 33865 | The Zariski topology is a ... |
| zartop 33866 | The Zariski topology is a ... |
| zartopon 33867 | The points of the Zariski ... |
| zar0ring 33868 | The Zariski Topology of th... |
| zart0 33869 | The Zariski topology is T_... |
| zarmxt1 33870 | The Zariski topology restr... |
| zarcmplem 33871 | Lemma for ~ zarcmp . (Con... |
| zarcmp 33872 | The Zariski topology is co... |
| rspectps 33873 | The spectrum of a ring ` R... |
| rhmpreimacnlem 33874 | Lemma for ~ rhmpreimacn . ... |
| rhmpreimacn 33875 | The function mapping a pri... |
| metidval 33880 | Value of the metric identi... |
| metidss 33881 | As a relation, the metric ... |
| metidv 33882 | ` A ` and ` B ` identify b... |
| metideq 33883 | Basic property of the metr... |
| metider 33884 | The metric identification ... |
| pstmval 33885 | Value of the metric induce... |
| pstmfval 33886 | Function value of the metr... |
| pstmxmet 33887 | The metric induced by a ps... |
| hauseqcn 33888 | In a Hausdorff topology, t... |
| elunitge0 33889 | An element of the closed u... |
| unitssxrge0 33890 | The closed unit interval i... |
| unitdivcld 33891 | Necessary conditions for a... |
| iistmd 33892 | The closed unit interval f... |
| unicls 33893 | The union of the closed se... |
| tpr2tp 33894 | The usual topology on ` ( ... |
| tpr2uni 33895 | The usual topology on ` ( ... |
| xpinpreima 33896 | Rewrite the cartesian prod... |
| xpinpreima2 33897 | Rewrite the cartesian prod... |
| sqsscirc1 33898 | The complex square of side... |
| sqsscirc2 33899 | The complex square of side... |
| cnre2csqlem 33900 | Lemma for ~ cnre2csqima . ... |
| cnre2csqima 33901 | Image of a centered square... |
| tpr2rico 33902 | For any point of an open s... |
| cnvordtrestixx 33903 | The restriction of the 'gr... |
| prsdm 33904 | Domain of the relation of ... |
| prsrn 33905 | Range of the relation of a... |
| prsss 33906 | Relation of a subproset. ... |
| prsssdm 33907 | Domain of a subproset rela... |
| ordtprsval 33908 | Value of the order topolog... |
| ordtprsuni 33909 | Value of the order topolog... |
| ordtcnvNEW 33910 | The order dual generates t... |
| ordtrestNEW 33911 | The subspace topology of a... |
| ordtrest2NEWlem 33912 | Lemma for ~ ordtrest2NEW .... |
| ordtrest2NEW 33913 | An interval-closed set ` A... |
| ordtconnlem1 33914 | Connectedness in the order... |
| ordtconn 33915 | Connectedness in the order... |
| mndpluscn 33916 | A mapping that is both a h... |
| mhmhmeotmd 33917 | Deduce a Topological Monoi... |
| rmulccn 33918 | Multiplication by a real c... |
| raddcn 33919 | Addition in the real numbe... |
| xrmulc1cn 33920 | The operation multiplying ... |
| fmcncfil 33921 | The image of a Cauchy filt... |
| xrge0hmph 33922 | The extended nonnegative r... |
| xrge0iifcnv 33923 | Define a bijection from ` ... |
| xrge0iifcv 33924 | The defined function's val... |
| xrge0iifiso 33925 | The defined bijection from... |
| xrge0iifhmeo 33926 | Expose a homeomorphism fro... |
| xrge0iifhom 33927 | The defined function from ... |
| xrge0iif1 33928 | Condition for the defined ... |
| xrge0iifmhm 33929 | The defined function from ... |
| xrge0pluscn 33930 | The addition operation of ... |
| xrge0mulc1cn 33931 | The operation multiplying ... |
| xrge0tps 33932 | The extended nonnegative r... |
| xrge0topn 33933 | The topology of the extend... |
| xrge0haus 33934 | The topology of the extend... |
| xrge0tmd 33935 | The extended nonnegative r... |
| xrge0tmdALT 33936 | Alternate proof of ~ xrge0... |
| lmlim 33937 | Relate a limit in a given ... |
| lmlimxrge0 33938 | Relate a limit in the nonn... |
| rge0scvg 33939 | Implication of convergence... |
| fsumcvg4 33940 | A serie with finite suppor... |
| pnfneige0 33941 | A neighborhood of ` +oo ` ... |
| lmxrge0 33942 | Express "sequence ` F ` co... |
| lmdvg 33943 | If a monotonic sequence of... |
| lmdvglim 33944 | If a monotonic real number... |
| pl1cn 33945 | A univariate polynomial is... |
| zringnm 33948 | The norm (function) for a ... |
| zzsnm 33949 | The norm of the ring of th... |
| zlm0 33950 | Zero of a ` ZZ ` -module. ... |
| zlm1 33951 | Unity element of a ` ZZ ` ... |
| zlmds 33952 | Distance in a ` ZZ ` -modu... |
| zlmtset 33953 | Topology in a ` ZZ ` -modu... |
| zlmnm 33954 | Norm of a ` ZZ ` -module (... |
| zhmnrg 33955 | The ` ZZ ` -module built f... |
| nmmulg 33956 | The norm of a group produc... |
| zrhnm 33957 | The norm of the image by `... |
| cnzh 33958 | The ` ZZ ` -module of ` CC... |
| rezh 33959 | The ` ZZ ` -module of ` RR... |
| qqhval 33962 | Value of the canonical hom... |
| zrhf1ker 33963 | The kernel of the homomorp... |
| zrhchr 33964 | The kernel of the homomorp... |
| zrhker 33965 | The kernel of the homomorp... |
| zrhunitpreima 33966 | The preimage by ` ZRHom ` ... |
| elzrhunit 33967 | Condition for the image by... |
| zrhneg 33968 | The canonical homomorphism... |
| zrhcntr 33969 | The canonical representati... |
| elzdif0 33970 | Lemma for ~ qqhval2 . (Co... |
| qqhval2lem 33971 | Lemma for ~ qqhval2 . (Co... |
| qqhval2 33972 | Value of the canonical hom... |
| qqhvval 33973 | Value of the canonical hom... |
| qqh0 33974 | The image of ` 0 ` by the ... |
| qqh1 33975 | The image of ` 1 ` by the ... |
| qqhf 33976 | ` QQHom ` as a function. ... |
| qqhvq 33977 | The image of a quotient by... |
| qqhghm 33978 | The ` QQHom ` homomorphism... |
| qqhrhm 33979 | The ` QQHom ` homomorphism... |
| qqhnm 33980 | The norm of the image by `... |
| qqhcn 33981 | The ` QQHom ` homomorphism... |
| qqhucn 33982 | The ` QQHom ` homomorphism... |
| rrhval 33986 | Value of the canonical hom... |
| rrhcn 33987 | If the topology of ` R ` i... |
| rrhf 33988 | If the topology of ` R ` i... |
| isrrext 33990 | Express the property " ` R... |
| rrextnrg 33991 | An extension of ` RR ` is ... |
| rrextdrg 33992 | An extension of ` RR ` is ... |
| rrextnlm 33993 | The norm of an extension o... |
| rrextchr 33994 | The ring characteristic of... |
| rrextcusp 33995 | An extension of ` RR ` is ... |
| rrexttps 33996 | An extension of ` RR ` is ... |
| rrexthaus 33997 | The topology of an extensi... |
| rrextust 33998 | The uniformity of an exten... |
| rerrext 33999 | The field of the real numb... |
| cnrrext 34000 | The field of the complex n... |
| qqtopn 34001 | The topology of the field ... |
| rrhfe 34002 | If ` R ` is an extension o... |
| rrhcne 34003 | If ` R ` is an extension o... |
| rrhqima 34004 | The ` RRHom ` homomorphism... |
| rrh0 34005 | The image of ` 0 ` by the ... |
| xrhval 34008 | The value of the embedding... |
| zrhre 34009 | The ` ZRHom ` homomorphism... |
| qqhre 34010 | The ` QQHom ` homomorphism... |
| rrhre 34011 | The ` RRHom ` homomorphism... |
| relmntop 34014 | Manifold is a relation. (... |
| ismntoplly 34015 | Property of being a manifo... |
| ismntop 34016 | Property of being a manifo... |
| esumex 34019 | An extended sum is a set b... |
| esumcl 34020 | Closure for extended sum i... |
| esumeq12dvaf 34021 | Equality deduction for ext... |
| esumeq12dva 34022 | Equality deduction for ext... |
| esumeq12d 34023 | Equality deduction for ext... |
| esumeq1 34024 | Equality theorem for an ex... |
| esumeq1d 34025 | Equality theorem for an ex... |
| esumeq2 34026 | Equality theorem for exten... |
| esumeq2d 34027 | Equality deduction for ext... |
| esumeq2dv 34028 | Equality deduction for ext... |
| esumeq2sdv 34029 | Equality deduction for ext... |
| nfesum1 34030 | Bound-variable hypothesis ... |
| nfesum2 34031 | Bound-variable hypothesis ... |
| cbvesum 34032 | Change bound variable in a... |
| cbvesumv 34033 | Change bound variable in a... |
| esumid 34034 | Identify the extended sum ... |
| esumgsum 34035 | A finite extended sum is t... |
| esumval 34036 | Develop the value of the e... |
| esumel 34037 | The extended sum is a limi... |
| esumnul 34038 | Extended sum over the empt... |
| esum0 34039 | Extended sum of zero. (Co... |
| esumf1o 34040 | Re-index an extended sum u... |
| esumc 34041 | Convert from the collectio... |
| esumrnmpt 34042 | Rewrite an extended sum in... |
| esumsplit 34043 | Split an extended sum into... |
| esummono 34044 | Extended sum is monotonic.... |
| esumpad 34045 | Extend an extended sum by ... |
| esumpad2 34046 | Remove zeroes from an exte... |
| esumadd 34047 | Addition of infinite sums.... |
| esumle 34048 | If all of the terms of an ... |
| gsumesum 34049 | Relate a group sum on ` ( ... |
| esumlub 34050 | The extended sum is the lo... |
| esumaddf 34051 | Addition of infinite sums.... |
| esumlef 34052 | If all of the terms of an ... |
| esumcst 34053 | The extended sum of a cons... |
| esumsnf 34054 | The extended sum of a sing... |
| esumsn 34055 | The extended sum of a sing... |
| esumpr 34056 | Extended sum over a pair. ... |
| esumpr2 34057 | Extended sum over a pair, ... |
| esumrnmpt2 34058 | Rewrite an extended sum in... |
| esumfzf 34059 | Formulating a partial exte... |
| esumfsup 34060 | Formulating an extended su... |
| esumfsupre 34061 | Formulating an extended su... |
| esumss 34062 | Change the index set to a ... |
| esumpinfval 34063 | The value of the extended ... |
| esumpfinvallem 34064 | Lemma for ~ esumpfinval . ... |
| esumpfinval 34065 | The value of the extended ... |
| esumpfinvalf 34066 | Same as ~ esumpfinval , mi... |
| esumpinfsum 34067 | The value of the extended ... |
| esumpcvgval 34068 | The value of the extended ... |
| esumpmono 34069 | The partial sums in an ext... |
| esumcocn 34070 | Lemma for ~ esummulc2 and ... |
| esummulc1 34071 | An extended sum multiplied... |
| esummulc2 34072 | An extended sum multiplied... |
| esumdivc 34073 | An extended sum divided by... |
| hashf2 34074 | Lemma for ~ hasheuni . (C... |
| hasheuni 34075 | The cardinality of a disjo... |
| esumcvg 34076 | The sequence of partial su... |
| esumcvg2 34077 | Simpler version of ~ esumc... |
| esumcvgsum 34078 | The value of the extended ... |
| esumsup 34079 | Express an extended sum as... |
| esumgect 34080 | "Send ` n ` to ` +oo ` " i... |
| esumcvgre 34081 | All terms of a converging ... |
| esum2dlem 34082 | Lemma for ~ esum2d (finite... |
| esum2d 34083 | Write a double extended su... |
| esumiun 34084 | Sum over a nonnecessarily ... |
| ofceq 34087 | Equality theorem for funct... |
| ofcfval 34088 | Value of an operation appl... |
| ofcval 34089 | Evaluate a function/consta... |
| ofcfn 34090 | The function operation pro... |
| ofcfeqd2 34091 | Equality theorem for funct... |
| ofcfval3 34092 | General value of ` ( F oFC... |
| ofcf 34093 | The function/constant oper... |
| ofcfval2 34094 | The function operation exp... |
| ofcfval4 34095 | The function/constant oper... |
| ofcc 34096 | Left operation by a consta... |
| ofcof 34097 | Relate function operation ... |
| sigaex 34100 | Lemma for ~ issiga and ~ i... |
| sigaval 34101 | The set of sigma-algebra w... |
| issiga 34102 | An alternative definition ... |
| isrnsiga 34103 | The property of being a si... |
| 0elsiga 34104 | A sigma-algebra contains t... |
| baselsiga 34105 | A sigma-algebra contains i... |
| sigasspw 34106 | A sigma-algebra is a set o... |
| sigaclcu 34107 | A sigma-algebra is closed ... |
| sigaclcuni 34108 | A sigma-algebra is closed ... |
| sigaclfu 34109 | A sigma-algebra is closed ... |
| sigaclcu2 34110 | A sigma-algebra is closed ... |
| sigaclfu2 34111 | A sigma-algebra is closed ... |
| sigaclcu3 34112 | A sigma-algebra is closed ... |
| issgon 34113 | Property of being a sigma-... |
| sgon 34114 | A sigma-algebra is a sigma... |
| elsigass 34115 | An element of a sigma-alge... |
| elrnsiga 34116 | Dropping the base informat... |
| isrnsigau 34117 | The property of being a si... |
| unielsiga 34118 | A sigma-algebra contains i... |
| dmvlsiga 34119 | Lebesgue-measurable subset... |
| pwsiga 34120 | Any power set forms a sigm... |
| prsiga 34121 | The smallest possible sigm... |
| sigaclci 34122 | A sigma-algebra is closed ... |
| difelsiga 34123 | A sigma-algebra is closed ... |
| unelsiga 34124 | A sigma-algebra is closed ... |
| inelsiga 34125 | A sigma-algebra is closed ... |
| sigainb 34126 | Building a sigma-algebra f... |
| insiga 34127 | The intersection of a coll... |
| sigagenval 34130 | Value of the generated sig... |
| sigagensiga 34131 | A generated sigma-algebra ... |
| sgsiga 34132 | A generated sigma-algebra ... |
| unisg 34133 | The sigma-algebra generate... |
| dmsigagen 34134 | A sigma-algebra can be gen... |
| sssigagen 34135 | A set is a subset of the s... |
| sssigagen2 34136 | A subset of the generating... |
| elsigagen 34137 | Any element of a set is al... |
| elsigagen2 34138 | Any countable union of ele... |
| sigagenss 34139 | The generated sigma-algebr... |
| sigagenss2 34140 | Sufficient condition for i... |
| sigagenid 34141 | The sigma-algebra generate... |
| ispisys 34142 | The property of being a pi... |
| ispisys2 34143 | The property of being a pi... |
| inelpisys 34144 | Pi-systems are closed unde... |
| sigapisys 34145 | All sigma-algebras are pi-... |
| isldsys 34146 | The property of being a la... |
| pwldsys 34147 | The power set of the unive... |
| unelldsys 34148 | Lambda-systems are closed ... |
| sigaldsys 34149 | All sigma-algebras are lam... |
| ldsysgenld 34150 | The intersection of all la... |
| sigapildsyslem 34151 | Lemma for ~ sigapildsys . ... |
| sigapildsys 34152 | Sigma-algebra are exactly ... |
| ldgenpisyslem1 34153 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem2 34154 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem3 34155 | Lemma for ~ ldgenpisys . ... |
| ldgenpisys 34156 | The lambda system ` E ` ge... |
| dynkin 34157 | Dynkin's lambda-pi theorem... |
| isros 34158 | The property of being a ri... |
| rossspw 34159 | A ring of sets is a collec... |
| 0elros 34160 | A ring of sets contains th... |
| unelros 34161 | A ring of sets is closed u... |
| difelros 34162 | A ring of sets is closed u... |
| inelros 34163 | A ring of sets is closed u... |
| fiunelros 34164 | A ring of sets is closed u... |
| issros 34165 | The property of being a se... |
| srossspw 34166 | A semiring of sets is a co... |
| 0elsros 34167 | A semiring of sets contain... |
| inelsros 34168 | A semiring of sets is clos... |
| diffiunisros 34169 | In semiring of sets, compl... |
| rossros 34170 | Rings of sets are semiring... |
| brsiga 34173 | The Borel Algebra on real ... |
| brsigarn 34174 | The Borel Algebra is a sig... |
| brsigasspwrn 34175 | The Borel Algebra is a set... |
| unibrsiga 34176 | The union of the Borel Alg... |
| cldssbrsiga 34177 | A Borel Algebra contains a... |
| sxval 34180 | Value of the product sigma... |
| sxsiga 34181 | A product sigma-algebra is... |
| sxsigon 34182 | A product sigma-algebra is... |
| sxuni 34183 | The base set of a product ... |
| elsx 34184 | The cartesian product of t... |
| measbase 34187 | The base set of a measure ... |
| measval 34188 | The value of the ` measure... |
| ismeas 34189 | The property of being a me... |
| isrnmeas 34190 | The property of being a me... |
| dmmeas 34191 | The domain of a measure is... |
| measbasedom 34192 | The base set of a measure ... |
| measfrge0 34193 | A measure is a function ov... |
| measfn 34194 | A measure is a function on... |
| measvxrge0 34195 | The values of a measure ar... |
| measvnul 34196 | The measure of the empty s... |
| measge0 34197 | A measure is nonnegative. ... |
| measle0 34198 | If the measure of a given ... |
| measvun 34199 | The measure of a countable... |
| measxun2 34200 | The measure the union of t... |
| measun 34201 | The measure the union of t... |
| measvunilem 34202 | Lemma for ~ measvuni . (C... |
| measvunilem0 34203 | Lemma for ~ measvuni . (C... |
| measvuni 34204 | The measure of a countable... |
| measssd 34205 | A measure is monotone with... |
| measunl 34206 | A measure is sub-additive ... |
| measiuns 34207 | The measure of the union o... |
| measiun 34208 | A measure is sub-additive.... |
| meascnbl 34209 | A measure is continuous fr... |
| measinblem 34210 | Lemma for ~ measinb . (Co... |
| measinb 34211 | Building a measure restric... |
| measres 34212 | Building a measure restric... |
| measinb2 34213 | Building a measure restric... |
| measdivcst 34214 | Division of a measure by a... |
| measdivcstALTV 34215 | Alternate version of ~ mea... |
| cntmeas 34216 | The Counting measure is a ... |
| pwcntmeas 34217 | The counting measure is a ... |
| cntnevol 34218 | Counting and Lebesgue meas... |
| voliune 34219 | The Lebesgue measure funct... |
| volfiniune 34220 | The Lebesgue measure funct... |
| volmeas 34221 | The Lebesgue measure is a ... |
| ddeval1 34224 | Value of the delta measure... |
| ddeval0 34225 | Value of the delta measure... |
| ddemeas 34226 | The Dirac delta measure is... |
| relae 34230 | 'almost everywhere' is a r... |
| brae 34231 | 'almost everywhere' relati... |
| braew 34232 | 'almost everywhere' relati... |
| truae 34233 | A truth holds almost every... |
| aean 34234 | A conjunction holds almost... |
| faeval 34236 | Value of the 'almost every... |
| relfae 34237 | The 'almost everywhere' bu... |
| brfae 34238 | 'almost everywhere' relati... |
| ismbfm 34241 | The predicate " ` F ` is a... |
| elunirnmbfm 34242 | The property of being a me... |
| mbfmfun 34243 | A measurable function is a... |
| mbfmf 34244 | A measurable function as a... |
| isanmbfmOLD 34245 | Obsolete version of ~ isan... |
| mbfmcnvima 34246 | The preimage by a measurab... |
| isanmbfm 34247 | The predicate to be a meas... |
| mbfmbfmOLD 34248 | A measurable function to a... |
| mbfmbfm 34249 | A measurable function to a... |
| mbfmcst 34250 | A constant function is mea... |
| 1stmbfm 34251 | The first projection map i... |
| 2ndmbfm 34252 | The second projection map ... |
| imambfm 34253 | If the sigma-algebra in th... |
| cnmbfm 34254 | A continuous function is m... |
| mbfmco 34255 | The composition of two mea... |
| mbfmco2 34256 | The pair building of two m... |
| mbfmvolf 34257 | Measurable functions with ... |
| elmbfmvol2 34258 | Measurable functions with ... |
| mbfmcnt 34259 | All functions are measurab... |
| br2base 34260 | The base set for the gener... |
| dya2ub 34261 | An upper bound for a dyadi... |
| sxbrsigalem0 34262 | The closed half-spaces of ... |
| sxbrsigalem3 34263 | The sigma-algebra generate... |
| dya2iocival 34264 | The function ` I ` returns... |
| dya2iocress 34265 | Dyadic intervals are subse... |
| dya2iocbrsiga 34266 | Dyadic intervals are Borel... |
| dya2icobrsiga 34267 | Dyadic intervals are Borel... |
| dya2icoseg 34268 | For any point and any clos... |
| dya2icoseg2 34269 | For any point and any open... |
| dya2iocrfn 34270 | The function returning dya... |
| dya2iocct 34271 | The dyadic rectangle set i... |
| dya2iocnrect 34272 | For any point of an open r... |
| dya2iocnei 34273 | For any point of an open s... |
| dya2iocuni 34274 | Every open set of ` ( RR X... |
| dya2iocucvr 34275 | The dyadic rectangular set... |
| sxbrsigalem1 34276 | The Borel algebra on ` ( R... |
| sxbrsigalem2 34277 | The sigma-algebra generate... |
| sxbrsigalem4 34278 | The Borel algebra on ` ( R... |
| sxbrsigalem5 34279 | First direction for ~ sxbr... |
| sxbrsigalem6 34280 | First direction for ~ sxbr... |
| sxbrsiga 34281 | The product sigma-algebra ... |
| omsval 34284 | Value of the function mapp... |
| omsfval 34285 | Value of the outer measure... |
| omscl 34286 | A closure lemma for the co... |
| omsf 34287 | A constructed outer measur... |
| oms0 34288 | A constructed outer measur... |
| omsmon 34289 | A constructed outer measur... |
| omssubaddlem 34290 | For any small margin ` E `... |
| omssubadd 34291 | A constructed outer measur... |
| carsgval 34294 | Value of the Caratheodory ... |
| carsgcl 34295 | Closure of the Caratheodor... |
| elcarsg 34296 | Property of being a Carath... |
| baselcarsg 34297 | The universe set, ` O ` , ... |
| 0elcarsg 34298 | The empty set is Caratheod... |
| carsguni 34299 | The union of all Caratheod... |
| elcarsgss 34300 | Caratheodory measurable se... |
| difelcarsg 34301 | The Caratheodory measurabl... |
| inelcarsg 34302 | The Caratheodory measurabl... |
| unelcarsg 34303 | The Caratheodory-measurabl... |
| difelcarsg2 34304 | The Caratheodory-measurabl... |
| carsgmon 34305 | Utility lemma: Apply mono... |
| carsgsigalem 34306 | Lemma for the following th... |
| fiunelcarsg 34307 | The Caratheodory measurabl... |
| carsgclctunlem1 34308 | Lemma for ~ carsgclctun . ... |
| carsggect 34309 | The outer measure is count... |
| carsgclctunlem2 34310 | Lemma for ~ carsgclctun . ... |
| carsgclctunlem3 34311 | Lemma for ~ carsgclctun . ... |
| carsgclctun 34312 | The Caratheodory measurabl... |
| carsgsiga 34313 | The Caratheodory measurabl... |
| omsmeas 34314 | The restriction of a const... |
| pmeasmono 34315 | This theorem's hypotheses ... |
| pmeasadd 34316 | A premeasure on a ring of ... |
| itgeq12dv 34317 | Equality theorem for an in... |
| sitgval 34323 | Value of the simple functi... |
| issibf 34324 | The predicate " ` F ` is a... |
| sibf0 34325 | The constant zero function... |
| sibfmbl 34326 | A simple function is measu... |
| sibff 34327 | A simple function is a fun... |
| sibfrn 34328 | A simple function has fini... |
| sibfima 34329 | Any preimage of a singleto... |
| sibfinima 34330 | The measure of the interse... |
| sibfof 34331 | Applying function operatio... |
| sitgfval 34332 | Value of the Bochner integ... |
| sitgclg 34333 | Closure of the Bochner int... |
| sitgclbn 34334 | Closure of the Bochner int... |
| sitgclcn 34335 | Closure of the Bochner int... |
| sitgclre 34336 | Closure of the Bochner int... |
| sitg0 34337 | The integral of the consta... |
| sitgf 34338 | The integral for simple fu... |
| sitgaddlemb 34339 | Lemma for * sitgadd . (Co... |
| sitmval 34340 | Value of the simple functi... |
| sitmfval 34341 | Value of the integral dist... |
| sitmcl 34342 | Closure of the integral di... |
| sitmf 34343 | The integral metric as a f... |
| oddpwdc 34345 | Lemma for ~ eulerpart . T... |
| oddpwdcv 34346 | Lemma for ~ eulerpart : va... |
| eulerpartlemsv1 34347 | Lemma for ~ eulerpart . V... |
| eulerpartlemelr 34348 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv2 34349 | Lemma for ~ eulerpart . V... |
| eulerpartlemsf 34350 | Lemma for ~ eulerpart . (... |
| eulerpartlems 34351 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv3 34352 | Lemma for ~ eulerpart . V... |
| eulerpartlemgc 34353 | Lemma for ~ eulerpart . (... |
| eulerpartleme 34354 | Lemma for ~ eulerpart . (... |
| eulerpartlemv 34355 | Lemma for ~ eulerpart . (... |
| eulerpartlemo 34356 | Lemma for ~ eulerpart : ` ... |
| eulerpartlemd 34357 | Lemma for ~ eulerpart : ` ... |
| eulerpartlem1 34358 | Lemma for ~ eulerpart . (... |
| eulerpartlemb 34359 | Lemma for ~ eulerpart . T... |
| eulerpartlemt0 34360 | Lemma for ~ eulerpart . (... |
| eulerpartlemf 34361 | Lemma for ~ eulerpart : O... |
| eulerpartlemt 34362 | Lemma for ~ eulerpart . (... |
| eulerpartgbij 34363 | Lemma for ~ eulerpart : T... |
| eulerpartlemgv 34364 | Lemma for ~ eulerpart : va... |
| eulerpartlemr 34365 | Lemma for ~ eulerpart . (... |
| eulerpartlemmf 34366 | Lemma for ~ eulerpart . (... |
| eulerpartlemgvv 34367 | Lemma for ~ eulerpart : va... |
| eulerpartlemgu 34368 | Lemma for ~ eulerpart : R... |
| eulerpartlemgh 34369 | Lemma for ~ eulerpart : T... |
| eulerpartlemgf 34370 | Lemma for ~ eulerpart : I... |
| eulerpartlemgs2 34371 | Lemma for ~ eulerpart : T... |
| eulerpartlemn 34372 | Lemma for ~ eulerpart . (... |
| eulerpart 34373 | Euler's theorem on partiti... |
| subiwrd 34376 | Lemma for ~ sseqp1 . (Con... |
| subiwrdlen 34377 | Length of a subword of an ... |
| iwrdsplit 34378 | Lemma for ~ sseqp1 . (Con... |
| sseqval 34379 | Value of the strong sequen... |
| sseqfv1 34380 | Value of the strong sequen... |
| sseqfn 34381 | A strong recursive sequenc... |
| sseqmw 34382 | Lemma for ~ sseqf amd ~ ss... |
| sseqf 34383 | A strong recursive sequenc... |
| sseqfres 34384 | The first elements in the ... |
| sseqfv2 34385 | Value of the strong sequen... |
| sseqp1 34386 | Value of the strong sequen... |
| fiblem 34389 | Lemma for ~ fib0 , ~ fib1 ... |
| fib0 34390 | Value of the Fibonacci seq... |
| fib1 34391 | Value of the Fibonacci seq... |
| fibp1 34392 | Value of the Fibonacci seq... |
| fib2 34393 | Value of the Fibonacci seq... |
| fib3 34394 | Value of the Fibonacci seq... |
| fib4 34395 | Value of the Fibonacci seq... |
| fib5 34396 | Value of the Fibonacci seq... |
| fib6 34397 | Value of the Fibonacci seq... |
| elprob 34400 | The property of being a pr... |
| domprobmeas 34401 | A probability measure is a... |
| domprobsiga 34402 | The domain of a probabilit... |
| probtot 34403 | The probability of the uni... |
| prob01 34404 | A probability is an elemen... |
| probnul 34405 | The probability of the emp... |
| unveldomd 34406 | The universe is an element... |
| unveldom 34407 | The universe is an element... |
| nuleldmp 34408 | The empty set is an elemen... |
| probcun 34409 | The probability of the uni... |
| probun 34410 | The probability of the uni... |
| probdif 34411 | The probability of the dif... |
| probinc 34412 | A probability law is incre... |
| probdsb 34413 | The probability of the com... |
| probmeasd 34414 | A probability measure is a... |
| probvalrnd 34415 | The value of a probability... |
| probtotrnd 34416 | The probability of the uni... |
| totprobd 34417 | Law of total probability, ... |
| totprob 34418 | Law of total probability. ... |
| probfinmeasb 34419 | Build a probability measur... |
| probfinmeasbALTV 34420 | Alternate version of ~ pro... |
| probmeasb 34421 | Build a probability from a... |
| cndprobval 34424 | The value of the condition... |
| cndprobin 34425 | An identity linking condit... |
| cndprob01 34426 | The conditional probabilit... |
| cndprobtot 34427 | The conditional probabilit... |
| cndprobnul 34428 | The conditional probabilit... |
| cndprobprob 34429 | The conditional probabilit... |
| bayesth 34430 | Bayes Theorem. (Contribut... |
| rrvmbfm 34433 | A real-valued random varia... |
| isrrvv 34434 | Elementhood to the set of ... |
| rrvvf 34435 | A real-valued random varia... |
| rrvfn 34436 | A real-valued random varia... |
| rrvdm 34437 | The domain of a random var... |
| rrvrnss 34438 | The range of a random vari... |
| rrvf2 34439 | A real-valued random varia... |
| rrvdmss 34440 | The domain of a random var... |
| rrvfinvima 34441 | For a real-value random va... |
| 0rrv 34442 | The constant function equa... |
| rrvadd 34443 | The sum of two random vari... |
| rrvmulc 34444 | A random variable multipli... |
| rrvsum 34445 | An indexed sum of random v... |
| boolesineq 34446 | Boole's inequality (union ... |
| orvcval 34449 | Value of the preimage mapp... |
| orvcval2 34450 | Another way to express the... |
| elorvc 34451 | Elementhood of a preimage.... |
| orvcval4 34452 | The value of the preimage ... |
| orvcoel 34453 | If the relation produces o... |
| orvccel 34454 | If the relation produces c... |
| elorrvc 34455 | Elementhood of a preimage ... |
| orrvcval4 34456 | The value of the preimage ... |
| orrvcoel 34457 | If the relation produces o... |
| orrvccel 34458 | If the relation produces c... |
| orvcgteel 34459 | Preimage maps produced by ... |
| orvcelval 34460 | Preimage maps produced by ... |
| orvcelel 34461 | Preimage maps produced by ... |
| dstrvval 34462 | The value of the distribut... |
| dstrvprob 34463 | The distribution of a rand... |
| orvclteel 34464 | Preimage maps produced by ... |
| dstfrvel 34465 | Elementhood of preimage ma... |
| dstfrvunirn 34466 | The limit of all preimage ... |
| orvclteinc 34467 | Preimage maps produced by ... |
| dstfrvinc 34468 | A cumulative distribution ... |
| dstfrvclim1 34469 | The limit of the cumulativ... |
| coinfliplem 34470 | Division in the extended r... |
| coinflipprob 34471 | The ` P ` we defined for c... |
| coinflipspace 34472 | The space of our coin-flip... |
| coinflipuniv 34473 | The universe of our coin-f... |
| coinfliprv 34474 | The ` X ` we defined for c... |
| coinflippv 34475 | The probability of heads i... |
| coinflippvt 34476 | The probability of tails i... |
| ballotlemoex 34477 | ` O ` is a set. (Contribu... |
| ballotlem1 34478 | The size of the universe i... |
| ballotlemelo 34479 | Elementhood in ` O ` . (C... |
| ballotlem2 34480 | The probability that the f... |
| ballotlemfval 34481 | The value of ` F ` . (Con... |
| ballotlemfelz 34482 | ` ( F `` C ) ` has values ... |
| ballotlemfp1 34483 | If the ` J ` th ballot is ... |
| ballotlemfc0 34484 | ` F ` takes value 0 betwee... |
| ballotlemfcc 34485 | ` F ` takes value 0 betwee... |
| ballotlemfmpn 34486 | ` ( F `` C ) ` finishes co... |
| ballotlemfval0 34487 | ` ( F `` C ) ` always star... |
| ballotleme 34488 | Elements of ` E ` . (Cont... |
| ballotlemodife 34489 | Elements of ` ( O \ E ) ` ... |
| ballotlem4 34490 | If the first pick is a vot... |
| ballotlem5 34491 | If A is not ahead througho... |
| ballotlemi 34492 | Value of ` I ` for a given... |
| ballotlemiex 34493 | Properties of ` ( I `` C )... |
| ballotlemi1 34494 | The first tie cannot be re... |
| ballotlemii 34495 | The first tie cannot be re... |
| ballotlemsup 34496 | The set of zeroes of ` F `... |
| ballotlemimin 34497 | ` ( I `` C ) ` is the firs... |
| ballotlemic 34498 | If the first vote is for B... |
| ballotlem1c 34499 | If the first vote is for A... |
| ballotlemsval 34500 | Value of ` S ` . (Contrib... |
| ballotlemsv 34501 | Value of ` S ` evaluated a... |
| ballotlemsgt1 34502 | ` S ` maps values less tha... |
| ballotlemsdom 34503 | Domain of ` S ` for a give... |
| ballotlemsel1i 34504 | The range ` ( 1 ... ( I ``... |
| ballotlemsf1o 34505 | The defined ` S ` is a bij... |
| ballotlemsi 34506 | The image by ` S ` of the ... |
| ballotlemsima 34507 | The image by ` S ` of an i... |
| ballotlemieq 34508 | If two countings share the... |
| ballotlemrval 34509 | Value of ` R ` . (Contrib... |
| ballotlemscr 34510 | The image of ` ( R `` C ) ... |
| ballotlemrv 34511 | Value of ` R ` evaluated a... |
| ballotlemrv1 34512 | Value of ` R ` before the ... |
| ballotlemrv2 34513 | Value of ` R ` after the t... |
| ballotlemro 34514 | Range of ` R ` is included... |
| ballotlemgval 34515 | Expand the value of ` .^ `... |
| ballotlemgun 34516 | A property of the defined ... |
| ballotlemfg 34517 | Express the value of ` ( F... |
| ballotlemfrc 34518 | Express the value of ` ( F... |
| ballotlemfrci 34519 | Reverse counting preserves... |
| ballotlemfrceq 34520 | Value of ` F ` for a rever... |
| ballotlemfrcn0 34521 | Value of ` F ` for a rever... |
| ballotlemrc 34522 | Range of ` R ` . (Contrib... |
| ballotlemirc 34523 | Applying ` R ` does not ch... |
| ballotlemrinv0 34524 | Lemma for ~ ballotlemrinv ... |
| ballotlemrinv 34525 | ` R ` is its own inverse :... |
| ballotlem1ri 34526 | When the vote on the first... |
| ballotlem7 34527 | ` R ` is a bijection betwe... |
| ballotlem8 34528 | There are as many counting... |
| ballotth 34529 | Bertrand's ballot problem ... |
| fzssfzo 34530 | Condition for an integer i... |
| gsumncl 34531 | Closure of a group sum in ... |
| gsumnunsn 34532 | Closure of a group sum in ... |
| ccatmulgnn0dir 34533 | Concatenation of words fol... |
| ofcccat 34534 | Letterwise operations on w... |
| ofcs1 34535 | Letterwise operations on a... |
| ofcs2 34536 | Letterwise operations on a... |
| plymul02 34537 | Product of a polynomial wi... |
| plymulx0 34538 | Coefficients of a polynomi... |
| plymulx 34539 | Coefficients of a polynomi... |
| plyrecld 34540 | Closure of a polynomial wi... |
| signsplypnf 34541 | The quotient of a polynomi... |
| signsply0 34542 | Lemma for the rule of sign... |
| signspval 34543 | The value of the skipping ... |
| signsw0glem 34544 | Neutral element property o... |
| signswbase 34545 | The base of ` W ` is the u... |
| signswplusg 34546 | The operation of ` W ` . ... |
| signsw0g 34547 | The neutral element of ` W... |
| signswmnd 34548 | ` W ` is a monoid structur... |
| signswrid 34549 | The zero-skipping operatio... |
| signswlid 34550 | The zero-skipping operatio... |
| signswn0 34551 | The zero-skipping operatio... |
| signswch 34552 | The zero-skipping operatio... |
| signslema 34553 | Computational part of ~~? ... |
| signstfv 34554 | Value of the zero-skipping... |
| signstfval 34555 | Value of the zero-skipping... |
| signstcl 34556 | Closure of the zero skippi... |
| signstf 34557 | The zero skipping sign wor... |
| signstlen 34558 | Length of the zero skippin... |
| signstf0 34559 | Sign of a single letter wo... |
| signstfvn 34560 | Zero-skipping sign in a wo... |
| signsvtn0 34561 | If the last letter is nonz... |
| signstfvp 34562 | Zero-skipping sign in a wo... |
| signstfvneq0 34563 | In case the first letter i... |
| signstfvcl 34564 | Closure of the zero skippi... |
| signstfvc 34565 | Zero-skipping sign in a wo... |
| signstres 34566 | Restriction of a zero skip... |
| signstfveq0a 34567 | Lemma for ~ signstfveq0 . ... |
| signstfveq0 34568 | In case the last letter is... |
| signsvvfval 34569 | The value of ` V ` , which... |
| signsvvf 34570 | ` V ` is a function. (Con... |
| signsvf0 34571 | There is no change of sign... |
| signsvf1 34572 | In a single-letter word, w... |
| signsvfn 34573 | Number of changes in a wor... |
| signsvtp 34574 | Adding a letter of the sam... |
| signsvtn 34575 | Adding a letter of a diffe... |
| signsvfpn 34576 | Adding a letter of the sam... |
| signsvfnn 34577 | Adding a letter of a diffe... |
| signlem0 34578 | Adding a zero as the highe... |
| signshf 34579 | ` H ` , corresponding to t... |
| signshwrd 34580 | ` H ` , corresponding to t... |
| signshlen 34581 | Length of ` H ` , correspo... |
| signshnz 34582 | ` H ` is not the empty wor... |
| iblidicc 34583 | The identity function is i... |
| rpsqrtcn 34584 | Continuity of the real pos... |
| divsqrtid 34585 | A real number divided by i... |
| cxpcncf1 34586 | The power function on comp... |
| efmul2picn 34587 | Multiplying by ` ( _i x. (... |
| fct2relem 34588 | Lemma for ~ ftc2re . (Con... |
| ftc2re 34589 | The Fundamental Theorem of... |
| fdvposlt 34590 | Functions with a positive ... |
| fdvneggt 34591 | Functions with a negative ... |
| fdvposle 34592 | Functions with a nonnegati... |
| fdvnegge 34593 | Functions with a nonpositi... |
| prodfzo03 34594 | A product of three factors... |
| actfunsnf1o 34595 | The action ` F ` of extend... |
| actfunsnrndisj 34596 | The action ` F ` of extend... |
| itgexpif 34597 | The basis for the circle m... |
| fsum2dsub 34598 | Lemma for ~ breprexp - Re-... |
| reprval 34601 | Value of the representatio... |
| repr0 34602 | There is exactly one repre... |
| reprf 34603 | Members of the representat... |
| reprsum 34604 | Sums of values of the memb... |
| reprle 34605 | Upper bound to the terms i... |
| reprsuc 34606 | Express the representation... |
| reprfi 34607 | Bounded representations ar... |
| reprss 34608 | Representations with terms... |
| reprinrn 34609 | Representations with term ... |
| reprlt 34610 | There are no representatio... |
| hashreprin 34611 | Express a sum of represent... |
| reprgt 34612 | There are no representatio... |
| reprinfz1 34613 | For the representation of ... |
| reprfi2 34614 | Corollary of ~ reprinfz1 .... |
| reprfz1 34615 | Corollary of ~ reprinfz1 .... |
| hashrepr 34616 | Develop the number of repr... |
| reprpmtf1o 34617 | Transposing ` 0 ` and ` X ... |
| reprdifc 34618 | Express the representation... |
| chpvalz 34619 | Value of the second Chebys... |
| chtvalz 34620 | Value of the Chebyshev fun... |
| breprexplema 34621 | Lemma for ~ breprexp (indu... |
| breprexplemb 34622 | Lemma for ~ breprexp (clos... |
| breprexplemc 34623 | Lemma for ~ breprexp (indu... |
| breprexp 34624 | Express the ` S ` th power... |
| breprexpnat 34625 | Express the ` S ` th power... |
| vtsval 34628 | Value of the Vinogradov tr... |
| vtscl 34629 | Closure of the Vinogradov ... |
| vtsprod 34630 | Express the Vinogradov tri... |
| circlemeth 34631 | The Hardy, Littlewood and ... |
| circlemethnat 34632 | The Hardy, Littlewood and ... |
| circlevma 34633 | The Circle Method, where t... |
| circlemethhgt 34634 | The circle method, where t... |
| hgt750lemc 34638 | An upper bound to the summ... |
| hgt750lemd 34639 | An upper bound to the summ... |
| hgt749d 34640 | A deduction version of ~ a... |
| logdivsqrle 34641 | Conditions for ` ( ( log `... |
| hgt750lem 34642 | Lemma for ~ tgoldbachgtd .... |
| hgt750lem2 34643 | Decimal multiplication gal... |
| hgt750lemf 34644 | Lemma for the statement 7.... |
| hgt750lemg 34645 | Lemma for the statement 7.... |
| oddprm2 34646 | Two ways to write the set ... |
| hgt750lemb 34647 | An upper bound on the cont... |
| hgt750lema 34648 | An upper bound on the cont... |
| hgt750leme 34649 | An upper bound on the cont... |
| tgoldbachgnn 34650 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtde 34651 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtda 34652 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtd 34653 | Odd integers greater than ... |
| tgoldbachgt 34654 | Odd integers greater than ... |
| istrkg2d 34657 | Property of fulfilling dim... |
| axtglowdim2ALTV 34658 | Alternate version of ~ axt... |
| axtgupdim2ALTV 34659 | Alternate version of ~ axt... |
| afsval 34662 | Value of the AFS relation ... |
| brafs 34663 | Binary relation form of th... |
| tg5segofs 34664 | Rephrase ~ axtg5seg using ... |
| lpadval 34667 | Value of the ` leftpad ` f... |
| lpadlem1 34668 | Lemma for the ` leftpad ` ... |
| lpadlem3 34669 | Lemma for ~ lpadlen1 . (C... |
| lpadlen1 34670 | Length of a left-padded wo... |
| lpadlem2 34671 | Lemma for the ` leftpad ` ... |
| lpadlen2 34672 | Length of a left-padded wo... |
| lpadmax 34673 | Length of a left-padded wo... |
| lpadleft 34674 | The contents of prefix of ... |
| lpadright 34675 | The suffix of a left-padde... |
| bnj170 34688 | ` /\ ` -manipulation. (Co... |
| bnj240 34689 | ` /\ ` -manipulation. (Co... |
| bnj248 34690 | ` /\ ` -manipulation. (Co... |
| bnj250 34691 | ` /\ ` -manipulation. (Co... |
| bnj251 34692 | ` /\ ` -manipulation. (Co... |
| bnj252 34693 | ` /\ ` -manipulation. (Co... |
| bnj253 34694 | ` /\ ` -manipulation. (Co... |
| bnj255 34695 | ` /\ ` -manipulation. (Co... |
| bnj256 34696 | ` /\ ` -manipulation. (Co... |
| bnj257 34697 | ` /\ ` -manipulation. (Co... |
| bnj258 34698 | ` /\ ` -manipulation. (Co... |
| bnj268 34699 | ` /\ ` -manipulation. (Co... |
| bnj290 34700 | ` /\ ` -manipulation. (Co... |
| bnj291 34701 | ` /\ ` -manipulation. (Co... |
| bnj312 34702 | ` /\ ` -manipulation. (Co... |
| bnj334 34703 | ` /\ ` -manipulation. (Co... |
| bnj345 34704 | ` /\ ` -manipulation. (Co... |
| bnj422 34705 | ` /\ ` -manipulation. (Co... |
| bnj432 34706 | ` /\ ` -manipulation. (Co... |
| bnj446 34707 | ` /\ ` -manipulation. (Co... |
| bnj23 34708 | First-order logic and set ... |
| bnj31 34709 | First-order logic and set ... |
| bnj62 34710 | First-order logic and set ... |
| bnj89 34711 | First-order logic and set ... |
| bnj90 34712 | First-order logic and set ... |
| bnj101 34713 | First-order logic and set ... |
| bnj105 34714 | First-order logic and set ... |
| bnj115 34715 | First-order logic and set ... |
| bnj132 34716 | First-order logic and set ... |
| bnj133 34717 | First-order logic and set ... |
| bnj156 34718 | First-order logic and set ... |
| bnj158 34719 | First-order logic and set ... |
| bnj168 34720 | First-order logic and set ... |
| bnj206 34721 | First-order logic and set ... |
| bnj216 34722 | First-order logic and set ... |
| bnj219 34723 | First-order logic and set ... |
| bnj226 34724 | First-order logic and set ... |
| bnj228 34725 | First-order logic and set ... |
| bnj519 34726 | First-order logic and set ... |
| bnj524 34727 | First-order logic and set ... |
| bnj525 34728 | First-order logic and set ... |
| bnj534 34729 | First-order logic and set ... |
| bnj538 34730 | First-order logic and set ... |
| bnj529 34731 | First-order logic and set ... |
| bnj551 34732 | First-order logic and set ... |
| bnj563 34733 | First-order logic and set ... |
| bnj564 34734 | First-order logic and set ... |
| bnj593 34735 | First-order logic and set ... |
| bnj596 34736 | First-order logic and set ... |
| bnj610 34737 | Pass from equality ( ` x =... |
| bnj642 34738 | ` /\ ` -manipulation. (Co... |
| bnj643 34739 | ` /\ ` -manipulation. (Co... |
| bnj645 34740 | ` /\ ` -manipulation. (Co... |
| bnj658 34741 | ` /\ ` -manipulation. (Co... |
| bnj667 34742 | ` /\ ` -manipulation. (Co... |
| bnj705 34743 | ` /\ ` -manipulation. (Co... |
| bnj706 34744 | ` /\ ` -manipulation. (Co... |
| bnj707 34745 | ` /\ ` -manipulation. (Co... |
| bnj708 34746 | ` /\ ` -manipulation. (Co... |
| bnj721 34747 | ` /\ ` -manipulation. (Co... |
| bnj832 34748 | ` /\ ` -manipulation. (Co... |
| bnj835 34749 | ` /\ ` -manipulation. (Co... |
| bnj836 34750 | ` /\ ` -manipulation. (Co... |
| bnj837 34751 | ` /\ ` -manipulation. (Co... |
| bnj769 34752 | ` /\ ` -manipulation. (Co... |
| bnj770 34753 | ` /\ ` -manipulation. (Co... |
| bnj771 34754 | ` /\ ` -manipulation. (Co... |
| bnj887 34755 | ` /\ ` -manipulation. (Co... |
| bnj918 34756 | First-order logic and set ... |
| bnj919 34757 | First-order logic and set ... |
| bnj923 34758 | First-order logic and set ... |
| bnj927 34759 | First-order logic and set ... |
| bnj931 34760 | First-order logic and set ... |
| bnj937 34761 | First-order logic and set ... |
| bnj941 34762 | First-order logic and set ... |
| bnj945 34763 | Technical lemma for ~ bnj6... |
| bnj946 34764 | First-order logic and set ... |
| bnj951 34765 | ` /\ ` -manipulation. (Co... |
| bnj956 34766 | First-order logic and set ... |
| bnj976 34767 | First-order logic and set ... |
| bnj982 34768 | First-order logic and set ... |
| bnj1019 34769 | First-order logic and set ... |
| bnj1023 34770 | First-order logic and set ... |
| bnj1095 34771 | First-order logic and set ... |
| bnj1096 34772 | First-order logic and set ... |
| bnj1098 34773 | First-order logic and set ... |
| bnj1101 34774 | First-order logic and set ... |
| bnj1113 34775 | First-order logic and set ... |
| bnj1109 34776 | First-order logic and set ... |
| bnj1131 34777 | First-order logic and set ... |
| bnj1138 34778 | First-order logic and set ... |
| bnj1142 34779 | First-order logic and set ... |
| bnj1143 34780 | First-order logic and set ... |
| bnj1146 34781 | First-order logic and set ... |
| bnj1149 34782 | First-order logic and set ... |
| bnj1185 34783 | First-order logic and set ... |
| bnj1196 34784 | First-order logic and set ... |
| bnj1198 34785 | First-order logic and set ... |
| bnj1209 34786 | First-order logic and set ... |
| bnj1211 34787 | First-order logic and set ... |
| bnj1213 34788 | First-order logic and set ... |
| bnj1212 34789 | First-order logic and set ... |
| bnj1219 34790 | First-order logic and set ... |
| bnj1224 34791 | First-order logic and set ... |
| bnj1230 34792 | First-order logic and set ... |
| bnj1232 34793 | First-order logic and set ... |
| bnj1235 34794 | First-order logic and set ... |
| bnj1239 34795 | First-order logic and set ... |
| bnj1238 34796 | First-order logic and set ... |
| bnj1241 34797 | First-order logic and set ... |
| bnj1247 34798 | First-order logic and set ... |
| bnj1254 34799 | First-order logic and set ... |
| bnj1262 34800 | First-order logic and set ... |
| bnj1266 34801 | First-order logic and set ... |
| bnj1265 34802 | First-order logic and set ... |
| bnj1275 34803 | First-order logic and set ... |
| bnj1276 34804 | First-order logic and set ... |
| bnj1292 34805 | First-order logic and set ... |
| bnj1293 34806 | First-order logic and set ... |
| bnj1294 34807 | First-order logic and set ... |
| bnj1299 34808 | First-order logic and set ... |
| bnj1304 34809 | First-order logic and set ... |
| bnj1316 34810 | First-order logic and set ... |
| bnj1317 34811 | First-order logic and set ... |
| bnj1322 34812 | First-order logic and set ... |
| bnj1340 34813 | First-order logic and set ... |
| bnj1345 34814 | First-order logic and set ... |
| bnj1350 34815 | First-order logic and set ... |
| bnj1351 34816 | First-order logic and set ... |
| bnj1352 34817 | First-order logic and set ... |
| bnj1361 34818 | First-order logic and set ... |
| bnj1366 34819 | First-order logic and set ... |
| bnj1379 34820 | First-order logic and set ... |
| bnj1383 34821 | First-order logic and set ... |
| bnj1385 34822 | First-order logic and set ... |
| bnj1386 34823 | First-order logic and set ... |
| bnj1397 34824 | First-order logic and set ... |
| bnj1400 34825 | First-order logic and set ... |
| bnj1405 34826 | First-order logic and set ... |
| bnj1422 34827 | First-order logic and set ... |
| bnj1424 34828 | First-order logic and set ... |
| bnj1436 34829 | First-order logic and set ... |
| bnj1441 34830 | First-order logic and set ... |
| bnj1441g 34831 | First-order logic and set ... |
| bnj1454 34832 | First-order logic and set ... |
| bnj1459 34833 | First-order logic and set ... |
| bnj1464 34834 | Conversion of implicit sub... |
| bnj1465 34835 | First-order logic and set ... |
| bnj1468 34836 | Conversion of implicit sub... |
| bnj1476 34837 | First-order logic and set ... |
| bnj1502 34838 | First-order logic and set ... |
| bnj1503 34839 | First-order logic and set ... |
| bnj1517 34840 | First-order logic and set ... |
| bnj1521 34841 | First-order logic and set ... |
| bnj1533 34842 | First-order logic and set ... |
| bnj1534 34843 | First-order logic and set ... |
| bnj1536 34844 | First-order logic and set ... |
| bnj1538 34845 | First-order logic and set ... |
| bnj1541 34846 | First-order logic and set ... |
| bnj1542 34847 | First-order logic and set ... |
| bnj110 34848 | Well-founded induction res... |
| bnj157 34849 | Well-founded induction res... |
| bnj66 34850 | Technical lemma for ~ bnj6... |
| bnj91 34851 | First-order logic and set ... |
| bnj92 34852 | First-order logic and set ... |
| bnj93 34853 | Technical lemma for ~ bnj9... |
| bnj95 34854 | Technical lemma for ~ bnj1... |
| bnj96 34855 | Technical lemma for ~ bnj1... |
| bnj97 34856 | Technical lemma for ~ bnj1... |
| bnj98 34857 | Technical lemma for ~ bnj1... |
| bnj106 34858 | First-order logic and set ... |
| bnj118 34859 | First-order logic and set ... |
| bnj121 34860 | First-order logic and set ... |
| bnj124 34861 | Technical lemma for ~ bnj1... |
| bnj125 34862 | Technical lemma for ~ bnj1... |
| bnj126 34863 | Technical lemma for ~ bnj1... |
| bnj130 34864 | Technical lemma for ~ bnj1... |
| bnj149 34865 | Technical lemma for ~ bnj1... |
| bnj150 34866 | Technical lemma for ~ bnj1... |
| bnj151 34867 | Technical lemma for ~ bnj1... |
| bnj154 34868 | Technical lemma for ~ bnj1... |
| bnj155 34869 | Technical lemma for ~ bnj1... |
| bnj153 34870 | Technical lemma for ~ bnj8... |
| bnj207 34871 | Technical lemma for ~ bnj8... |
| bnj213 34872 | First-order logic and set ... |
| bnj222 34873 | Technical lemma for ~ bnj2... |
| bnj229 34874 | Technical lemma for ~ bnj5... |
| bnj517 34875 | Technical lemma for ~ bnj5... |
| bnj518 34876 | Technical lemma for ~ bnj8... |
| bnj523 34877 | Technical lemma for ~ bnj8... |
| bnj526 34878 | Technical lemma for ~ bnj8... |
| bnj528 34879 | Technical lemma for ~ bnj8... |
| bnj535 34880 | Technical lemma for ~ bnj8... |
| bnj539 34881 | Technical lemma for ~ bnj8... |
| bnj540 34882 | Technical lemma for ~ bnj8... |
| bnj543 34883 | Technical lemma for ~ bnj8... |
| bnj544 34884 | Technical lemma for ~ bnj8... |
| bnj545 34885 | Technical lemma for ~ bnj8... |
| bnj546 34886 | Technical lemma for ~ bnj8... |
| bnj548 34887 | Technical lemma for ~ bnj8... |
| bnj553 34888 | Technical lemma for ~ bnj8... |
| bnj554 34889 | Technical lemma for ~ bnj8... |
| bnj556 34890 | Technical lemma for ~ bnj8... |
| bnj557 34891 | Technical lemma for ~ bnj8... |
| bnj558 34892 | Technical lemma for ~ bnj8... |
| bnj561 34893 | Technical lemma for ~ bnj8... |
| bnj562 34894 | Technical lemma for ~ bnj8... |
| bnj570 34895 | Technical lemma for ~ bnj8... |
| bnj571 34896 | Technical lemma for ~ bnj8... |
| bnj605 34897 | Technical lemma. This lem... |
| bnj581 34898 | Technical lemma for ~ bnj5... |
| bnj589 34899 | Technical lemma for ~ bnj8... |
| bnj590 34900 | Technical lemma for ~ bnj8... |
| bnj591 34901 | Technical lemma for ~ bnj8... |
| bnj594 34902 | Technical lemma for ~ bnj8... |
| bnj580 34903 | Technical lemma for ~ bnj5... |
| bnj579 34904 | Technical lemma for ~ bnj8... |
| bnj602 34905 | Equality theorem for the `... |
| bnj607 34906 | Technical lemma for ~ bnj8... |
| bnj609 34907 | Technical lemma for ~ bnj8... |
| bnj611 34908 | Technical lemma for ~ bnj8... |
| bnj600 34909 | Technical lemma for ~ bnj8... |
| bnj601 34910 | Technical lemma for ~ bnj8... |
| bnj852 34911 | Technical lemma for ~ bnj6... |
| bnj864 34912 | Technical lemma for ~ bnj6... |
| bnj865 34913 | Technical lemma for ~ bnj6... |
| bnj873 34914 | Technical lemma for ~ bnj6... |
| bnj849 34915 | Technical lemma for ~ bnj6... |
| bnj882 34916 | Definition (using hypothes... |
| bnj18eq1 34917 | Equality theorem for trans... |
| bnj893 34918 | Property of ` _trCl ` . U... |
| bnj900 34919 | Technical lemma for ~ bnj6... |
| bnj906 34920 | Property of ` _trCl ` . (... |
| bnj908 34921 | Technical lemma for ~ bnj6... |
| bnj911 34922 | Technical lemma for ~ bnj6... |
| bnj916 34923 | Technical lemma for ~ bnj6... |
| bnj917 34924 | Technical lemma for ~ bnj6... |
| bnj934 34925 | Technical lemma for ~ bnj6... |
| bnj929 34926 | Technical lemma for ~ bnj6... |
| bnj938 34927 | Technical lemma for ~ bnj6... |
| bnj944 34928 | Technical lemma for ~ bnj6... |
| bnj953 34929 | Technical lemma for ~ bnj6... |
| bnj958 34930 | Technical lemma for ~ bnj6... |
| bnj1000 34931 | Technical lemma for ~ bnj8... |
| bnj965 34932 | Technical lemma for ~ bnj8... |
| bnj964 34933 | Technical lemma for ~ bnj6... |
| bnj966 34934 | Technical lemma for ~ bnj6... |
| bnj967 34935 | Technical lemma for ~ bnj6... |
| bnj969 34936 | Technical lemma for ~ bnj6... |
| bnj970 34937 | Technical lemma for ~ bnj6... |
| bnj910 34938 | Technical lemma for ~ bnj6... |
| bnj978 34939 | Technical lemma for ~ bnj6... |
| bnj981 34940 | Technical lemma for ~ bnj6... |
| bnj983 34941 | Technical lemma for ~ bnj6... |
| bnj984 34942 | Technical lemma for ~ bnj6... |
| bnj985v 34943 | Version of ~ bnj985 with a... |
| bnj985 34944 | Technical lemma for ~ bnj6... |
| bnj986 34945 | Technical lemma for ~ bnj6... |
| bnj996 34946 | Technical lemma for ~ bnj6... |
| bnj998 34947 | Technical lemma for ~ bnj6... |
| bnj999 34948 | Technical lemma for ~ bnj6... |
| bnj1001 34949 | Technical lemma for ~ bnj6... |
| bnj1006 34950 | Technical lemma for ~ bnj6... |
| bnj1014 34951 | Technical lemma for ~ bnj6... |
| bnj1015 34952 | Technical lemma for ~ bnj6... |
| bnj1018g 34953 | Version of ~ bnj1018 with ... |
| bnj1018 34954 | Technical lemma for ~ bnj6... |
| bnj1020 34955 | Technical lemma for ~ bnj6... |
| bnj1021 34956 | Technical lemma for ~ bnj6... |
| bnj907 34957 | Technical lemma for ~ bnj6... |
| bnj1029 34958 | Property of ` _trCl ` . (... |
| bnj1033 34959 | Technical lemma for ~ bnj6... |
| bnj1034 34960 | Technical lemma for ~ bnj6... |
| bnj1039 34961 | Technical lemma for ~ bnj6... |
| bnj1040 34962 | Technical lemma for ~ bnj6... |
| bnj1047 34963 | Technical lemma for ~ bnj6... |
| bnj1049 34964 | Technical lemma for ~ bnj6... |
| bnj1052 34965 | Technical lemma for ~ bnj6... |
| bnj1053 34966 | Technical lemma for ~ bnj6... |
| bnj1071 34967 | Technical lemma for ~ bnj6... |
| bnj1083 34968 | Technical lemma for ~ bnj6... |
| bnj1090 34969 | Technical lemma for ~ bnj6... |
| bnj1093 34970 | Technical lemma for ~ bnj6... |
| bnj1097 34971 | Technical lemma for ~ bnj6... |
| bnj1110 34972 | Technical lemma for ~ bnj6... |
| bnj1112 34973 | Technical lemma for ~ bnj6... |
| bnj1118 34974 | Technical lemma for ~ bnj6... |
| bnj1121 34975 | Technical lemma for ~ bnj6... |
| bnj1123 34976 | Technical lemma for ~ bnj6... |
| bnj1030 34977 | Technical lemma for ~ bnj6... |
| bnj1124 34978 | Property of ` _trCl ` . (... |
| bnj1133 34979 | Technical lemma for ~ bnj6... |
| bnj1128 34980 | Technical lemma for ~ bnj6... |
| bnj1127 34981 | Property of ` _trCl ` . (... |
| bnj1125 34982 | Property of ` _trCl ` . (... |
| bnj1145 34983 | Technical lemma for ~ bnj6... |
| bnj1147 34984 | Property of ` _trCl ` . (... |
| bnj1137 34985 | Property of ` _trCl ` . (... |
| bnj1148 34986 | Property of ` _pred ` . (... |
| bnj1136 34987 | Technical lemma for ~ bnj6... |
| bnj1152 34988 | Technical lemma for ~ bnj6... |
| bnj1154 34989 | Property of ` Fr ` . (Con... |
| bnj1171 34990 | Technical lemma for ~ bnj6... |
| bnj1172 34991 | Technical lemma for ~ bnj6... |
| bnj1173 34992 | Technical lemma for ~ bnj6... |
| bnj1174 34993 | Technical lemma for ~ bnj6... |
| bnj1175 34994 | Technical lemma for ~ bnj6... |
| bnj1176 34995 | Technical lemma for ~ bnj6... |
| bnj1177 34996 | Technical lemma for ~ bnj6... |
| bnj1186 34997 | Technical lemma for ~ bnj6... |
| bnj1190 34998 | Technical lemma for ~ bnj6... |
| bnj1189 34999 | Technical lemma for ~ bnj6... |
| bnj69 35000 | Existence of a minimal ele... |
| bnj1228 35001 | Existence of a minimal ele... |
| bnj1204 35002 | Well-founded induction. T... |
| bnj1234 35003 | Technical lemma for ~ bnj6... |
| bnj1245 35004 | Technical lemma for ~ bnj6... |
| bnj1256 35005 | Technical lemma for ~ bnj6... |
| bnj1259 35006 | Technical lemma for ~ bnj6... |
| bnj1253 35007 | Technical lemma for ~ bnj6... |
| bnj1279 35008 | Technical lemma for ~ bnj6... |
| bnj1286 35009 | Technical lemma for ~ bnj6... |
| bnj1280 35010 | Technical lemma for ~ bnj6... |
| bnj1296 35011 | Technical lemma for ~ bnj6... |
| bnj1309 35012 | Technical lemma for ~ bnj6... |
| bnj1307 35013 | Technical lemma for ~ bnj6... |
| bnj1311 35014 | Technical lemma for ~ bnj6... |
| bnj1318 35015 | Technical lemma for ~ bnj6... |
| bnj1326 35016 | Technical lemma for ~ bnj6... |
| bnj1321 35017 | Technical lemma for ~ bnj6... |
| bnj1364 35018 | Property of ` _FrSe ` . (... |
| bnj1371 35019 | Technical lemma for ~ bnj6... |
| bnj1373 35020 | Technical lemma for ~ bnj6... |
| bnj1374 35021 | Technical lemma for ~ bnj6... |
| bnj1384 35022 | Technical lemma for ~ bnj6... |
| bnj1388 35023 | Technical lemma for ~ bnj6... |
| bnj1398 35024 | Technical lemma for ~ bnj6... |
| bnj1413 35025 | Property of ` _trCl ` . (... |
| bnj1408 35026 | Technical lemma for ~ bnj1... |
| bnj1414 35027 | Property of ` _trCl ` . (... |
| bnj1415 35028 | Technical lemma for ~ bnj6... |
| bnj1416 35029 | Technical lemma for ~ bnj6... |
| bnj1418 35030 | Property of ` _pred ` . (... |
| bnj1417 35031 | Technical lemma for ~ bnj6... |
| bnj1421 35032 | Technical lemma for ~ bnj6... |
| bnj1444 35033 | Technical lemma for ~ bnj6... |
| bnj1445 35034 | Technical lemma for ~ bnj6... |
| bnj1446 35035 | Technical lemma for ~ bnj6... |
| bnj1447 35036 | Technical lemma for ~ bnj6... |
| bnj1448 35037 | Technical lemma for ~ bnj6... |
| bnj1449 35038 | Technical lemma for ~ bnj6... |
| bnj1442 35039 | Technical lemma for ~ bnj6... |
| bnj1450 35040 | Technical lemma for ~ bnj6... |
| bnj1423 35041 | Technical lemma for ~ bnj6... |
| bnj1452 35042 | Technical lemma for ~ bnj6... |
| bnj1466 35043 | Technical lemma for ~ bnj6... |
| bnj1467 35044 | Technical lemma for ~ bnj6... |
| bnj1463 35045 | Technical lemma for ~ bnj6... |
| bnj1489 35046 | Technical lemma for ~ bnj6... |
| bnj1491 35047 | Technical lemma for ~ bnj6... |
| bnj1312 35048 | Technical lemma for ~ bnj6... |
| bnj1493 35049 | Technical lemma for ~ bnj6... |
| bnj1497 35050 | Technical lemma for ~ bnj6... |
| bnj1498 35051 | Technical lemma for ~ bnj6... |
| bnj60 35052 | Well-founded recursion, pa... |
| bnj1514 35053 | Technical lemma for ~ bnj1... |
| bnj1518 35054 | Technical lemma for ~ bnj1... |
| bnj1519 35055 | Technical lemma for ~ bnj1... |
| bnj1520 35056 | Technical lemma for ~ bnj1... |
| bnj1501 35057 | Technical lemma for ~ bnj1... |
| bnj1500 35058 | Well-founded recursion, pa... |
| bnj1525 35059 | Technical lemma for ~ bnj1... |
| bnj1529 35060 | Technical lemma for ~ bnj1... |
| bnj1523 35061 | Technical lemma for ~ bnj1... |
| bnj1522 35062 | Well-founded recursion, pa... |
| nfan1c 35063 | Variant of ~ nfan and comm... |
| cbvex1v 35064 | Rule used to change bound ... |
| dvelimalcased 35065 | Eliminate a disjoint varia... |
| dvelimalcasei 35066 | Eliminate a disjoint varia... |
| dvelimexcased 35067 | Eliminate a disjoint varia... |
| dvelimexcasei 35068 | Eliminate a disjoint varia... |
| exdifsn 35069 | There exists an element in... |
| srcmpltd 35070 | If a statement is true for... |
| prsrcmpltd 35071 | If a statement is true for... |
| axsepg2 35072 | A generalization of ~ ax-s... |
| axsepg2ALT 35073 | Alternate proof of ~ axsep... |
| dff15 35074 | A one-to-one function in t... |
| f1resveqaeq 35075 | If a function restricted t... |
| f1resrcmplf1dlem 35076 | Lemma for ~ f1resrcmplf1d ... |
| f1resrcmplf1d 35077 | If a function's restrictio... |
| funen1cnv 35078 | If a function is equinumer... |
| fnrelpredd 35079 | A function that preserves ... |
| cardpred 35080 | The cardinality function p... |
| nummin 35081 | Every nonempty class of nu... |
| axnulg 35082 | A generalization of ~ ax-n... |
| axnulALT2 35083 | Alternate proof of ~ axnul... |
| prcinf 35084 | Any proper class is litera... |
| fineqvrep 35085 | If the Axiom of Infinity i... |
| fineqvpow 35086 | If the Axiom of Infinity i... |
| fineqvac 35087 | If the Axiom of Infinity i... |
| fineqvacALT 35088 | Shorter proof of ~ fineqva... |
| gblacfnacd 35089 | If ` G ` is a global choic... |
| onvf1odlem1 35090 | Lemma for ~ onvf1od . (Co... |
| onvf1odlem2 35091 | Lemma for ~ onvf1od . (Co... |
| onvf1odlem3 35092 | Lemma for ~ onvf1od . The... |
| onvf1odlem4 35093 | Lemma for ~ onvf1od . If ... |
| onvf1od 35094 | If ` G ` is a global choic... |
| vonf1owev 35095 | If ` F ` is a bijection fr... |
| wevgblacfn 35096 | If ` R ` is a well-orderin... |
| zltp1ne 35097 | Integer ordering relation.... |
| nnltp1ne 35098 | Positive integer ordering ... |
| nn0ltp1ne 35099 | Nonnegative integer orderi... |
| 0nn0m1nnn0 35100 | A number is zero if and on... |
| f1resfz0f1d 35101 | If a function with a seque... |
| fisshasheq 35102 | A finite set is equal to i... |
| revpfxsfxrev 35103 | The reverse of a prefix of... |
| swrdrevpfx 35104 | A subword expressed in ter... |
| lfuhgr 35105 | A hypergraph is loop-free ... |
| lfuhgr2 35106 | A hypergraph is loop-free ... |
| lfuhgr3 35107 | A hypergraph is loop-free ... |
| cplgredgex 35108 | Any two (distinct) vertice... |
| cusgredgex 35109 | Any two (distinct) vertice... |
| cusgredgex2 35110 | Any two distinct vertices ... |
| pfxwlk 35111 | A prefix of a walk is a wa... |
| revwlk 35112 | The reverse of a walk is a... |
| revwlkb 35113 | Two words represent a walk... |
| swrdwlk 35114 | Two matching subwords of a... |
| pthhashvtx 35115 | A graph containing a path ... |
| spthcycl 35116 | A walk is a trivial path i... |
| usgrgt2cycl 35117 | A non-trivial cycle in a s... |
| usgrcyclgt2v 35118 | A simple graph with a non-... |
| subgrwlk 35119 | If a walk exists in a subg... |
| subgrtrl 35120 | If a trail exists in a sub... |
| subgrpth 35121 | If a path exists in a subg... |
| subgrcycl 35122 | If a cycle exists in a sub... |
| cusgr3cyclex 35123 | Every complete simple grap... |
| loop1cycl 35124 | A hypergraph has a cycle o... |
| 2cycld 35125 | Construction of a 2-cycle ... |
| 2cycl2d 35126 | Construction of a 2-cycle ... |
| umgr2cycllem 35127 | Lemma for ~ umgr2cycl . (... |
| umgr2cycl 35128 | A multigraph with two dist... |
| dfacycgr1 35131 | An alternate definition of... |
| isacycgr 35132 | The property of being an a... |
| isacycgr1 35133 | The property of being an a... |
| acycgrcycl 35134 | Any cycle in an acyclic gr... |
| acycgr0v 35135 | A null graph (with no vert... |
| acycgr1v 35136 | A multigraph with one vert... |
| acycgr2v 35137 | A simple graph with two ve... |
| prclisacycgr 35138 | A proper class (representi... |
| acycgrislfgr 35139 | An acyclic hypergraph is a... |
| upgracycumgr 35140 | An acyclic pseudograph is ... |
| umgracycusgr 35141 | An acyclic multigraph is a... |
| upgracycusgr 35142 | An acyclic pseudograph is ... |
| cusgracyclt3v 35143 | A complete simple graph is... |
| pthacycspth 35144 | A path in an acyclic graph... |
| acycgrsubgr 35145 | The subgraph of an acyclic... |
| quartfull 35152 | The quartic equation, writ... |
| deranglem 35153 | Lemma for derangements. (... |
| derangval 35154 | Define the derangement fun... |
| derangf 35155 | The derangement number is ... |
| derang0 35156 | The derangement number of ... |
| derangsn 35157 | The derangement number of ... |
| derangenlem 35158 | One half of ~ derangen . ... |
| derangen 35159 | The derangement number is ... |
| subfacval 35160 | The subfactorial is define... |
| derangen2 35161 | Write the derangement numb... |
| subfacf 35162 | The subfactorial is a func... |
| subfaclefac 35163 | The subfactorial is less t... |
| subfac0 35164 | The subfactorial at zero. ... |
| subfac1 35165 | The subfactorial at one. ... |
| subfacp1lem1 35166 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem2a 35167 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem2b 35168 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem3 35169 | Lemma for ~ subfacp1 . In... |
| subfacp1lem4 35170 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem5 35171 | Lemma for ~ subfacp1 . In... |
| subfacp1lem6 35172 | Lemma for ~ subfacp1 . By... |
| subfacp1 35173 | A two-term recurrence for ... |
| subfacval2 35174 | A closed-form expression f... |
| subfaclim 35175 | The subfactorial converges... |
| subfacval3 35176 | Another closed form expres... |
| derangfmla 35177 | The derangements formula, ... |
| erdszelem1 35178 | Lemma for ~ erdsze . (Con... |
| erdszelem2 35179 | Lemma for ~ erdsze . (Con... |
| erdszelem3 35180 | Lemma for ~ erdsze . (Con... |
| erdszelem4 35181 | Lemma for ~ erdsze . (Con... |
| erdszelem5 35182 | Lemma for ~ erdsze . (Con... |
| erdszelem6 35183 | Lemma for ~ erdsze . (Con... |
| erdszelem7 35184 | Lemma for ~ erdsze . (Con... |
| erdszelem8 35185 | Lemma for ~ erdsze . (Con... |
| erdszelem9 35186 | Lemma for ~ erdsze . (Con... |
| erdszelem10 35187 | Lemma for ~ erdsze . (Con... |
| erdszelem11 35188 | Lemma for ~ erdsze . (Con... |
| erdsze 35189 | The Erdős-Szekeres th... |
| erdsze2lem1 35190 | Lemma for ~ erdsze2 . (Co... |
| erdsze2lem2 35191 | Lemma for ~ erdsze2 . (Co... |
| erdsze2 35192 | Generalize the statement o... |
| kur14lem1 35193 | Lemma for ~ kur14 . (Cont... |
| kur14lem2 35194 | Lemma for ~ kur14 . Write... |
| kur14lem3 35195 | Lemma for ~ kur14 . A clo... |
| kur14lem4 35196 | Lemma for ~ kur14 . Compl... |
| kur14lem5 35197 | Lemma for ~ kur14 . Closu... |
| kur14lem6 35198 | Lemma for ~ kur14 . If ` ... |
| kur14lem7 35199 | Lemma for ~ kur14 : main p... |
| kur14lem8 35200 | Lemma for ~ kur14 . Show ... |
| kur14lem9 35201 | Lemma for ~ kur14 . Since... |
| kur14lem10 35202 | Lemma for ~ kur14 . Disch... |
| kur14 35203 | Kuratowski's closure-compl... |
| ispconn 35210 | The property of being a pa... |
| pconncn 35211 | The property of being a pa... |
| pconntop 35212 | A simply connected space i... |
| issconn 35213 | The property of being a si... |
| sconnpconn 35214 | A simply connected space i... |
| sconntop 35215 | A simply connected space i... |
| sconnpht 35216 | A closed path in a simply ... |
| cnpconn 35217 | An image of a path-connect... |
| pconnconn 35218 | A path-connected space is ... |
| txpconn 35219 | The topological product of... |
| ptpconn 35220 | The topological product of... |
| indispconn 35221 | The indiscrete topology (o... |
| connpconn 35222 | A connected and locally pa... |
| qtoppconn 35223 | A quotient of a path-conne... |
| pconnpi1 35224 | All fundamental groups in ... |
| sconnpht2 35225 | Any two paths in a simply ... |
| sconnpi1 35226 | A path-connected topologic... |
| txsconnlem 35227 | Lemma for ~ txsconn . (Co... |
| txsconn 35228 | The topological product of... |
| cvxpconn 35229 | A convex subset of the com... |
| cvxsconn 35230 | A convex subset of the com... |
| blsconn 35231 | An open ball in the comple... |
| cnllysconn 35232 | The topology of the comple... |
| resconn 35233 | A subset of ` RR ` is simp... |
| ioosconn 35234 | An open interval is simply... |
| iccsconn 35235 | A closed interval is simpl... |
| retopsconn 35236 | The real numbers are simpl... |
| iccllysconn 35237 | A closed interval is local... |
| rellysconn 35238 | The real numbers are local... |
| iisconn 35239 | The unit interval is simpl... |
| iillysconn 35240 | The unit interval is local... |
| iinllyconn 35241 | The unit interval is local... |
| fncvm 35244 | Lemma for covering maps. ... |
| cvmscbv 35245 | Change bound variables in ... |
| iscvm 35246 | The property of being a co... |
| cvmtop1 35247 | Reverse closure for a cove... |
| cvmtop2 35248 | Reverse closure for a cove... |
| cvmcn 35249 | A covering map is a contin... |
| cvmcov 35250 | Property of a covering map... |
| cvmsrcl 35251 | Reverse closure for an eve... |
| cvmsi 35252 | One direction of ~ cvmsval... |
| cvmsval 35253 | Elementhood in the set ` S... |
| cvmsss 35254 | An even covering is a subs... |
| cvmsn0 35255 | An even covering is nonemp... |
| cvmsuni 35256 | An even covering of ` U ` ... |
| cvmsdisj 35257 | An even covering of ` U ` ... |
| cvmshmeo 35258 | Every element of an even c... |
| cvmsf1o 35259 | ` F ` , localized to an el... |
| cvmscld 35260 | The sets of an even coveri... |
| cvmsss2 35261 | An open subset of an evenl... |
| cvmcov2 35262 | The covering map property ... |
| cvmseu 35263 | Every element in ` U. T ` ... |
| cvmsiota 35264 | Identify the unique elemen... |
| cvmopnlem 35265 | Lemma for ~ cvmopn . (Con... |
| cvmfolem 35266 | Lemma for ~ cvmfo . (Cont... |
| cvmopn 35267 | A covering map is an open ... |
| cvmliftmolem1 35268 | Lemma for ~ cvmliftmo . (... |
| cvmliftmolem2 35269 | Lemma for ~ cvmliftmo . (... |
| cvmliftmoi 35270 | A lift of a continuous fun... |
| cvmliftmo 35271 | A lift of a continuous fun... |
| cvmliftlem1 35272 | Lemma for ~ cvmlift . In ... |
| cvmliftlem2 35273 | Lemma for ~ cvmlift . ` W ... |
| cvmliftlem3 35274 | Lemma for ~ cvmlift . Sin... |
| cvmliftlem4 35275 | Lemma for ~ cvmlift . The... |
| cvmliftlem5 35276 | Lemma for ~ cvmlift . Def... |
| cvmliftlem6 35277 | Lemma for ~ cvmlift . Ind... |
| cvmliftlem7 35278 | Lemma for ~ cvmlift . Pro... |
| cvmliftlem8 35279 | Lemma for ~ cvmlift . The... |
| cvmliftlem9 35280 | Lemma for ~ cvmlift . The... |
| cvmliftlem10 35281 | Lemma for ~ cvmlift . The... |
| cvmliftlem11 35282 | Lemma for ~ cvmlift . (Co... |
| cvmliftlem13 35283 | Lemma for ~ cvmlift . The... |
| cvmliftlem14 35284 | Lemma for ~ cvmlift . Put... |
| cvmliftlem15 35285 | Lemma for ~ cvmlift . Dis... |
| cvmlift 35286 | One of the important prope... |
| cvmfo 35287 | A covering map is an onto ... |
| cvmliftiota 35288 | Write out a function ` H `... |
| cvmlift2lem1 35289 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9a 35290 | Lemma for ~ cvmlift2 and ~... |
| cvmlift2lem2 35291 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem3 35292 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem4 35293 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem5 35294 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem6 35295 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem7 35296 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem8 35297 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9 35298 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem10 35299 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem11 35300 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem12 35301 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem13 35302 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2 35303 | A two-dimensional version ... |
| cvmliftphtlem 35304 | Lemma for ~ cvmliftpht . ... |
| cvmliftpht 35305 | If ` G ` and ` H ` are pat... |
| cvmlift3lem1 35306 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem2 35307 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem3 35308 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem4 35309 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem5 35310 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem6 35311 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem7 35312 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem8 35313 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem9 35314 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3 35315 | A general version of ~ cvm... |
| snmlff 35316 | The function ` F ` from ~ ... |
| snmlfval 35317 | The function ` F ` from ~ ... |
| snmlval 35318 | The property " ` A ` is si... |
| snmlflim 35319 | If ` A ` is simply normal,... |
| goel 35334 | A "Godel-set of membership... |
| goelel3xp 35335 | A "Godel-set of membership... |
| goeleq12bg 35336 | Two "Godel-set of membersh... |
| gonafv 35337 | The "Godel-set for the She... |
| goaleq12d 35338 | Equality of the "Godel-set... |
| gonanegoal 35339 | The Godel-set for the Shef... |
| satf 35340 | The satisfaction predicate... |
| satfsucom 35341 | The satisfaction predicate... |
| satfn 35342 | The satisfaction predicate... |
| satom 35343 | The satisfaction predicate... |
| satfvsucom 35344 | The satisfaction predicate... |
| satfv0 35345 | The value of the satisfact... |
| satfvsuclem1 35346 | Lemma 1 for ~ satfvsuc . ... |
| satfvsuclem2 35347 | Lemma 2 for ~ satfvsuc . ... |
| satfvsuc 35348 | The value of the satisfact... |
| satfv1lem 35349 | Lemma for ~ satfv1 . (Con... |
| satfv1 35350 | The value of the satisfact... |
| satfsschain 35351 | The binary relation of a s... |
| satfvsucsuc 35352 | The satisfaction predicate... |
| satfbrsuc 35353 | The binary relation of a s... |
| satfrel 35354 | The value of the satisfact... |
| satfdmlem 35355 | Lemma for ~ satfdm . (Con... |
| satfdm 35356 | The domain of the satisfac... |
| satfrnmapom 35357 | The range of the satisfact... |
| satfv0fun 35358 | The value of the satisfact... |
| satf0 35359 | The satisfaction predicate... |
| satf0sucom 35360 | The satisfaction predicate... |
| satf00 35361 | The value of the satisfact... |
| satf0suclem 35362 | Lemma for ~ satf0suc , ~ s... |
| satf0suc 35363 | The value of the satisfact... |
| satf0op 35364 | An element of a value of t... |
| satf0n0 35365 | The value of the satisfact... |
| sat1el2xp 35366 | The first component of an ... |
| fmlafv 35367 | The valid Godel formulas o... |
| fmla 35368 | The set of all valid Godel... |
| fmla0 35369 | The valid Godel formulas o... |
| fmla0xp 35370 | The valid Godel formulas o... |
| fmlasuc0 35371 | The valid Godel formulas o... |
| fmlafvel 35372 | A class is a valid Godel f... |
| fmlasuc 35373 | The valid Godel formulas o... |
| fmla1 35374 | The valid Godel formulas o... |
| isfmlasuc 35375 | The characterization of a ... |
| fmlasssuc 35376 | The Godel formulas of heig... |
| fmlaomn0 35377 | The empty set is not a God... |
| fmlan0 35378 | The empty set is not a God... |
| gonan0 35379 | The "Godel-set of NAND" is... |
| goaln0 35380 | The "Godel-set of universa... |
| gonarlem 35381 | Lemma for ~ gonar (inducti... |
| gonar 35382 | If the "Godel-set of NAND"... |
| goalrlem 35383 | Lemma for ~ goalr (inducti... |
| goalr 35384 | If the "Godel-set of unive... |
| fmla0disjsuc 35385 | The set of valid Godel for... |
| fmlasucdisj 35386 | The valid Godel formulas o... |
| satfdmfmla 35387 | The domain of the satisfac... |
| satffunlem 35388 | Lemma for ~ satffunlem1lem... |
| satffunlem1lem1 35389 | Lemma for ~ satffunlem1 . ... |
| satffunlem1lem2 35390 | Lemma 2 for ~ satffunlem1 ... |
| satffunlem2lem1 35391 | Lemma 1 for ~ satffunlem2 ... |
| dmopab3rexdif 35392 | The domain of an ordered p... |
| satffunlem2lem2 35393 | Lemma 2 for ~ satffunlem2 ... |
| satffunlem1 35394 | Lemma 1 for ~ satffun : in... |
| satffunlem2 35395 | Lemma 2 for ~ satffun : in... |
| satffun 35396 | The value of the satisfact... |
| satff 35397 | The satisfaction predicate... |
| satfun 35398 | The satisfaction predicate... |
| satfvel 35399 | An element of the value of... |
| satfv0fvfmla0 35400 | The value of the satisfact... |
| satefv 35401 | The simplified satisfactio... |
| sate0 35402 | The simplified satisfactio... |
| satef 35403 | The simplified satisfactio... |
| sate0fv0 35404 | A simplified satisfaction ... |
| satefvfmla0 35405 | The simplified satisfactio... |
| sategoelfvb 35406 | Characterization of a valu... |
| sategoelfv 35407 | Condition of a valuation `... |
| ex-sategoelel 35408 | Example of a valuation of ... |
| ex-sategoel 35409 | Instance of ~ sategoelfv f... |
| satfv1fvfmla1 35410 | The value of the satisfact... |
| 2goelgoanfmla1 35411 | Two Godel-sets of membersh... |
| satefvfmla1 35412 | The simplified satisfactio... |
| ex-sategoelelomsuc 35413 | Example of a valuation of ... |
| ex-sategoelel12 35414 | Example of a valuation of ... |
| prv 35415 | The "proves" relation on a... |
| elnanelprv 35416 | The wff ` ( A e. B -/\ B e... |
| prv0 35417 | Every wff encoded as ` U `... |
| prv1n 35418 | No wff encoded as a Godel-... |
| mvtval 35487 | The set of variable typeco... |
| mrexval 35488 | The set of "raw expression... |
| mexval 35489 | The set of expressions, wh... |
| mexval2 35490 | The set of expressions, wh... |
| mdvval 35491 | The set of disjoint variab... |
| mvrsval 35492 | The set of variables in an... |
| mvrsfpw 35493 | The set of variables in an... |
| mrsubffval 35494 | The substitution of some v... |
| mrsubfval 35495 | The substitution of some v... |
| mrsubval 35496 | The substitution of some v... |
| mrsubcv 35497 | The value of a substituted... |
| mrsubvr 35498 | The value of a substituted... |
| mrsubff 35499 | A substitution is a functi... |
| mrsubrn 35500 | Although it is defined for... |
| mrsubff1 35501 | When restricted to complet... |
| mrsubff1o 35502 | When restricted to complet... |
| mrsub0 35503 | The value of the substitut... |
| mrsubf 35504 | A substitution is a functi... |
| mrsubccat 35505 | Substitution distributes o... |
| mrsubcn 35506 | A substitution does not ch... |
| elmrsubrn 35507 | Characterization of the su... |
| mrsubco 35508 | The composition of two sub... |
| mrsubvrs 35509 | The set of variables in a ... |
| msubffval 35510 | A substitution applied to ... |
| msubfval 35511 | A substitution applied to ... |
| msubval 35512 | A substitution applied to ... |
| msubrsub 35513 | A substitution applied to ... |
| msubty 35514 | The type of a substituted ... |
| elmsubrn 35515 | Characterization of substi... |
| msubrn 35516 | Although it is defined for... |
| msubff 35517 | A substitution is a functi... |
| msubco 35518 | The composition of two sub... |
| msubf 35519 | A substitution is a functi... |
| mvhfval 35520 | Value of the function mapp... |
| mvhval 35521 | Value of the function mapp... |
| mpstval 35522 | A pre-statement is an orde... |
| elmpst 35523 | Property of being a pre-st... |
| msrfval 35524 | Value of the reduct of a p... |
| msrval 35525 | Value of the reduct of a p... |
| mpstssv 35526 | A pre-statement is an orde... |
| mpst123 35527 | Decompose a pre-statement ... |
| mpstrcl 35528 | The elements of a pre-stat... |
| msrf 35529 | The reduct of a pre-statem... |
| msrrcl 35530 | If ` X ` and ` Y ` have th... |
| mstaval 35531 | Value of the set of statem... |
| msrid 35532 | The reduct of a statement ... |
| msrfo 35533 | The reduct of a pre-statem... |
| mstapst 35534 | A statement is a pre-state... |
| elmsta 35535 | Property of being a statem... |
| ismfs 35536 | A formal system is a tuple... |
| mfsdisj 35537 | The constants and variable... |
| mtyf2 35538 | The type function maps var... |
| mtyf 35539 | The type function maps var... |
| mvtss 35540 | The set of variable typeco... |
| maxsta 35541 | An axiom is a statement. ... |
| mvtinf 35542 | Each variable typecode has... |
| msubff1 35543 | When restricted to complet... |
| msubff1o 35544 | When restricted to complet... |
| mvhf 35545 | The function mapping varia... |
| mvhf1 35546 | The function mapping varia... |
| msubvrs 35547 | The set of variables in a ... |
| mclsrcl 35548 | Reverse closure for the cl... |
| mclsssvlem 35549 | Lemma for ~ mclsssv . (Co... |
| mclsval 35550 | The function mapping varia... |
| mclsssv 35551 | The closure of a set of ex... |
| ssmclslem 35552 | Lemma for ~ ssmcls . (Con... |
| vhmcls 35553 | All variable hypotheses ar... |
| ssmcls 35554 | The original expressions a... |
| ss2mcls 35555 | The closure is monotonic u... |
| mclsax 35556 | The closure is closed unde... |
| mclsind 35557 | Induction theorem for clos... |
| mppspstlem 35558 | Lemma for ~ mppspst . (Co... |
| mppsval 35559 | Definition of a provable p... |
| elmpps 35560 | Definition of a provable p... |
| mppspst 35561 | A provable pre-statement i... |
| mthmval 35562 | A theorem is a pre-stateme... |
| elmthm 35563 | A theorem is a pre-stateme... |
| mthmi 35564 | A statement whose reduct i... |
| mthmsta 35565 | A theorem is a pre-stateme... |
| mppsthm 35566 | A provable pre-statement i... |
| mthmblem 35567 | Lemma for ~ mthmb . (Cont... |
| mthmb 35568 | If two statements have the... |
| mthmpps 35569 | Given a theorem, there is ... |
| mclsppslem 35570 | The closure is closed unde... |
| mclspps 35571 | The closure is closed unde... |
| rexxfr3d 35625 | Transfer existential quant... |
| rexxfr3dALT 35626 | Longer proof of ~ rexxfr3d... |
| rspssbasd 35627 | The span of a set of ring ... |
| ellcsrspsn 35628 | Membership in a left coset... |
| ply1divalg3 35629 | Uniqueness of polynomial r... |
| r1peuqusdeg1 35630 | Uniqueness of polynomial r... |
| problem1 35652 | Practice problem 1. Clues... |
| problem2 35653 | Practice problem 2. Clues... |
| problem3 35654 | Practice problem 3. Clues... |
| problem4 35655 | Practice problem 4. Clues... |
| problem5 35656 | Practice problem 5. Clues... |
| quad3 35657 | Variant of quadratic equat... |
| climuzcnv 35658 | Utility lemma to convert b... |
| sinccvglem 35659 | ` ( ( sin `` x ) / x ) ~~>... |
| sinccvg 35660 | ` ( ( sin `` x ) / x ) ~~>... |
| circum 35661 | The circumference of a cir... |
| elfzm12 35662 | Membership in a curtailed ... |
| nn0seqcvg 35663 | A strictly-decreasing nonn... |
| lediv2aALT 35664 | Division of both sides of ... |
| abs2sqlei 35665 | The absolute values of two... |
| abs2sqlti 35666 | The absolute values of two... |
| abs2sqle 35667 | The absolute values of two... |
| abs2sqlt 35668 | The absolute values of two... |
| abs2difi 35669 | Difference of absolute val... |
| abs2difabsi 35670 | Absolute value of differen... |
| 2thALT 35671 | Alternate proof of ~ 2th .... |
| orbi2iALT 35672 | Alternate proof of ~ orbi2... |
| pm3.48ALT 35673 | Alternate proof of ~ pm3.4... |
| 3jcadALT 35674 | Alternate proof of ~ 3jcad... |
| currybi 35675 | Biconditional version of C... |
| antnest 35676 | Suppose ` ph ` , ` ps ` ar... |
| antnestlaw3lem 35677 | Lemma for ~ antnestlaw3 . ... |
| antnestlaw1 35678 | A law of nested antecedent... |
| antnestlaw2 35679 | A law of nested antecedent... |
| antnestlaw3 35680 | A law of nested antecedent... |
| antnestALT 35681 | Alternative proof of ~ ant... |
| axextprim 35688 | ~ ax-ext without distinct ... |
| axrepprim 35689 | ~ ax-rep without distinct ... |
| axunprim 35690 | ~ ax-un without distinct v... |
| axpowprim 35691 | ~ ax-pow without distinct ... |
| axregprim 35692 | ~ ax-reg without distinct ... |
| axinfprim 35693 | ~ ax-inf without distinct ... |
| axacprim 35694 | ~ ax-ac without distinct v... |
| untelirr 35695 | We call a class "untanged"... |
| untuni 35696 | The union of a class is un... |
| untsucf 35697 | If a class is untangled, t... |
| unt0 35698 | The null set is untangled.... |
| untint 35699 | If there is an untangled e... |
| efrunt 35700 | If ` A ` is well-founded b... |
| untangtr 35701 | A transitive class is unta... |
| 3jaodd 35702 | Double deduction form of ~... |
| 3orit 35703 | Closed form of ~ 3ori . (... |
| biimpexp 35704 | A biconditional in the ant... |
| nepss 35705 | Two classes are unequal if... |
| 3ccased 35706 | Triple disjunction form of... |
| dfso3 35707 | Expansion of the definitio... |
| brtpid1 35708 | A binary relation involvin... |
| brtpid2 35709 | A binary relation involvin... |
| brtpid3 35710 | A binary relation involvin... |
| iota5f 35711 | A method for computing iot... |
| jath 35712 | Closed form of ~ ja . Pro... |
| xpab 35713 | Cartesian product of two c... |
| nnuni 35714 | The union of a finite ordi... |
| sqdivzi 35715 | Distribution of square ove... |
| supfz 35716 | The supremum of a finite s... |
| inffz 35717 | The infimum of a finite se... |
| fz0n 35718 | The sequence ` ( 0 ... ( N... |
| shftvalg 35719 | Value of a sequence shifte... |
| divcnvlin 35720 | Limit of the ratio of two ... |
| climlec3 35721 | Comparison of a constant t... |
| iexpire 35722 | ` _i ` raised to itself is... |
| bcneg1 35723 | The binomial coefficient o... |
| bcm1nt 35724 | The proportion of one bino... |
| bcprod 35725 | A product identity for bin... |
| bccolsum 35726 | A column-sum rule for bino... |
| iprodefisumlem 35727 | Lemma for ~ iprodefisum . ... |
| iprodefisum 35728 | Applying the exponential f... |
| iprodgam 35729 | An infinite product versio... |
| faclimlem1 35730 | Lemma for ~ faclim . Clos... |
| faclimlem2 35731 | Lemma for ~ faclim . Show... |
| faclimlem3 35732 | Lemma for ~ faclim . Alge... |
| faclim 35733 | An infinite product expres... |
| iprodfac 35734 | An infinite product expres... |
| faclim2 35735 | Another factorial limit du... |
| gcd32 35736 | Swap the second and third ... |
| gcdabsorb 35737 | Absorption law for gcd. (... |
| dftr6 35738 | A potential definition of ... |
| coep 35739 | Composition with the membe... |
| coepr 35740 | Composition with the conve... |
| dffr5 35741 | A quantifier-free definiti... |
| dfso2 35742 | Quantifier-free definition... |
| br8 35743 | Substitution for an eight-... |
| br6 35744 | Substitution for a six-pla... |
| br4 35745 | Substitution for a four-pl... |
| cnvco1 35746 | Another distributive law o... |
| cnvco2 35747 | Another distributive law o... |
| eldm3 35748 | Quantifier-free definition... |
| elrn3 35749 | Quantifier-free definition... |
| pocnv 35750 | The converse of a partial ... |
| socnv 35751 | The converse of a strict o... |
| elintfv 35752 | Membership in an intersect... |
| funpsstri 35753 | A condition for subset tri... |
| fundmpss 35754 | If a class ` F ` is a prop... |
| funsseq 35755 | Given two functions with e... |
| fununiq 35756 | The uniqueness condition o... |
| funbreq 35757 | An equality condition for ... |
| br1steq 35758 | Uniqueness condition for t... |
| br2ndeq 35759 | Uniqueness condition for t... |
| dfdm5 35760 | Definition of domain in te... |
| dfrn5 35761 | Definition of range in ter... |
| opelco3 35762 | Alternate way of saying th... |
| elima4 35763 | Quantifier-free expression... |
| fv1stcnv 35764 | The value of the converse ... |
| fv2ndcnv 35765 | The value of the converse ... |
| setinds 35766 | Principle of set induction... |
| setinds2f 35767 | ` _E ` induction schema, u... |
| setinds2 35768 | ` _E ` induction schema, u... |
| elpotr 35769 | A class of transitive sets... |
| dford5reg 35770 | Given ~ ax-reg , an ordina... |
| dfon2lem1 35771 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem2 35772 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem3 35773 | Lemma for ~ dfon2 . All s... |
| dfon2lem4 35774 | Lemma for ~ dfon2 . If tw... |
| dfon2lem5 35775 | Lemma for ~ dfon2 . Two s... |
| dfon2lem6 35776 | Lemma for ~ dfon2 . A tra... |
| dfon2lem7 35777 | Lemma for ~ dfon2 . All e... |
| dfon2lem8 35778 | Lemma for ~ dfon2 . The i... |
| dfon2lem9 35779 | Lemma for ~ dfon2 . A cla... |
| dfon2 35780 | ` On ` consists of all set... |
| rdgprc0 35781 | The value of the recursive... |
| rdgprc 35782 | The value of the recursive... |
| dfrdg2 35783 | Alternate definition of th... |
| dfrdg3 35784 | Generalization of ~ dfrdg2... |
| axextdfeq 35785 | A version of ~ ax-ext for ... |
| ax8dfeq 35786 | A version of ~ ax-8 for us... |
| axextdist 35787 | ~ ax-ext with distinctors ... |
| axextbdist 35788 | ~ axextb with distinctors ... |
| 19.12b 35789 | Version of ~ 19.12vv with ... |
| exnel 35790 | There is always a set not ... |
| distel 35791 | Distinctors in terms of me... |
| axextndbi 35792 | ~ axextnd as a bicondition... |
| hbntg 35793 | A more general form of ~ h... |
| hbimtg 35794 | A more general and closed ... |
| hbaltg 35795 | A more general and closed ... |
| hbng 35796 | A more general form of ~ h... |
| hbimg 35797 | A more general form of ~ h... |
| wsuceq123 35802 | Equality theorem for well-... |
| wsuceq1 35803 | Equality theorem for well-... |
| wsuceq2 35804 | Equality theorem for well-... |
| wsuceq3 35805 | Equality theorem for well-... |
| nfwsuc 35806 | Bound-variable hypothesis ... |
| wlimeq12 35807 | Equality theorem for the l... |
| wlimeq1 35808 | Equality theorem for the l... |
| wlimeq2 35809 | Equality theorem for the l... |
| nfwlim 35810 | Bound-variable hypothesis ... |
| elwlim 35811 | Membership in the limit cl... |
| wzel 35812 | The zero of a well-founded... |
| wsuclem 35813 | Lemma for the supremum pro... |
| wsucex 35814 | Existence theorem for well... |
| wsuccl 35815 | If ` X ` is a set with an ... |
| wsuclb 35816 | A well-founded successor i... |
| wlimss 35817 | The class of limit points ... |
| txpss3v 35866 | A tail Cartesian product i... |
| txprel 35867 | A tail Cartesian product i... |
| brtxp 35868 | Characterize a ternary rel... |
| brtxp2 35869 | The binary relation over a... |
| dfpprod2 35870 | Expanded definition of par... |
| pprodcnveq 35871 | A converse law for paralle... |
| pprodss4v 35872 | The parallel product is a ... |
| brpprod 35873 | Characterize a quaternary ... |
| brpprod3a 35874 | Condition for parallel pro... |
| brpprod3b 35875 | Condition for parallel pro... |
| relsset 35876 | The subset class is a bina... |
| brsset 35877 | For sets, the ` SSet ` bin... |
| idsset 35878 | ` _I ` is equal to the int... |
| eltrans 35879 | Membership in the class of... |
| dfon3 35880 | A quantifier-free definiti... |
| dfon4 35881 | Another quantifier-free de... |
| brtxpsd 35882 | Expansion of a common form... |
| brtxpsd2 35883 | Another common abbreviatio... |
| brtxpsd3 35884 | A third common abbreviatio... |
| relbigcup 35885 | The ` Bigcup ` relationshi... |
| brbigcup 35886 | Binary relation over ` Big... |
| dfbigcup2 35887 | ` Bigcup ` using maps-to n... |
| fobigcup 35888 | ` Bigcup ` maps the univer... |
| fnbigcup 35889 | ` Bigcup ` is a function o... |
| fvbigcup 35890 | For sets, ` Bigcup ` yield... |
| elfix 35891 | Membership in the fixpoint... |
| elfix2 35892 | Alternative membership in ... |
| dffix2 35893 | The fixpoints of a class i... |
| fixssdm 35894 | The fixpoints of a class a... |
| fixssrn 35895 | The fixpoints of a class a... |
| fixcnv 35896 | The fixpoints of a class a... |
| fixun 35897 | The fixpoint operator dist... |
| ellimits 35898 | Membership in the class of... |
| limitssson 35899 | The class of all limit ord... |
| dfom5b 35900 | A quantifier-free definiti... |
| sscoid 35901 | A condition for subset and... |
| dffun10 35902 | Another potential definiti... |
| elfuns 35903 | Membership in the class of... |
| elfunsg 35904 | Closed form of ~ elfuns . ... |
| brsingle 35905 | The binary relation form o... |
| elsingles 35906 | Membership in the class of... |
| fnsingle 35907 | The singleton relationship... |
| fvsingle 35908 | The value of the singleton... |
| dfsingles2 35909 | Alternate definition of th... |
| snelsingles 35910 | A singleton is a member of... |
| dfiota3 35911 | A definition of iota using... |
| dffv5 35912 | Another quantifier-free de... |
| unisnif 35913 | Express union of singleton... |
| brimage 35914 | Binary relation form of th... |
| brimageg 35915 | Closed form of ~ brimage .... |
| funimage 35916 | ` Image A ` is a function.... |
| fnimage 35917 | ` Image R ` is a function ... |
| imageval 35918 | The image functor in maps-... |
| fvimage 35919 | Value of the image functor... |
| brcart 35920 | Binary relation form of th... |
| brdomain 35921 | Binary relation form of th... |
| brrange 35922 | Binary relation form of th... |
| brdomaing 35923 | Closed form of ~ brdomain ... |
| brrangeg 35924 | Closed form of ~ brrange .... |
| brimg 35925 | Binary relation form of th... |
| brapply 35926 | Binary relation form of th... |
| brcup 35927 | Binary relation form of th... |
| brcap 35928 | Binary relation form of th... |
| brsuccf 35929 | Binary relation form of th... |
| funpartlem 35930 | Lemma for ~ funpartfun . ... |
| funpartfun 35931 | The functional part of ` F... |
| funpartss 35932 | The functional part of ` F... |
| funpartfv 35933 | The function value of the ... |
| fullfunfnv 35934 | The full functional part o... |
| fullfunfv 35935 | The function value of the ... |
| brfullfun 35936 | A binary relation form con... |
| brrestrict 35937 | Binary relation form of th... |
| dfrecs2 35938 | A quantifier-free definiti... |
| dfrdg4 35939 | A quantifier-free definiti... |
| dfint3 35940 | Quantifier-free definition... |
| imagesset 35941 | The Image functor applied ... |
| brub 35942 | Binary relation form of th... |
| brlb 35943 | Binary relation form of th... |
| altopex 35948 | Alternative ordered pairs ... |
| altopthsn 35949 | Two alternate ordered pair... |
| altopeq12 35950 | Equality for alternate ord... |
| altopeq1 35951 | Equality for alternate ord... |
| altopeq2 35952 | Equality for alternate ord... |
| altopth1 35953 | Equality of the first memb... |
| altopth2 35954 | Equality of the second mem... |
| altopthg 35955 | Alternate ordered pair the... |
| altopthbg 35956 | Alternate ordered pair the... |
| altopth 35957 | The alternate ordered pair... |
| altopthb 35958 | Alternate ordered pair the... |
| altopthc 35959 | Alternate ordered pair the... |
| altopthd 35960 | Alternate ordered pair the... |
| altxpeq1 35961 | Equality for alternate Car... |
| altxpeq2 35962 | Equality for alternate Car... |
| elaltxp 35963 | Membership in alternate Ca... |
| altopelaltxp 35964 | Alternate ordered pair mem... |
| altxpsspw 35965 | An inclusion rule for alte... |
| altxpexg 35966 | The alternate Cartesian pr... |
| rankaltopb 35967 | Compute the rank of an alt... |
| nfaltop 35968 | Bound-variable hypothesis ... |
| sbcaltop 35969 | Distribution of class subs... |
| cgrrflx2d 35972 | Deduction form of ~ axcgrr... |
| cgrtr4d 35973 | Deduction form of ~ axcgrt... |
| cgrtr4and 35974 | Deduction form of ~ axcgrt... |
| cgrrflx 35975 | Reflexivity law for congru... |
| cgrrflxd 35976 | Deduction form of ~ cgrrfl... |
| cgrcomim 35977 | Congruence commutes on the... |
| cgrcom 35978 | Congruence commutes betwee... |
| cgrcomand 35979 | Deduction form of ~ cgrcom... |
| cgrtr 35980 | Transitivity law for congr... |
| cgrtrand 35981 | Deduction form of ~ cgrtr ... |
| cgrtr3 35982 | Transitivity law for congr... |
| cgrtr3and 35983 | Deduction form of ~ cgrtr3... |
| cgrcoml 35984 | Congruence commutes on the... |
| cgrcomr 35985 | Congruence commutes on the... |
| cgrcomlr 35986 | Congruence commutes on bot... |
| cgrcomland 35987 | Deduction form of ~ cgrcom... |
| cgrcomrand 35988 | Deduction form of ~ cgrcom... |
| cgrcomlrand 35989 | Deduction form of ~ cgrcom... |
| cgrtriv 35990 | Degenerate segments are co... |
| cgrid2 35991 | Identity law for congruenc... |
| cgrdegen 35992 | Two congruent segments are... |
| brofs 35993 | Binary relation form of th... |
| 5segofs 35994 | Rephrase ~ ax5seg using th... |
| ofscom 35995 | The outer five segment pre... |
| cgrextend 35996 | Link congruence over a pai... |
| cgrextendand 35997 | Deduction form of ~ cgrext... |
| segconeq 35998 | Two points that satisfy th... |
| segconeu 35999 | Existential uniqueness ver... |
| btwntriv2 36000 | Betweenness always holds f... |
| btwncomim 36001 | Betweenness commutes. Imp... |
| btwncom 36002 | Betweenness commutes. (Co... |
| btwncomand 36003 | Deduction form of ~ btwnco... |
| btwntriv1 36004 | Betweenness always holds f... |
| btwnswapid 36005 | If you can swap the first ... |
| btwnswapid2 36006 | If you can swap arguments ... |
| btwnintr 36007 | Inner transitivity law for... |
| btwnexch3 36008 | Exchange the first endpoin... |
| btwnexch3and 36009 | Deduction form of ~ btwnex... |
| btwnouttr2 36010 | Outer transitivity law for... |
| btwnexch2 36011 | Exchange the outer point o... |
| btwnouttr 36012 | Outer transitivity law for... |
| btwnexch 36013 | Outer transitivity law for... |
| btwnexchand 36014 | Deduction form of ~ btwnex... |
| btwndiff 36015 | There is always a ` c ` di... |
| trisegint 36016 | A line segment between two... |
| funtransport 36019 | The ` TransportTo ` relati... |
| fvtransport 36020 | Calculate the value of the... |
| transportcl 36021 | Closure law for segment tr... |
| transportprops 36022 | Calculate the defining pro... |
| brifs 36031 | Binary relation form of th... |
| ifscgr 36032 | Inner five segment congrue... |
| cgrsub 36033 | Removing identical parts f... |
| brcgr3 36034 | Binary relation form of th... |
| cgr3permute3 36035 | Permutation law for three-... |
| cgr3permute1 36036 | Permutation law for three-... |
| cgr3permute2 36037 | Permutation law for three-... |
| cgr3permute4 36038 | Permutation law for three-... |
| cgr3permute5 36039 | Permutation law for three-... |
| cgr3tr4 36040 | Transitivity law for three... |
| cgr3com 36041 | Commutativity law for thre... |
| cgr3rflx 36042 | Identity law for three-pla... |
| cgrxfr 36043 | A line segment can be divi... |
| btwnxfr 36044 | A condition for extending ... |
| colinrel 36045 | Colinearity is a relations... |
| brcolinear2 36046 | Alternate colinearity bina... |
| brcolinear 36047 | The binary relation form o... |
| colinearex 36048 | The colinear predicate exi... |
| colineardim1 36049 | If ` A ` is colinear with ... |
| colinearperm1 36050 | Permutation law for coline... |
| colinearperm3 36051 | Permutation law for coline... |
| colinearperm2 36052 | Permutation law for coline... |
| colinearperm4 36053 | Permutation law for coline... |
| colinearperm5 36054 | Permutation law for coline... |
| colineartriv1 36055 | Trivial case of colinearit... |
| colineartriv2 36056 | Trivial case of colinearit... |
| btwncolinear1 36057 | Betweenness implies coline... |
| btwncolinear2 36058 | Betweenness implies coline... |
| btwncolinear3 36059 | Betweenness implies coline... |
| btwncolinear4 36060 | Betweenness implies coline... |
| btwncolinear5 36061 | Betweenness implies coline... |
| btwncolinear6 36062 | Betweenness implies coline... |
| colinearxfr 36063 | Transfer law for colineari... |
| lineext 36064 | Extend a line with a missi... |
| brofs2 36065 | Change some conditions for... |
| brifs2 36066 | Change some conditions for... |
| brfs 36067 | Binary relation form of th... |
| fscgr 36068 | Congruence law for the gen... |
| linecgr 36069 | Congruence rule for lines.... |
| linecgrand 36070 | Deduction form of ~ linecg... |
| lineid 36071 | Identity law for points on... |
| idinside 36072 | Law for finding a point in... |
| endofsegid 36073 | If ` A ` , ` B ` , and ` C... |
| endofsegidand 36074 | Deduction form of ~ endofs... |
| btwnconn1lem1 36075 | Lemma for ~ btwnconn1 . T... |
| btwnconn1lem2 36076 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem3 36077 | Lemma for ~ btwnconn1 . E... |
| btwnconn1lem4 36078 | Lemma for ~ btwnconn1 . A... |
| btwnconn1lem5 36079 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem6 36080 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem7 36081 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem8 36082 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem9 36083 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem10 36084 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem11 36085 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem12 36086 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem13 36087 | Lemma for ~ btwnconn1 . B... |
| btwnconn1lem14 36088 | Lemma for ~ btwnconn1 . F... |
| btwnconn1 36089 | Connectitivy law for betwe... |
| btwnconn2 36090 | Another connectivity law f... |
| btwnconn3 36091 | Inner connectivity law for... |
| midofsegid 36092 | If two points fall in the ... |
| segcon2 36093 | Generalization of ~ axsegc... |
| brsegle 36096 | Binary relation form of th... |
| brsegle2 36097 | Alternate characterization... |
| seglecgr12im 36098 | Substitution law for segme... |
| seglecgr12 36099 | Substitution law for segme... |
| seglerflx 36100 | Segment comparison is refl... |
| seglemin 36101 | Any segment is at least as... |
| segletr 36102 | Segment less than is trans... |
| segleantisym 36103 | Antisymmetry law for segme... |
| seglelin 36104 | Linearity law for segment ... |
| btwnsegle 36105 | If ` B ` falls between ` A... |
| colinbtwnle 36106 | Given three colinear point... |
| broutsideof 36109 | Binary relation form of ` ... |
| broutsideof2 36110 | Alternate form of ` Outsid... |
| outsidene1 36111 | Outsideness implies inequa... |
| outsidene2 36112 | Outsideness implies inequa... |
| btwnoutside 36113 | A principle linking outsid... |
| broutsideof3 36114 | Characterization of outsid... |
| outsideofrflx 36115 | Reflexivity of outsideness... |
| outsideofcom 36116 | Commutativity law for outs... |
| outsideoftr 36117 | Transitivity law for outsi... |
| outsideofeq 36118 | Uniqueness law for ` Outsi... |
| outsideofeu 36119 | Given a nondegenerate ray,... |
| outsidele 36120 | Relate ` OutsideOf ` to ` ... |
| outsideofcol 36121 | Outside of implies colinea... |
| funray 36128 | Show that the ` Ray ` rela... |
| fvray 36129 | Calculate the value of the... |
| funline 36130 | Show that the ` Line ` rel... |
| linedegen 36131 | When ` Line ` is applied w... |
| fvline 36132 | Calculate the value of the... |
| liness 36133 | A line is a subset of the ... |
| fvline2 36134 | Alternate definition of a ... |
| lineunray 36135 | A line is composed of a po... |
| lineelsb2 36136 | If ` S ` lies on ` P Q ` ,... |
| linerflx1 36137 | Reflexivity law for line m... |
| linecom 36138 | Commutativity law for line... |
| linerflx2 36139 | Reflexivity law for line m... |
| ellines 36140 | Membership in the set of a... |
| linethru 36141 | If ` A ` is a line contain... |
| hilbert1.1 36142 | There is a line through an... |
| hilbert1.2 36143 | There is at most one line ... |
| linethrueu 36144 | There is a unique line goi... |
| lineintmo 36145 | Two distinct lines interse... |
| fwddifval 36150 | Calculate the value of the... |
| fwddifnval 36151 | The value of the forward d... |
| fwddifn0 36152 | The value of the n-iterate... |
| fwddifnp1 36153 | The value of the n-iterate... |
| rankung 36154 | The rank of the union of t... |
| ranksng 36155 | The rank of a singleton. ... |
| rankelg 36156 | The membership relation is... |
| rankpwg 36157 | The rank of a power set. ... |
| rank0 36158 | The rank of the empty set ... |
| rankeq1o 36159 | The only set with rank ` 1... |
| elhf 36162 | Membership in the heredita... |
| elhf2 36163 | Alternate form of membersh... |
| elhf2g 36164 | Hereditarily finiteness vi... |
| 0hf 36165 | The empty set is a heredit... |
| hfun 36166 | The union of two HF sets i... |
| hfsn 36167 | The singleton of an HF set... |
| hfadj 36168 | Adjoining one HF element t... |
| hfelhf 36169 | Any member of an HF set is... |
| hftr 36170 | The class of all hereditar... |
| hfext 36171 | Extensionality for HF sets... |
| hfuni 36172 | The union of an HF set is ... |
| hfpw 36173 | The power class of an HF s... |
| hfninf 36174 | ` _om ` is not hereditaril... |
| rmoeqi 36175 | Equality inference for res... |
| rmoeqbii 36176 | Equality inference for res... |
| reueqi 36177 | Equality inference for res... |
| reueqbii 36178 | Equality inference for res... |
| sbceqbii 36179 | Formula-building inference... |
| disjeq1i 36180 | Equality theorem for disjo... |
| disjeq12i 36181 | Equality theorem for disjo... |
| rabeqbii 36182 | Equality theorem for restr... |
| iuneq12i 36183 | Equality theorem for index... |
| iineq1i 36184 | Equality theorem for index... |
| iineq12i 36185 | Equality theorem for index... |
| riotaeqbii 36186 | Equivalent wff's and equal... |
| riotaeqi 36187 | Equal domains yield equal ... |
| ixpeq1i 36188 | Equality inference for inf... |
| ixpeq12i 36189 | Equality inference for inf... |
| sumeq2si 36190 | Equality inference for sum... |
| sumeq12si 36191 | Equality inference for sum... |
| prodeq2si 36192 | Equality inference for pro... |
| prodeq12si 36193 | Equality inference for pro... |
| itgeq12i 36194 | Equality inference for an ... |
| itgeq1i 36195 | Equality inference for an ... |
| itgeq2i 36196 | Equality inference for an ... |
| ditgeq123i 36197 | Equality inference for the... |
| ditgeq12i 36198 | Equality inference for the... |
| ditgeq3i 36199 | Equality inference for the... |
| rmoeqdv 36200 | Formula-building rule for ... |
| rmoeqbidv 36201 | Formula-building rule for ... |
| sbequbidv 36202 | Deduction substituting bot... |
| disjeq12dv 36203 | Equality theorem for disjo... |
| ixpeq12dv 36204 | Equality theorem for infin... |
| sumeq12sdv 36205 | Equality deduction for sum... |
| prodeq12sdv 36206 | Equality deduction for pro... |
| itgeq12sdv 36207 | Equality theorem for an in... |
| itgeq2sdv 36208 | Equality theorem for an in... |
| ditgeq123dv 36209 | Equality theorem for the d... |
| ditgeq12d 36210 | Equality theorem for the d... |
| ditgeq3sdv 36211 | Equality theorem for the d... |
| in-ax8 36212 | A proof of ~ ax-8 that doe... |
| ss-ax8 36213 | A proof of ~ ax-8 that doe... |
| cbvralvw2 36214 | Change bound variable and ... |
| cbvrexvw2 36215 | Change bound variable and ... |
| cbvrmovw2 36216 | Change bound variable and ... |
| cbvreuvw2 36217 | Change bound variable and ... |
| cbvsbcvw2 36218 | Change bound variable of a... |
| cbvcsbvw2 36219 | Change bound variable of a... |
| cbviunvw2 36220 | Change bound variable and ... |
| cbviinvw2 36221 | Change bound variable and ... |
| cbvmptvw2 36222 | Change bound variable and ... |
| cbvdisjvw2 36223 | Change bound variable and ... |
| cbvriotavw2 36224 | Change bound variable and ... |
| cbvoprab1vw 36225 | Change the first bound var... |
| cbvoprab2vw 36226 | Change the second bound va... |
| cbvoprab123vw 36227 | Change all bound variables... |
| cbvoprab23vw 36228 | Change the second and thir... |
| cbvoprab13vw 36229 | Change the first and third... |
| cbvmpovw2 36230 | Change bound variables and... |
| cbvmpo1vw2 36231 | Change domains and the fir... |
| cbvmpo2vw2 36232 | Change domains and the sec... |
| cbvixpvw2 36233 | Change bound variable and ... |
| cbvsumvw2 36234 | Change bound variable and ... |
| cbvprodvw2 36235 | Change bound variable and ... |
| cbvitgvw2 36236 | Change bound variable and ... |
| cbvditgvw2 36237 | Change bound variable and ... |
| cbvmodavw 36238 | Change bound variable in t... |
| cbveudavw 36239 | Change bound variable in t... |
| cbvrmodavw 36240 | Change bound variable in t... |
| cbvreudavw 36241 | Change bound variable in t... |
| cbvsbdavw 36242 | Change bound variable in p... |
| cbvsbdavw2 36243 | Change bound variable in p... |
| cbvabdavw 36244 | Change bound variable in c... |
| cbvsbcdavw 36245 | Change bound variable of a... |
| cbvsbcdavw2 36246 | Change bound variable of a... |
| cbvcsbdavw 36247 | Change bound variable of a... |
| cbvcsbdavw2 36248 | Change bound variable of a... |
| cbvrabdavw 36249 | Change bound variable in r... |
| cbviundavw 36250 | Change bound variable in i... |
| cbviindavw 36251 | Change bound variable in i... |
| cbvopab1davw 36252 | Change the first bound var... |
| cbvopab2davw 36253 | Change the second bound va... |
| cbvopabdavw 36254 | Change bound variables in ... |
| cbvmptdavw 36255 | Change bound variable in a... |
| cbvdisjdavw 36256 | Change bound variable in a... |
| cbviotadavw 36257 | Change bound variable in a... |
| cbvriotadavw 36258 | Change bound variable in a... |
| cbvoprab1davw 36259 | Change the first bound var... |
| cbvoprab2davw 36260 | Change the second bound va... |
| cbvoprab3davw 36261 | Change the third bound var... |
| cbvoprab123davw 36262 | Change all bound variables... |
| cbvoprab12davw 36263 | Change the first and secon... |
| cbvoprab23davw 36264 | Change the second and thir... |
| cbvoprab13davw 36265 | Change the first and third... |
| cbvixpdavw 36266 | Change bound variable in a... |
| cbvsumdavw 36267 | Change bound variable in a... |
| cbvproddavw 36268 | Change bound variable in a... |
| cbvitgdavw 36269 | Change bound variable in a... |
| cbvditgdavw 36270 | Change bound variable in a... |
| cbvrmodavw2 36271 | Change bound variable and ... |
| cbvreudavw2 36272 | Change bound variable and ... |
| cbvrabdavw2 36273 | Change bound variable and ... |
| cbviundavw2 36274 | Change bound variable and ... |
| cbviindavw2 36275 | Change bound variable and ... |
| cbvmptdavw2 36276 | Change bound variable and ... |
| cbvdisjdavw2 36277 | Change bound variable and ... |
| cbvriotadavw2 36278 | Change bound variable and ... |
| cbvmpodavw2 36279 | Change bound variable and ... |
| cbvmpo1davw2 36280 | Change first bound variabl... |
| cbvmpo2davw2 36281 | Change second bound variab... |
| cbvixpdavw2 36282 | Change bound variable and ... |
| cbvsumdavw2 36283 | Change bound variable and ... |
| cbvproddavw2 36284 | Change bound variable and ... |
| cbvitgdavw2 36285 | Change bound variable and ... |
| cbvditgdavw2 36286 | Change bound variable and ... |
| mpomulnzcnf 36287 | Multiplication maps nonzer... |
| a1i14 36288 | Add two antecedents to a w... |
| a1i24 36289 | Add two antecedents to a w... |
| exp5d 36290 | An exportation inference. ... |
| exp5g 36291 | An exportation inference. ... |
| exp5k 36292 | An exportation inference. ... |
| exp56 36293 | An exportation inference. ... |
| exp58 36294 | An exportation inference. ... |
| exp510 36295 | An exportation inference. ... |
| exp511 36296 | An exportation inference. ... |
| exp512 36297 | An exportation inference. ... |
| 3com12d 36298 | Commutation in consequent.... |
| imp5p 36299 | A triple importation infer... |
| imp5q 36300 | A triple importation infer... |
| ecase13d 36301 | Deduction for elimination ... |
| subtr 36302 | Transitivity of implicit s... |
| subtr2 36303 | Transitivity of implicit s... |
| trer 36304 | A relation intersected wit... |
| elicc3 36305 | An equivalent membership c... |
| finminlem 36306 | A useful lemma about finit... |
| gtinf 36307 | Any number greater than an... |
| opnrebl 36308 | A set is open in the stand... |
| opnrebl2 36309 | A set is open in the stand... |
| nn0prpwlem 36310 | Lemma for ~ nn0prpw . Use... |
| nn0prpw 36311 | Two nonnegative integers a... |
| topbnd 36312 | Two equivalent expressions... |
| opnbnd 36313 | A set is open iff it is di... |
| cldbnd 36314 | A set is closed iff it con... |
| ntruni 36315 | A union of interiors is a ... |
| clsun 36316 | A pairwise union of closur... |
| clsint2 36317 | The closure of an intersec... |
| opnregcld 36318 | A set is regularly closed ... |
| cldregopn 36319 | A set if regularly open if... |
| neiin 36320 | Two neighborhoods intersec... |
| hmeoclda 36321 | Homeomorphisms preserve cl... |
| hmeocldb 36322 | Homeomorphisms preserve cl... |
| ivthALT 36323 | An alternate proof of the ... |
| fnerel 36326 | Fineness is a relation. (... |
| isfne 36327 | The predicate " ` B ` is f... |
| isfne4 36328 | The predicate " ` B ` is f... |
| isfne4b 36329 | A condition for a topology... |
| isfne2 36330 | The predicate " ` B ` is f... |
| isfne3 36331 | The predicate " ` B ` is f... |
| fnebas 36332 | A finer cover covers the s... |
| fnetg 36333 | A finer cover generates a ... |
| fnessex 36334 | If ` B ` is finer than ` A... |
| fneuni 36335 | If ` B ` is finer than ` A... |
| fneint 36336 | If a cover is finer than a... |
| fness 36337 | A cover is finer than its ... |
| fneref 36338 | Reflexivity of the finenes... |
| fnetr 36339 | Transitivity of the finene... |
| fneval 36340 | Two covers are finer than ... |
| fneer 36341 | Fineness intersected with ... |
| topfne 36342 | Fineness for covers corres... |
| topfneec 36343 | A cover is equivalent to a... |
| topfneec2 36344 | A topology is precisely id... |
| fnessref 36345 | A cover is finer iff it ha... |
| refssfne 36346 | A cover is a refinement if... |
| neibastop1 36347 | A collection of neighborho... |
| neibastop2lem 36348 | Lemma for ~ neibastop2 . ... |
| neibastop2 36349 | In the topology generated ... |
| neibastop3 36350 | The topology generated by ... |
| topmtcl 36351 | The meet of a collection o... |
| topmeet 36352 | Two equivalent formulation... |
| topjoin 36353 | Two equivalent formulation... |
| fnemeet1 36354 | The meet of a collection o... |
| fnemeet2 36355 | The meet of equivalence cl... |
| fnejoin1 36356 | Join of equivalence classe... |
| fnejoin2 36357 | Join of equivalence classe... |
| fgmin 36358 | Minimality property of a g... |
| neifg 36359 | The neighborhood filter of... |
| tailfval 36360 | The tail function for a di... |
| tailval 36361 | The tail of an element in ... |
| eltail 36362 | An element of a tail. (Co... |
| tailf 36363 | The tail function of a dir... |
| tailini 36364 | A tail contains its initia... |
| tailfb 36365 | The collection of tails of... |
| filnetlem1 36366 | Lemma for ~ filnet . Chan... |
| filnetlem2 36367 | Lemma for ~ filnet . The ... |
| filnetlem3 36368 | Lemma for ~ filnet . (Con... |
| filnetlem4 36369 | Lemma for ~ filnet . (Con... |
| filnet 36370 | A filter has the same conv... |
| tb-ax1 36371 | The first of three axioms ... |
| tb-ax2 36372 | The second of three axioms... |
| tb-ax3 36373 | The third of three axioms ... |
| tbsyl 36374 | The weak syllogism from Ta... |
| re1ax2lem 36375 | Lemma for ~ re1ax2 . (Con... |
| re1ax2 36376 | ~ ax-2 rederived from the ... |
| naim1 36377 | Constructor theorem for ` ... |
| naim2 36378 | Constructor theorem for ` ... |
| naim1i 36379 | Constructor rule for ` -/\... |
| naim2i 36380 | Constructor rule for ` -/\... |
| naim12i 36381 | Constructor rule for ` -/\... |
| nabi1i 36382 | Constructor rule for ` -/\... |
| nabi2i 36383 | Constructor rule for ` -/\... |
| nabi12i 36384 | Constructor rule for ` -/\... |
| df3nandALT1 36387 | The double nand expressed ... |
| df3nandALT2 36388 | The double nand expressed ... |
| andnand1 36389 | Double and in terms of dou... |
| imnand2 36390 | An ` -> ` nand relation. ... |
| nalfal 36391 | Not all sets hold ` F. ` a... |
| nexntru 36392 | There does not exist a set... |
| nexfal 36393 | There does not exist a set... |
| neufal 36394 | There does not exist exact... |
| neutru 36395 | There does not exist exact... |
| nmotru 36396 | There does not exist at mo... |
| mofal 36397 | There exist at most one se... |
| nrmo 36398 | "At most one" restricted e... |
| meran1 36399 | A single axiom for proposi... |
| meran2 36400 | A single axiom for proposi... |
| meran3 36401 | A single axiom for proposi... |
| waj-ax 36402 | A single axiom for proposi... |
| lukshef-ax2 36403 | A single axiom for proposi... |
| arg-ax 36404 | A single axiom for proposi... |
| negsym1 36405 | In the paper "On Variable ... |
| imsym1 36406 | A symmetry with ` -> ` . ... |
| bisym1 36407 | A symmetry with ` <-> ` . ... |
| consym1 36408 | A symmetry with ` /\ ` . ... |
| dissym1 36409 | A symmetry with ` \/ ` . ... |
| nandsym1 36410 | A symmetry with ` -/\ ` . ... |
| unisym1 36411 | A symmetry with ` A. ` . ... |
| exisym1 36412 | A symmetry with ` E. ` . ... |
| unqsym1 36413 | A symmetry with ` E! ` . ... |
| amosym1 36414 | A symmetry with ` E* ` . ... |
| subsym1 36415 | A symmetry with ` [ x / y ... |
| ontopbas 36416 | An ordinal number is a top... |
| onsstopbas 36417 | The class of ordinal numbe... |
| onpsstopbas 36418 | The class of ordinal numbe... |
| ontgval 36419 | The topology generated fro... |
| ontgsucval 36420 | The topology generated fro... |
| onsuctop 36421 | A successor ordinal number... |
| onsuctopon 36422 | One of the topologies on a... |
| ordtoplem 36423 | Membership of the class of... |
| ordtop 36424 | An ordinal is a topology i... |
| onsucconni 36425 | A successor ordinal number... |
| onsucconn 36426 | A successor ordinal number... |
| ordtopconn 36427 | An ordinal topology is con... |
| onintopssconn 36428 | An ordinal topology is con... |
| onsuct0 36429 | A successor ordinal number... |
| ordtopt0 36430 | An ordinal topology is T_0... |
| onsucsuccmpi 36431 | The successor of a success... |
| onsucsuccmp 36432 | The successor of a success... |
| limsucncmpi 36433 | The successor of a limit o... |
| limsucncmp 36434 | The successor of a limit o... |
| ordcmp 36435 | An ordinal topology is com... |
| ssoninhaus 36436 | The ordinal topologies ` 1... |
| onint1 36437 | The ordinal T_1 spaces are... |
| oninhaus 36438 | The ordinal Hausdorff spac... |
| fveleq 36439 | Please add description her... |
| findfvcl 36440 | Please add description her... |
| findreccl 36441 | Please add description her... |
| findabrcl 36442 | Please add description her... |
| nnssi2 36443 | Convert a theorem for real... |
| nnssi3 36444 | Convert a theorem for real... |
| nndivsub 36445 | Please add description her... |
| nndivlub 36446 | A factor of a positive int... |
| ee7.2aOLD 36449 | Lemma for Euclid's Element... |
| weiunlem1 36450 | Lemma for ~ weiunpo , ~ we... |
| weiunlem2 36451 | Lemma for ~ weiunpo , ~ we... |
| weiunfrlem 36452 | Lemma for ~ weiunfr . (Co... |
| weiunpo 36453 | A partial ordering on an i... |
| weiunso 36454 | A strict ordering on an in... |
| weiunfr 36455 | A well-founded relation on... |
| weiunse 36456 | The relation constructed i... |
| weiunwe 36457 | A well-ordering on an inde... |
| numiunnum 36458 | An indexed union of sets i... |
| dnival 36459 | Value of the "distance to ... |
| dnicld1 36460 | Closure theorem for the "d... |
| dnicld2 36461 | Closure theorem for the "d... |
| dnif 36462 | The "distance to nearest i... |
| dnizeq0 36463 | The distance to nearest in... |
| dnizphlfeqhlf 36464 | The distance to nearest in... |
| rddif2 36465 | Variant of ~ rddif . (Con... |
| dnibndlem1 36466 | Lemma for ~ dnibnd . (Con... |
| dnibndlem2 36467 | Lemma for ~ dnibnd . (Con... |
| dnibndlem3 36468 | Lemma for ~ dnibnd . (Con... |
| dnibndlem4 36469 | Lemma for ~ dnibnd . (Con... |
| dnibndlem5 36470 | Lemma for ~ dnibnd . (Con... |
| dnibndlem6 36471 | Lemma for ~ dnibnd . (Con... |
| dnibndlem7 36472 | Lemma for ~ dnibnd . (Con... |
| dnibndlem8 36473 | Lemma for ~ dnibnd . (Con... |
| dnibndlem9 36474 | Lemma for ~ dnibnd . (Con... |
| dnibndlem10 36475 | Lemma for ~ dnibnd . (Con... |
| dnibndlem11 36476 | Lemma for ~ dnibnd . (Con... |
| dnibndlem12 36477 | Lemma for ~ dnibnd . (Con... |
| dnibndlem13 36478 | Lemma for ~ dnibnd . (Con... |
| dnibnd 36479 | The "distance to nearest i... |
| dnicn 36480 | The "distance to nearest i... |
| knoppcnlem1 36481 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem2 36482 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem3 36483 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem4 36484 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem5 36485 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem6 36486 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem7 36487 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem8 36488 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem9 36489 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem10 36490 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem11 36491 | Lemma for ~ knoppcn . (Co... |
| knoppcn 36492 | The continuous nowhere dif... |
| knoppcld 36493 | Closure theorem for Knopp'... |
| unblimceq0lem 36494 | Lemma for ~ unblimceq0 . ... |
| unblimceq0 36495 | If ` F ` is unbounded near... |
| unbdqndv1 36496 | If the difference quotient... |
| unbdqndv2lem1 36497 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2lem2 36498 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2 36499 | Variant of ~ unbdqndv1 wit... |
| knoppndvlem1 36500 | Lemma for ~ knoppndv . (C... |
| knoppndvlem2 36501 | Lemma for ~ knoppndv . (C... |
| knoppndvlem3 36502 | Lemma for ~ knoppndv . (C... |
| knoppndvlem4 36503 | Lemma for ~ knoppndv . (C... |
| knoppndvlem5 36504 | Lemma for ~ knoppndv . (C... |
| knoppndvlem6 36505 | Lemma for ~ knoppndv . (C... |
| knoppndvlem7 36506 | Lemma for ~ knoppndv . (C... |
| knoppndvlem8 36507 | Lemma for ~ knoppndv . (C... |
| knoppndvlem9 36508 | Lemma for ~ knoppndv . (C... |
| knoppndvlem10 36509 | Lemma for ~ knoppndv . (C... |
| knoppndvlem11 36510 | Lemma for ~ knoppndv . (C... |
| knoppndvlem12 36511 | Lemma for ~ knoppndv . (C... |
| knoppndvlem13 36512 | Lemma for ~ knoppndv . (C... |
| knoppndvlem14 36513 | Lemma for ~ knoppndv . (C... |
| knoppndvlem15 36514 | Lemma for ~ knoppndv . (C... |
| knoppndvlem16 36515 | Lemma for ~ knoppndv . (C... |
| knoppndvlem17 36516 | Lemma for ~ knoppndv . (C... |
| knoppndvlem18 36517 | Lemma for ~ knoppndv . (C... |
| knoppndvlem19 36518 | Lemma for ~ knoppndv . (C... |
| knoppndvlem20 36519 | Lemma for ~ knoppndv . (C... |
| knoppndvlem21 36520 | Lemma for ~ knoppndv . (C... |
| knoppndvlem22 36521 | Lemma for ~ knoppndv . (C... |
| knoppndv 36522 | The continuous nowhere dif... |
| knoppf 36523 | Knopp's function is a func... |
| knoppcn2 36524 | Variant of ~ knoppcn with ... |
| cnndvlem1 36525 | Lemma for ~ cnndv . (Cont... |
| cnndvlem2 36526 | Lemma for ~ cnndv . (Cont... |
| cnndv 36527 | There exists a continuous ... |
| bj-mp2c 36528 | A double _modus ponens_ in... |
| bj-mp2d 36529 | A double _modus ponens_ in... |
| bj-0 36530 | A syntactic theorem. See ... |
| bj-1 36531 | In this proof, the use of ... |
| bj-a1k 36532 | Weakening of ~ ax-1 . As ... |
| bj-poni 36533 | Inference associated with ... |
| bj-nnclav 36534 | When ` F. ` is substituted... |
| bj-nnclavi 36535 | Inference associated with ... |
| bj-nnclavc 36536 | Commuted form of ~ bj-nncl... |
| bj-nnclavci 36537 | Inference associated with ... |
| bj-jarrii 36538 | Inference associated with ... |
| bj-imim21 36539 | The propositional function... |
| bj-imim21i 36540 | Inference associated with ... |
| bj-peircestab 36541 | Over minimal implicational... |
| bj-stabpeirce 36542 | This minimal implicational... |
| bj-syl66ib 36543 | A mixed syllogism inferenc... |
| bj-orim2 36544 | Proof of ~ orim2 from the ... |
| bj-currypeirce 36545 | Curry's axiom ~ curryax (a... |
| bj-peircecurry 36546 | Peirce's axiom ~ peirce im... |
| bj-animbi 36547 | Conjunction in terms of im... |
| bj-currypara 36548 | Curry's paradox. Note tha... |
| bj-con2com 36549 | A commuted form of the con... |
| bj-con2comi 36550 | Inference associated with ... |
| bj-nimn 36551 | If a formula is true, then... |
| bj-nimni 36552 | Inference associated with ... |
| bj-peircei 36553 | Inference associated with ... |
| bj-looinvi 36554 | Inference associated with ... |
| bj-looinvii 36555 | Inference associated with ... |
| bj-mt2bi 36556 | Version of ~ mt2 where the... |
| bj-ntrufal 36557 | The negation of a theorem ... |
| bj-fal 36558 | Shortening of ~ fal using ... |
| bj-jaoi1 36559 | Shortens ~ orfa2 (58>53), ... |
| bj-jaoi2 36560 | Shortens ~ consensus (110>... |
| bj-dfbi4 36561 | Alternate definition of th... |
| bj-dfbi5 36562 | Alternate definition of th... |
| bj-dfbi6 36563 | Alternate definition of th... |
| bj-bijust0ALT 36564 | Alternate proof of ~ bijus... |
| bj-bijust00 36565 | A self-implication does no... |
| bj-consensus 36566 | Version of ~ consensus exp... |
| bj-consensusALT 36567 | Alternate proof of ~ bj-co... |
| bj-df-ifc 36568 | Candidate definition for t... |
| bj-dfif 36569 | Alternate definition of th... |
| bj-ififc 36570 | A biconditional connecting... |
| bj-imbi12 36571 | Uncurried (imported) form ... |
| bj-falor 36572 | Dual of ~ truan (which has... |
| bj-falor2 36573 | Dual of ~ truan . (Contri... |
| bj-bibibi 36574 | A property of the bicondit... |
| bj-imn3ani 36575 | Duplication of ~ bnj1224 .... |
| bj-andnotim 36576 | Two ways of expressing a c... |
| bj-bi3ant 36577 | This used to be in the mai... |
| bj-bisym 36578 | This used to be in the mai... |
| bj-bixor 36579 | Equivalence of two ternary... |
| bj-axdd2 36580 | This implication, proved u... |
| bj-axd2d 36581 | This implication, proved u... |
| bj-axtd 36582 | This implication, proved f... |
| bj-gl4 36583 | In a normal modal logic, t... |
| bj-axc4 36584 | Over minimal calculus, the... |
| prvlem1 36589 | An elementary property of ... |
| prvlem2 36590 | An elementary property of ... |
| bj-babygodel 36591 | See the section header com... |
| bj-babylob 36592 | See the section header com... |
| bj-godellob 36593 | Proof of Gödel's theo... |
| bj-genr 36594 | Generalization rule on the... |
| bj-genl 36595 | Generalization rule on the... |
| bj-genan 36596 | Generalization rule on a c... |
| bj-mpgs 36597 | From a closed form theorem... |
| bj-2alim 36598 | Closed form of ~ 2alimi . ... |
| bj-2exim 36599 | Closed form of ~ 2eximi . ... |
| bj-alanim 36600 | Closed form of ~ alanimi .... |
| bj-2albi 36601 | Closed form of ~ 2albii . ... |
| bj-notalbii 36602 | Equivalence of universal q... |
| bj-2exbi 36603 | Closed form of ~ 2exbii . ... |
| bj-3exbi 36604 | Closed form of ~ 3exbii . ... |
| bj-sylggt 36605 | Stronger form of ~ sylgt ,... |
| bj-sylgt2 36606 | Uncurried (imported) form ... |
| bj-alrimg 36607 | The general form of the *a... |
| bj-alrimd 36608 | A slightly more general ~ ... |
| bj-sylget 36609 | Dual statement of ~ sylgt ... |
| bj-sylget2 36610 | Uncurried (imported) form ... |
| bj-exlimg 36611 | The general form of the *e... |
| bj-sylge 36612 | Dual statement of ~ sylg (... |
| bj-exlimd 36613 | A slightly more general ~ ... |
| bj-nfimexal 36614 | A weak from of nonfreeness... |
| bj-alexim 36615 | Closed form of ~ aleximi .... |
| bj-nexdh 36616 | Closed form of ~ nexdh (ac... |
| bj-nexdh2 36617 | Uncurried (imported) form ... |
| bj-hbxfrbi 36618 | Closed form of ~ hbxfrbi .... |
| bj-hbyfrbi 36619 | Version of ~ bj-hbxfrbi wi... |
| bj-exalim 36620 | Distribute quantifiers ove... |
| bj-exalimi 36621 | An inference for distribut... |
| bj-exalims 36622 | Distributing quantifiers o... |
| bj-exalimsi 36623 | An inference for distribut... |
| bj-ax12ig 36624 | A lemma used to prove a we... |
| bj-ax12i 36625 | A weakening of ~ bj-ax12ig... |
| bj-nfimt 36626 | Closed form of ~ nfim and ... |
| bj-cbvalimt 36627 | A lemma in closed form use... |
| bj-cbveximt 36628 | A lemma in closed form use... |
| bj-eximALT 36629 | Alternate proof of ~ exim ... |
| bj-aleximiALT 36630 | Alternate proof of ~ alexi... |
| bj-eximcom 36631 | A commuted form of ~ exim ... |
| bj-ax12wlem 36632 | A lemma used to prove a we... |
| bj-cbvalim 36633 | A lemma used to prove ~ bj... |
| bj-cbvexim 36634 | A lemma used to prove ~ bj... |
| bj-cbvalimi 36635 | An equality-free general i... |
| bj-cbveximi 36636 | An equality-free general i... |
| bj-cbval 36637 | Changing a bound variable ... |
| bj-cbvex 36638 | Changing a bound variable ... |
| bj-ssbeq 36641 | Substitution in an equalit... |
| bj-ssblem1 36642 | A lemma for the definiens ... |
| bj-ssblem2 36643 | An instance of ~ ax-11 pro... |
| bj-ax12v 36644 | A weaker form of ~ ax-12 a... |
| bj-ax12 36645 | Remove a DV condition from... |
| bj-ax12ssb 36646 | Axiom ~ bj-ax12 expressed ... |
| bj-19.41al 36647 | Special case of ~ 19.41 pr... |
| bj-equsexval 36648 | Special case of ~ equsexv ... |
| bj-subst 36649 | Proof of ~ sbalex from cor... |
| bj-ssbid2 36650 | A special case of ~ sbequ2... |
| bj-ssbid2ALT 36651 | Alternate proof of ~ bj-ss... |
| bj-ssbid1 36652 | A special case of ~ sbequ1... |
| bj-ssbid1ALT 36653 | Alternate proof of ~ bj-ss... |
| bj-ax6elem1 36654 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6elem2 36655 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6e 36656 | Proof of ~ ax6e (hence ~ a... |
| bj-spimvwt 36657 | Closed form of ~ spimvw . ... |
| bj-spnfw 36658 | Theorem close to a closed ... |
| bj-cbvexiw 36659 | Change bound variable. Th... |
| bj-cbvexivw 36660 | Change bound variable. Th... |
| bj-modald 36661 | A short form of the axiom ... |
| bj-denot 36662 | A weakening of ~ ax-6 and ... |
| bj-eqs 36663 | A lemma for substitutions,... |
| bj-cbvexw 36664 | Change bound variable. Th... |
| bj-ax12w 36665 | The general statement that... |
| bj-ax89 36666 | A theorem which could be u... |
| bj-cleljusti 36667 | One direction of ~ cleljus... |
| bj-alcomexcom 36668 | Commutation of two existen... |
| bj-hbalt 36669 | Closed form of ~ hbal . W... |
| axc11n11 36670 | Proof of ~ axc11n from { ~... |
| axc11n11r 36671 | Proof of ~ axc11n from { ~... |
| bj-axc16g16 36672 | Proof of ~ axc16g from { ~... |
| bj-ax12v3 36673 | A weak version of ~ ax-12 ... |
| bj-ax12v3ALT 36674 | Alternate proof of ~ bj-ax... |
| bj-sb 36675 | A weak variant of ~ sbid2 ... |
| bj-modalbe 36676 | The predicate-calculus ver... |
| bj-spst 36677 | Closed form of ~ sps . On... |
| bj-19.21bit 36678 | Closed form of ~ 19.21bi .... |
| bj-19.23bit 36679 | Closed form of ~ 19.23bi .... |
| bj-nexrt 36680 | Closed form of ~ nexr . C... |
| bj-alrim 36681 | Closed form of ~ alrimi . ... |
| bj-alrim2 36682 | Uncurried (imported) form ... |
| bj-nfdt0 36683 | A theorem close to a close... |
| bj-nfdt 36684 | Closed form of ~ nf5d and ... |
| bj-nexdt 36685 | Closed form of ~ nexd . (... |
| bj-nexdvt 36686 | Closed form of ~ nexdv . ... |
| bj-alexbiex 36687 | Adding a second quantifier... |
| bj-exexbiex 36688 | Adding a second quantifier... |
| bj-alalbial 36689 | Adding a second quantifier... |
| bj-exalbial 36690 | Adding a second quantifier... |
| bj-19.9htbi 36691 | Strengthening ~ 19.9ht by ... |
| bj-hbntbi 36692 | Strengthening ~ hbnt by re... |
| bj-biexal1 36693 | A general FOL biconditiona... |
| bj-biexal2 36694 | When ` ph ` is substituted... |
| bj-biexal3 36695 | When ` ph ` is substituted... |
| bj-bialal 36696 | When ` ph ` is substituted... |
| bj-biexex 36697 | When ` ph ` is substituted... |
| bj-hbext 36698 | Closed form of ~ hbex . (... |
| bj-nfalt 36699 | Closed form of ~ nfal . (... |
| bj-nfext 36700 | Closed form of ~ nfex . (... |
| bj-eeanvw 36701 | Version of ~ exdistrv with... |
| bj-modal4 36702 | First-order logic form of ... |
| bj-modal4e 36703 | First-order logic form of ... |
| bj-modalb 36704 | A short form of the axiom ... |
| bj-wnf1 36705 | When ` ph ` is substituted... |
| bj-wnf2 36706 | When ` ph ` is substituted... |
| bj-wnfanf 36707 | When ` ph ` is substituted... |
| bj-wnfenf 36708 | When ` ph ` is substituted... |
| bj-substax12 36709 | Equivalent form of the axi... |
| bj-substw 36710 | Weak form of the LHS of ~ ... |
| bj-nnfbi 36713 | If two formulas are equiva... |
| bj-nnfbd 36714 | If two formulas are equiva... |
| bj-nnfbii 36715 | If two formulas are equiva... |
| bj-nnfa 36716 | Nonfreeness implies the eq... |
| bj-nnfad 36717 | Nonfreeness implies the eq... |
| bj-nnfai 36718 | Nonfreeness implies the eq... |
| bj-nnfe 36719 | Nonfreeness implies the eq... |
| bj-nnfed 36720 | Nonfreeness implies the eq... |
| bj-nnfei 36721 | Nonfreeness implies the eq... |
| bj-nnfea 36722 | Nonfreeness implies the eq... |
| bj-nnfead 36723 | Nonfreeness implies the eq... |
| bj-nnfeai 36724 | Nonfreeness implies the eq... |
| bj-dfnnf2 36725 | Alternate definition of ~ ... |
| bj-nnfnfTEMP 36726 | New nonfreeness implies ol... |
| bj-wnfnf 36727 | When ` ph ` is substituted... |
| bj-nnfnt 36728 | A variable is nonfree in a... |
| bj-nnftht 36729 | A variable is nonfree in a... |
| bj-nnfth 36730 | A variable is nonfree in a... |
| bj-nnfnth 36731 | A variable is nonfree in t... |
| bj-nnfim1 36732 | A consequence of nonfreene... |
| bj-nnfim2 36733 | A consequence of nonfreene... |
| bj-nnfim 36734 | Nonfreeness in the anteced... |
| bj-nnfimd 36735 | Nonfreeness in the anteced... |
| bj-nnfan 36736 | Nonfreeness in both conjun... |
| bj-nnfand 36737 | Nonfreeness in both conjun... |
| bj-nnfor 36738 | Nonfreeness in both disjun... |
| bj-nnford 36739 | Nonfreeness in both disjun... |
| bj-nnfbit 36740 | Nonfreeness in both sides ... |
| bj-nnfbid 36741 | Nonfreeness in both sides ... |
| bj-nnfv 36742 | A non-occurring variable i... |
| bj-nnf-alrim 36743 | Proof of the closed form o... |
| bj-nnf-exlim 36744 | Proof of the closed form o... |
| bj-dfnnf3 36745 | Alternate definition of no... |
| bj-nfnnfTEMP 36746 | New nonfreeness is equival... |
| bj-nnfa1 36747 | See ~ nfa1 . (Contributed... |
| bj-nnfe1 36748 | See ~ nfe1 . (Contributed... |
| bj-19.12 36749 | See ~ 19.12 . Could be la... |
| bj-nnflemaa 36750 | One of four lemmas for non... |
| bj-nnflemee 36751 | One of four lemmas for non... |
| bj-nnflemae 36752 | One of four lemmas for non... |
| bj-nnflemea 36753 | One of four lemmas for non... |
| bj-nnfalt 36754 | See ~ nfal and ~ bj-nfalt ... |
| bj-nnfext 36755 | See ~ nfex and ~ bj-nfext ... |
| bj-stdpc5t 36756 | Alias of ~ bj-nnf-alrim fo... |
| bj-19.21t 36757 | Statement ~ 19.21t proved ... |
| bj-19.23t 36758 | Statement ~ 19.23t proved ... |
| bj-19.36im 36759 | One direction of ~ 19.36 f... |
| bj-19.37im 36760 | One direction of ~ 19.37 f... |
| bj-19.42t 36761 | Closed form of ~ 19.42 fro... |
| bj-19.41t 36762 | Closed form of ~ 19.41 fro... |
| bj-sbft 36763 | Version of ~ sbft using ` ... |
| bj-pm11.53vw 36764 | Version of ~ pm11.53v with... |
| bj-pm11.53v 36765 | Version of ~ pm11.53v with... |
| bj-pm11.53a 36766 | A variant of ~ pm11.53v . ... |
| bj-equsvt 36767 | A variant of ~ equsv . (C... |
| bj-equsalvwd 36768 | Variant of ~ equsalvw . (... |
| bj-equsexvwd 36769 | Variant of ~ equsexvw . (... |
| bj-sbievwd 36770 | Variant of ~ sbievw . (Co... |
| bj-axc10 36771 | Alternate proof of ~ axc10... |
| bj-alequex 36772 | A fol lemma. See ~ aleque... |
| bj-spimt2 36773 | A step in the proof of ~ s... |
| bj-cbv3ta 36774 | Closed form of ~ cbv3 . (... |
| bj-cbv3tb 36775 | Closed form of ~ cbv3 . (... |
| bj-hbsb3t 36776 | A theorem close to a close... |
| bj-hbsb3 36777 | Shorter proof of ~ hbsb3 .... |
| bj-nfs1t 36778 | A theorem close to a close... |
| bj-nfs1t2 36779 | A theorem close to a close... |
| bj-nfs1 36780 | Shorter proof of ~ nfs1 (t... |
| bj-axc10v 36781 | Version of ~ axc10 with a ... |
| bj-spimtv 36782 | Version of ~ spimt with a ... |
| bj-cbv3hv2 36783 | Version of ~ cbv3h with tw... |
| bj-cbv1hv 36784 | Version of ~ cbv1h with a ... |
| bj-cbv2hv 36785 | Version of ~ cbv2h with a ... |
| bj-cbv2v 36786 | Version of ~ cbv2 with a d... |
| bj-cbvaldv 36787 | Version of ~ cbvald with a... |
| bj-cbvexdv 36788 | Version of ~ cbvexd with a... |
| bj-cbval2vv 36789 | Version of ~ cbval2vv with... |
| bj-cbvex2vv 36790 | Version of ~ cbvex2vv with... |
| bj-cbvaldvav 36791 | Version of ~ cbvaldva with... |
| bj-cbvexdvav 36792 | Version of ~ cbvexdva with... |
| bj-cbvex4vv 36793 | Version of ~ cbvex4v with ... |
| bj-equsalhv 36794 | Version of ~ equsalh with ... |
| bj-axc11nv 36795 | Version of ~ axc11n with a... |
| bj-aecomsv 36796 | Version of ~ aecoms with a... |
| bj-axc11v 36797 | Version of ~ axc11 with a ... |
| bj-drnf2v 36798 | Version of ~ drnf2 with a ... |
| bj-equs45fv 36799 | Version of ~ equs45f with ... |
| bj-hbs1 36800 | Version of ~ hbsb2 with a ... |
| bj-nfs1v 36801 | Version of ~ nfsb2 with a ... |
| bj-hbsb2av 36802 | Version of ~ hbsb2a with a... |
| bj-hbsb3v 36803 | Version of ~ hbsb3 with a ... |
| bj-nfsab1 36804 | Remove dependency on ~ ax-... |
| bj-dtrucor2v 36805 | Version of ~ dtrucor2 with... |
| bj-hbaeb2 36806 | Biconditional version of a... |
| bj-hbaeb 36807 | Biconditional version of ~... |
| bj-hbnaeb 36808 | Biconditional version of ~... |
| bj-dvv 36809 | A special instance of ~ bj... |
| bj-equsal1t 36810 | Duplication of ~ wl-equsal... |
| bj-equsal1ti 36811 | Inference associated with ... |
| bj-equsal1 36812 | One direction of ~ equsal ... |
| bj-equsal2 36813 | One direction of ~ equsal ... |
| bj-equsal 36814 | Shorter proof of ~ equsal ... |
| stdpc5t 36815 | Closed form of ~ stdpc5 . ... |
| bj-stdpc5 36816 | More direct proof of ~ std... |
| 2stdpc5 36817 | A double ~ stdpc5 (one dir... |
| bj-19.21t0 36818 | Proof of ~ 19.21t from ~ s... |
| exlimii 36819 | Inference associated with ... |
| ax11-pm 36820 | Proof of ~ ax-11 similar t... |
| ax6er 36821 | Commuted form of ~ ax6e . ... |
| exlimiieq1 36822 | Inferring a theorem when i... |
| exlimiieq2 36823 | Inferring a theorem when i... |
| ax11-pm2 36824 | Proof of ~ ax-11 from the ... |
| bj-sbsb 36825 | Biconditional showing two ... |
| bj-dfsb2 36826 | Alternate (dual) definitio... |
| bj-sbf3 36827 | Substitution has no effect... |
| bj-sbf4 36828 | Substitution has no effect... |
| bj-eu3f 36829 | Version of ~ eu3v where th... |
| bj-sblem1 36830 | Lemma for substitution. (... |
| bj-sblem2 36831 | Lemma for substitution. (... |
| bj-sblem 36832 | Lemma for substitution. (... |
| bj-sbievw1 36833 | Lemma for substitution. (... |
| bj-sbievw2 36834 | Lemma for substitution. (... |
| bj-sbievw 36835 | Lemma for substitution. C... |
| bj-sbievv 36836 | Version of ~ sbie with a s... |
| bj-moeub 36837 | Uniqueness is equivalent t... |
| bj-sbidmOLD 36838 | Obsolete proof of ~ sbidm ... |
| bj-dvelimdv 36839 | Deduction form of ~ dvelim... |
| bj-dvelimdv1 36840 | Curried (exported) form of... |
| bj-dvelimv 36841 | A version of ~ dvelim usin... |
| bj-nfeel2 36842 | Nonfreeness in a membershi... |
| bj-axc14nf 36843 | Proof of a version of ~ ax... |
| bj-axc14 36844 | Alternate proof of ~ axc14... |
| mobidvALT 36845 | Alternate proof of ~ mobid... |
| sbn1ALT 36846 | Alternate proof of ~ sbn1 ... |
| eliminable1 36847 | A theorem used to prove th... |
| eliminable2a 36848 | A theorem used to prove th... |
| eliminable2b 36849 | A theorem used to prove th... |
| eliminable2c 36850 | A theorem used to prove th... |
| eliminable3a 36851 | A theorem used to prove th... |
| eliminable3b 36852 | A theorem used to prove th... |
| eliminable-velab 36853 | A theorem used to prove th... |
| eliminable-veqab 36854 | A theorem used to prove th... |
| eliminable-abeqv 36855 | A theorem used to prove th... |
| eliminable-abeqab 36856 | A theorem used to prove th... |
| eliminable-abelv 36857 | A theorem used to prove th... |
| eliminable-abelab 36858 | A theorem used to prove th... |
| bj-denoteslem 36859 | Duplicate of ~ issettru an... |
| bj-denotesALTV 36860 | Moved to main as ~ iseqset... |
| bj-issettruALTV 36861 | Moved to main as ~ issettr... |
| bj-elabtru 36862 | This is as close as we can... |
| bj-issetwt 36863 | Closed form of ~ bj-issetw... |
| bj-issetw 36864 | The closest one can get to... |
| bj-issetiv 36865 | Version of ~ bj-isseti wit... |
| bj-isseti 36866 | Version of ~ isseti with a... |
| bj-ralvw 36867 | A weak version of ~ ralv n... |
| bj-rexvw 36868 | A weak version of ~ rexv n... |
| bj-rababw 36869 | A weak version of ~ rabab ... |
| bj-rexcom4bv 36870 | Version of ~ rexcom4b and ... |
| bj-rexcom4b 36871 | Remove from ~ rexcom4b dep... |
| bj-ceqsalt0 36872 | The FOL content of ~ ceqsa... |
| bj-ceqsalt1 36873 | The FOL content of ~ ceqsa... |
| bj-ceqsalt 36874 | Remove from ~ ceqsalt depe... |
| bj-ceqsaltv 36875 | Version of ~ bj-ceqsalt wi... |
| bj-ceqsalg0 36876 | The FOL content of ~ ceqsa... |
| bj-ceqsalg 36877 | Remove from ~ ceqsalg depe... |
| bj-ceqsalgALT 36878 | Alternate proof of ~ bj-ce... |
| bj-ceqsalgv 36879 | Version of ~ bj-ceqsalg wi... |
| bj-ceqsalgvALT 36880 | Alternate proof of ~ bj-ce... |
| bj-ceqsal 36881 | Remove from ~ ceqsal depen... |
| bj-ceqsalv 36882 | Remove from ~ ceqsalv depe... |
| bj-spcimdv 36883 | Remove from ~ spcimdv depe... |
| bj-spcimdvv 36884 | Remove from ~ spcimdv depe... |
| elelb 36885 | Equivalence between two co... |
| bj-pwvrelb 36886 | Characterization of the el... |
| bj-nfcsym 36887 | The nonfreeness quantifier... |
| bj-sbeqALT 36888 | Substitution in an equalit... |
| bj-sbeq 36889 | Distribute proper substitu... |
| bj-sbceqgALT 36890 | Distribute proper substitu... |
| bj-csbsnlem 36891 | Lemma for ~ bj-csbsn (in t... |
| bj-csbsn 36892 | Substitution in a singleto... |
| bj-sbel1 36893 | Version of ~ sbcel1g when ... |
| bj-abv 36894 | The class of sets verifyin... |
| bj-abvALT 36895 | Alternate version of ~ bj-... |
| bj-ab0 36896 | The class of sets verifyin... |
| bj-abf 36897 | Shorter proof of ~ abf (wh... |
| bj-csbprc 36898 | More direct proof of ~ csb... |
| bj-exlimvmpi 36899 | A Fol lemma ( ~ exlimiv fo... |
| bj-exlimmpi 36900 | Lemma for ~ bj-vtoclg1f1 (... |
| bj-exlimmpbi 36901 | Lemma for theorems of the ... |
| bj-exlimmpbir 36902 | Lemma for theorems of the ... |
| bj-vtoclf 36903 | Remove dependency on ~ ax-... |
| bj-vtocl 36904 | Remove dependency on ~ ax-... |
| bj-vtoclg1f1 36905 | The FOL content of ~ vtocl... |
| bj-vtoclg1f 36906 | Reprove ~ vtoclg1f from ~ ... |
| bj-vtoclg1fv 36907 | Version of ~ bj-vtoclg1f w... |
| bj-vtoclg 36908 | A version of ~ vtoclg with... |
| bj-rabeqbid 36909 | Version of ~ rabeqbidv wit... |
| bj-seex 36910 | Version of ~ seex with a d... |
| bj-nfcf 36911 | Version of ~ df-nfc with a... |
| bj-zfauscl 36912 | General version of ~ zfaus... |
| bj-elabd2ALT 36913 | Alternate proof of ~ elabd... |
| bj-unrab 36914 | Generalization of ~ unrab ... |
| bj-inrab 36915 | Generalization of ~ inrab ... |
| bj-inrab2 36916 | Shorter proof of ~ inrab .... |
| bj-inrab3 36917 | Generalization of ~ dfrab3... |
| bj-rabtr 36918 | Restricted class abstracti... |
| bj-rabtrALT 36919 | Alternate proof of ~ bj-ra... |
| bj-rabtrAUTO 36920 | Proof of ~ bj-rabtr found ... |
| bj-gabss 36923 | Inclusion of generalized c... |
| bj-gabssd 36924 | Inclusion of generalized c... |
| bj-gabeqd 36925 | Equality of generalized cl... |
| bj-gabeqis 36926 | Equality of generalized cl... |
| bj-elgab 36927 | Elements of a generalized ... |
| bj-gabima 36928 | Generalized class abstract... |
| bj-ru1 36931 | A version of Russell's par... |
| bj-ru 36932 | Remove dependency on ~ ax-... |
| currysetlem 36933 | Lemma for ~ currysetlem , ... |
| curryset 36934 | Curry's paradox in set the... |
| currysetlem1 36935 | Lemma for ~ currysetALT . ... |
| currysetlem2 36936 | Lemma for ~ currysetALT . ... |
| currysetlem3 36937 | Lemma for ~ currysetALT . ... |
| currysetALT 36938 | Alternate proof of ~ curry... |
| bj-n0i 36939 | Inference associated with ... |
| bj-disjsn01 36940 | Disjointness of the single... |
| bj-0nel1 36941 | The empty set does not bel... |
| bj-1nel0 36942 | ` 1o ` does not belong to ... |
| bj-xpimasn 36943 | The image of a singleton, ... |
| bj-xpima1sn 36944 | The image of a singleton b... |
| bj-xpima1snALT 36945 | Alternate proof of ~ bj-xp... |
| bj-xpima2sn 36946 | The image of a singleton b... |
| bj-xpnzex 36947 | If the first factor of a p... |
| bj-xpexg2 36948 | Curried (exported) form of... |
| bj-xpnzexb 36949 | If the first factor of a p... |
| bj-cleq 36950 | Substitution property for ... |
| bj-snsetex 36951 | The class of sets "whose s... |
| bj-clexab 36952 | Sethood of certain classes... |
| bj-sngleq 36955 | Substitution property for ... |
| bj-elsngl 36956 | Characterization of the el... |
| bj-snglc 36957 | Characterization of the el... |
| bj-snglss 36958 | The singletonization of a ... |
| bj-0nelsngl 36959 | The empty set is not a mem... |
| bj-snglinv 36960 | Inverse of singletonizatio... |
| bj-snglex 36961 | A class is a set if and on... |
| bj-tageq 36964 | Substitution property for ... |
| bj-eltag 36965 | Characterization of the el... |
| bj-0eltag 36966 | The empty set belongs to t... |
| bj-tagn0 36967 | The tagging of a class is ... |
| bj-tagss 36968 | The tagging of a class is ... |
| bj-snglsstag 36969 | The singletonization is in... |
| bj-sngltagi 36970 | The singletonization is in... |
| bj-sngltag 36971 | The singletonization and t... |
| bj-tagci 36972 | Characterization of the el... |
| bj-tagcg 36973 | Characterization of the el... |
| bj-taginv 36974 | Inverse of tagging. (Cont... |
| bj-tagex 36975 | A class is a set if and on... |
| bj-xtageq 36976 | The products of a given cl... |
| bj-xtagex 36977 | The product of a set and t... |
| bj-projeq 36980 | Substitution property for ... |
| bj-projeq2 36981 | Substitution property for ... |
| bj-projun 36982 | The class projection on a ... |
| bj-projex 36983 | Sethood of the class proje... |
| bj-projval 36984 | Value of the class project... |
| bj-1upleq 36987 | Substitution property for ... |
| bj-pr1eq 36990 | Substitution property for ... |
| bj-pr1un 36991 | The first projection prese... |
| bj-pr1val 36992 | Value of the first project... |
| bj-pr11val 36993 | Value of the first project... |
| bj-pr1ex 36994 | Sethood of the first proje... |
| bj-1uplth 36995 | The characteristic propert... |
| bj-1uplex 36996 | A monuple is a set if and ... |
| bj-1upln0 36997 | A monuple is nonempty. (C... |
| bj-2upleq 37000 | Substitution property for ... |
| bj-pr21val 37001 | Value of the first project... |
| bj-pr2eq 37004 | Substitution property for ... |
| bj-pr2un 37005 | The second projection pres... |
| bj-pr2val 37006 | Value of the second projec... |
| bj-pr22val 37007 | Value of the second projec... |
| bj-pr2ex 37008 | Sethood of the second proj... |
| bj-2uplth 37009 | The characteristic propert... |
| bj-2uplex 37010 | A couple is a set if and o... |
| bj-2upln0 37011 | A couple is nonempty. (Co... |
| bj-2upln1upl 37012 | A couple is never equal to... |
| bj-rcleqf 37013 | Relative version of ~ cleq... |
| bj-rcleq 37014 | Relative version of ~ dfcl... |
| bj-reabeq 37015 | Relative form of ~ eqabb .... |
| bj-disj2r 37016 | Relative version of ~ ssdi... |
| bj-sscon 37017 | Contraposition law for rel... |
| bj-abex 37018 | Two ways of stating that t... |
| bj-clex 37019 | Two ways of stating that a... |
| bj-axsn 37020 | Two ways of stating the ax... |
| bj-snexg 37022 | A singleton built on a set... |
| bj-snex 37023 | A singleton is a set. See... |
| bj-axbun 37024 | Two ways of stating the ax... |
| bj-unexg 37026 | Existence of binary unions... |
| bj-prexg 37027 | Existence of unordered pai... |
| bj-prex 37028 | Existence of unordered pai... |
| bj-axadj 37029 | Two ways of stating the ax... |
| bj-adjg1 37031 | Existence of the result of... |
| bj-snfromadj 37032 | Singleton from adjunction ... |
| bj-prfromadj 37033 | Unordered pair from adjunc... |
| bj-adjfrombun 37034 | Adjunction from singleton ... |
| eleq2w2ALT 37035 | Alternate proof of ~ eleq2... |
| bj-clel3gALT 37036 | Alternate proof of ~ clel3... |
| bj-pw0ALT 37037 | Alternate proof of ~ pw0 .... |
| bj-sselpwuni 37038 | Quantitative version of ~ ... |
| bj-unirel 37039 | Quantitative version of ~ ... |
| bj-elpwg 37040 | If the intersection of two... |
| bj-velpwALT 37041 | This theorem ~ bj-velpwALT... |
| bj-elpwgALT 37042 | Alternate proof of ~ elpwg... |
| bj-vjust 37043 | Justification theorem for ... |
| bj-nul 37044 | Two formulations of the ax... |
| bj-nuliota 37045 | Definition of the empty se... |
| bj-nuliotaALT 37046 | Alternate proof of ~ bj-nu... |
| bj-vtoclgfALT 37047 | Alternate proof of ~ vtocl... |
| bj-elsn12g 37048 | Join of ~ elsng and ~ elsn... |
| bj-elsnb 37049 | Biconditional version of ~... |
| bj-pwcfsdom 37050 | Remove hypothesis from ~ p... |
| bj-grur1 37051 | Remove hypothesis from ~ g... |
| bj-bm1.3ii 37052 | The extension of a predica... |
| bj-dfid2ALT 37053 | Alternate version of ~ dfi... |
| bj-0nelopab 37054 | The empty set is never an ... |
| bj-brrelex12ALT 37055 | Two classes related by a b... |
| bj-epelg 37056 | The membership relation an... |
| bj-epelb 37057 | Two classes are related by... |
| bj-nsnid 37058 | A set does not contain the... |
| bj-rdg0gALT 37059 | Alternate proof of ~ rdg0g... |
| bj-evaleq 37060 | Equality theorem for the `... |
| bj-evalfun 37061 | The evaluation at a class ... |
| bj-evalfn 37062 | The evaluation at a class ... |
| bj-evalval 37063 | Value of the evaluation at... |
| bj-evalid 37064 | The evaluation at a set of... |
| bj-ndxarg 37065 | Proof of ~ ndxarg from ~ b... |
| bj-evalidval 37066 | Closed general form of ~ s... |
| bj-rest00 37069 | An elementwise intersectio... |
| bj-restsn 37070 | An elementwise intersectio... |
| bj-restsnss 37071 | Special case of ~ bj-rests... |
| bj-restsnss2 37072 | Special case of ~ bj-rests... |
| bj-restsn0 37073 | An elementwise intersectio... |
| bj-restsn10 37074 | Special case of ~ bj-rests... |
| bj-restsnid 37075 | The elementwise intersecti... |
| bj-rest10 37076 | An elementwise intersectio... |
| bj-rest10b 37077 | Alternate version of ~ bj-... |
| bj-restn0 37078 | An elementwise intersectio... |
| bj-restn0b 37079 | Alternate version of ~ bj-... |
| bj-restpw 37080 | The elementwise intersecti... |
| bj-rest0 37081 | An elementwise intersectio... |
| bj-restb 37082 | An elementwise intersectio... |
| bj-restv 37083 | An elementwise intersectio... |
| bj-resta 37084 | An elementwise intersectio... |
| bj-restuni 37085 | The union of an elementwis... |
| bj-restuni2 37086 | The union of an elementwis... |
| bj-restreg 37087 | A reformulation of the axi... |
| bj-raldifsn 37088 | All elements in a set sati... |
| bj-0int 37089 | If ` A ` is a collection o... |
| bj-mooreset 37090 | A Moore collection is a se... |
| bj-ismoore 37093 | Characterization of Moore ... |
| bj-ismoored0 37094 | Necessary condition to be ... |
| bj-ismoored 37095 | Necessary condition to be ... |
| bj-ismoored2 37096 | Necessary condition to be ... |
| bj-ismooredr 37097 | Sufficient condition to be... |
| bj-ismooredr2 37098 | Sufficient condition to be... |
| bj-discrmoore 37099 | The powerclass ` ~P A ` is... |
| bj-0nmoore 37100 | The empty set is not a Moo... |
| bj-snmoore 37101 | A singleton is a Moore col... |
| bj-snmooreb 37102 | A singleton is a Moore col... |
| bj-prmoore 37103 | A pair formed of two neste... |
| bj-0nelmpt 37104 | The empty set is not an el... |
| bj-mptval 37105 | Value of a function given ... |
| bj-dfmpoa 37106 | An equivalent definition o... |
| bj-mpomptALT 37107 | Alternate proof of ~ mpomp... |
| setsstrset 37124 | Relation between ~ df-sets... |
| bj-nfald 37125 | Variant of ~ nfald . (Con... |
| bj-nfexd 37126 | Variant of ~ nfexd . (Con... |
| copsex2d 37127 | Implicit substitution dedu... |
| copsex2b 37128 | Biconditional form of ~ co... |
| opelopabd 37129 | Membership of an ordere pa... |
| opelopabb 37130 | Membership of an ordered p... |
| opelopabbv 37131 | Membership of an ordered p... |
| bj-opelrelex 37132 | The coordinates of an orde... |
| bj-opelresdm 37133 | If an ordered pair is in a... |
| bj-brresdm 37134 | If two classes are related... |
| brabd0 37135 | Expressing that two sets a... |
| brabd 37136 | Expressing that two sets a... |
| bj-brab2a1 37137 | "Unbounded" version of ~ b... |
| bj-opabssvv 37138 | A variant of ~ relopabiv (... |
| bj-funidres 37139 | The restricted identity re... |
| bj-opelidb 37140 | Characterization of the or... |
| bj-opelidb1 37141 | Characterization of the or... |
| bj-inexeqex 37142 | Lemma for ~ bj-opelid (but... |
| bj-elsn0 37143 | If the intersection of two... |
| bj-opelid 37144 | Characterization of the or... |
| bj-ideqg 37145 | Characterization of the cl... |
| bj-ideqgALT 37146 | Alternate proof of ~ bj-id... |
| bj-ideqb 37147 | Characterization of classe... |
| bj-idres 37148 | Alternate expression for t... |
| bj-opelidres 37149 | Characterization of the or... |
| bj-idreseq 37150 | Sufficient condition for t... |
| bj-idreseqb 37151 | Characterization for two c... |
| bj-ideqg1 37152 | For sets, the identity rel... |
| bj-ideqg1ALT 37153 | Alternate proof of bj-ideq... |
| bj-opelidb1ALT 37154 | Characterization of the co... |
| bj-elid3 37155 | Characterization of the co... |
| bj-elid4 37156 | Characterization of the el... |
| bj-elid5 37157 | Characterization of the el... |
| bj-elid6 37158 | Characterization of the el... |
| bj-elid7 37159 | Characterization of the el... |
| bj-diagval 37162 | Value of the functionalize... |
| bj-diagval2 37163 | Value of the functionalize... |
| bj-eldiag 37164 | Characterization of the el... |
| bj-eldiag2 37165 | Characterization of the el... |
| bj-imdirvallem 37168 | Lemma for ~ bj-imdirval an... |
| bj-imdirval 37169 | Value of the functionalize... |
| bj-imdirval2lem 37170 | Lemma for ~ bj-imdirval2 a... |
| bj-imdirval2 37171 | Value of the functionalize... |
| bj-imdirval3 37172 | Value of the functionalize... |
| bj-imdiridlem 37173 | Lemma for ~ bj-imdirid and... |
| bj-imdirid 37174 | Functorial property of the... |
| bj-opelopabid 37175 | Membership in an ordered-p... |
| bj-opabco 37176 | Composition of ordered-pai... |
| bj-xpcossxp 37177 | The composition of two Car... |
| bj-imdirco 37178 | Functorial property of the... |
| bj-iminvval 37181 | Value of the functionalize... |
| bj-iminvval2 37182 | Value of the functionalize... |
| bj-iminvid 37183 | Functorial property of the... |
| bj-inftyexpitaufo 37190 | The function ` inftyexpita... |
| bj-inftyexpitaudisj 37193 | An element of the circle a... |
| bj-inftyexpiinv 37196 | Utility theorem for the in... |
| bj-inftyexpiinj 37197 | Injectivity of the paramet... |
| bj-inftyexpidisj 37198 | An element of the circle a... |
| bj-ccinftydisj 37201 | The circle at infinity is ... |
| bj-elccinfty 37202 | A lemma for infinite exten... |
| bj-ccssccbar 37205 | Complex numbers are extend... |
| bj-ccinftyssccbar 37206 | Infinite extended complex ... |
| bj-pinftyccb 37209 | The class ` pinfty ` is an... |
| bj-pinftynrr 37210 | The extended complex numbe... |
| bj-minftyccb 37213 | The class ` minfty ` is an... |
| bj-minftynrr 37214 | The extended complex numbe... |
| bj-pinftynminfty 37215 | The extended complex numbe... |
| bj-rrhatsscchat 37224 | The real projective line i... |
| bj-imafv 37239 | If the direct image of a s... |
| bj-funun 37240 | Value of a function expres... |
| bj-fununsn1 37241 | Value of a function expres... |
| bj-fununsn2 37242 | Value of a function expres... |
| bj-fvsnun1 37243 | The value of a function wi... |
| bj-fvsnun2 37244 | The value of a function wi... |
| bj-fvmptunsn1 37245 | Value of a function expres... |
| bj-fvmptunsn2 37246 | Value of a function expres... |
| bj-iomnnom 37247 | The canonical bijection fr... |
| bj-smgrpssmgm 37256 | Semigroups are magmas. (C... |
| bj-smgrpssmgmel 37257 | Semigroups are magmas (ele... |
| bj-mndsssmgrp 37258 | Monoids are semigroups. (... |
| bj-mndsssmgrpel 37259 | Monoids are semigroups (el... |
| bj-cmnssmnd 37260 | Commutative monoids are mo... |
| bj-cmnssmndel 37261 | Commutative monoids are mo... |
| bj-grpssmnd 37262 | Groups are monoids. (Cont... |
| bj-grpssmndel 37263 | Groups are monoids (elemen... |
| bj-ablssgrp 37264 | Abelian groups are groups.... |
| bj-ablssgrpel 37265 | Abelian groups are groups ... |
| bj-ablsscmn 37266 | Abelian groups are commuta... |
| bj-ablsscmnel 37267 | Abelian groups are commuta... |
| bj-modssabl 37268 | (The additive groups of) m... |
| bj-vecssmod 37269 | Vector spaces are modules.... |
| bj-vecssmodel 37270 | Vector spaces are modules ... |
| bj-finsumval0 37273 | Value of a finite sum. (C... |
| bj-fvimacnv0 37274 | Variant of ~ fvimacnv wher... |
| bj-isvec 37275 | The predicate "is a vector... |
| bj-fldssdrng 37276 | Fields are division rings.... |
| bj-flddrng 37277 | Fields are division rings ... |
| bj-rrdrg 37278 | The field of real numbers ... |
| bj-isclm 37279 | The predicate "is a subcom... |
| bj-isrvec 37282 | The predicate "is a real v... |
| bj-rvecmod 37283 | Real vector spaces are mod... |
| bj-rvecssmod 37284 | Real vector spaces are mod... |
| bj-rvecrr 37285 | The field of scalars of a ... |
| bj-isrvecd 37286 | The predicate "is a real v... |
| bj-rvecvec 37287 | Real vector spaces are vec... |
| bj-isrvec2 37288 | The predicate "is a real v... |
| bj-rvecssvec 37289 | Real vector spaces are vec... |
| bj-rveccmod 37290 | Real vector spaces are sub... |
| bj-rvecsscmod 37291 | Real vector spaces are sub... |
| bj-rvecsscvec 37292 | Real vector spaces are sub... |
| bj-rveccvec 37293 | Real vector spaces are sub... |
| bj-rvecssabl 37294 | (The additive groups of) r... |
| bj-rvecabl 37295 | (The additive groups of) r... |
| bj-subcom 37296 | A consequence of commutati... |
| bj-lineqi 37297 | Solution of a (scalar) lin... |
| bj-bary1lem 37298 | Lemma for ~ bj-bary1 : exp... |
| bj-bary1lem1 37299 | Lemma for ~ bj-bary1 : com... |
| bj-bary1 37300 | Barycentric coordinates in... |
| bj-endval 37303 | Value of the monoid of end... |
| bj-endbase 37304 | Base set of the monoid of ... |
| bj-endcomp 37305 | Composition law of the mon... |
| bj-endmnd 37306 | The monoid of endomorphism... |
| taupilem3 37307 | Lemma for tau-related theo... |
| taupilemrplb 37308 | A set of positive reals ha... |
| taupilem1 37309 | Lemma for ~ taupi . A pos... |
| taupilem2 37310 | Lemma for ~ taupi . The s... |
| taupi 37311 | Relationship between ` _ta... |
| dfgcd3 37312 | Alternate definition of th... |
| irrdifflemf 37313 | Lemma for ~ irrdiff . The... |
| irrdiff 37314 | The irrationals are exactl... |
| iccioo01 37315 | The closed unit interval i... |
| csbrecsg 37316 | Move class substitution in... |
| csbrdgg 37317 | Move class substitution in... |
| csboprabg 37318 | Move class substitution in... |
| csbmpo123 37319 | Move class substitution in... |
| con1bii2 37320 | A contraposition inference... |
| con2bii2 37321 | A contraposition inference... |
| vtoclefex 37322 | Implicit substitution of a... |
| rnmptsn 37323 | The range of a function ma... |
| f1omptsnlem 37324 | This is the core of the pr... |
| f1omptsn 37325 | A function mapping to sing... |
| mptsnunlem 37326 | This is the core of the pr... |
| mptsnun 37327 | A class ` B ` is equal to ... |
| dissneqlem 37328 | This is the core of the pr... |
| dissneq 37329 | Any topology that contains... |
| exlimim 37330 | Closed form of ~ exlimimd ... |
| exlimimd 37331 | Existential elimination ru... |
| exellim 37332 | Closed form of ~ exellimdd... |
| exellimddv 37333 | Eliminate an antecedent wh... |
| topdifinfindis 37334 | Part of Exercise 3 of [Mun... |
| topdifinffinlem 37335 | This is the core of the pr... |
| topdifinffin 37336 | Part of Exercise 3 of [Mun... |
| topdifinf 37337 | Part of Exercise 3 of [Mun... |
| topdifinfeq 37338 | Two different ways of defi... |
| icorempo 37339 | Closed-below, open-above i... |
| icoreresf 37340 | Closed-below, open-above i... |
| icoreval 37341 | Value of the closed-below,... |
| icoreelrnab 37342 | Elementhood in the set of ... |
| isbasisrelowllem1 37343 | Lemma for ~ isbasisrelowl ... |
| isbasisrelowllem2 37344 | Lemma for ~ isbasisrelowl ... |
| icoreclin 37345 | The set of closed-below, o... |
| isbasisrelowl 37346 | The set of all closed-belo... |
| icoreunrn 37347 | The union of all closed-be... |
| istoprelowl 37348 | The set of all closed-belo... |
| icoreelrn 37349 | A class abstraction which ... |
| iooelexlt 37350 | An element of an open inte... |
| relowlssretop 37351 | The lower limit topology o... |
| relowlpssretop 37352 | The lower limit topology o... |
| sucneqond 37353 | Inequality of an ordinal s... |
| sucneqoni 37354 | Inequality of an ordinal s... |
| onsucuni3 37355 | If an ordinal number has a... |
| 1oequni2o 37356 | The ordinal number ` 1o ` ... |
| rdgsucuni 37357 | If an ordinal number has a... |
| rdgeqoa 37358 | If a recursive function wi... |
| elxp8 37359 | Membership in a Cartesian ... |
| cbveud 37360 | Deduction used to change b... |
| cbvreud 37361 | Deduction used to change b... |
| difunieq 37362 | The difference of unions i... |
| inunissunidif 37363 | Theorem about subsets of t... |
| rdgellim 37364 | Elementhood in a recursive... |
| rdglimss 37365 | A recursive definition at ... |
| rdgssun 37366 | In a recursive definition ... |
| exrecfnlem 37367 | Lemma for ~ exrecfn . (Co... |
| exrecfn 37368 | Theorem about the existenc... |
| exrecfnpw 37369 | For any base set, a set wh... |
| finorwe 37370 | If the Axiom of Infinity i... |
| dffinxpf 37373 | This theorem is the same a... |
| finxpeq1 37374 | Equality theorem for Carte... |
| finxpeq2 37375 | Equality theorem for Carte... |
| csbfinxpg 37376 | Distribute proper substitu... |
| finxpreclem1 37377 | Lemma for ` ^^ ` recursion... |
| finxpreclem2 37378 | Lemma for ` ^^ ` recursion... |
| finxp0 37379 | The value of Cartesian exp... |
| finxp1o 37380 | The value of Cartesian exp... |
| finxpreclem3 37381 | Lemma for ` ^^ ` recursion... |
| finxpreclem4 37382 | Lemma for ` ^^ ` recursion... |
| finxpreclem5 37383 | Lemma for ` ^^ ` recursion... |
| finxpreclem6 37384 | Lemma for ` ^^ ` recursion... |
| finxpsuclem 37385 | Lemma for ~ finxpsuc . (C... |
| finxpsuc 37386 | The value of Cartesian exp... |
| finxp2o 37387 | The value of Cartesian exp... |
| finxp3o 37388 | The value of Cartesian exp... |
| finxpnom 37389 | Cartesian exponentiation w... |
| finxp00 37390 | Cartesian exponentiation o... |
| iunctb2 37391 | Using the axiom of countab... |
| domalom 37392 | A class which dominates ev... |
| isinf2 37393 | The converse of ~ isinf . ... |
| ctbssinf 37394 | Using the axiom of choice,... |
| ralssiun 37395 | The index set of an indexe... |
| nlpineqsn 37396 | For every point ` p ` of a... |
| nlpfvineqsn 37397 | Given a subset ` A ` of ` ... |
| fvineqsnf1 37398 | A theorem about functions ... |
| fvineqsneu 37399 | A theorem about functions ... |
| fvineqsneq 37400 | A theorem about functions ... |
| pibp16 37401 | Property P000016 of pi-bas... |
| pibp19 37402 | Property P000019 of pi-bas... |
| pibp21 37403 | Property P000021 of pi-bas... |
| pibt1 37404 | Theorem T000001 of pi-base... |
| pibt2 37405 | Theorem T000002 of pi-base... |
| wl-section-prop 37406 | Intuitionistic logic is no... |
| wl-section-boot 37410 | In this section, I provide... |
| wl-luk-imim1i 37411 | Inference adding common co... |
| wl-luk-syl 37412 | An inference version of th... |
| wl-luk-imtrid 37413 | A syllogism rule of infere... |
| wl-luk-pm2.18d 37414 | Deduction based on reducti... |
| wl-luk-con4i 37415 | Inference rule. Copy of ~... |
| wl-luk-pm2.24i 37416 | Inference rule. Copy of ~... |
| wl-luk-a1i 37417 | Inference rule. Copy of ~... |
| wl-luk-mpi 37418 | A nested _modus ponens_ in... |
| wl-luk-imim2i 37419 | Inference adding common an... |
| wl-luk-imtrdi 37420 | A syllogism rule of infere... |
| wl-luk-ax3 37421 | ~ ax-3 proved from Lukasie... |
| wl-luk-ax1 37422 | ~ ax-1 proved from Lukasie... |
| wl-luk-pm2.27 37423 | This theorem, called "Asse... |
| wl-luk-com12 37424 | Inference that swaps (comm... |
| wl-luk-pm2.21 37425 | From a wff and its negatio... |
| wl-luk-con1i 37426 | A contraposition inference... |
| wl-luk-ja 37427 | Inference joining the ante... |
| wl-luk-imim2 37428 | A closed form of syllogism... |
| wl-luk-a1d 37429 | Deduction introducing an e... |
| wl-luk-ax2 37430 | ~ ax-2 proved from Lukasie... |
| wl-luk-id 37431 | Principle of identity. Th... |
| wl-luk-notnotr 37432 | Converse of double negatio... |
| wl-luk-pm2.04 37433 | Swap antecedents. Theorem... |
| wl-section-impchain 37434 | An implication like ` ( ps... |
| wl-impchain-mp-x 37435 | This series of theorems pr... |
| wl-impchain-mp-0 37436 | This theorem is the start ... |
| wl-impchain-mp-1 37437 | This theorem is in fact a ... |
| wl-impchain-mp-2 37438 | This theorem is in fact a ... |
| wl-impchain-com-1.x 37439 | It is often convenient to ... |
| wl-impchain-com-1.1 37440 | A degenerate form of antec... |
| wl-impchain-com-1.2 37441 | This theorem is in fact a ... |
| wl-impchain-com-1.3 37442 | This theorem is in fact a ... |
| wl-impchain-com-1.4 37443 | This theorem is in fact a ... |
| wl-impchain-com-n.m 37444 | This series of theorems al... |
| wl-impchain-com-2.3 37445 | This theorem is in fact a ... |
| wl-impchain-com-2.4 37446 | This theorem is in fact a ... |
| wl-impchain-com-3.2.1 37447 | This theorem is in fact a ... |
| wl-impchain-a1-x 37448 | If an implication chain is... |
| wl-impchain-a1-1 37449 | Inference rule, a copy of ... |
| wl-impchain-a1-2 37450 | Inference rule, a copy of ... |
| wl-impchain-a1-3 37451 | Inference rule, a copy of ... |
| wl-ifp-ncond1 37452 | If one case of an ` if- ` ... |
| wl-ifp-ncond2 37453 | If one case of an ` if- ` ... |
| wl-ifpimpr 37454 | If one case of an ` if- ` ... |
| wl-ifp4impr 37455 | If one case of an ` if- ` ... |
| wl-df-3xor 37456 | Alternative definition of ... |
| wl-df3xor2 37457 | Alternative definition of ... |
| wl-df3xor3 37458 | Alternative form of ~ wl-d... |
| wl-3xortru 37459 | If the first input is true... |
| wl-3xorfal 37460 | If the first input is fals... |
| wl-3xorbi 37461 | Triple xor can be replaced... |
| wl-3xorbi2 37462 | Alternative form of ~ wl-3... |
| wl-3xorbi123d 37463 | Equivalence theorem for tr... |
| wl-3xorbi123i 37464 | Equivalence theorem for tr... |
| wl-3xorrot 37465 | Rotation law for triple xo... |
| wl-3xorcoma 37466 | Commutative law for triple... |
| wl-3xorcomb 37467 | Commutative law for triple... |
| wl-3xornot1 37468 | Flipping the first input f... |
| wl-3xornot 37469 | Triple xor distributes ove... |
| wl-1xor 37470 | In the recursive scheme ... |
| wl-2xor 37471 | In the recursive scheme ... |
| wl-df-3mintru2 37472 | Alternative definition of ... |
| wl-df2-3mintru2 37473 | The adder carry in disjunc... |
| wl-df3-3mintru2 37474 | The adder carry in conjunc... |
| wl-df4-3mintru2 37475 | An alternative definition ... |
| wl-1mintru1 37476 | Using the recursion formul... |
| wl-1mintru2 37477 | Using the recursion formul... |
| wl-2mintru1 37478 | Using the recursion formul... |
| wl-2mintru2 37479 | Using the recursion formul... |
| wl-df3maxtru1 37480 | Assuming "(n+1)-maxtru1" `... |
| wl-ax13lem1 37482 | A version of ~ ax-wl-13v w... |
| wl-cleq-0 37483 |
Disclaimer: |
| wl-cleq-1 37484 |
Disclaimer: |
| wl-cleq-2 37485 |
Disclaimer: |
| wl-cleq-3 37486 |
Disclaimer: |
| wl-cleq-4 37487 |
Disclaimer: |
| wl-cleq-5 37488 |
Disclaimer: |
| wl-cleq-6 37489 |
Disclaimer: |
| wl-df-clab 37492 | Disclaimer: The material ... |
| wl-isseteq 37493 | A class equal to a set var... |
| wl-ax12v2cl 37494 | The class version of ~ ax1... |
| wl-mps 37495 | Replacing a nested consequ... |
| wl-syls1 37496 | Replacing a nested consequ... |
| wl-syls2 37497 | Replacing a nested anteced... |
| wl-embant 37498 | A true wff can always be a... |
| wl-orel12 37499 | In a conjunctive normal fo... |
| wl-cases2-dnf 37500 | A particular instance of ~... |
| wl-cbvmotv 37501 | Change bound variable. Us... |
| wl-moteq 37502 | Change bound variable. Us... |
| wl-motae 37503 | Change bound variable. Us... |
| wl-moae 37504 | Two ways to express "at mo... |
| wl-euae 37505 | Two ways to express "exact... |
| wl-nax6im 37506 | The following series of th... |
| wl-hbae1 37507 | This specialization of ~ h... |
| wl-naevhba1v 37508 | An instance of ~ hbn1w app... |
| wl-spae 37509 | Prove an instance of ~ sp ... |
| wl-speqv 37510 | Under the assumption ` -. ... |
| wl-19.8eqv 37511 | Under the assumption ` -. ... |
| wl-19.2reqv 37512 | Under the assumption ` -. ... |
| wl-nfalv 37513 | If ` x ` is not present in... |
| wl-nfimf1 37514 | An antecedent is irrelevan... |
| wl-nfae1 37515 | Unlike ~ nfae , this speci... |
| wl-nfnae1 37516 | Unlike ~ nfnae , this spec... |
| wl-aetr 37517 | A transitive law for varia... |
| wl-axc11r 37518 | Same as ~ axc11r , but usi... |
| wl-dral1d 37519 | A version of ~ dral1 with ... |
| wl-cbvalnaed 37520 | ~ wl-cbvalnae with a conte... |
| wl-cbvalnae 37521 | A more general version of ... |
| wl-exeq 37522 | The semantics of ` E. x y ... |
| wl-aleq 37523 | The semantics of ` A. x y ... |
| wl-nfeqfb 37524 | Extend ~ nfeqf to an equiv... |
| wl-nfs1t 37525 | If ` y ` is not free in ` ... |
| wl-equsalvw 37526 | Version of ~ equsalv with ... |
| wl-equsald 37527 | Deduction version of ~ equ... |
| wl-equsaldv 37528 | Deduction version of ~ equ... |
| wl-equsal 37529 | A useful equivalence relat... |
| wl-equsal1t 37530 | The expression ` x = y ` i... |
| wl-equsalcom 37531 | This simple equivalence ea... |
| wl-equsal1i 37532 | The antecedent ` x = y ` i... |
| wl-sbid2ft 37533 | A more general version of ... |
| wl-cbvalsbi 37534 | Change bounded variables i... |
| wl-sbrimt 37535 | Substitution with a variab... |
| wl-sblimt 37536 | Substitution with a variab... |
| wl-sb9v 37537 | Commutation of quantificat... |
| wl-sb8ft 37538 | Substitution of variable i... |
| wl-sb8eft 37539 | Substitution of variable i... |
| wl-sb8t 37540 | Substitution of variable i... |
| wl-sb8et 37541 | Substitution of variable i... |
| wl-sbhbt 37542 | Closed form of ~ sbhb . C... |
| wl-sbnf1 37543 | Two ways expressing that `... |
| wl-equsb3 37544 | ~ equsb3 with a distinctor... |
| wl-equsb4 37545 | Substitution applied to an... |
| wl-2sb6d 37546 | Version of ~ 2sb6 with a c... |
| wl-sbcom2d-lem1 37547 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d-lem2 37548 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d 37549 | Version of ~ sbcom2 with a... |
| wl-sbalnae 37550 | A theorem used in eliminat... |
| wl-sbal1 37551 | A theorem used in eliminat... |
| wl-sbal2 37552 | Move quantifier in and out... |
| wl-2spsbbi 37553 | ~ spsbbi applied twice. (... |
| wl-lem-exsb 37554 | This theorem provides a ba... |
| wl-lem-nexmo 37555 | This theorem provides a ba... |
| wl-lem-moexsb 37556 | The antecedent ` A. x ( ph... |
| wl-alanbii 37557 | This theorem extends ~ ala... |
| wl-mo2df 37558 | Version of ~ mof with a co... |
| wl-mo2tf 37559 | Closed form of ~ mof with ... |
| wl-eudf 37560 | Version of ~ eu6 with a co... |
| wl-eutf 37561 | Closed form of ~ eu6 with ... |
| wl-euequf 37562 | ~ euequ proved with a dist... |
| wl-mo2t 37563 | Closed form of ~ mof . (C... |
| wl-mo3t 37564 | Closed form of ~ mo3 . (C... |
| wl-nfsbtv 37565 | Closed form of ~ nfsbv . ... |
| wl-sb8eut 37566 | Substitution of variable i... |
| wl-sb8eutv 37567 | Substitution of variable i... |
| wl-sb8mot 37568 | Substitution of variable i... |
| wl-sb8motv 37569 | Substitution of variable i... |
| wl-issetft 37570 | A closed form of ~ issetf ... |
| wl-axc11rc11 37571 | Proving ~ axc11r from ~ ax... |
| wl-ax11-lem1 37573 | A transitive law for varia... |
| wl-ax11-lem2 37574 | Lemma. (Contributed by Wo... |
| wl-ax11-lem3 37575 | Lemma. (Contributed by Wo... |
| wl-ax11-lem4 37576 | Lemma. (Contributed by Wo... |
| wl-ax11-lem5 37577 | Lemma. (Contributed by Wo... |
| wl-ax11-lem6 37578 | Lemma. (Contributed by Wo... |
| wl-ax11-lem7 37579 | Lemma. (Contributed by Wo... |
| wl-ax11-lem8 37580 | Lemma. (Contributed by Wo... |
| wl-ax11-lem9 37581 | The easy part when ` x ` c... |
| wl-ax11-lem10 37582 | We now have prepared every... |
| wl-clabv 37583 | Variant of ~ df-clab , whe... |
| wl-dfclab 37584 | Rederive ~ df-clab from ~ ... |
| wl-clabtv 37585 | Using class abstraction in... |
| wl-clabt 37586 | Using class abstraction in... |
| rabiun 37587 | Abstraction restricted to ... |
| iundif1 37588 | Indexed union of class dif... |
| imadifss 37589 | The difference of images i... |
| cureq 37590 | Equality theorem for curry... |
| unceq 37591 | Equality theorem for uncur... |
| curf 37592 | Functional property of cur... |
| uncf 37593 | Functional property of unc... |
| curfv 37594 | Value of currying. (Contr... |
| uncov 37595 | Value of uncurrying. (Con... |
| curunc 37596 | Currying of uncurrying. (... |
| unccur 37597 | Uncurrying of currying. (... |
| phpreu 37598 | Theorem related to pigeonh... |
| finixpnum 37599 | A finite Cartesian product... |
| fin2solem 37600 | Lemma for ~ fin2so . (Con... |
| fin2so 37601 | Any totally ordered Tarski... |
| ltflcei 37602 | Theorem to move the floor ... |
| leceifl 37603 | Theorem to move the floor ... |
| sin2h 37604 | Half-angle rule for sine. ... |
| cos2h 37605 | Half-angle rule for cosine... |
| tan2h 37606 | Half-angle rule for tangen... |
| lindsadd 37607 | In a vector space, the uni... |
| lindsdom 37608 | A linearly independent set... |
| lindsenlbs 37609 | A maximal linearly indepen... |
| matunitlindflem1 37610 | One direction of ~ matunit... |
| matunitlindflem2 37611 | One direction of ~ matunit... |
| matunitlindf 37612 | A matrix over a field is i... |
| ptrest 37613 | Expressing a restriction o... |
| ptrecube 37614 | Any point in an open set o... |
| poimirlem1 37615 | Lemma for ~ poimir - the v... |
| poimirlem2 37616 | Lemma for ~ poimir - conse... |
| poimirlem3 37617 | Lemma for ~ poimir to add ... |
| poimirlem4 37618 | Lemma for ~ poimir connect... |
| poimirlem5 37619 | Lemma for ~ poimir to esta... |
| poimirlem6 37620 | Lemma for ~ poimir establi... |
| poimirlem7 37621 | Lemma for ~ poimir , simil... |
| poimirlem8 37622 | Lemma for ~ poimir , estab... |
| poimirlem9 37623 | Lemma for ~ poimir , estab... |
| poimirlem10 37624 | Lemma for ~ poimir establi... |
| poimirlem11 37625 | Lemma for ~ poimir connect... |
| poimirlem12 37626 | Lemma for ~ poimir connect... |
| poimirlem13 37627 | Lemma for ~ poimir - for a... |
| poimirlem14 37628 | Lemma for ~ poimir - for a... |
| poimirlem15 37629 | Lemma for ~ poimir , that ... |
| poimirlem16 37630 | Lemma for ~ poimir establi... |
| poimirlem17 37631 | Lemma for ~ poimir establi... |
| poimirlem18 37632 | Lemma for ~ poimir stating... |
| poimirlem19 37633 | Lemma for ~ poimir establi... |
| poimirlem20 37634 | Lemma for ~ poimir establi... |
| poimirlem21 37635 | Lemma for ~ poimir stating... |
| poimirlem22 37636 | Lemma for ~ poimir , that ... |
| poimirlem23 37637 | Lemma for ~ poimir , two w... |
| poimirlem24 37638 | Lemma for ~ poimir , two w... |
| poimirlem25 37639 | Lemma for ~ poimir stating... |
| poimirlem26 37640 | Lemma for ~ poimir showing... |
| poimirlem27 37641 | Lemma for ~ poimir showing... |
| poimirlem28 37642 | Lemma for ~ poimir , a var... |
| poimirlem29 37643 | Lemma for ~ poimir connect... |
| poimirlem30 37644 | Lemma for ~ poimir combini... |
| poimirlem31 37645 | Lemma for ~ poimir , assig... |
| poimirlem32 37646 | Lemma for ~ poimir , combi... |
| poimir 37647 | Poincare-Miranda theorem. ... |
| broucube 37648 | Brouwer - or as Kulpa call... |
| heicant 37649 | Heine-Cantor theorem: a co... |
| opnmbllem0 37650 | Lemma for ~ ismblfin ; cou... |
| mblfinlem1 37651 | Lemma for ~ ismblfin , ord... |
| mblfinlem2 37652 | Lemma for ~ ismblfin , eff... |
| mblfinlem3 37653 | The difference between two... |
| mblfinlem4 37654 | Backward direction of ~ is... |
| ismblfin 37655 | Measurability in terms of ... |
| ovoliunnfl 37656 | ~ ovoliun is incompatible ... |
| ex-ovoliunnfl 37657 | Demonstration of ~ ovoliun... |
| voliunnfl 37658 | ~ voliun is incompatible w... |
| volsupnfl 37659 | ~ volsup is incompatible w... |
| mbfresfi 37660 | Measurability of a piecewi... |
| mbfposadd 37661 | If the sum of two measurab... |
| cnambfre 37662 | A real-valued, a.e. contin... |
| dvtanlem 37663 | Lemma for ~ dvtan - the do... |
| dvtan 37664 | Derivative of tangent. (C... |
| itg2addnclem 37665 | An alternate expression fo... |
| itg2addnclem2 37666 | Lemma for ~ itg2addnc . T... |
| itg2addnclem3 37667 | Lemma incomprehensible in ... |
| itg2addnc 37668 | Alternate proof of ~ itg2a... |
| itg2gt0cn 37669 | ~ itg2gt0 holds on functio... |
| ibladdnclem 37670 | Lemma for ~ ibladdnc ; cf ... |
| ibladdnc 37671 | Choice-free analogue of ~ ... |
| itgaddnclem1 37672 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnclem2 37673 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnc 37674 | Choice-free analogue of ~ ... |
| iblsubnc 37675 | Choice-free analogue of ~ ... |
| itgsubnc 37676 | Choice-free analogue of ~ ... |
| iblabsnclem 37677 | Lemma for ~ iblabsnc ; cf.... |
| iblabsnc 37678 | Choice-free analogue of ~ ... |
| iblmulc2nc 37679 | Choice-free analogue of ~ ... |
| itgmulc2nclem1 37680 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nclem2 37681 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nc 37682 | Choice-free analogue of ~ ... |
| itgabsnc 37683 | Choice-free analogue of ~ ... |
| itggt0cn 37684 | ~ itggt0 holds for continu... |
| ftc1cnnclem 37685 | Lemma for ~ ftc1cnnc ; cf.... |
| ftc1cnnc 37686 | Choice-free proof of ~ ftc... |
| ftc1anclem1 37687 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem2 37688 | Lemma for ~ ftc1anc - rest... |
| ftc1anclem3 37689 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem4 37690 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem5 37691 | Lemma for ~ ftc1anc , the ... |
| ftc1anclem6 37692 | Lemma for ~ ftc1anc - cons... |
| ftc1anclem7 37693 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem8 37694 | Lemma for ~ ftc1anc . (Co... |
| ftc1anc 37695 | ~ ftc1a holds for function... |
| ftc2nc 37696 | Choice-free proof of ~ ftc... |
| asindmre 37697 | Real part of domain of dif... |
| dvasin 37698 | Derivative of arcsine. (C... |
| dvacos 37699 | Derivative of arccosine. ... |
| dvreasin 37700 | Real derivative of arcsine... |
| dvreacos 37701 | Real derivative of arccosi... |
| areacirclem1 37702 | Antiderivative of cross-se... |
| areacirclem2 37703 | Endpoint-inclusive continu... |
| areacirclem3 37704 | Integrability of cross-sec... |
| areacirclem4 37705 | Endpoint-inclusive continu... |
| areacirclem5 37706 | Finding the cross-section ... |
| areacirc 37707 | The area of a circle of ra... |
| unirep 37708 | Define a quantity whose de... |
| cover2 37709 | Two ways of expressing the... |
| cover2g 37710 | Two ways of expressing the... |
| brabg2 37711 | Relation by a binary relat... |
| opelopab3 37712 | Ordered pair membership in... |
| cocanfo 37713 | Cancellation of a surjecti... |
| brresi2 37714 | Restriction of a binary re... |
| fnopabeqd 37715 | Equality deduction for fun... |
| fvopabf4g 37716 | Function value of an opera... |
| fnopabco 37717 | Composition of a function ... |
| opropabco 37718 | Composition of an operator... |
| cocnv 37719 | Composition with a functio... |
| f1ocan1fv 37720 | Cancel a composition by a ... |
| f1ocan2fv 37721 | Cancel a composition by th... |
| inixp 37722 | Intersection of Cartesian ... |
| upixp 37723 | Universal property of the ... |
| abrexdom 37724 | An indexed set is dominate... |
| abrexdom2 37725 | An indexed set is dominate... |
| ac6gf 37726 | Axiom of Choice. (Contrib... |
| indexa 37727 | If for every element of an... |
| indexdom 37728 | If for every element of an... |
| frinfm 37729 | A subset of a well-founded... |
| welb 37730 | A nonempty subset of a wel... |
| supex2g 37731 | Existence of supremum. (C... |
| supclt 37732 | Closure of supremum. (Con... |
| supubt 37733 | Upper bound property of su... |
| filbcmb 37734 | Combine a finite set of lo... |
| fzmul 37735 | Membership of a product in... |
| sdclem2 37736 | Lemma for ~ sdc . (Contri... |
| sdclem1 37737 | Lemma for ~ sdc . (Contri... |
| sdc 37738 | Strong dependent choice. ... |
| fdc 37739 | Finite version of dependen... |
| fdc1 37740 | Variant of ~ fdc with no s... |
| seqpo 37741 | Two ways to say that a seq... |
| incsequz 37742 | An increasing sequence of ... |
| incsequz2 37743 | An increasing sequence of ... |
| nnubfi 37744 | A bounded above set of pos... |
| nninfnub 37745 | An infinite set of positiv... |
| subspopn 37746 | An open set is open in the... |
| neificl 37747 | Neighborhoods are closed u... |
| lpss2 37748 | Limit points of a subset a... |
| metf1o 37749 | Use a bijection with a met... |
| blssp 37750 | A ball in the subspace met... |
| mettrifi 37751 | Generalized triangle inequ... |
| lmclim2 37752 | A sequence in a metric spa... |
| geomcau 37753 | If the distance between co... |
| caures 37754 | The restriction of a Cauch... |
| caushft 37755 | A shifted Cauchy sequence ... |
| constcncf 37756 | A constant function is a c... |
| cnres2 37757 | The restriction of a conti... |
| cnresima 37758 | A continuous function is c... |
| cncfres 37759 | A continuous function on c... |
| istotbnd 37763 | The predicate "is a totall... |
| istotbnd2 37764 | The predicate "is a totall... |
| istotbnd3 37765 | A metric space is totally ... |
| totbndmet 37766 | The predicate "totally bou... |
| 0totbnd 37767 | The metric (there is only ... |
| sstotbnd2 37768 | Condition for a subset of ... |
| sstotbnd 37769 | Condition for a subset of ... |
| sstotbnd3 37770 | Use a net that is not nece... |
| totbndss 37771 | A subset of a totally boun... |
| equivtotbnd 37772 | If the metric ` M ` is "st... |
| isbnd 37774 | The predicate "is a bounde... |
| bndmet 37775 | A bounded metric space is ... |
| isbndx 37776 | A "bounded extended metric... |
| isbnd2 37777 | The predicate "is a bounde... |
| isbnd3 37778 | A metric space is bounded ... |
| isbnd3b 37779 | A metric space is bounded ... |
| bndss 37780 | A subset of a bounded metr... |
| blbnd 37781 | A ball is bounded. (Contr... |
| ssbnd 37782 | A subset of a metric space... |
| totbndbnd 37783 | A totally bounded metric s... |
| equivbnd 37784 | If the metric ` M ` is "st... |
| bnd2lem 37785 | Lemma for ~ equivbnd2 and ... |
| equivbnd2 37786 | If balls are totally bound... |
| prdsbnd 37787 | The product metric over fi... |
| prdstotbnd 37788 | The product metric over fi... |
| prdsbnd2 37789 | If balls are totally bound... |
| cntotbnd 37790 | A subset of the complex nu... |
| cnpwstotbnd 37791 | A subset of ` A ^ I ` , wh... |
| ismtyval 37794 | The set of isometries betw... |
| isismty 37795 | The condition "is an isome... |
| ismtycnv 37796 | The inverse of an isometry... |
| ismtyima 37797 | The image of a ball under ... |
| ismtyhmeolem 37798 | Lemma for ~ ismtyhmeo . (... |
| ismtyhmeo 37799 | An isometry is a homeomorp... |
| ismtybndlem 37800 | Lemma for ~ ismtybnd . (C... |
| ismtybnd 37801 | Isometries preserve bounde... |
| ismtyres 37802 | A restriction of an isomet... |
| heibor1lem 37803 | Lemma for ~ heibor1 . A c... |
| heibor1 37804 | One half of ~ heibor , tha... |
| heiborlem1 37805 | Lemma for ~ heibor . We w... |
| heiborlem2 37806 | Lemma for ~ heibor . Subs... |
| heiborlem3 37807 | Lemma for ~ heibor . Usin... |
| heiborlem4 37808 | Lemma for ~ heibor . Usin... |
| heiborlem5 37809 | Lemma for ~ heibor . The ... |
| heiborlem6 37810 | Lemma for ~ heibor . Sinc... |
| heiborlem7 37811 | Lemma for ~ heibor . Sinc... |
| heiborlem8 37812 | Lemma for ~ heibor . The ... |
| heiborlem9 37813 | Lemma for ~ heibor . Disc... |
| heiborlem10 37814 | Lemma for ~ heibor . The ... |
| heibor 37815 | Generalized Heine-Borel Th... |
| bfplem1 37816 | Lemma for ~ bfp . The seq... |
| bfplem2 37817 | Lemma for ~ bfp . Using t... |
| bfp 37818 | Banach fixed point theorem... |
| rrnval 37821 | The n-dimensional Euclidea... |
| rrnmval 37822 | The value of the Euclidean... |
| rrnmet 37823 | Euclidean space is a metri... |
| rrndstprj1 37824 | The distance between two p... |
| rrndstprj2 37825 | Bound on the distance betw... |
| rrncmslem 37826 | Lemma for ~ rrncms . (Con... |
| rrncms 37827 | Euclidean space is complet... |
| repwsmet 37828 | The supremum metric on ` R... |
| rrnequiv 37829 | The supremum metric on ` R... |
| rrntotbnd 37830 | A set in Euclidean space i... |
| rrnheibor 37831 | Heine-Borel theorem for Eu... |
| ismrer1 37832 | An isometry between ` RR `... |
| reheibor 37833 | Heine-Borel theorem for re... |
| iccbnd 37834 | A closed interval in ` RR ... |
| icccmpALT 37835 | A closed interval in ` RR ... |
| isass 37840 | The predicate "is an assoc... |
| isexid 37841 | The predicate ` G ` has a ... |
| ismgmOLD 37844 | Obsolete version of ~ ismg... |
| clmgmOLD 37845 | Obsolete version of ~ mgmc... |
| opidonOLD 37846 | Obsolete version of ~ mndp... |
| rngopidOLD 37847 | Obsolete version of ~ mndp... |
| opidon2OLD 37848 | Obsolete version of ~ mndp... |
| isexid2 37849 | If ` G e. ( Magma i^i ExId... |
| exidu1 37850 | Uniqueness of the left and... |
| idrval 37851 | The value of the identity ... |
| iorlid 37852 | A magma right and left ide... |
| cmpidelt 37853 | A magma right and left ide... |
| smgrpismgmOLD 37856 | Obsolete version of ~ sgrp... |
| issmgrpOLD 37857 | Obsolete version of ~ issg... |
| smgrpmgm 37858 | A semigroup is a magma. (... |
| smgrpassOLD 37859 | Obsolete version of ~ sgrp... |
| mndoissmgrpOLD 37862 | Obsolete version of ~ mnds... |
| mndoisexid 37863 | A monoid has an identity e... |
| mndoismgmOLD 37864 | Obsolete version of ~ mndm... |
| mndomgmid 37865 | A monoid is a magma with a... |
| ismndo 37866 | The predicate "is a monoid... |
| ismndo1 37867 | The predicate "is a monoid... |
| ismndo2 37868 | The predicate "is a monoid... |
| grpomndo 37869 | A group is a monoid. (Con... |
| exidcl 37870 | Closure of the binary oper... |
| exidreslem 37871 | Lemma for ~ exidres and ~ ... |
| exidres 37872 | The restriction of a binar... |
| exidresid 37873 | The restriction of a binar... |
| ablo4pnp 37874 | A commutative/associative ... |
| grpoeqdivid 37875 | Two group elements are equ... |
| grposnOLD 37876 | The group operation for th... |
| elghomlem1OLD 37879 | Obsolete as of 15-Mar-2020... |
| elghomlem2OLD 37880 | Obsolete as of 15-Mar-2020... |
| elghomOLD 37881 | Obsolete version of ~ isgh... |
| ghomlinOLD 37882 | Obsolete version of ~ ghml... |
| ghomidOLD 37883 | Obsolete version of ~ ghmi... |
| ghomf 37884 | Mapping property of a grou... |
| ghomco 37885 | The composition of two gro... |
| ghomdiv 37886 | Group homomorphisms preser... |
| grpokerinj 37887 | A group homomorphism is in... |
| relrngo 37890 | The class of all unital ri... |
| isrngo 37891 | The predicate "is a (unita... |
| isrngod 37892 | Conditions that determine ... |
| rngoi 37893 | The properties of a unital... |
| rngosm 37894 | Functionality of the multi... |
| rngocl 37895 | Closure of the multiplicat... |
| rngoid 37896 | The multiplication operati... |
| rngoideu 37897 | The unity element of a rin... |
| rngodi 37898 | Distributive law for the m... |
| rngodir 37899 | Distributive law for the m... |
| rngoass 37900 | Associative law for the mu... |
| rngo2 37901 | A ring element plus itself... |
| rngoablo 37902 | A ring's addition operatio... |
| rngoablo2 37903 | In a unital ring the addit... |
| rngogrpo 37904 | A ring's addition operatio... |
| rngone0 37905 | The base set of a ring is ... |
| rngogcl 37906 | Closure law for the additi... |
| rngocom 37907 | The addition operation of ... |
| rngoaass 37908 | The addition operation of ... |
| rngoa32 37909 | The addition operation of ... |
| rngoa4 37910 | Rearrangement of 4 terms i... |
| rngorcan 37911 | Right cancellation law for... |
| rngolcan 37912 | Left cancellation law for ... |
| rngo0cl 37913 | A ring has an additive ide... |
| rngo0rid 37914 | The additive identity of a... |
| rngo0lid 37915 | The additive identity of a... |
| rngolz 37916 | The zero of a unital ring ... |
| rngorz 37917 | The zero of a unital ring ... |
| rngosn3 37918 | Obsolete as of 25-Jan-2020... |
| rngosn4 37919 | Obsolete as of 25-Jan-2020... |
| rngosn6 37920 | Obsolete as of 25-Jan-2020... |
| rngonegcl 37921 | A ring is closed under neg... |
| rngoaddneg1 37922 | Adding the negative in a r... |
| rngoaddneg2 37923 | Adding the negative in a r... |
| rngosub 37924 | Subtraction in a ring, in ... |
| rngmgmbs4 37925 | The range of an internal o... |
| rngodm1dm2 37926 | In a unital ring the domai... |
| rngorn1 37927 | In a unital ring the range... |
| rngorn1eq 37928 | In a unital ring the range... |
| rngomndo 37929 | In a unital ring the multi... |
| rngoidmlem 37930 | The unity element of a rin... |
| rngolidm 37931 | The unity element of a rin... |
| rngoridm 37932 | The unity element of a rin... |
| rngo1cl 37933 | The unity element of a rin... |
| rngoueqz 37934 | Obsolete as of 23-Jan-2020... |
| rngonegmn1l 37935 | Negation in a ring is the ... |
| rngonegmn1r 37936 | Negation in a ring is the ... |
| rngoneglmul 37937 | Negation of a product in a... |
| rngonegrmul 37938 | Negation of a product in a... |
| rngosubdi 37939 | Ring multiplication distri... |
| rngosubdir 37940 | Ring multiplication distri... |
| zerdivemp1x 37941 | In a unital ring a left in... |
| isdivrngo 37944 | The predicate "is a divisi... |
| drngoi 37945 | The properties of a divisi... |
| gidsn 37946 | Obsolete as of 23-Jan-2020... |
| zrdivrng 37947 | The zero ring is not a div... |
| dvrunz 37948 | In a division ring the rin... |
| isgrpda 37949 | Properties that determine ... |
| isdrngo1 37950 | The predicate "is a divisi... |
| divrngcl 37951 | The product of two nonzero... |
| isdrngo2 37952 | A division ring is a ring ... |
| isdrngo3 37953 | A division ring is a ring ... |
| rngohomval 37958 | The set of ring homomorphi... |
| isrngohom 37959 | The predicate "is a ring h... |
| rngohomf 37960 | A ring homomorphism is a f... |
| rngohomcl 37961 | Closure law for a ring hom... |
| rngohom1 37962 | A ring homomorphism preser... |
| rngohomadd 37963 | Ring homomorphisms preserv... |
| rngohommul 37964 | Ring homomorphisms preserv... |
| rngogrphom 37965 | A ring homomorphism is a g... |
| rngohom0 37966 | A ring homomorphism preser... |
| rngohomsub 37967 | Ring homomorphisms preserv... |
| rngohomco 37968 | The composition of two rin... |
| rngokerinj 37969 | A ring homomorphism is inj... |
| rngoisoval 37971 | The set of ring isomorphis... |
| isrngoiso 37972 | The predicate "is a ring i... |
| rngoiso1o 37973 | A ring isomorphism is a bi... |
| rngoisohom 37974 | A ring isomorphism is a ri... |
| rngoisocnv 37975 | The inverse of a ring isom... |
| rngoisoco 37976 | The composition of two rin... |
| isriscg 37978 | The ring isomorphism relat... |
| isrisc 37979 | The ring isomorphism relat... |
| risc 37980 | The ring isomorphism relat... |
| risci 37981 | Determine that two rings a... |
| riscer 37982 | Ring isomorphism is an equ... |
| iscom2 37989 | A device to add commutativ... |
| iscrngo 37990 | The predicate "is a commut... |
| iscrngo2 37991 | The predicate "is a commut... |
| iscringd 37992 | Conditions that determine ... |
| flddivrng 37993 | A field is a division ring... |
| crngorngo 37994 | A commutative ring is a ri... |
| crngocom 37995 | The multiplication operati... |
| crngm23 37996 | Commutative/associative la... |
| crngm4 37997 | Commutative/associative la... |
| fldcrngo 37998 | A field is a commutative r... |
| isfld2 37999 | The predicate "is a field"... |
| crngohomfo 38000 | The image of a homomorphis... |
| idlval 38007 | The class of ideals of a r... |
| isidl 38008 | The predicate "is an ideal... |
| isidlc 38009 | The predicate "is an ideal... |
| idlss 38010 | An ideal of ` R ` is a sub... |
| idlcl 38011 | An element of an ideal is ... |
| idl0cl 38012 | An ideal contains ` 0 ` . ... |
| idladdcl 38013 | An ideal is closed under a... |
| idllmulcl 38014 | An ideal is closed under m... |
| idlrmulcl 38015 | An ideal is closed under m... |
| idlnegcl 38016 | An ideal is closed under n... |
| idlsubcl 38017 | An ideal is closed under s... |
| rngoidl 38018 | A ring ` R ` is an ` R ` i... |
| 0idl 38019 | The set containing only ` ... |
| 1idl 38020 | Two ways of expressing the... |
| 0rngo 38021 | In a ring, ` 0 = 1 ` iff t... |
| divrngidl 38022 | The only ideals in a divis... |
| intidl 38023 | The intersection of a none... |
| inidl 38024 | The intersection of two id... |
| unichnidl 38025 | The union of a nonempty ch... |
| keridl 38026 | The kernel of a ring homom... |
| pridlval 38027 | The class of prime ideals ... |
| ispridl 38028 | The predicate "is a prime ... |
| pridlidl 38029 | A prime ideal is an ideal.... |
| pridlnr 38030 | A prime ideal is a proper ... |
| pridl 38031 | The main property of a pri... |
| ispridl2 38032 | A condition that shows an ... |
| maxidlval 38033 | The set of maximal ideals ... |
| ismaxidl 38034 | The predicate "is a maxima... |
| maxidlidl 38035 | A maximal ideal is an idea... |
| maxidlnr 38036 | A maximal ideal is proper.... |
| maxidlmax 38037 | A maximal ideal is a maxim... |
| maxidln1 38038 | One is not contained in an... |
| maxidln0 38039 | A ring with a maximal idea... |
| isprrngo 38044 | The predicate "is a prime ... |
| prrngorngo 38045 | A prime ring is a ring. (... |
| smprngopr 38046 | A simple ring (one whose o... |
| divrngpr 38047 | A division ring is a prime... |
| isdmn 38048 | The predicate "is a domain... |
| isdmn2 38049 | The predicate "is a domain... |
| dmncrng 38050 | A domain is a commutative ... |
| dmnrngo 38051 | A domain is a ring. (Cont... |
| flddmn 38052 | A field is a domain. (Con... |
| igenval 38055 | The ideal generated by a s... |
| igenss 38056 | A set is a subset of the i... |
| igenidl 38057 | The ideal generated by a s... |
| igenmin 38058 | The ideal generated by a s... |
| igenidl2 38059 | The ideal generated by an ... |
| igenval2 38060 | The ideal generated by a s... |
| prnc 38061 | A principal ideal (an idea... |
| isfldidl 38062 | Determine if a ring is a f... |
| isfldidl2 38063 | Determine if a ring is a f... |
| ispridlc 38064 | The predicate "is a prime ... |
| pridlc 38065 | Property of a prime ideal ... |
| pridlc2 38066 | Property of a prime ideal ... |
| pridlc3 38067 | Property of a prime ideal ... |
| isdmn3 38068 | The predicate "is a domain... |
| dmnnzd 38069 | A domain has no zero-divis... |
| dmncan1 38070 | Cancellation law for domai... |
| dmncan2 38071 | Cancellation law for domai... |
| efald2 38072 | A proof by contradiction. ... |
| notbinot1 38073 | Simplification rule of neg... |
| bicontr 38074 | Biconditional of its own n... |
| impor 38075 | An equivalent formula for ... |
| orfa 38076 | The falsum ` F. ` can be r... |
| notbinot2 38077 | Commutation rule between n... |
| biimpor 38078 | A rewriting rule for bicon... |
| orfa1 38079 | Add a contradicting disjun... |
| orfa2 38080 | Remove a contradicting dis... |
| bifald 38081 | Infer the equivalence to a... |
| orsild 38082 | A lemma for not-or-not eli... |
| orsird 38083 | A lemma for not-or-not eli... |
| cnf1dd 38084 | A lemma for Conjunctive No... |
| cnf2dd 38085 | A lemma for Conjunctive No... |
| cnfn1dd 38086 | A lemma for Conjunctive No... |
| cnfn2dd 38087 | A lemma for Conjunctive No... |
| or32dd 38088 | A rearrangement of disjunc... |
| notornotel1 38089 | A lemma for not-or-not eli... |
| notornotel2 38090 | A lemma for not-or-not eli... |
| contrd 38091 | A proof by contradiction, ... |
| an12i 38092 | An inference from commutin... |
| exmid2 38093 | An excluded middle law. (... |
| selconj 38094 | An inference for selecting... |
| truconj 38095 | Add true as a conjunct. (... |
| orel 38096 | An inference for disjuncti... |
| negel 38097 | An inference for negation ... |
| botel 38098 | An inference for bottom el... |
| tradd 38099 | Add top ad a conjunct. (C... |
| gm-sbtru 38100 | Substitution does not chan... |
| sbfal 38101 | Substitution does not chan... |
| sbcani 38102 | Distribution of class subs... |
| sbcori 38103 | Distribution of class subs... |
| sbcimi 38104 | Distribution of class subs... |
| sbcni 38105 | Move class substitution in... |
| sbali 38106 | Discard class substitution... |
| sbexi 38107 | Discard class substitution... |
| sbcalf 38108 | Move universal quantifier ... |
| sbcexf 38109 | Move existential quantifie... |
| sbcalfi 38110 | Move universal quantifier ... |
| sbcexfi 38111 | Move existential quantifie... |
| spsbcdi 38112 | A lemma for eliminating a ... |
| alrimii 38113 | A lemma for introducing a ... |
| spesbcdi 38114 | A lemma for introducing an... |
| exlimddvf 38115 | A lemma for eliminating an... |
| exlimddvfi 38116 | A lemma for eliminating an... |
| sbceq1ddi 38117 | A lemma for eliminating in... |
| sbccom2lem 38118 | Lemma for ~ sbccom2 . (Co... |
| sbccom2 38119 | Commutative law for double... |
| sbccom2f 38120 | Commutative law for double... |
| sbccom2fi 38121 | Commutative law for double... |
| csbcom2fi 38122 | Commutative law for double... |
| fald 38123 | Refutation of falsity, in ... |
| tsim1 38124 | A Tseitin axiom for logica... |
| tsim2 38125 | A Tseitin axiom for logica... |
| tsim3 38126 | A Tseitin axiom for logica... |
| tsbi1 38127 | A Tseitin axiom for logica... |
| tsbi2 38128 | A Tseitin axiom for logica... |
| tsbi3 38129 | A Tseitin axiom for logica... |
| tsbi4 38130 | A Tseitin axiom for logica... |
| tsxo1 38131 | A Tseitin axiom for logica... |
| tsxo2 38132 | A Tseitin axiom for logica... |
| tsxo3 38133 | A Tseitin axiom for logica... |
| tsxo4 38134 | A Tseitin axiom for logica... |
| tsan1 38135 | A Tseitin axiom for logica... |
| tsan2 38136 | A Tseitin axiom for logica... |
| tsan3 38137 | A Tseitin axiom for logica... |
| tsna1 38138 | A Tseitin axiom for logica... |
| tsna2 38139 | A Tseitin axiom for logica... |
| tsna3 38140 | A Tseitin axiom for logica... |
| tsor1 38141 | A Tseitin axiom for logica... |
| tsor2 38142 | A Tseitin axiom for logica... |
| tsor3 38143 | A Tseitin axiom for logica... |
| ts3an1 38144 | A Tseitin axiom for triple... |
| ts3an2 38145 | A Tseitin axiom for triple... |
| ts3an3 38146 | A Tseitin axiom for triple... |
| ts3or1 38147 | A Tseitin axiom for triple... |
| ts3or2 38148 | A Tseitin axiom for triple... |
| ts3or3 38149 | A Tseitin axiom for triple... |
| iuneq2f 38150 | Equality deduction for ind... |
| rabeq12f 38151 | Equality deduction for res... |
| csbeq12 38152 | Equality deduction for sub... |
| sbeqi 38153 | Equality deduction for sub... |
| ralbi12f 38154 | Equality deduction for res... |
| oprabbi 38155 | Equality deduction for cla... |
| mpobi123f 38156 | Equality deduction for map... |
| iuneq12f 38157 | Equality deduction for ind... |
| iineq12f 38158 | Equality deduction for ind... |
| opabbi 38159 | Equality deduction for cla... |
| mptbi12f 38160 | Equality deduction for map... |
| orcomdd 38161 | Commutativity of logic dis... |
| scottexf 38162 | A version of ~ scottex wit... |
| scott0f 38163 | A version of ~ scott0 with... |
| scottn0f 38164 | A version of ~ scott0f wit... |
| ac6s3f 38165 | Generalization of the Axio... |
| ac6s6 38166 | Generalization of the Axio... |
| ac6s6f 38167 | Generalization of the Axio... |
| el2v1 38211 | New way ( ~ elv , and the ... |
| el3v1 38212 | New way ( ~ elv , and the ... |
| el3v2 38213 | New way ( ~ elv , and the ... |
| el3v12 38214 | New way ( ~ elv , and the ... |
| el3v13 38215 | New way ( ~ elv , and the ... |
| el3v23 38216 | New way ( ~ elv , and the ... |
| anan 38217 | Multiple commutations in c... |
| triantru3 38218 | A wff is equivalent to its... |
| biorfd 38219 | A wff is equivalent to its... |
| eqbrtr 38220 | Substitution of equal clas... |
| eqbrb 38221 | Substitution of equal clas... |
| eqeltr 38222 | Substitution of equal clas... |
| eqelb 38223 | Substitution of equal clas... |
| eqeqan2d 38224 | Implication of introducing... |
| suceqsneq 38225 | One-to-one relationship be... |
| sucdifsn2 38226 | Absorption of union with a... |
| sucdifsn 38227 | The difference between the... |
| disjresin 38228 | The restriction to a disjo... |
| disjresdisj 38229 | The intersection of restri... |
| disjresdif 38230 | The difference between res... |
| disjresundif 38231 | Lemma for ~ ressucdifsn2 .... |
| ressucdifsn2 38232 | The difference between res... |
| ressucdifsn 38233 | The difference between res... |
| inres2 38234 | Two ways of expressing the... |
| coideq 38235 | Equality theorem for compo... |
| nexmo1 38236 | If there is no case where ... |
| eqab2 38237 | Implication of a class abs... |
| r2alan 38238 | Double restricted universa... |
| ssrabi 38239 | Inference of restricted ab... |
| rabimbieq 38240 | Restricted equivalent wff'... |
| abeqin 38241 | Intersection with class ab... |
| abeqinbi 38242 | Intersection with class ab... |
| rabeqel 38243 | Class element of a restric... |
| eqrelf 38244 | The equality connective be... |
| br1cnvinxp 38245 | Binary relation on the con... |
| releleccnv 38246 | Elementhood in a converse ... |
| releccnveq 38247 | Equality of converse ` R `... |
| opelvvdif 38248 | Negated elementhood of ord... |
| vvdifopab 38249 | Ordered-pair class abstrac... |
| brvdif 38250 | Binary relation with unive... |
| brvdif2 38251 | Binary relation with unive... |
| brvvdif 38252 | Binary relation with the c... |
| brvbrvvdif 38253 | Binary relation with the c... |
| brcnvep 38254 | The converse of the binary... |
| elecALTV 38255 | Elementhood in the ` R ` -... |
| brcnvepres 38256 | Restricted converse epsilo... |
| brres2 38257 | Binary relation on a restr... |
| br1cnvres 38258 | Binary relation on the con... |
| eldmres 38259 | Elementhood in the domain ... |
| elrnres 38260 | Element of the range of a ... |
| eldmressnALTV 38261 | Element of the domain of a... |
| elrnressn 38262 | Element of the range of a ... |
| eldm4 38263 | Elementhood in a domain. ... |
| eldmres2 38264 | Elementhood in the domain ... |
| eldmres3 38265 | Elementhood in the domain ... |
| eceq1i 38266 | Equality theorem for ` C `... |
| ecres 38267 | Restricted coset of ` B ` ... |
| eccnvepres 38268 | Restricted converse epsilo... |
| eleccnvep 38269 | Elementhood in the convers... |
| eccnvep 38270 | The converse epsilon coset... |
| extep 38271 | Property of epsilon relati... |
| disjeccnvep 38272 | Property of the epsilon re... |
| eccnvepres2 38273 | The restricted converse ep... |
| eccnvepres3 38274 | Condition for a restricted... |
| eldmqsres 38275 | Elementhood in a restricte... |
| eldmqsres2 38276 | Elementhood in a restricte... |
| qsss1 38277 | Subclass theorem for quoti... |
| qseq1i 38278 | Equality theorem for quoti... |
| brinxprnres 38279 | Binary relation on a restr... |
| inxprnres 38280 | Restriction of a class as ... |
| dfres4 38281 | Alternate definition of th... |
| exan3 38282 | Equivalent expressions wit... |
| exanres 38283 | Equivalent expressions wit... |
| exanres3 38284 | Equivalent expressions wit... |
| exanres2 38285 | Equivalent expressions wit... |
| cnvepres 38286 | Restricted converse epsilo... |
| eqrel2 38287 | Equality of relations. (C... |
| rncnv 38288 | Range of converse is the d... |
| dfdm6 38289 | Alternate definition of do... |
| dfrn6 38290 | Alternate definition of ra... |
| rncnvepres 38291 | The range of the restricte... |
| dmecd 38292 | Equality of the coset of `... |
| dmec2d 38293 | Equality of the coset of `... |
| brid 38294 | Property of the identity b... |
| ideq2 38295 | For sets, the identity bin... |
| idresssidinxp 38296 | Condition for the identity... |
| idreseqidinxp 38297 | Condition for the identity... |
| extid 38298 | Property of identity relat... |
| inxpss 38299 | Two ways to say that an in... |
| idinxpss 38300 | Two ways to say that an in... |
| ref5 38301 | Two ways to say that an in... |
| inxpss3 38302 | Two ways to say that an in... |
| inxpss2 38303 | Two ways to say that inter... |
| inxpssidinxp 38304 | Two ways to say that inter... |
| idinxpssinxp 38305 | Two ways to say that inter... |
| idinxpssinxp2 38306 | Identity intersection with... |
| idinxpssinxp3 38307 | Identity intersection with... |
| idinxpssinxp4 38308 | Identity intersection with... |
| relcnveq3 38309 | Two ways of saying a relat... |
| relcnveq 38310 | Two ways of saying a relat... |
| relcnveq2 38311 | Two ways of saying a relat... |
| relcnveq4 38312 | Two ways of saying a relat... |
| qsresid 38313 | Simplification of a specia... |
| n0elqs 38314 | Two ways of expressing tha... |
| n0elqs2 38315 | Two ways of expressing tha... |
| rnresequniqs 38316 | The range of a restriction... |
| n0el2 38317 | Two ways of expressing tha... |
| cnvepresex 38318 | Sethood condition for the ... |
| cnvepima 38319 | The image of converse epsi... |
| inex3 38320 | Sufficient condition for t... |
| inxpex 38321 | Sufficient condition for a... |
| eqres 38322 | Converting a class constan... |
| brrabga 38323 | The law of concretion for ... |
| brcnvrabga 38324 | The law of concretion for ... |
| opideq 38325 | Equality conditions for or... |
| iss2 38326 | A subclass of the identity... |
| eldmcnv 38327 | Elementhood in a domain of... |
| dfrel5 38328 | Alternate definition of th... |
| dfrel6 38329 | Alternate definition of th... |
| cnvresrn 38330 | Converse restricted to ran... |
| relssinxpdmrn 38331 | Subset of restriction, spe... |
| cnvref4 38332 | Two ways to say that a rel... |
| cnvref5 38333 | Two ways to say that a rel... |
| ecin0 38334 | Two ways of saying that th... |
| ecinn0 38335 | Two ways of saying that th... |
| ineleq 38336 | Equivalence of restricted ... |
| inecmo 38337 | Equivalence of a double re... |
| inecmo2 38338 | Equivalence of a double re... |
| ineccnvmo 38339 | Equivalence of a double re... |
| alrmomorn 38340 | Equivalence of an "at most... |
| alrmomodm 38341 | Equivalence of an "at most... |
| ineccnvmo2 38342 | Equivalence of a double un... |
| inecmo3 38343 | Equivalence of a double un... |
| moeu2 38344 | Uniqueness is equivalent t... |
| mopickr 38345 | "At most one" picks a vari... |
| moantr 38346 | Sufficient condition for t... |
| brabidgaw 38347 | The law of concretion for ... |
| brabidga 38348 | The law of concretion for ... |
| inxp2 38349 | Intersection with a Cartes... |
| opabf 38350 | A class abstraction of a c... |
| ec0 38351 | The empty-coset of a class... |
| brcnvin 38352 | Intersection with a conver... |
| xrnss3v 38354 | A range Cartesian product ... |
| xrnrel 38355 | A range Cartesian product ... |
| brxrn 38356 | Characterize a ternary rel... |
| brxrn2 38357 | A characterization of the ... |
| dfxrn2 38358 | Alternate definition of th... |
| brxrncnvep 38359 | The range product with con... |
| dmxrn 38360 | Domain of the range produc... |
| dmcnvep 38361 | Domain of converse epsilon... |
| dmxrncnvep 38362 | Domain of the range produc... |
| xrneq1 38363 | Equality theorem for the r... |
| xrneq1i 38364 | Equality theorem for the r... |
| xrneq1d 38365 | Equality theorem for the r... |
| xrneq2 38366 | Equality theorem for the r... |
| xrneq2i 38367 | Equality theorem for the r... |
| xrneq2d 38368 | Equality theorem for the r... |
| xrneq12 38369 | Equality theorem for the r... |
| xrneq12i 38370 | Equality theorem for the r... |
| xrneq12d 38371 | Equality theorem for the r... |
| elecxrn 38372 | Elementhood in the ` ( R |... |
| ecxrn 38373 | The ` ( R |X. S ) ` -coset... |
| disjressuc2 38374 | Double restricted quantifi... |
| disjecxrn 38375 | Two ways of saying that ` ... |
| disjecxrncnvep 38376 | Two ways of saying that co... |
| disjsuc2 38377 | Double restricted quantifi... |
| xrninxp 38378 | Intersection of a range Ca... |
| xrninxp2 38379 | Intersection of a range Ca... |
| xrninxpex 38380 | Sufficient condition for t... |
| inxpxrn 38381 | Two ways to express the in... |
| br1cnvxrn2 38382 | The converse of a binary r... |
| elec1cnvxrn2 38383 | Elementhood in the convers... |
| rnxrn 38384 | Range of the range Cartesi... |
| rnxrnres 38385 | Range of a range Cartesian... |
| rnxrncnvepres 38386 | Range of a range Cartesian... |
| rnxrnidres 38387 | Range of a range Cartesian... |
| xrnres 38388 | Two ways to express restri... |
| xrnres2 38389 | Two ways to express restri... |
| xrnres3 38390 | Two ways to express restri... |
| xrnres4 38391 | Two ways to express restri... |
| xrnresex 38392 | Sufficient condition for a... |
| xrnidresex 38393 | Sufficient condition for a... |
| xrncnvepresex 38394 | Sufficient condition for a... |
| dmxrncnvepres 38395 | Domain of the range produc... |
| eldmxrncnvepres 38396 | Element of the domain of t... |
| eldmxrncnvepres2 38397 | Element of the domain of t... |
| eceldmqsxrncnvepres 38398 | An ` ( R |X. ( ` ' E | ` A... |
| eceldmqsxrncnvepres2 38399 | An ` ( R |X. ( ` ' E | ` A... |
| brin2 38400 | Binary relation on an inte... |
| brin3 38401 | Binary relation on an inte... |
| dfcoss2 38404 | Alternate definition of th... |
| dfcoss3 38405 | Alternate definition of th... |
| dfcoss4 38406 | Alternate definition of th... |
| cosscnv 38407 | Class of cosets by the con... |
| coss1cnvres 38408 | Class of cosets by the con... |
| coss2cnvepres 38409 | Special case of ~ coss1cnv... |
| cossex 38410 | If ` A ` is a set then the... |
| cosscnvex 38411 | If ` A ` is a set then the... |
| 1cosscnvepresex 38412 | Sufficient condition for a... |
| 1cossxrncnvepresex 38413 | Sufficient condition for a... |
| relcoss 38414 | Cosets by ` R ` is a relat... |
| relcoels 38415 | Coelements on ` A ` is a r... |
| cossss 38416 | Subclass theorem for the c... |
| cosseq 38417 | Equality theorem for the c... |
| cosseqi 38418 | Equality theorem for the c... |
| cosseqd 38419 | Equality theorem for the c... |
| 1cossres 38420 | The class of cosets by a r... |
| dfcoels 38421 | Alternate definition of th... |
| brcoss 38422 | ` A ` and ` B ` are cosets... |
| brcoss2 38423 | Alternate form of the ` A ... |
| brcoss3 38424 | Alternate form of the ` A ... |
| brcosscnvcoss 38425 | For sets, the ` A ` and ` ... |
| brcoels 38426 | ` B ` and ` C ` are coelem... |
| cocossss 38427 | Two ways of saying that co... |
| cnvcosseq 38428 | The converse of cosets by ... |
| br2coss 38429 | Cosets by ` ,~ R ` binary ... |
| br1cossres 38430 | ` B ` and ` C ` are cosets... |
| br1cossres2 38431 | ` B ` and ` C ` are cosets... |
| brressn 38432 | Binary relation on a restr... |
| ressn2 38433 | A class ' R ' restricted t... |
| refressn 38434 | Any class ' R ' restricted... |
| antisymressn 38435 | Every class ' R ' restrict... |
| trressn 38436 | Any class ' R ' restricted... |
| relbrcoss 38437 | ` A ` and ` B ` are cosets... |
| br1cossinres 38438 | ` B ` and ` C ` are cosets... |
| br1cossxrnres 38439 | ` <. B , C >. ` and ` <. D... |
| br1cossinidres 38440 | ` B ` and ` C ` are cosets... |
| br1cossincnvepres 38441 | ` B ` and ` C ` are cosets... |
| br1cossxrnidres 38442 | ` <. B , C >. ` and ` <. D... |
| br1cossxrncnvepres 38443 | ` <. B , C >. ` and ` <. D... |
| dmcoss3 38444 | The domain of cosets is th... |
| dmcoss2 38445 | The domain of cosets is th... |
| rncossdmcoss 38446 | The range of cosets is the... |
| dm1cosscnvepres 38447 | The domain of cosets of th... |
| dmcoels 38448 | The domain of coelements i... |
| eldmcoss 38449 | Elementhood in the domain ... |
| eldmcoss2 38450 | Elementhood in the domain ... |
| eldm1cossres 38451 | Elementhood in the domain ... |
| eldm1cossres2 38452 | Elementhood in the domain ... |
| refrelcosslem 38453 | Lemma for the left side of... |
| refrelcoss3 38454 | The class of cosets by ` R... |
| refrelcoss2 38455 | The class of cosets by ` R... |
| symrelcoss3 38456 | The class of cosets by ` R... |
| symrelcoss2 38457 | The class of cosets by ` R... |
| cossssid 38458 | Equivalent expressions for... |
| cossssid2 38459 | Equivalent expressions for... |
| cossssid3 38460 | Equivalent expressions for... |
| cossssid4 38461 | Equivalent expressions for... |
| cossssid5 38462 | Equivalent expressions for... |
| brcosscnv 38463 | ` A ` and ` B ` are cosets... |
| brcosscnv2 38464 | ` A ` and ` B ` are cosets... |
| br1cosscnvxrn 38465 | ` A ` and ` B ` are cosets... |
| 1cosscnvxrn 38466 | Cosets by the converse ran... |
| cosscnvssid3 38467 | Equivalent expressions for... |
| cosscnvssid4 38468 | Equivalent expressions for... |
| cosscnvssid5 38469 | Equivalent expressions for... |
| coss0 38470 | Cosets by the empty set ar... |
| cossid 38471 | Cosets by the identity rel... |
| cosscnvid 38472 | Cosets by the converse ide... |
| trcoss 38473 | Sufficient condition for t... |
| eleccossin 38474 | Two ways of saying that th... |
| trcoss2 38475 | Equivalent expressions for... |
| elrels2 38477 | The element of the relatio... |
| elrelsrel 38478 | The element of the relatio... |
| elrelsrelim 38479 | The element of the relatio... |
| elrels5 38480 | Equivalent expressions for... |
| elrels6 38481 | Equivalent expressions for... |
| elrelscnveq3 38482 | Two ways of saying a relat... |
| elrelscnveq 38483 | Two ways of saying a relat... |
| elrelscnveq2 38484 | Two ways of saying a relat... |
| elrelscnveq4 38485 | Two ways of saying a relat... |
| cnvelrels 38486 | The converse of a set is a... |
| cosselrels 38487 | Cosets of sets are element... |
| cosscnvelrels 38488 | Cosets of converse sets ar... |
| dfssr2 38490 | Alternate definition of th... |
| relssr 38491 | The subset relation is a r... |
| brssr 38492 | The subset relation and su... |
| brssrid 38493 | Any set is a subset of its... |
| issetssr 38494 | Two ways of expressing set... |
| brssrres 38495 | Restricted subset binary r... |
| br1cnvssrres 38496 | Restricted converse subset... |
| brcnvssr 38497 | The converse of a subset r... |
| brcnvssrid 38498 | Any set is a converse subs... |
| br1cossxrncnvssrres 38499 | ` <. B , C >. ` and ` <. D... |
| extssr 38500 | Property of subset relatio... |
| dfrefrels2 38504 | Alternate definition of th... |
| dfrefrels3 38505 | Alternate definition of th... |
| dfrefrel2 38506 | Alternate definition of th... |
| dfrefrel3 38507 | Alternate definition of th... |
| dfrefrel5 38508 | Alternate definition of th... |
| elrefrels2 38509 | Element of the class of re... |
| elrefrels3 38510 | Element of the class of re... |
| elrefrelsrel 38511 | For sets, being an element... |
| refreleq 38512 | Equality theorem for refle... |
| refrelid 38513 | Identity relation is refle... |
| refrelcoss 38514 | The class of cosets by ` R... |
| refrelressn 38515 | Any class ' R ' restricted... |
| dfcnvrefrels2 38519 | Alternate definition of th... |
| dfcnvrefrels3 38520 | Alternate definition of th... |
| dfcnvrefrel2 38521 | Alternate definition of th... |
| dfcnvrefrel3 38522 | Alternate definition of th... |
| dfcnvrefrel4 38523 | Alternate definition of th... |
| dfcnvrefrel5 38524 | Alternate definition of th... |
| elcnvrefrels2 38525 | Element of the class of co... |
| elcnvrefrels3 38526 | Element of the class of co... |
| elcnvrefrelsrel 38527 | For sets, being an element... |
| cnvrefrelcoss2 38528 | Necessary and sufficient c... |
| cosselcnvrefrels2 38529 | Necessary and sufficient c... |
| cosselcnvrefrels3 38530 | Necessary and sufficient c... |
| cosselcnvrefrels4 38531 | Necessary and sufficient c... |
| cosselcnvrefrels5 38532 | Necessary and sufficient c... |
| dfsymrels2 38536 | Alternate definition of th... |
| dfsymrels3 38537 | Alternate definition of th... |
| dfsymrels4 38538 | Alternate definition of th... |
| dfsymrels5 38539 | Alternate definition of th... |
| dfsymrel2 38540 | Alternate definition of th... |
| dfsymrel3 38541 | Alternate definition of th... |
| dfsymrel4 38542 | Alternate definition of th... |
| dfsymrel5 38543 | Alternate definition of th... |
| elsymrels2 38544 | Element of the class of sy... |
| elsymrels3 38545 | Element of the class of sy... |
| elsymrels4 38546 | Element of the class of sy... |
| elsymrels5 38547 | Element of the class of sy... |
| elsymrelsrel 38548 | For sets, being an element... |
| symreleq 38549 | Equality theorem for symme... |
| symrelim 38550 | Symmetric relation implies... |
| symrelcoss 38551 | The class of cosets by ` R... |
| idsymrel 38552 | The identity relation is s... |
| epnsymrel 38553 | The membership (epsilon) r... |
| symrefref2 38554 | Symmetry is a sufficient c... |
| symrefref3 38555 | Symmetry is a sufficient c... |
| refsymrels2 38556 | Elements of the class of r... |
| refsymrels3 38557 | Elements of the class of r... |
| refsymrel2 38558 | A relation which is reflex... |
| refsymrel3 38559 | A relation which is reflex... |
| elrefsymrels2 38560 | Elements of the class of r... |
| elrefsymrels3 38561 | Elements of the class of r... |
| elrefsymrelsrel 38562 | For sets, being an element... |
| dftrrels2 38566 | Alternate definition of th... |
| dftrrels3 38567 | Alternate definition of th... |
| dftrrel2 38568 | Alternate definition of th... |
| dftrrel3 38569 | Alternate definition of th... |
| eltrrels2 38570 | Element of the class of tr... |
| eltrrels3 38571 | Element of the class of tr... |
| eltrrelsrel 38572 | For sets, being an element... |
| trreleq 38573 | Equality theorem for the t... |
| trrelressn 38574 | Any class ' R ' restricted... |
| dfeqvrels2 38579 | Alternate definition of th... |
| dfeqvrels3 38580 | Alternate definition of th... |
| dfeqvrel2 38581 | Alternate definition of th... |
| dfeqvrel3 38582 | Alternate definition of th... |
| eleqvrels2 38583 | Element of the class of eq... |
| eleqvrels3 38584 | Element of the class of eq... |
| eleqvrelsrel 38585 | For sets, being an element... |
| elcoeleqvrels 38586 | Elementhood in the coeleme... |
| elcoeleqvrelsrel 38587 | For sets, being an element... |
| eqvrelrel 38588 | An equivalence relation is... |
| eqvrelrefrel 38589 | An equivalence relation is... |
| eqvrelsymrel 38590 | An equivalence relation is... |
| eqvreltrrel 38591 | An equivalence relation is... |
| eqvrelim 38592 | Equivalence relation impli... |
| eqvreleq 38593 | Equality theorem for equiv... |
| eqvreleqi 38594 | Equality theorem for equiv... |
| eqvreleqd 38595 | Equality theorem for equiv... |
| eqvrelsym 38596 | An equivalence relation is... |
| eqvrelsymb 38597 | An equivalence relation is... |
| eqvreltr 38598 | An equivalence relation is... |
| eqvreltrd 38599 | A transitivity relation fo... |
| eqvreltr4d 38600 | A transitivity relation fo... |
| eqvrelref 38601 | An equivalence relation is... |
| eqvrelth 38602 | Basic property of equivale... |
| eqvrelcl 38603 | Elementhood in the field o... |
| eqvrelthi 38604 | Basic property of equivale... |
| eqvreldisj 38605 | Equivalence classes do not... |
| qsdisjALTV 38606 | Elements of a quotient set... |
| eqvrelqsel 38607 | If an element of a quotien... |
| eqvrelcoss 38608 | Two ways to express equiva... |
| eqvrelcoss3 38609 | Two ways to express equiva... |
| eqvrelcoss2 38610 | Two ways to express equiva... |
| eqvrelcoss4 38611 | Two ways to express equiva... |
| dfcoeleqvrels 38612 | Alternate definition of th... |
| dfcoeleqvrel 38613 | Alternate definition of th... |
| brredunds 38617 | Binary relation on the cla... |
| brredundsredund 38618 | For sets, binary relation ... |
| redundss3 38619 | Implication of redundancy ... |
| redundeq1 38620 | Equivalence of redundancy ... |
| redundpim3 38621 | Implication of redundancy ... |
| redundpbi1 38622 | Equivalence of redundancy ... |
| refrelsredund4 38623 | The naive version of the c... |
| refrelsredund2 38624 | The naive version of the c... |
| refrelsredund3 38625 | The naive version of the c... |
| refrelredund4 38626 | The naive version of the d... |
| refrelredund2 38627 | The naive version of the d... |
| refrelredund3 38628 | The naive version of the d... |
| dmqseq 38631 | Equality theorem for domai... |
| dmqseqi 38632 | Equality theorem for domai... |
| dmqseqd 38633 | Equality theorem for domai... |
| dmqseqeq1 38634 | Equality theorem for domai... |
| dmqseqeq1i 38635 | Equality theorem for domai... |
| dmqseqeq1d 38636 | Equality theorem for domai... |
| brdmqss 38637 | The domain quotient binary... |
| brdmqssqs 38638 | If ` A ` and ` R ` are set... |
| n0eldmqs 38639 | The empty set is not an el... |
| qseq 38640 | The quotient set equal to ... |
| n0eldmqseq 38641 | The empty set is not an el... |
| n0elim 38642 | Implication of that the em... |
| n0el3 38643 | Two ways of expressing tha... |
| cnvepresdmqss 38644 | The domain quotient binary... |
| cnvepresdmqs 38645 | The domain quotient predic... |
| unidmqs 38646 | The range of a relation is... |
| unidmqseq 38647 | The union of the domain qu... |
| dmqseqim 38648 | If the domain quotient of ... |
| dmqseqim2 38649 | Lemma for ~ erimeq2 . (Co... |
| releldmqs 38650 | Elementhood in the domain ... |
| eldmqs1cossres 38651 | Elementhood in the domain ... |
| releldmqscoss 38652 | Elementhood in the domain ... |
| dmqscoelseq 38653 | Two ways to express the eq... |
| dmqs1cosscnvepreseq 38654 | Two ways to express the eq... |
| brers 38659 | Binary equivalence relatio... |
| dferALTV2 38660 | Equivalence relation with ... |
| erALTVeq1 38661 | Equality theorem for equiv... |
| erALTVeq1i 38662 | Equality theorem for equiv... |
| erALTVeq1d 38663 | Equality theorem for equiv... |
| dfcomember 38664 | Alternate definition of th... |
| dfcomember2 38665 | Alternate definition of th... |
| dfcomember3 38666 | Alternate definition of th... |
| eqvreldmqs 38667 | Two ways to express comemb... |
| eqvreldmqs2 38668 | Two ways to express comemb... |
| brerser 38669 | Binary equivalence relatio... |
| erimeq2 38670 | Equivalence relation on it... |
| erimeq 38671 | Equivalence relation on it... |
| dffunsALTV 38675 | Alternate definition of th... |
| dffunsALTV2 38676 | Alternate definition of th... |
| dffunsALTV3 38677 | Alternate definition of th... |
| dffunsALTV4 38678 | Alternate definition of th... |
| dffunsALTV5 38679 | Alternate definition of th... |
| dffunALTV2 38680 | Alternate definition of th... |
| dffunALTV3 38681 | Alternate definition of th... |
| dffunALTV4 38682 | Alternate definition of th... |
| dffunALTV5 38683 | Alternate definition of th... |
| elfunsALTV 38684 | Elementhood in the class o... |
| elfunsALTV2 38685 | Elementhood in the class o... |
| elfunsALTV3 38686 | Elementhood in the class o... |
| elfunsALTV4 38687 | Elementhood in the class o... |
| elfunsALTV5 38688 | Elementhood in the class o... |
| elfunsALTVfunALTV 38689 | The element of the class o... |
| funALTVfun 38690 | Our definition of the func... |
| funALTVss 38691 | Subclass theorem for funct... |
| funALTVeq 38692 | Equality theorem for funct... |
| funALTVeqi 38693 | Equality inference for the... |
| funALTVeqd 38694 | Equality deduction for the... |
| dfdisjs 38700 | Alternate definition of th... |
| dfdisjs2 38701 | Alternate definition of th... |
| dfdisjs3 38702 | Alternate definition of th... |
| dfdisjs4 38703 | Alternate definition of th... |
| dfdisjs5 38704 | Alternate definition of th... |
| dfdisjALTV 38705 | Alternate definition of th... |
| dfdisjALTV2 38706 | Alternate definition of th... |
| dfdisjALTV3 38707 | Alternate definition of th... |
| dfdisjALTV4 38708 | Alternate definition of th... |
| dfdisjALTV5 38709 | Alternate definition of th... |
| dfeldisj2 38710 | Alternate definition of th... |
| dfeldisj3 38711 | Alternate definition of th... |
| dfeldisj4 38712 | Alternate definition of th... |
| dfeldisj5 38713 | Alternate definition of th... |
| eldisjs 38714 | Elementhood in the class o... |
| eldisjs2 38715 | Elementhood in the class o... |
| eldisjs3 38716 | Elementhood in the class o... |
| eldisjs4 38717 | Elementhood in the class o... |
| eldisjs5 38718 | Elementhood in the class o... |
| eldisjsdisj 38719 | The element of the class o... |
| eleldisjs 38720 | Elementhood in the disjoin... |
| eleldisjseldisj 38721 | The element of the disjoin... |
| disjrel 38722 | Disjoint relation is a rel... |
| disjss 38723 | Subclass theorem for disjo... |
| disjssi 38724 | Subclass theorem for disjo... |
| disjssd 38725 | Subclass theorem for disjo... |
| disjeq 38726 | Equality theorem for disjo... |
| disjeqi 38727 | Equality theorem for disjo... |
| disjeqd 38728 | Equality theorem for disjo... |
| disjdmqseqeq1 38729 | Lemma for the equality the... |
| eldisjss 38730 | Subclass theorem for disjo... |
| eldisjssi 38731 | Subclass theorem for disjo... |
| eldisjssd 38732 | Subclass theorem for disjo... |
| eldisjeq 38733 | Equality theorem for disjo... |
| eldisjeqi 38734 | Equality theorem for disjo... |
| eldisjeqd 38735 | Equality theorem for disjo... |
| disjres 38736 | Disjoint restriction. (Co... |
| eldisjn0elb 38737 | Two forms of disjoint elem... |
| disjxrn 38738 | Two ways of saying that a ... |
| disjxrnres5 38739 | Disjoint range Cartesian p... |
| disjorimxrn 38740 | Disjointness condition for... |
| disjimxrn 38741 | Disjointness condition for... |
| disjimres 38742 | Disjointness condition for... |
| disjimin 38743 | Disjointness condition for... |
| disjiminres 38744 | Disjointness condition for... |
| disjimxrnres 38745 | Disjointness condition for... |
| disjALTV0 38746 | The null class is disjoint... |
| disjALTVid 38747 | The class of identity rela... |
| disjALTVidres 38748 | The class of identity rela... |
| disjALTVinidres 38749 | The intersection with rest... |
| disjALTVxrnidres 38750 | The class of range Cartesi... |
| disjsuc 38751 | Disjoint range Cartesian p... |
| dfantisymrel4 38753 | Alternate definition of th... |
| dfantisymrel5 38754 | Alternate definition of th... |
| antisymrelres 38755 | (Contributed by Peter Mazs... |
| antisymrelressn 38756 | (Contributed by Peter Mazs... |
| dfpart2 38761 | Alternate definition of th... |
| dfmembpart2 38762 | Alternate definition of th... |
| brparts 38763 | Binary partitions relation... |
| brparts2 38764 | Binary partitions relation... |
| brpartspart 38765 | Binary partition and the p... |
| parteq1 38766 | Equality theorem for parti... |
| parteq2 38767 | Equality theorem for parti... |
| parteq12 38768 | Equality theorem for parti... |
| parteq1i 38769 | Equality theorem for parti... |
| parteq1d 38770 | Equality theorem for parti... |
| partsuc2 38771 | Property of the partition.... |
| partsuc 38772 | Property of the partition.... |
| disjim 38773 | The "Divide et Aequivalere... |
| disjimi 38774 | Every disjoint relation ge... |
| detlem 38775 | If a relation is disjoint,... |
| eldisjim 38776 | If the elements of ` A ` a... |
| eldisjim2 38777 | Alternate form of ~ eldisj... |
| eqvrel0 38778 | The null class is an equiv... |
| det0 38779 | The cosets by the null cla... |
| eqvrelcoss0 38780 | The cosets by the null cla... |
| eqvrelid 38781 | The identity relation is a... |
| eqvrel1cossidres 38782 | The cosets by a restricted... |
| eqvrel1cossinidres 38783 | The cosets by an intersect... |
| eqvrel1cossxrnidres 38784 | The cosets by a range Cart... |
| detid 38785 | The cosets by the identity... |
| eqvrelcossid 38786 | The cosets by the identity... |
| detidres 38787 | The cosets by the restrict... |
| detinidres 38788 | The cosets by the intersec... |
| detxrnidres 38789 | The cosets by the range Ca... |
| disjlem14 38790 | Lemma for ~ disjdmqseq , ~... |
| disjlem17 38791 | Lemma for ~ disjdmqseq , ~... |
| disjlem18 38792 | Lemma for ~ disjdmqseq , ~... |
| disjlem19 38793 | Lemma for ~ disjdmqseq , ~... |
| disjdmqsss 38794 | Lemma for ~ disjdmqseq via... |
| disjdmqscossss 38795 | Lemma for ~ disjdmqseq via... |
| disjdmqs 38796 | If a relation is disjoint,... |
| disjdmqseq 38797 | If a relation is disjoint,... |
| eldisjn0el 38798 | Special case of ~ disjdmqs... |
| partim2 38799 | Disjoint relation on its n... |
| partim 38800 | Partition implies equivale... |
| partimeq 38801 | Partition implies that the... |
| eldisjlem19 38802 | Special case of ~ disjlem1... |
| membpartlem19 38803 | Together with ~ disjlem19 ... |
| petlem 38804 | If you can prove that the ... |
| petlemi 38805 | If you can prove disjointn... |
| pet02 38806 | Class ` A ` is a partition... |
| pet0 38807 | Class ` A ` is a partition... |
| petid2 38808 | Class ` A ` is a partition... |
| petid 38809 | A class is a partition by ... |
| petidres2 38810 | Class ` A ` is a partition... |
| petidres 38811 | A class is a partition by ... |
| petinidres2 38812 | Class ` A ` is a partition... |
| petinidres 38813 | A class is a partition by ... |
| petxrnidres2 38814 | Class ` A ` is a partition... |
| petxrnidres 38815 | A class is a partition by ... |
| eqvreldisj1 38816 | The elements of the quotie... |
| eqvreldisj2 38817 | The elements of the quotie... |
| eqvreldisj3 38818 | The elements of the quotie... |
| eqvreldisj4 38819 | Intersection with the conv... |
| eqvreldisj5 38820 | Range Cartesian product wi... |
| eqvrelqseqdisj2 38821 | Implication of ~ eqvreldis... |
| fences3 38822 | Implication of ~ eqvrelqse... |
| eqvrelqseqdisj3 38823 | Implication of ~ eqvreldis... |
| eqvrelqseqdisj4 38824 | Lemma for ~ petincnvepres2... |
| eqvrelqseqdisj5 38825 | Lemma for the Partition-Eq... |
| mainer 38826 | The Main Theorem of Equiva... |
| partimcomember 38827 | Partition with general ` R... |
| mpet3 38828 | Member Partition-Equivalen... |
| cpet2 38829 | The conventional form of t... |
| cpet 38830 | The conventional form of M... |
| mpet 38831 | Member Partition-Equivalen... |
| mpet2 38832 | Member Partition-Equivalen... |
| mpets2 38833 | Member Partition-Equivalen... |
| mpets 38834 | Member Partition-Equivalen... |
| mainpart 38835 | Partition with general ` R... |
| fences 38836 | The Theorem of Fences by E... |
| fences2 38837 | The Theorem of Fences by E... |
| mainer2 38838 | The Main Theorem of Equiva... |
| mainerim 38839 | Every equivalence relation... |
| petincnvepres2 38840 | A partition-equivalence th... |
| petincnvepres 38841 | The shortest form of a par... |
| pet2 38842 | Partition-Equivalence Theo... |
| pet 38843 | Partition-Equivalence Theo... |
| pets 38844 | Partition-Equivalence Theo... |
| dmqsblocks 38845 | If the ~ pet span ` ( R |X... |
| prtlem60 38846 | Lemma for ~ prter3 . (Con... |
| bicomdd 38847 | Commute two sides of a bic... |
| jca2r 38848 | Inference conjoining the c... |
| jca3 38849 | Inference conjoining the c... |
| prtlem70 38850 | Lemma for ~ prter3 : a rea... |
| ibdr 38851 | Reverse of ~ ibd . (Contr... |
| prtlem100 38852 | Lemma for ~ prter3 . (Con... |
| prtlem5 38853 | Lemma for ~ prter1 , ~ prt... |
| prtlem80 38854 | Lemma for ~ prter2 . (Con... |
| brabsb2 38855 | A closed form of ~ brabsb ... |
| eqbrrdv2 38856 | Other version of ~ eqbrrdi... |
| prtlem9 38857 | Lemma for ~ prter3 . (Con... |
| prtlem10 38858 | Lemma for ~ prter3 . (Con... |
| prtlem11 38859 | Lemma for ~ prter2 . (Con... |
| prtlem12 38860 | Lemma for ~ prtex and ~ pr... |
| prtlem13 38861 | Lemma for ~ prter1 , ~ prt... |
| prtlem16 38862 | Lemma for ~ prtex , ~ prte... |
| prtlem400 38863 | Lemma for ~ prter2 and als... |
| erprt 38866 | The quotient set of an equ... |
| prtlem14 38867 | Lemma for ~ prter1 , ~ prt... |
| prtlem15 38868 | Lemma for ~ prter1 and ~ p... |
| prtlem17 38869 | Lemma for ~ prter2 . (Con... |
| prtlem18 38870 | Lemma for ~ prter2 . (Con... |
| prtlem19 38871 | Lemma for ~ prter2 . (Con... |
| prter1 38872 | Every partition generates ... |
| prtex 38873 | The equivalence relation g... |
| prter2 38874 | The quotient set of the eq... |
| prter3 38875 | For every partition there ... |
| axc5 38886 | This theorem repeats ~ sp ... |
| ax4fromc4 38887 | Rederivation of Axiom ~ ax... |
| ax10fromc7 38888 | Rederivation of Axiom ~ ax... |
| ax6fromc10 38889 | Rederivation of Axiom ~ ax... |
| hba1-o 38890 | The setvar ` x ` is not fr... |
| axc4i-o 38891 | Inference version of ~ ax-... |
| equid1 38892 | Proof of ~ equid from our ... |
| equcomi1 38893 | Proof of ~ equcomi from ~ ... |
| aecom-o 38894 | Commutation law for identi... |
| aecoms-o 38895 | A commutation rule for ide... |
| hbae-o 38896 | All variables are effectiv... |
| dral1-o 38897 | Formula-building lemma for... |
| ax12fromc15 38898 | Rederivation of Axiom ~ ax... |
| ax13fromc9 38899 | Derive ~ ax-13 from ~ ax-c... |
| ax5ALT 38900 | Axiom to quantify a variab... |
| sps-o 38901 | Generalization of antecede... |
| hbequid 38902 | Bound-variable hypothesis ... |
| nfequid-o 38903 | Bound-variable hypothesis ... |
| axc5c7 38904 | Proof of a single axiom th... |
| axc5c7toc5 38905 | Rederivation of ~ ax-c5 fr... |
| axc5c7toc7 38906 | Rederivation of ~ ax-c7 fr... |
| axc711 38907 | Proof of a single axiom th... |
| nfa1-o 38908 | ` x ` is not free in ` A. ... |
| axc711toc7 38909 | Rederivation of ~ ax-c7 fr... |
| axc711to11 38910 | Rederivation of ~ ax-11 fr... |
| axc5c711 38911 | Proof of a single axiom th... |
| axc5c711toc5 38912 | Rederivation of ~ ax-c5 fr... |
| axc5c711toc7 38913 | Rederivation of ~ ax-c7 fr... |
| axc5c711to11 38914 | Rederivation of ~ ax-11 fr... |
| equidqe 38915 | ~ equid with existential q... |
| axc5sp1 38916 | A special case of ~ ax-c5 ... |
| equidq 38917 | ~ equid with universal qua... |
| equid1ALT 38918 | Alternate proof of ~ equid... |
| axc11nfromc11 38919 | Rederivation of ~ ax-c11n ... |
| naecoms-o 38920 | A commutation rule for dis... |
| hbnae-o 38921 | All variables are effectiv... |
| dvelimf-o 38922 | Proof of ~ dvelimh that us... |
| dral2-o 38923 | Formula-building lemma for... |
| aev-o 38924 | A "distinctor elimination"... |
| ax5eq 38925 | Theorem to add distinct qu... |
| dveeq2-o 38926 | Quantifier introduction wh... |
| axc16g-o 38927 | A generalization of Axiom ... |
| dveeq1-o 38928 | Quantifier introduction wh... |
| dveeq1-o16 38929 | Version of ~ dveeq1 using ... |
| ax5el 38930 | Theorem to add distinct qu... |
| axc11n-16 38931 | This theorem shows that, g... |
| dveel2ALT 38932 | Alternate proof of ~ dveel... |
| ax12f 38933 | Basis step for constructin... |
| ax12eq 38934 | Basis step for constructin... |
| ax12el 38935 | Basis step for constructin... |
| ax12indn 38936 | Induction step for constru... |
| ax12indi 38937 | Induction step for constru... |
| ax12indalem 38938 | Lemma for ~ ax12inda2 and ... |
| ax12inda2ALT 38939 | Alternate proof of ~ ax12i... |
| ax12inda2 38940 | Induction step for constru... |
| ax12inda 38941 | Induction step for constru... |
| ax12v2-o 38942 | Rederivation of ~ ax-c15 f... |
| ax12a2-o 38943 | Derive ~ ax-c15 from a hyp... |
| axc11-o 38944 | Show that ~ ax-c11 can be ... |
| fsumshftd 38945 | Index shift of a finite su... |
| riotaclbgBAD 38947 | Closure of restricted iota... |
| riotaclbBAD 38948 | Closure of restricted iota... |
| riotasvd 38949 | Deduction version of ~ rio... |
| riotasv2d 38950 | Value of description binde... |
| riotasv2s 38951 | The value of description b... |
| riotasv 38952 | Value of description binde... |
| riotasv3d 38953 | A property ` ch ` holding ... |
| elimhyps 38954 | A version of ~ elimhyp usi... |
| dedths 38955 | A version of weak deductio... |
| renegclALT 38956 | Closure law for negative o... |
| elimhyps2 38957 | Generalization of ~ elimhy... |
| dedths2 38958 | Generalization of ~ dedths... |
| nfcxfrdf 38959 | A utility lemma to transfe... |
| nfded 38960 | A deduction theorem that c... |
| nfded2 38961 | A deduction theorem that c... |
| nfunidALT2 38962 | Deduction version of ~ nfu... |
| nfunidALT 38963 | Deduction version of ~ nfu... |
| nfopdALT 38964 | Deduction version of bound... |
| cnaddcom 38965 | Recover the commutative la... |
| toycom 38966 | Show the commutative law f... |
| lshpset 38971 | The set of all hyperplanes... |
| islshp 38972 | The predicate "is a hyperp... |
| islshpsm 38973 | Hyperplane properties expr... |
| lshplss 38974 | A hyperplane is a subspace... |
| lshpne 38975 | A hyperplane is not equal ... |
| lshpnel 38976 | A hyperplane's generating ... |
| lshpnelb 38977 | The subspace sum of a hype... |
| lshpnel2N 38978 | Condition that determines ... |
| lshpne0 38979 | The member of the span in ... |
| lshpdisj 38980 | A hyperplane and the span ... |
| lshpcmp 38981 | If two hyperplanes are com... |
| lshpinN 38982 | The intersection of two di... |
| lsatset 38983 | The set of all 1-dim subsp... |
| islsat 38984 | The predicate "is a 1-dim ... |
| lsatlspsn2 38985 | The span of a nonzero sing... |
| lsatlspsn 38986 | The span of a nonzero sing... |
| islsati 38987 | A 1-dim subspace (atom) (o... |
| lsateln0 38988 | A 1-dim subspace (atom) (o... |
| lsatlss 38989 | The set of 1-dim subspaces... |
| lsatlssel 38990 | An atom is a subspace. (C... |
| lsatssv 38991 | An atom is a set of vector... |
| lsatn0 38992 | A 1-dim subspace (atom) of... |
| lsatspn0 38993 | The span of a vector is an... |
| lsator0sp 38994 | The span of a vector is ei... |
| lsatssn0 38995 | A subspace (or any class) ... |
| lsatcmp 38996 | If two atoms are comparabl... |
| lsatcmp2 38997 | If an atom is included in ... |
| lsatel 38998 | A nonzero vector in an ato... |
| lsatelbN 38999 | A nonzero vector in an ato... |
| lsat2el 39000 | Two atoms sharing a nonzer... |
| lsmsat 39001 | Convert comparison of atom... |
| lsatfixedN 39002 | Show equality with the spa... |
| lsmsatcv 39003 | Subspace sum has the cover... |
| lssatomic 39004 | The lattice of subspaces i... |
| lssats 39005 | The lattice of subspaces i... |
| lpssat 39006 | Two subspaces in a proper ... |
| lrelat 39007 | Subspaces are relatively a... |
| lssatle 39008 | The ordering of two subspa... |
| lssat 39009 | Two subspaces in a proper ... |
| islshpat 39010 | Hyperplane properties expr... |
| lcvfbr 39013 | The covers relation for a ... |
| lcvbr 39014 | The covers relation for a ... |
| lcvbr2 39015 | The covers relation for a ... |
| lcvbr3 39016 | The covers relation for a ... |
| lcvpss 39017 | The covers relation implie... |
| lcvnbtwn 39018 | The covers relation implie... |
| lcvntr 39019 | The covers relation is not... |
| lcvnbtwn2 39020 | The covers relation implie... |
| lcvnbtwn3 39021 | The covers relation implie... |
| lsmcv2 39022 | Subspace sum has the cover... |
| lcvat 39023 | If a subspace covers anoth... |
| lsatcv0 39024 | An atom covers the zero su... |
| lsatcveq0 39025 | A subspace covered by an a... |
| lsat0cv 39026 | A subspace is an atom iff ... |
| lcvexchlem1 39027 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem2 39028 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem3 39029 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem4 39030 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem5 39031 | Lemma for ~ lcvexch . (Co... |
| lcvexch 39032 | Subspaces satisfy the exch... |
| lcvp 39033 | Covering property of Defin... |
| lcv1 39034 | Covering property of a sub... |
| lcv2 39035 | Covering property of a sub... |
| lsatexch 39036 | The atom exchange property... |
| lsatnle 39037 | The meet of a subspace and... |
| lsatnem0 39038 | The meet of distinct atoms... |
| lsatexch1 39039 | The atom exch1ange propert... |
| lsatcv0eq 39040 | If the sum of two atoms co... |
| lsatcv1 39041 | Two atoms covering the zer... |
| lsatcvatlem 39042 | Lemma for ~ lsatcvat . (C... |
| lsatcvat 39043 | A nonzero subspace less th... |
| lsatcvat2 39044 | A subspace covered by the ... |
| lsatcvat3 39045 | A condition implying that ... |
| islshpcv 39046 | Hyperplane properties expr... |
| l1cvpat 39047 | A subspace covered by the ... |
| l1cvat 39048 | Create an atom under an el... |
| lshpat 39049 | Create an atom under a hyp... |
| lflset 39052 | The set of linear function... |
| islfl 39053 | The predicate "is a linear... |
| lfli 39054 | Property of a linear funct... |
| islfld 39055 | Properties that determine ... |
| lflf 39056 | A linear functional is a f... |
| lflcl 39057 | A linear functional value ... |
| lfl0 39058 | A linear functional is zer... |
| lfladd 39059 | Property of a linear funct... |
| lflsub 39060 | Property of a linear funct... |
| lflmul 39061 | Property of a linear funct... |
| lfl0f 39062 | The zero function is a fun... |
| lfl1 39063 | A nonzero functional has a... |
| lfladdcl 39064 | Closure of addition of two... |
| lfladdcom 39065 | Commutativity of functiona... |
| lfladdass 39066 | Associativity of functiona... |
| lfladd0l 39067 | Functional addition with t... |
| lflnegcl 39068 | Closure of the negative of... |
| lflnegl 39069 | A functional plus its nega... |
| lflvscl 39070 | Closure of a scalar produc... |
| lflvsdi1 39071 | Distributive law for (righ... |
| lflvsdi2 39072 | Reverse distributive law f... |
| lflvsdi2a 39073 | Reverse distributive law f... |
| lflvsass 39074 | Associative law for (right... |
| lfl0sc 39075 | The (right vector space) s... |
| lflsc0N 39076 | The scalar product with th... |
| lfl1sc 39077 | The (right vector space) s... |
| lkrfval 39080 | The kernel of a functional... |
| lkrval 39081 | Value of the kernel of a f... |
| ellkr 39082 | Membership in the kernel o... |
| lkrval2 39083 | Value of the kernel of a f... |
| ellkr2 39084 | Membership in the kernel o... |
| lkrcl 39085 | A member of the kernel of ... |
| lkrf0 39086 | The value of a functional ... |
| lkr0f 39087 | The kernel of the zero fun... |
| lkrlss 39088 | The kernel of a linear fun... |
| lkrssv 39089 | The kernel of a linear fun... |
| lkrsc 39090 | The kernel of a nonzero sc... |
| lkrscss 39091 | The kernel of a scalar pro... |
| eqlkr 39092 | Two functionals with the s... |
| eqlkr2 39093 | Two functionals with the s... |
| eqlkr3 39094 | Two functionals with the s... |
| lkrlsp 39095 | The subspace sum of a kern... |
| lkrlsp2 39096 | The subspace sum of a kern... |
| lkrlsp3 39097 | The subspace sum of a kern... |
| lkrshp 39098 | The kernel of a nonzero fu... |
| lkrshp3 39099 | The kernels of nonzero fun... |
| lkrshpor 39100 | The kernel of a functional... |
| lkrshp4 39101 | A kernel is a hyperplane i... |
| lshpsmreu 39102 | Lemma for ~ lshpkrex . Sh... |
| lshpkrlem1 39103 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem2 39104 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem3 39105 | Lemma for ~ lshpkrex . De... |
| lshpkrlem4 39106 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem5 39107 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem6 39108 | Lemma for ~ lshpkrex . Sh... |
| lshpkrcl 39109 | The set ` G ` defined by h... |
| lshpkr 39110 | The kernel of functional `... |
| lshpkrex 39111 | There exists a functional ... |
| lshpset2N 39112 | The set of all hyperplanes... |
| islshpkrN 39113 | The predicate "is a hyperp... |
| lfl1dim 39114 | Equivalent expressions for... |
| lfl1dim2N 39115 | Equivalent expressions for... |
| ldualset 39118 | Define the (left) dual of ... |
| ldualvbase 39119 | The vectors of a dual spac... |
| ldualelvbase 39120 | Utility theorem for conver... |
| ldualfvadd 39121 | Vector addition in the dua... |
| ldualvadd 39122 | Vector addition in the dua... |
| ldualvaddcl 39123 | The value of vector additi... |
| ldualvaddval 39124 | The value of the value of ... |
| ldualsca 39125 | The ring of scalars of the... |
| ldualsbase 39126 | Base set of scalar ring fo... |
| ldualsaddN 39127 | Scalar addition for the du... |
| ldualsmul 39128 | Scalar multiplication for ... |
| ldualfvs 39129 | Scalar product operation f... |
| ldualvs 39130 | Scalar product operation v... |
| ldualvsval 39131 | Value of scalar product op... |
| ldualvscl 39132 | The scalar product operati... |
| ldualvaddcom 39133 | Commutative law for vector... |
| ldualvsass 39134 | Associative law for scalar... |
| ldualvsass2 39135 | Associative law for scalar... |
| ldualvsdi1 39136 | Distributive law for scala... |
| ldualvsdi2 39137 | Reverse distributive law f... |
| ldualgrplem 39138 | Lemma for ~ ldualgrp . (C... |
| ldualgrp 39139 | The dual of a vector space... |
| ldual0 39140 | The zero scalar of the dua... |
| ldual1 39141 | The unit scalar of the dua... |
| ldualneg 39142 | The negative of a scalar o... |
| ldual0v 39143 | The zero vector of the dua... |
| ldual0vcl 39144 | The dual zero vector is a ... |
| lduallmodlem 39145 | Lemma for ~ lduallmod . (... |
| lduallmod 39146 | The dual of a left module ... |
| lduallvec 39147 | The dual of a left vector ... |
| ldualvsub 39148 | The value of vector subtra... |
| ldualvsubcl 39149 | Closure of vector subtract... |
| ldualvsubval 39150 | The value of the value of ... |
| ldualssvscl 39151 | Closure of scalar product ... |
| ldualssvsubcl 39152 | Closure of vector subtract... |
| ldual0vs 39153 | Scalar zero times a functi... |
| lkr0f2 39154 | The kernel of the zero fun... |
| lduallkr3 39155 | The kernels of nonzero fun... |
| lkrpssN 39156 | Proper subset relation bet... |
| lkrin 39157 | Intersection of the kernel... |
| eqlkr4 39158 | Two functionals with the s... |
| ldual1dim 39159 | Equivalent expressions for... |
| ldualkrsc 39160 | The kernel of a nonzero sc... |
| lkrss 39161 | The kernel of a scalar pro... |
| lkrss2N 39162 | Two functionals with kerne... |
| lkreqN 39163 | Proportional functionals h... |
| lkrlspeqN 39164 | Condition for colinear fun... |
| isopos 39173 | The predicate "is an ortho... |
| opposet 39174 | Every orthoposet is a pose... |
| oposlem 39175 | Lemma for orthoposet prope... |
| op01dm 39176 | Conditions necessary for z... |
| op0cl 39177 | An orthoposet has a zero e... |
| op1cl 39178 | An orthoposet has a unity ... |
| op0le 39179 | Orthoposet zero is less th... |
| ople0 39180 | An element less than or eq... |
| opnlen0 39181 | An element not less than a... |
| lub0N 39182 | The least upper bound of t... |
| opltn0 39183 | A lattice element greater ... |
| ople1 39184 | Any element is less than t... |
| op1le 39185 | If the orthoposet unity is... |
| glb0N 39186 | The greatest lower bound o... |
| opoccl 39187 | Closure of orthocomplement... |
| opococ 39188 | Double negative law for or... |
| opcon3b 39189 | Contraposition law for ort... |
| opcon2b 39190 | Orthocomplement contraposi... |
| opcon1b 39191 | Orthocomplement contraposi... |
| oplecon3 39192 | Contraposition law for ort... |
| oplecon3b 39193 | Contraposition law for ort... |
| oplecon1b 39194 | Contraposition law for str... |
| opoc1 39195 | Orthocomplement of orthopo... |
| opoc0 39196 | Orthocomplement of orthopo... |
| opltcon3b 39197 | Contraposition law for str... |
| opltcon1b 39198 | Contraposition law for str... |
| opltcon2b 39199 | Contraposition law for str... |
| opexmid 39200 | Law of excluded middle for... |
| opnoncon 39201 | Law of contradiction for o... |
| riotaocN 39202 | The orthocomplement of the... |
| cmtfvalN 39203 | Value of commutes relation... |
| cmtvalN 39204 | Equivalence for commutes r... |
| isolat 39205 | The predicate "is an ortho... |
| ollat 39206 | An ortholattice is a latti... |
| olop 39207 | An ortholattice is an orth... |
| olposN 39208 | An ortholattice is a poset... |
| isolatiN 39209 | Properties that determine ... |
| oldmm1 39210 | De Morgan's law for meet i... |
| oldmm2 39211 | De Morgan's law for meet i... |
| oldmm3N 39212 | De Morgan's law for meet i... |
| oldmm4 39213 | De Morgan's law for meet i... |
| oldmj1 39214 | De Morgan's law for join i... |
| oldmj2 39215 | De Morgan's law for join i... |
| oldmj3 39216 | De Morgan's law for join i... |
| oldmj4 39217 | De Morgan's law for join i... |
| olj01 39218 | An ortholattice element jo... |
| olj02 39219 | An ortholattice element jo... |
| olm11 39220 | The meet of an ortholattic... |
| olm12 39221 | The meet of an ortholattic... |
| latmassOLD 39222 | Ortholattice meet is assoc... |
| latm12 39223 | A rearrangement of lattice... |
| latm32 39224 | A rearrangement of lattice... |
| latmrot 39225 | Rotate lattice meet of 3 c... |
| latm4 39226 | Rearrangement of lattice m... |
| latmmdiN 39227 | Lattice meet distributes o... |
| latmmdir 39228 | Lattice meet distributes o... |
| olm01 39229 | Meet with lattice zero is ... |
| olm02 39230 | Meet with lattice zero is ... |
| isoml 39231 | The predicate "is an ortho... |
| isomliN 39232 | Properties that determine ... |
| omlol 39233 | An orthomodular lattice is... |
| omlop 39234 | An orthomodular lattice is... |
| omllat 39235 | An orthomodular lattice is... |
| omllaw 39236 | The orthomodular law. (Co... |
| omllaw2N 39237 | Variation of orthomodular ... |
| omllaw3 39238 | Orthomodular law equivalen... |
| omllaw4 39239 | Orthomodular law equivalen... |
| omllaw5N 39240 | The orthomodular law. Rem... |
| cmtcomlemN 39241 | Lemma for ~ cmtcomN . ( ~... |
| cmtcomN 39242 | Commutation is symmetric. ... |
| cmt2N 39243 | Commutation with orthocomp... |
| cmt3N 39244 | Commutation with orthocomp... |
| cmt4N 39245 | Commutation with orthocomp... |
| cmtbr2N 39246 | Alternate definition of th... |
| cmtbr3N 39247 | Alternate definition for t... |
| cmtbr4N 39248 | Alternate definition for t... |
| lecmtN 39249 | Ordered elements commute. ... |
| cmtidN 39250 | Any element commutes with ... |
| omlfh1N 39251 | Foulis-Holland Theorem, pa... |
| omlfh3N 39252 | Foulis-Holland Theorem, pa... |
| omlmod1i2N 39253 | Analogue of modular law ~ ... |
| omlspjN 39254 | Contraction of a Sasaki pr... |
| cvrfval 39261 | Value of covers relation "... |
| cvrval 39262 | Binary relation expressing... |
| cvrlt 39263 | The covers relation implie... |
| cvrnbtwn 39264 | There is no element betwee... |
| ncvr1 39265 | No element covers the latt... |
| cvrletrN 39266 | Property of an element abo... |
| cvrval2 39267 | Binary relation expressing... |
| cvrnbtwn2 39268 | The covers relation implie... |
| cvrnbtwn3 39269 | The covers relation implie... |
| cvrcon3b 39270 | Contraposition law for the... |
| cvrle 39271 | The covers relation implie... |
| cvrnbtwn4 39272 | The covers relation implie... |
| cvrnle 39273 | The covers relation implie... |
| cvrne 39274 | The covers relation implie... |
| cvrnrefN 39275 | The covers relation is not... |
| cvrcmp 39276 | If two lattice elements th... |
| cvrcmp2 39277 | If two lattice elements co... |
| pats 39278 | The set of atoms in a pose... |
| isat 39279 | The predicate "is an atom"... |
| isat2 39280 | The predicate "is an atom"... |
| atcvr0 39281 | An atom covers zero. ( ~ ... |
| atbase 39282 | An atom is a member of the... |
| atssbase 39283 | The set of atoms is a subs... |
| 0ltat 39284 | An atom is greater than ze... |
| leatb 39285 | A poset element less than ... |
| leat 39286 | A poset element less than ... |
| leat2 39287 | A nonzero poset element le... |
| leat3 39288 | A poset element less than ... |
| meetat 39289 | The meet of any element wi... |
| meetat2 39290 | The meet of any element wi... |
| isatl 39292 | The predicate "is an atomi... |
| atllat 39293 | An atomic lattice is a lat... |
| atlpos 39294 | An atomic lattice is a pos... |
| atl0dm 39295 | Condition necessary for ze... |
| atl0cl 39296 | An atomic lattice has a ze... |
| atl0le 39297 | Orthoposet zero is less th... |
| atlle0 39298 | An element less than or eq... |
| atlltn0 39299 | A lattice element greater ... |
| isat3 39300 | The predicate "is an atom"... |
| atn0 39301 | An atom is not zero. ( ~ ... |
| atnle0 39302 | An atom is not less than o... |
| atlen0 39303 | A lattice element is nonze... |
| atcmp 39304 | If two atoms are comparabl... |
| atncmp 39305 | Frequently-used variation ... |
| atnlt 39306 | Two atoms cannot satisfy t... |
| atcvreq0 39307 | An element covered by an a... |
| atncvrN 39308 | Two atoms cannot satisfy t... |
| atlex 39309 | Every nonzero element of a... |
| atnle 39310 | Two ways of expressing "an... |
| atnem0 39311 | The meet of distinct atoms... |
| atlatmstc 39312 | An atomic, complete, ortho... |
| atlatle 39313 | The ordering of two Hilber... |
| atlrelat1 39314 | An atomistic lattice with ... |
| iscvlat 39316 | The predicate "is an atomi... |
| iscvlat2N 39317 | The predicate "is an atomi... |
| cvlatl 39318 | An atomic lattice with the... |
| cvllat 39319 | An atomic lattice with the... |
| cvlposN 39320 | An atomic lattice with the... |
| cvlexch1 39321 | An atomic covering lattice... |
| cvlexch2 39322 | An atomic covering lattice... |
| cvlexchb1 39323 | An atomic covering lattice... |
| cvlexchb2 39324 | An atomic covering lattice... |
| cvlexch3 39325 | An atomic covering lattice... |
| cvlexch4N 39326 | An atomic covering lattice... |
| cvlatexchb1 39327 | A version of ~ cvlexchb1 f... |
| cvlatexchb2 39328 | A version of ~ cvlexchb2 f... |
| cvlatexch1 39329 | Atom exchange property. (... |
| cvlatexch2 39330 | Atom exchange property. (... |
| cvlatexch3 39331 | Atom exchange property. (... |
| cvlcvr1 39332 | The covering property. Pr... |
| cvlcvrp 39333 | A Hilbert lattice satisfie... |
| cvlatcvr1 39334 | An atom is covered by its ... |
| cvlatcvr2 39335 | An atom is covered by its ... |
| cvlsupr2 39336 | Two equivalent ways of exp... |
| cvlsupr3 39337 | Two equivalent ways of exp... |
| cvlsupr4 39338 | Consequence of superpositi... |
| cvlsupr5 39339 | Consequence of superpositi... |
| cvlsupr6 39340 | Consequence of superpositi... |
| cvlsupr7 39341 | Consequence of superpositi... |
| cvlsupr8 39342 | Consequence of superpositi... |
| ishlat1 39345 | The predicate "is a Hilber... |
| ishlat2 39346 | The predicate "is a Hilber... |
| ishlat3N 39347 | The predicate "is a Hilber... |
| ishlatiN 39348 | Properties that determine ... |
| hlomcmcv 39349 | A Hilbert lattice is ortho... |
| hloml 39350 | A Hilbert lattice is ortho... |
| hlclat 39351 | A Hilbert lattice is compl... |
| hlcvl 39352 | A Hilbert lattice is an at... |
| hlatl 39353 | A Hilbert lattice is atomi... |
| hlol 39354 | A Hilbert lattice is an or... |
| hlop 39355 | A Hilbert lattice is an or... |
| hllat 39356 | A Hilbert lattice is a lat... |
| hllatd 39357 | Deduction form of ~ hllat ... |
| hlomcmat 39358 | A Hilbert lattice is ortho... |
| hlpos 39359 | A Hilbert lattice is a pos... |
| hlatjcl 39360 | Closure of join operation.... |
| hlatjcom 39361 | Commutatitivity of join op... |
| hlatjidm 39362 | Idempotence of join operat... |
| hlatjass 39363 | Lattice join is associativ... |
| hlatj12 39364 | Swap 1st and 2nd members o... |
| hlatj32 39365 | Swap 2nd and 3rd members o... |
| hlatjrot 39366 | Rotate lattice join of 3 c... |
| hlatj4 39367 | Rearrangement of lattice j... |
| hlatlej1 39368 | A join's first argument is... |
| hlatlej2 39369 | A join's second argument i... |
| glbconN 39370 | De Morgan's law for GLB an... |
| glbconNOLD 39371 | Obsolete version of ~ glbc... |
| glbconxN 39372 | De Morgan's law for GLB an... |
| atnlej1 39373 | If an atom is not less tha... |
| atnlej2 39374 | If an atom is not less tha... |
| hlsuprexch 39375 | A Hilbert lattice has the ... |
| hlexch1 39376 | A Hilbert lattice has the ... |
| hlexch2 39377 | A Hilbert lattice has the ... |
| hlexchb1 39378 | A Hilbert lattice has the ... |
| hlexchb2 39379 | A Hilbert lattice has the ... |
| hlsupr 39380 | A Hilbert lattice has the ... |
| hlsupr2 39381 | A Hilbert lattice has the ... |
| hlhgt4 39382 | A Hilbert lattice has a he... |
| hlhgt2 39383 | A Hilbert lattice has a he... |
| hl0lt1N 39384 | Lattice 0 is less than lat... |
| hlexch3 39385 | A Hilbert lattice has the ... |
| hlexch4N 39386 | A Hilbert lattice has the ... |
| hlatexchb1 39387 | A version of ~ hlexchb1 fo... |
| hlatexchb2 39388 | A version of ~ hlexchb2 fo... |
| hlatexch1 39389 | Atom exchange property. (... |
| hlatexch2 39390 | Atom exchange property. (... |
| hlatmstcOLDN 39391 | An atomic, complete, ortho... |
| hlatle 39392 | The ordering of two Hilber... |
| hlateq 39393 | The equality of two Hilber... |
| hlrelat1 39394 | An atomistic lattice with ... |
| hlrelat5N 39395 | An atomistic lattice with ... |
| hlrelat 39396 | A Hilbert lattice is relat... |
| hlrelat2 39397 | A consequence of relative ... |
| exatleN 39398 | A condition for an atom to... |
| hl2at 39399 | A Hilbert lattice has at l... |
| atex 39400 | At least one atom exists. ... |
| intnatN 39401 | If the intersection with a... |
| 2llnne2N 39402 | Condition implying that tw... |
| 2llnneN 39403 | Condition implying that tw... |
| cvr1 39404 | A Hilbert lattice has the ... |
| cvr2N 39405 | Less-than and covers equiv... |
| hlrelat3 39406 | The Hilbert lattice is rel... |
| cvrval3 39407 | Binary relation expressing... |
| cvrval4N 39408 | Binary relation expressing... |
| cvrval5 39409 | Binary relation expressing... |
| cvrp 39410 | A Hilbert lattice satisfie... |
| atcvr1 39411 | An atom is covered by its ... |
| atcvr2 39412 | An atom is covered by its ... |
| cvrexchlem 39413 | Lemma for ~ cvrexch . ( ~... |
| cvrexch 39414 | A Hilbert lattice satisfie... |
| cvratlem 39415 | Lemma for ~ cvrat . ( ~ a... |
| cvrat 39416 | A nonzero Hilbert lattice ... |
| ltltncvr 39417 | A chained strong ordering ... |
| ltcvrntr 39418 | Non-transitive condition f... |
| cvrntr 39419 | The covers relation is not... |
| atcvr0eq 39420 | The covers relation is not... |
| lnnat 39421 | A line (the join of two di... |
| atcvrj0 39422 | Two atoms covering the zer... |
| cvrat2 39423 | A Hilbert lattice element ... |
| atcvrneN 39424 | Inequality derived from at... |
| atcvrj1 39425 | Condition for an atom to b... |
| atcvrj2b 39426 | Condition for an atom to b... |
| atcvrj2 39427 | Condition for an atom to b... |
| atleneN 39428 | Inequality derived from at... |
| atltcvr 39429 | An equivalence of less-tha... |
| atle 39430 | Any nonzero element has an... |
| atlt 39431 | Two atoms are unequal iff ... |
| atlelt 39432 | Transfer less-than relatio... |
| 2atlt 39433 | Given an atom less than an... |
| atexchcvrN 39434 | Atom exchange property. V... |
| atexchltN 39435 | Atom exchange property. V... |
| cvrat3 39436 | A condition implying that ... |
| cvrat4 39437 | A condition implying exist... |
| cvrat42 39438 | Commuted version of ~ cvra... |
| 2atjm 39439 | The meet of a line (expres... |
| atbtwn 39440 | Property of a 3rd atom ` R... |
| atbtwnexOLDN 39441 | There exists a 3rd atom ` ... |
| atbtwnex 39442 | Given atoms ` P ` in ` X `... |
| 3noncolr2 39443 | Two ways to express 3 non-... |
| 3noncolr1N 39444 | Two ways to express 3 non-... |
| hlatcon3 39445 | Atom exchange combined wit... |
| hlatcon2 39446 | Atom exchange combined wit... |
| 4noncolr3 39447 | A way to express 4 non-col... |
| 4noncolr2 39448 | A way to express 4 non-col... |
| 4noncolr1 39449 | A way to express 4 non-col... |
| athgt 39450 | A Hilbert lattice, whose h... |
| 3dim0 39451 | There exists a 3-dimension... |
| 3dimlem1 39452 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem2 39453 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3a 39454 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem3 39455 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3OLDN 39456 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4a 39457 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem4 39458 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4OLDN 39459 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1lem5 39460 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1 39461 | Construct a 3-dimensional ... |
| 3dim2 39462 | Construct 2 new layers on ... |
| 3dim3 39463 | Construct a new layer on t... |
| 2dim 39464 | Generate a height-3 elemen... |
| 1dimN 39465 | An atom is covered by a he... |
| 1cvrco 39466 | The orthocomplement of an ... |
| 1cvratex 39467 | There exists an atom less ... |
| 1cvratlt 39468 | An atom less than or equal... |
| 1cvrjat 39469 | An element covered by the ... |
| 1cvrat 39470 | Create an atom under an el... |
| ps-1 39471 | The join of two atoms ` R ... |
| ps-2 39472 | Lattice analogue for the p... |
| 2atjlej 39473 | Two atoms are different if... |
| hlatexch3N 39474 | Rearrange join of atoms in... |
| hlatexch4 39475 | Exchange 2 atoms. (Contri... |
| ps-2b 39476 | Variation of projective ge... |
| 3atlem1 39477 | Lemma for ~ 3at . (Contri... |
| 3atlem2 39478 | Lemma for ~ 3at . (Contri... |
| 3atlem3 39479 | Lemma for ~ 3at . (Contri... |
| 3atlem4 39480 | Lemma for ~ 3at . (Contri... |
| 3atlem5 39481 | Lemma for ~ 3at . (Contri... |
| 3atlem6 39482 | Lemma for ~ 3at . (Contri... |
| 3atlem7 39483 | Lemma for ~ 3at . (Contri... |
| 3at 39484 | Any three non-colinear ato... |
| llnset 39499 | The set of lattice lines i... |
| islln 39500 | The predicate "is a lattic... |
| islln4 39501 | The predicate "is a lattic... |
| llni 39502 | Condition implying a latti... |
| llnbase 39503 | A lattice line is a lattic... |
| islln3 39504 | The predicate "is a lattic... |
| islln2 39505 | The predicate "is a lattic... |
| llni2 39506 | The join of two different ... |
| llnnleat 39507 | An atom cannot majorize a ... |
| llnneat 39508 | A lattice line is not an a... |
| 2atneat 39509 | The join of two distinct a... |
| llnn0 39510 | A lattice line is nonzero.... |
| islln2a 39511 | The predicate "is a lattic... |
| llnle 39512 | Any element greater than 0... |
| atcvrlln2 39513 | An atom under a line is co... |
| atcvrlln 39514 | An element covering an ato... |
| llnexatN 39515 | Given an atom on a line, t... |
| llncmp 39516 | If two lattice lines are c... |
| llnnlt 39517 | Two lattice lines cannot s... |
| 2llnmat 39518 | Two intersecting lines int... |
| 2at0mat0 39519 | Special case of ~ 2atmat0 ... |
| 2atmat0 39520 | The meet of two unequal li... |
| 2atm 39521 | An atom majorized by two d... |
| ps-2c 39522 | Variation of projective ge... |
| lplnset 39523 | The set of lattice planes ... |
| islpln 39524 | The predicate "is a lattic... |
| islpln4 39525 | The predicate "is a lattic... |
| lplni 39526 | Condition implying a latti... |
| islpln3 39527 | The predicate "is a lattic... |
| lplnbase 39528 | A lattice plane is a latti... |
| islpln5 39529 | The predicate "is a lattic... |
| islpln2 39530 | The predicate "is a lattic... |
| lplni2 39531 | The join of 3 different at... |
| lvolex3N 39532 | There is an atom outside o... |
| llnmlplnN 39533 | The intersection of a line... |
| lplnle 39534 | Any element greater than 0... |
| lplnnle2at 39535 | A lattice line (or atom) c... |
| lplnnleat 39536 | A lattice plane cannot maj... |
| lplnnlelln 39537 | A lattice plane is not les... |
| 2atnelpln 39538 | The join of two atoms is n... |
| lplnneat 39539 | No lattice plane is an ato... |
| lplnnelln 39540 | No lattice plane is a latt... |
| lplnn0N 39541 | A lattice plane is nonzero... |
| islpln2a 39542 | The predicate "is a lattic... |
| islpln2ah 39543 | The predicate "is a lattic... |
| lplnriaN 39544 | Property of a lattice plan... |
| lplnribN 39545 | Property of a lattice plan... |
| lplnric 39546 | Property of a lattice plan... |
| lplnri1 39547 | Property of a lattice plan... |
| lplnri2N 39548 | Property of a lattice plan... |
| lplnri3N 39549 | Property of a lattice plan... |
| lplnllnneN 39550 | Two lattice lines defined ... |
| llncvrlpln2 39551 | A lattice line under a lat... |
| llncvrlpln 39552 | An element covering a latt... |
| 2lplnmN 39553 | If the join of two lattice... |
| 2llnmj 39554 | The meet of two lattice li... |
| 2atmat 39555 | The meet of two intersecti... |
| lplncmp 39556 | If two lattice planes are ... |
| lplnexatN 39557 | Given a lattice line on a ... |
| lplnexllnN 39558 | Given an atom on a lattice... |
| lplnnlt 39559 | Two lattice planes cannot ... |
| 2llnjaN 39560 | The join of two different ... |
| 2llnjN 39561 | The join of two different ... |
| 2llnm2N 39562 | The meet of two different ... |
| 2llnm3N 39563 | Two lattice lines in a lat... |
| 2llnm4 39564 | Two lattice lines that maj... |
| 2llnmeqat 39565 | An atom equals the interse... |
| lvolset 39566 | The set of 3-dim lattice v... |
| islvol 39567 | The predicate "is a 3-dim ... |
| islvol4 39568 | The predicate "is a 3-dim ... |
| lvoli 39569 | Condition implying a 3-dim... |
| islvol3 39570 | The predicate "is a 3-dim ... |
| lvoli3 39571 | Condition implying a 3-dim... |
| lvolbase 39572 | A 3-dim lattice volume is ... |
| islvol5 39573 | The predicate "is a 3-dim ... |
| islvol2 39574 | The predicate "is a 3-dim ... |
| lvoli2 39575 | The join of 4 different at... |
| lvolnle3at 39576 | A lattice plane (or lattic... |
| lvolnleat 39577 | An atom cannot majorize a ... |
| lvolnlelln 39578 | A lattice line cannot majo... |
| lvolnlelpln 39579 | A lattice plane cannot maj... |
| 3atnelvolN 39580 | The join of 3 atoms is not... |
| 2atnelvolN 39581 | The join of two atoms is n... |
| lvolneatN 39582 | No lattice volume is an at... |
| lvolnelln 39583 | No lattice volume is a lat... |
| lvolnelpln 39584 | No lattice volume is a lat... |
| lvoln0N 39585 | A lattice volume is nonzer... |
| islvol2aN 39586 | The predicate "is a lattic... |
| 4atlem0a 39587 | Lemma for ~ 4at . (Contri... |
| 4atlem0ae 39588 | Lemma for ~ 4at . (Contri... |
| 4atlem0be 39589 | Lemma for ~ 4at . (Contri... |
| 4atlem3 39590 | Lemma for ~ 4at . Break i... |
| 4atlem3a 39591 | Lemma for ~ 4at . Break i... |
| 4atlem3b 39592 | Lemma for ~ 4at . Break i... |
| 4atlem4a 39593 | Lemma for ~ 4at . Frequen... |
| 4atlem4b 39594 | Lemma for ~ 4at . Frequen... |
| 4atlem4c 39595 | Lemma for ~ 4at . Frequen... |
| 4atlem4d 39596 | Lemma for ~ 4at . Frequen... |
| 4atlem9 39597 | Lemma for ~ 4at . Substit... |
| 4atlem10a 39598 | Lemma for ~ 4at . Substit... |
| 4atlem10b 39599 | Lemma for ~ 4at . Substit... |
| 4atlem10 39600 | Lemma for ~ 4at . Combine... |
| 4atlem11a 39601 | Lemma for ~ 4at . Substit... |
| 4atlem11b 39602 | Lemma for ~ 4at . Substit... |
| 4atlem11 39603 | Lemma for ~ 4at . Combine... |
| 4atlem12a 39604 | Lemma for ~ 4at . Substit... |
| 4atlem12b 39605 | Lemma for ~ 4at . Substit... |
| 4atlem12 39606 | Lemma for ~ 4at . Combine... |
| 4at 39607 | Four atoms determine a lat... |
| 4at2 39608 | Four atoms determine a lat... |
| lplncvrlvol2 39609 | A lattice line under a lat... |
| lplncvrlvol 39610 | An element covering a latt... |
| lvolcmp 39611 | If two lattice planes are ... |
| lvolnltN 39612 | Two lattice volumes cannot... |
| 2lplnja 39613 | The join of two different ... |
| 2lplnj 39614 | The join of two different ... |
| 2lplnm2N 39615 | The meet of two different ... |
| 2lplnmj 39616 | The meet of two lattice pl... |
| dalemkehl 39617 | Lemma for ~ dath . Freque... |
| dalemkelat 39618 | Lemma for ~ dath . Freque... |
| dalemkeop 39619 | Lemma for ~ dath . Freque... |
| dalempea 39620 | Lemma for ~ dath . Freque... |
| dalemqea 39621 | Lemma for ~ dath . Freque... |
| dalemrea 39622 | Lemma for ~ dath . Freque... |
| dalemsea 39623 | Lemma for ~ dath . Freque... |
| dalemtea 39624 | Lemma for ~ dath . Freque... |
| dalemuea 39625 | Lemma for ~ dath . Freque... |
| dalemyeo 39626 | Lemma for ~ dath . Freque... |
| dalemzeo 39627 | Lemma for ~ dath . Freque... |
| dalemclpjs 39628 | Lemma for ~ dath . Freque... |
| dalemclqjt 39629 | Lemma for ~ dath . Freque... |
| dalemclrju 39630 | Lemma for ~ dath . Freque... |
| dalem-clpjq 39631 | Lemma for ~ dath . Freque... |
| dalemceb 39632 | Lemma for ~ dath . Freque... |
| dalempeb 39633 | Lemma for ~ dath . Freque... |
| dalemqeb 39634 | Lemma for ~ dath . Freque... |
| dalemreb 39635 | Lemma for ~ dath . Freque... |
| dalemseb 39636 | Lemma for ~ dath . Freque... |
| dalemteb 39637 | Lemma for ~ dath . Freque... |
| dalemueb 39638 | Lemma for ~ dath . Freque... |
| dalempjqeb 39639 | Lemma for ~ dath . Freque... |
| dalemsjteb 39640 | Lemma for ~ dath . Freque... |
| dalemtjueb 39641 | Lemma for ~ dath . Freque... |
| dalemqrprot 39642 | Lemma for ~ dath . Freque... |
| dalemyeb 39643 | Lemma for ~ dath . Freque... |
| dalemcnes 39644 | Lemma for ~ dath . Freque... |
| dalempnes 39645 | Lemma for ~ dath . Freque... |
| dalemqnet 39646 | Lemma for ~ dath . Freque... |
| dalempjsen 39647 | Lemma for ~ dath . Freque... |
| dalemply 39648 | Lemma for ~ dath . Freque... |
| dalemsly 39649 | Lemma for ~ dath . Freque... |
| dalemswapyz 39650 | Lemma for ~ dath . Swap t... |
| dalemrot 39651 | Lemma for ~ dath . Rotate... |
| dalemrotyz 39652 | Lemma for ~ dath . Rotate... |
| dalem1 39653 | Lemma for ~ dath . Show t... |
| dalemcea 39654 | Lemma for ~ dath . Freque... |
| dalem2 39655 | Lemma for ~ dath . Show t... |
| dalemdea 39656 | Lemma for ~ dath . Freque... |
| dalemeea 39657 | Lemma for ~ dath . Freque... |
| dalem3 39658 | Lemma for ~ dalemdnee . (... |
| dalem4 39659 | Lemma for ~ dalemdnee . (... |
| dalemdnee 39660 | Lemma for ~ dath . Axis o... |
| dalem5 39661 | Lemma for ~ dath . Atom `... |
| dalem6 39662 | Lemma for ~ dath . Analog... |
| dalem7 39663 | Lemma for ~ dath . Analog... |
| dalem8 39664 | Lemma for ~ dath . Plane ... |
| dalem-cly 39665 | Lemma for ~ dalem9 . Cent... |
| dalem9 39666 | Lemma for ~ dath . Since ... |
| dalem10 39667 | Lemma for ~ dath . Atom `... |
| dalem11 39668 | Lemma for ~ dath . Analog... |
| dalem12 39669 | Lemma for ~ dath . Analog... |
| dalem13 39670 | Lemma for ~ dalem14 . (Co... |
| dalem14 39671 | Lemma for ~ dath . Planes... |
| dalem15 39672 | Lemma for ~ dath . The ax... |
| dalem16 39673 | Lemma for ~ dath . The at... |
| dalem17 39674 | Lemma for ~ dath . When p... |
| dalem18 39675 | Lemma for ~ dath . Show t... |
| dalem19 39676 | Lemma for ~ dath . Show t... |
| dalemccea 39677 | Lemma for ~ dath . Freque... |
| dalemddea 39678 | Lemma for ~ dath . Freque... |
| dalem-ccly 39679 | Lemma for ~ dath . Freque... |
| dalem-ddly 39680 | Lemma for ~ dath . Freque... |
| dalemccnedd 39681 | Lemma for ~ dath . Freque... |
| dalemclccjdd 39682 | Lemma for ~ dath . Freque... |
| dalemcceb 39683 | Lemma for ~ dath . Freque... |
| dalemswapyzps 39684 | Lemma for ~ dath . Swap t... |
| dalemrotps 39685 | Lemma for ~ dath . Rotate... |
| dalemcjden 39686 | Lemma for ~ dath . Show t... |
| dalem20 39687 | Lemma for ~ dath . Show t... |
| dalem21 39688 | Lemma for ~ dath . Show t... |
| dalem22 39689 | Lemma for ~ dath . Show t... |
| dalem23 39690 | Lemma for ~ dath . Show t... |
| dalem24 39691 | Lemma for ~ dath . Show t... |
| dalem25 39692 | Lemma for ~ dath . Show t... |
| dalem27 39693 | Lemma for ~ dath . Show t... |
| dalem28 39694 | Lemma for ~ dath . Lemma ... |
| dalem29 39695 | Lemma for ~ dath . Analog... |
| dalem30 39696 | Lemma for ~ dath . Analog... |
| dalem31N 39697 | Lemma for ~ dath . Analog... |
| dalem32 39698 | Lemma for ~ dath . Analog... |
| dalem33 39699 | Lemma for ~ dath . Analog... |
| dalem34 39700 | Lemma for ~ dath . Analog... |
| dalem35 39701 | Lemma for ~ dath . Analog... |
| dalem36 39702 | Lemma for ~ dath . Analog... |
| dalem37 39703 | Lemma for ~ dath . Analog... |
| dalem38 39704 | Lemma for ~ dath . Plane ... |
| dalem39 39705 | Lemma for ~ dath . Auxili... |
| dalem40 39706 | Lemma for ~ dath . Analog... |
| dalem41 39707 | Lemma for ~ dath . (Contr... |
| dalem42 39708 | Lemma for ~ dath . Auxili... |
| dalem43 39709 | Lemma for ~ dath . Planes... |
| dalem44 39710 | Lemma for ~ dath . Dummy ... |
| dalem45 39711 | Lemma for ~ dath . Dummy ... |
| dalem46 39712 | Lemma for ~ dath . Analog... |
| dalem47 39713 | Lemma for ~ dath . Analog... |
| dalem48 39714 | Lemma for ~ dath . Analog... |
| dalem49 39715 | Lemma for ~ dath . Analog... |
| dalem50 39716 | Lemma for ~ dath . Analog... |
| dalem51 39717 | Lemma for ~ dath . Constr... |
| dalem52 39718 | Lemma for ~ dath . Lines ... |
| dalem53 39719 | Lemma for ~ dath . The au... |
| dalem54 39720 | Lemma for ~ dath . Line `... |
| dalem55 39721 | Lemma for ~ dath . Lines ... |
| dalem56 39722 | Lemma for ~ dath . Analog... |
| dalem57 39723 | Lemma for ~ dath . Axis o... |
| dalem58 39724 | Lemma for ~ dath . Analog... |
| dalem59 39725 | Lemma for ~ dath . Analog... |
| dalem60 39726 | Lemma for ~ dath . ` B ` i... |
| dalem61 39727 | Lemma for ~ dath . Show t... |
| dalem62 39728 | Lemma for ~ dath . Elimin... |
| dalem63 39729 | Lemma for ~ dath . Combin... |
| dath 39730 | Desargues's theorem of pro... |
| dath2 39731 | Version of Desargues's the... |
| lineset 39732 | The set of lines in a Hilb... |
| isline 39733 | The predicate "is a line".... |
| islinei 39734 | Condition implying "is a l... |
| pointsetN 39735 | The set of points in a Hil... |
| ispointN 39736 | The predicate "is a point"... |
| atpointN 39737 | The singleton of an atom i... |
| psubspset 39738 | The set of projective subs... |
| ispsubsp 39739 | The predicate "is a projec... |
| ispsubsp2 39740 | The predicate "is a projec... |
| psubspi 39741 | Property of a projective s... |
| psubspi2N 39742 | Property of a projective s... |
| 0psubN 39743 | The empty set is a project... |
| snatpsubN 39744 | The singleton of an atom i... |
| pointpsubN 39745 | A point (singleton of an a... |
| linepsubN 39746 | A line is a projective sub... |
| atpsubN 39747 | The set of all atoms is a ... |
| psubssat 39748 | A projective subspace cons... |
| psubatN 39749 | A member of a projective s... |
| pmapfval 39750 | The projective map of a Hi... |
| pmapval 39751 | Value of the projective ma... |
| elpmap 39752 | Member of a projective map... |
| pmapssat 39753 | The projective map of a Hi... |
| pmapssbaN 39754 | A weakening of ~ pmapssat ... |
| pmaple 39755 | The projective map of a Hi... |
| pmap11 39756 | The projective map of a Hi... |
| pmapat 39757 | The projective map of an a... |
| elpmapat 39758 | Member of the projective m... |
| pmap0 39759 | Value of the projective ma... |
| pmapeq0 39760 | A projective map value is ... |
| pmap1N 39761 | Value of the projective ma... |
| pmapsub 39762 | The projective map of a Hi... |
| pmapglbx 39763 | The projective map of the ... |
| pmapglb 39764 | The projective map of the ... |
| pmapglb2N 39765 | The projective map of the ... |
| pmapglb2xN 39766 | The projective map of the ... |
| pmapmeet 39767 | The projective map of a me... |
| isline2 39768 | Definition of line in term... |
| linepmap 39769 | A line described with a pr... |
| isline3 39770 | Definition of line in term... |
| isline4N 39771 | Definition of line in term... |
| lneq2at 39772 | A line equals the join of ... |
| lnatexN 39773 | There is an atom in a line... |
| lnjatN 39774 | Given an atom in a line, t... |
| lncvrelatN 39775 | A lattice element covered ... |
| lncvrat 39776 | A line covers the atoms it... |
| lncmp 39777 | If two lines are comparabl... |
| 2lnat 39778 | Two intersecting lines int... |
| 2atm2atN 39779 | Two joins with a common at... |
| 2llnma1b 39780 | Generalization of ~ 2llnma... |
| 2llnma1 39781 | Two different intersecting... |
| 2llnma3r 39782 | Two different intersecting... |
| 2llnma2 39783 | Two different intersecting... |
| 2llnma2rN 39784 | Two different intersecting... |
| cdlema1N 39785 | A condition for required f... |
| cdlema2N 39786 | A condition for required f... |
| cdlemblem 39787 | Lemma for ~ cdlemb . (Con... |
| cdlemb 39788 | Given two atoms not less t... |
| paddfval 39791 | Projective subspace sum op... |
| paddval 39792 | Projective subspace sum op... |
| elpadd 39793 | Member of a projective sub... |
| elpaddn0 39794 | Member of projective subsp... |
| paddvaln0N 39795 | Projective subspace sum op... |
| elpaddri 39796 | Condition implying members... |
| elpaddatriN 39797 | Condition implying members... |
| elpaddat 39798 | Membership in a projective... |
| elpaddatiN 39799 | Consequence of membership ... |
| elpadd2at 39800 | Membership in a projective... |
| elpadd2at2 39801 | Membership in a projective... |
| paddunssN 39802 | Projective subspace sum in... |
| elpadd0 39803 | Member of projective subsp... |
| paddval0 39804 | Projective subspace sum wi... |
| padd01 39805 | Projective subspace sum wi... |
| padd02 39806 | Projective subspace sum wi... |
| paddcom 39807 | Projective subspace sum co... |
| paddssat 39808 | A projective subspace sum ... |
| sspadd1 39809 | A projective subspace sum ... |
| sspadd2 39810 | A projective subspace sum ... |
| paddss1 39811 | Subset law for projective ... |
| paddss2 39812 | Subset law for projective ... |
| paddss12 39813 | Subset law for projective ... |
| paddasslem1 39814 | Lemma for ~ paddass . (Co... |
| paddasslem2 39815 | Lemma for ~ paddass . (Co... |
| paddasslem3 39816 | Lemma for ~ paddass . Res... |
| paddasslem4 39817 | Lemma for ~ paddass . Com... |
| paddasslem5 39818 | Lemma for ~ paddass . Sho... |
| paddasslem6 39819 | Lemma for ~ paddass . (Co... |
| paddasslem7 39820 | Lemma for ~ paddass . Com... |
| paddasslem8 39821 | Lemma for ~ paddass . (Co... |
| paddasslem9 39822 | Lemma for ~ paddass . Com... |
| paddasslem10 39823 | Lemma for ~ paddass . Use... |
| paddasslem11 39824 | Lemma for ~ paddass . The... |
| paddasslem12 39825 | Lemma for ~ paddass . The... |
| paddasslem13 39826 | Lemma for ~ paddass . The... |
| paddasslem14 39827 | Lemma for ~ paddass . Rem... |
| paddasslem15 39828 | Lemma for ~ paddass . Use... |
| paddasslem16 39829 | Lemma for ~ paddass . Use... |
| paddasslem17 39830 | Lemma for ~ paddass . The... |
| paddasslem18 39831 | Lemma for ~ paddass . Com... |
| paddass 39832 | Projective subspace sum is... |
| padd12N 39833 | Commutative/associative la... |
| padd4N 39834 | Rearrangement of 4 terms i... |
| paddidm 39835 | Projective subspace sum is... |
| paddclN 39836 | The projective sum of two ... |
| paddssw1 39837 | Subset law for projective ... |
| paddssw2 39838 | Subset law for projective ... |
| paddss 39839 | Subset law for projective ... |
| pmodlem1 39840 | Lemma for ~ pmod1i . (Con... |
| pmodlem2 39841 | Lemma for ~ pmod1i . (Con... |
| pmod1i 39842 | The modular law holds in a... |
| pmod2iN 39843 | Dual of the modular law. ... |
| pmodN 39844 | The modular law for projec... |
| pmodl42N 39845 | Lemma derived from modular... |
| pmapjoin 39846 | The projective map of the ... |
| pmapjat1 39847 | The projective map of the ... |
| pmapjat2 39848 | The projective map of the ... |
| pmapjlln1 39849 | The projective map of the ... |
| hlmod1i 39850 | A version of the modular l... |
| atmod1i1 39851 | Version of modular law ~ p... |
| atmod1i1m 39852 | Version of modular law ~ p... |
| atmod1i2 39853 | Version of modular law ~ p... |
| llnmod1i2 39854 | Version of modular law ~ p... |
| atmod2i1 39855 | Version of modular law ~ p... |
| atmod2i2 39856 | Version of modular law ~ p... |
| llnmod2i2 39857 | Version of modular law ~ p... |
| atmod3i1 39858 | Version of modular law tha... |
| atmod3i2 39859 | Version of modular law tha... |
| atmod4i1 39860 | Version of modular law tha... |
| atmod4i2 39861 | Version of modular law tha... |
| llnexchb2lem 39862 | Lemma for ~ llnexchb2 . (... |
| llnexchb2 39863 | Line exchange property (co... |
| llnexch2N 39864 | Line exchange property (co... |
| dalawlem1 39865 | Lemma for ~ dalaw . Speci... |
| dalawlem2 39866 | Lemma for ~ dalaw . Utili... |
| dalawlem3 39867 | Lemma for ~ dalaw . First... |
| dalawlem4 39868 | Lemma for ~ dalaw . Secon... |
| dalawlem5 39869 | Lemma for ~ dalaw . Speci... |
| dalawlem6 39870 | Lemma for ~ dalaw . First... |
| dalawlem7 39871 | Lemma for ~ dalaw . Secon... |
| dalawlem8 39872 | Lemma for ~ dalaw . Speci... |
| dalawlem9 39873 | Lemma for ~ dalaw . Speci... |
| dalawlem10 39874 | Lemma for ~ dalaw . Combi... |
| dalawlem11 39875 | Lemma for ~ dalaw . First... |
| dalawlem12 39876 | Lemma for ~ dalaw . Secon... |
| dalawlem13 39877 | Lemma for ~ dalaw . Speci... |
| dalawlem14 39878 | Lemma for ~ dalaw . Combi... |
| dalawlem15 39879 | Lemma for ~ dalaw . Swap ... |
| dalaw 39880 | Desargues's law, derived f... |
| pclfvalN 39883 | The projective subspace cl... |
| pclvalN 39884 | Value of the projective su... |
| pclclN 39885 | Closure of the projective ... |
| elpclN 39886 | Membership in the projecti... |
| elpcliN 39887 | Implication of membership ... |
| pclssN 39888 | Ordering is preserved by s... |
| pclssidN 39889 | A set of atoms is included... |
| pclidN 39890 | The projective subspace cl... |
| pclbtwnN 39891 | A projective subspace sand... |
| pclunN 39892 | The projective subspace cl... |
| pclun2N 39893 | The projective subspace cl... |
| pclfinN 39894 | The projective subspace cl... |
| pclcmpatN 39895 | The set of projective subs... |
| polfvalN 39898 | The projective subspace po... |
| polvalN 39899 | Value of the projective su... |
| polval2N 39900 | Alternate expression for v... |
| polsubN 39901 | The polarity of a set of a... |
| polssatN 39902 | The polarity of a set of a... |
| pol0N 39903 | The polarity of the empty ... |
| pol1N 39904 | The polarity of the whole ... |
| 2pol0N 39905 | The closed subspace closur... |
| polpmapN 39906 | The polarity of a projecti... |
| 2polpmapN 39907 | Double polarity of a proje... |
| 2polvalN 39908 | Value of double polarity. ... |
| 2polssN 39909 | A set of atoms is a subset... |
| 3polN 39910 | Triple polarity cancels to... |
| polcon3N 39911 | Contraposition law for pol... |
| 2polcon4bN 39912 | Contraposition law for pol... |
| polcon2N 39913 | Contraposition law for pol... |
| polcon2bN 39914 | Contraposition law for pol... |
| pclss2polN 39915 | The projective subspace cl... |
| pcl0N 39916 | The projective subspace cl... |
| pcl0bN 39917 | The projective subspace cl... |
| pmaplubN 39918 | The LUB of a projective ma... |
| sspmaplubN 39919 | A set of atoms is a subset... |
| 2pmaplubN 39920 | Double projective map of a... |
| paddunN 39921 | The closure of the project... |
| poldmj1N 39922 | De Morgan's law for polari... |
| pmapj2N 39923 | The projective map of the ... |
| pmapocjN 39924 | The projective map of the ... |
| polatN 39925 | The polarity of the single... |
| 2polatN 39926 | Double polarity of the sin... |
| pnonsingN 39927 | The intersection of a set ... |
| psubclsetN 39930 | The set of closed projecti... |
| ispsubclN 39931 | The predicate "is a closed... |
| psubcliN 39932 | Property of a closed proje... |
| psubcli2N 39933 | Property of a closed proje... |
| psubclsubN 39934 | A closed projective subspa... |
| psubclssatN 39935 | A closed projective subspa... |
| pmapidclN 39936 | Projective map of the LUB ... |
| 0psubclN 39937 | The empty set is a closed ... |
| 1psubclN 39938 | The set of all atoms is a ... |
| atpsubclN 39939 | A point (singleton of an a... |
| pmapsubclN 39940 | A projective map value is ... |
| ispsubcl2N 39941 | Alternate predicate for "i... |
| psubclinN 39942 | The intersection of two cl... |
| paddatclN 39943 | The projective sum of a cl... |
| pclfinclN 39944 | The projective subspace cl... |
| linepsubclN 39945 | A line is a closed project... |
| polsubclN 39946 | A polarity is a closed pro... |
| poml4N 39947 | Orthomodular law for proje... |
| poml5N 39948 | Orthomodular law for proje... |
| poml6N 39949 | Orthomodular law for proje... |
| osumcllem1N 39950 | Lemma for ~ osumclN . (Co... |
| osumcllem2N 39951 | Lemma for ~ osumclN . (Co... |
| osumcllem3N 39952 | Lemma for ~ osumclN . (Co... |
| osumcllem4N 39953 | Lemma for ~ osumclN . (Co... |
| osumcllem5N 39954 | Lemma for ~ osumclN . (Co... |
| osumcllem6N 39955 | Lemma for ~ osumclN . Use... |
| osumcllem7N 39956 | Lemma for ~ osumclN . (Co... |
| osumcllem8N 39957 | Lemma for ~ osumclN . (Co... |
| osumcllem9N 39958 | Lemma for ~ osumclN . (Co... |
| osumcllem10N 39959 | Lemma for ~ osumclN . Con... |
| osumcllem11N 39960 | Lemma for ~ osumclN . (Co... |
| osumclN 39961 | Closure of orthogonal sum.... |
| pmapojoinN 39962 | For orthogonal elements, p... |
| pexmidN 39963 | Excluded middle law for cl... |
| pexmidlem1N 39964 | Lemma for ~ pexmidN . Hol... |
| pexmidlem2N 39965 | Lemma for ~ pexmidN . (Co... |
| pexmidlem3N 39966 | Lemma for ~ pexmidN . Use... |
| pexmidlem4N 39967 | Lemma for ~ pexmidN . (Co... |
| pexmidlem5N 39968 | Lemma for ~ pexmidN . (Co... |
| pexmidlem6N 39969 | Lemma for ~ pexmidN . (Co... |
| pexmidlem7N 39970 | Lemma for ~ pexmidN . Con... |
| pexmidlem8N 39971 | Lemma for ~ pexmidN . The... |
| pexmidALTN 39972 | Excluded middle law for cl... |
| pl42lem1N 39973 | Lemma for ~ pl42N . (Cont... |
| pl42lem2N 39974 | Lemma for ~ pl42N . (Cont... |
| pl42lem3N 39975 | Lemma for ~ pl42N . (Cont... |
| pl42lem4N 39976 | Lemma for ~ pl42N . (Cont... |
| pl42N 39977 | Law holding in a Hilbert l... |
| watfvalN 39986 | The W atoms function. (Co... |
| watvalN 39987 | Value of the W atoms funct... |
| iswatN 39988 | The predicate "is a W atom... |
| lhpset 39989 | The set of co-atoms (latti... |
| islhp 39990 | The predicate "is a co-ato... |
| islhp2 39991 | The predicate "is a co-ato... |
| lhpbase 39992 | A co-atom is a member of t... |
| lhp1cvr 39993 | The lattice unity covers a... |
| lhplt 39994 | An atom under a co-atom is... |
| lhp2lt 39995 | The join of two atoms unde... |
| lhpexlt 39996 | There exists an atom less ... |
| lhp0lt 39997 | A co-atom is greater than ... |
| lhpn0 39998 | A co-atom is nonzero. TOD... |
| lhpexle 39999 | There exists an atom under... |
| lhpexnle 40000 | There exists an atom not u... |
| lhpexle1lem 40001 | Lemma for ~ lhpexle1 and o... |
| lhpexle1 40002 | There exists an atom under... |
| lhpexle2lem 40003 | Lemma for ~ lhpexle2 . (C... |
| lhpexle2 40004 | There exists atom under a ... |
| lhpexle3lem 40005 | There exists atom under a ... |
| lhpexle3 40006 | There exists atom under a ... |
| lhpex2leN 40007 | There exist at least two d... |
| lhpoc 40008 | The orthocomplement of a c... |
| lhpoc2N 40009 | The orthocomplement of an ... |
| lhpocnle 40010 | The orthocomplement of a c... |
| lhpocat 40011 | The orthocomplement of a c... |
| lhpocnel 40012 | The orthocomplement of a c... |
| lhpocnel2 40013 | The orthocomplement of a c... |
| lhpjat1 40014 | The join of a co-atom (hyp... |
| lhpjat2 40015 | The join of a co-atom (hyp... |
| lhpj1 40016 | The join of a co-atom (hyp... |
| lhpmcvr 40017 | The meet of a lattice hype... |
| lhpmcvr2 40018 | Alternate way to express t... |
| lhpmcvr3 40019 | Specialization of ~ lhpmcv... |
| lhpmcvr4N 40020 | Specialization of ~ lhpmcv... |
| lhpmcvr5N 40021 | Specialization of ~ lhpmcv... |
| lhpmcvr6N 40022 | Specialization of ~ lhpmcv... |
| lhpm0atN 40023 | If the meet of a lattice h... |
| lhpmat 40024 | An element covered by the ... |
| lhpmatb 40025 | An element covered by the ... |
| lhp2at0 40026 | Join and meet with differe... |
| lhp2atnle 40027 | Inequality for 2 different... |
| lhp2atne 40028 | Inequality for joins with ... |
| lhp2at0nle 40029 | Inequality for 2 different... |
| lhp2at0ne 40030 | Inequality for joins with ... |
| lhpelim 40031 | Eliminate an atom not unde... |
| lhpmod2i2 40032 | Modular law for hyperplane... |
| lhpmod6i1 40033 | Modular law for hyperplane... |
| lhprelat3N 40034 | The Hilbert lattice is rel... |
| cdlemb2 40035 | Given two atoms not under ... |
| lhple 40036 | Property of a lattice elem... |
| lhpat 40037 | Create an atom under a co-... |
| lhpat4N 40038 | Property of an atom under ... |
| lhpat2 40039 | Create an atom under a co-... |
| lhpat3 40040 | There is only one atom und... |
| 4atexlemk 40041 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemw 40042 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempw 40043 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemp 40044 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemq 40045 | Lemma for ~ 4atexlem7 . (... |
| 4atexlems 40046 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemt 40047 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemutvt 40048 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempnq 40049 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnslpq 40050 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkl 40051 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkc 40052 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemwb 40053 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempsb 40054 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemqtb 40055 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempns 40056 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemswapqr 40057 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemu 40058 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemv 40059 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemunv 40060 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemtlw 40061 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemntlpq 40062 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemc 40063 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnclw 40064 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex2 40065 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemcnd 40066 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex4 40067 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemex6 40068 | Lemma for ~ 4atexlem7 . (... |
| 4atexlem7 40069 | Whenever there are at leas... |
| 4atex 40070 | Whenever there are at leas... |
| 4atex2 40071 | More general version of ~ ... |
| 4atex2-0aOLDN 40072 | Same as ~ 4atex2 except th... |
| 4atex2-0bOLDN 40073 | Same as ~ 4atex2 except th... |
| 4atex2-0cOLDN 40074 | Same as ~ 4atex2 except th... |
| 4atex3 40075 | More general version of ~ ... |
| lautset 40076 | The set of lattice automor... |
| islaut 40077 | The predicate "is a lattic... |
| lautle 40078 | Less-than or equal propert... |
| laut1o 40079 | A lattice automorphism is ... |
| laut11 40080 | One-to-one property of a l... |
| lautcl 40081 | A lattice automorphism val... |
| lautcnvclN 40082 | Reverse closure of a latti... |
| lautcnvle 40083 | Less-than or equal propert... |
| lautcnv 40084 | The converse of a lattice ... |
| lautlt 40085 | Less-than property of a la... |
| lautcvr 40086 | Covering property of a lat... |
| lautj 40087 | Meet property of a lattice... |
| lautm 40088 | Meet property of a lattice... |
| lauteq 40089 | A lattice automorphism arg... |
| idlaut 40090 | The identity function is a... |
| lautco 40091 | The composition of two lat... |
| pautsetN 40092 | The set of projective auto... |
| ispautN 40093 | The predicate "is a projec... |
| ldilfset 40102 | The mapping from fiducial ... |
| ldilset 40103 | The set of lattice dilatio... |
| isldil 40104 | The predicate "is a lattic... |
| ldillaut 40105 | A lattice dilation is an a... |
| ldil1o 40106 | A lattice dilation is a on... |
| ldilval 40107 | Value of a lattice dilatio... |
| idldil 40108 | The identity function is a... |
| ldilcnv 40109 | The converse of a lattice ... |
| ldilco 40110 | The composition of two lat... |
| ltrnfset 40111 | The set of all lattice tra... |
| ltrnset 40112 | The set of lattice transla... |
| isltrn 40113 | The predicate "is a lattic... |
| isltrn2N 40114 | The predicate "is a lattic... |
| ltrnu 40115 | Uniqueness property of a l... |
| ltrnldil 40116 | A lattice translation is a... |
| ltrnlaut 40117 | A lattice translation is a... |
| ltrn1o 40118 | A lattice translation is a... |
| ltrncl 40119 | Closure of a lattice trans... |
| ltrn11 40120 | One-to-one property of a l... |
| ltrncnvnid 40121 | If a translation is differ... |
| ltrncoidN 40122 | Two translations are equal... |
| ltrnle 40123 | Less-than or equal propert... |
| ltrncnvleN 40124 | Less-than or equal propert... |
| ltrnm 40125 | Lattice translation of a m... |
| ltrnj 40126 | Lattice translation of a m... |
| ltrncvr 40127 | Covering property of a lat... |
| ltrnval1 40128 | Value of a lattice transla... |
| ltrnid 40129 | A lattice translation is t... |
| ltrnnid 40130 | If a lattice translation i... |
| ltrnatb 40131 | The lattice translation of... |
| ltrncnvatb 40132 | The converse of the lattic... |
| ltrnel 40133 | The lattice translation of... |
| ltrnat 40134 | The lattice translation of... |
| ltrncnvat 40135 | The converse of the lattic... |
| ltrncnvel 40136 | The converse of the lattic... |
| ltrncoelN 40137 | Composition of lattice tra... |
| ltrncoat 40138 | Composition of lattice tra... |
| ltrncoval 40139 | Two ways to express value ... |
| ltrncnv 40140 | The converse of a lattice ... |
| ltrn11at 40141 | Frequently used one-to-one... |
| ltrneq2 40142 | The equality of two transl... |
| ltrneq 40143 | The equality of two transl... |
| idltrn 40144 | The identity function is a... |
| ltrnmw 40145 | Property of lattice transl... |
| dilfsetN 40146 | The mapping from fiducial ... |
| dilsetN 40147 | The set of dilations for a... |
| isdilN 40148 | The predicate "is a dilati... |
| trnfsetN 40149 | The mapping from fiducial ... |
| trnsetN 40150 | The set of translations fo... |
| istrnN 40151 | The predicate "is a transl... |
| trlfset 40154 | The set of all traces of l... |
| trlset 40155 | The set of traces of latti... |
| trlval 40156 | The value of the trace of ... |
| trlval2 40157 | The value of the trace of ... |
| trlcl 40158 | Closure of the trace of a ... |
| trlcnv 40159 | The trace of the converse ... |
| trljat1 40160 | The value of a translation... |
| trljat2 40161 | The value of a translation... |
| trljat3 40162 | The value of a translation... |
| trlat 40163 | If an atom differs from it... |
| trl0 40164 | If an atom not under the f... |
| trlator0 40165 | The trace of a lattice tra... |
| trlatn0 40166 | The trace of a lattice tra... |
| trlnidat 40167 | The trace of a lattice tra... |
| ltrnnidn 40168 | If a lattice translation i... |
| ltrnideq 40169 | Property of the identity l... |
| trlid0 40170 | The trace of the identity ... |
| trlnidatb 40171 | A lattice translation is n... |
| trlid0b 40172 | A lattice translation is t... |
| trlnid 40173 | Different translations wit... |
| ltrn2ateq 40174 | Property of the equality o... |
| ltrnateq 40175 | If any atom (under ` W ` )... |
| ltrnatneq 40176 | If any atom (under ` W ` )... |
| ltrnatlw 40177 | If the value of an atom eq... |
| trlle 40178 | The trace of a lattice tra... |
| trlne 40179 | The trace of a lattice tra... |
| trlnle 40180 | The atom not under the fid... |
| trlval3 40181 | The value of the trace of ... |
| trlval4 40182 | The value of the trace of ... |
| trlval5 40183 | The value of the trace of ... |
| arglem1N 40184 | Lemma for Desargues's law.... |
| cdlemc1 40185 | Part of proof of Lemma C i... |
| cdlemc2 40186 | Part of proof of Lemma C i... |
| cdlemc3 40187 | Part of proof of Lemma C i... |
| cdlemc4 40188 | Part of proof of Lemma C i... |
| cdlemc5 40189 | Lemma for ~ cdlemc . (Con... |
| cdlemc6 40190 | Lemma for ~ cdlemc . (Con... |
| cdlemc 40191 | Lemma C in [Crawley] p. 11... |
| cdlemd1 40192 | Part of proof of Lemma D i... |
| cdlemd2 40193 | Part of proof of Lemma D i... |
| cdlemd3 40194 | Part of proof of Lemma D i... |
| cdlemd4 40195 | Part of proof of Lemma D i... |
| cdlemd5 40196 | Part of proof of Lemma D i... |
| cdlemd6 40197 | Part of proof of Lemma D i... |
| cdlemd7 40198 | Part of proof of Lemma D i... |
| cdlemd8 40199 | Part of proof of Lemma D i... |
| cdlemd9 40200 | Part of proof of Lemma D i... |
| cdlemd 40201 | If two translations agree ... |
| ltrneq3 40202 | Two translations agree at ... |
| cdleme00a 40203 | Part of proof of Lemma E i... |
| cdleme0aa 40204 | Part of proof of Lemma E i... |
| cdleme0a 40205 | Part of proof of Lemma E i... |
| cdleme0b 40206 | Part of proof of Lemma E i... |
| cdleme0c 40207 | Part of proof of Lemma E i... |
| cdleme0cp 40208 | Part of proof of Lemma E i... |
| cdleme0cq 40209 | Part of proof of Lemma E i... |
| cdleme0dN 40210 | Part of proof of Lemma E i... |
| cdleme0e 40211 | Part of proof of Lemma E i... |
| cdleme0fN 40212 | Part of proof of Lemma E i... |
| cdleme0gN 40213 | Part of proof of Lemma E i... |
| cdlemeulpq 40214 | Part of proof of Lemma E i... |
| cdleme01N 40215 | Part of proof of Lemma E i... |
| cdleme02N 40216 | Part of proof of Lemma E i... |
| cdleme0ex1N 40217 | Part of proof of Lemma E i... |
| cdleme0ex2N 40218 | Part of proof of Lemma E i... |
| cdleme0moN 40219 | Part of proof of Lemma E i... |
| cdleme1b 40220 | Part of proof of Lemma E i... |
| cdleme1 40221 | Part of proof of Lemma E i... |
| cdleme2 40222 | Part of proof of Lemma E i... |
| cdleme3b 40223 | Part of proof of Lemma E i... |
| cdleme3c 40224 | Part of proof of Lemma E i... |
| cdleme3d 40225 | Part of proof of Lemma E i... |
| cdleme3e 40226 | Part of proof of Lemma E i... |
| cdleme3fN 40227 | Part of proof of Lemma E i... |
| cdleme3g 40228 | Part of proof of Lemma E i... |
| cdleme3h 40229 | Part of proof of Lemma E i... |
| cdleme3fa 40230 | Part of proof of Lemma E i... |
| cdleme3 40231 | Part of proof of Lemma E i... |
| cdleme4 40232 | Part of proof of Lemma E i... |
| cdleme4a 40233 | Part of proof of Lemma E i... |
| cdleme5 40234 | Part of proof of Lemma E i... |
| cdleme6 40235 | Part of proof of Lemma E i... |
| cdleme7aa 40236 | Part of proof of Lemma E i... |
| cdleme7a 40237 | Part of proof of Lemma E i... |
| cdleme7b 40238 | Part of proof of Lemma E i... |
| cdleme7c 40239 | Part of proof of Lemma E i... |
| cdleme7d 40240 | Part of proof of Lemma E i... |
| cdleme7e 40241 | Part of proof of Lemma E i... |
| cdleme7ga 40242 | Part of proof of Lemma E i... |
| cdleme7 40243 | Part of proof of Lemma E i... |
| cdleme8 40244 | Part of proof of Lemma E i... |
| cdleme9a 40245 | Part of proof of Lemma E i... |
| cdleme9b 40246 | Utility lemma for Lemma E ... |
| cdleme9 40247 | Part of proof of Lemma E i... |
| cdleme10 40248 | Part of proof of Lemma E i... |
| cdleme8tN 40249 | Part of proof of Lemma E i... |
| cdleme9taN 40250 | Part of proof of Lemma E i... |
| cdleme9tN 40251 | Part of proof of Lemma E i... |
| cdleme10tN 40252 | Part of proof of Lemma E i... |
| cdleme16aN 40253 | Part of proof of Lemma E i... |
| cdleme11a 40254 | Part of proof of Lemma E i... |
| cdleme11c 40255 | Part of proof of Lemma E i... |
| cdleme11dN 40256 | Part of proof of Lemma E i... |
| cdleme11e 40257 | Part of proof of Lemma E i... |
| cdleme11fN 40258 | Part of proof of Lemma E i... |
| cdleme11g 40259 | Part of proof of Lemma E i... |
| cdleme11h 40260 | Part of proof of Lemma E i... |
| cdleme11j 40261 | Part of proof of Lemma E i... |
| cdleme11k 40262 | Part of proof of Lemma E i... |
| cdleme11l 40263 | Part of proof of Lemma E i... |
| cdleme11 40264 | Part of proof of Lemma E i... |
| cdleme12 40265 | Part of proof of Lemma E i... |
| cdleme13 40266 | Part of proof of Lemma E i... |
| cdleme14 40267 | Part of proof of Lemma E i... |
| cdleme15a 40268 | Part of proof of Lemma E i... |
| cdleme15b 40269 | Part of proof of Lemma E i... |
| cdleme15c 40270 | Part of proof of Lemma E i... |
| cdleme15d 40271 | Part of proof of Lemma E i... |
| cdleme15 40272 | Part of proof of Lemma E i... |
| cdleme16b 40273 | Part of proof of Lemma E i... |
| cdleme16c 40274 | Part of proof of Lemma E i... |
| cdleme16d 40275 | Part of proof of Lemma E i... |
| cdleme16e 40276 | Part of proof of Lemma E i... |
| cdleme16f 40277 | Part of proof of Lemma E i... |
| cdleme16g 40278 | Part of proof of Lemma E i... |
| cdleme16 40279 | Part of proof of Lemma E i... |
| cdleme17a 40280 | Part of proof of Lemma E i... |
| cdleme17b 40281 | Lemma leading to ~ cdleme1... |
| cdleme17c 40282 | Part of proof of Lemma E i... |
| cdleme17d1 40283 | Part of proof of Lemma E i... |
| cdleme0nex 40284 | Part of proof of Lemma E i... |
| cdleme18a 40285 | Part of proof of Lemma E i... |
| cdleme18b 40286 | Part of proof of Lemma E i... |
| cdleme18c 40287 | Part of proof of Lemma E i... |
| cdleme22gb 40288 | Utility lemma for Lemma E ... |
| cdleme18d 40289 | Part of proof of Lemma E i... |
| cdlemesner 40290 | Part of proof of Lemma E i... |
| cdlemedb 40291 | Part of proof of Lemma E i... |
| cdlemeda 40292 | Part of proof of Lemma E i... |
| cdlemednpq 40293 | Part of proof of Lemma E i... |
| cdlemednuN 40294 | Part of proof of Lemma E i... |
| cdleme20zN 40295 | Part of proof of Lemma E i... |
| cdleme20y 40296 | Part of proof of Lemma E i... |
| cdleme19a 40297 | Part of proof of Lemma E i... |
| cdleme19b 40298 | Part of proof of Lemma E i... |
| cdleme19c 40299 | Part of proof of Lemma E i... |
| cdleme19d 40300 | Part of proof of Lemma E i... |
| cdleme19e 40301 | Part of proof of Lemma E i... |
| cdleme19f 40302 | Part of proof of Lemma E i... |
| cdleme20aN 40303 | Part of proof of Lemma E i... |
| cdleme20bN 40304 | Part of proof of Lemma E i... |
| cdleme20c 40305 | Part of proof of Lemma E i... |
| cdleme20d 40306 | Part of proof of Lemma E i... |
| cdleme20e 40307 | Part of proof of Lemma E i... |
| cdleme20f 40308 | Part of proof of Lemma E i... |
| cdleme20g 40309 | Part of proof of Lemma E i... |
| cdleme20h 40310 | Part of proof of Lemma E i... |
| cdleme20i 40311 | Part of proof of Lemma E i... |
| cdleme20j 40312 | Part of proof of Lemma E i... |
| cdleme20k 40313 | Part of proof of Lemma E i... |
| cdleme20l1 40314 | Part of proof of Lemma E i... |
| cdleme20l2 40315 | Part of proof of Lemma E i... |
| cdleme20l 40316 | Part of proof of Lemma E i... |
| cdleme20m 40317 | Part of proof of Lemma E i... |
| cdleme20 40318 | Combine ~ cdleme19f and ~ ... |
| cdleme21a 40319 | Part of proof of Lemma E i... |
| cdleme21b 40320 | Part of proof of Lemma E i... |
| cdleme21c 40321 | Part of proof of Lemma E i... |
| cdleme21at 40322 | Part of proof of Lemma E i... |
| cdleme21ct 40323 | Part of proof of Lemma E i... |
| cdleme21d 40324 | Part of proof of Lemma E i... |
| cdleme21e 40325 | Part of proof of Lemma E i... |
| cdleme21f 40326 | Part of proof of Lemma E i... |
| cdleme21g 40327 | Part of proof of Lemma E i... |
| cdleme21h 40328 | Part of proof of Lemma E i... |
| cdleme21i 40329 | Part of proof of Lemma E i... |
| cdleme21j 40330 | Combine ~ cdleme20 and ~ c... |
| cdleme21 40331 | Part of proof of Lemma E i... |
| cdleme21k 40332 | Eliminate ` S =/= T ` cond... |
| cdleme22aa 40333 | Part of proof of Lemma E i... |
| cdleme22a 40334 | Part of proof of Lemma E i... |
| cdleme22b 40335 | Part of proof of Lemma E i... |
| cdleme22cN 40336 | Part of proof of Lemma E i... |
| cdleme22d 40337 | Part of proof of Lemma E i... |
| cdleme22e 40338 | Part of proof of Lemma E i... |
| cdleme22eALTN 40339 | Part of proof of Lemma E i... |
| cdleme22f 40340 | Part of proof of Lemma E i... |
| cdleme22f2 40341 | Part of proof of Lemma E i... |
| cdleme22g 40342 | Part of proof of Lemma E i... |
| cdleme23a 40343 | Part of proof of Lemma E i... |
| cdleme23b 40344 | Part of proof of Lemma E i... |
| cdleme23c 40345 | Part of proof of Lemma E i... |
| cdleme24 40346 | Quantified version of ~ cd... |
| cdleme25a 40347 | Lemma for ~ cdleme25b . (... |
| cdleme25b 40348 | Transform ~ cdleme24 . TO... |
| cdleme25c 40349 | Transform ~ cdleme25b . (... |
| cdleme25dN 40350 | Transform ~ cdleme25c . (... |
| cdleme25cl 40351 | Show closure of the unique... |
| cdleme25cv 40352 | Change bound variables in ... |
| cdleme26e 40353 | Part of proof of Lemma E i... |
| cdleme26ee 40354 | Part of proof of Lemma E i... |
| cdleme26eALTN 40355 | Part of proof of Lemma E i... |
| cdleme26fALTN 40356 | Part of proof of Lemma E i... |
| cdleme26f 40357 | Part of proof of Lemma E i... |
| cdleme26f2ALTN 40358 | Part of proof of Lemma E i... |
| cdleme26f2 40359 | Part of proof of Lemma E i... |
| cdleme27cl 40360 | Part of proof of Lemma E i... |
| cdleme27a 40361 | Part of proof of Lemma E i... |
| cdleme27b 40362 | Lemma for ~ cdleme27N . (... |
| cdleme27N 40363 | Part of proof of Lemma E i... |
| cdleme28a 40364 | Lemma for ~ cdleme25b . T... |
| cdleme28b 40365 | Lemma for ~ cdleme25b . T... |
| cdleme28c 40366 | Part of proof of Lemma E i... |
| cdleme28 40367 | Quantified version of ~ cd... |
| cdleme29ex 40368 | Lemma for ~ cdleme29b . (... |
| cdleme29b 40369 | Transform ~ cdleme28 . (C... |
| cdleme29c 40370 | Transform ~ cdleme28b . (... |
| cdleme29cl 40371 | Show closure of the unique... |
| cdleme30a 40372 | Part of proof of Lemma E i... |
| cdleme31so 40373 | Part of proof of Lemma E i... |
| cdleme31sn 40374 | Part of proof of Lemma E i... |
| cdleme31sn1 40375 | Part of proof of Lemma E i... |
| cdleme31se 40376 | Part of proof of Lemma D i... |
| cdleme31se2 40377 | Part of proof of Lemma D i... |
| cdleme31sc 40378 | Part of proof of Lemma E i... |
| cdleme31sde 40379 | Part of proof of Lemma D i... |
| cdleme31snd 40380 | Part of proof of Lemma D i... |
| cdleme31sdnN 40381 | Part of proof of Lemma E i... |
| cdleme31sn1c 40382 | Part of proof of Lemma E i... |
| cdleme31sn2 40383 | Part of proof of Lemma E i... |
| cdleme31fv 40384 | Part of proof of Lemma E i... |
| cdleme31fv1 40385 | Part of proof of Lemma E i... |
| cdleme31fv1s 40386 | Part of proof of Lemma E i... |
| cdleme31fv2 40387 | Part of proof of Lemma E i... |
| cdleme31id 40388 | Part of proof of Lemma E i... |
| cdlemefrs29pre00 40389 | ***START OF VALUE AT ATOM ... |
| cdlemefrs29bpre0 40390 | TODO fix comment. (Contri... |
| cdlemefrs29bpre1 40391 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29cpre1 40392 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29clN 40393 | TODO: NOT USED? Show clo... |
| cdlemefrs32fva 40394 | Part of proof of Lemma E i... |
| cdlemefrs32fva1 40395 | Part of proof of Lemma E i... |
| cdlemefr29exN 40396 | Lemma for ~ cdlemefs29bpre... |
| cdlemefr27cl 40397 | Part of proof of Lemma E i... |
| cdlemefr32sn2aw 40398 | Show that ` [_ R / s ]_ N ... |
| cdlemefr32snb 40399 | Show closure of ` [_ R / s... |
| cdlemefr29bpre0N 40400 | TODO fix comment. (Contri... |
| cdlemefr29clN 40401 | Show closure of the unique... |
| cdleme43frv1snN 40402 | Value of ` [_ R / s ]_ N `... |
| cdlemefr32fvaN 40403 | Part of proof of Lemma E i... |
| cdlemefr32fva1 40404 | Part of proof of Lemma E i... |
| cdlemefr31fv1 40405 | Value of ` ( F `` R ) ` wh... |
| cdlemefs29pre00N 40406 | FIX COMMENT. TODO: see if ... |
| cdlemefs27cl 40407 | Part of proof of Lemma E i... |
| cdlemefs32sn1aw 40408 | Show that ` [_ R / s ]_ N ... |
| cdlemefs32snb 40409 | Show closure of ` [_ R / s... |
| cdlemefs29bpre0N 40410 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29bpre1N 40411 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29cpre1N 40412 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29clN 40413 | Show closure of the unique... |
| cdleme43fsv1snlem 40414 | Value of ` [_ R / s ]_ N `... |
| cdleme43fsv1sn 40415 | Value of ` [_ R / s ]_ N `... |
| cdlemefs32fvaN 40416 | Part of proof of Lemma E i... |
| cdlemefs32fva1 40417 | Part of proof of Lemma E i... |
| cdlemefs31fv1 40418 | Value of ` ( F `` R ) ` wh... |
| cdlemefr44 40419 | Value of f(r) when r is an... |
| cdlemefs44 40420 | Value of f_s(r) when r is ... |
| cdlemefr45 40421 | Value of f(r) when r is an... |
| cdlemefr45e 40422 | Explicit expansion of ~ cd... |
| cdlemefs45 40423 | Value of f_s(r) when r is ... |
| cdlemefs45ee 40424 | Explicit expansion of ~ cd... |
| cdlemefs45eN 40425 | Explicit expansion of ~ cd... |
| cdleme32sn1awN 40426 | Show that ` [_ R / s ]_ N ... |
| cdleme41sn3a 40427 | Show that ` [_ R / s ]_ N ... |
| cdleme32sn2awN 40428 | Show that ` [_ R / s ]_ N ... |
| cdleme32snaw 40429 | Show that ` [_ R / s ]_ N ... |
| cdleme32snb 40430 | Show closure of ` [_ R / s... |
| cdleme32fva 40431 | Part of proof of Lemma D i... |
| cdleme32fva1 40432 | Part of proof of Lemma D i... |
| cdleme32fvaw 40433 | Show that ` ( F `` R ) ` i... |
| cdleme32fvcl 40434 | Part of proof of Lemma D i... |
| cdleme32a 40435 | Part of proof of Lemma D i... |
| cdleme32b 40436 | Part of proof of Lemma D i... |
| cdleme32c 40437 | Part of proof of Lemma D i... |
| cdleme32d 40438 | Part of proof of Lemma D i... |
| cdleme32e 40439 | Part of proof of Lemma D i... |
| cdleme32f 40440 | Part of proof of Lemma D i... |
| cdleme32le 40441 | Part of proof of Lemma D i... |
| cdleme35a 40442 | Part of proof of Lemma E i... |
| cdleme35fnpq 40443 | Part of proof of Lemma E i... |
| cdleme35b 40444 | Part of proof of Lemma E i... |
| cdleme35c 40445 | Part of proof of Lemma E i... |
| cdleme35d 40446 | Part of proof of Lemma E i... |
| cdleme35e 40447 | Part of proof of Lemma E i... |
| cdleme35f 40448 | Part of proof of Lemma E i... |
| cdleme35g 40449 | Part of proof of Lemma E i... |
| cdleme35h 40450 | Part of proof of Lemma E i... |
| cdleme35h2 40451 | Part of proof of Lemma E i... |
| cdleme35sn2aw 40452 | Part of proof of Lemma E i... |
| cdleme35sn3a 40453 | Part of proof of Lemma E i... |
| cdleme36a 40454 | Part of proof of Lemma E i... |
| cdleme36m 40455 | Part of proof of Lemma E i... |
| cdleme37m 40456 | Part of proof of Lemma E i... |
| cdleme38m 40457 | Part of proof of Lemma E i... |
| cdleme38n 40458 | Part of proof of Lemma E i... |
| cdleme39a 40459 | Part of proof of Lemma E i... |
| cdleme39n 40460 | Part of proof of Lemma E i... |
| cdleme40m 40461 | Part of proof of Lemma E i... |
| cdleme40n 40462 | Part of proof of Lemma E i... |
| cdleme40v 40463 | Part of proof of Lemma E i... |
| cdleme40w 40464 | Part of proof of Lemma E i... |
| cdleme42a 40465 | Part of proof of Lemma E i... |
| cdleme42c 40466 | Part of proof of Lemma E i... |
| cdleme42d 40467 | Part of proof of Lemma E i... |
| cdleme41sn3aw 40468 | Part of proof of Lemma E i... |
| cdleme41sn4aw 40469 | Part of proof of Lemma E i... |
| cdleme41snaw 40470 | Part of proof of Lemma E i... |
| cdleme41fva11 40471 | Part of proof of Lemma E i... |
| cdleme42b 40472 | Part of proof of Lemma E i... |
| cdleme42e 40473 | Part of proof of Lemma E i... |
| cdleme42f 40474 | Part of proof of Lemma E i... |
| cdleme42g 40475 | Part of proof of Lemma E i... |
| cdleme42h 40476 | Part of proof of Lemma E i... |
| cdleme42i 40477 | Part of proof of Lemma E i... |
| cdleme42k 40478 | Part of proof of Lemma E i... |
| cdleme42ke 40479 | Part of proof of Lemma E i... |
| cdleme42keg 40480 | Part of proof of Lemma E i... |
| cdleme42mN 40481 | Part of proof of Lemma E i... |
| cdleme42mgN 40482 | Part of proof of Lemma E i... |
| cdleme43aN 40483 | Part of proof of Lemma E i... |
| cdleme43bN 40484 | Lemma for Lemma E in [Craw... |
| cdleme43cN 40485 | Part of proof of Lemma E i... |
| cdleme43dN 40486 | Part of proof of Lemma E i... |
| cdleme46f2g2 40487 | Conversion for ` G ` to re... |
| cdleme46f2g1 40488 | Conversion for ` G ` to re... |
| cdleme17d2 40489 | Part of proof of Lemma E i... |
| cdleme17d3 40490 | TODO: FIX COMMENT. (Contr... |
| cdleme17d4 40491 | TODO: FIX COMMENT. (Contr... |
| cdleme17d 40492 | Part of proof of Lemma E i... |
| cdleme48fv 40493 | Part of proof of Lemma D i... |
| cdleme48fvg 40494 | Remove ` P =/= Q ` conditi... |
| cdleme46fvaw 40495 | Show that ` ( F `` R ) ` i... |
| cdleme48bw 40496 | TODO: fix comment. TODO: ... |
| cdleme48b 40497 | TODO: fix comment. (Contr... |
| cdleme46frvlpq 40498 | Show that ` ( F `` S ) ` i... |
| cdleme46fsvlpq 40499 | Show that ` ( F `` R ) ` i... |
| cdlemeg46fvcl 40500 | TODO: fix comment. (Contr... |
| cdleme4gfv 40501 | Part of proof of Lemma D i... |
| cdlemeg47b 40502 | TODO: FIX COMMENT. (Contr... |
| cdlemeg47rv 40503 | Value of g_s(r) when r is ... |
| cdlemeg47rv2 40504 | Value of g_s(r) when r is ... |
| cdlemeg49le 40505 | Part of proof of Lemma D i... |
| cdlemeg46bOLDN 40506 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46c 40507 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46rvOLDN 40508 | Value of g_s(r) when r is ... |
| cdlemeg46rv2OLDN 40509 | Value of g_s(r) when r is ... |
| cdlemeg46fvaw 40510 | Show that ` ( F `` R ) ` i... |
| cdlemeg46nlpq 40511 | Show that ` ( G `` S ) ` i... |
| cdlemeg46ngfr 40512 | TODO FIX COMMENT g(f(s))=s... |
| cdlemeg46nfgr 40513 | TODO FIX COMMENT f(g(s))=s... |
| cdlemeg46sfg 40514 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fjgN 40515 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46rjgN 40516 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46fjv 40517 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fsfv 40518 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46frv 40519 | TODO FIX COMMENT. (f(r) ` ... |
| cdlemeg46v1v2 40520 | TODO FIX COMMENT v_1 = v_2... |
| cdlemeg46vrg 40521 | TODO FIX COMMENT v_1 ` <_ ... |
| cdlemeg46rgv 40522 | TODO FIX COMMENT r ` <_ ` ... |
| cdlemeg46req 40523 | TODO FIX COMMENT r = (v_1 ... |
| cdlemeg46gfv 40524 | TODO FIX COMMENT p. 115 pe... |
| cdlemeg46gfr 40525 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gfre 40526 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gf 40527 | TODO FIX COMMENT Eliminate... |
| cdlemeg46fgN 40528 | TODO FIX COMMENT p. 116 pe... |
| cdleme48d 40529 | TODO: fix comment. (Contr... |
| cdleme48gfv1 40530 | TODO: fix comment. (Contr... |
| cdleme48gfv 40531 | TODO: fix comment. (Contr... |
| cdleme48fgv 40532 | TODO: fix comment. (Contr... |
| cdlemeg49lebilem 40533 | Part of proof of Lemma D i... |
| cdleme50lebi 40534 | Part of proof of Lemma D i... |
| cdleme50eq 40535 | Part of proof of Lemma D i... |
| cdleme50f 40536 | Part of proof of Lemma D i... |
| cdleme50f1 40537 | Part of proof of Lemma D i... |
| cdleme50rnlem 40538 | Part of proof of Lemma D i... |
| cdleme50rn 40539 | Part of proof of Lemma D i... |
| cdleme50f1o 40540 | Part of proof of Lemma D i... |
| cdleme50laut 40541 | Part of proof of Lemma D i... |
| cdleme50ldil 40542 | Part of proof of Lemma D i... |
| cdleme50trn1 40543 | Part of proof that ` F ` i... |
| cdleme50trn2a 40544 | Part of proof that ` F ` i... |
| cdleme50trn2 40545 | Part of proof that ` F ` i... |
| cdleme50trn12 40546 | Part of proof that ` F ` i... |
| cdleme50trn3 40547 | Part of proof that ` F ` i... |
| cdleme50trn123 40548 | Part of proof that ` F ` i... |
| cdleme51finvfvN 40549 | Part of proof of Lemma E i... |
| cdleme51finvN 40550 | Part of proof of Lemma E i... |
| cdleme50ltrn 40551 | Part of proof of Lemma E i... |
| cdleme51finvtrN 40552 | Part of proof of Lemma E i... |
| cdleme50ex 40553 | Part of Lemma E in [Crawle... |
| cdleme 40554 | Lemma E in [Crawley] p. 11... |
| cdlemf1 40555 | Part of Lemma F in [Crawle... |
| cdlemf2 40556 | Part of Lemma F in [Crawle... |
| cdlemf 40557 | Lemma F in [Crawley] p. 11... |
| cdlemfnid 40558 | ~ cdlemf with additional c... |
| cdlemftr3 40559 | Special case of ~ cdlemf s... |
| cdlemftr2 40560 | Special case of ~ cdlemf s... |
| cdlemftr1 40561 | Part of proof of Lemma G o... |
| cdlemftr0 40562 | Special case of ~ cdlemf s... |
| trlord 40563 | The ordering of two Hilber... |
| cdlemg1a 40564 | Shorter expression for ` G... |
| cdlemg1b2 40565 | This theorem can be used t... |
| cdlemg1idlemN 40566 | Lemma for ~ cdlemg1idN . ... |
| cdlemg1fvawlemN 40567 | Lemma for ~ ltrniotafvawN ... |
| cdlemg1ltrnlem 40568 | Lemma for ~ ltrniotacl . ... |
| cdlemg1finvtrlemN 40569 | Lemma for ~ ltrniotacnvN .... |
| cdlemg1bOLDN 40570 | This theorem can be used t... |
| cdlemg1idN 40571 | Version of ~ cdleme31id wi... |
| ltrniotafvawN 40572 | Version of ~ cdleme46fvaw ... |
| ltrniotacl 40573 | Version of ~ cdleme50ltrn ... |
| ltrniotacnvN 40574 | Version of ~ cdleme51finvt... |
| ltrniotaval 40575 | Value of the unique transl... |
| ltrniotacnvval 40576 | Converse value of the uniq... |
| ltrniotaidvalN 40577 | Value of the unique transl... |
| ltrniotavalbN 40578 | Value of the unique transl... |
| cdlemeiota 40579 | A translation is uniquely ... |
| cdlemg1ci2 40580 | Any function of the form o... |
| cdlemg1cN 40581 | Any translation belongs to... |
| cdlemg1cex 40582 | Any translation is one of ... |
| cdlemg2cN 40583 | Any translation belongs to... |
| cdlemg2dN 40584 | This theorem can be used t... |
| cdlemg2cex 40585 | Any translation is one of ... |
| cdlemg2ce 40586 | Utility theorem to elimina... |
| cdlemg2jlemOLDN 40587 | Part of proof of Lemma E i... |
| cdlemg2fvlem 40588 | Lemma for ~ cdlemg2fv . (... |
| cdlemg2klem 40589 | ~ cdleme42keg with simpler... |
| cdlemg2idN 40590 | Version of ~ cdleme31id wi... |
| cdlemg3a 40591 | Part of proof of Lemma G i... |
| cdlemg2jOLDN 40592 | TODO: Replace this with ~... |
| cdlemg2fv 40593 | Value of a translation in ... |
| cdlemg2fv2 40594 | Value of a translation in ... |
| cdlemg2k 40595 | ~ cdleme42keg with simpler... |
| cdlemg2kq 40596 | ~ cdlemg2k with ` P ` and ... |
| cdlemg2l 40597 | TODO: FIX COMMENT. (Contr... |
| cdlemg2m 40598 | TODO: FIX COMMENT. (Contr... |
| cdlemg5 40599 | TODO: Is there a simpler ... |
| cdlemb3 40600 | Given two atoms not under ... |
| cdlemg7fvbwN 40601 | Properties of a translatio... |
| cdlemg4a 40602 | TODO: FIX COMMENT If fg(p... |
| cdlemg4b1 40603 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b2 40604 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b12 40605 | TODO: FIX COMMENT. (Contr... |
| cdlemg4c 40606 | TODO: FIX COMMENT. (Contr... |
| cdlemg4d 40607 | TODO: FIX COMMENT. (Contr... |
| cdlemg4e 40608 | TODO: FIX COMMENT. (Contr... |
| cdlemg4f 40609 | TODO: FIX COMMENT. (Contr... |
| cdlemg4g 40610 | TODO: FIX COMMENT. (Contr... |
| cdlemg4 40611 | TODO: FIX COMMENT. (Contr... |
| cdlemg6a 40612 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6b 40613 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6c 40614 | TODO: FIX COMMENT. (Contr... |
| cdlemg6d 40615 | TODO: FIX COMMENT. (Contr... |
| cdlemg6e 40616 | TODO: FIX COMMENT. (Contr... |
| cdlemg6 40617 | TODO: FIX COMMENT. (Contr... |
| cdlemg7fvN 40618 | Value of a translation com... |
| cdlemg7aN 40619 | TODO: FIX COMMENT. (Contr... |
| cdlemg7N 40620 | TODO: FIX COMMENT. (Contr... |
| cdlemg8a 40621 | TODO: FIX COMMENT. (Contr... |
| cdlemg8b 40622 | TODO: FIX COMMENT. (Contr... |
| cdlemg8c 40623 | TODO: FIX COMMENT. (Contr... |
| cdlemg8d 40624 | TODO: FIX COMMENT. (Contr... |
| cdlemg8 40625 | TODO: FIX COMMENT. (Contr... |
| cdlemg9a 40626 | TODO: FIX COMMENT. (Contr... |
| cdlemg9b 40627 | The triples ` <. P , ( F `... |
| cdlemg9 40628 | The triples ` <. P , ( F `... |
| cdlemg10b 40629 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10bALTN 40630 | TODO: FIX COMMENT. TODO: ... |
| cdlemg11a 40631 | TODO: FIX COMMENT. (Contr... |
| cdlemg11aq 40632 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10c 40633 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10a 40634 | TODO: FIX COMMENT. (Contr... |
| cdlemg10 40635 | TODO: FIX COMMENT. (Contr... |
| cdlemg11b 40636 | TODO: FIX COMMENT. (Contr... |
| cdlemg12a 40637 | TODO: FIX COMMENT. (Contr... |
| cdlemg12b 40638 | The triples ` <. P , ( F `... |
| cdlemg12c 40639 | The triples ` <. P , ( F `... |
| cdlemg12d 40640 | TODO: FIX COMMENT. (Contr... |
| cdlemg12e 40641 | TODO: FIX COMMENT. (Contr... |
| cdlemg12f 40642 | TODO: FIX COMMENT. (Contr... |
| cdlemg12g 40643 | TODO: FIX COMMENT. TODO: ... |
| cdlemg12 40644 | TODO: FIX COMMENT. (Contr... |
| cdlemg13a 40645 | TODO: FIX COMMENT. (Contr... |
| cdlemg13 40646 | TODO: FIX COMMENT. (Contr... |
| cdlemg14f 40647 | TODO: FIX COMMENT. (Contr... |
| cdlemg14g 40648 | TODO: FIX COMMENT. (Contr... |
| cdlemg15a 40649 | Eliminate the ` ( F `` P )... |
| cdlemg15 40650 | Eliminate the ` ( (... |
| cdlemg16 40651 | Part of proof of Lemma G o... |
| cdlemg16ALTN 40652 | This version of ~ cdlemg16... |
| cdlemg16z 40653 | Eliminate ` ( ( F `... |
| cdlemg16zz 40654 | Eliminate ` P =/= Q ` from... |
| cdlemg17a 40655 | TODO: FIX COMMENT. (Contr... |
| cdlemg17b 40656 | Part of proof of Lemma G i... |
| cdlemg17dN 40657 | TODO: fix comment. (Contr... |
| cdlemg17dALTN 40658 | Same as ~ cdlemg17dN with ... |
| cdlemg17e 40659 | TODO: fix comment. (Contr... |
| cdlemg17f 40660 | TODO: fix comment. (Contr... |
| cdlemg17g 40661 | TODO: fix comment. (Contr... |
| cdlemg17h 40662 | TODO: fix comment. (Contr... |
| cdlemg17i 40663 | TODO: fix comment. (Contr... |
| cdlemg17ir 40664 | TODO: fix comment. (Contr... |
| cdlemg17j 40665 | TODO: fix comment. (Contr... |
| cdlemg17pq 40666 | Utility theorem for swappi... |
| cdlemg17bq 40667 | ~ cdlemg17b with ` P ` and... |
| cdlemg17iqN 40668 | ~ cdlemg17i with ` P ` and... |
| cdlemg17irq 40669 | ~ cdlemg17ir with ` P ` an... |
| cdlemg17jq 40670 | ~ cdlemg17j with ` P ` and... |
| cdlemg17 40671 | Part of Lemma G of [Crawle... |
| cdlemg18a 40672 | Show two lines are differe... |
| cdlemg18b 40673 | Lemma for ~ cdlemg18c . T... |
| cdlemg18c 40674 | Show two lines intersect a... |
| cdlemg18d 40675 | Show two lines intersect a... |
| cdlemg18 40676 | Show two lines intersect a... |
| cdlemg19a 40677 | Show two lines intersect a... |
| cdlemg19 40678 | Show two lines intersect a... |
| cdlemg20 40679 | Show two lines intersect a... |
| cdlemg21 40680 | Version of cdlemg19 with `... |
| cdlemg22 40681 | ~ cdlemg21 with ` ( F `` P... |
| cdlemg24 40682 | Combine ~ cdlemg16z and ~ ... |
| cdlemg37 40683 | Use ~ cdlemg8 to eliminate... |
| cdlemg25zz 40684 | ~ cdlemg16zz restated for ... |
| cdlemg26zz 40685 | ~ cdlemg16zz restated for ... |
| cdlemg27a 40686 | For use with case when ` (... |
| cdlemg28a 40687 | Part of proof of Lemma G o... |
| cdlemg31b0N 40688 | TODO: Fix comment. (Cont... |
| cdlemg31b0a 40689 | TODO: Fix comment. (Cont... |
| cdlemg27b 40690 | TODO: Fix comment. (Cont... |
| cdlemg31a 40691 | TODO: fix comment. (Contr... |
| cdlemg31b 40692 | TODO: fix comment. (Contr... |
| cdlemg31c 40693 | Show that when ` N ` is an... |
| cdlemg31d 40694 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33b0 40695 | TODO: Fix comment. (Cont... |
| cdlemg33c0 40696 | TODO: Fix comment. (Cont... |
| cdlemg28b 40697 | Part of proof of Lemma G o... |
| cdlemg28 40698 | Part of proof of Lemma G o... |
| cdlemg29 40699 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33a 40700 | TODO: Fix comment. (Cont... |
| cdlemg33b 40701 | TODO: Fix comment. (Cont... |
| cdlemg33c 40702 | TODO: Fix comment. (Cont... |
| cdlemg33d 40703 | TODO: Fix comment. (Cont... |
| cdlemg33e 40704 | TODO: Fix comment. (Cont... |
| cdlemg33 40705 | Combine ~ cdlemg33b , ~ cd... |
| cdlemg34 40706 | Use cdlemg33 to eliminate ... |
| cdlemg35 40707 | TODO: Fix comment. TODO:... |
| cdlemg36 40708 | Use cdlemg35 to eliminate ... |
| cdlemg38 40709 | Use ~ cdlemg37 to eliminat... |
| cdlemg39 40710 | Eliminate ` =/= ` conditio... |
| cdlemg40 40711 | Eliminate ` P =/= Q ` cond... |
| cdlemg41 40712 | Convert ~ cdlemg40 to func... |
| ltrnco 40713 | The composition of two tra... |
| trlcocnv 40714 | Swap the arguments of the ... |
| trlcoabs 40715 | Absorption into a composit... |
| trlcoabs2N 40716 | Absorption of the trace of... |
| trlcoat 40717 | The trace of a composition... |
| trlcocnvat 40718 | Commonly used special case... |
| trlconid 40719 | The composition of two dif... |
| trlcolem 40720 | Lemma for ~ trlco . (Cont... |
| trlco 40721 | The trace of a composition... |
| trlcone 40722 | If two translations have d... |
| cdlemg42 40723 | Part of proof of Lemma G o... |
| cdlemg43 40724 | Part of proof of Lemma G o... |
| cdlemg44a 40725 | Part of proof of Lemma G o... |
| cdlemg44b 40726 | Eliminate ` ( F `` P ) =/=... |
| cdlemg44 40727 | Part of proof of Lemma G o... |
| cdlemg47a 40728 | TODO: fix comment. TODO: ... |
| cdlemg46 40729 | Part of proof of Lemma G o... |
| cdlemg47 40730 | Part of proof of Lemma G o... |
| cdlemg48 40731 | Eliminate ` h ` from ~ cdl... |
| ltrncom 40732 | Composition is commutative... |
| ltrnco4 40733 | Rearrange a composition of... |
| trljco 40734 | Trace joined with trace of... |
| trljco2 40735 | Trace joined with trace of... |
| tgrpfset 40738 | The translation group maps... |
| tgrpset 40739 | The translation group for ... |
| tgrpbase 40740 | The base set of the transl... |
| tgrpopr 40741 | The group operation of the... |
| tgrpov 40742 | The group operation value ... |
| tgrpgrplem 40743 | Lemma for ~ tgrpgrp . (Co... |
| tgrpgrp 40744 | The translation group is a... |
| tgrpabl 40745 | The translation group is a... |
| tendofset 40752 | The set of all trace-prese... |
| tendoset 40753 | The set of trace-preservin... |
| istendo 40754 | The predicate "is a trace-... |
| tendotp 40755 | Trace-preserving property ... |
| istendod 40756 | Deduce the predicate "is a... |
| tendof 40757 | Functionality of a trace-p... |
| tendoeq1 40758 | Condition determining equa... |
| tendovalco 40759 | Value of composition of tr... |
| tendocoval 40760 | Value of composition of en... |
| tendocl 40761 | Closure of a trace-preserv... |
| tendoco2 40762 | Distribution of compositio... |
| tendoidcl 40763 | The identity is a trace-pr... |
| tendo1mul 40764 | Multiplicative identity mu... |
| tendo1mulr 40765 | Multiplicative identity mu... |
| tendococl 40766 | The composition of two tra... |
| tendoid 40767 | The identity value of a tr... |
| tendoeq2 40768 | Condition determining equa... |
| tendoplcbv 40769 | Define sum operation for t... |
| tendopl 40770 | Value of endomorphism sum ... |
| tendopl2 40771 | Value of result of endomor... |
| tendoplcl2 40772 | Value of result of endomor... |
| tendoplco2 40773 | Value of result of endomor... |
| tendopltp 40774 | Trace-preserving property ... |
| tendoplcl 40775 | Endomorphism sum is a trac... |
| tendoplcom 40776 | The endomorphism sum opera... |
| tendoplass 40777 | The endomorphism sum opera... |
| tendodi1 40778 | Endomorphism composition d... |
| tendodi2 40779 | Endomorphism composition d... |
| tendo0cbv 40780 | Define additive identity f... |
| tendo02 40781 | Value of additive identity... |
| tendo0co2 40782 | The additive identity trac... |
| tendo0tp 40783 | Trace-preserving property ... |
| tendo0cl 40784 | The additive identity is a... |
| tendo0pl 40785 | Property of the additive i... |
| tendo0plr 40786 | Property of the additive i... |
| tendoicbv 40787 | Define inverse function fo... |
| tendoi 40788 | Value of inverse endomorph... |
| tendoi2 40789 | Value of additive inverse ... |
| tendoicl 40790 | Closure of the additive in... |
| tendoipl 40791 | Property of the additive i... |
| tendoipl2 40792 | Property of the additive i... |
| erngfset 40793 | The division rings on trac... |
| erngset 40794 | The division ring on trace... |
| erngbase 40795 | The base set of the divisi... |
| erngfplus 40796 | Ring addition operation. ... |
| erngplus 40797 | Ring addition operation. ... |
| erngplus2 40798 | Ring addition operation. ... |
| erngfmul 40799 | Ring multiplication operat... |
| erngmul 40800 | Ring addition operation. ... |
| erngfset-rN 40801 | The division rings on trac... |
| erngset-rN 40802 | The division ring on trace... |
| erngbase-rN 40803 | The base set of the divisi... |
| erngfplus-rN 40804 | Ring addition operation. ... |
| erngplus-rN 40805 | Ring addition operation. ... |
| erngplus2-rN 40806 | Ring addition operation. ... |
| erngfmul-rN 40807 | Ring multiplication operat... |
| erngmul-rN 40808 | Ring addition operation. ... |
| cdlemh1 40809 | Part of proof of Lemma H o... |
| cdlemh2 40810 | Part of proof of Lemma H o... |
| cdlemh 40811 | Lemma H of [Crawley] p. 11... |
| cdlemi1 40812 | Part of proof of Lemma I o... |
| cdlemi2 40813 | Part of proof of Lemma I o... |
| cdlemi 40814 | Lemma I of [Crawley] p. 11... |
| cdlemj1 40815 | Part of proof of Lemma J o... |
| cdlemj2 40816 | Part of proof of Lemma J o... |
| cdlemj3 40817 | Part of proof of Lemma J o... |
| tendocan 40818 | Cancellation law: if the v... |
| tendoid0 40819 | A trace-preserving endomor... |
| tendo0mul 40820 | Additive identity multipli... |
| tendo0mulr 40821 | Additive identity multipli... |
| tendo1ne0 40822 | The identity (unity) is no... |
| tendoconid 40823 | The composition (product) ... |
| tendotr 40824 | The trace of the value of ... |
| cdlemk1 40825 | Part of proof of Lemma K o... |
| cdlemk2 40826 | Part of proof of Lemma K o... |
| cdlemk3 40827 | Part of proof of Lemma K o... |
| cdlemk4 40828 | Part of proof of Lemma K o... |
| cdlemk5a 40829 | Part of proof of Lemma K o... |
| cdlemk5 40830 | Part of proof of Lemma K o... |
| cdlemk6 40831 | Part of proof of Lemma K o... |
| cdlemk8 40832 | Part of proof of Lemma K o... |
| cdlemk9 40833 | Part of proof of Lemma K o... |
| cdlemk9bN 40834 | Part of proof of Lemma K o... |
| cdlemki 40835 | Part of proof of Lemma K o... |
| cdlemkvcl 40836 | Part of proof of Lemma K o... |
| cdlemk10 40837 | Part of proof of Lemma K o... |
| cdlemksv 40838 | Part of proof of Lemma K o... |
| cdlemksel 40839 | Part of proof of Lemma K o... |
| cdlemksat 40840 | Part of proof of Lemma K o... |
| cdlemksv2 40841 | Part of proof of Lemma K o... |
| cdlemk7 40842 | Part of proof of Lemma K o... |
| cdlemk11 40843 | Part of proof of Lemma K o... |
| cdlemk12 40844 | Part of proof of Lemma K o... |
| cdlemkoatnle 40845 | Utility lemma. (Contribut... |
| cdlemk13 40846 | Part of proof of Lemma K o... |
| cdlemkole 40847 | Utility lemma. (Contribut... |
| cdlemk14 40848 | Part of proof of Lemma K o... |
| cdlemk15 40849 | Part of proof of Lemma K o... |
| cdlemk16a 40850 | Part of proof of Lemma K o... |
| cdlemk16 40851 | Part of proof of Lemma K o... |
| cdlemk17 40852 | Part of proof of Lemma K o... |
| cdlemk1u 40853 | Part of proof of Lemma K o... |
| cdlemk5auN 40854 | Part of proof of Lemma K o... |
| cdlemk5u 40855 | Part of proof of Lemma K o... |
| cdlemk6u 40856 | Part of proof of Lemma K o... |
| cdlemkj 40857 | Part of proof of Lemma K o... |
| cdlemkuvN 40858 | Part of proof of Lemma K o... |
| cdlemkuel 40859 | Part of proof of Lemma K o... |
| cdlemkuat 40860 | Part of proof of Lemma K o... |
| cdlemkuv2 40861 | Part of proof of Lemma K o... |
| cdlemk18 40862 | Part of proof of Lemma K o... |
| cdlemk19 40863 | Part of proof of Lemma K o... |
| cdlemk7u 40864 | Part of proof of Lemma K o... |
| cdlemk11u 40865 | Part of proof of Lemma K o... |
| cdlemk12u 40866 | Part of proof of Lemma K o... |
| cdlemk21N 40867 | Part of proof of Lemma K o... |
| cdlemk20 40868 | Part of proof of Lemma K o... |
| cdlemkoatnle-2N 40869 | Utility lemma. (Contribut... |
| cdlemk13-2N 40870 | Part of proof of Lemma K o... |
| cdlemkole-2N 40871 | Utility lemma. (Contribut... |
| cdlemk14-2N 40872 | Part of proof of Lemma K o... |
| cdlemk15-2N 40873 | Part of proof of Lemma K o... |
| cdlemk16-2N 40874 | Part of proof of Lemma K o... |
| cdlemk17-2N 40875 | Part of proof of Lemma K o... |
| cdlemkj-2N 40876 | Part of proof of Lemma K o... |
| cdlemkuv-2N 40877 | Part of proof of Lemma K o... |
| cdlemkuel-2N 40878 | Part of proof of Lemma K o... |
| cdlemkuv2-2 40879 | Part of proof of Lemma K o... |
| cdlemk18-2N 40880 | Part of proof of Lemma K o... |
| cdlemk19-2N 40881 | Part of proof of Lemma K o... |
| cdlemk7u-2N 40882 | Part of proof of Lemma K o... |
| cdlemk11u-2N 40883 | Part of proof of Lemma K o... |
| cdlemk12u-2N 40884 | Part of proof of Lemma K o... |
| cdlemk21-2N 40885 | Part of proof of Lemma K o... |
| cdlemk20-2N 40886 | Part of proof of Lemma K o... |
| cdlemk22 40887 | Part of proof of Lemma K o... |
| cdlemk30 40888 | Part of proof of Lemma K o... |
| cdlemkuu 40889 | Convert between function a... |
| cdlemk31 40890 | Part of proof of Lemma K o... |
| cdlemk32 40891 | Part of proof of Lemma K o... |
| cdlemkuel-3 40892 | Part of proof of Lemma K o... |
| cdlemkuv2-3N 40893 | Part of proof of Lemma K o... |
| cdlemk18-3N 40894 | Part of proof of Lemma K o... |
| cdlemk22-3 40895 | Part of proof of Lemma K o... |
| cdlemk23-3 40896 | Part of proof of Lemma K o... |
| cdlemk24-3 40897 | Part of proof of Lemma K o... |
| cdlemk25-3 40898 | Part of proof of Lemma K o... |
| cdlemk26b-3 40899 | Part of proof of Lemma K o... |
| cdlemk26-3 40900 | Part of proof of Lemma K o... |
| cdlemk27-3 40901 | Part of proof of Lemma K o... |
| cdlemk28-3 40902 | Part of proof of Lemma K o... |
| cdlemk33N 40903 | Part of proof of Lemma K o... |
| cdlemk34 40904 | Part of proof of Lemma K o... |
| cdlemk29-3 40905 | Part of proof of Lemma K o... |
| cdlemk35 40906 | Part of proof of Lemma K o... |
| cdlemk36 40907 | Part of proof of Lemma K o... |
| cdlemk37 40908 | Part of proof of Lemma K o... |
| cdlemk38 40909 | Part of proof of Lemma K o... |
| cdlemk39 40910 | Part of proof of Lemma K o... |
| cdlemk40 40911 | TODO: fix comment. (Contr... |
| cdlemk40t 40912 | TODO: fix comment. (Contr... |
| cdlemk40f 40913 | TODO: fix comment. (Contr... |
| cdlemk41 40914 | Part of proof of Lemma K o... |
| cdlemkfid1N 40915 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid1 40916 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid2N 40917 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid2 40918 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid3N 40919 | TODO: is this useful or sh... |
| cdlemky 40920 | Part of proof of Lemma K o... |
| cdlemkyu 40921 | Convert between function a... |
| cdlemkyuu 40922 | ~ cdlemkyu with some hypot... |
| cdlemk11ta 40923 | Part of proof of Lemma K o... |
| cdlemk19ylem 40924 | Lemma for ~ cdlemk19y . (... |
| cdlemk11tb 40925 | Part of proof of Lemma K o... |
| cdlemk19y 40926 | ~ cdlemk19 with simpler hy... |
| cdlemkid3N 40927 | Lemma for ~ cdlemkid . (C... |
| cdlemkid4 40928 | Lemma for ~ cdlemkid . (C... |
| cdlemkid5 40929 | Lemma for ~ cdlemkid . (C... |
| cdlemkid 40930 | The value of the tau funct... |
| cdlemk35s 40931 | Substitution version of ~ ... |
| cdlemk35s-id 40932 | Substitution version of ~ ... |
| cdlemk39s 40933 | Substitution version of ~ ... |
| cdlemk39s-id 40934 | Substitution version of ~ ... |
| cdlemk42 40935 | Part of proof of Lemma K o... |
| cdlemk19xlem 40936 | Lemma for ~ cdlemk19x . (... |
| cdlemk19x 40937 | ~ cdlemk19 with simpler hy... |
| cdlemk42yN 40938 | Part of proof of Lemma K o... |
| cdlemk11tc 40939 | Part of proof of Lemma K o... |
| cdlemk11t 40940 | Part of proof of Lemma K o... |
| cdlemk45 40941 | Part of proof of Lemma K o... |
| cdlemk46 40942 | Part of proof of Lemma K o... |
| cdlemk47 40943 | Part of proof of Lemma K o... |
| cdlemk48 40944 | Part of proof of Lemma K o... |
| cdlemk49 40945 | Part of proof of Lemma K o... |
| cdlemk50 40946 | Part of proof of Lemma K o... |
| cdlemk51 40947 | Part of proof of Lemma K o... |
| cdlemk52 40948 | Part of proof of Lemma K o... |
| cdlemk53a 40949 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53b 40950 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53 40951 | Part of proof of Lemma K o... |
| cdlemk54 40952 | Part of proof of Lemma K o... |
| cdlemk55a 40953 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55b 40954 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55 40955 | Part of proof of Lemma K o... |
| cdlemkyyN 40956 | Part of proof of Lemma K o... |
| cdlemk43N 40957 | Part of proof of Lemma K o... |
| cdlemk35u 40958 | Substitution version of ~ ... |
| cdlemk55u1 40959 | Lemma for ~ cdlemk55u . (... |
| cdlemk55u 40960 | Part of proof of Lemma K o... |
| cdlemk39u1 40961 | Lemma for ~ cdlemk39u . (... |
| cdlemk39u 40962 | Part of proof of Lemma K o... |
| cdlemk19u1 40963 | ~ cdlemk19 with simpler hy... |
| cdlemk19u 40964 | Part of Lemma K of [Crawle... |
| cdlemk56 40965 | Part of Lemma K of [Crawle... |
| cdlemk19w 40966 | Use a fixed element to eli... |
| cdlemk56w 40967 | Use a fixed element to eli... |
| cdlemk 40968 | Lemma K of [Crawley] p. 11... |
| tendoex 40969 | Generalization of Lemma K ... |
| cdleml1N 40970 | Part of proof of Lemma L o... |
| cdleml2N 40971 | Part of proof of Lemma L o... |
| cdleml3N 40972 | Part of proof of Lemma L o... |
| cdleml4N 40973 | Part of proof of Lemma L o... |
| cdleml5N 40974 | Part of proof of Lemma L o... |
| cdleml6 40975 | Part of proof of Lemma L o... |
| cdleml7 40976 | Part of proof of Lemma L o... |
| cdleml8 40977 | Part of proof of Lemma L o... |
| cdleml9 40978 | Part of proof of Lemma L o... |
| dva1dim 40979 | Two expressions for the 1-... |
| dvhb1dimN 40980 | Two expressions for the 1-... |
| erng1lem 40981 | Value of the endomorphism ... |
| erngdvlem1 40982 | Lemma for ~ eringring . (... |
| erngdvlem2N 40983 | Lemma for ~ eringring . (... |
| erngdvlem3 40984 | Lemma for ~ eringring . (... |
| erngdvlem4 40985 | Lemma for ~ erngdv . (Con... |
| eringring 40986 | An endomorphism ring is a ... |
| erngdv 40987 | An endomorphism ring is a ... |
| erng0g 40988 | The division ring zero of ... |
| erng1r 40989 | The division ring unity of... |
| erngdvlem1-rN 40990 | Lemma for ~ eringring . (... |
| erngdvlem2-rN 40991 | Lemma for ~ eringring . (... |
| erngdvlem3-rN 40992 | Lemma for ~ eringring . (... |
| erngdvlem4-rN 40993 | Lemma for ~ erngdv . (Con... |
| erngring-rN 40994 | An endomorphism ring is a ... |
| erngdv-rN 40995 | An endomorphism ring is a ... |
| dvafset 40998 | The constructed partial ve... |
| dvaset 40999 | The constructed partial ve... |
| dvasca 41000 | The ring base set of the c... |
| dvabase 41001 | The ring base set of the c... |
| dvafplusg 41002 | Ring addition operation fo... |
| dvaplusg 41003 | Ring addition operation fo... |
| dvaplusgv 41004 | Ring addition operation fo... |
| dvafmulr 41005 | Ring multiplication operat... |
| dvamulr 41006 | Ring multiplication operat... |
| dvavbase 41007 | The vectors (vector base s... |
| dvafvadd 41008 | The vector sum operation f... |
| dvavadd 41009 | Ring addition operation fo... |
| dvafvsca 41010 | Ring addition operation fo... |
| dvavsca 41011 | Ring addition operation fo... |
| tendospcl 41012 | Closure of endomorphism sc... |
| tendospass 41013 | Associative law for endomo... |
| tendospdi1 41014 | Forward distributive law f... |
| tendocnv 41015 | Converse of a trace-preser... |
| tendospdi2 41016 | Reverse distributive law f... |
| tendospcanN 41017 | Cancellation law for trace... |
| dvaabl 41018 | The constructed partial ve... |
| dvalveclem 41019 | Lemma for ~ dvalvec . (Co... |
| dvalvec 41020 | The constructed partial ve... |
| dva0g 41021 | The zero vector of partial... |
| diaffval 41024 | The partial isomorphism A ... |
| diafval 41025 | The partial isomorphism A ... |
| diaval 41026 | The partial isomorphism A ... |
| diaelval 41027 | Member of the partial isom... |
| diafn 41028 | Functionality and domain o... |
| diadm 41029 | Domain of the partial isom... |
| diaeldm 41030 | Member of domain of the pa... |
| diadmclN 41031 | A member of domain of the ... |
| diadmleN 41032 | A member of domain of the ... |
| dian0 41033 | The value of the partial i... |
| dia0eldmN 41034 | The lattice zero belongs t... |
| dia1eldmN 41035 | The fiducial hyperplane (t... |
| diass 41036 | The value of the partial i... |
| diael 41037 | A member of the value of t... |
| diatrl 41038 | Trace of a member of the p... |
| diaelrnN 41039 | Any value of the partial i... |
| dialss 41040 | The value of partial isomo... |
| diaord 41041 | The partial isomorphism A ... |
| dia11N 41042 | The partial isomorphism A ... |
| diaf11N 41043 | The partial isomorphism A ... |
| diaclN 41044 | Closure of partial isomorp... |
| diacnvclN 41045 | Closure of partial isomorp... |
| dia0 41046 | The value of the partial i... |
| dia1N 41047 | The value of the partial i... |
| dia1elN 41048 | The largest subspace in th... |
| diaglbN 41049 | Partial isomorphism A of a... |
| diameetN 41050 | Partial isomorphism A of a... |
| diainN 41051 | Inverse partial isomorphis... |
| diaintclN 41052 | The intersection of partia... |
| diasslssN 41053 | The partial isomorphism A ... |
| diassdvaN 41054 | The partial isomorphism A ... |
| dia1dim 41055 | Two expressions for the 1-... |
| dia1dim2 41056 | Two expressions for a 1-di... |
| dia1dimid 41057 | A vector (translation) bel... |
| dia2dimlem1 41058 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem2 41059 | Lemma for ~ dia2dim . Def... |
| dia2dimlem3 41060 | Lemma for ~ dia2dim . Def... |
| dia2dimlem4 41061 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem5 41062 | Lemma for ~ dia2dim . The... |
| dia2dimlem6 41063 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem7 41064 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem8 41065 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem9 41066 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem10 41067 | Lemma for ~ dia2dim . Con... |
| dia2dimlem11 41068 | Lemma for ~ dia2dim . Con... |
| dia2dimlem12 41069 | Lemma for ~ dia2dim . Obt... |
| dia2dimlem13 41070 | Lemma for ~ dia2dim . Eli... |
| dia2dim 41071 | A two-dimensional subspace... |
| dvhfset 41074 | The constructed full vecto... |
| dvhset 41075 | The constructed full vecto... |
| dvhsca 41076 | The ring of scalars of the... |
| dvhbase 41077 | The ring base set of the c... |
| dvhfplusr 41078 | Ring addition operation fo... |
| dvhfmulr 41079 | Ring multiplication operat... |
| dvhmulr 41080 | Ring multiplication operat... |
| dvhvbase 41081 | The vectors (vector base s... |
| dvhelvbasei 41082 | Vector membership in the c... |
| dvhvaddcbv 41083 | Change bound variables to ... |
| dvhvaddval 41084 | The vector sum operation f... |
| dvhfvadd 41085 | The vector sum operation f... |
| dvhvadd 41086 | The vector sum operation f... |
| dvhopvadd 41087 | The vector sum operation f... |
| dvhopvadd2 41088 | The vector sum operation f... |
| dvhvaddcl 41089 | Closure of the vector sum ... |
| dvhvaddcomN 41090 | Commutativity of vector su... |
| dvhvaddass 41091 | Associativity of vector su... |
| dvhvscacbv 41092 | Change bound variables to ... |
| dvhvscaval 41093 | The scalar product operati... |
| dvhfvsca 41094 | Scalar product operation f... |
| dvhvsca 41095 | Scalar product operation f... |
| dvhopvsca 41096 | Scalar product operation f... |
| dvhvscacl 41097 | Closure of the scalar prod... |
| tendoinvcl 41098 | Closure of multiplicative ... |
| tendolinv 41099 | Left multiplicative invers... |
| tendorinv 41100 | Right multiplicative inver... |
| dvhgrp 41101 | The full vector space ` U ... |
| dvhlveclem 41102 | Lemma for ~ dvhlvec . TOD... |
| dvhlvec 41103 | The full vector space ` U ... |
| dvhlmod 41104 | The full vector space ` U ... |
| dvh0g 41105 | The zero vector of vector ... |
| dvheveccl 41106 | Properties of a unit vecto... |
| dvhopclN 41107 | Closure of a ` DVecH ` vec... |
| dvhopaddN 41108 | Sum of ` DVecH ` vectors e... |
| dvhopspN 41109 | Scalar product of ` DVecH ... |
| dvhopN 41110 | Decompose a ` DVecH ` vect... |
| dvhopellsm 41111 | Ordered pair membership in... |
| cdlemm10N 41112 | The image of the map ` G `... |
| docaffvalN 41115 | Subspace orthocomplement f... |
| docafvalN 41116 | Subspace orthocomplement f... |
| docavalN 41117 | Subspace orthocomplement f... |
| docaclN 41118 | Closure of subspace orthoc... |
| diaocN 41119 | Value of partial isomorphi... |
| doca2N 41120 | Double orthocomplement of ... |
| doca3N 41121 | Double orthocomplement of ... |
| dvadiaN 41122 | Any closed subspace is a m... |
| diarnN 41123 | Partial isomorphism A maps... |
| diaf1oN 41124 | The partial isomorphism A ... |
| djaffvalN 41127 | Subspace join for ` DVecA ... |
| djafvalN 41128 | Subspace join for ` DVecA ... |
| djavalN 41129 | Subspace join for ` DVecA ... |
| djaclN 41130 | Closure of subspace join f... |
| djajN 41131 | Transfer lattice join to `... |
| dibffval 41134 | The partial isomorphism B ... |
| dibfval 41135 | The partial isomorphism B ... |
| dibval 41136 | The partial isomorphism B ... |
| dibopelvalN 41137 | Member of the partial isom... |
| dibval2 41138 | Value of the partial isomo... |
| dibopelval2 41139 | Member of the partial isom... |
| dibval3N 41140 | Value of the partial isomo... |
| dibelval3 41141 | Member of the partial isom... |
| dibopelval3 41142 | Member of the partial isom... |
| dibelval1st 41143 | Membership in value of the... |
| dibelval1st1 41144 | Membership in value of the... |
| dibelval1st2N 41145 | Membership in value of the... |
| dibelval2nd 41146 | Membership in value of the... |
| dibn0 41147 | The value of the partial i... |
| dibfna 41148 | Functionality and domain o... |
| dibdiadm 41149 | Domain of the partial isom... |
| dibfnN 41150 | Functionality and domain o... |
| dibdmN 41151 | Domain of the partial isom... |
| dibeldmN 41152 | Member of domain of the pa... |
| dibord 41153 | The isomorphism B for a la... |
| dib11N 41154 | The isomorphism B for a la... |
| dibf11N 41155 | The partial isomorphism A ... |
| dibclN 41156 | Closure of partial isomorp... |
| dibvalrel 41157 | The value of partial isomo... |
| dib0 41158 | The value of partial isomo... |
| dib1dim 41159 | Two expressions for the 1-... |
| dibglbN 41160 | Partial isomorphism B of a... |
| dibintclN 41161 | The intersection of partia... |
| dib1dim2 41162 | Two expressions for a 1-di... |
| dibss 41163 | The partial isomorphism B ... |
| diblss 41164 | The value of partial isomo... |
| diblsmopel 41165 | Membership in subspace sum... |
| dicffval 41168 | The partial isomorphism C ... |
| dicfval 41169 | The partial isomorphism C ... |
| dicval 41170 | The partial isomorphism C ... |
| dicopelval 41171 | Membership in value of the... |
| dicelvalN 41172 | Membership in value of the... |
| dicval2 41173 | The partial isomorphism C ... |
| dicelval3 41174 | Member of the partial isom... |
| dicopelval2 41175 | Membership in value of the... |
| dicelval2N 41176 | Membership in value of the... |
| dicfnN 41177 | Functionality and domain o... |
| dicdmN 41178 | Domain of the partial isom... |
| dicvalrelN 41179 | The value of partial isomo... |
| dicssdvh 41180 | The partial isomorphism C ... |
| dicelval1sta 41181 | Membership in value of the... |
| dicelval1stN 41182 | Membership in value of the... |
| dicelval2nd 41183 | Membership in value of the... |
| dicvaddcl 41184 | Membership in value of the... |
| dicvscacl 41185 | Membership in value of the... |
| dicn0 41186 | The value of the partial i... |
| diclss 41187 | The value of partial isomo... |
| diclspsn 41188 | The value of isomorphism C... |
| cdlemn2 41189 | Part of proof of Lemma N o... |
| cdlemn2a 41190 | Part of proof of Lemma N o... |
| cdlemn3 41191 | Part of proof of Lemma N o... |
| cdlemn4 41192 | Part of proof of Lemma N o... |
| cdlemn4a 41193 | Part of proof of Lemma N o... |
| cdlemn5pre 41194 | Part of proof of Lemma N o... |
| cdlemn5 41195 | Part of proof of Lemma N o... |
| cdlemn6 41196 | Part of proof of Lemma N o... |
| cdlemn7 41197 | Part of proof of Lemma N o... |
| cdlemn8 41198 | Part of proof of Lemma N o... |
| cdlemn9 41199 | Part of proof of Lemma N o... |
| cdlemn10 41200 | Part of proof of Lemma N o... |
| cdlemn11a 41201 | Part of proof of Lemma N o... |
| cdlemn11b 41202 | Part of proof of Lemma N o... |
| cdlemn11c 41203 | Part of proof of Lemma N o... |
| cdlemn11pre 41204 | Part of proof of Lemma N o... |
| cdlemn11 41205 | Part of proof of Lemma N o... |
| cdlemn 41206 | Lemma N of [Crawley] p. 12... |
| dihordlem6 41207 | Part of proof of Lemma N o... |
| dihordlem7 41208 | Part of proof of Lemma N o... |
| dihordlem7b 41209 | Part of proof of Lemma N o... |
| dihjustlem 41210 | Part of proof after Lemma ... |
| dihjust 41211 | Part of proof after Lemma ... |
| dihord1 41212 | Part of proof after Lemma ... |
| dihord2a 41213 | Part of proof after Lemma ... |
| dihord2b 41214 | Part of proof after Lemma ... |
| dihord2cN 41215 | Part of proof after Lemma ... |
| dihord11b 41216 | Part of proof after Lemma ... |
| dihord10 41217 | Part of proof after Lemma ... |
| dihord11c 41218 | Part of proof after Lemma ... |
| dihord2pre 41219 | Part of proof after Lemma ... |
| dihord2pre2 41220 | Part of proof after Lemma ... |
| dihord2 41221 | Part of proof after Lemma ... |
| dihffval 41224 | The isomorphism H for a la... |
| dihfval 41225 | Isomorphism H for a lattic... |
| dihval 41226 | Value of isomorphism H for... |
| dihvalc 41227 | Value of isomorphism H for... |
| dihlsscpre 41228 | Closure of isomorphism H f... |
| dihvalcqpre 41229 | Value of isomorphism H for... |
| dihvalcq 41230 | Value of isomorphism H for... |
| dihvalb 41231 | Value of isomorphism H for... |
| dihopelvalbN 41232 | Ordered pair member of the... |
| dihvalcqat 41233 | Value of isomorphism H for... |
| dih1dimb 41234 | Two expressions for a 1-di... |
| dih1dimb2 41235 | Isomorphism H at an atom u... |
| dih1dimc 41236 | Isomorphism H at an atom n... |
| dib2dim 41237 | Extend ~ dia2dim to partia... |
| dih2dimb 41238 | Extend ~ dib2dim to isomor... |
| dih2dimbALTN 41239 | Extend ~ dia2dim to isomor... |
| dihopelvalcqat 41240 | Ordered pair member of the... |
| dihvalcq2 41241 | Value of isomorphism H for... |
| dihopelvalcpre 41242 | Member of value of isomorp... |
| dihopelvalc 41243 | Member of value of isomorp... |
| dihlss 41244 | The value of isomorphism H... |
| dihss 41245 | The value of isomorphism H... |
| dihssxp 41246 | An isomorphism H value is ... |
| dihopcl 41247 | Closure of an ordered pair... |
| xihopellsmN 41248 | Ordered pair membership in... |
| dihopellsm 41249 | Ordered pair membership in... |
| dihord6apre 41250 | Part of proof that isomorp... |
| dihord3 41251 | The isomorphism H for a la... |
| dihord4 41252 | The isomorphism H for a la... |
| dihord5b 41253 | Part of proof that isomorp... |
| dihord6b 41254 | Part of proof that isomorp... |
| dihord6a 41255 | Part of proof that isomorp... |
| dihord5apre 41256 | Part of proof that isomorp... |
| dihord5a 41257 | Part of proof that isomorp... |
| dihord 41258 | The isomorphism H is order... |
| dih11 41259 | The isomorphism H is one-t... |
| dihf11lem 41260 | Functionality of the isomo... |
| dihf11 41261 | The isomorphism H for a la... |
| dihfn 41262 | Functionality and domain o... |
| dihdm 41263 | Domain of isomorphism H. (... |
| dihcl 41264 | Closure of isomorphism H. ... |
| dihcnvcl 41265 | Closure of isomorphism H c... |
| dihcnvid1 41266 | The converse isomorphism o... |
| dihcnvid2 41267 | The isomorphism of a conve... |
| dihcnvord 41268 | Ordering property for conv... |
| dihcnv11 41269 | The converse of isomorphis... |
| dihsslss 41270 | The isomorphism H maps to ... |
| dihrnlss 41271 | The isomorphism H maps to ... |
| dihrnss 41272 | The isomorphism H maps to ... |
| dihvalrel 41273 | The value of isomorphism H... |
| dih0 41274 | The value of isomorphism H... |
| dih0bN 41275 | A lattice element is zero ... |
| dih0vbN 41276 | A vector is zero iff its s... |
| dih0cnv 41277 | The isomorphism H converse... |
| dih0rn 41278 | The zero subspace belongs ... |
| dih0sb 41279 | A subspace is zero iff the... |
| dih1 41280 | The value of isomorphism H... |
| dih1rn 41281 | The full vector space belo... |
| dih1cnv 41282 | The isomorphism H converse... |
| dihwN 41283 | Value of isomorphism H at ... |
| dihmeetlem1N 41284 | Isomorphism H of a conjunc... |
| dihglblem5apreN 41285 | A conjunction property of ... |
| dihglblem5aN 41286 | A conjunction property of ... |
| dihglblem2aN 41287 | Lemma for isomorphism H of... |
| dihglblem2N 41288 | The GLB of a set of lattic... |
| dihglblem3N 41289 | Isomorphism H of a lattice... |
| dihglblem3aN 41290 | Isomorphism H of a lattice... |
| dihglblem4 41291 | Isomorphism H of a lattice... |
| dihglblem5 41292 | Isomorphism H of a lattice... |
| dihmeetlem2N 41293 | Isomorphism H of a conjunc... |
| dihglbcpreN 41294 | Isomorphism H of a lattice... |
| dihglbcN 41295 | Isomorphism H of a lattice... |
| dihmeetcN 41296 | Isomorphism H of a lattice... |
| dihmeetbN 41297 | Isomorphism H of a lattice... |
| dihmeetbclemN 41298 | Lemma for isomorphism H of... |
| dihmeetlem3N 41299 | Lemma for isomorphism H of... |
| dihmeetlem4preN 41300 | Lemma for isomorphism H of... |
| dihmeetlem4N 41301 | Lemma for isomorphism H of... |
| dihmeetlem5 41302 | Part of proof that isomorp... |
| dihmeetlem6 41303 | Lemma for isomorphism H of... |
| dihmeetlem7N 41304 | Lemma for isomorphism H of... |
| dihjatc1 41305 | Lemma for isomorphism H of... |
| dihjatc2N 41306 | Isomorphism H of join with... |
| dihjatc3 41307 | Isomorphism H of join with... |
| dihmeetlem8N 41308 | Lemma for isomorphism H of... |
| dihmeetlem9N 41309 | Lemma for isomorphism H of... |
| dihmeetlem10N 41310 | Lemma for isomorphism H of... |
| dihmeetlem11N 41311 | Lemma for isomorphism H of... |
| dihmeetlem12N 41312 | Lemma for isomorphism H of... |
| dihmeetlem13N 41313 | Lemma for isomorphism H of... |
| dihmeetlem14N 41314 | Lemma for isomorphism H of... |
| dihmeetlem15N 41315 | Lemma for isomorphism H of... |
| dihmeetlem16N 41316 | Lemma for isomorphism H of... |
| dihmeetlem17N 41317 | Lemma for isomorphism H of... |
| dihmeetlem18N 41318 | Lemma for isomorphism H of... |
| dihmeetlem19N 41319 | Lemma for isomorphism H of... |
| dihmeetlem20N 41320 | Lemma for isomorphism H of... |
| dihmeetALTN 41321 | Isomorphism H of a lattice... |
| dih1dimatlem0 41322 | Lemma for ~ dih1dimat . (... |
| dih1dimatlem 41323 | Lemma for ~ dih1dimat . (... |
| dih1dimat 41324 | Any 1-dimensional subspace... |
| dihlsprn 41325 | The span of a vector belon... |
| dihlspsnssN 41326 | A subspace included in a 1... |
| dihlspsnat 41327 | The inverse isomorphism H ... |
| dihatlat 41328 | The isomorphism H of an at... |
| dihat 41329 | There exists at least one ... |
| dihpN 41330 | The value of isomorphism H... |
| dihlatat 41331 | The reverse isomorphism H ... |
| dihatexv 41332 | There is a nonzero vector ... |
| dihatexv2 41333 | There is a nonzero vector ... |
| dihglblem6 41334 | Isomorphism H of a lattice... |
| dihglb 41335 | Isomorphism H of a lattice... |
| dihglb2 41336 | Isomorphism H of a lattice... |
| dihmeet 41337 | Isomorphism H of a lattice... |
| dihintcl 41338 | The intersection of closed... |
| dihmeetcl 41339 | Closure of closed subspace... |
| dihmeet2 41340 | Reverse isomorphism H of a... |
| dochffval 41343 | Subspace orthocomplement f... |
| dochfval 41344 | Subspace orthocomplement f... |
| dochval 41345 | Subspace orthocomplement f... |
| dochval2 41346 | Subspace orthocomplement f... |
| dochcl 41347 | Closure of subspace orthoc... |
| dochlss 41348 | A subspace orthocomplement... |
| dochssv 41349 | A subspace orthocomplement... |
| dochfN 41350 | Domain and codomain of the... |
| dochvalr 41351 | Orthocomplement of a close... |
| doch0 41352 | Orthocomplement of the zer... |
| doch1 41353 | Orthocomplement of the uni... |
| dochoc0 41354 | The zero subspace is close... |
| dochoc1 41355 | The unit subspace (all vec... |
| dochvalr2 41356 | Orthocomplement of a close... |
| dochvalr3 41357 | Orthocomplement of a close... |
| doch2val2 41358 | Double orthocomplement for... |
| dochss 41359 | Subset law for orthocomple... |
| dochocss 41360 | Double negative law for or... |
| dochoc 41361 | Double negative law for or... |
| dochsscl 41362 | If a set of vectors is inc... |
| dochoccl 41363 | A set of vectors is closed... |
| dochord 41364 | Ordering law for orthocomp... |
| dochord2N 41365 | Ordering law for orthocomp... |
| dochord3 41366 | Ordering law for orthocomp... |
| doch11 41367 | Orthocomplement is one-to-... |
| dochsordN 41368 | Strict ordering law for or... |
| dochn0nv 41369 | An orthocomplement is nonz... |
| dihoml4c 41370 | Version of ~ dihoml4 with ... |
| dihoml4 41371 | Orthomodular law for const... |
| dochspss 41372 | The span of a set of vecto... |
| dochocsp 41373 | The span of an orthocomple... |
| dochspocN 41374 | The span of an orthocomple... |
| dochocsn 41375 | The double orthocomplement... |
| dochsncom 41376 | Swap vectors in an orthoco... |
| dochsat 41377 | The double orthocomplement... |
| dochshpncl 41378 | If a hyperplane is not clo... |
| dochlkr 41379 | Equivalent conditions for ... |
| dochkrshp 41380 | The closure of a kernel is... |
| dochkrshp2 41381 | Properties of the closure ... |
| dochkrshp3 41382 | Properties of the closure ... |
| dochkrshp4 41383 | Properties of the closure ... |
| dochdmj1 41384 | De Morgan-like law for sub... |
| dochnoncon 41385 | Law of noncontradiction. ... |
| dochnel2 41386 | A nonzero member of a subs... |
| dochnel 41387 | A nonzero vector doesn't b... |
| djhffval 41390 | Subspace join for ` DVecH ... |
| djhfval 41391 | Subspace join for ` DVecH ... |
| djhval 41392 | Subspace join for ` DVecH ... |
| djhval2 41393 | Value of subspace join for... |
| djhcl 41394 | Closure of subspace join f... |
| djhlj 41395 | Transfer lattice join to `... |
| djhljjN 41396 | Lattice join in terms of `... |
| djhjlj 41397 | ` DVecH ` vector space clo... |
| djhj 41398 | ` DVecH ` vector space clo... |
| djhcom 41399 | Subspace join commutes. (... |
| djhspss 41400 | Subspace span of union is ... |
| djhsumss 41401 | Subspace sum is a subset o... |
| dihsumssj 41402 | The subspace sum of two is... |
| djhunssN 41403 | Subspace union is a subset... |
| dochdmm1 41404 | De Morgan-like law for clo... |
| djhexmid 41405 | Excluded middle property o... |
| djh01 41406 | Closed subspace join with ... |
| djh02 41407 | Closed subspace join with ... |
| djhlsmcl 41408 | A closed subspace sum equa... |
| djhcvat42 41409 | A covering property. ( ~ ... |
| dihjatb 41410 | Isomorphism H of lattice j... |
| dihjatc 41411 | Isomorphism H of lattice j... |
| dihjatcclem1 41412 | Lemma for isomorphism H of... |
| dihjatcclem2 41413 | Lemma for isomorphism H of... |
| dihjatcclem3 41414 | Lemma for ~ dihjatcc . (C... |
| dihjatcclem4 41415 | Lemma for isomorphism H of... |
| dihjatcc 41416 | Isomorphism H of lattice j... |
| dihjat 41417 | Isomorphism H of lattice j... |
| dihprrnlem1N 41418 | Lemma for ~ dihprrn , show... |
| dihprrnlem2 41419 | Lemma for ~ dihprrn . (Co... |
| dihprrn 41420 | The span of a vector pair ... |
| djhlsmat 41421 | The sum of two subspace at... |
| dihjat1lem 41422 | Subspace sum of a closed s... |
| dihjat1 41423 | Subspace sum of a closed s... |
| dihsmsprn 41424 | Subspace sum of a closed s... |
| dihjat2 41425 | The subspace sum of a clos... |
| dihjat3 41426 | Isomorphism H of lattice j... |
| dihjat4 41427 | Transfer the subspace sum ... |
| dihjat6 41428 | Transfer the subspace sum ... |
| dihsmsnrn 41429 | The subspace sum of two si... |
| dihsmatrn 41430 | The subspace sum of a clos... |
| dihjat5N 41431 | Transfer lattice join with... |
| dvh4dimat 41432 | There is an atom that is o... |
| dvh3dimatN 41433 | There is an atom that is o... |
| dvh2dimatN 41434 | Given an atom, there exist... |
| dvh1dimat 41435 | There exists an atom. (Co... |
| dvh1dim 41436 | There exists a nonzero vec... |
| dvh4dimlem 41437 | Lemma for ~ dvh4dimN . (C... |
| dvhdimlem 41438 | Lemma for ~ dvh2dim and ~ ... |
| dvh2dim 41439 | There is a vector that is ... |
| dvh3dim 41440 | There is a vector that is ... |
| dvh4dimN 41441 | There is a vector that is ... |
| dvh3dim2 41442 | There is a vector that is ... |
| dvh3dim3N 41443 | There is a vector that is ... |
| dochsnnz 41444 | The orthocomplement of a s... |
| dochsatshp 41445 | The orthocomplement of a s... |
| dochsatshpb 41446 | The orthocomplement of a s... |
| dochsnshp 41447 | The orthocomplement of a n... |
| dochshpsat 41448 | A hyperplane is closed iff... |
| dochkrsat 41449 | The orthocomplement of a k... |
| dochkrsat2 41450 | The orthocomplement of a k... |
| dochsat0 41451 | The orthocomplement of a k... |
| dochkrsm 41452 | The subspace sum of a clos... |
| dochexmidat 41453 | Special case of excluded m... |
| dochexmidlem1 41454 | Lemma for ~ dochexmid . H... |
| dochexmidlem2 41455 | Lemma for ~ dochexmid . (... |
| dochexmidlem3 41456 | Lemma for ~ dochexmid . U... |
| dochexmidlem4 41457 | Lemma for ~ dochexmid . (... |
| dochexmidlem5 41458 | Lemma for ~ dochexmid . (... |
| dochexmidlem6 41459 | Lemma for ~ dochexmid . (... |
| dochexmidlem7 41460 | Lemma for ~ dochexmid . C... |
| dochexmidlem8 41461 | Lemma for ~ dochexmid . T... |
| dochexmid 41462 | Excluded middle law for cl... |
| dochsnkrlem1 41463 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem2 41464 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem3 41465 | Lemma for ~ dochsnkr . (C... |
| dochsnkr 41466 | A (closed) kernel expresse... |
| dochsnkr2 41467 | Kernel of the explicit fun... |
| dochsnkr2cl 41468 | The ` X ` determining func... |
| dochflcl 41469 | Closure of the explicit fu... |
| dochfl1 41470 | The value of the explicit ... |
| dochfln0 41471 | The value of a functional ... |
| dochkr1 41472 | A nonzero functional has a... |
| dochkr1OLDN 41473 | A nonzero functional has a... |
| lpolsetN 41476 | The set of polarities of a... |
| islpolN 41477 | The predicate "is a polari... |
| islpoldN 41478 | Properties that determine ... |
| lpolfN 41479 | Functionality of a polarit... |
| lpolvN 41480 | The polarity of the whole ... |
| lpolconN 41481 | Contraposition property of... |
| lpolsatN 41482 | The polarity of an atomic ... |
| lpolpolsatN 41483 | Property of a polarity. (... |
| dochpolN 41484 | The subspace orthocompleme... |
| lcfl1lem 41485 | Property of a functional w... |
| lcfl1 41486 | Property of a functional w... |
| lcfl2 41487 | Property of a functional w... |
| lcfl3 41488 | Property of a functional w... |
| lcfl4N 41489 | Property of a functional w... |
| lcfl5 41490 | Property of a functional w... |
| lcfl5a 41491 | Property of a functional w... |
| lcfl6lem 41492 | Lemma for ~ lcfl6 . A fun... |
| lcfl7lem 41493 | Lemma for ~ lcfl7N . If t... |
| lcfl6 41494 | Property of a functional w... |
| lcfl7N 41495 | Property of a functional w... |
| lcfl8 41496 | Property of a functional w... |
| lcfl8a 41497 | Property of a functional w... |
| lcfl8b 41498 | Property of a nonzero func... |
| lcfl9a 41499 | Property implying that a f... |
| lclkrlem1 41500 | The set of functionals hav... |
| lclkrlem2a 41501 | Lemma for ~ lclkr . Use ~... |
| lclkrlem2b 41502 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2c 41503 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2d 41504 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2e 41505 | Lemma for ~ lclkr . The k... |
| lclkrlem2f 41506 | Lemma for ~ lclkr . Const... |
| lclkrlem2g 41507 | Lemma for ~ lclkr . Compa... |
| lclkrlem2h 41508 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2i 41509 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2j 41510 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2k 41511 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2l 41512 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2m 41513 | Lemma for ~ lclkr . Const... |
| lclkrlem2n 41514 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2o 41515 | Lemma for ~ lclkr . When ... |
| lclkrlem2p 41516 | Lemma for ~ lclkr . When ... |
| lclkrlem2q 41517 | Lemma for ~ lclkr . The s... |
| lclkrlem2r 41518 | Lemma for ~ lclkr . When ... |
| lclkrlem2s 41519 | Lemma for ~ lclkr . Thus,... |
| lclkrlem2t 41520 | Lemma for ~ lclkr . We el... |
| lclkrlem2u 41521 | Lemma for ~ lclkr . ~ lclk... |
| lclkrlem2v 41522 | Lemma for ~ lclkr . When ... |
| lclkrlem2w 41523 | Lemma for ~ lclkr . This ... |
| lclkrlem2x 41524 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2y 41525 | Lemma for ~ lclkr . Resta... |
| lclkrlem2 41526 | The set of functionals hav... |
| lclkr 41527 | The set of functionals wit... |
| lcfls1lem 41528 | Property of a functional w... |
| lcfls1N 41529 | Property of a functional w... |
| lcfls1c 41530 | Property of a functional w... |
| lclkrslem1 41531 | The set of functionals hav... |
| lclkrslem2 41532 | The set of functionals hav... |
| lclkrs 41533 | The set of functionals hav... |
| lclkrs2 41534 | The set of functionals wit... |
| lcfrvalsnN 41535 | Reconstruction from the du... |
| lcfrlem1 41536 | Lemma for ~ lcfr . Note t... |
| lcfrlem2 41537 | Lemma for ~ lcfr . (Contr... |
| lcfrlem3 41538 | Lemma for ~ lcfr . (Contr... |
| lcfrlem4 41539 | Lemma for ~ lcfr . (Contr... |
| lcfrlem5 41540 | Lemma for ~ lcfr . The se... |
| lcfrlem6 41541 | Lemma for ~ lcfr . Closur... |
| lcfrlem7 41542 | Lemma for ~ lcfr . Closur... |
| lcfrlem8 41543 | Lemma for ~ lcf1o and ~ lc... |
| lcfrlem9 41544 | Lemma for ~ lcf1o . (This... |
| lcf1o 41545 | Define a function ` J ` th... |
| lcfrlem10 41546 | Lemma for ~ lcfr . (Contr... |
| lcfrlem11 41547 | Lemma for ~ lcfr . (Contr... |
| lcfrlem12N 41548 | Lemma for ~ lcfr . (Contr... |
| lcfrlem13 41549 | Lemma for ~ lcfr . (Contr... |
| lcfrlem14 41550 | Lemma for ~ lcfr . (Contr... |
| lcfrlem15 41551 | Lemma for ~ lcfr . (Contr... |
| lcfrlem16 41552 | Lemma for ~ lcfr . (Contr... |
| lcfrlem17 41553 | Lemma for ~ lcfr . Condit... |
| lcfrlem18 41554 | Lemma for ~ lcfr . (Contr... |
| lcfrlem19 41555 | Lemma for ~ lcfr . (Contr... |
| lcfrlem20 41556 | Lemma for ~ lcfr . (Contr... |
| lcfrlem21 41557 | Lemma for ~ lcfr . (Contr... |
| lcfrlem22 41558 | Lemma for ~ lcfr . (Contr... |
| lcfrlem23 41559 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem24 41560 | Lemma for ~ lcfr . (Contr... |
| lcfrlem25 41561 | Lemma for ~ lcfr . Specia... |
| lcfrlem26 41562 | Lemma for ~ lcfr . Specia... |
| lcfrlem27 41563 | Lemma for ~ lcfr . Specia... |
| lcfrlem28 41564 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem29 41565 | Lemma for ~ lcfr . (Contr... |
| lcfrlem30 41566 | Lemma for ~ lcfr . (Contr... |
| lcfrlem31 41567 | Lemma for ~ lcfr . (Contr... |
| lcfrlem32 41568 | Lemma for ~ lcfr . (Contr... |
| lcfrlem33 41569 | Lemma for ~ lcfr . (Contr... |
| lcfrlem34 41570 | Lemma for ~ lcfr . (Contr... |
| lcfrlem35 41571 | Lemma for ~ lcfr . (Contr... |
| lcfrlem36 41572 | Lemma for ~ lcfr . (Contr... |
| lcfrlem37 41573 | Lemma for ~ lcfr . (Contr... |
| lcfrlem38 41574 | Lemma for ~ lcfr . Combin... |
| lcfrlem39 41575 | Lemma for ~ lcfr . Elimin... |
| lcfrlem40 41576 | Lemma for ~ lcfr . Elimin... |
| lcfrlem41 41577 | Lemma for ~ lcfr . Elimin... |
| lcfrlem42 41578 | Lemma for ~ lcfr . Elimin... |
| lcfr 41579 | Reconstruction of a subspa... |
| lcdfval 41582 | Dual vector space of funct... |
| lcdval 41583 | Dual vector space of funct... |
| lcdval2 41584 | Dual vector space of funct... |
| lcdlvec 41585 | The dual vector space of f... |
| lcdlmod 41586 | The dual vector space of f... |
| lcdvbase 41587 | Vector base set of a dual ... |
| lcdvbasess 41588 | The vector base set of the... |
| lcdvbaselfl 41589 | A vector in the base set o... |
| lcdvbasecl 41590 | Closure of the value of a ... |
| lcdvadd 41591 | Vector addition for the cl... |
| lcdvaddval 41592 | The value of the value of ... |
| lcdsca 41593 | The ring of scalars of the... |
| lcdsbase 41594 | Base set of scalar ring fo... |
| lcdsadd 41595 | Scalar addition for the cl... |
| lcdsmul 41596 | Scalar multiplication for ... |
| lcdvs 41597 | Scalar product for the clo... |
| lcdvsval 41598 | Value of scalar product op... |
| lcdvscl 41599 | The scalar product operati... |
| lcdlssvscl 41600 | Closure of scalar product ... |
| lcdvsass 41601 | Associative law for scalar... |
| lcd0 41602 | The zero scalar of the clo... |
| lcd1 41603 | The unit scalar of the clo... |
| lcdneg 41604 | The unit scalar of the clo... |
| lcd0v 41605 | The zero functional in the... |
| lcd0v2 41606 | The zero functional in the... |
| lcd0vvalN 41607 | Value of the zero function... |
| lcd0vcl 41608 | Closure of the zero functi... |
| lcd0vs 41609 | A scalar zero times a func... |
| lcdvs0N 41610 | A scalar times the zero fu... |
| lcdvsub 41611 | The value of vector subtra... |
| lcdvsubval 41612 | The value of the value of ... |
| lcdlss 41613 | Subspaces of a dual vector... |
| lcdlss2N 41614 | Subspaces of a dual vector... |
| lcdlsp 41615 | Span in the set of functio... |
| lcdlkreqN 41616 | Colinear functionals have ... |
| lcdlkreq2N 41617 | Colinear functionals have ... |
| mapdffval 41620 | Projectivity from vector s... |
| mapdfval 41621 | Projectivity from vector s... |
| mapdval 41622 | Value of projectivity from... |
| mapdvalc 41623 | Value of projectivity from... |
| mapdval2N 41624 | Value of projectivity from... |
| mapdval3N 41625 | Value of projectivity from... |
| mapdval4N 41626 | Value of projectivity from... |
| mapdval5N 41627 | Value of projectivity from... |
| mapdordlem1a 41628 | Lemma for ~ mapdord . (Co... |
| mapdordlem1bN 41629 | Lemma for ~ mapdord . (Co... |
| mapdordlem1 41630 | Lemma for ~ mapdord . (Co... |
| mapdordlem2 41631 | Lemma for ~ mapdord . Ord... |
| mapdord 41632 | Ordering property of the m... |
| mapd11 41633 | The map defined by ~ df-ma... |
| mapddlssN 41634 | The mapping of a subspace ... |
| mapdsn 41635 | Value of the map defined b... |
| mapdsn2 41636 | Value of the map defined b... |
| mapdsn3 41637 | Value of the map defined b... |
| mapd1dim2lem1N 41638 | Value of the map defined b... |
| mapdrvallem2 41639 | Lemma for ~ mapdrval . TO... |
| mapdrvallem3 41640 | Lemma for ~ mapdrval . (C... |
| mapdrval 41641 | Given a dual subspace ` R ... |
| mapd1o 41642 | The map defined by ~ df-ma... |
| mapdrn 41643 | Range of the map defined b... |
| mapdunirnN 41644 | Union of the range of the ... |
| mapdrn2 41645 | Range of the map defined b... |
| mapdcnvcl 41646 | Closure of the converse of... |
| mapdcl 41647 | Closure the value of the m... |
| mapdcnvid1N 41648 | Converse of the value of t... |
| mapdsord 41649 | Strong ordering property o... |
| mapdcl2 41650 | The mapping of a subspace ... |
| mapdcnvid2 41651 | Value of the converse of t... |
| mapdcnvordN 41652 | Ordering property of the c... |
| mapdcnv11N 41653 | The converse of the map de... |
| mapdcv 41654 | Covering property of the c... |
| mapdincl 41655 | Closure of dual subspace i... |
| mapdin 41656 | Subspace intersection is p... |
| mapdlsmcl 41657 | Closure of dual subspace s... |
| mapdlsm 41658 | Subspace sum is preserved ... |
| mapd0 41659 | Projectivity map of the ze... |
| mapdcnvatN 41660 | Atoms are preserved by the... |
| mapdat 41661 | Atoms are preserved by the... |
| mapdspex 41662 | The map of a span equals t... |
| mapdn0 41663 | Transfer nonzero property ... |
| mapdncol 41664 | Transfer non-colinearity f... |
| mapdindp 41665 | Transfer (part of) vector ... |
| mapdpglem1 41666 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2 41667 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2a 41668 | Lemma for ~ mapdpg . (Con... |
| mapdpglem3 41669 | Lemma for ~ mapdpg . Baer... |
| mapdpglem4N 41670 | Lemma for ~ mapdpg . (Con... |
| mapdpglem5N 41671 | Lemma for ~ mapdpg . (Con... |
| mapdpglem6 41672 | Lemma for ~ mapdpg . Baer... |
| mapdpglem8 41673 | Lemma for ~ mapdpg . Baer... |
| mapdpglem9 41674 | Lemma for ~ mapdpg . Baer... |
| mapdpglem10 41675 | Lemma for ~ mapdpg . Baer... |
| mapdpglem11 41676 | Lemma for ~ mapdpg . (Con... |
| mapdpglem12 41677 | Lemma for ~ mapdpg . TODO... |
| mapdpglem13 41678 | Lemma for ~ mapdpg . (Con... |
| mapdpglem14 41679 | Lemma for ~ mapdpg . (Con... |
| mapdpglem15 41680 | Lemma for ~ mapdpg . (Con... |
| mapdpglem16 41681 | Lemma for ~ mapdpg . Baer... |
| mapdpglem17N 41682 | Lemma for ~ mapdpg . Baer... |
| mapdpglem18 41683 | Lemma for ~ mapdpg . Baer... |
| mapdpglem19 41684 | Lemma for ~ mapdpg . Baer... |
| mapdpglem20 41685 | Lemma for ~ mapdpg . Baer... |
| mapdpglem21 41686 | Lemma for ~ mapdpg . (Con... |
| mapdpglem22 41687 | Lemma for ~ mapdpg . Baer... |
| mapdpglem23 41688 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30a 41689 | Lemma for ~ mapdpg . (Con... |
| mapdpglem30b 41690 | Lemma for ~ mapdpg . (Con... |
| mapdpglem25 41691 | Lemma for ~ mapdpg . Baer... |
| mapdpglem26 41692 | Lemma for ~ mapdpg . Baer... |
| mapdpglem27 41693 | Lemma for ~ mapdpg . Baer... |
| mapdpglem29 41694 | Lemma for ~ mapdpg . Baer... |
| mapdpglem28 41695 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30 41696 | Lemma for ~ mapdpg . Baer... |
| mapdpglem31 41697 | Lemma for ~ mapdpg . Baer... |
| mapdpglem24 41698 | Lemma for ~ mapdpg . Exis... |
| mapdpglem32 41699 | Lemma for ~ mapdpg . Uniq... |
| mapdpg 41700 | Part 1 of proof of the fir... |
| baerlem3lem1 41701 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem1 41702 | Lemma for ~ baerlem5a . (... |
| baerlem5blem1 41703 | Lemma for ~ baerlem5b . (... |
| baerlem3lem2 41704 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem2 41705 | Lemma for ~ baerlem5a . (... |
| baerlem5blem2 41706 | Lemma for ~ baerlem5b . (... |
| baerlem3 41707 | An equality that holds whe... |
| baerlem5a 41708 | An equality that holds whe... |
| baerlem5b 41709 | An equality that holds whe... |
| baerlem5amN 41710 | An equality that holds whe... |
| baerlem5bmN 41711 | An equality that holds whe... |
| baerlem5abmN 41712 | An equality that holds whe... |
| mapdindp0 41713 | Vector independence lemma.... |
| mapdindp1 41714 | Vector independence lemma.... |
| mapdindp2 41715 | Vector independence lemma.... |
| mapdindp3 41716 | Vector independence lemma.... |
| mapdindp4 41717 | Vector independence lemma.... |
| mapdhval 41718 | Lemmma for ~~? mapdh . (C... |
| mapdhval0 41719 | Lemmma for ~~? mapdh . (C... |
| mapdhval2 41720 | Lemmma for ~~? mapdh . (C... |
| mapdhcl 41721 | Lemmma for ~~? mapdh . (C... |
| mapdheq 41722 | Lemmma for ~~? mapdh . Th... |
| mapdheq2 41723 | Lemmma for ~~? mapdh . On... |
| mapdheq2biN 41724 | Lemmma for ~~? mapdh . Pa... |
| mapdheq4lem 41725 | Lemma for ~ mapdheq4 . Pa... |
| mapdheq4 41726 | Lemma for ~~? mapdh . Par... |
| mapdh6lem1N 41727 | Lemma for ~ mapdh6N . Par... |
| mapdh6lem2N 41728 | Lemma for ~ mapdh6N . Par... |
| mapdh6aN 41729 | Lemma for ~ mapdh6N . Par... |
| mapdh6b0N 41730 | Lemmma for ~ mapdh6N . (C... |
| mapdh6bN 41731 | Lemmma for ~ mapdh6N . (C... |
| mapdh6cN 41732 | Lemmma for ~ mapdh6N . (C... |
| mapdh6dN 41733 | Lemmma for ~ mapdh6N . (C... |
| mapdh6eN 41734 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6fN 41735 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6gN 41736 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6hN 41737 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6iN 41738 | Lemmma for ~ mapdh6N . El... |
| mapdh6jN 41739 | Lemmma for ~ mapdh6N . El... |
| mapdh6kN 41740 | Lemmma for ~ mapdh6N . El... |
| mapdh6N 41741 | Part (6) of [Baer] p. 47 l... |
| mapdh7eN 41742 | Part (7) of [Baer] p. 48 l... |
| mapdh7cN 41743 | Part (7) of [Baer] p. 48 l... |
| mapdh7dN 41744 | Part (7) of [Baer] p. 48 l... |
| mapdh7fN 41745 | Part (7) of [Baer] p. 48 l... |
| mapdh75e 41746 | Part (7) of [Baer] p. 48 l... |
| mapdh75cN 41747 | Part (7) of [Baer] p. 48 l... |
| mapdh75d 41748 | Part (7) of [Baer] p. 48 l... |
| mapdh75fN 41749 | Part (7) of [Baer] p. 48 l... |
| hvmapffval 41752 | Map from nonzero vectors t... |
| hvmapfval 41753 | Map from nonzero vectors t... |
| hvmapval 41754 | Value of map from nonzero ... |
| hvmapvalvalN 41755 | Value of value of map (i.e... |
| hvmapidN 41756 | The value of the vector to... |
| hvmap1o 41757 | The vector to functional m... |
| hvmapclN 41758 | Closure of the vector to f... |
| hvmap1o2 41759 | The vector to functional m... |
| hvmapcl2 41760 | Closure of the vector to f... |
| hvmaplfl 41761 | The vector to functional m... |
| hvmaplkr 41762 | Kernel of the vector to fu... |
| mapdhvmap 41763 | Relationship between ` map... |
| lspindp5 41764 | Obtain an independent vect... |
| hdmaplem1 41765 | Lemma to convert a frequen... |
| hdmaplem2N 41766 | Lemma to convert a frequen... |
| hdmaplem3 41767 | Lemma to convert a frequen... |
| hdmaplem4 41768 | Lemma to convert a frequen... |
| mapdh8a 41769 | Part of Part (8) in [Baer]... |
| mapdh8aa 41770 | Part of Part (8) in [Baer]... |
| mapdh8ab 41771 | Part of Part (8) in [Baer]... |
| mapdh8ac 41772 | Part of Part (8) in [Baer]... |
| mapdh8ad 41773 | Part of Part (8) in [Baer]... |
| mapdh8b 41774 | Part of Part (8) in [Baer]... |
| mapdh8c 41775 | Part of Part (8) in [Baer]... |
| mapdh8d0N 41776 | Part of Part (8) in [Baer]... |
| mapdh8d 41777 | Part of Part (8) in [Baer]... |
| mapdh8e 41778 | Part of Part (8) in [Baer]... |
| mapdh8g 41779 | Part of Part (8) in [Baer]... |
| mapdh8i 41780 | Part of Part (8) in [Baer]... |
| mapdh8j 41781 | Part of Part (8) in [Baer]... |
| mapdh8 41782 | Part (8) in [Baer] p. 48. ... |
| mapdh9a 41783 | Lemma for part (9) in [Bae... |
| mapdh9aOLDN 41784 | Lemma for part (9) in [Bae... |
| hdmap1ffval 41789 | Preliminary map from vecto... |
| hdmap1fval 41790 | Preliminary map from vecto... |
| hdmap1vallem 41791 | Value of preliminary map f... |
| hdmap1val 41792 | Value of preliminary map f... |
| hdmap1val0 41793 | Value of preliminary map f... |
| hdmap1val2 41794 | Value of preliminary map f... |
| hdmap1eq 41795 | The defining equation for ... |
| hdmap1cbv 41796 | Frequently used lemma to c... |
| hdmap1valc 41797 | Connect the value of the p... |
| hdmap1cl 41798 | Convert closure theorem ~ ... |
| hdmap1eq2 41799 | Convert ~ mapdheq2 to use ... |
| hdmap1eq4N 41800 | Convert ~ mapdheq4 to use ... |
| hdmap1l6lem1 41801 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6lem2 41802 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6a 41803 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6b0N 41804 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6b 41805 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6c 41806 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6d 41807 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6e 41808 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6f 41809 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6g 41810 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6h 41811 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6i 41812 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6j 41813 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6k 41814 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6 41815 | Part (6) of [Baer] p. 47 l... |
| hdmap1eulem 41816 | Lemma for ~ hdmap1eu . TO... |
| hdmap1eulemOLDN 41817 | Lemma for ~ hdmap1euOLDN .... |
| hdmap1eu 41818 | Convert ~ mapdh9a to use t... |
| hdmap1euOLDN 41819 | Convert ~ mapdh9aOLDN to u... |
| hdmapffval 41820 | Map from vectors to functi... |
| hdmapfval 41821 | Map from vectors to functi... |
| hdmapval 41822 | Value of map from vectors ... |
| hdmapfnN 41823 | Functionality of map from ... |
| hdmapcl 41824 | Closure of map from vector... |
| hdmapval2lem 41825 | Lemma for ~ hdmapval2 . (... |
| hdmapval2 41826 | Value of map from vectors ... |
| hdmapval0 41827 | Value of map from vectors ... |
| hdmapeveclem 41828 | Lemma for ~ hdmapevec . T... |
| hdmapevec 41829 | Value of map from vectors ... |
| hdmapevec2 41830 | The inner product of the r... |
| hdmapval3lemN 41831 | Value of map from vectors ... |
| hdmapval3N 41832 | Value of map from vectors ... |
| hdmap10lem 41833 | Lemma for ~ hdmap10 . (Co... |
| hdmap10 41834 | Part 10 in [Baer] p. 48 li... |
| hdmap11lem1 41835 | Lemma for ~ hdmapadd . (C... |
| hdmap11lem2 41836 | Lemma for ~ hdmapadd . (C... |
| hdmapadd 41837 | Part 11 in [Baer] p. 48 li... |
| hdmapeq0 41838 | Part of proof of part 12 i... |
| hdmapnzcl 41839 | Nonzero vector closure of ... |
| hdmapneg 41840 | Part of proof of part 12 i... |
| hdmapsub 41841 | Part of proof of part 12 i... |
| hdmap11 41842 | Part of proof of part 12 i... |
| hdmaprnlem1N 41843 | Part of proof of part 12 i... |
| hdmaprnlem3N 41844 | Part of proof of part 12 i... |
| hdmaprnlem3uN 41845 | Part of proof of part 12 i... |
| hdmaprnlem4tN 41846 | Lemma for ~ hdmaprnN . TO... |
| hdmaprnlem4N 41847 | Part of proof of part 12 i... |
| hdmaprnlem6N 41848 | Part of proof of part 12 i... |
| hdmaprnlem7N 41849 | Part of proof of part 12 i... |
| hdmaprnlem8N 41850 | Part of proof of part 12 i... |
| hdmaprnlem9N 41851 | Part of proof of part 12 i... |
| hdmaprnlem3eN 41852 | Lemma for ~ hdmaprnN . (C... |
| hdmaprnlem10N 41853 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem11N 41854 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem15N 41855 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem16N 41856 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem17N 41857 | Lemma for ~ hdmaprnN . In... |
| hdmaprnN 41858 | Part of proof of part 12 i... |
| hdmapf1oN 41859 | Part 12 in [Baer] p. 49. ... |
| hdmap14lem1a 41860 | Prior to part 14 in [Baer]... |
| hdmap14lem2a 41861 | Prior to part 14 in [Baer]... |
| hdmap14lem1 41862 | Prior to part 14 in [Baer]... |
| hdmap14lem2N 41863 | Prior to part 14 in [Baer]... |
| hdmap14lem3 41864 | Prior to part 14 in [Baer]... |
| hdmap14lem4a 41865 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem4 41866 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem6 41867 | Case where ` F ` is zero. ... |
| hdmap14lem7 41868 | Combine cases of ` F ` . ... |
| hdmap14lem8 41869 | Part of proof of part 14 i... |
| hdmap14lem9 41870 | Part of proof of part 14 i... |
| hdmap14lem10 41871 | Part of proof of part 14 i... |
| hdmap14lem11 41872 | Part of proof of part 14 i... |
| hdmap14lem12 41873 | Lemma for proof of part 14... |
| hdmap14lem13 41874 | Lemma for proof of part 14... |
| hdmap14lem14 41875 | Part of proof of part 14 i... |
| hdmap14lem15 41876 | Part of proof of part 14 i... |
| hgmapffval 41879 | Map from the scalar divisi... |
| hgmapfval 41880 | Map from the scalar divisi... |
| hgmapval 41881 | Value of map from the scal... |
| hgmapfnN 41882 | Functionality of scalar si... |
| hgmapcl 41883 | Closure of scalar sigma ma... |
| hgmapdcl 41884 | Closure of the vector spac... |
| hgmapvs 41885 | Part 15 of [Baer] p. 50 li... |
| hgmapval0 41886 | Value of the scalar sigma ... |
| hgmapval1 41887 | Value of the scalar sigma ... |
| hgmapadd 41888 | Part 15 of [Baer] p. 50 li... |
| hgmapmul 41889 | Part 15 of [Baer] p. 50 li... |
| hgmaprnlem1N 41890 | Lemma for ~ hgmaprnN . (C... |
| hgmaprnlem2N 41891 | Lemma for ~ hgmaprnN . Pa... |
| hgmaprnlem3N 41892 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem4N 41893 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem5N 41894 | Lemma for ~ hgmaprnN . El... |
| hgmaprnN 41895 | Part of proof of part 16 i... |
| hgmap11 41896 | The scalar sigma map is on... |
| hgmapf1oN 41897 | The scalar sigma map is a ... |
| hgmapeq0 41898 | The scalar sigma map is ze... |
| hdmapipcl 41899 | The inner product (Hermiti... |
| hdmapln1 41900 | Linearity property that wi... |
| hdmaplna1 41901 | Additive property of first... |
| hdmaplns1 41902 | Subtraction property of fi... |
| hdmaplnm1 41903 | Multiplicative property of... |
| hdmaplna2 41904 | Additive property of secon... |
| hdmapglnm2 41905 | g-linear property of secon... |
| hdmapgln2 41906 | g-linear property that wil... |
| hdmaplkr 41907 | Kernel of the vector to du... |
| hdmapellkr 41908 | Membership in the kernel (... |
| hdmapip0 41909 | Zero property that will be... |
| hdmapip1 41910 | Construct a proportional v... |
| hdmapip0com 41911 | Commutation property of Ba... |
| hdmapinvlem1 41912 | Line 27 in [Baer] p. 110. ... |
| hdmapinvlem2 41913 | Line 28 in [Baer] p. 110, ... |
| hdmapinvlem3 41914 | Line 30 in [Baer] p. 110, ... |
| hdmapinvlem4 41915 | Part 1.1 of Proposition 1 ... |
| hdmapglem5 41916 | Part 1.2 in [Baer] p. 110 ... |
| hgmapvvlem1 41917 | Involution property of sca... |
| hgmapvvlem2 41918 | Lemma for ~ hgmapvv . Eli... |
| hgmapvvlem3 41919 | Lemma for ~ hgmapvv . Eli... |
| hgmapvv 41920 | Value of a double involuti... |
| hdmapglem7a 41921 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7b 41922 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7 41923 | Lemma for ~ hdmapg . Line... |
| hdmapg 41924 | Apply the scalar sigma fun... |
| hdmapoc 41925 | Express our constructed or... |
| hlhilset 41928 | The final Hilbert space co... |
| hlhilsca 41929 | The scalar of the final co... |
| hlhilbase 41930 | The base set of the final ... |
| hlhilplus 41931 | The vector addition for th... |
| hlhilslem 41932 | Lemma for ~ hlhilsbase etc... |
| hlhilsbase 41933 | The scalar base set of the... |
| hlhilsplus 41934 | Scalar addition for the fi... |
| hlhilsmul 41935 | Scalar multiplication for ... |
| hlhilsbase2 41936 | The scalar base set of the... |
| hlhilsplus2 41937 | Scalar addition for the fi... |
| hlhilsmul2 41938 | Scalar multiplication for ... |
| hlhils0 41939 | The scalar ring zero for t... |
| hlhils1N 41940 | The scalar ring unity for ... |
| hlhilvsca 41941 | The scalar product for the... |
| hlhilip 41942 | Inner product operation fo... |
| hlhilipval 41943 | Value of inner product ope... |
| hlhilnvl 41944 | The involution operation o... |
| hlhillvec 41945 | The final constructed Hilb... |
| hlhildrng 41946 | The star division ring for... |
| hlhilsrnglem 41947 | Lemma for ~ hlhilsrng . (... |
| hlhilsrng 41948 | The star division ring for... |
| hlhil0 41949 | The zero vector for the fi... |
| hlhillsm 41950 | The vector sum operation f... |
| hlhilocv 41951 | The orthocomplement for th... |
| hlhillcs 41952 | The closed subspaces of th... |
| hlhilphllem 41953 | Lemma for ~ hlhil . (Cont... |
| hlhilhillem 41954 | Lemma for ~ hlhil . (Cont... |
| hlathil 41955 | Construction of a Hilbert ... |
| iscsrg 41958 | A commutative semiring is ... |
| rhmzrhval 41959 | Evaluation of integers acr... |
| zndvdchrrhm 41960 | Construction of a ring hom... |
| relogbcld 41961 | Closure of the general log... |
| relogbexpd 41962 | Identity law for general l... |
| relogbzexpd 41963 | Power law for the general ... |
| logblebd 41964 | The general logarithm is m... |
| uzindd 41965 | Induction on the upper int... |
| fzadd2d 41966 | Membership of a sum in a f... |
| zltp1led 41967 | Integer ordering relation,... |
| fzne2d 41968 | Elementhood in a finite se... |
| eqfnfv2d2 41969 | Equality of functions is d... |
| fzsplitnd 41970 | Split a finite interval of... |
| fzsplitnr 41971 | Split a finite interval of... |
| addassnni 41972 | Associative law for additi... |
| addcomnni 41973 | Commutative law for additi... |
| mulassnni 41974 | Associative law for multip... |
| mulcomnni 41975 | Commutative law for multip... |
| gcdcomnni 41976 | Commutative law for gcd. ... |
| gcdnegnni 41977 | Negation invariance for gc... |
| neggcdnni 41978 | Negation invariance for gc... |
| bccl2d 41979 | Closure of the binomial co... |
| recbothd 41980 | Take reciprocal on both si... |
| gcdmultiplei 41981 | The GCD of a multiple of a... |
| gcdaddmzz2nni 41982 | Adding a multiple of one o... |
| gcdaddmzz2nncomi 41983 | Adding a multiple of one o... |
| gcdnncli 41984 | Closure of the gcd operato... |
| muldvds1d 41985 | If a product divides an in... |
| muldvds2d 41986 | If a product divides an in... |
| nndivdvdsd 41987 | A positive integer divides... |
| nnproddivdvdsd 41988 | A product of natural numbe... |
| coprmdvds2d 41989 | If an integer is divisible... |
| imadomfi 41990 | An image of a function und... |
| 12gcd5e1 41991 | The gcd of 12 and 5 is 1. ... |
| 60gcd6e6 41992 | The gcd of 60 and 6 is 6. ... |
| 60gcd7e1 41993 | The gcd of 60 and 7 is 1. ... |
| 420gcd8e4 41994 | The gcd of 420 and 8 is 4.... |
| lcmeprodgcdi 41995 | Calculate the least common... |
| 12lcm5e60 41996 | The lcm of 12 and 5 is 60.... |
| 60lcm6e60 41997 | The lcm of 60 and 6 is 60.... |
| 60lcm7e420 41998 | The lcm of 60 and 7 is 420... |
| 420lcm8e840 41999 | The lcm of 420 and 8 is 84... |
| lcmfunnnd 42000 | Useful equation to calcula... |
| lcm1un 42001 | Least common multiple of n... |
| lcm2un 42002 | Least common multiple of n... |
| lcm3un 42003 | Least common multiple of n... |
| lcm4un 42004 | Least common multiple of n... |
| lcm5un 42005 | Least common multiple of n... |
| lcm6un 42006 | Least common multiple of n... |
| lcm7un 42007 | Least common multiple of n... |
| lcm8un 42008 | Least common multiple of n... |
| 3factsumint1 42009 | Move constants out of inte... |
| 3factsumint2 42010 | Move constants out of inte... |
| 3factsumint3 42011 | Move constants out of inte... |
| 3factsumint4 42012 | Move constants out of inte... |
| 3factsumint 42013 | Helpful equation for lcm i... |
| resopunitintvd 42014 | Restrict continuous functi... |
| resclunitintvd 42015 | Restrict continuous functi... |
| resdvopclptsd 42016 | Restrict derivative on uni... |
| lcmineqlem1 42017 | Part of lcm inequality lem... |
| lcmineqlem2 42018 | Part of lcm inequality lem... |
| lcmineqlem3 42019 | Part of lcm inequality lem... |
| lcmineqlem4 42020 | Part of lcm inequality lem... |
| lcmineqlem5 42021 | Technical lemma for recipr... |
| lcmineqlem6 42022 | Part of lcm inequality lem... |
| lcmineqlem7 42023 | Derivative of 1-x for chai... |
| lcmineqlem8 42024 | Derivative of (1-x)^(N-M).... |
| lcmineqlem9 42025 | (1-x)^(N-M) is continuous.... |
| lcmineqlem10 42026 | Induction step of ~ lcmine... |
| lcmineqlem11 42027 | Induction step, continuati... |
| lcmineqlem12 42028 | Base case for induction. ... |
| lcmineqlem13 42029 | Induction proof for lcm in... |
| lcmineqlem14 42030 | Technical lemma for inequa... |
| lcmineqlem15 42031 | F times the least common m... |
| lcmineqlem16 42032 | Technical divisibility lem... |
| lcmineqlem17 42033 | Inequality of 2^{2n}. (Co... |
| lcmineqlem18 42034 | Technical lemma to shift f... |
| lcmineqlem19 42035 | Dividing implies inequalit... |
| lcmineqlem20 42036 | Inequality for lcm lemma. ... |
| lcmineqlem21 42037 | The lcm inequality lemma w... |
| lcmineqlem22 42038 | The lcm inequality lemma w... |
| lcmineqlem23 42039 | Penultimate step to the lc... |
| lcmineqlem 42040 | The least common multiple ... |
| 3exp7 42041 | 3 to the power of 7 equals... |
| 3lexlogpow5ineq1 42042 | First inequality in inequa... |
| 3lexlogpow5ineq2 42043 | Second inequality in inequ... |
| 3lexlogpow5ineq4 42044 | Sharper logarithm inequali... |
| 3lexlogpow5ineq3 42045 | Combined inequality chain ... |
| 3lexlogpow2ineq1 42046 | Result for bound in AKS in... |
| 3lexlogpow2ineq2 42047 | Result for bound in AKS in... |
| 3lexlogpow5ineq5 42048 | Result for bound in AKS in... |
| intlewftc 42049 | Inequality inference by in... |
| aks4d1lem1 42050 | Technical lemma to reduce ... |
| aks4d1p1p1 42051 | Exponential law for finite... |
| dvrelog2 42052 | The derivative of the loga... |
| dvrelog3 42053 | The derivative of the loga... |
| dvrelog2b 42054 | Derivative of the binary l... |
| 0nonelalab 42055 | Technical lemma for open i... |
| dvrelogpow2b 42056 | Derivative of the power of... |
| aks4d1p1p3 42057 | Bound of a ceiling of the ... |
| aks4d1p1p2 42058 | Rewrite ` A ` in more suit... |
| aks4d1p1p4 42059 | Technical step for inequal... |
| dvle2 42060 | Collapsed ~ dvle . (Contr... |
| aks4d1p1p6 42061 | Inequality lift to differe... |
| aks4d1p1p7 42062 | Bound of intermediary of i... |
| aks4d1p1p5 42063 | Show inequality for existe... |
| aks4d1p1 42064 | Show inequality for existe... |
| aks4d1p2 42065 | Technical lemma for existe... |
| aks4d1p3 42066 | There exists a small enoug... |
| aks4d1p4 42067 | There exists a small enoug... |
| aks4d1p5 42068 | Show that ` N ` and ` R ` ... |
| aks4d1p6 42069 | The maximal prime power ex... |
| aks4d1p7d1 42070 | Technical step in AKS lemm... |
| aks4d1p7 42071 | Technical step in AKS lemm... |
| aks4d1p8d1 42072 | If a prime divides one num... |
| aks4d1p8d2 42073 | Any prime power dividing a... |
| aks4d1p8d3 42074 | The remainder of a divisio... |
| aks4d1p8 42075 | Show that ` N ` and ` R ` ... |
| aks4d1p9 42076 | Show that the order is bou... |
| aks4d1 42077 | Lemma 4.1 from ~ https://w... |
| fldhmf1 42078 | A field homomorphism is in... |
| isprimroot 42081 | The value of a primitive r... |
| isprimroot2 42082 | Alternative way of creatin... |
| mndmolinv 42083 | An element of a monoid tha... |
| linvh 42084 | If an element has a unique... |
| primrootsunit1 42085 | Primitive roots have left ... |
| primrootsunit 42086 | Primitive roots have left ... |
| primrootscoprmpow 42087 | Coprime powers of primitiv... |
| posbezout 42088 | Bezout's identity restrict... |
| primrootscoprf 42089 | Coprime powers of primitiv... |
| primrootscoprbij 42090 | A bijection between coprim... |
| primrootscoprbij2 42091 | A bijection between coprim... |
| remexz 42092 | Division with rest. (Cont... |
| primrootlekpowne0 42093 | There is no smaller power ... |
| primrootspoweq0 42094 | The power of a ` R ` -th p... |
| aks6d1c1p1 42095 | Definition of the introspe... |
| aks6d1c1p1rcl 42096 | Reverse closure of the int... |
| aks6d1c1p2 42097 | ` P ` and linear factors a... |
| aks6d1c1p3 42098 | In a field with a Frobeniu... |
| aks6d1c1p4 42099 | The product of polynomials... |
| aks6d1c1p5 42100 | The product of exponents i... |
| aks6d1c1p7 42101 | ` X ` is introspective to ... |
| aks6d1c1p6 42102 | If a polynomials ` F ` is ... |
| aks6d1c1p8 42103 | If a number ` E ` is intro... |
| aks6d1c1 42104 | Claim 1 of Theorem 6.1 ~ h... |
| evl1gprodd 42105 | Polynomial evaluation buil... |
| aks6d1c2p1 42106 | In the AKS-theorem the sub... |
| aks6d1c2p2 42107 | Injective condition for co... |
| hashscontpowcl 42108 | Closure of E for ~ https:/... |
| hashscontpow1 42109 | Helper lemma for to prove ... |
| hashscontpow 42110 | If a set contains all ` N ... |
| aks6d1c3 42111 | Claim 3 of Theorem 6.1 of ... |
| aks6d1c4 42112 | Claim 4 of Theorem 6.1 of ... |
| aks6d1c1rh 42113 | Claim 1 of AKS primality p... |
| aks6d1c2lem3 42114 | Lemma for ~ aks6d1c2 to si... |
| aks6d1c2lem4 42115 | Claim 2 of Theorem 6.1 AKS... |
| hashnexinj 42116 | If the number of elements ... |
| hashnexinjle 42117 | If the number of elements ... |
| aks6d1c2 42118 | Claim 2 of Theorem 6.1 of ... |
| rspcsbnea 42119 | Special case related to ~ ... |
| idomnnzpownz 42120 | A non-zero power in an int... |
| idomnnzgmulnz 42121 | A finite product of non-ze... |
| ringexp0nn 42122 | Zero to the power of a pos... |
| aks6d1c5lem0 42123 | Lemma for Claim 5 of Theor... |
| aks6d1c5lem1 42124 | Lemma for claim 5, evaluat... |
| aks6d1c5lem3 42125 | Lemma for Claim 5, polynom... |
| aks6d1c5lem2 42126 | Lemma for Claim 5, contrad... |
| aks6d1c5 42127 | Claim 5 of Theorem 6.1 ~ h... |
| deg1gprod 42128 | Degree multiplication is a... |
| deg1pow 42129 | Exact degree of a power of... |
| 5bc2eq10 42130 | The value of 5 choose 2. ... |
| facp2 42131 | The factorial of a success... |
| 2np3bcnp1 42132 | Part of induction step for... |
| 2ap1caineq 42133 | Inequality for Theorem 6.6... |
| sticksstones1 42134 | Different strictly monoton... |
| sticksstones2 42135 | The range function on stri... |
| sticksstones3 42136 | The range function on stri... |
| sticksstones4 42137 | Equinumerosity lemma for s... |
| sticksstones5 42138 | Count the number of strict... |
| sticksstones6 42139 | Function induces an order ... |
| sticksstones7 42140 | Closure property of sticks... |
| sticksstones8 42141 | Establish mapping between ... |
| sticksstones9 42142 | Establish mapping between ... |
| sticksstones10 42143 | Establish mapping between ... |
| sticksstones11 42144 | Establish bijective mappin... |
| sticksstones12a 42145 | Establish bijective mappin... |
| sticksstones12 42146 | Establish bijective mappin... |
| sticksstones13 42147 | Establish bijective mappin... |
| sticksstones14 42148 | Sticks and stones with def... |
| sticksstones15 42149 | Sticks and stones with alm... |
| sticksstones16 42150 | Sticks and stones with col... |
| sticksstones17 42151 | Extend sticks and stones t... |
| sticksstones18 42152 | Extend sticks and stones t... |
| sticksstones19 42153 | Extend sticks and stones t... |
| sticksstones20 42154 | Lift sticks and stones to ... |
| sticksstones21 42155 | Lift sticks and stones to ... |
| sticksstones22 42156 | Non-exhaustive sticks and ... |
| sticksstones23 42157 | Non-exhaustive sticks and ... |
| aks6d1c6lem1 42158 | Lemma for claim 6, deduce ... |
| aks6d1c6lem2 42159 | Every primitive root is ro... |
| aks6d1c6lem3 42160 | Claim 6 of Theorem 6.1 of ... |
| aks6d1c6lem4 42161 | Claim 6 of Theorem 6.1 of ... |
| aks6d1c6isolem1 42162 | Lemma to construct the map... |
| aks6d1c6isolem2 42163 | Lemma to construct the gro... |
| aks6d1c6isolem3 42164 | The preimage of a map send... |
| aks6d1c6lem5 42165 | Eliminate the size hypothe... |
| bcled 42166 | Inequality for binomial co... |
| bcle2d 42167 | Inequality for binomial co... |
| aks6d1c7lem1 42168 | The last set of inequaliti... |
| aks6d1c7lem2 42169 | Contradiction to Claim 2 a... |
| aks6d1c7lem3 42170 | Remove lots of hypotheses ... |
| aks6d1c7lem4 42171 | In the AKS algorithm there... |
| aks6d1c7 42172 | ` N ` is a prime power if ... |
| rhmqusspan 42173 | Ring homomorphism out of a... |
| aks5lem1 42174 | Section 5 of ~ https://www... |
| aks5lem2 42175 | Lemma for section 5 ~ http... |
| ply1asclzrhval 42176 | Transfer results from alge... |
| aks5lem3a 42177 | Lemma for AKS section 5. ... |
| aks5lem4a 42178 | Lemma for AKS section 5, r... |
| aks5lem5a 42179 | Lemma for AKS, section 5, ... |
| aks5lem6 42180 | Connect results of section... |
| indstrd 42181 | Strong induction, deductio... |
| grpods 42182 | Relate sums of elements of... |
| unitscyglem1 42183 | Lemma for unitscyg. (Cont... |
| unitscyglem2 42184 | Lemma for unitscyg. (Cont... |
| unitscyglem3 42185 | Lemma for unitscyg. (Cont... |
| unitscyglem4 42186 | Lemma for unitscyg (Contri... |
| unitscyglem5 42187 | Lemma for unitscyg (Contri... |
| aks5lem7 42188 | Lemma for aks5. We clean ... |
| aks5lem8 42189 | Lemma for aks5. Clean up ... |
| exfinfldd 42191 | For any prime ` P ` and an... |
| aks5 42192 | The AKS Primality test, gi... |
| jarrii 42193 | Inference associated with ... |
| intnanrt 42194 | Introduction of conjunct i... |
| ioin9i8 42195 | Miscellaneous inference cr... |
| jaodd 42196 | Double deduction form of ~... |
| syl3an12 42197 | A double syllogism inferen... |
| exbiii 42198 | Inference associated with ... |
| sbtd 42199 | A true statement is true u... |
| sbor2 42200 | One direction of ~ sbor , ... |
| sbalexi 42201 | Inference form of ~ sbalex... |
| 19.9dev 42202 | ~ 19.9d in the case of an ... |
| 3rspcedvd 42203 | Triple application of ~ rs... |
| sn-axrep5v 42204 | A condensed form of ~ axre... |
| sn-axprlem3 42205 | ~ axprlem3 using only Tars... |
| sn-exelALT 42206 | Alternate proof of ~ exel ... |
| ss2ab1 42207 | Class abstractions in a su... |
| ssabdv 42208 | Deduction of abstraction s... |
| sn-iotalem 42209 | An unused lemma showing th... |
| sn-iotalemcor 42210 | Corollary of ~ sn-iotalem ... |
| abbi1sn 42211 | Originally part of ~ uniab... |
| brif2 42212 | Move a relation inside and... |
| brif12 42213 | Move a relation inside and... |
| pssexg 42214 | The proper subset of a set... |
| pssn0 42215 | A proper superset is nonem... |
| psspwb 42216 | Classes are proper subclas... |
| xppss12 42217 | Proper subset theorem for ... |
| elpwbi 42218 | Membership in a power set,... |
| imaopab 42219 | The image of a class of or... |
| eqresfnbd 42220 | Property of being the rest... |
| f1o2d2 42221 | Sufficient condition for a... |
| fmpocos 42222 | Composition of two functio... |
| ovmpogad 42223 | Value of an operation give... |
| ofun 42224 | A function operation of un... |
| dfqs2 42225 | Alternate definition of qu... |
| dfqs3 42226 | Alternate definition of qu... |
| qseq12d 42227 | Equality theorem for quoti... |
| qsalrel 42228 | The quotient set is equal ... |
| elmapssresd 42229 | A restricted mapping is a ... |
| supinf 42230 | The supremum is the infimu... |
| mapcod 42231 | Compose two mappings. (Co... |
| fisdomnn 42232 | A finite set is dominated ... |
| ltex 42233 | The less-than relation is ... |
| leex 42234 | The less-than-or-equal-to ... |
| subex 42235 | The subtraction operation ... |
| absex 42236 | The absolute value functio... |
| cjex 42237 | The conjugate function is ... |
| fzosumm1 42238 | Separate out the last term... |
| ccatcan2d 42239 | Cancellation law for conca... |
| c0exALT 42240 | Alternate proof of ~ c0ex ... |
| 0cnALT3 42241 | Alternate proof of ~ 0cn u... |
| elre0re 42242 | Specialized version of ~ 0... |
| 1t1e1ALT 42243 | Alternate proof of ~ 1t1e1... |
| lttrii 42244 | 'Less than' is transitive.... |
| remulcan2d 42245 | ~ mulcan2d for real number... |
| readdridaddlidd 42246 | Given some real number ` B... |
| 1p3e4 42247 | 1 + 3 = 4. (Contributed b... |
| 5ne0 42248 | The number 5 is nonzero. ... |
| 6ne0 42249 | The number 6 is nonzero. ... |
| 7ne0 42250 | The number 7 is nonzero. ... |
| 8ne0 42251 | The number 8 is nonzero. ... |
| 9ne0 42252 | The number 9 is nonzero. ... |
| sn-1ne2 42253 | A proof of ~ 1ne2 without ... |
| nnn1suc 42254 | A positive integer that is... |
| nnadd1com 42255 | Addition with 1 is commuta... |
| nnaddcom 42256 | Addition is commutative fo... |
| nnaddcomli 42257 | Version of ~ addcomli for ... |
| nnadddir 42258 | Right-distributivity for n... |
| nnmul1com 42259 | Multiplication with 1 is c... |
| nnmulcom 42260 | Multiplication is commutat... |
| readdrcl2d 42261 | Reverse closure for additi... |
| mvrrsubd 42262 | Move a subtraction in the ... |
| laddrotrd 42263 | Rotate the variables right... |
| raddswap12d 42264 | Swap the first two variabl... |
| lsubrotld 42265 | Rotate the variables left ... |
| rsubrotld 42266 | Rotate the variables left ... |
| lsubswap23d 42267 | Swap the second and third ... |
| addsubeq4com 42268 | Relation between sums and ... |
| sqsumi 42269 | A sum squared. (Contribut... |
| negn0nposznnd 42270 | Lemma for ~ dffltz . (Con... |
| sqmid3api 42271 | Value of the square of the... |
| decaddcom 42272 | Commute ones place in addi... |
| sqn5i 42273 | The square of a number end... |
| sqn5ii 42274 | The square of a number end... |
| decpmulnc 42275 | Partial products algorithm... |
| decpmul 42276 | Partial products algorithm... |
| sqdeccom12 42277 | The square of a number in ... |
| sq3deccom12 42278 | Variant of ~ sqdeccom12 wi... |
| 4t5e20 42279 | 4 times 5 equals 20. (Con... |
| 3rdpwhole 42280 | A third of a number plus t... |
| sq4 42281 | The square of 4 is 16. (C... |
| sq5 42282 | The square of 5 is 25. (C... |
| sq6 42283 | The square of 6 is 36. (C... |
| sq7 42284 | The square of 7 is 49. (C... |
| sq8 42285 | The square of 8 is 64. (C... |
| sq9 42286 | The square of 9 is 81. (C... |
| rpsscn 42287 | The positive reals are a s... |
| 4rp 42288 | 4 is a positive real. (Co... |
| 6rp 42289 | 6 is a positive real. (Co... |
| 7rp 42290 | 7 is a positive real. (Co... |
| 8rp 42291 | 8 is a positive real. (Co... |
| 9rp 42292 | 9 is a positive real. (Co... |
| 235t711 42293 | Calculate a product by lon... |
| ex-decpmul 42294 | Example usage of ~ decpmul... |
| eluzp1 42295 | Membership in a successor ... |
| sn-eluzp1l 42296 | Shorter proof of ~ eluzp1l... |
| fz1sumconst 42297 | The sum of ` N ` constant ... |
| fz1sump1 42298 | Add one more term to a sum... |
| oddnumth 42299 | The Odd Number Theorem. T... |
| nicomachus 42300 | Nicomachus's Theorem. The... |
| sumcubes 42301 | The sum of the first ` N `... |
| ine1 42302 | ` _i ` is not 1. (Contrib... |
| 0tie0 42303 | 0 times ` _i ` equals 0. ... |
| it1ei 42304 | ` _i ` times 1 equals ` _i... |
| 1tiei 42305 | 1 times ` _i ` equals ` _i... |
| itrere 42306 | ` _i ` times a real is rea... |
| retire 42307 | A real times ` _i ` is rea... |
| iocioodisjd 42308 | Adjacent intervals where t... |
| rpabsid 42309 | A positive real is its own... |
| oexpreposd 42310 | Lemma for ~ dffltz . For ... |
| explt1d 42311 | A nonnegative real number ... |
| expeq1d 42312 | A nonnegative real number ... |
| expeqidd 42313 | A nonnegative real number ... |
| exp11d 42314 | ~ exp11nnd for nonzero int... |
| 0dvds0 42315 | 0 divides 0. (Contributed... |
| absdvdsabsb 42316 | Divisibility is invariant ... |
| gcdnn0id 42317 | The ` gcd ` of a nonnegati... |
| gcdle1d 42318 | The greatest common diviso... |
| gcdle2d 42319 | The greatest common diviso... |
| dvdsexpad 42320 | Deduction associated with ... |
| dvdsexpnn 42321 | ~ dvdssqlem generalized to... |
| dvdsexpnn0 42322 | ~ dvdsexpnn generalized to... |
| dvdsexpb 42323 | ~ dvdssq generalized to po... |
| posqsqznn 42324 | When a positive rational s... |
| zdivgd 42325 | Two ways to express " ` N ... |
| efsubd 42326 | Difference of exponents la... |
| ef11d 42327 | General condition for the ... |
| logccne0d 42328 | The logarithm isn't 0 if i... |
| cxp112d 42329 | General condition for comp... |
| cxp111d 42330 | General condition for comp... |
| cxpi11d 42331 | ` _i ` to the powers of ` ... |
| logne0d 42332 | Deduction form of ~ logne0... |
| rxp112d 42333 | Real exponentiation is one... |
| log11d 42334 | The natural logarithm is o... |
| rplog11d 42335 | The natural logarithm is o... |
| rxp11d 42336 | Real exponentiation is one... |
| tanhalfpim 42337 | The tangent of ` _pi / 2 `... |
| sinpim 42338 | Sine of a number subtracte... |
| cospim 42339 | Cosine of a number subtrac... |
| tan3rdpi 42340 | The tangent of ` _pi / 3 `... |
| sin2t3rdpi 42341 | The sine of ` 2 x. ( _pi /... |
| cos2t3rdpi 42342 | The cosine of ` 2 x. ( _pi... |
| sin4t3rdpi 42343 | The sine of ` 4 x. ( _pi /... |
| cos4t3rdpi 42344 | The cosine of ` 4 x. ( _pi... |
| asin1half 42345 | The arcsine of ` 1 / 2 ` i... |
| acos1half 42346 | The arccosine of ` 1 / 2 `... |
| dvun 42347 | Condition for the union of... |
| redvmptabs 42348 | The derivative of the abso... |
| readvrec2 42349 | The antiderivative of 1/x ... |
| readvrec 42350 | For real numbers, the anti... |
| resuppsinopn 42351 | The support of sin ( ~ df-... |
| readvcot 42352 | Real antiderivative of cot... |
| resubval 42355 | Value of real subtraction,... |
| renegeulemv 42356 | Lemma for ~ renegeu and si... |
| renegeulem 42357 | Lemma for ~ renegeu and si... |
| renegeu 42358 | Existential uniqueness of ... |
| rernegcl 42359 | Closure law for negative r... |
| renegadd 42360 | Relationship between real ... |
| renegid 42361 | Addition of a real number ... |
| reneg0addlid 42362 | Negative zero is a left ad... |
| resubeulem1 42363 | Lemma for ~ resubeu . A v... |
| resubeulem2 42364 | Lemma for ~ resubeu . A v... |
| resubeu 42365 | Existential uniqueness of ... |
| rersubcl 42366 | Closure for real subtracti... |
| resubadd 42367 | Relation between real subt... |
| resubaddd 42368 | Relationship between subtr... |
| resubf 42369 | Real subtraction is an ope... |
| repncan2 42370 | Addition and subtraction o... |
| repncan3 42371 | Addition and subtraction o... |
| readdsub 42372 | Law for addition and subtr... |
| reladdrsub 42373 | Move LHS of a sum into RHS... |
| reltsub1 42374 | Subtraction from both side... |
| reltsubadd2 42375 | 'Less than' relationship b... |
| resubcan2 42376 | Cancellation law for real ... |
| resubsub4 42377 | Law for double subtraction... |
| rennncan2 42378 | Cancellation law for real ... |
| renpncan3 42379 | Cancellation law for real ... |
| repnpcan 42380 | Cancellation law for addit... |
| reppncan 42381 | Cancellation law for mixed... |
| resubidaddlidlem 42382 | Lemma for ~ resubidaddlid ... |
| resubidaddlid 42383 | Any real number subtracted... |
| resubdi 42384 | Distribution of multiplica... |
| re1m1e0m0 42385 | Equality of two left-addit... |
| sn-00idlem1 42386 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem2 42387 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem3 42388 | Lemma for ~ sn-00id . (Co... |
| sn-00id 42389 | ~ 00id proven without ~ ax... |
| re0m0e0 42390 | Real number version of ~ 0... |
| readdlid 42391 | Real number version of ~ a... |
| sn-addlid 42392 | ~ addlid without ~ ax-mulc... |
| remul02 42393 | Real number version of ~ m... |
| sn-0ne2 42394 | ~ 0ne2 without ~ ax-mulcom... |
| remul01 42395 | Real number version of ~ m... |
| sn-remul0ord 42396 | A product is zero iff one ... |
| resubid 42397 | Subtraction of a real numb... |
| readdrid 42398 | Real number version of ~ a... |
| resubid1 42399 | Real number version of ~ s... |
| renegneg 42400 | A real number is equal to ... |
| readdcan2 42401 | Commuted version of ~ read... |
| renegid2 42402 | Commuted version of ~ rene... |
| remulneg2d 42403 | Product with negative is n... |
| sn-it0e0 42404 | Proof of ~ it0e0 without ~... |
| sn-negex12 42405 | A combination of ~ cnegex ... |
| sn-negex 42406 | Proof of ~ cnegex without ... |
| sn-negex2 42407 | Proof of ~ cnegex2 without... |
| sn-addcand 42408 | ~ addcand without ~ ax-mul... |
| sn-addrid 42409 | ~ addrid without ~ ax-mulc... |
| sn-addcan2d 42410 | ~ addcan2d without ~ ax-mu... |
| reixi 42411 | ~ ixi without ~ ax-mulcom ... |
| rei4 42412 | ~ i4 without ~ ax-mulcom .... |
| sn-addid0 42413 | A number that sums to itse... |
| sn-mul01 42414 | ~ mul01 without ~ ax-mulco... |
| sn-subeu 42415 | ~ negeu without ~ ax-mulco... |
| sn-subcl 42416 | ~ subcl without ~ ax-mulco... |
| sn-subf 42417 | ~ subf without ~ ax-mulcom... |
| resubeqsub 42418 | Equivalence between real s... |
| subresre 42419 | Subtraction restricted to ... |
| addinvcom 42420 | A number commutes with its... |
| remulinvcom 42421 | A left multiplicative inve... |
| remullid 42422 | Commuted version of ~ ax-1... |
| sn-1ticom 42423 | Lemma for ~ sn-mullid and ... |
| sn-mullid 42424 | ~ mullid without ~ ax-mulc... |
| sn-it1ei 42425 | ~ it1ei without ~ ax-mulco... |
| ipiiie0 42426 | The multiplicative inverse... |
| remulcand 42427 | Commuted version of ~ remu... |
| redivvald 42430 | Value of real division, wh... |
| rediveud 42431 | Existential uniqueness of ... |
| sn-redivcld 42432 | Closure law for real divis... |
| redivmuld 42433 | Relationship between divis... |
| redivcan2d 42434 | A cancellation law for div... |
| redivcan3d 42435 | A cancellation law for div... |
| sn-rereccld 42436 | Closure law for reciprocal... |
| rerecid 42437 | Multiplication of a number... |
| rerecid2 42438 | Multiplication of a number... |
| sn-0tie0 42439 | Lemma for ~ sn-mul02 . Co... |
| sn-mul02 42440 | ~ mul02 without ~ ax-mulco... |
| sn-ltaddpos 42441 | ~ ltaddpos without ~ ax-mu... |
| sn-ltaddneg 42442 | ~ ltaddneg without ~ ax-mu... |
| reposdif 42443 | Comparison of two numbers ... |
| relt0neg1 42444 | Comparison of a real and i... |
| relt0neg2 42445 | Comparison of a real and i... |
| sn-addlt0d 42446 | The sum of negative number... |
| sn-addgt0d 42447 | The sum of positive number... |
| sn-nnne0 42448 | ~ nnne0 without ~ ax-mulco... |
| reelznn0nn 42449 | ~ elznn0nn restated using ... |
| nn0addcom 42450 | Addition is commutative fo... |
| zaddcomlem 42451 | Lemma for ~ zaddcom . (Co... |
| zaddcom 42452 | Addition is commutative fo... |
| renegmulnnass 42453 | Move multiplication by a n... |
| nn0mulcom 42454 | Multiplication is commutat... |
| zmulcomlem 42455 | Lemma for ~ zmulcom . (Co... |
| zmulcom 42456 | Multiplication is commutat... |
| mulgt0con1dlem 42457 | Lemma for ~ mulgt0con1d . ... |
| mulgt0con1d 42458 | Counterpart to ~ mulgt0con... |
| mulgt0con2d 42459 | Lemma for ~ mulgt0b1d and ... |
| mulgt0b1d 42460 | Biconditional, deductive f... |
| sn-ltmul2d 42461 | ~ ltmul2d without ~ ax-mul... |
| sn-ltmulgt11d 42462 | ~ ltmulgt11d without ~ ax-... |
| sn-0lt1 42463 | ~ 0lt1 without ~ ax-mulcom... |
| sn-ltp1 42464 | ~ ltp1 without ~ ax-mulcom... |
| sn-recgt0d 42465 | The reciprocal of a positi... |
| mulgt0b2d 42466 | Biconditional, deductive f... |
| sn-mulgt1d 42467 | ~ mulgt1d without ~ ax-mul... |
| reneg1lt0 42468 | Negative one is a negative... |
| sn-reclt0d 42469 | The reciprocal of a negati... |
| mulltgt0d 42470 | Negative times positive is... |
| mullt0b1d 42471 | When the first term is neg... |
| mullt0b2d 42472 | When the second term is ne... |
| sn-mullt0d 42473 | The product of two negativ... |
| sn-msqgt0d 42474 | A nonzero square is positi... |
| sn-inelr 42475 | ~ inelr without ~ ax-mulco... |
| sn-itrere 42476 | ` _i ` times a real is rea... |
| sn-retire 42477 | Commuted version of ~ sn-i... |
| cnreeu 42478 | The reals in the expressio... |
| sn-sup2 42479 | ~ sup2 with exactly the sa... |
| sn-sup3d 42480 | ~ sup3 without ~ ax-mulcom... |
| sn-suprcld 42481 | ~ suprcld without ~ ax-mul... |
| sn-suprubd 42482 | ~ suprubd without ~ ax-mul... |
| sn-base0 42483 | Avoid axioms in ~ base0 by... |
| nelsubginvcld 42484 | The inverse of a non-subgr... |
| nelsubgcld 42485 | A non-subgroup-member plus... |
| nelsubgsubcld 42486 | A non-subgroup-member minu... |
| rnasclg 42487 | The set of injected scalar... |
| frlmfielbas 42488 | The vectors of a finite fr... |
| frlmfzwrd 42489 | A vector of a module with ... |
| frlmfzowrd 42490 | A vector of a module with ... |
| frlmfzolen 42491 | The dimension of a vector ... |
| frlmfzowrdb 42492 | The vectors of a module wi... |
| frlmfzoccat 42493 | The concatenation of two v... |
| frlmvscadiccat 42494 | Scalar multiplication dist... |
| grpasscan2d 42495 | An associative cancellatio... |
| grpcominv1 42496 | If two elements commute, t... |
| grpcominv2 42497 | If two elements commute, t... |
| finsubmsubg 42498 | A submonoid of a finite gr... |
| opprmndb 42499 | A class is a monoid if and... |
| opprgrpb 42500 | A class is a group if and ... |
| opprablb 42501 | A class is an Abelian grou... |
| imacrhmcl 42502 | The image of a commutative... |
| rimrcl1 42503 | Reverse closure of a ring ... |
| rimrcl2 42504 | Reverse closure of a ring ... |
| rimcnv 42505 | The converse of a ring iso... |
| rimco 42506 | The composition of ring is... |
| ricsym 42507 | Ring isomorphism is symmet... |
| rictr 42508 | Ring isomorphism is transi... |
| riccrng1 42509 | Ring isomorphism preserves... |
| riccrng 42510 | A ring is commutative if a... |
| domnexpgn0cl 42511 | In a domain, a (nonnegativ... |
| drnginvrn0d 42512 | A multiplicative inverse i... |
| drngmullcan 42513 | Cancellation of a nonzero ... |
| drngmulrcan 42514 | Cancellation of a nonzero ... |
| drnginvmuld 42515 | Inverse of a nonzero produ... |
| ricdrng1 42516 | A ring isomorphism maps a ... |
| ricdrng 42517 | A ring is a division ring ... |
| ricfld 42518 | A ring is a field if and o... |
| asclf1 42519 | Two ways of saying the sca... |
| abvexp 42520 | Move exponentiation in and... |
| fimgmcyclem 42521 | Lemma for ~ fimgmcyc . (C... |
| fimgmcyc 42522 | Version of ~ odcl2 for fin... |
| fidomncyc 42523 | Version of ~ odcl2 for mul... |
| fiabv 42524 | In a finite domain (a fini... |
| lvecgrp 42525 | A vector space is a group.... |
| lvecring 42526 | The scalar component of a ... |
| frlm0vald 42527 | All coordinates of the zer... |
| frlmsnic 42528 | Given a free module with a... |
| uvccl 42529 | A unit vector is a vector.... |
| uvcn0 42530 | A unit vector is nonzero. ... |
| pwselbasr 42531 | The reverse direction of ~... |
| pwsgprod 42532 | Finite products in a power... |
| psrmnd 42533 | The ring of power series i... |
| psrbagres 42534 | Restrict a bag of variable... |
| mplcrngd 42535 | The polynomial ring is a c... |
| mplsubrgcl 42536 | An element of a polynomial... |
| mhmcopsr 42537 | The composition of a monoi... |
| mhmcoaddpsr 42538 | Show that the ring homomor... |
| rhmcomulpsr 42539 | Show that the ring homomor... |
| rhmpsr 42540 | Provide a ring homomorphis... |
| rhmpsr1 42541 | Provide a ring homomorphis... |
| mplascl0 42542 | The zero scalar as a polyn... |
| mplascl1 42543 | The one scalar as a polyno... |
| mplmapghm 42544 | The function ` H ` mapping... |
| evl0 42545 | The zero polynomial evalua... |
| evlscl 42546 | A polynomial over the ring... |
| evlsval3 42547 | Give a formula for the pol... |
| evlsvval 42548 | Give a formula for the eva... |
| evlsvvvallem 42549 | Lemma for ~ evlsvvval akin... |
| evlsvvvallem2 42550 | Lemma for theorems using ~... |
| evlsvvval 42551 | Give a formula for the eva... |
| evlsscaval 42552 | Polynomial evaluation buil... |
| evlsvarval 42553 | Polynomial evaluation buil... |
| evlsbagval 42554 | Polynomial evaluation buil... |
| evlsexpval 42555 | Polynomial evaluation buil... |
| evlsaddval 42556 | Polynomial evaluation buil... |
| evlsmulval 42557 | Polynomial evaluation buil... |
| evlsmaprhm 42558 | The function ` F ` mapping... |
| evlsevl 42559 | Evaluation in a subring is... |
| evlcl 42560 | A polynomial over the ring... |
| evlvvval 42561 | Give a formula for the eva... |
| evlvvvallem 42562 | Lemma for theorems using ~... |
| evladdval 42563 | Polynomial evaluation buil... |
| evlmulval 42564 | Polynomial evaluation buil... |
| selvcllem1 42565 | ` T ` is an associative al... |
| selvcllem2 42566 | ` D ` is a ring homomorphi... |
| selvcllem3 42567 | The third argument passed ... |
| selvcllemh 42568 | Apply the third argument (... |
| selvcllem4 42569 | The fourth argument passed... |
| selvcllem5 42570 | The fifth argument passed ... |
| selvcl 42571 | Closure of the "variable s... |
| selvval2 42572 | Value of the "variable sel... |
| selvvvval 42573 | Recover the original polyn... |
| evlselvlem 42574 | Lemma for ~ evlselv . Use... |
| evlselv 42575 | Evaluating a selection of ... |
| selvadd 42576 | The "variable selection" f... |
| selvmul 42577 | The "variable selection" f... |
| fsuppind 42578 | Induction on functions ` F... |
| fsuppssindlem1 42579 | Lemma for ~ fsuppssind . ... |
| fsuppssindlem2 42580 | Lemma for ~ fsuppssind . ... |
| fsuppssind 42581 | Induction on functions ` F... |
| mhpind 42582 | The homogeneous polynomial... |
| evlsmhpvvval 42583 | Give a formula for the eva... |
| mhphflem 42584 | Lemma for ~ mhphf . Add s... |
| mhphf 42585 | A homogeneous polynomial d... |
| mhphf2 42586 | A homogeneous polynomial d... |
| mhphf3 42587 | A homogeneous polynomial d... |
| mhphf4 42588 | A homogeneous polynomial d... |
| prjspval 42591 | Value of the projective sp... |
| prjsprel 42592 | Utility theorem regarding ... |
| prjspertr 42593 | The relation in ` PrjSp ` ... |
| prjsperref 42594 | The relation in ` PrjSp ` ... |
| prjspersym 42595 | The relation in ` PrjSp ` ... |
| prjsper 42596 | The relation used to defin... |
| prjspreln0 42597 | Two nonzero vectors are eq... |
| prjspvs 42598 | A nonzero multiple of a ve... |
| prjsprellsp 42599 | Two vectors are equivalent... |
| prjspeclsp 42600 | The vectors equivalent to ... |
| prjspval2 42601 | Alternate definition of pr... |
| prjspnval 42604 | Value of the n-dimensional... |
| prjspnerlem 42605 | A lemma showing that the e... |
| prjspnval2 42606 | Value of the n-dimensional... |
| prjspner 42607 | The relation used to defin... |
| prjspnvs 42608 | A nonzero multiple of a ve... |
| prjspnssbas 42609 | A projective point spans a... |
| prjspnn0 42610 | A projective point is none... |
| 0prjspnlem 42611 | Lemma for ~ 0prjspn . The... |
| prjspnfv01 42612 | Any vector is equivalent t... |
| prjspner01 42613 | Any vector is equivalent t... |
| prjspner1 42614 | Two vectors whose zeroth c... |
| 0prjspnrel 42615 | In the zero-dimensional pr... |
| 0prjspn 42616 | A zero-dimensional project... |
| prjcrvfval 42619 | Value of the projective cu... |
| prjcrvval 42620 | Value of the projective cu... |
| prjcrv0 42621 | The "curve" (zero set) cor... |
| dffltz 42622 | Fermat's Last Theorem (FLT... |
| fltmul 42623 | A counterexample to FLT st... |
| fltdiv 42624 | A counterexample to FLT st... |
| flt0 42625 | A counterexample for FLT d... |
| fltdvdsabdvdsc 42626 | Any factor of both ` A ` a... |
| fltabcoprmex 42627 | A counterexample to FLT im... |
| fltaccoprm 42628 | A counterexample to FLT wi... |
| fltbccoprm 42629 | A counterexample to FLT wi... |
| fltabcoprm 42630 | A counterexample to FLT wi... |
| infdesc 42631 | Infinite descent. The hyp... |
| fltne 42632 | If a counterexample to FLT... |
| flt4lem 42633 | Raising a number to the fo... |
| flt4lem1 42634 | Satisfy the antecedent use... |
| flt4lem2 42635 | If ` A ` is even, ` B ` is... |
| flt4lem3 42636 | Equivalent to ~ pythagtrip... |
| flt4lem4 42637 | If the product of two copr... |
| flt4lem5 42638 | In the context of the lemm... |
| flt4lem5elem 42639 | Version of ~ fltaccoprm an... |
| flt4lem5a 42640 | Part 1 of Equation 1 of ... |
| flt4lem5b 42641 | Part 2 of Equation 1 of ... |
| flt4lem5c 42642 | Part 2 of Equation 2 of ... |
| flt4lem5d 42643 | Part 3 of Equation 2 of ... |
| flt4lem5e 42644 | Satisfy the hypotheses of ... |
| flt4lem5f 42645 | Final equation of ~... |
| flt4lem6 42646 | Remove shared factors in a... |
| flt4lem7 42647 | Convert ~ flt4lem5f into a... |
| nna4b4nsq 42648 | Strengthening of Fermat's ... |
| fltltc 42649 | ` ( C ^ N ) ` is the large... |
| fltnltalem 42650 | Lemma for ~ fltnlta . A l... |
| fltnlta 42651 | In a Fermat counterexample... |
| iddii 42652 | Version of ~ a1ii with the... |
| bicomdALT 42653 | Alternate proof of ~ bicom... |
| alan 42654 | Alias for ~ 19.26 for easi... |
| exor 42655 | Alias for ~ 19.43 for easi... |
| rexor 42656 | Alias for ~ r19.43 for eas... |
| ruvALT 42657 | Alternate proof of ~ ruv w... |
| sn-wcdeq 42658 | Alternative to ~ wcdeq and... |
| sq45 42659 | 45 squared is 2025. (Cont... |
| sum9cubes 42660 | The sum of the first nine ... |
| sn-isghm 42661 | Longer proof of ~ isghm , ... |
| aprilfools2025 42662 | An abuse of notation. (Co... |
| nfa1w 42663 | Replace ~ ax-10 in ~ nfa1 ... |
| eu6w 42664 | Replace ~ ax-10 , ~ ax-12 ... |
| abbibw 42665 | Replace ~ ax-10 , ~ ax-11 ... |
| absnw 42666 | Replace ~ ax-10 , ~ ax-11 ... |
| euabsn2w 42667 | Replace ~ ax-10 , ~ ax-11 ... |
| sn-tz6.12-2 42668 | ~ tz6.12-2 without ~ ax-10... |
| cu3addd 42669 | Cube of sum of three numbe... |
| negexpidd 42670 | The sum of a real number t... |
| rexlimdv3d 42671 | An extended version of ~ r... |
| 3cubeslem1 42672 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem2 42673 | Lemma for ~ 3cubes . Used... |
| 3cubeslem3l 42674 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3r 42675 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3 42676 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem4 42677 | Lemma for ~ 3cubes . This... |
| 3cubes 42678 | Every rational number is a... |
| rntrclfvOAI 42679 | The range of the transitiv... |
| moxfr 42680 | Transfer at-most-one betwe... |
| imaiinfv 42681 | Indexed intersection of an... |
| elrfi 42682 | Elementhood in a set of re... |
| elrfirn 42683 | Elementhood in a set of re... |
| elrfirn2 42684 | Elementhood in a set of re... |
| cmpfiiin 42685 | In a compact topology, a s... |
| ismrcd1 42686 | Any function from the subs... |
| ismrcd2 42687 | Second half of ~ ismrcd1 .... |
| istopclsd 42688 | A closure function which s... |
| ismrc 42689 | A function is a Moore clos... |
| isnacs 42692 | Expand definition of Noeth... |
| nacsfg 42693 | In a Noetherian-type closu... |
| isnacs2 42694 | Express Noetherian-type cl... |
| mrefg2 42695 | Slight variation on finite... |
| mrefg3 42696 | Slight variation on finite... |
| nacsacs 42697 | A closure system of Noethe... |
| isnacs3 42698 | A choice-free order equiva... |
| incssnn0 42699 | Transitivity induction of ... |
| nacsfix 42700 | An increasing sequence of ... |
| constmap 42701 | A constant (represented wi... |
| mapco2g 42702 | Renaming indices in a tupl... |
| mapco2 42703 | Post-composition (renaming... |
| mapfzcons 42704 | Extending a one-based mapp... |
| mapfzcons1 42705 | Recover prefix mapping fro... |
| mapfzcons1cl 42706 | A nonempty mapping has a p... |
| mapfzcons2 42707 | Recover added element from... |
| mptfcl 42708 | Interpret range of a maps-... |
| mzpclval 42713 | Substitution lemma for ` m... |
| elmzpcl 42714 | Double substitution lemma ... |
| mzpclall 42715 | The set of all functions w... |
| mzpcln0 42716 | Corollary of ~ mzpclall : ... |
| mzpcl1 42717 | Defining property 1 of a p... |
| mzpcl2 42718 | Defining property 2 of a p... |
| mzpcl34 42719 | Defining properties 3 and ... |
| mzpval 42720 | Value of the ` mzPoly ` fu... |
| dmmzp 42721 | ` mzPoly ` is defined for ... |
| mzpincl 42722 | Polynomial closedness is a... |
| mzpconst 42723 | Constant functions are pol... |
| mzpf 42724 | A polynomial function is a... |
| mzpproj 42725 | A projection function is p... |
| mzpadd 42726 | The pointwise sum of two p... |
| mzpmul 42727 | The pointwise product of t... |
| mzpconstmpt 42728 | A constant function expres... |
| mzpaddmpt 42729 | Sum of polynomial function... |
| mzpmulmpt 42730 | Product of polynomial func... |
| mzpsubmpt 42731 | The difference of two poly... |
| mzpnegmpt 42732 | Negation of a polynomial f... |
| mzpexpmpt 42733 | Raise a polynomial functio... |
| mzpindd 42734 | "Structural" induction to ... |
| mzpmfp 42735 | Relationship between multi... |
| mzpsubst 42736 | Substituting polynomials f... |
| mzprename 42737 | Simplified version of ~ mz... |
| mzpresrename 42738 | A polynomial is a polynomi... |
| mzpcompact2lem 42739 | Lemma for ~ mzpcompact2 . ... |
| mzpcompact2 42740 | Polynomials are finitary o... |
| coeq0i 42741 | ~ coeq0 but without explic... |
| fzsplit1nn0 42742 | Split a finite 1-based set... |
| eldiophb 42745 | Initial expression of Diop... |
| eldioph 42746 | Condition for a set to be ... |
| diophrw 42747 | Renaming and adding unused... |
| eldioph2lem1 42748 | Lemma for ~ eldioph2 . Co... |
| eldioph2lem2 42749 | Lemma for ~ eldioph2 . Co... |
| eldioph2 42750 | Construct a Diophantine se... |
| eldioph2b 42751 | While Diophantine sets wer... |
| eldiophelnn0 42752 | Remove antecedent on ` B `... |
| eldioph3b 42753 | Define Diophantine sets in... |
| eldioph3 42754 | Inference version of ~ eld... |
| ellz1 42755 | Membership in a lower set ... |
| lzunuz 42756 | The union of a lower set o... |
| fz1eqin 42757 | Express a one-based finite... |
| lzenom 42758 | Lower integers are countab... |
| elmapresaunres2 42759 | ~ fresaunres2 transposed t... |
| diophin 42760 | If two sets are Diophantin... |
| diophun 42761 | If two sets are Diophantin... |
| eldiophss 42762 | Diophantine sets are sets ... |
| diophrex 42763 | Projecting a Diophantine s... |
| eq0rabdioph 42764 | This is the first of a num... |
| eqrabdioph 42765 | Diophantine set builder fo... |
| 0dioph 42766 | The null set is Diophantin... |
| vdioph 42767 | The "universal" set (as la... |
| anrabdioph 42768 | Diophantine set builder fo... |
| orrabdioph 42769 | Diophantine set builder fo... |
| 3anrabdioph 42770 | Diophantine set builder fo... |
| 3orrabdioph 42771 | Diophantine set builder fo... |
| 2sbcrex 42772 | Exchange an existential qu... |
| sbcrexgOLD 42773 | Interchange class substitu... |
| 2sbcrexOLD 42774 | Exchange an existential qu... |
| sbc2rex 42775 | Exchange a substitution wi... |
| sbc2rexgOLD 42776 | Exchange a substitution wi... |
| sbc4rex 42777 | Exchange a substitution wi... |
| sbc4rexgOLD 42778 | Exchange a substitution wi... |
| sbcrot3 42779 | Rotate a sequence of three... |
| sbcrot5 42780 | Rotate a sequence of five ... |
| sbccomieg 42781 | Commute two explicit subst... |
| rexrabdioph 42782 | Diophantine set builder fo... |
| rexfrabdioph 42783 | Diophantine set builder fo... |
| 2rexfrabdioph 42784 | Diophantine set builder fo... |
| 3rexfrabdioph 42785 | Diophantine set builder fo... |
| 4rexfrabdioph 42786 | Diophantine set builder fo... |
| 6rexfrabdioph 42787 | Diophantine set builder fo... |
| 7rexfrabdioph 42788 | Diophantine set builder fo... |
| rabdiophlem1 42789 | Lemma for arithmetic dioph... |
| rabdiophlem2 42790 | Lemma for arithmetic dioph... |
| elnn0rabdioph 42791 | Diophantine set builder fo... |
| rexzrexnn0 42792 | Rewrite an existential qua... |
| lerabdioph 42793 | Diophantine set builder fo... |
| eluzrabdioph 42794 | Diophantine set builder fo... |
| elnnrabdioph 42795 | Diophantine set builder fo... |
| ltrabdioph 42796 | Diophantine set builder fo... |
| nerabdioph 42797 | Diophantine set builder fo... |
| dvdsrabdioph 42798 | Divisibility is a Diophant... |
| eldioph4b 42799 | Membership in ` Dioph ` ex... |
| eldioph4i 42800 | Forward-only version of ~ ... |
| diophren 42801 | Change variables in a Diop... |
| rabrenfdioph 42802 | Change variable numbers in... |
| rabren3dioph 42803 | Change variable numbers in... |
| fphpd 42804 | Pigeonhole principle expre... |
| fphpdo 42805 | Pigeonhole principle for s... |
| ctbnfien 42806 | An infinite subset of a co... |
| fiphp3d 42807 | Infinite pigeonhole princi... |
| rencldnfilem 42808 | Lemma for ~ rencldnfi . (... |
| rencldnfi 42809 | A set of real numbers whic... |
| irrapxlem1 42810 | Lemma for ~ irrapx1 . Div... |
| irrapxlem2 42811 | Lemma for ~ irrapx1 . Two... |
| irrapxlem3 42812 | Lemma for ~ irrapx1 . By ... |
| irrapxlem4 42813 | Lemma for ~ irrapx1 . Eli... |
| irrapxlem5 42814 | Lemma for ~ irrapx1 . Swi... |
| irrapxlem6 42815 | Lemma for ~ irrapx1 . Exp... |
| irrapx1 42816 | Dirichlet's approximation ... |
| pellexlem1 42817 | Lemma for ~ pellex . Arit... |
| pellexlem2 42818 | Lemma for ~ pellex . Arit... |
| pellexlem3 42819 | Lemma for ~ pellex . To e... |
| pellexlem4 42820 | Lemma for ~ pellex . Invo... |
| pellexlem5 42821 | Lemma for ~ pellex . Invo... |
| pellexlem6 42822 | Lemma for ~ pellex . Doin... |
| pellex 42823 | Every Pell equation has a ... |
| pell1qrval 42834 | Value of the set of first-... |
| elpell1qr 42835 | Membership in a first-quad... |
| pell14qrval 42836 | Value of the set of positi... |
| elpell14qr 42837 | Membership in the set of p... |
| pell1234qrval 42838 | Value of the set of genera... |
| elpell1234qr 42839 | Membership in the set of g... |
| pell1234qrre 42840 | General Pell solutions are... |
| pell1234qrne0 42841 | No solution to a Pell equa... |
| pell1234qrreccl 42842 | General solutions of the P... |
| pell1234qrmulcl 42843 | General solutions of the P... |
| pell14qrss1234 42844 | A positive Pell solution i... |
| pell14qrre 42845 | A positive Pell solution i... |
| pell14qrne0 42846 | A positive Pell solution i... |
| pell14qrgt0 42847 | A positive Pell solution i... |
| pell14qrrp 42848 | A positive Pell solution i... |
| pell1234qrdich 42849 | A general Pell solution is... |
| elpell14qr2 42850 | A number is a positive Pel... |
| pell14qrmulcl 42851 | Positive Pell solutions ar... |
| pell14qrreccl 42852 | Positive Pell solutions ar... |
| pell14qrdivcl 42853 | Positive Pell solutions ar... |
| pell14qrexpclnn0 42854 | Lemma for ~ pell14qrexpcl ... |
| pell14qrexpcl 42855 | Positive Pell solutions ar... |
| pell1qrss14 42856 | First-quadrant Pell soluti... |
| pell14qrdich 42857 | A positive Pell solution i... |
| pell1qrge1 42858 | A Pell solution in the fir... |
| pell1qr1 42859 | 1 is a Pell solution and i... |
| elpell1qr2 42860 | The first quadrant solutio... |
| pell1qrgaplem 42861 | Lemma for ~ pell1qrgap . ... |
| pell1qrgap 42862 | First-quadrant Pell soluti... |
| pell14qrgap 42863 | Positive Pell solutions ar... |
| pell14qrgapw 42864 | Positive Pell solutions ar... |
| pellqrexplicit 42865 | Condition for a calculated... |
| infmrgelbi 42866 | Any lower bound of a nonem... |
| pellqrex 42867 | There is a nontrivial solu... |
| pellfundval 42868 | Value of the fundamental s... |
| pellfundre 42869 | The fundamental solution o... |
| pellfundge 42870 | Lower bound on the fundame... |
| pellfundgt1 42871 | Weak lower bound on the Pe... |
| pellfundlb 42872 | A nontrivial first quadran... |
| pellfundglb 42873 | If a real is larger than t... |
| pellfundex 42874 | The fundamental solution a... |
| pellfund14gap 42875 | There are no solutions bet... |
| pellfundrp 42876 | The fundamental Pell solut... |
| pellfundne1 42877 | The fundamental Pell solut... |
| reglogcl 42878 | General logarithm is a rea... |
| reglogltb 42879 | General logarithm preserve... |
| reglogleb 42880 | General logarithm preserve... |
| reglogmul 42881 | Multiplication law for gen... |
| reglogexp 42882 | Power law for general log.... |
| reglogbas 42883 | General log of the base is... |
| reglog1 42884 | General log of 1 is 0. (C... |
| reglogexpbas 42885 | General log of a power of ... |
| pellfund14 42886 | Every positive Pell soluti... |
| pellfund14b 42887 | The positive Pell solution... |
| rmxfval 42892 | Value of the X sequence. ... |
| rmyfval 42893 | Value of the Y sequence. ... |
| rmspecsqrtnq 42894 | The discriminant used to d... |
| rmspecnonsq 42895 | The discriminant used to d... |
| qirropth 42896 | This lemma implements the ... |
| rmspecfund 42897 | The base of exponent used ... |
| rmxyelqirr 42898 | The solutions used to cons... |
| rmxyelqirrOLD 42899 | Obsolete version of ~ rmxy... |
| rmxypairf1o 42900 | The function used to extra... |
| rmxyelxp 42901 | Lemma for ~ frmx and ~ frm... |
| frmx 42902 | The X sequence is a nonneg... |
| frmy 42903 | The Y sequence is an integ... |
| rmxyval 42904 | Main definition of the X a... |
| rmspecpos 42905 | The discriminant used to d... |
| rmxycomplete 42906 | The X and Y sequences take... |
| rmxynorm 42907 | The X and Y sequences defi... |
| rmbaserp 42908 | The base of exponentiation... |
| rmxyneg 42909 | Negation law for X and Y s... |
| rmxyadd 42910 | Addition formula for X and... |
| rmxy1 42911 | Value of the X and Y seque... |
| rmxy0 42912 | Value of the X and Y seque... |
| rmxneg 42913 | Negation law (even functio... |
| rmx0 42914 | Value of X sequence at 0. ... |
| rmx1 42915 | Value of X sequence at 1. ... |
| rmxadd 42916 | Addition formula for X seq... |
| rmyneg 42917 | Negation formula for Y seq... |
| rmy0 42918 | Value of Y sequence at 0. ... |
| rmy1 42919 | Value of Y sequence at 1. ... |
| rmyadd 42920 | Addition formula for Y seq... |
| rmxp1 42921 | Special addition-of-1 form... |
| rmyp1 42922 | Special addition of 1 form... |
| rmxm1 42923 | Subtraction of 1 formula f... |
| rmym1 42924 | Subtraction of 1 formula f... |
| rmxluc 42925 | The X sequence is a Lucas ... |
| rmyluc 42926 | The Y sequence is a Lucas ... |
| rmyluc2 42927 | Lucas sequence property of... |
| rmxdbl 42928 | "Double-angle formula" for... |
| rmydbl 42929 | "Double-angle formula" for... |
| monotuz 42930 | A function defined on an u... |
| monotoddzzfi 42931 | A function which is odd an... |
| monotoddzz 42932 | A function (given implicit... |
| oddcomabszz 42933 | An odd function which take... |
| 2nn0ind 42934 | Induction on nonnegative i... |
| zindbi 42935 | Inductively transfer a pro... |
| rmxypos 42936 | For all nonnegative indice... |
| ltrmynn0 42937 | The Y-sequence is strictly... |
| ltrmxnn0 42938 | The X-sequence is strictly... |
| lermxnn0 42939 | The X-sequence is monotoni... |
| rmxnn 42940 | The X-sequence is defined ... |
| ltrmy 42941 | The Y-sequence is strictly... |
| rmyeq0 42942 | Y is zero only at zero. (... |
| rmyeq 42943 | Y is one-to-one. (Contrib... |
| lermy 42944 | Y is monotonic (non-strict... |
| rmynn 42945 | ` rmY ` is positive for po... |
| rmynn0 42946 | ` rmY ` is nonnegative for... |
| rmyabs 42947 | ` rmY ` commutes with ` ab... |
| jm2.24nn 42948 | X(n) is strictly greater t... |
| jm2.17a 42949 | First half of lemma 2.17 o... |
| jm2.17b 42950 | Weak form of the second ha... |
| jm2.17c 42951 | Second half of lemma 2.17 ... |
| jm2.24 42952 | Lemma 2.24 of [JonesMatija... |
| rmygeid 42953 | Y(n) increases faster than... |
| congtr 42954 | A wff of the form ` A || (... |
| congadd 42955 | If two pairs of numbers ar... |
| congmul 42956 | If two pairs of numbers ar... |
| congsym 42957 | Congruence mod ` A ` is a ... |
| congneg 42958 | If two integers are congru... |
| congsub 42959 | If two pairs of numbers ar... |
| congid 42960 | Every integer is congruent... |
| mzpcong 42961 | Polynomials commute with c... |
| congrep 42962 | Every integer is congruent... |
| congabseq 42963 | If two integers are congru... |
| acongid 42964 | A wff like that in this th... |
| acongsym 42965 | Symmetry of alternating co... |
| acongneg2 42966 | Negate right side of alter... |
| acongtr 42967 | Transitivity of alternatin... |
| acongeq12d 42968 | Substitution deduction for... |
| acongrep 42969 | Every integer is alternati... |
| fzmaxdif 42970 | Bound on the difference be... |
| fzneg 42971 | Reflection of a finite ran... |
| acongeq 42972 | Two numbers in the fundame... |
| dvdsacongtr 42973 | Alternating congruence pas... |
| coprmdvdsb 42974 | Multiplication by a coprim... |
| modabsdifz 42975 | Divisibility in terms of m... |
| dvdsabsmod0 42976 | Divisibility in terms of m... |
| jm2.18 42977 | Theorem 2.18 of [JonesMati... |
| jm2.19lem1 42978 | Lemma for ~ jm2.19 . X an... |
| jm2.19lem2 42979 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem3 42980 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem4 42981 | Lemma for ~ jm2.19 . Exte... |
| jm2.19 42982 | Lemma 2.19 of [JonesMatija... |
| jm2.21 42983 | Lemma for ~ jm2.20nn . Ex... |
| jm2.22 42984 | Lemma for ~ jm2.20nn . Ap... |
| jm2.23 42985 | Lemma for ~ jm2.20nn . Tr... |
| jm2.20nn 42986 | Lemma 2.20 of [JonesMatija... |
| jm2.25lem1 42987 | Lemma for ~ jm2.26 . (Con... |
| jm2.25 42988 | Lemma for ~ jm2.26 . Rema... |
| jm2.26a 42989 | Lemma for ~ jm2.26 . Reve... |
| jm2.26lem3 42990 | Lemma for ~ jm2.26 . Use ... |
| jm2.26 42991 | Lemma 2.26 of [JonesMatija... |
| jm2.15nn0 42992 | Lemma 2.15 of [JonesMatija... |
| jm2.16nn0 42993 | Lemma 2.16 of [JonesMatija... |
| jm2.27a 42994 | Lemma for ~ jm2.27 . Reve... |
| jm2.27b 42995 | Lemma for ~ jm2.27 . Expa... |
| jm2.27c 42996 | Lemma for ~ jm2.27 . Forw... |
| jm2.27 42997 | Lemma 2.27 of [JonesMatija... |
| jm2.27dlem1 42998 | Lemma for ~ rmydioph . Su... |
| jm2.27dlem2 42999 | Lemma for ~ rmydioph . Th... |
| jm2.27dlem3 43000 | Lemma for ~ rmydioph . In... |
| jm2.27dlem4 43001 | Lemma for ~ rmydioph . In... |
| jm2.27dlem5 43002 | Lemma for ~ rmydioph . Us... |
| rmydioph 43003 | ~ jm2.27 restated in terms... |
| rmxdiophlem 43004 | X can be expressed in term... |
| rmxdioph 43005 | X is a Diophantine functio... |
| jm3.1lem1 43006 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem2 43007 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem3 43008 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1 43009 | Diophantine expression for... |
| expdiophlem1 43010 | Lemma for ~ expdioph . Fu... |
| expdiophlem2 43011 | Lemma for ~ expdioph . Ex... |
| expdioph 43012 | The exponential function i... |
| setindtr 43013 | Set induction for sets con... |
| setindtrs 43014 | Set induction scheme witho... |
| dford3lem1 43015 | Lemma for ~ dford3 . (Con... |
| dford3lem2 43016 | Lemma for ~ dford3 . (Con... |
| dford3 43017 | Ordinals are precisely the... |
| dford4 43018 | ~ dford3 expressed in prim... |
| wopprc 43019 | Unrelated: Wiener pairs t... |
| rpnnen3lem 43020 | Lemma for ~ rpnnen3 . (Co... |
| rpnnen3 43021 | Dedekind cut injection of ... |
| axac10 43022 | Characterization of choice... |
| harinf 43023 | The Hartogs number of an i... |
| wdom2d2 43024 | Deduction for weak dominan... |
| ttac 43025 | Tarski's theorem about cho... |
| pw2f1ocnv 43026 | Define a bijection between... |
| pw2f1o2 43027 | Define a bijection between... |
| pw2f1o2val 43028 | Function value of the ~ pw... |
| pw2f1o2val2 43029 | Membership in a mapped set... |
| limsuc2 43030 | Limit ordinals in the sens... |
| wepwsolem 43031 | Transfer an ordering on ch... |
| wepwso 43032 | A well-ordering induces a ... |
| dnnumch1 43033 | Define an enumeration of a... |
| dnnumch2 43034 | Define an enumeration (wea... |
| dnnumch3lem 43035 | Value of the ordinal injec... |
| dnnumch3 43036 | Define an injection from a... |
| dnwech 43037 | Define a well-ordering fro... |
| fnwe2val 43038 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem1 43039 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem2 43040 | Lemma for ~ fnwe2 . An el... |
| fnwe2lem3 43041 | Lemma for ~ fnwe2 . Trich... |
| fnwe2 43042 | A well-ordering can be con... |
| aomclem1 43043 | Lemma for ~ dfac11 . This... |
| aomclem2 43044 | Lemma for ~ dfac11 . Succ... |
| aomclem3 43045 | Lemma for ~ dfac11 . Succ... |
| aomclem4 43046 | Lemma for ~ dfac11 . Limi... |
| aomclem5 43047 | Lemma for ~ dfac11 . Comb... |
| aomclem6 43048 | Lemma for ~ dfac11 . Tran... |
| aomclem7 43049 | Lemma for ~ dfac11 . ` ( R... |
| aomclem8 43050 | Lemma for ~ dfac11 . Perf... |
| dfac11 43051 | The right-hand side of thi... |
| kelac1 43052 | Kelley's choice, basic for... |
| kelac2lem 43053 | Lemma for ~ kelac2 and ~ d... |
| kelac2 43054 | Kelley's choice, most comm... |
| dfac21 43055 | Tychonoff's theorem is a c... |
| islmodfg 43058 | Property of a finitely gen... |
| islssfg 43059 | Property of a finitely gen... |
| islssfg2 43060 | Property of a finitely gen... |
| islssfgi 43061 | Finitely spanned subspaces... |
| fglmod 43062 | Finitely generated left mo... |
| lsmfgcl 43063 | The sum of two finitely ge... |
| islnm 43066 | Property of being a Noethe... |
| islnm2 43067 | Property of being a Noethe... |
| lnmlmod 43068 | A Noetherian left module i... |
| lnmlssfg 43069 | A submodule of Noetherian ... |
| lnmlsslnm 43070 | All submodules of a Noethe... |
| lnmfg 43071 | A Noetherian left module i... |
| kercvrlsm 43072 | The domain of a linear fun... |
| lmhmfgima 43073 | A homomorphism maps finite... |
| lnmepi 43074 | Epimorphic images of Noeth... |
| lmhmfgsplit 43075 | If the kernel and range of... |
| lmhmlnmsplit 43076 | If the kernel and range of... |
| lnmlmic 43077 | Noetherian is an invariant... |
| pwssplit4 43078 | Splitting for structure po... |
| filnm 43079 | Finite left modules are No... |
| pwslnmlem0 43080 | Zeroeth powers are Noether... |
| pwslnmlem1 43081 | First powers are Noetheria... |
| pwslnmlem2 43082 | A sum of powers is Noether... |
| pwslnm 43083 | Finite powers of Noetheria... |
| unxpwdom3 43084 | Weaker version of ~ unxpwd... |
| pwfi2f1o 43085 | The ~ pw2f1o bijection rel... |
| pwfi2en 43086 | Finitely supported indicat... |
| frlmpwfi 43087 | Formal linear combinations... |
| gicabl 43088 | Being Abelian is a group i... |
| imasgim 43089 | A relabeling of the elemen... |
| isnumbasgrplem1 43090 | A set which is equipollent... |
| harn0 43091 | The Hartogs number of a se... |
| numinfctb 43092 | A numerable infinite set c... |
| isnumbasgrplem2 43093 | If the (to be thought of a... |
| isnumbasgrplem3 43094 | Every nonempty numerable s... |
| isnumbasabl 43095 | A set is numerable iff it ... |
| isnumbasgrp 43096 | A set is numerable iff it ... |
| dfacbasgrp 43097 | A choice equivalent in abs... |
| islnr 43100 | Property of a left-Noether... |
| lnrring 43101 | Left-Noetherian rings are ... |
| lnrlnm 43102 | Left-Noetherian rings have... |
| islnr2 43103 | Property of being a left-N... |
| islnr3 43104 | Relate left-Noetherian rin... |
| lnr2i 43105 | Given an ideal in a left-N... |
| lpirlnr 43106 | Left principal ideal rings... |
| lnrfrlm 43107 | Finite-dimensional free mo... |
| lnrfg 43108 | Finitely-generated modules... |
| lnrfgtr 43109 | A submodule of a finitely ... |
| hbtlem1 43112 | Value of the leading coeff... |
| hbtlem2 43113 | Leading coefficient ideals... |
| hbtlem7 43114 | Functionality of leading c... |
| hbtlem4 43115 | The leading ideal function... |
| hbtlem3 43116 | The leading ideal function... |
| hbtlem5 43117 | The leading ideal function... |
| hbtlem6 43118 | There is a finite set of p... |
| hbt 43119 | The Hilbert Basis Theorem ... |
| dgrsub2 43124 | Subtracting two polynomial... |
| elmnc 43125 | Property of a monic polyno... |
| mncply 43126 | A monic polynomial is a po... |
| mnccoe 43127 | A monic polynomial has lea... |
| mncn0 43128 | A monic polynomial is not ... |
| dgraaval 43133 | Value of the degree functi... |
| dgraalem 43134 | Properties of the degree o... |
| dgraacl 43135 | Closure of the degree func... |
| dgraaf 43136 | Degree function on algebra... |
| dgraaub 43137 | Upper bound on degree of a... |
| dgraa0p 43138 | A rational polynomial of d... |
| mpaaeu 43139 | An algebraic number has ex... |
| mpaaval 43140 | Value of the minimal polyn... |
| mpaalem 43141 | Properties of the minimal ... |
| mpaacl 43142 | Minimal polynomial is a po... |
| mpaadgr 43143 | Minimal polynomial has deg... |
| mpaaroot 43144 | The minimal polynomial of ... |
| mpaamn 43145 | Minimal polynomial is moni... |
| itgoval 43150 | Value of the integral-over... |
| aaitgo 43151 | The standard algebraic num... |
| itgoss 43152 | An integral element is int... |
| itgocn 43153 | All integral elements are ... |
| cnsrexpcl 43154 | Exponentiation is closed i... |
| fsumcnsrcl 43155 | Finite sums are closed in ... |
| cnsrplycl 43156 | Polynomials are closed in ... |
| rgspnid 43157 | The span of a subring is i... |
| rngunsnply 43158 | Adjoining one element to a... |
| flcidc 43159 | Finite linear combinations... |
| algstr 43162 | Lemma to shorten proofs of... |
| algbase 43163 | The base set of a construc... |
| algaddg 43164 | The additive operation of ... |
| algmulr 43165 | The multiplicative operati... |
| algsca 43166 | The set of scalars of a co... |
| algvsca 43167 | The scalar product operati... |
| mendval 43168 | Value of the module endomo... |
| mendbas 43169 | Base set of the module end... |
| mendplusgfval 43170 | Addition in the module end... |
| mendplusg 43171 | A specific addition in the... |
| mendmulrfval 43172 | Multiplication in the modu... |
| mendmulr 43173 | A specific multiplication ... |
| mendsca 43174 | The module endomorphism al... |
| mendvscafval 43175 | Scalar multiplication in t... |
| mendvsca 43176 | A specific scalar multipli... |
| mendring 43177 | The module endomorphism al... |
| mendlmod 43178 | The module endomorphism al... |
| mendassa 43179 | The module endomorphism al... |
| idomodle 43180 | Limit on the number of ` N... |
| fiuneneq 43181 | Two finite sets of equal s... |
| idomsubgmo 43182 | The units of an integral d... |
| proot1mul 43183 | Any primitive ` N ` -th ro... |
| proot1hash 43184 | If an integral domain has ... |
| proot1ex 43185 | The complex field has prim... |
| mon1psubm 43188 | Monic polynomials are a mu... |
| deg1mhm 43189 | Homomorphic property of th... |
| cytpfn 43190 | Functionality of the cyclo... |
| cytpval 43191 | Substitutions for the Nth ... |
| fgraphopab 43192 | Express a function as a su... |
| fgraphxp 43193 | Express a function as a su... |
| hausgraph 43194 | The graph of a continuous ... |
| r1sssucd 43199 | Deductive form of ~ r1sssu... |
| iocunico 43200 | Split an open interval int... |
| iocinico 43201 | The intersection of two se... |
| iocmbl 43202 | An open-below, closed-abov... |
| cnioobibld 43203 | A bounded, continuous func... |
| arearect 43204 | The area of a rectangle wh... |
| areaquad 43205 | The area of a quadrilatera... |
| uniel 43206 | Two ways to say a union is... |
| unielss 43207 | Two ways to say the union ... |
| unielid 43208 | Two ways to say the union ... |
| ssunib 43209 | Two ways to say a class is... |
| rp-intrabeq 43210 | Equality theorem for supre... |
| rp-unirabeq 43211 | Equality theorem for infim... |
| onmaxnelsup 43212 | Two ways to say the maximu... |
| onsupneqmaxlim0 43213 | If the supremum of a class... |
| onsupcl2 43214 | The supremum of a set of o... |
| onuniintrab 43215 | The union of a set of ordi... |
| onintunirab 43216 | The intersection of a non-... |
| onsupnmax 43217 | If the union of a class of... |
| onsupuni 43218 | The supremum of a set of o... |
| onsupuni2 43219 | The supremum of a set of o... |
| onsupintrab 43220 | The supremum of a set of o... |
| onsupintrab2 43221 | The supremum of a set of o... |
| onsupcl3 43222 | The supremum of a set of o... |
| onsupex3 43223 | The supremum of a set of o... |
| onuniintrab2 43224 | The union of a set of ordi... |
| oninfint 43225 | The infimum of a non-empty... |
| oninfunirab 43226 | The infimum of a non-empty... |
| oninfcl2 43227 | The infimum of a non-empty... |
| onsupmaxb 43228 | The union of a class of or... |
| onexgt 43229 | For any ordinal, there is ... |
| onexomgt 43230 | For any ordinal, there is ... |
| omlimcl2 43231 | The product of a limit ord... |
| onexlimgt 43232 | For any ordinal, there is ... |
| onexoegt 43233 | For any ordinal, there is ... |
| oninfex2 43234 | The infimum of a non-empty... |
| onsupeqmax 43235 | Condition when the supremu... |
| onsupeqnmax 43236 | Condition when the supremu... |
| onsuplub 43237 | The supremum of a set of o... |
| onsupnub 43238 | An upper bound of a set of... |
| onfisupcl 43239 | Sufficient condition when ... |
| onelord 43240 | Every element of a ordinal... |
| onepsuc 43241 | Every ordinal is less than... |
| epsoon 43242 | The ordinals are strictly ... |
| epirron 43243 | The strict order on the or... |
| oneptr 43244 | The strict order on the or... |
| oneltr 43245 | The elementhood relation o... |
| oneptri 43246 | The strict, complete (line... |
| ordeldif 43247 | Membership in the differen... |
| ordeldifsucon 43248 | Membership in the differen... |
| ordeldif1o 43249 | Membership in the differen... |
| ordne0gt0 43250 | Ordinal zero is less than ... |
| ondif1i 43251 | Ordinal zero is less than ... |
| onsucelab 43252 | The successor of every ord... |
| dflim6 43253 | A limit ordinal is a non-z... |
| limnsuc 43254 | A limit ordinal is not an ... |
| onsucss 43255 | If one ordinal is less tha... |
| ordnexbtwnsuc 43256 | For any distinct pair of o... |
| orddif0suc 43257 | For any distinct pair of o... |
| onsucf1lem 43258 | For ordinals, the successo... |
| onsucf1olem 43259 | The successor operation is... |
| onsucrn 43260 | The successor operation is... |
| onsucf1o 43261 | The successor operation is... |
| dflim7 43262 | A limit ordinal is a non-z... |
| onov0suclim 43263 | Compactly express rules fo... |
| oa0suclim 43264 | Closed form expression of ... |
| om0suclim 43265 | Closed form expression of ... |
| oe0suclim 43266 | Closed form expression of ... |
| oaomoecl 43267 | The operations of addition... |
| onsupsucismax 43268 | If the union of a set of o... |
| onsssupeqcond 43269 | If for every element of a ... |
| limexissup 43270 | An ordinal which is a limi... |
| limiun 43271 | A limit ordinal is the uni... |
| limexissupab 43272 | An ordinal which is a limi... |
| om1om1r 43273 | Ordinal one is both a left... |
| oe0rif 43274 | Ordinal zero raised to any... |
| oasubex 43275 | While subtraction can't be... |
| nnamecl 43276 | Natural numbers are closed... |
| onsucwordi 43277 | The successor operation pr... |
| oalim2cl 43278 | The ordinal sum of any ord... |
| oaltublim 43279 | Given ` C ` is a limit ord... |
| oaordi3 43280 | Ordinal addition of the sa... |
| oaord3 43281 | When the same ordinal is a... |
| 1oaomeqom 43282 | Ordinal one plus omega is ... |
| oaabsb 43283 | The right addend absorbs t... |
| oaordnrex 43284 | When omega is added on the... |
| oaordnr 43285 | When the same ordinal is a... |
| omge1 43286 | Any non-zero ordinal produ... |
| omge2 43287 | Any non-zero ordinal produ... |
| omlim2 43288 | The non-zero product with ... |
| omord2lim 43289 | Given a limit ordinal, the... |
| omord2i 43290 | Ordinal multiplication of ... |
| omord2com 43291 | When the same non-zero ord... |
| 2omomeqom 43292 | Ordinal two times omega is... |
| omnord1ex 43293 | When omega is multiplied o... |
| omnord1 43294 | When the same non-zero ord... |
| oege1 43295 | Any non-zero ordinal power... |
| oege2 43296 | Any power of an ordinal at... |
| rp-oelim2 43297 | The power of an ordinal at... |
| oeord2lim 43298 | Given a limit ordinal, the... |
| oeord2i 43299 | Ordinal exponentiation of ... |
| oeord2com 43300 | When the same base at leas... |
| nnoeomeqom 43301 | Any natural number at leas... |
| df3o2 43302 | Ordinal 3 is the unordered... |
| df3o3 43303 | Ordinal 3, fully expanded.... |
| oenord1ex 43304 | When ordinals two and thre... |
| oenord1 43305 | When two ordinals (both at... |
| oaomoencom 43306 | Ordinal addition, multipli... |
| oenassex 43307 | Ordinal two raised to two ... |
| oenass 43308 | Ordinal exponentiation is ... |
| cantnftermord 43309 | For terms of the form of a... |
| cantnfub 43310 | Given a finite number of t... |
| cantnfub2 43311 | Given a finite number of t... |
| bropabg 43312 | Equivalence for two classe... |
| cantnfresb 43313 | A Cantor normal form which... |
| cantnf2 43314 | For every ordinal, ` A ` ,... |
| oawordex2 43315 | If ` C ` is between ` A ` ... |
| nnawordexg 43316 | If an ordinal, ` B ` , is ... |
| succlg 43317 | Closure law for ordinal su... |
| dflim5 43318 | A limit ordinal is either ... |
| oacl2g 43319 | Closure law for ordinal ad... |
| onmcl 43320 | If an ordinal is less than... |
| omabs2 43321 | Ordinal multiplication by ... |
| omcl2 43322 | Closure law for ordinal mu... |
| omcl3g 43323 | Closure law for ordinal mu... |
| ordsssucb 43324 | An ordinal number is less ... |
| tfsconcatlem 43325 | Lemma for ~ tfsconcatun . ... |
| tfsconcatun 43326 | The concatenation of two t... |
| tfsconcatfn 43327 | The concatenation of two t... |
| tfsconcatfv1 43328 | An early value of the conc... |
| tfsconcatfv2 43329 | A latter value of the conc... |
| tfsconcatfv 43330 | The value of the concatena... |
| tfsconcatrn 43331 | The range of the concatena... |
| tfsconcatfo 43332 | The concatenation of two t... |
| tfsconcatb0 43333 | The concatentation with th... |
| tfsconcat0i 43334 | The concatentation with th... |
| tfsconcat0b 43335 | The concatentation with th... |
| tfsconcat00 43336 | The concatentation of two ... |
| tfsconcatrev 43337 | If the domain of a transfi... |
| tfsconcatrnss12 43338 | The range of the concatena... |
| tfsconcatrnss 43339 | The concatenation of trans... |
| tfsconcatrnsson 43340 | The concatenation of trans... |
| tfsnfin 43341 | A transfinite sequence is ... |
| rp-tfslim 43342 | The limit of a sequence of... |
| ofoafg 43343 | Addition operator for func... |
| ofoaf 43344 | Addition operator for func... |
| ofoafo 43345 | Addition operator for func... |
| ofoacl 43346 | Closure law for component ... |
| ofoaid1 43347 | Identity law for component... |
| ofoaid2 43348 | Identity law for component... |
| ofoaass 43349 | Component-wise addition of... |
| ofoacom 43350 | Component-wise addition of... |
| naddcnff 43351 | Addition operator for Cant... |
| naddcnffn 43352 | Addition operator for Cant... |
| naddcnffo 43353 | Addition of Cantor normal ... |
| naddcnfcl 43354 | Closure law for component-... |
| naddcnfcom 43355 | Component-wise ordinal add... |
| naddcnfid1 43356 | Identity law for component... |
| naddcnfid2 43357 | Identity law for component... |
| naddcnfass 43358 | Component-wise addition of... |
| onsucunifi 43359 | The successor to the union... |
| sucunisn 43360 | The successor to the union... |
| onsucunipr 43361 | The successor to the union... |
| onsucunitp 43362 | The successor to the union... |
| oaun3lem1 43363 | The class of all ordinal s... |
| oaun3lem2 43364 | The class of all ordinal s... |
| oaun3lem3 43365 | The class of all ordinal s... |
| oaun3lem4 43366 | The class of all ordinal s... |
| rp-abid 43367 | Two ways to express a clas... |
| oadif1lem 43368 | Express the set difference... |
| oadif1 43369 | Express the set difference... |
| oaun2 43370 | Ordinal addition as a unio... |
| oaun3 43371 | Ordinal addition as a unio... |
| naddov4 43372 | Alternate expression for n... |
| nadd2rabtr 43373 | The set of ordinals which ... |
| nadd2rabord 43374 | The set of ordinals which ... |
| nadd2rabex 43375 | The class of ordinals whic... |
| nadd2rabon 43376 | The set of ordinals which ... |
| nadd1rabtr 43377 | The set of ordinals which ... |
| nadd1rabord 43378 | The set of ordinals which ... |
| nadd1rabex 43379 | The class of ordinals whic... |
| nadd1rabon 43380 | The set of ordinals which ... |
| nadd1suc 43381 | Natural addition with 1 is... |
| naddass1 43382 | Natural addition of ordina... |
| naddgeoa 43383 | Natural addition results i... |
| naddonnn 43384 | Natural addition with a na... |
| naddwordnexlem0 43385 | When ` A ` is the sum of a... |
| naddwordnexlem1 43386 | When ` A ` is the sum of a... |
| naddwordnexlem2 43387 | When ` A ` is the sum of a... |
| naddwordnexlem3 43388 | When ` A ` is the sum of a... |
| oawordex3 43389 | When ` A ` is the sum of a... |
| naddwordnexlem4 43390 | When ` A ` is the sum of a... |
| ordsssucim 43391 | If an ordinal is less than... |
| insucid 43392 | The intersection of a clas... |
| om2 43393 | Two ways to double an ordi... |
| oaltom 43394 | Multiplication eventually ... |
| oe2 43395 | Two ways to square an ordi... |
| omltoe 43396 | Exponentiation eventually ... |
| abeqabi 43397 | Generalized condition for ... |
| abpr 43398 | Condition for a class abst... |
| abtp 43399 | Condition for a class abst... |
| ralopabb 43400 | Restricted universal quant... |
| fpwfvss 43401 | Functions into a powerset ... |
| sdomne0 43402 | A class that strictly domi... |
| sdomne0d 43403 | A class that strictly domi... |
| safesnsupfiss 43404 | If ` B ` is a finite subse... |
| safesnsupfiub 43405 | If ` B ` is a finite subse... |
| safesnsupfidom1o 43406 | If ` B ` is a finite subse... |
| safesnsupfilb 43407 | If ` B ` is a finite subse... |
| isoeq145d 43408 | Equality deduction for iso... |
| resisoeq45d 43409 | Equality deduction for equ... |
| negslem1 43410 | An equivalence between ide... |
| nvocnvb 43411 | Equivalence to saying the ... |
| rp-brsslt 43412 | Binary relation form of a ... |
| nla0002 43413 | Extending a linear order t... |
| nla0003 43414 | Extending a linear order t... |
| nla0001 43415 | Extending a linear order t... |
| faosnf0.11b 43416 | ` B ` is called a non-limi... |
| dfno2 43417 | A surreal number, in the f... |
| onnog 43418 | Every ordinal maps to a su... |
| onnobdayg 43419 | Every ordinal maps to a su... |
| bdaybndex 43420 | Bounds formed from the bir... |
| bdaybndbday 43421 | Bounds formed from the bir... |
| onno 43422 | Every ordinal maps to a su... |
| onnoi 43423 | Every ordinal maps to a su... |
| 0no 43424 | Ordinal zero maps to a sur... |
| 1no 43425 | Ordinal one maps to a surr... |
| 2no 43426 | Ordinal two maps to a surr... |
| 3no 43427 | Ordinal three maps to a su... |
| 4no 43428 | Ordinal four maps to a sur... |
| fnimafnex 43429 | The functional image of a ... |
| nlimsuc 43430 | A successor is not a limit... |
| nlim1NEW 43431 | 1 is not a limit ordinal. ... |
| nlim2NEW 43432 | 2 is not a limit ordinal. ... |
| nlim3 43433 | 3 is not a limit ordinal. ... |
| nlim4 43434 | 4 is not a limit ordinal. ... |
| oa1un 43435 | Given ` A e. On ` , let ` ... |
| oa1cl 43436 | ` A +o 1o ` is in ` On ` .... |
| 0finon 43437 | 0 is a finite ordinal. Se... |
| 1finon 43438 | 1 is a finite ordinal. Se... |
| 2finon 43439 | 2 is a finite ordinal. Se... |
| 3finon 43440 | 3 is a finite ordinal. Se... |
| 4finon 43441 | 4 is a finite ordinal. Se... |
| finona1cl 43442 | The finite ordinals are cl... |
| finonex 43443 | The finite ordinals are a ... |
| fzunt 43444 | Union of two adjacent fini... |
| fzuntd 43445 | Union of two adjacent fini... |
| fzunt1d 43446 | Union of two overlapping f... |
| fzuntgd 43447 | Union of two adjacent or o... |
| ifpan123g 43448 | Conjunction of conditional... |
| ifpan23 43449 | Conjunction of conditional... |
| ifpdfor2 43450 | Define or in terms of cond... |
| ifporcor 43451 | Corollary of commutation o... |
| ifpdfan2 43452 | Define and with conditiona... |
| ifpancor 43453 | Corollary of commutation o... |
| ifpdfor 43454 | Define or in terms of cond... |
| ifpdfan 43455 | Define and with conditiona... |
| ifpbi2 43456 | Equivalence theorem for co... |
| ifpbi3 43457 | Equivalence theorem for co... |
| ifpim1 43458 | Restate implication as con... |
| ifpnot 43459 | Restate negated wff as con... |
| ifpid2 43460 | Restate wff as conditional... |
| ifpim2 43461 | Restate implication as con... |
| ifpbi23 43462 | Equivalence theorem for co... |
| ifpbiidcor 43463 | Restatement of ~ biid . (... |
| ifpbicor 43464 | Corollary of commutation o... |
| ifpxorcor 43465 | Corollary of commutation o... |
| ifpbi1 43466 | Equivalence theorem for co... |
| ifpnot23 43467 | Negation of conditional lo... |
| ifpnotnotb 43468 | Factor conditional logic o... |
| ifpnorcor 43469 | Corollary of commutation o... |
| ifpnancor 43470 | Corollary of commutation o... |
| ifpnot23b 43471 | Negation of conditional lo... |
| ifpbiidcor2 43472 | Restatement of ~ biid . (... |
| ifpnot23c 43473 | Negation of conditional lo... |
| ifpnot23d 43474 | Negation of conditional lo... |
| ifpdfnan 43475 | Define nand as conditional... |
| ifpdfxor 43476 | Define xor as conditional ... |
| ifpbi12 43477 | Equivalence theorem for co... |
| ifpbi13 43478 | Equivalence theorem for co... |
| ifpbi123 43479 | Equivalence theorem for co... |
| ifpidg 43480 | Restate wff as conditional... |
| ifpid3g 43481 | Restate wff as conditional... |
| ifpid2g 43482 | Restate wff as conditional... |
| ifpid1g 43483 | Restate wff as conditional... |
| ifpim23g 43484 | Restate implication as con... |
| ifpim3 43485 | Restate implication as con... |
| ifpnim1 43486 | Restate negated implicatio... |
| ifpim4 43487 | Restate implication as con... |
| ifpnim2 43488 | Restate negated implicatio... |
| ifpim123g 43489 | Implication of conditional... |
| ifpim1g 43490 | Implication of conditional... |
| ifp1bi 43491 | Substitute the first eleme... |
| ifpbi1b 43492 | When the first variable is... |
| ifpimimb 43493 | Factor conditional logic o... |
| ifpororb 43494 | Factor conditional logic o... |
| ifpananb 43495 | Factor conditional logic o... |
| ifpnannanb 43496 | Factor conditional logic o... |
| ifpor123g 43497 | Disjunction of conditional... |
| ifpimim 43498 | Consequnce of implication.... |
| ifpbibib 43499 | Factor conditional logic o... |
| ifpxorxorb 43500 | Factor conditional logic o... |
| rp-fakeimass 43501 | A special case where impli... |
| rp-fakeanorass 43502 | A special case where a mix... |
| rp-fakeoranass 43503 | A special case where a mix... |
| rp-fakeinunass 43504 | A special case where a mix... |
| rp-fakeuninass 43505 | A special case where a mix... |
| rp-isfinite5 43506 | A set is said to be finite... |
| rp-isfinite6 43507 | A set is said to be finite... |
| intabssd 43508 | When for each element ` y ... |
| eu0 43509 | There is only one empty se... |
| epelon2 43510 | Over the ordinal numbers, ... |
| ontric3g 43511 | For all ` x , y e. On ` , ... |
| dfsucon 43512 | ` A ` is called a successo... |
| snen1g 43513 | A singleton is equinumerou... |
| snen1el 43514 | A singleton is equinumerou... |
| sn1dom 43515 | A singleton is dominated b... |
| pr2dom 43516 | An unordered pair is domin... |
| tr3dom 43517 | An unordered triple is dom... |
| ensucne0 43518 | A class equinumerous to a ... |
| ensucne0OLD 43519 | A class equinumerous to a ... |
| dfom6 43520 | Let ` _om ` be defined to ... |
| infordmin 43521 | ` _om ` is the smallest in... |
| iscard4 43522 | Two ways to express the pr... |
| minregex 43523 | Given any cardinal number ... |
| minregex2 43524 | Given any cardinal number ... |
| iscard5 43525 | Two ways to express the pr... |
| elrncard 43526 | Let us define a cardinal n... |
| harval3 43527 | ` ( har `` A ) ` is the le... |
| harval3on 43528 | For any ordinal number ` A... |
| omssrncard 43529 | All natural numbers are ca... |
| 0iscard 43530 | 0 is a cardinal number. (... |
| 1iscard 43531 | 1 is a cardinal number. (... |
| omiscard 43532 | ` _om ` is a cardinal numb... |
| sucomisnotcard 43533 | ` _om +o 1o ` is not a car... |
| nna1iscard 43534 | For any natural number, th... |
| har2o 43535 | The least cardinal greater... |
| en2pr 43536 | A class is equinumerous to... |
| pr2cv 43537 | If an unordered pair is eq... |
| pr2el1 43538 | If an unordered pair is eq... |
| pr2cv1 43539 | If an unordered pair is eq... |
| pr2el2 43540 | If an unordered pair is eq... |
| pr2cv2 43541 | If an unordered pair is eq... |
| pren2 43542 | An unordered pair is equin... |
| pr2eldif1 43543 | If an unordered pair is eq... |
| pr2eldif2 43544 | If an unordered pair is eq... |
| pren2d 43545 | A pair of two distinct set... |
| aleph1min 43546 | ` ( aleph `` 1o ) ` is the... |
| alephiso2 43547 | ` aleph ` is a strictly or... |
| alephiso3 43548 | ` aleph ` is a strictly or... |
| pwelg 43549 | The powerclass is an eleme... |
| pwinfig 43550 | The powerclass of an infin... |
| pwinfi2 43551 | The powerclass of an infin... |
| pwinfi3 43552 | The powerclass of an infin... |
| pwinfi 43553 | The powerclass of an infin... |
| fipjust 43554 | A definition of the finite... |
| cllem0 43555 | The class of all sets with... |
| superficl 43556 | The class of all supersets... |
| superuncl 43557 | The class of all supersets... |
| ssficl 43558 | The class of all subsets o... |
| ssuncl 43559 | The class of all subsets o... |
| ssdifcl 43560 | The class of all subsets o... |
| sssymdifcl 43561 | The class of all subsets o... |
| fiinfi 43562 | If two classes have the fi... |
| rababg 43563 | Condition when restricted ... |
| elinintab 43564 | Two ways of saying a set i... |
| elmapintrab 43565 | Two ways to say a set is a... |
| elinintrab 43566 | Two ways of saying a set i... |
| inintabss 43567 | Upper bound on intersectio... |
| inintabd 43568 | Value of the intersection ... |
| xpinintabd 43569 | Value of the intersection ... |
| relintabex 43570 | If the intersection of a c... |
| elcnvcnvintab 43571 | Two ways of saying a set i... |
| relintab 43572 | Value of the intersection ... |
| nonrel 43573 | A non-relation is equal to... |
| elnonrel 43574 | Only an ordered pair where... |
| cnvssb 43575 | Subclass theorem for conve... |
| relnonrel 43576 | The non-relation part of a... |
| cnvnonrel 43577 | The converse of the non-re... |
| brnonrel 43578 | A non-relation cannot rela... |
| dmnonrel 43579 | The domain of the non-rela... |
| rnnonrel 43580 | The range of the non-relat... |
| resnonrel 43581 | A restriction of the non-r... |
| imanonrel 43582 | An image under the non-rel... |
| cononrel1 43583 | Composition with the non-r... |
| cononrel2 43584 | Composition with the non-r... |
| elmapintab 43585 | Two ways to say a set is a... |
| fvnonrel 43586 | The function value of any ... |
| elinlem 43587 | Two ways to say a set is a... |
| elcnvcnvlem 43588 | Two ways to say a set is a... |
| cnvcnvintabd 43589 | Value of the relationship ... |
| elcnvlem 43590 | Two ways to say a set is a... |
| elcnvintab 43591 | Two ways of saying a set i... |
| cnvintabd 43592 | Value of the converse of t... |
| undmrnresiss 43593 | Two ways of saying the ide... |
| reflexg 43594 | Two ways of saying a relat... |
| cnvssco 43595 | A condition weaker than re... |
| refimssco 43596 | Reflexive relations are su... |
| cleq2lem 43597 | Equality implies bijection... |
| cbvcllem 43598 | Change of bound variable i... |
| clublem 43599 | If a superset ` Y ` of ` X... |
| clss2lem 43600 | The closure of a property ... |
| dfid7 43601 | Definition of identity rel... |
| mptrcllem 43602 | Show two versions of a clo... |
| cotrintab 43603 | The intersection of a clas... |
| rclexi 43604 | The reflexive closure of a... |
| rtrclexlem 43605 | Existence of relation impl... |
| rtrclex 43606 | The reflexive-transitive c... |
| trclubgNEW 43607 | If a relation exists then ... |
| trclubNEW 43608 | If a relation exists then ... |
| trclexi 43609 | The transitive closure of ... |
| rtrclexi 43610 | The reflexive-transitive c... |
| clrellem 43611 | When the property ` ps ` h... |
| clcnvlem 43612 | When ` A ` , an upper boun... |
| cnvtrucl0 43613 | The converse of the trivia... |
| cnvrcl0 43614 | The converse of the reflex... |
| cnvtrcl0 43615 | The converse of the transi... |
| dmtrcl 43616 | The domain of the transiti... |
| rntrcl 43617 | The range of the transitiv... |
| dfrtrcl5 43618 | Definition of reflexive-tr... |
| trcleq2lemRP 43619 | Equality implies bijection... |
| sqrtcvallem1 43620 | Two ways of saying a compl... |
| reabsifneg 43621 | Alternate expression for t... |
| reabsifnpos 43622 | Alternate expression for t... |
| reabsifpos 43623 | Alternate expression for t... |
| reabsifnneg 43624 | Alternate expression for t... |
| reabssgn 43625 | Alternate expression for t... |
| sqrtcvallem2 43626 | Equivalent to saying that ... |
| sqrtcvallem3 43627 | Equivalent to saying that ... |
| sqrtcvallem4 43628 | Equivalent to saying that ... |
| sqrtcvallem5 43629 | Equivalent to saying that ... |
| sqrtcval 43630 | Explicit formula for the c... |
| sqrtcval2 43631 | Explicit formula for the c... |
| resqrtval 43632 | Real part of the complex s... |
| imsqrtval 43633 | Imaginary part of the comp... |
| resqrtvalex 43634 | Example for ~ resqrtval . ... |
| imsqrtvalex 43635 | Example for ~ imsqrtval . ... |
| al3im 43636 | Version of ~ ax-4 for a ne... |
| intima0 43637 | Two ways of expressing the... |
| elimaint 43638 | Element of image of inters... |
| cnviun 43639 | Converse of indexed union.... |
| imaiun1 43640 | The image of an indexed un... |
| coiun1 43641 | Composition with an indexe... |
| elintima 43642 | Element of intersection of... |
| intimass 43643 | The image under the inters... |
| intimass2 43644 | The image under the inters... |
| intimag 43645 | Requirement for the image ... |
| intimasn 43646 | Two ways to express the im... |
| intimasn2 43647 | Two ways to express the im... |
| ss2iundf 43648 | Subclass theorem for index... |
| ss2iundv 43649 | Subclass theorem for index... |
| cbviuneq12df 43650 | Rule used to change the bo... |
| cbviuneq12dv 43651 | Rule used to change the bo... |
| conrel1d 43652 | Deduction about compositio... |
| conrel2d 43653 | Deduction about compositio... |
| trrelind 43654 | The intersection of transi... |
| xpintrreld 43655 | The intersection of a tran... |
| restrreld 43656 | The restriction of a trans... |
| trrelsuperreldg 43657 | Concrete construction of a... |
| trficl 43658 | The class of all transitiv... |
| cnvtrrel 43659 | The converse of a transiti... |
| trrelsuperrel2dg 43660 | Concrete construction of a... |
| dfrcl2 43663 | Reflexive closure of a rel... |
| dfrcl3 43664 | Reflexive closure of a rel... |
| dfrcl4 43665 | Reflexive closure of a rel... |
| relexp2 43666 | A set operated on by the r... |
| relexpnul 43667 | If the domain and range of... |
| eliunov2 43668 | Membership in the indexed ... |
| eltrclrec 43669 | Membership in the indexed ... |
| elrtrclrec 43670 | Membership in the indexed ... |
| briunov2 43671 | Two classes related by the... |
| brmptiunrelexpd 43672 | If two elements are connec... |
| fvmptiunrelexplb0d 43673 | If the indexed union range... |
| fvmptiunrelexplb0da 43674 | If the indexed union range... |
| fvmptiunrelexplb1d 43675 | If the indexed union range... |
| brfvid 43676 | If two elements are connec... |
| brfvidRP 43677 | If two elements are connec... |
| fvilbd 43678 | A set is a subset of its i... |
| fvilbdRP 43679 | A set is a subset of its i... |
| brfvrcld 43680 | If two elements are connec... |
| brfvrcld2 43681 | If two elements are connec... |
| fvrcllb0d 43682 | A restriction of the ident... |
| fvrcllb0da 43683 | A restriction of the ident... |
| fvrcllb1d 43684 | A set is a subset of its i... |
| brtrclrec 43685 | Two classes related by the... |
| brrtrclrec 43686 | Two classes related by the... |
| briunov2uz 43687 | Two classes related by the... |
| eliunov2uz 43688 | Membership in the indexed ... |
| ov2ssiunov2 43689 | Any particular operator va... |
| relexp0eq 43690 | The zeroth power of relati... |
| iunrelexp0 43691 | Simplification of zeroth p... |
| relexpxpnnidm 43692 | Any positive power of a Ca... |
| relexpiidm 43693 | Any power of any restricti... |
| relexpss1d 43694 | The relational power of a ... |
| comptiunov2i 43695 | The composition two indexe... |
| corclrcl 43696 | The reflexive closure is i... |
| iunrelexpmin1 43697 | The indexed union of relat... |
| relexpmulnn 43698 | With exponents limited to ... |
| relexpmulg 43699 | With ordered exponents, th... |
| trclrelexplem 43700 | The union of relational po... |
| iunrelexpmin2 43701 | The indexed union of relat... |
| relexp01min 43702 | With exponents limited to ... |
| relexp1idm 43703 | Repeated raising a relatio... |
| relexp0idm 43704 | Repeated raising a relatio... |
| relexp0a 43705 | Absorption law for zeroth ... |
| relexpxpmin 43706 | The composition of powers ... |
| relexpaddss 43707 | The composition of two pow... |
| iunrelexpuztr 43708 | The indexed union of relat... |
| dftrcl3 43709 | Transitive closure of a re... |
| brfvtrcld 43710 | If two elements are connec... |
| fvtrcllb1d 43711 | A set is a subset of its i... |
| trclfvcom 43712 | The transitive closure of ... |
| cnvtrclfv 43713 | The converse of the transi... |
| cotrcltrcl 43714 | The transitive closure is ... |
| trclimalb2 43715 | Lower bound for image unde... |
| brtrclfv2 43716 | Two ways to indicate two e... |
| trclfvdecomr 43717 | The transitive closure of ... |
| trclfvdecoml 43718 | The transitive closure of ... |
| dmtrclfvRP 43719 | The domain of the transiti... |
| rntrclfvRP 43720 | The range of the transitiv... |
| rntrclfv 43721 | The range of the transitiv... |
| dfrtrcl3 43722 | Reflexive-transitive closu... |
| brfvrtrcld 43723 | If two elements are connec... |
| fvrtrcllb0d 43724 | A restriction of the ident... |
| fvrtrcllb0da 43725 | A restriction of the ident... |
| fvrtrcllb1d 43726 | A set is a subset of its i... |
| dfrtrcl4 43727 | Reflexive-transitive closu... |
| corcltrcl 43728 | The composition of the ref... |
| cortrcltrcl 43729 | Composition with the refle... |
| corclrtrcl 43730 | Composition with the refle... |
| cotrclrcl 43731 | The composition of the ref... |
| cortrclrcl 43732 | Composition with the refle... |
| cotrclrtrcl 43733 | Composition with the refle... |
| cortrclrtrcl 43734 | The reflexive-transitive c... |
| frege77d 43735 | If the images of both ` { ... |
| frege81d 43736 | If the image of ` U ` is a... |
| frege83d 43737 | If the image of the union ... |
| frege96d 43738 | If ` C ` follows ` A ` in ... |
| frege87d 43739 | If the images of both ` { ... |
| frege91d 43740 | If ` B ` follows ` A ` in ... |
| frege97d 43741 | If ` A ` contains all elem... |
| frege98d 43742 | If ` C ` follows ` A ` and... |
| frege102d 43743 | If either ` A ` and ` C ` ... |
| frege106d 43744 | If ` B ` follows ` A ` in ... |
| frege108d 43745 | If either ` A ` and ` C ` ... |
| frege109d 43746 | If ` A ` contains all elem... |
| frege114d 43747 | If either ` R ` relates ` ... |
| frege111d 43748 | If either ` A ` and ` C ` ... |
| frege122d 43749 | If ` F ` is a function, ` ... |
| frege124d 43750 | If ` F ` is a function, ` ... |
| frege126d 43751 | If ` F ` is a function, ` ... |
| frege129d 43752 | If ` F ` is a function and... |
| frege131d 43753 | If ` F ` is a function and... |
| frege133d 43754 | If ` F ` is a function and... |
| dfxor4 43755 | Express exclusive-or in te... |
| dfxor5 43756 | Express exclusive-or in te... |
| df3or2 43757 | Express triple-or in terms... |
| df3an2 43758 | Express triple-and in term... |
| nev 43759 | Express that not every set... |
| 0pssin 43760 | Express that an intersecti... |
| dfhe2 43763 | The property of relation `... |
| dfhe3 43764 | The property of relation `... |
| heeq12 43765 | Equality law for relations... |
| heeq1 43766 | Equality law for relations... |
| heeq2 43767 | Equality law for relations... |
| sbcheg 43768 | Distribute proper substitu... |
| hess 43769 | Subclass law for relations... |
| xphe 43770 | Any Cartesian product is h... |
| 0he 43771 | The empty relation is here... |
| 0heALT 43772 | The empty relation is here... |
| he0 43773 | Any relation is hereditary... |
| unhe1 43774 | The union of two relations... |
| snhesn 43775 | Any singleton is hereditar... |
| idhe 43776 | The identity relation is h... |
| psshepw 43777 | The relation between sets ... |
| sshepw 43778 | The relation between sets ... |
| rp-simp2-frege 43781 | Simplification of triple c... |
| rp-simp2 43782 | Simplification of triple c... |
| rp-frege3g 43783 | Add antecedent to ~ ax-fre... |
| frege3 43784 | Add antecedent to ~ ax-fre... |
| rp-misc1-frege 43785 | Double-use of ~ ax-frege2 ... |
| rp-frege24 43786 | Introducing an embedded an... |
| rp-frege4g 43787 | Deduction related to distr... |
| frege4 43788 | Special case of closed for... |
| frege5 43789 | A closed form of ~ syl . ... |
| rp-7frege 43790 | Distribute antecedent and ... |
| rp-4frege 43791 | Elimination of a nested an... |
| rp-6frege 43792 | Elimination of a nested an... |
| rp-8frege 43793 | Eliminate antecedent when ... |
| rp-frege25 43794 | Closed form for ~ a1dd . ... |
| frege6 43795 | A closed form of ~ imim2d ... |
| axfrege8 43796 | Swap antecedents. Identic... |
| frege7 43797 | A closed form of ~ syl6 . ... |
| frege26 43799 | Identical to ~ idd . Prop... |
| frege27 43800 | We cannot (at the same tim... |
| frege9 43801 | Closed form of ~ syl with ... |
| frege12 43802 | A closed form of ~ com23 .... |
| frege11 43803 | Elimination of a nested an... |
| frege24 43804 | Closed form for ~ a1d . D... |
| frege16 43805 | A closed form of ~ com34 .... |
| frege25 43806 | Closed form for ~ a1dd . ... |
| frege18 43807 | Closed form of a syllogism... |
| frege22 43808 | A closed form of ~ com45 .... |
| frege10 43809 | Result commuting anteceden... |
| frege17 43810 | A closed form of ~ com3l .... |
| frege13 43811 | A closed form of ~ com3r .... |
| frege14 43812 | Closed form of a deduction... |
| frege19 43813 | A closed form of ~ syl6 . ... |
| frege23 43814 | Syllogism followed by rota... |
| frege15 43815 | A closed form of ~ com4r .... |
| frege21 43816 | Replace antecedent in ante... |
| frege20 43817 | A closed form of ~ syl8 . ... |
| axfrege28 43818 | Contraposition. Identical... |
| frege29 43820 | Closed form of ~ con3d . ... |
| frege30 43821 | Commuted, closed form of ~... |
| axfrege31 43822 | Identical to ~ notnotr . ... |
| frege32 43824 | Deduce ~ con1 from ~ con3 ... |
| frege33 43825 | If ` ph ` or ` ps ` takes ... |
| frege34 43826 | If as a consequence of the... |
| frege35 43827 | Commuted, closed form of ~... |
| frege36 43828 | The case in which ` ps ` i... |
| frege37 43829 | If ` ch ` is a necessary c... |
| frege38 43830 | Identical to ~ pm2.21 . P... |
| frege39 43831 | Syllogism between ~ pm2.18... |
| frege40 43832 | Anything implies ~ pm2.18 ... |
| axfrege41 43833 | Identical to ~ notnot . A... |
| frege42 43835 | Not not ~ id . Propositio... |
| frege43 43836 | If there is a choice only ... |
| frege44 43837 | Similar to a commuted ~ pm... |
| frege45 43838 | Deduce ~ pm2.6 from ~ con1... |
| frege46 43839 | If ` ps ` holds when ` ph ... |
| frege47 43840 | Deduce consequence follows... |
| frege48 43841 | Closed form of syllogism w... |
| frege49 43842 | Closed form of deduction w... |
| frege50 43843 | Closed form of ~ jaoi . P... |
| frege51 43844 | Compare with ~ jaod . Pro... |
| axfrege52a 43845 | Justification for ~ ax-fre... |
| frege52aid 43847 | The case when the content ... |
| frege53aid 43848 | Specialization of ~ frege5... |
| frege53a 43849 | Lemma for ~ frege55a . Pr... |
| axfrege54a 43850 | Justification for ~ ax-fre... |
| frege54cor0a 43852 | Synonym for logical equiva... |
| frege54cor1a 43853 | Reflexive equality. (Cont... |
| frege55aid 43854 | Lemma for ~ frege57aid . ... |
| frege55lem1a 43855 | Necessary deduction regard... |
| frege55lem2a 43856 | Core proof of Proposition ... |
| frege55a 43857 | Proposition 55 of [Frege18... |
| frege55cor1a 43858 | Proposition 55 of [Frege18... |
| frege56aid 43859 | Lemma for ~ frege57aid . ... |
| frege56a 43860 | Proposition 56 of [Frege18... |
| frege57aid 43861 | This is the all important ... |
| frege57a 43862 | Analogue of ~ frege57aid .... |
| axfrege58a 43863 | Identical to ~ anifp . Ju... |
| frege58acor 43865 | Lemma for ~ frege59a . (C... |
| frege59a 43866 | A kind of Aristotelian inf... |
| frege60a 43867 | Swap antecedents of ~ ax-f... |
| frege61a 43868 | Lemma for ~ frege65a . Pr... |
| frege62a 43869 | A kind of Aristotelian inf... |
| frege63a 43870 | Proposition 63 of [Frege18... |
| frege64a 43871 | Lemma for ~ frege65a . Pr... |
| frege65a 43872 | A kind of Aristotelian inf... |
| frege66a 43873 | Swap antecedents of ~ freg... |
| frege67a 43874 | Lemma for ~ frege68a . Pr... |
| frege68a 43875 | Combination of applying a ... |
| axfrege52c 43876 | Justification for ~ ax-fre... |
| frege52b 43878 | The case when the content ... |
| frege53b 43879 | Lemma for frege102 (via ~ ... |
| axfrege54c 43880 | Reflexive equality of clas... |
| frege54b 43882 | Reflexive equality of sets... |
| frege54cor1b 43883 | Reflexive equality. (Cont... |
| frege55lem1b 43884 | Necessary deduction regard... |
| frege55lem2b 43885 | Lemma for ~ frege55b . Co... |
| frege55b 43886 | Lemma for ~ frege57b . Pr... |
| frege56b 43887 | Lemma for ~ frege57b . Pr... |
| frege57b 43888 | Analogue of ~ frege57aid .... |
| axfrege58b 43889 | If ` A. x ph ` is affirmed... |
| frege58bid 43891 | If ` A. x ph ` is affirmed... |
| frege58bcor 43892 | Lemma for ~ frege59b . (C... |
| frege59b 43893 | A kind of Aristotelian inf... |
| frege60b 43894 | Swap antecedents of ~ ax-f... |
| frege61b 43895 | Lemma for ~ frege65b . Pr... |
| frege62b 43896 | A kind of Aristotelian inf... |
| frege63b 43897 | Lemma for ~ frege91 . Pro... |
| frege64b 43898 | Lemma for ~ frege65b . Pr... |
| frege65b 43899 | A kind of Aristotelian inf... |
| frege66b 43900 | Swap antecedents of ~ freg... |
| frege67b 43901 | Lemma for ~ frege68b . Pr... |
| frege68b 43902 | Combination of applying a ... |
| frege53c 43903 | Proposition 53 of [Frege18... |
| frege54cor1c 43904 | Reflexive equality. (Cont... |
| frege55lem1c 43905 | Necessary deduction regard... |
| frege55lem2c 43906 | Core proof of Proposition ... |
| frege55c 43907 | Proposition 55 of [Frege18... |
| frege56c 43908 | Lemma for ~ frege57c . Pr... |
| frege57c 43909 | Swap order of implication ... |
| frege58c 43910 | Principle related to ~ sp ... |
| frege59c 43911 | A kind of Aristotelian inf... |
| frege60c 43912 | Swap antecedents of ~ freg... |
| frege61c 43913 | Lemma for ~ frege65c . Pr... |
| frege62c 43914 | A kind of Aristotelian inf... |
| frege63c 43915 | Analogue of ~ frege63b . ... |
| frege64c 43916 | Lemma for ~ frege65c . Pr... |
| frege65c 43917 | A kind of Aristotelian inf... |
| frege66c 43918 | Swap antecedents of ~ freg... |
| frege67c 43919 | Lemma for ~ frege68c . Pr... |
| frege68c 43920 | Combination of applying a ... |
| dffrege69 43921 | If from the proposition th... |
| frege70 43922 | Lemma for ~ frege72 . Pro... |
| frege71 43923 | Lemma for ~ frege72 . Pro... |
| frege72 43924 | If property ` A ` is hered... |
| frege73 43925 | Lemma for ~ frege87 . Pro... |
| frege74 43926 | If ` X ` has a property ` ... |
| frege75 43927 | If from the proposition th... |
| dffrege76 43928 | If from the two propositio... |
| frege77 43929 | If ` Y ` follows ` X ` in ... |
| frege78 43930 | Commuted form of ~ frege77... |
| frege79 43931 | Distributed form of ~ freg... |
| frege80 43932 | Add additional condition t... |
| frege81 43933 | If ` X ` has a property ` ... |
| frege82 43934 | Closed-form deduction base... |
| frege83 43935 | Apply commuted form of ~ f... |
| frege84 43936 | Commuted form of ~ frege81... |
| frege85 43937 | Commuted form of ~ frege77... |
| frege86 43938 | Conclusion about element o... |
| frege87 43939 | If ` Z ` is a result of an... |
| frege88 43940 | Commuted form of ~ frege87... |
| frege89 43941 | One direction of ~ dffrege... |
| frege90 43942 | Add antecedent to ~ frege8... |
| frege91 43943 | Every result of an applica... |
| frege92 43944 | Inference from ~ frege91 .... |
| frege93 43945 | Necessary condition for tw... |
| frege94 43946 | Looking one past a pair re... |
| frege95 43947 | Looking one past a pair re... |
| frege96 43948 | Every result of an applica... |
| frege97 43949 | The property of following ... |
| frege98 43950 | If ` Y ` follows ` X ` and... |
| dffrege99 43951 | If ` Z ` is identical with... |
| frege100 43952 | One direction of ~ dffrege... |
| frege101 43953 | Lemma for ~ frege102 . Pr... |
| frege102 43954 | If ` Z ` belongs to the ` ... |
| frege103 43955 | Proposition 103 of [Frege1... |
| frege104 43956 | Proposition 104 of [Frege1... |
| frege105 43957 | Proposition 105 of [Frege1... |
| frege106 43958 | Whatever follows ` X ` in ... |
| frege107 43959 | Proposition 107 of [Frege1... |
| frege108 43960 | If ` Y ` belongs to the ` ... |
| frege109 43961 | The property of belonging ... |
| frege110 43962 | Proposition 110 of [Frege1... |
| frege111 43963 | If ` Y ` belongs to the ` ... |
| frege112 43964 | Identity implies belonging... |
| frege113 43965 | Proposition 113 of [Frege1... |
| frege114 43966 | If ` X ` belongs to the ` ... |
| dffrege115 43967 | If from the circumstance t... |
| frege116 43968 | One direction of ~ dffrege... |
| frege117 43969 | Lemma for ~ frege118 . Pr... |
| frege118 43970 | Simplified application of ... |
| frege119 43971 | Lemma for ~ frege120 . Pr... |
| frege120 43972 | Simplified application of ... |
| frege121 43973 | Lemma for ~ frege122 . Pr... |
| frege122 43974 | If ` X ` is a result of an... |
| frege123 43975 | Lemma for ~ frege124 . Pr... |
| frege124 43976 | If ` X ` is a result of an... |
| frege125 43977 | Lemma for ~ frege126 . Pr... |
| frege126 43978 | If ` M ` follows ` Y ` in ... |
| frege127 43979 | Communte antecedents of ~ ... |
| frege128 43980 | Lemma for ~ frege129 . Pr... |
| frege129 43981 | If the procedure ` R ` is ... |
| frege130 43982 | Lemma for ~ frege131 . Pr... |
| frege131 43983 | If the procedure ` R ` is ... |
| frege132 43984 | Lemma for ~ frege133 . Pr... |
| frege133 43985 | If the procedure ` R ` is ... |
| enrelmap 43986 | The set of all possible re... |
| enrelmapr 43987 | The set of all possible re... |
| enmappw 43988 | The set of all mappings fr... |
| enmappwid 43989 | The set of all mappings fr... |
| rfovd 43990 | Value of the operator, ` (... |
| rfovfvd 43991 | Value of the operator, ` (... |
| rfovfvfvd 43992 | Value of the operator, ` (... |
| rfovcnvf1od 43993 | Properties of the operator... |
| rfovcnvd 43994 | Value of the converse of t... |
| rfovf1od 43995 | The value of the operator,... |
| rfovcnvfvd 43996 | Value of the converse of t... |
| fsovd 43997 | Value of the operator, ` (... |
| fsovrfovd 43998 | The operator which gives a... |
| fsovfvd 43999 | Value of the operator, ` (... |
| fsovfvfvd 44000 | Value of the operator, ` (... |
| fsovfd 44001 | The operator, ` ( A O B ) ... |
| fsovcnvlem 44002 | The ` O ` operator, which ... |
| fsovcnvd 44003 | The value of the converse ... |
| fsovcnvfvd 44004 | The value of the converse ... |
| fsovf1od 44005 | The value of ` ( A O B ) `... |
| dssmapfvd 44006 | Value of the duality opera... |
| dssmapfv2d 44007 | Value of the duality opera... |
| dssmapfv3d 44008 | Value of the duality opera... |
| dssmapnvod 44009 | For any base set ` B ` the... |
| dssmapf1od 44010 | For any base set ` B ` the... |
| dssmap2d 44011 | For any base set ` B ` the... |
| or3or 44012 | Decompose disjunction into... |
| andi3or 44013 | Distribute over triple dis... |
| uneqsn 44014 | If a union of classes is e... |
| brfvimex 44015 | If a binary relation holds... |
| brovmptimex 44016 | If a binary relation holds... |
| brovmptimex1 44017 | If a binary relation holds... |
| brovmptimex2 44018 | If a binary relation holds... |
| brcoffn 44019 | Conditions allowing the de... |
| brcofffn 44020 | Conditions allowing the de... |
| brco2f1o 44021 | Conditions allowing the de... |
| brco3f1o 44022 | Conditions allowing the de... |
| ntrclsbex 44023 | If (pseudo-)interior and (... |
| ntrclsrcomplex 44024 | The relative complement of... |
| neik0imk0p 44025 | Kuratowski's K0 axiom impl... |
| ntrk2imkb 44026 | If an interior function is... |
| ntrkbimka 44027 | If the interiors of disjoi... |
| ntrk0kbimka 44028 | If the interiors of disjoi... |
| clsk3nimkb 44029 | If the base set is not emp... |
| clsk1indlem0 44030 | The ansatz closure functio... |
| clsk1indlem2 44031 | The ansatz closure functio... |
| clsk1indlem3 44032 | The ansatz closure functio... |
| clsk1indlem4 44033 | The ansatz closure functio... |
| clsk1indlem1 44034 | The ansatz closure functio... |
| clsk1independent 44035 | For generalized closure fu... |
| neik0pk1imk0 44036 | Kuratowski's K0' and K1 ax... |
| isotone1 44037 | Two different ways to say ... |
| isotone2 44038 | Two different ways to say ... |
| ntrk1k3eqk13 44039 | An interior function is bo... |
| ntrclsf1o 44040 | If (pseudo-)interior and (... |
| ntrclsnvobr 44041 | If (pseudo-)interior and (... |
| ntrclsiex 44042 | If (pseudo-)interior and (... |
| ntrclskex 44043 | If (pseudo-)interior and (... |
| ntrclsfv1 44044 | If (pseudo-)interior and (... |
| ntrclsfv2 44045 | If (pseudo-)interior and (... |
| ntrclselnel1 44046 | If (pseudo-)interior and (... |
| ntrclselnel2 44047 | If (pseudo-)interior and (... |
| ntrclsfv 44048 | The value of the interior ... |
| ntrclsfveq1 44049 | If interior and closure fu... |
| ntrclsfveq2 44050 | If interior and closure fu... |
| ntrclsfveq 44051 | If interior and closure fu... |
| ntrclsss 44052 | If interior and closure fu... |
| ntrclsneine0lem 44053 | If (pseudo-)interior and (... |
| ntrclsneine0 44054 | If (pseudo-)interior and (... |
| ntrclscls00 44055 | If (pseudo-)interior and (... |
| ntrclsiso 44056 | If (pseudo-)interior and (... |
| ntrclsk2 44057 | An interior function is co... |
| ntrclskb 44058 | The interiors of disjoint ... |
| ntrclsk3 44059 | The intersection of interi... |
| ntrclsk13 44060 | The interior of the inters... |
| ntrclsk4 44061 | Idempotence of the interio... |
| ntrneibex 44062 | If (pseudo-)interior and (... |
| ntrneircomplex 44063 | The relative complement of... |
| ntrneif1o 44064 | If (pseudo-)interior and (... |
| ntrneiiex 44065 | If (pseudo-)interior and (... |
| ntrneinex 44066 | If (pseudo-)interior and (... |
| ntrneicnv 44067 | If (pseudo-)interior and (... |
| ntrneifv1 44068 | If (pseudo-)interior and (... |
| ntrneifv2 44069 | If (pseudo-)interior and (... |
| ntrneiel 44070 | If (pseudo-)interior and (... |
| ntrneifv3 44071 | The value of the neighbors... |
| ntrneineine0lem 44072 | If (pseudo-)interior and (... |
| ntrneineine1lem 44073 | If (pseudo-)interior and (... |
| ntrneifv4 44074 | The value of the interior ... |
| ntrneiel2 44075 | Membership in iterated int... |
| ntrneineine0 44076 | If (pseudo-)interior and (... |
| ntrneineine1 44077 | If (pseudo-)interior and (... |
| ntrneicls00 44078 | If (pseudo-)interior and (... |
| ntrneicls11 44079 | If (pseudo-)interior and (... |
| ntrneiiso 44080 | If (pseudo-)interior and (... |
| ntrneik2 44081 | An interior function is co... |
| ntrneix2 44082 | An interior (closure) func... |
| ntrneikb 44083 | The interiors of disjoint ... |
| ntrneixb 44084 | The interiors (closures) o... |
| ntrneik3 44085 | The intersection of interi... |
| ntrneix3 44086 | The closure of the union o... |
| ntrneik13 44087 | The interior of the inters... |
| ntrneix13 44088 | The closure of the union o... |
| ntrneik4w 44089 | Idempotence of the interio... |
| ntrneik4 44090 | Idempotence of the interio... |
| clsneibex 44091 | If (pseudo-)closure and (p... |
| clsneircomplex 44092 | The relative complement of... |
| clsneif1o 44093 | If a (pseudo-)closure func... |
| clsneicnv 44094 | If a (pseudo-)closure func... |
| clsneikex 44095 | If closure and neighborhoo... |
| clsneinex 44096 | If closure and neighborhoo... |
| clsneiel1 44097 | If a (pseudo-)closure func... |
| clsneiel2 44098 | If a (pseudo-)closure func... |
| clsneifv3 44099 | Value of the neighborhoods... |
| clsneifv4 44100 | Value of the closure (inte... |
| neicvgbex 44101 | If (pseudo-)neighborhood a... |
| neicvgrcomplex 44102 | The relative complement of... |
| neicvgf1o 44103 | If neighborhood and conver... |
| neicvgnvo 44104 | If neighborhood and conver... |
| neicvgnvor 44105 | If neighborhood and conver... |
| neicvgmex 44106 | If the neighborhoods and c... |
| neicvgnex 44107 | If the neighborhoods and c... |
| neicvgel1 44108 | A subset being an element ... |
| neicvgel2 44109 | The complement of a subset... |
| neicvgfv 44110 | The value of the neighborh... |
| ntrrn 44111 | The range of the interior ... |
| ntrf 44112 | The interior function of a... |
| ntrf2 44113 | The interior function is a... |
| ntrelmap 44114 | The interior function is a... |
| clsf2 44115 | The closure function is a ... |
| clselmap 44116 | The closure function is a ... |
| dssmapntrcls 44117 | The interior and closure o... |
| dssmapclsntr 44118 | The closure and interior o... |
| gneispa 44119 | Each point ` p ` of the ne... |
| gneispb 44120 | Given a neighborhood ` N `... |
| gneispace2 44121 | The predicate that ` F ` i... |
| gneispace3 44122 | The predicate that ` F ` i... |
| gneispace 44123 | The predicate that ` F ` i... |
| gneispacef 44124 | A generic neighborhood spa... |
| gneispacef2 44125 | A generic neighborhood spa... |
| gneispacefun 44126 | A generic neighborhood spa... |
| gneispacern 44127 | A generic neighborhood spa... |
| gneispacern2 44128 | A generic neighborhood spa... |
| gneispace0nelrn 44129 | A generic neighborhood spa... |
| gneispace0nelrn2 44130 | A generic neighborhood spa... |
| gneispace0nelrn3 44131 | A generic neighborhood spa... |
| gneispaceel 44132 | Every neighborhood of a po... |
| gneispaceel2 44133 | Every neighborhood of a po... |
| gneispacess 44134 | All supersets of a neighbo... |
| gneispacess2 44135 | All supersets of a neighbo... |
| k0004lem1 44136 | Application of ~ ssin to r... |
| k0004lem2 44137 | A mapping with a particula... |
| k0004lem3 44138 | When the value of a mappin... |
| k0004val 44139 | The topological simplex of... |
| k0004ss1 44140 | The topological simplex of... |
| k0004ss2 44141 | The topological simplex of... |
| k0004ss3 44142 | The topological simplex of... |
| k0004val0 44143 | The topological simplex of... |
| inductionexd 44144 | Simple induction example. ... |
| wwlemuld 44145 | Natural deduction form of ... |
| leeq1d 44146 | Specialization of ~ breq1d... |
| leeq2d 44147 | Specialization of ~ breq2d... |
| absmulrposd 44148 | Specialization of absmuld ... |
| imadisjld 44149 | Natural dduction form of o... |
| wnefimgd 44150 | The image of a mapping fro... |
| fco2d 44151 | Natural deduction form of ... |
| wfximgfd 44152 | The value of a function on... |
| extoimad 44153 | If |f(x)| <= C for all x t... |
| imo72b2lem0 44154 | Lemma for ~ imo72b2 . (Co... |
| suprleubrd 44155 | Natural deduction form of ... |
| imo72b2lem2 44156 | Lemma for ~ imo72b2 . (Co... |
| suprlubrd 44157 | Natural deduction form of ... |
| imo72b2lem1 44158 | Lemma for ~ imo72b2 . (Co... |
| lemuldiv3d 44159 | 'Less than or equal to' re... |
| lemuldiv4d 44160 | 'Less than or equal to' re... |
| imo72b2 44161 | IMO 1972 B2. (14th Intern... |
| int-addcomd 44162 | AdditionCommutativity gene... |
| int-addassocd 44163 | AdditionAssociativity gene... |
| int-addsimpd 44164 | AdditionSimplification gen... |
| int-mulcomd 44165 | MultiplicationCommutativit... |
| int-mulassocd 44166 | MultiplicationAssociativit... |
| int-mulsimpd 44167 | MultiplicationSimplificati... |
| int-leftdistd 44168 | AdditionMultiplicationLeft... |
| int-rightdistd 44169 | AdditionMultiplicationRigh... |
| int-sqdefd 44170 | SquareDefinition generator... |
| int-mul11d 44171 | First MultiplicationOne ge... |
| int-mul12d 44172 | Second MultiplicationOne g... |
| int-add01d 44173 | First AdditionZero generat... |
| int-add02d 44174 | Second AdditionZero genera... |
| int-sqgeq0d 44175 | SquareGEQZero generator ru... |
| int-eqprincd 44176 | PrincipleOfEquality genera... |
| int-eqtransd 44177 | EqualityTransitivity gener... |
| int-eqmvtd 44178 | EquMoveTerm generator rule... |
| int-eqineqd 44179 | EquivalenceImpliesDoubleIn... |
| int-ineqmvtd 44180 | IneqMoveTerm generator rul... |
| int-ineq1stprincd 44181 | FirstPrincipleOfInequality... |
| int-ineq2ndprincd 44182 | SecondPrincipleOfInequalit... |
| int-ineqtransd 44183 | InequalityTransitivity gen... |
| unitadd 44184 | Theorem used in conjunctio... |
| gsumws3 44185 | Valuation of a length 3 wo... |
| gsumws4 44186 | Valuation of a length 4 wo... |
| amgm2d 44187 | Arithmetic-geometric mean ... |
| amgm3d 44188 | Arithmetic-geometric mean ... |
| amgm4d 44189 | Arithmetic-geometric mean ... |
| spALT 44190 | ~ sp can be proven from th... |
| elnelneqd 44191 | Two classes are not equal ... |
| elnelneq2d 44192 | Two classes are not equal ... |
| rr-spce 44193 | Prove an existential. (Co... |
| rexlimdvaacbv 44194 | Unpack a restricted existe... |
| rexlimddvcbvw 44195 | Unpack a restricted existe... |
| rexlimddvcbv 44196 | Unpack a restricted existe... |
| rr-elrnmpt3d 44197 | Elementhood in an image se... |
| rr-phpd 44198 | Equivalent of ~ php withou... |
| tfindsd 44199 | Deduction associated with ... |
| mnringvald 44202 | Value of the monoid ring f... |
| mnringnmulrd 44203 | Components of a monoid rin... |
| mnringbased 44204 | The base set of a monoid r... |
| mnringbaserd 44205 | The base set of a monoid r... |
| mnringelbased 44206 | Membership in the base set... |
| mnringbasefd 44207 | Elements of a monoid ring ... |
| mnringbasefsuppd 44208 | Elements of a monoid ring ... |
| mnringaddgd 44209 | The additive operation of ... |
| mnring0gd 44210 | The additive identity of a... |
| mnring0g2d 44211 | The additive identity of a... |
| mnringmulrd 44212 | The ring product of a mono... |
| mnringscad 44213 | The scalar ring of a monoi... |
| mnringvscad 44214 | The scalar product of a mo... |
| mnringlmodd 44215 | Monoid rings are left modu... |
| mnringmulrvald 44216 | Value of multiplication in... |
| mnringmulrcld 44217 | Monoid rings are closed un... |
| gru0eld 44218 | A nonempty Grothendieck un... |
| grusucd 44219 | Grothendieck universes are... |
| r1rankcld 44220 | Any rank of the cumulative... |
| grur1cld 44221 | Grothendieck universes are... |
| grurankcld 44222 | Grothendieck universes are... |
| grurankrcld 44223 | If a Grothendieck universe... |
| scotteqd 44226 | Equality theorem for the S... |
| scotteq 44227 | Closed form of ~ scotteqd ... |
| nfscott 44228 | Bound-variable hypothesis ... |
| scottabf 44229 | Value of the Scott operati... |
| scottab 44230 | Value of the Scott operati... |
| scottabes 44231 | Value of the Scott operati... |
| scottss 44232 | Scott's trick produces a s... |
| elscottab 44233 | An element of the output o... |
| scottex2 44234 | ~ scottex expressed using ... |
| scotteld 44235 | The Scott operation sends ... |
| scottelrankd 44236 | Property of a Scott's tric... |
| scottrankd 44237 | Rank of a nonempty Scott's... |
| gruscottcld 44238 | If a Grothendieck universe... |
| dfcoll2 44241 | Alternate definition of th... |
| colleq12d 44242 | Equality theorem for the c... |
| colleq1 44243 | Equality theorem for the c... |
| colleq2 44244 | Equality theorem for the c... |
| nfcoll 44245 | Bound-variable hypothesis ... |
| collexd 44246 | The output of the collecti... |
| cpcolld 44247 | Property of the collection... |
| cpcoll2d 44248 | ~ cpcolld with an extra ex... |
| grucollcld 44249 | A Grothendieck universe co... |
| ismnu 44250 | The hypothesis of this the... |
| mnuop123d 44251 | Operations of a minimal un... |
| mnussd 44252 | Minimal universes are clos... |
| mnuss2d 44253 | ~ mnussd with arguments pr... |
| mnu0eld 44254 | A nonempty minimal univers... |
| mnuop23d 44255 | Second and third operation... |
| mnupwd 44256 | Minimal universes are clos... |
| mnusnd 44257 | Minimal universes are clos... |
| mnuprssd 44258 | A minimal universe contain... |
| mnuprss2d 44259 | Special case of ~ mnuprssd... |
| mnuop3d 44260 | Third operation of a minim... |
| mnuprdlem1 44261 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem2 44262 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem3 44263 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem4 44264 | Lemma for ~ mnuprd . Gene... |
| mnuprd 44265 | Minimal universes are clos... |
| mnuunid 44266 | Minimal universes are clos... |
| mnuund 44267 | Minimal universes are clos... |
| mnutrcld 44268 | Minimal universes contain ... |
| mnutrd 44269 | Minimal universes are tran... |
| mnurndlem1 44270 | Lemma for ~ mnurnd . (Con... |
| mnurndlem2 44271 | Lemma for ~ mnurnd . Dedu... |
| mnurnd 44272 | Minimal universes contain ... |
| mnugrud 44273 | Minimal universes are Grot... |
| grumnudlem 44274 | Lemma for ~ grumnud . (Co... |
| grumnud 44275 | Grothendieck universes are... |
| grumnueq 44276 | The class of Grothendieck ... |
| expandan 44277 | Expand conjunction to prim... |
| expandexn 44278 | Expand an existential quan... |
| expandral 44279 | Expand a restricted univer... |
| expandrexn 44280 | Expand a restricted existe... |
| expandrex 44281 | Expand a restricted existe... |
| expanduniss 44282 | Expand ` U. A C_ B ` to pr... |
| ismnuprim 44283 | Express the predicate on `... |
| rr-grothprimbi 44284 | Express "every set is cont... |
| inagrud 44285 | Inaccessible levels of the... |
| inaex 44286 | Assuming the Tarski-Grothe... |
| gruex 44287 | Assuming the Tarski-Grothe... |
| rr-groth 44288 | An equivalent of ~ ax-grot... |
| rr-grothprim 44289 | An equivalent of ~ ax-grot... |
| ismnushort 44290 | Express the predicate on `... |
| dfuniv2 44291 | Alternative definition of ... |
| rr-grothshortbi 44292 | Express "every set is cont... |
| rr-grothshort 44293 | A shorter equivalent of ~ ... |
| nanorxor 44294 | 'nand' is equivalent to th... |
| undisjrab 44295 | Union of two disjoint rest... |
| iso0 44296 | The empty set is an ` R , ... |
| ssrecnpr 44297 | ` RR ` is a subset of both... |
| seff 44298 | Let set ` S ` be the real ... |
| sblpnf 44299 | The infinity ball in the a... |
| prmunb2 44300 | The primes are unbounded. ... |
| dvgrat 44301 | Ratio test for divergence ... |
| cvgdvgrat 44302 | Ratio test for convergence... |
| radcnvrat 44303 | Let ` L ` be the limit, if... |
| reldvds 44304 | The divides relation is in... |
| nznngen 44305 | All positive integers in t... |
| nzss 44306 | The set of multiples of _m... |
| nzin 44307 | The intersection of the se... |
| nzprmdif 44308 | Subtract one prime's multi... |
| hashnzfz 44309 | Special case of ~ hashdvds... |
| hashnzfz2 44310 | Special case of ~ hashnzfz... |
| hashnzfzclim 44311 | As the upper bound ` K ` o... |
| caofcan 44312 | Transfer a cancellation la... |
| ofsubid 44313 | Function analogue of ~ sub... |
| ofmul12 44314 | Function analogue of ~ mul... |
| ofdivrec 44315 | Function analogue of ~ div... |
| ofdivcan4 44316 | Function analogue of ~ div... |
| ofdivdiv2 44317 | Function analogue of ~ div... |
| lhe4.4ex1a 44318 | Example of the Fundamental... |
| dvsconst 44319 | Derivative of a constant f... |
| dvsid 44320 | Derivative of the identity... |
| dvsef 44321 | Derivative of the exponent... |
| expgrowthi 44322 | Exponential growth and dec... |
| dvconstbi 44323 | The derivative of a functi... |
| expgrowth 44324 | Exponential growth and dec... |
| bccval 44327 | Value of the generalized b... |
| bcccl 44328 | Closure of the generalized... |
| bcc0 44329 | The generalized binomial c... |
| bccp1k 44330 | Generalized binomial coeff... |
| bccm1k 44331 | Generalized binomial coeff... |
| bccn0 44332 | Generalized binomial coeff... |
| bccn1 44333 | Generalized binomial coeff... |
| bccbc 44334 | The binomial coefficient a... |
| uzmptshftfval 44335 | When ` F ` is a maps-to fu... |
| dvradcnv2 44336 | The radius of convergence ... |
| binomcxplemwb 44337 | Lemma for ~ binomcxp . Th... |
| binomcxplemnn0 44338 | Lemma for ~ binomcxp . Wh... |
| binomcxplemrat 44339 | Lemma for ~ binomcxp . As... |
| binomcxplemfrat 44340 | Lemma for ~ binomcxp . ~ b... |
| binomcxplemradcnv 44341 | Lemma for ~ binomcxp . By... |
| binomcxplemdvbinom 44342 | Lemma for ~ binomcxp . By... |
| binomcxplemcvg 44343 | Lemma for ~ binomcxp . Th... |
| binomcxplemdvsum 44344 | Lemma for ~ binomcxp . Th... |
| binomcxplemnotnn0 44345 | Lemma for ~ binomcxp . Wh... |
| binomcxp 44346 | Generalize the binomial th... |
| pm10.12 44347 | Theorem *10.12 in [Whitehe... |
| pm10.14 44348 | Theorem *10.14 in [Whitehe... |
| pm10.251 44349 | Theorem *10.251 in [Whiteh... |
| pm10.252 44350 | Theorem *10.252 in [Whiteh... |
| pm10.253 44351 | Theorem *10.253 in [Whiteh... |
| albitr 44352 | Theorem *10.301 in [Whiteh... |
| pm10.42 44353 | Theorem *10.42 in [Whitehe... |
| pm10.52 44354 | Theorem *10.52 in [Whitehe... |
| pm10.53 44355 | Theorem *10.53 in [Whitehe... |
| pm10.541 44356 | Theorem *10.541 in [Whiteh... |
| pm10.542 44357 | Theorem *10.542 in [Whiteh... |
| pm10.55 44358 | Theorem *10.55 in [Whitehe... |
| pm10.56 44359 | Theorem *10.56 in [Whitehe... |
| pm10.57 44360 | Theorem *10.57 in [Whitehe... |
| 2alanimi 44361 | Removes two universal quan... |
| 2al2imi 44362 | Removes two universal quan... |
| pm11.11 44363 | Theorem *11.11 in [Whitehe... |
| pm11.12 44364 | Theorem *11.12 in [Whitehe... |
| 19.21vv 44365 | Compare Theorem *11.3 in [... |
| 2alim 44366 | Theorem *11.32 in [Whitehe... |
| 2albi 44367 | Theorem *11.33 in [Whitehe... |
| 2exim 44368 | Theorem *11.34 in [Whitehe... |
| 2exbi 44369 | Theorem *11.341 in [Whiteh... |
| spsbce-2 44370 | Theorem *11.36 in [Whitehe... |
| 19.33-2 44371 | Theorem *11.421 in [Whiteh... |
| 19.36vv 44372 | Theorem *11.43 in [Whitehe... |
| 19.31vv 44373 | Theorem *11.44 in [Whitehe... |
| 19.37vv 44374 | Theorem *11.46 in [Whitehe... |
| 19.28vv 44375 | Theorem *11.47 in [Whitehe... |
| pm11.52 44376 | Theorem *11.52 in [Whitehe... |
| aaanv 44377 | Theorem *11.56 in [Whitehe... |
| pm11.57 44378 | Theorem *11.57 in [Whitehe... |
| pm11.58 44379 | Theorem *11.58 in [Whitehe... |
| pm11.59 44380 | Theorem *11.59 in [Whitehe... |
| pm11.6 44381 | Theorem *11.6 in [Whitehea... |
| pm11.61 44382 | Theorem *11.61 in [Whitehe... |
| pm11.62 44383 | Theorem *11.62 in [Whitehe... |
| pm11.63 44384 | Theorem *11.63 in [Whitehe... |
| pm11.7 44385 | Theorem *11.7 in [Whitehea... |
| pm11.71 44386 | Theorem *11.71 in [Whitehe... |
| sbeqal1 44387 | If ` x = y ` always implie... |
| sbeqal1i 44388 | Suppose you know ` x = y `... |
| sbeqal2i 44389 | If ` x = y ` implies ` x =... |
| axc5c4c711 44390 | Proof of a theorem that ca... |
| axc5c4c711toc5 44391 | Rederivation of ~ sp from ... |
| axc5c4c711toc4 44392 | Rederivation of ~ axc4 fro... |
| axc5c4c711toc7 44393 | Rederivation of ~ axc7 fro... |
| axc5c4c711to11 44394 | Rederivation of ~ ax-11 fr... |
| axc11next 44395 | This theorem shows that, g... |
| pm13.13a 44396 | One result of theorem *13.... |
| pm13.13b 44397 | Theorem *13.13 in [Whitehe... |
| pm13.14 44398 | Theorem *13.14 in [Whitehe... |
| pm13.192 44399 | Theorem *13.192 in [Whiteh... |
| pm13.193 44400 | Theorem *13.193 in [Whiteh... |
| pm13.194 44401 | Theorem *13.194 in [Whiteh... |
| pm13.195 44402 | Theorem *13.195 in [Whiteh... |
| pm13.196a 44403 | Theorem *13.196 in [Whiteh... |
| 2sbc6g 44404 | Theorem *13.21 in [Whitehe... |
| 2sbc5g 44405 | Theorem *13.22 in [Whitehe... |
| iotain 44406 | Equivalence between two di... |
| iotaexeu 44407 | The iota class exists. Th... |
| iotasbc 44408 | Definition *14.01 in [Whit... |
| iotasbc2 44409 | Theorem *14.111 in [Whiteh... |
| pm14.12 44410 | Theorem *14.12 in [Whitehe... |
| pm14.122a 44411 | Theorem *14.122 in [Whiteh... |
| pm14.122b 44412 | Theorem *14.122 in [Whiteh... |
| pm14.122c 44413 | Theorem *14.122 in [Whiteh... |
| pm14.123a 44414 | Theorem *14.123 in [Whiteh... |
| pm14.123b 44415 | Theorem *14.123 in [Whiteh... |
| pm14.123c 44416 | Theorem *14.123 in [Whiteh... |
| pm14.18 44417 | Theorem *14.18 in [Whitehe... |
| iotaequ 44418 | Theorem *14.2 in [Whitehea... |
| iotavalb 44419 | Theorem *14.202 in [Whiteh... |
| iotasbc5 44420 | Theorem *14.205 in [Whiteh... |
| pm14.24 44421 | Theorem *14.24 in [Whitehe... |
| iotavalsb 44422 | Theorem *14.242 in [Whiteh... |
| sbiota1 44423 | Theorem *14.25 in [Whitehe... |
| sbaniota 44424 | Theorem *14.26 in [Whitehe... |
| eubiOLD 44425 | Obsolete proof of ~ eubi a... |
| iotasbcq 44426 | Theorem *14.272 in [Whiteh... |
| elnev 44427 | Any set that contains one ... |
| rusbcALT 44428 | A version of Russell's par... |
| compeq 44429 | Equality between two ways ... |
| compne 44430 | The complement of ` A ` is... |
| compab 44431 | Two ways of saying "the co... |
| conss2 44432 | Contrapositive law for sub... |
| conss1 44433 | Contrapositive law for sub... |
| ralbidar 44434 | More general form of ~ ral... |
| rexbidar 44435 | More general form of ~ rex... |
| dropab1 44436 | Theorem to aid use of the ... |
| dropab2 44437 | Theorem to aid use of the ... |
| ipo0 44438 | If the identity relation p... |
| ifr0 44439 | A class that is founded by... |
| ordpss 44440 | ~ ordelpss with an anteced... |
| fvsb 44441 | Explicit substitution of a... |
| fveqsb 44442 | Implicit substitution of a... |
| xpexb 44443 | A Cartesian product exists... |
| trelpss 44444 | An element of a transitive... |
| addcomgi 44445 | Generalization of commutat... |
| addrval 44455 | Value of the operation of ... |
| subrval 44456 | Value of the operation of ... |
| mulvval 44457 | Value of the operation of ... |
| addrfv 44458 | Vector addition at a value... |
| subrfv 44459 | Vector subtraction at a va... |
| mulvfv 44460 | Scalar multiplication at a... |
| addrfn 44461 | Vector addition produces a... |
| subrfn 44462 | Vector subtraction produce... |
| mulvfn 44463 | Scalar multiplication prod... |
| addrcom 44464 | Vector addition is commuta... |
| idiALT 44468 | Placeholder for ~ idi . T... |
| exbir 44469 | Exportation implication al... |
| 3impexpbicom 44470 | Version of ~ 3impexp where... |
| 3impexpbicomi 44471 | Inference associated with ... |
| bi1imp 44472 | Importation inference simi... |
| bi2imp 44473 | Importation inference simi... |
| bi3impb 44474 | Similar to ~ 3impb with im... |
| bi3impa 44475 | Similar to ~ 3impa with im... |
| bi23impib 44476 | ~ 3impib with the inner im... |
| bi13impib 44477 | ~ 3impib with the outer im... |
| bi123impib 44478 | ~ 3impib with the implicat... |
| bi13impia 44479 | ~ 3impia with the outer im... |
| bi123impia 44480 | ~ 3impia with the implicat... |
| bi33imp12 44481 | ~ 3imp with innermost impl... |
| bi13imp23 44482 | ~ 3imp with outermost impl... |
| bi13imp2 44483 | Similar to ~ 3imp except t... |
| bi12imp3 44484 | Similar to ~ 3imp except a... |
| bi23imp1 44485 | Similar to ~ 3imp except a... |
| bi123imp0 44486 | Similar to ~ 3imp except a... |
| 4animp1 44487 | A single hypothesis unific... |
| 4an31 44488 | A rearrangement of conjunc... |
| 4an4132 44489 | A rearrangement of conjunc... |
| expcomdg 44490 | Biconditional form of ~ ex... |
| iidn3 44491 | ~ idn3 without virtual ded... |
| ee222 44492 | ~ e222 without virtual ded... |
| ee3bir 44493 | Right-biconditional form o... |
| ee13 44494 | ~ e13 without virtual dedu... |
| ee121 44495 | ~ e121 without virtual ded... |
| ee122 44496 | ~ e122 without virtual ded... |
| ee333 44497 | ~ e333 without virtual ded... |
| ee323 44498 | ~ e323 without virtual ded... |
| 3ornot23 44499 | If the second and third di... |
| orbi1r 44500 | ~ orbi1 with order of disj... |
| 3orbi123 44501 | ~ pm4.39 with a 3-conjunct... |
| syl5imp 44502 | Closed form of ~ syl5 . D... |
| impexpd 44503 | The following User's Proof... |
| com3rgbi 44504 | The following User's Proof... |
| impexpdcom 44505 | The following User's Proof... |
| ee1111 44506 | Non-virtual deduction form... |
| pm2.43bgbi 44507 | Logical equivalence of a 2... |
| pm2.43cbi 44508 | Logical equivalence of a 3... |
| ee233 44509 | Non-virtual deduction form... |
| imbi13 44510 | Join three logical equival... |
| ee33 44511 | Non-virtual deduction form... |
| con5 44512 | Biconditional contrapositi... |
| con5i 44513 | Inference form of ~ con5 .... |
| exlimexi 44514 | Inference similar to Theor... |
| sb5ALT 44515 | Equivalence for substituti... |
| eexinst01 44516 | ~ exinst01 without virtual... |
| eexinst11 44517 | ~ exinst11 without virtual... |
| vk15.4j 44518 | Excercise 4j of Unit 15 of... |
| notnotrALT 44519 | Converse of double negatio... |
| con3ALT2 44520 | Contraposition. Alternate... |
| ssralv2 44521 | Quantification restricted ... |
| sbc3or 44522 | ~ sbcor with a 3-disjuncts... |
| alrim3con13v 44523 | Closed form of ~ alrimi wi... |
| rspsbc2 44524 | ~ rspsbc with two quantify... |
| sbcoreleleq 44525 | Substitution of a setvar v... |
| tratrb 44526 | If a class is transitive a... |
| ordelordALT 44527 | An element of an ordinal c... |
| sbcim2g 44528 | Distribution of class subs... |
| sbcbi 44529 | Implication form of ~ sbcb... |
| trsbc 44530 | Formula-building inference... |
| truniALT 44531 | The union of a class of tr... |
| onfrALTlem5 44532 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem4 44533 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem3 44534 | Lemma for ~ onfrALT . (Co... |
| ggen31 44535 | ~ gen31 without virtual de... |
| onfrALTlem2 44536 | Lemma for ~ onfrALT . (Co... |
| cbvexsv 44537 | A theorem pertaining to th... |
| onfrALTlem1 44538 | Lemma for ~ onfrALT . (Co... |
| onfrALT 44539 | The membership relation is... |
| 19.41rg 44540 | Closed form of right-to-le... |
| opelopab4 44541 | Ordered pair membership in... |
| 2pm13.193 44542 | ~ pm13.193 for two variabl... |
| hbntal 44543 | A closed form of ~ hbn . ~... |
| hbimpg 44544 | A closed form of ~ hbim . ... |
| hbalg 44545 | Closed form of ~ hbal . D... |
| hbexg 44546 | Closed form of ~ nfex . D... |
| ax6e2eq 44547 | Alternate form of ~ ax6e f... |
| ax6e2nd 44548 | If at least two sets exist... |
| ax6e2ndeq 44549 | "At least two sets exist" ... |
| 2sb5nd 44550 | Equivalence for double sub... |
| 2uasbanh 44551 | Distribute the unabbreviat... |
| 2uasban 44552 | Distribute the unabbreviat... |
| e2ebind 44553 | Absorption of an existenti... |
| elpwgded 44554 | ~ elpwgdedVD in convention... |
| trelded 44555 | Deduction form of ~ trel .... |
| jaoded 44556 | Deduction form of ~ jao . ... |
| sbtT 44557 | A substitution into a theo... |
| not12an2impnot1 44558 | If a double conjunction is... |
| in1 44561 | Inference form of ~ df-vd1... |
| iin1 44562 | ~ in1 without virtual dedu... |
| dfvd1ir 44563 | Inference form of ~ df-vd1... |
| idn1 44564 | Virtual deduction identity... |
| dfvd1imp 44565 | Left-to-right part of defi... |
| dfvd1impr 44566 | Right-to-left part of defi... |
| dfvd2 44569 | Definition of a 2-hypothes... |
| dfvd2an 44572 | Definition of a 2-hypothes... |
| dfvd2ani 44573 | Inference form of ~ dfvd2a... |
| dfvd2anir 44574 | Right-to-left inference fo... |
| dfvd2i 44575 | Inference form of ~ dfvd2 ... |
| dfvd2ir 44576 | Right-to-left inference fo... |
| dfvd3 44581 | Definition of a 3-hypothes... |
| dfvd3i 44582 | Inference form of ~ dfvd3 ... |
| dfvd3ir 44583 | Right-to-left inference fo... |
| dfvd3an 44584 | Definition of a 3-hypothes... |
| dfvd3ani 44585 | Inference form of ~ dfvd3a... |
| dfvd3anir 44586 | Right-to-left inference fo... |
| vd01 44587 | A virtual hypothesis virtu... |
| vd02 44588 | Two virtual hypotheses vir... |
| vd03 44589 | A theorem is virtually inf... |
| vd12 44590 | A virtual deduction with 1... |
| vd13 44591 | A virtual deduction with 1... |
| vd23 44592 | A virtual deduction with 2... |
| dfvd2imp 44593 | The virtual deduction form... |
| dfvd2impr 44594 | A 2-antecedent nested impl... |
| in2 44595 | The virtual deduction intr... |
| int2 44596 | The virtual deduction intr... |
| iin2 44597 | ~ in2 without virtual dedu... |
| in2an 44598 | The virtual deduction intr... |
| in3 44599 | The virtual deduction intr... |
| iin3 44600 | ~ in3 without virtual dedu... |
| in3an 44601 | The virtual deduction intr... |
| int3 44602 | The virtual deduction intr... |
| idn2 44603 | Virtual deduction identity... |
| iden2 44604 | Virtual deduction identity... |
| idn3 44605 | Virtual deduction identity... |
| gen11 44606 | Virtual deduction generali... |
| gen11nv 44607 | Virtual deduction generali... |
| gen12 44608 | Virtual deduction generali... |
| gen21 44609 | Virtual deduction generali... |
| gen21nv 44610 | Virtual deduction form of ... |
| gen31 44611 | Virtual deduction generali... |
| gen22 44612 | Virtual deduction generali... |
| ggen22 44613 | ~ gen22 without virtual de... |
| exinst 44614 | Existential Instantiation.... |
| exinst01 44615 | Existential Instantiation.... |
| exinst11 44616 | Existential Instantiation.... |
| e1a 44617 | A Virtual deduction elimin... |
| el1 44618 | A Virtual deduction elimin... |
| e1bi 44619 | Biconditional form of ~ e1... |
| e1bir 44620 | Right biconditional form o... |
| e2 44621 | A virtual deduction elimin... |
| e2bi 44622 | Biconditional form of ~ e2... |
| e2bir 44623 | Right biconditional form o... |
| ee223 44624 | ~ e223 without virtual ded... |
| e223 44625 | A virtual deduction elimin... |
| e222 44626 | A virtual deduction elimin... |
| e220 44627 | A virtual deduction elimin... |
| ee220 44628 | ~ e220 without virtual ded... |
| e202 44629 | A virtual deduction elimin... |
| ee202 44630 | ~ e202 without virtual ded... |
| e022 44631 | A virtual deduction elimin... |
| ee022 44632 | ~ e022 without virtual ded... |
| e002 44633 | A virtual deduction elimin... |
| ee002 44634 | ~ e002 without virtual ded... |
| e020 44635 | A virtual deduction elimin... |
| ee020 44636 | ~ e020 without virtual ded... |
| e200 44637 | A virtual deduction elimin... |
| ee200 44638 | ~ e200 without virtual ded... |
| e221 44639 | A virtual deduction elimin... |
| ee221 44640 | ~ e221 without virtual ded... |
| e212 44641 | A virtual deduction elimin... |
| ee212 44642 | ~ e212 without virtual ded... |
| e122 44643 | A virtual deduction elimin... |
| e112 44644 | A virtual deduction elimin... |
| ee112 44645 | ~ e112 without virtual ded... |
| e121 44646 | A virtual deduction elimin... |
| e211 44647 | A virtual deduction elimin... |
| ee211 44648 | ~ e211 without virtual ded... |
| e210 44649 | A virtual deduction elimin... |
| ee210 44650 | ~ e210 without virtual ded... |
| e201 44651 | A virtual deduction elimin... |
| ee201 44652 | ~ e201 without virtual ded... |
| e120 44653 | A virtual deduction elimin... |
| ee120 44654 | Virtual deduction rule ~ e... |
| e021 44655 | A virtual deduction elimin... |
| ee021 44656 | ~ e021 without virtual ded... |
| e012 44657 | A virtual deduction elimin... |
| ee012 44658 | ~ e012 without virtual ded... |
| e102 44659 | A virtual deduction elimin... |
| ee102 44660 | ~ e102 without virtual ded... |
| e22 44661 | A virtual deduction elimin... |
| e22an 44662 | Conjunction form of ~ e22 ... |
| ee22an 44663 | ~ e22an without virtual de... |
| e111 44664 | A virtual deduction elimin... |
| e1111 44665 | A virtual deduction elimin... |
| e110 44666 | A virtual deduction elimin... |
| ee110 44667 | ~ e110 without virtual ded... |
| e101 44668 | A virtual deduction elimin... |
| ee101 44669 | ~ e101 without virtual ded... |
| e011 44670 | A virtual deduction elimin... |
| ee011 44671 | ~ e011 without virtual ded... |
| e100 44672 | A virtual deduction elimin... |
| ee100 44673 | ~ e100 without virtual ded... |
| e010 44674 | A virtual deduction elimin... |
| ee010 44675 | ~ e010 without virtual ded... |
| e001 44676 | A virtual deduction elimin... |
| ee001 44677 | ~ e001 without virtual ded... |
| e11 44678 | A virtual deduction elimin... |
| e11an 44679 | Conjunction form of ~ e11 ... |
| ee11an 44680 | ~ e11an without virtual de... |
| e01 44681 | A virtual deduction elimin... |
| e01an 44682 | Conjunction form of ~ e01 ... |
| ee01an 44683 | ~ e01an without virtual de... |
| e10 44684 | A virtual deduction elimin... |
| e10an 44685 | Conjunction form of ~ e10 ... |
| ee10an 44686 | ~ e10an without virtual de... |
| e02 44687 | A virtual deduction elimin... |
| e02an 44688 | Conjunction form of ~ e02 ... |
| ee02an 44689 | ~ e02an without virtual de... |
| eel021old 44690 | ~ el021old without virtual... |
| el021old 44691 | A virtual deduction elimin... |
| eel000cT 44692 | An elimination deduction. ... |
| eel0TT 44693 | An elimination deduction. ... |
| eelT00 44694 | An elimination deduction. ... |
| eelTTT 44695 | An elimination deduction. ... |
| eelT11 44696 | An elimination deduction. ... |
| eelT1 44697 | Syllogism inference combin... |
| eelT12 44698 | An elimination deduction. ... |
| eelTT1 44699 | An elimination deduction. ... |
| eelT01 44700 | An elimination deduction. ... |
| eel0T1 44701 | An elimination deduction. ... |
| eel12131 44702 | An elimination deduction. ... |
| eel2131 44703 | ~ syl2an with antecedents ... |
| eel3132 44704 | ~ syl2an with antecedents ... |
| eel0321old 44705 | ~ el0321old without virtua... |
| el0321old 44706 | A virtual deduction elimin... |
| eel2122old 44707 | ~ el2122old without virtua... |
| el2122old 44708 | A virtual deduction elimin... |
| eel0000 44709 | Elimination rule similar t... |
| eel00001 44710 | An elimination deduction. ... |
| eel00000 44711 | Elimination rule similar ~... |
| eel11111 44712 | Five-hypothesis eliminatio... |
| e12 44713 | A virtual deduction elimin... |
| e12an 44714 | Conjunction form of ~ e12 ... |
| el12 44715 | Virtual deduction form of ... |
| e20 44716 | A virtual deduction elimin... |
| e20an 44717 | Conjunction form of ~ e20 ... |
| ee20an 44718 | ~ e20an without virtual de... |
| e21 44719 | A virtual deduction elimin... |
| e21an 44720 | Conjunction form of ~ e21 ... |
| ee21an 44721 | ~ e21an without virtual de... |
| e333 44722 | A virtual deduction elimin... |
| e33 44723 | A virtual deduction elimin... |
| e33an 44724 | Conjunction form of ~ e33 ... |
| ee33an 44725 | ~ e33an without virtual de... |
| e3 44726 | Meta-connective form of ~ ... |
| e3bi 44727 | Biconditional form of ~ e3... |
| e3bir 44728 | Right biconditional form o... |
| e03 44729 | A virtual deduction elimin... |
| ee03 44730 | ~ e03 without virtual dedu... |
| e03an 44731 | Conjunction form of ~ e03 ... |
| ee03an 44732 | Conjunction form of ~ ee03... |
| e30 44733 | A virtual deduction elimin... |
| ee30 44734 | ~ e30 without virtual dedu... |
| e30an 44735 | A virtual deduction elimin... |
| ee30an 44736 | Conjunction form of ~ ee30... |
| e13 44737 | A virtual deduction elimin... |
| e13an 44738 | A virtual deduction elimin... |
| ee13an 44739 | ~ e13an without virtual de... |
| e31 44740 | A virtual deduction elimin... |
| ee31 44741 | ~ e31 without virtual dedu... |
| e31an 44742 | A virtual deduction elimin... |
| ee31an 44743 | ~ e31an without virtual de... |
| e23 44744 | A virtual deduction elimin... |
| e23an 44745 | A virtual deduction elimin... |
| ee23an 44746 | ~ e23an without virtual de... |
| e32 44747 | A virtual deduction elimin... |
| ee32 44748 | ~ e32 without virtual dedu... |
| e32an 44749 | A virtual deduction elimin... |
| ee32an 44750 | ~ e33an without virtual de... |
| e123 44751 | A virtual deduction elimin... |
| ee123 44752 | ~ e123 without virtual ded... |
| el123 44753 | A virtual deduction elimin... |
| e233 44754 | A virtual deduction elimin... |
| e323 44755 | A virtual deduction elimin... |
| e000 44756 | A virtual deduction elimin... |
| e00 44757 | Elimination rule identical... |
| e00an 44758 | Elimination rule identical... |
| eel00cT 44759 | An elimination deduction. ... |
| eelTT 44760 | An elimination deduction. ... |
| e0a 44761 | Elimination rule identical... |
| eelT 44762 | An elimination deduction. ... |
| eel0cT 44763 | An elimination deduction. ... |
| eelT0 44764 | An elimination deduction. ... |
| e0bi 44765 | Elimination rule identical... |
| e0bir 44766 | Elimination rule identical... |
| uun0.1 44767 | Convention notation form o... |
| un0.1 44768 | ` T. ` is the constant tru... |
| uunT1 44769 | A deduction unionizing a n... |
| uunT1p1 44770 | A deduction unionizing a n... |
| uunT21 44771 | A deduction unionizing a n... |
| uun121 44772 | A deduction unionizing a n... |
| uun121p1 44773 | A deduction unionizing a n... |
| uun132 44774 | A deduction unionizing a n... |
| uun132p1 44775 | A deduction unionizing a n... |
| anabss7p1 44776 | A deduction unionizing a n... |
| un10 44777 | A unionizing deduction. (... |
| un01 44778 | A unionizing deduction. (... |
| un2122 44779 | A deduction unionizing a n... |
| uun2131 44780 | A deduction unionizing a n... |
| uun2131p1 44781 | A deduction unionizing a n... |
| uunTT1 44782 | A deduction unionizing a n... |
| uunTT1p1 44783 | A deduction unionizing a n... |
| uunTT1p2 44784 | A deduction unionizing a n... |
| uunT11 44785 | A deduction unionizing a n... |
| uunT11p1 44786 | A deduction unionizing a n... |
| uunT11p2 44787 | A deduction unionizing a n... |
| uunT12 44788 | A deduction unionizing a n... |
| uunT12p1 44789 | A deduction unionizing a n... |
| uunT12p2 44790 | A deduction unionizing a n... |
| uunT12p3 44791 | A deduction unionizing a n... |
| uunT12p4 44792 | A deduction unionizing a n... |
| uunT12p5 44793 | A deduction unionizing a n... |
| uun111 44794 | A deduction unionizing a n... |
| 3anidm12p1 44795 | A deduction unionizing a n... |
| 3anidm12p2 44796 | A deduction unionizing a n... |
| uun123 44797 | A deduction unionizing a n... |
| uun123p1 44798 | A deduction unionizing a n... |
| uun123p2 44799 | A deduction unionizing a n... |
| uun123p3 44800 | A deduction unionizing a n... |
| uun123p4 44801 | A deduction unionizing a n... |
| uun2221 44802 | A deduction unionizing a n... |
| uun2221p1 44803 | A deduction unionizing a n... |
| uun2221p2 44804 | A deduction unionizing a n... |
| 3impdirp1 44805 | A deduction unionizing a n... |
| 3impcombi 44806 | A 1-hypothesis proposition... |
| trsspwALT 44807 | Virtual deduction proof of... |
| trsspwALT2 44808 | Virtual deduction proof of... |
| trsspwALT3 44809 | Short predicate calculus p... |
| sspwtr 44810 | Virtual deduction proof of... |
| sspwtrALT 44811 | Virtual deduction proof of... |
| sspwtrALT2 44812 | Short predicate calculus p... |
| pwtrVD 44813 | Virtual deduction proof of... |
| pwtrrVD 44814 | Virtual deduction proof of... |
| suctrALT 44815 | The successor of a transit... |
| snssiALTVD 44816 | Virtual deduction proof of... |
| snssiALT 44817 | If a class is an element o... |
| snsslVD 44818 | Virtual deduction proof of... |
| snssl 44819 | If a singleton is a subcla... |
| snelpwrVD 44820 | Virtual deduction proof of... |
| unipwrVD 44821 | Virtual deduction proof of... |
| unipwr 44822 | A class is a subclass of t... |
| sstrALT2VD 44823 | Virtual deduction proof of... |
| sstrALT2 44824 | Virtual deduction proof of... |
| suctrALT2VD 44825 | Virtual deduction proof of... |
| suctrALT2 44826 | Virtual deduction proof of... |
| elex2VD 44827 | Virtual deduction proof of... |
| elex22VD 44828 | Virtual deduction proof of... |
| eqsbc2VD 44829 | Virtual deduction proof of... |
| zfregs2VD 44830 | Virtual deduction proof of... |
| tpid3gVD 44831 | Virtual deduction proof of... |
| en3lplem1VD 44832 | Virtual deduction proof of... |
| en3lplem2VD 44833 | Virtual deduction proof of... |
| en3lpVD 44834 | Virtual deduction proof of... |
| simplbi2VD 44835 | Virtual deduction proof of... |
| 3ornot23VD 44836 | Virtual deduction proof of... |
| orbi1rVD 44837 | Virtual deduction proof of... |
| bitr3VD 44838 | Virtual deduction proof of... |
| 3orbi123VD 44839 | Virtual deduction proof of... |
| sbc3orgVD 44840 | Virtual deduction proof of... |
| 19.21a3con13vVD 44841 | Virtual deduction proof of... |
| exbirVD 44842 | Virtual deduction proof of... |
| exbiriVD 44843 | Virtual deduction proof of... |
| rspsbc2VD 44844 | Virtual deduction proof of... |
| 3impexpVD 44845 | Virtual deduction proof of... |
| 3impexpbicomVD 44846 | Virtual deduction proof of... |
| 3impexpbicomiVD 44847 | Virtual deduction proof of... |
| sbcoreleleqVD 44848 | Virtual deduction proof of... |
| hbra2VD 44849 | Virtual deduction proof of... |
| tratrbVD 44850 | Virtual deduction proof of... |
| al2imVD 44851 | Virtual deduction proof of... |
| syl5impVD 44852 | Virtual deduction proof of... |
| idiVD 44853 | Virtual deduction proof of... |
| ancomstVD 44854 | Closed form of ~ ancoms . ... |
| ssralv2VD 44855 | Quantification restricted ... |
| ordelordALTVD 44856 | An element of an ordinal c... |
| equncomVD 44857 | If a class equals the unio... |
| equncomiVD 44858 | Inference form of ~ equnco... |
| sucidALTVD 44859 | A set belongs to its succe... |
| sucidALT 44860 | A set belongs to its succe... |
| sucidVD 44861 | A set belongs to its succe... |
| imbi12VD 44862 | Implication form of ~ imbi... |
| imbi13VD 44863 | Join three logical equival... |
| sbcim2gVD 44864 | Distribution of class subs... |
| sbcbiVD 44865 | Implication form of ~ sbcb... |
| trsbcVD 44866 | Formula-building inference... |
| truniALTVD 44867 | The union of a class of tr... |
| ee33VD 44868 | Non-virtual deduction form... |
| trintALTVD 44869 | The intersection of a clas... |
| trintALT 44870 | The intersection of a clas... |
| undif3VD 44871 | The first equality of Exer... |
| sbcssgVD 44872 | Virtual deduction proof of... |
| csbingVD 44873 | Virtual deduction proof of... |
| onfrALTlem5VD 44874 | Virtual deduction proof of... |
| onfrALTlem4VD 44875 | Virtual deduction proof of... |
| onfrALTlem3VD 44876 | Virtual deduction proof of... |
| simplbi2comtVD 44877 | Virtual deduction proof of... |
| onfrALTlem2VD 44878 | Virtual deduction proof of... |
| onfrALTlem1VD 44879 | Virtual deduction proof of... |
| onfrALTVD 44880 | Virtual deduction proof of... |
| csbeq2gVD 44881 | Virtual deduction proof of... |
| csbsngVD 44882 | Virtual deduction proof of... |
| csbxpgVD 44883 | Virtual deduction proof of... |
| csbresgVD 44884 | Virtual deduction proof of... |
| csbrngVD 44885 | Virtual deduction proof of... |
| csbima12gALTVD 44886 | Virtual deduction proof of... |
| csbunigVD 44887 | Virtual deduction proof of... |
| csbfv12gALTVD 44888 | Virtual deduction proof of... |
| con5VD 44889 | Virtual deduction proof of... |
| relopabVD 44890 | Virtual deduction proof of... |
| 19.41rgVD 44891 | Virtual deduction proof of... |
| 2pm13.193VD 44892 | Virtual deduction proof of... |
| hbimpgVD 44893 | Virtual deduction proof of... |
| hbalgVD 44894 | Virtual deduction proof of... |
| hbexgVD 44895 | Virtual deduction proof of... |
| ax6e2eqVD 44896 | The following User's Proof... |
| ax6e2ndVD 44897 | The following User's Proof... |
| ax6e2ndeqVD 44898 | The following User's Proof... |
| 2sb5ndVD 44899 | The following User's Proof... |
| 2uasbanhVD 44900 | The following User's Proof... |
| e2ebindVD 44901 | The following User's Proof... |
| sb5ALTVD 44902 | The following User's Proof... |
| vk15.4jVD 44903 | The following User's Proof... |
| notnotrALTVD 44904 | The following User's Proof... |
| con3ALTVD 44905 | The following User's Proof... |
| elpwgdedVD 44906 | Membership in a power clas... |
| sspwimp 44907 | If a class is a subclass o... |
| sspwimpVD 44908 | The following User's Proof... |
| sspwimpcf 44909 | If a class is a subclass o... |
| sspwimpcfVD 44910 | The following User's Proof... |
| suctrALTcf 44911 | The successor of a transit... |
| suctrALTcfVD 44912 | The following User's Proof... |
| suctrALT3 44913 | The successor of a transit... |
| sspwimpALT 44914 | If a class is a subclass o... |
| unisnALT 44915 | A set equals the union of ... |
| notnotrALT2 44916 | Converse of double negatio... |
| sspwimpALT2 44917 | If a class is a subclass o... |
| e2ebindALT 44918 | Absorption of an existenti... |
| ax6e2ndALT 44919 | If at least two sets exist... |
| ax6e2ndeqALT 44920 | "At least two sets exist" ... |
| 2sb5ndALT 44921 | Equivalence for double sub... |
| chordthmALT 44922 | The intersecting chords th... |
| isosctrlem1ALT 44923 | Lemma for ~ isosctr . Thi... |
| iunconnlem2 44924 | The indexed union of conne... |
| iunconnALT 44925 | The indexed union of conne... |
| sineq0ALT 44926 | A complex number whose sin... |
| rspesbcd 44927 | Restricted quantifier vers... |
| rext0 44928 | Nonempty existential quant... |
| dfbi1ALTa 44929 | Version of ~ dfbi1ALT usin... |
| simprimi 44930 | Inference associated with ... |
| dfbi1ALTb 44931 | Further shorten ~ dfbi1ALT... |
| relpeq1 44934 | Equality theorem for relat... |
| relpeq2 44935 | Equality theorem for relat... |
| relpeq3 44936 | Equality theorem for relat... |
| relpeq4 44937 | Equality theorem for relat... |
| relpeq5 44938 | Equality theorem for relat... |
| nfrelp 44939 | Bound-variable hypothesis ... |
| relpf 44940 | A relation-preserving func... |
| relprel 44941 | A relation-preserving func... |
| relpmin 44942 | A preimage of a minimal el... |
| relpfrlem 44943 | Lemma for ~ relpfr . Prov... |
| relpfr 44944 | If the image of a set unde... |
| orbitex 44945 | Orbits exist. Given a set... |
| orbitinit 44946 | A set is contained in its ... |
| orbitcl 44947 | The orbit under a function... |
| orbitclmpt 44948 | Version of ~ orbitcl using... |
| trwf 44949 | The class of well-founded ... |
| rankrelp 44950 | The rank function preserve... |
| wffr 44951 | The class of well-founded ... |
| trfr 44952 | A transitive class well-fo... |
| tcfr 44953 | A set is well-founded if a... |
| xpwf 44954 | The Cartesian product of t... |
| dmwf 44955 | The domain of a well-found... |
| rnwf 44956 | The range of a well-founde... |
| relwf 44957 | A relation is a well-found... |
| ralabso 44958 | Simplification of restrict... |
| rexabso 44959 | Simplification of restrict... |
| ralabsod 44960 | Deduction form of ~ ralabs... |
| rexabsod 44961 | Deduction form of ~ rexabs... |
| ralabsobidv 44962 | Formula-building lemma for... |
| rexabsobidv 44963 | Formula-building lemma for... |
| ssabso 44964 | The notion " ` x ` is a su... |
| disjabso 44965 | Disjointness is absolute f... |
| n0abso 44966 | Nonemptiness is absolute f... |
| traxext 44967 | A transitive class models ... |
| modelaxreplem1 44968 | Lemma for ~ modelaxrep . ... |
| modelaxreplem2 44969 | Lemma for ~ modelaxrep . ... |
| modelaxreplem3 44970 | Lemma for ~ modelaxrep . ... |
| modelaxrep 44971 | Conditions which guarantee... |
| ssclaxsep 44972 | A class that is closed und... |
| 0elaxnul 44973 | A class that contains the ... |
| pwclaxpow 44974 | Suppose ` M ` is a transit... |
| prclaxpr 44975 | A class that is closed und... |
| uniclaxun 44976 | A class that is closed und... |
| sswfaxreg 44977 | A subclass of the class of... |
| omssaxinf2 44978 | A class that contains all ... |
| omelaxinf2 44979 | A transitive class that co... |
| dfac5prim 44980 | ~ dfac5 expanded into prim... |
| ac8prim 44981 | ~ ac8 expanded into primit... |
| modelac8prim 44982 | If ` M ` is a transitive c... |
| wfaxext 44983 | The class of well-founded ... |
| wfaxrep 44984 | The class of well-founded ... |
| wfaxsep 44985 | The class of well-founded ... |
| wfaxnul 44986 | The class of well-founded ... |
| wfaxpow 44987 | The class of well-founded ... |
| wfaxpr 44988 | The class of well-founded ... |
| wfaxun 44989 | The class of well-founded ... |
| wfaxreg 44990 | The class of well-founded ... |
| wfaxinf2 44991 | The class of well-founded ... |
| wfac8prim 44992 | The class of well-founded ... |
| brpermmodel 44993 | The membership relation in... |
| brpermmodelcnv 44994 | Ordinary membership expres... |
| permaxext 44995 | The Axiom of Extensionalit... |
| permaxrep 44996 | The Axiom of Replacement ~... |
| permaxsep 44997 | The Axiom of Separation ~ ... |
| permaxnul 44998 | The Null Set Axiom ~ ax-nu... |
| permaxpow 44999 | The Axiom of Power Sets ~ ... |
| permaxpr 45000 | The Axiom of Pairing ~ ax-... |
| permaxun 45001 | The Axiom of Union ~ ax-un... |
| permaxinf2lem 45002 | Lemma for ~ permaxinf2 . ... |
| permaxinf2 45003 | The Axiom of Infinity ~ ax... |
| permac8prim 45004 | The Axiom of Choice ~ ac8p... |
| nregmodelf1o 45005 | Define a permutation ` F `... |
| nregmodellem 45006 | Lemma for ~ nregmodel . (... |
| nregmodel 45007 | The Axiom of Regularity ~ ... |
| nregmodelaxext 45008 | The Axiom of Extensionalit... |
| evth2f 45009 | A version of ~ evth2 using... |
| elunif 45010 | A version of ~ eluni using... |
| rzalf 45011 | A version of ~ rzal using ... |
| fvelrnbf 45012 | A version of ~ fvelrnb usi... |
| rfcnpre1 45013 | If F is a continuous funct... |
| ubelsupr 45014 | If U belongs to A and U is... |
| fsumcnf 45015 | A finite sum of functions ... |
| mulltgt0 45016 | The product of a negative ... |
| rspcegf 45017 | A version of ~ rspcev usin... |
| rabexgf 45018 | A version of ~ rabexg usin... |
| fcnre 45019 | A function continuous with... |
| sumsnd 45020 | A sum of a singleton is th... |
| evthf 45021 | A version of ~ evth using ... |
| cnfex 45022 | The class of continuous fu... |
| fnchoice 45023 | For a finite set, a choice... |
| refsumcn 45024 | A finite sum of continuous... |
| rfcnpre2 45025 | If ` F ` is a continuous f... |
| cncmpmax 45026 | When the hypothesis for th... |
| rfcnpre3 45027 | If F is a continuous funct... |
| rfcnpre4 45028 | If F is a continuous funct... |
| sumpair 45029 | Sum of two distinct comple... |
| rfcnnnub 45030 | Given a real continuous fu... |
| refsum2cnlem1 45031 | This is the core Lemma for... |
| refsum2cn 45032 | The sum of two continuus r... |
| adantlllr 45033 | Deduction adding a conjunc... |
| 3adantlr3 45034 | Deduction adding a conjunc... |
| 3adantll2 45035 | Deduction adding a conjunc... |
| 3adantll3 45036 | Deduction adding a conjunc... |
| ssnel 45037 | If not element of a set, t... |
| sncldre 45038 | A singleton is closed w.r.... |
| n0p 45039 | A polynomial with a nonzer... |
| pm2.65ni 45040 | Inference rule for proof b... |
| iuneq2df 45041 | Equality deduction for ind... |
| nnfoctb 45042 | There exists a mapping fro... |
| elpwinss 45043 | An element of the powerset... |
| unidmex 45044 | If ` F ` is a set, then ` ... |
| ndisj2 45045 | A non-disjointness conditi... |
| zenom 45046 | The set of integer numbers... |
| uzwo4 45047 | Well-ordering principle: a... |
| unisn0 45048 | The union of the singleton... |
| ssin0 45049 | If two classes are disjoin... |
| inabs3 45050 | Absorption law for interse... |
| pwpwuni 45051 | Relationship between power... |
| disjiun2 45052 | In a disjoint collection, ... |
| 0pwfi 45053 | The empty set is in any po... |
| ssinss2d 45054 | Intersection preserves sub... |
| zct 45055 | The set of integer numbers... |
| pwfin0 45056 | A finite set always belong... |
| uzct 45057 | An upper integer set is co... |
| iunxsnf 45058 | A singleton index picks ou... |
| fiiuncl 45059 | If a set is closed under t... |
| iunp1 45060 | The addition of the next s... |
| fiunicl 45061 | If a set is closed under t... |
| ixpeq2d 45062 | Equality theorem for infin... |
| disjxp1 45063 | The sets of a cartesian pr... |
| disjsnxp 45064 | The sets in the cartesian ... |
| eliind 45065 | Membership in indexed inte... |
| rspcef 45066 | Restricted existential spe... |
| ixpssmapc 45067 | An infinite Cartesian prod... |
| elintd 45068 | Membership in class inters... |
| ssdf 45069 | A sufficient condition for... |
| brneqtrd 45070 | Substitution of equal clas... |
| ssnct 45071 | A set containing an uncoun... |
| ssuniint 45072 | Sufficient condition for b... |
| elintdv 45073 | Membership in class inters... |
| ssd 45074 | A sufficient condition for... |
| ralimralim 45075 | Introducing any antecedent... |
| snelmap 45076 | Membership of the element ... |
| xrnmnfpnf 45077 | An extended real that is n... |
| nelrnmpt 45078 | Non-membership in the rang... |
| iuneq1i 45079 | Equality theorem for index... |
| nssrex 45080 | Negation of subclass relat... |
| ssinc 45081 | Inclusion relation for a m... |
| ssdec 45082 | Inclusion relation for a m... |
| elixpconstg 45083 | Membership in an infinite ... |
| iineq1d 45084 | Equality theorem for index... |
| metpsmet 45085 | A metric is a pseudometric... |
| ixpssixp 45086 | Subclass theorem for infin... |
| ballss3 45087 | A sufficient condition for... |
| iunincfi 45088 | Given a sequence of increa... |
| nsstr 45089 | If it's not a subclass, it... |
| rexanuz3 45090 | Combine two different uppe... |
| cbvmpo2 45091 | Rule to change the second ... |
| cbvmpo1 45092 | Rule to change the first b... |
| eliuniin 45093 | Indexed union of indexed i... |
| ssabf 45094 | Subclass of a class abstra... |
| pssnssi 45095 | A proper subclass does not... |
| rabidim2 45096 | Membership in a restricted... |
| eluni2f 45097 | Membership in class union.... |
| eliin2f 45098 | Membership in indexed inte... |
| nssd 45099 | Negation of subclass relat... |
| iineq12dv 45100 | Equality deduction for ind... |
| supxrcld 45101 | The supremum of an arbitra... |
| elrestd 45102 | A sufficient condition for... |
| eliuniincex 45103 | Counterexample to show tha... |
| eliincex 45104 | Counterexample to show tha... |
| eliinid 45105 | Membership in an indexed i... |
| abssf 45106 | Class abstraction in a sub... |
| supxrubd 45107 | A member of a set of exten... |
| ssrabf 45108 | Subclass of a restricted c... |
| ssrabdf 45109 | Subclass of a restricted c... |
| eliin2 45110 | Membership in indexed inte... |
| ssrab2f 45111 | Subclass relation for a re... |
| restuni3 45112 | The underlying set of a su... |
| rabssf 45113 | Restricted class abstracti... |
| eliuniin2 45114 | Indexed union of indexed i... |
| restuni4 45115 | The underlying set of a su... |
| restuni6 45116 | The underlying set of a su... |
| restuni5 45117 | The underlying set of a su... |
| unirestss 45118 | The union of an elementwis... |
| iniin1 45119 | Indexed intersection of in... |
| iniin2 45120 | Indexed intersection of in... |
| cbvrabv2 45121 | A more general version of ... |
| cbvrabv2w 45122 | A more general version of ... |
| iinssiin 45123 | Subset implication for an ... |
| eliind2 45124 | Membership in indexed inte... |
| iinssd 45125 | Subset implication for an ... |
| rabbida2 45126 | Equivalent wff's yield equ... |
| iinexd 45127 | The existence of an indexe... |
| rabexf 45128 | Separation Scheme in terms... |
| rabbida3 45129 | Equivalent wff's yield equ... |
| r19.36vf 45130 | Restricted quantifier vers... |
| raleqd 45131 | Equality deduction for res... |
| iinssf 45132 | Subset implication for an ... |
| iinssdf 45133 | Subset implication for an ... |
| resabs2i 45134 | Absorption law for restric... |
| ssdf2 45135 | A sufficient condition for... |
| rabssd 45136 | Restricted class abstracti... |
| rexnegd 45137 | Minus a real number. (Con... |
| rexlimd3 45138 | * Inference from Theorem 1... |
| nel1nelini 45139 | Membership in an intersect... |
| nel2nelini 45140 | Membership in an intersect... |
| eliunid 45141 | Membership in indexed unio... |
| reximdd 45142 | Deduction from Theorem 19.... |
| inopnd 45143 | The intersection of two op... |
| ss2rabdf 45144 | Deduction of restricted ab... |
| restopn3 45145 | If ` A ` is open, then ` A... |
| restopnssd 45146 | A topology restricted to a... |
| restsubel 45147 | A subset belongs in the sp... |
| toprestsubel 45148 | A subset is open in the to... |
| rabidd 45149 | An "identity" law of concr... |
| iunssdf 45150 | Subset theorem for an inde... |
| iinss2d 45151 | Subset implication for an ... |
| r19.3rzf 45152 | Restricted quantification ... |
| r19.28zf 45153 | Restricted quantifier vers... |
| iindif2f 45154 | Indexed intersection of cl... |
| ralfal 45155 | Two ways of expressing emp... |
| archd 45156 | Archimedean property of re... |
| nimnbi 45157 | If an implication is false... |
| nimnbi2 45158 | If an implication is false... |
| notbicom 45159 | Commutative law for the ne... |
| rexeqif 45160 | Equality inference for res... |
| rspced 45161 | Restricted existential spe... |
| fnresdmss 45162 | A function does not change... |
| fmptsnxp 45163 | Maps-to notation and Carte... |
| fvmpt2bd 45164 | Value of a function given ... |
| rnmptfi 45165 | The range of a function wi... |
| fresin2 45166 | Restriction of a function ... |
| ffi 45167 | A function with finite dom... |
| suprnmpt 45168 | An explicit bound for the ... |
| rnffi 45169 | The range of a function wi... |
| mptelpm 45170 | A function in maps-to nota... |
| rnmptpr 45171 | Range of a function define... |
| resmpti 45172 | Restriction of the mapping... |
| founiiun 45173 | Union expressed as an inde... |
| rnresun 45174 | Distribution law for range... |
| elrnmptf 45175 | The range of a function in... |
| rnmptssrn 45176 | Inclusion relation for two... |
| disjf1 45177 | A 1 to 1 mapping built fro... |
| rnsnf 45178 | The range of a function wh... |
| wessf1ornlem 45179 | Given a function ` F ` on ... |
| wessf1orn 45180 | Given a function ` F ` on ... |
| nelrnres 45181 | If ` A ` is not in the ran... |
| disjrnmpt2 45182 | Disjointness of the range ... |
| elrnmpt1sf 45183 | Elementhood in an image se... |
| founiiun0 45184 | Union expressed as an inde... |
| disjf1o 45185 | A bijection built from dis... |
| disjinfi 45186 | Only a finite number of di... |
| fvovco 45187 | Value of the composition o... |
| ssnnf1octb 45188 | There exists a bijection b... |
| nnf1oxpnn 45189 | There is a bijection betwe... |
| rnmptssd 45190 | The range of a function gi... |
| projf1o 45191 | A biijection from a set to... |
| fvmap 45192 | Function value for a membe... |
| fvixp2 45193 | Projection of a factor of ... |
| choicefi 45194 | For a finite set, a choice... |
| mpct 45195 | The exponentiation of a co... |
| cnmetcoval 45196 | Value of the distance func... |
| fcomptss 45197 | Express composition of two... |
| elmapsnd 45198 | Membership in a set expone... |
| mapss2 45199 | Subset inheritance for set... |
| fsneq 45200 | Equality condition for two... |
| difmap 45201 | Difference of two sets exp... |
| unirnmap 45202 | Given a subset of a set ex... |
| inmap 45203 | Intersection of two sets e... |
| fcoss 45204 | Composition of two mapping... |
| fsneqrn 45205 | Equality condition for two... |
| difmapsn 45206 | Difference of two sets exp... |
| mapssbi 45207 | Subset inheritance for set... |
| unirnmapsn 45208 | Equality theorem for a sub... |
| iunmapss 45209 | The indexed union of set e... |
| ssmapsn 45210 | A subset ` C ` of a set ex... |
| iunmapsn 45211 | The indexed union of set e... |
| absfico 45212 | Mapping domain and codomai... |
| icof 45213 | The set of left-closed rig... |
| elpmrn 45214 | The range of a partial fun... |
| imaexi 45215 | The image of a set is a se... |
| axccdom 45216 | Relax the constraint on ax... |
| dmmptdff 45217 | The domain of the mapping ... |
| dmmptdf 45218 | The domain of the mapping ... |
| elpmi2 45219 | The domain of a partial fu... |
| dmrelrnrel 45220 | A relation preserving func... |
| fvcod 45221 | Value of a function compos... |
| elrnmpoid 45222 | Membership in the range of... |
| axccd 45223 | An alternative version of ... |
| axccd2 45224 | An alternative version of ... |
| feqresmptf 45225 | Express a restricted funct... |
| dmmptssf 45226 | The domain of a mapping is... |
| dmmptdf2 45227 | The domain of the mapping ... |
| dmuz 45228 | Domain of the upper intege... |
| fmptd2f 45229 | Domain and codomain of the... |
| mpteq1df 45230 | An equality theorem for th... |
| mptexf 45231 | If the domain of a functio... |
| fvmpt4 45232 | Value of a function given ... |
| fmptf 45233 | Functionality of the mappi... |
| resimass 45234 | The image of a restriction... |
| mptssid 45235 | The mapping operation expr... |
| mptfnd 45236 | The maps-to notation defin... |
| rnmptlb 45237 | Boundness below of the ran... |
| rnmptbddlem 45238 | Boundness of the range of ... |
| rnmptbdd 45239 | Boundness of the range of ... |
| funimaeq 45240 | Membership relation for th... |
| rnmptssf 45241 | The range of a function gi... |
| rnmptbd2lem 45242 | Boundness below of the ran... |
| rnmptbd2 45243 | Boundness below of the ran... |
| infnsuprnmpt 45244 | The indexed infimum of rea... |
| suprclrnmpt 45245 | Closure of the indexed sup... |
| suprubrnmpt2 45246 | A member of a nonempty ind... |
| suprubrnmpt 45247 | A member of a nonempty ind... |
| rnmptssdf 45248 | The range of a function gi... |
| rnmptbdlem 45249 | Boundness above of the ran... |
| rnmptbd 45250 | Boundness above of the ran... |
| rnmptss2 45251 | The range of a function gi... |
| elmptima 45252 | The image of a function in... |
| ralrnmpt3 45253 | A restricted quantifier ov... |
| rnmptssbi 45254 | The range of a function gi... |
| imass2d 45255 | Subset theorem for image. ... |
| imassmpt 45256 | Membership relation for th... |
| fpmd 45257 | A total function is a part... |
| fconst7 45258 | An alternative way to expr... |
| fnmptif 45259 | Functionality and domain o... |
| dmmptif 45260 | Domain of the mapping oper... |
| mpteq2dfa 45261 | Slightly more general equa... |
| dmmpt1 45262 | The domain of the mapping ... |
| fmptff 45263 | Functionality of the mappi... |
| fvmptelcdmf 45264 | The value of a function at... |
| fmptdff 45265 | A version of ~ fmptd using... |
| fvmpt2df 45266 | Deduction version of ~ fvm... |
| rn1st 45267 | The range of a function wi... |
| rnmptssff 45268 | The range of a function gi... |
| rnmptssdff 45269 | The range of a function gi... |
| fvmpt4d 45270 | Value of a function given ... |
| sub2times 45271 | Subtracting from a number,... |
| nnxrd 45272 | A natural number is an ext... |
| nnxr 45273 | A natural number is an ext... |
| abssubrp 45274 | The distance of two distin... |
| elfzfzo 45275 | Relationship between membe... |
| oddfl 45276 | Odd number representation ... |
| abscosbd 45277 | Bound for the absolute val... |
| mul13d 45278 | Commutative/associative la... |
| negpilt0 45279 | Negative ` _pi ` is negati... |
| dstregt0 45280 | A complex number ` A ` tha... |
| subadd4b 45281 | Rearrangement of 4 terms i... |
| xrlttri5d 45282 | Not equal and not larger i... |
| zltlesub 45283 | If an integer ` N ` is les... |
| divlt0gt0d 45284 | The ratio of a negative nu... |
| subsub23d 45285 | Swap subtrahend and result... |
| 2timesgt 45286 | Double of a positive real ... |
| reopn 45287 | The reals are open with re... |
| sub31 45288 | Swap the first and third t... |
| nnne1ge2 45289 | A positive integer which i... |
| lefldiveq 45290 | A closed enough, smaller r... |
| negsubdi3d 45291 | Distribution of negative o... |
| ltdiv2dd 45292 | Division of a positive num... |
| abssinbd 45293 | Bound for the absolute val... |
| halffl 45294 | Floor of ` ( 1 / 2 ) ` . ... |
| monoords 45295 | Ordering relation for a st... |
| hashssle 45296 | The size of a subset of a ... |
| lttri5d 45297 | Not equal and not larger i... |
| fzisoeu 45298 | A finite ordered set has a... |
| lt3addmuld 45299 | If three real numbers are ... |
| absnpncan2d 45300 | Triangular inequality, com... |
| fperiodmullem 45301 | A function with period ` T... |
| fperiodmul 45302 | A function with period T i... |
| upbdrech 45303 | Choice of an upper bound f... |
| lt4addmuld 45304 | If four real numbers are l... |
| absnpncan3d 45305 | Triangular inequality, com... |
| upbdrech2 45306 | Choice of an upper bound f... |
| ssfiunibd 45307 | A finite union of bounded ... |
| fzdifsuc2 45308 | Remove a successor from th... |
| fzsscn 45309 | A finite sequence of integ... |
| divcan8d 45310 | A cancellation law for div... |
| dmmcand 45311 | Cancellation law for divis... |
| fzssre 45312 | A finite sequence of integ... |
| bccld 45313 | A binomial coefficient, in... |
| fzssnn0 45314 | A finite set of sequential... |
| xreqle 45315 | Equality implies 'less tha... |
| xaddlidd 45316 | ` 0 ` is a left identity f... |
| xadd0ge 45317 | A number is less than or e... |
| xrgtned 45318 | 'Greater than' implies not... |
| xrleneltd 45319 | 'Less than or equal to' an... |
| xaddcomd 45320 | The extended real addition... |
| supxrre3 45321 | The supremum of a nonempty... |
| uzfissfz 45322 | For any finite subset of t... |
| xleadd2d 45323 | Addition of extended reals... |
| suprltrp 45324 | The supremum of a nonempty... |
| xleadd1d 45325 | Addition of extended reals... |
| xreqled 45326 | Equality implies 'less tha... |
| xrgepnfd 45327 | An extended real greater t... |
| xrge0nemnfd 45328 | A nonnegative extended rea... |
| supxrgere 45329 | If a real number can be ap... |
| iuneqfzuzlem 45330 | Lemma for ~ iuneqfzuz : he... |
| iuneqfzuz 45331 | If two unions indexed by u... |
| xle2addd 45332 | Adding both side of two in... |
| supxrgelem 45333 | If an extended real number... |
| supxrge 45334 | If an extended real number... |
| suplesup 45335 | If any element of ` A ` ca... |
| infxrglb 45336 | The infimum of a set of ex... |
| xadd0ge2 45337 | A number is less than or e... |
| nepnfltpnf 45338 | An extended real that is n... |
| ltadd12dd 45339 | Addition to both sides of ... |
| nemnftgtmnft 45340 | An extended real that is n... |
| xrgtso 45341 | 'Greater than' is a strict... |
| rpex 45342 | The positive reals form a ... |
| xrge0ge0 45343 | A nonnegative extended rea... |
| xrssre 45344 | A subset of extended reals... |
| ssuzfz 45345 | A finite subset of the upp... |
| absfun 45346 | The absolute value is a fu... |
| infrpge 45347 | The infimum of a nonempty,... |
| xrlexaddrp 45348 | If an extended real number... |
| supsubc 45349 | The supremum function dist... |
| xralrple2 45350 | Show that ` A ` is less th... |
| nnuzdisj 45351 | The first ` N ` elements o... |
| ltdivgt1 45352 | Divsion by a number greate... |
| xrltned 45353 | 'Less than' implies not eq... |
| nnsplit 45354 | Express the set of positiv... |
| divdiv3d 45355 | Division into a fraction. ... |
| abslt2sqd 45356 | Comparison of the square o... |
| qenom 45357 | The set of rational number... |
| qct 45358 | The set of rational number... |
| xrltnled 45359 | 'Less than' in terms of 'l... |
| lenlteq 45360 | 'less than or equal to' bu... |
| xrred 45361 | An extended real that is n... |
| rr2sscn2 45362 | The cartesian square of ` ... |
| infxr 45363 | The infimum of a set of ex... |
| infxrunb2 45364 | The infimum of an unbounde... |
| infxrbnd2 45365 | The infimum of a bounded-b... |
| infleinflem1 45366 | Lemma for ~ infleinf , cas... |
| infleinflem2 45367 | Lemma for ~ infleinf , whe... |
| infleinf 45368 | If any element of ` B ` ca... |
| xralrple4 45369 | Show that ` A ` is less th... |
| xralrple3 45370 | Show that ` A ` is less th... |
| eluzelzd 45371 | A member of an upper set o... |
| suplesup2 45372 | If any element of ` A ` is... |
| recnnltrp 45373 | ` N ` is a natural number ... |
| nnn0 45374 | The set of positive intege... |
| fzct 45375 | A finite set of sequential... |
| rpgtrecnn 45376 | Any positive real number i... |
| fzossuz 45377 | A half-open integer interv... |
| infxrrefi 45378 | The real and extended real... |
| xrralrecnnle 45379 | Show that ` A ` is less th... |
| fzoct 45380 | A finite set of sequential... |
| frexr 45381 | A function taking real val... |
| nnrecrp 45382 | The reciprocal of a positi... |
| reclt0d 45383 | The reciprocal of a negati... |
| lt0neg1dd 45384 | If a number is negative, i... |
| infxrcld 45385 | The infimum of an arbitrar... |
| xrralrecnnge 45386 | Show that ` A ` is less th... |
| reclt0 45387 | The reciprocal of a negati... |
| ltmulneg 45388 | Multiplying by a negative ... |
| allbutfi 45389 | For all but finitely many.... |
| ltdiv23neg 45390 | Swap denominator with othe... |
| xreqnltd 45391 | A consequence of trichotom... |
| mnfnre2 45392 | Minus infinity is not a re... |
| zssxr 45393 | The integers are a subset ... |
| fisupclrnmpt 45394 | A nonempty finite indexed ... |
| supxrunb3 45395 | The supremum of an unbound... |
| elfzod 45396 | Membership in a half-open ... |
| fimaxre4 45397 | A nonempty finite set of r... |
| ren0 45398 | The set of reals is nonemp... |
| eluzelz2 45399 | A member of an upper set o... |
| resabs2d 45400 | Absorption law for restric... |
| uzid2 45401 | Membership of the least me... |
| supxrleubrnmpt 45402 | The supremum of a nonempty... |
| uzssre2 45403 | An upper set of integers i... |
| uzssd 45404 | Subset relationship for tw... |
| eluzd 45405 | Membership in an upper set... |
| infxrlbrnmpt2 45406 | A member of a nonempty ind... |
| xrre4 45407 | An extended real is real i... |
| uz0 45408 | The upper integers functio... |
| eluzelz2d 45409 | A member of an upper set o... |
| infleinf2 45410 | If any element in ` B ` is... |
| unb2ltle 45411 | "Unbounded below" expresse... |
| uzidd2 45412 | Membership of the least me... |
| uzssd2 45413 | Subset relationship for tw... |
| rexabslelem 45414 | An indexed set of absolute... |
| rexabsle 45415 | An indexed set of absolute... |
| allbutfiinf 45416 | Given a "for all but finit... |
| supxrrernmpt 45417 | The real and extended real... |
| suprleubrnmpt 45418 | The supremum of a nonempty... |
| infrnmptle 45419 | An indexed infimum of exte... |
| infxrunb3 45420 | The infimum of an unbounde... |
| uzn0d 45421 | The upper integers are all... |
| uzssd3 45422 | Subset relationship for tw... |
| rexabsle2 45423 | An indexed set of absolute... |
| infxrunb3rnmpt 45424 | The infimum of an unbounde... |
| supxrre3rnmpt 45425 | The indexed supremum of a ... |
| uzublem 45426 | A set of reals, indexed by... |
| uzub 45427 | A set of reals, indexed by... |
| ssrexr 45428 | A subset of the reals is a... |
| supxrmnf2 45429 | Removing minus infinity fr... |
| supxrcli 45430 | The supremum of an arbitra... |
| uzid3 45431 | Membership of the least me... |
| infxrlesupxr 45432 | The supremum of a nonempty... |
| xnegeqd 45433 | Equality of two extended n... |
| xnegrecl 45434 | The extended real negative... |
| xnegnegi 45435 | Extended real version of ~... |
| xnegeqi 45436 | Equality of two extended n... |
| nfxnegd 45437 | Deduction version of ~ nfx... |
| xnegnegd 45438 | Extended real version of ~... |
| uzred 45439 | An upper integer is a real... |
| xnegcli 45440 | Closure of extended real n... |
| supminfrnmpt 45441 | The indexed supremum of a ... |
| infxrpnf 45442 | Adding plus infinity to a ... |
| infxrrnmptcl 45443 | The infimum of an arbitrar... |
| leneg2d 45444 | Negative of one side of 'l... |
| supxrltinfxr 45445 | The supremum of the empty ... |
| max1d 45446 | A number is less than or e... |
| supxrleubrnmptf 45447 | The supremum of a nonempty... |
| nleltd 45448 | 'Not less than or equal to... |
| zxrd 45449 | An integer is an extended ... |
| infxrgelbrnmpt 45450 | The infimum of an indexed ... |
| rphalfltd 45451 | Half of a positive real is... |
| uzssz2 45452 | An upper set of integers i... |
| leneg3d 45453 | Negative of one side of 'l... |
| max2d 45454 | A number is less than or e... |
| uzn0bi 45455 | The upper integers functio... |
| xnegrecl2 45456 | If the extended real negat... |
| nfxneg 45457 | Bound-variable hypothesis ... |
| uzxrd 45458 | An upper integer is an ext... |
| infxrpnf2 45459 | Removing plus infinity fro... |
| supminfxr 45460 | The extended real suprema ... |
| infrpgernmpt 45461 | The infimum of a nonempty,... |
| xnegre 45462 | An extended real is real i... |
| xnegrecl2d 45463 | If the extended real negat... |
| uzxr 45464 | An upper integer is an ext... |
| supminfxr2 45465 | The extended real suprema ... |
| xnegred 45466 | An extended real is real i... |
| supminfxrrnmpt 45467 | The indexed supremum of a ... |
| min1d 45468 | The minimum of two numbers... |
| min2d 45469 | The minimum of two numbers... |
| xrnpnfmnf 45470 | An extended real that is n... |
| uzsscn 45471 | An upper set of integers i... |
| absimnre 45472 | The absolute value of the ... |
| uzsscn2 45473 | An upper set of integers i... |
| xrtgcntopre 45474 | The standard topologies on... |
| absimlere 45475 | The absolute value of the ... |
| rpssxr 45476 | The positive reals are a s... |
| monoordxrv 45477 | Ordering relation for a mo... |
| monoordxr 45478 | Ordering relation for a mo... |
| monoord2xrv 45479 | Ordering relation for a mo... |
| monoord2xr 45480 | Ordering relation for a mo... |
| xrpnf 45481 | An extended real is plus i... |
| xlenegcon1 45482 | Extended real version of ~... |
| xlenegcon2 45483 | Extended real version of ~... |
| pimxrneun 45484 | The preimage of a set of e... |
| caucvgbf 45485 | A function is convergent i... |
| cvgcau 45486 | A convergent function is C... |
| cvgcaule 45487 | A convergent function is C... |
| rexanuz2nf 45488 | A simple counterexample re... |
| gtnelioc 45489 | A real number larger than ... |
| ioossioc 45490 | An open interval is a subs... |
| ioondisj2 45491 | A condition for two open i... |
| ioondisj1 45492 | A condition for two open i... |
| ioogtlb 45493 | An element of a closed int... |
| evthiccabs 45494 | Extreme Value Theorem on y... |
| ltnelicc 45495 | A real number smaller than... |
| eliood 45496 | Membership in an open real... |
| iooabslt 45497 | An upper bound for the dis... |
| gtnelicc 45498 | A real number greater than... |
| iooinlbub 45499 | An open interval has empty... |
| iocgtlb 45500 | An element of a left-open ... |
| iocleub 45501 | An element of a left-open ... |
| eliccd 45502 | Membership in a closed rea... |
| eliccre 45503 | A member of a closed inter... |
| eliooshift 45504 | Element of an open interva... |
| eliocd 45505 | Membership in a left-open ... |
| icoltub 45506 | An element of a left-close... |
| eliocre 45507 | A member of a left-open ri... |
| iooltub 45508 | An element of an open inte... |
| ioontr 45509 | The interior of an interva... |
| snunioo1 45510 | The closure of one end of ... |
| lbioc 45511 | A left-open right-closed i... |
| ioomidp 45512 | The midpoint is an element... |
| iccdifioo 45513 | If the open inverval is re... |
| iccdifprioo 45514 | An open interval is the cl... |
| ioossioobi 45515 | Biconditional form of ~ io... |
| iccshift 45516 | A closed interval shifted ... |
| iccsuble 45517 | An upper bound to the dist... |
| iocopn 45518 | A left-open right-closed i... |
| eliccelioc 45519 | Membership in a closed int... |
| iooshift 45520 | An open interval shifted b... |
| iccintsng 45521 | Intersection of two adiace... |
| icoiccdif 45522 | Left-closed right-open int... |
| icoopn 45523 | A left-closed right-open i... |
| icoub 45524 | A left-closed, right-open ... |
| eliccxrd 45525 | Membership in a closed rea... |
| pnfel0pnf 45526 | ` +oo ` is a nonnegative e... |
| eliccnelico 45527 | An element of a closed int... |
| eliccelicod 45528 | A member of a closed inter... |
| ge0xrre 45529 | A nonnegative extended rea... |
| ge0lere 45530 | A nonnegative extended Rea... |
| elicores 45531 | Membership in a left-close... |
| inficc 45532 | The infimum of a nonempty ... |
| qinioo 45533 | The rational numbers are d... |
| lenelioc 45534 | A real number smaller than... |
| ioonct 45535 | A nonempty open interval i... |
| xrgtnelicc 45536 | A real number greater than... |
| iccdificc 45537 | The difference of two clos... |
| iocnct 45538 | A nonempty left-open, righ... |
| iccnct 45539 | A closed interval, with mo... |
| iooiinicc 45540 | A closed interval expresse... |
| iccgelbd 45541 | An element of a closed int... |
| iooltubd 45542 | An element of an open inte... |
| icoltubd 45543 | An element of a left-close... |
| qelioo 45544 | The rational numbers are d... |
| tgqioo2 45545 | Every open set of reals is... |
| iccleubd 45546 | An element of a closed int... |
| elioored 45547 | A member of an open interv... |
| ioogtlbd 45548 | An element of a closed int... |
| ioofun 45549 | ` (,) ` is a function. (C... |
| icomnfinre 45550 | A left-closed, right-open,... |
| sqrlearg 45551 | The square compared with i... |
| ressiocsup 45552 | If the supremum belongs to... |
| ressioosup 45553 | If the supremum does not b... |
| iooiinioc 45554 | A left-open, right-closed ... |
| ressiooinf 45555 | If the infimum does not be... |
| iocleubd 45556 | An element of a left-open ... |
| uzinico 45557 | An upper interval of integ... |
| preimaiocmnf 45558 | Preimage of a right-closed... |
| uzinico2 45559 | An upper interval of integ... |
| uzinico3 45560 | An upper interval of integ... |
| dmico 45561 | The domain of the closed-b... |
| ndmico 45562 | The closed-below, open-abo... |
| uzubioo 45563 | The upper integers are unb... |
| uzubico 45564 | The upper integers are unb... |
| uzubioo2 45565 | The upper integers are unb... |
| uzubico2 45566 | The upper integers are unb... |
| iocgtlbd 45567 | An element of a left-open ... |
| xrtgioo2 45568 | The topology on the extend... |
| fsummulc1f 45569 | Closure of a finite sum of... |
| fsumnncl 45570 | Closure of a nonempty, fin... |
| fsumge0cl 45571 | The finite sum of nonnegat... |
| fsumf1of 45572 | Re-index a finite sum usin... |
| fsumiunss 45573 | Sum over a disjoint indexe... |
| fsumreclf 45574 | Closure of a finite sum of... |
| fsumlessf 45575 | A shorter sum of nonnegati... |
| fsumsupp0 45576 | Finite sum of function val... |
| fsumsermpt 45577 | A finite sum expressed in ... |
| fmul01 45578 | Multiplying a finite numbe... |
| fmulcl 45579 | If ' Y ' is closed under t... |
| fmuldfeqlem1 45580 | induction step for the pro... |
| fmuldfeq 45581 | X and Z are two equivalent... |
| fmul01lt1lem1 45582 | Given a finite multiplicat... |
| fmul01lt1lem2 45583 | Given a finite multiplicat... |
| fmul01lt1 45584 | Given a finite multiplicat... |
| cncfmptss 45585 | A continuous complex funct... |
| rrpsscn 45586 | The positive reals are a s... |
| mulc1cncfg 45587 | A version of ~ mulc1cncf u... |
| infrglb 45588 | The infimum of a nonempty ... |
| expcnfg 45589 | If ` F ` is a complex cont... |
| prodeq2ad 45590 | Equality deduction for pro... |
| fprodsplit1 45591 | Separate out a term in a f... |
| fprodexp 45592 | Positive integer exponenti... |
| fprodabs2 45593 | The absolute value of a fi... |
| fprod0 45594 | A finite product with a ze... |
| mccllem 45595 | * Induction step for ~ mcc... |
| mccl 45596 | A multinomial coefficient,... |
| fprodcnlem 45597 | A finite product of functi... |
| fprodcn 45598 | A finite product of functi... |
| clim1fr1 45599 | A class of sequences of fr... |
| isumneg 45600 | Negation of a converging s... |
| climrec 45601 | Limit of the reciprocal of... |
| climmulf 45602 | A version of ~ climmul usi... |
| climexp 45603 | The limit of natural power... |
| climinf 45604 | A bounded monotonic noninc... |
| climsuselem1 45605 | The subsequence index ` I ... |
| climsuse 45606 | A subsequence ` G ` of a c... |
| climrecf 45607 | A version of ~ climrec usi... |
| climneg 45608 | Complex limit of the negat... |
| climinff 45609 | A version of ~ climinf usi... |
| climdivf 45610 | Limit of the ratio of two ... |
| climreeq 45611 | If ` F ` is a real functio... |
| ellimciota 45612 | An explicit value for the ... |
| climaddf 45613 | A version of ~ climadd usi... |
| mullimc 45614 | Limit of the product of tw... |
| ellimcabssub0 45615 | An equivalent condition fo... |
| limcdm0 45616 | If a function has empty do... |
| islptre 45617 | An equivalence condition f... |
| limccog 45618 | Limit of the composition o... |
| limciccioolb 45619 | The limit of a function at... |
| climf 45620 | Express the predicate: Th... |
| mullimcf 45621 | Limit of the multiplicatio... |
| constlimc 45622 | Limit of constant function... |
| rexlim2d 45623 | Inference removing two res... |
| idlimc 45624 | Limit of the identity func... |
| divcnvg 45625 | The sequence of reciprocal... |
| limcperiod 45626 | If ` F ` is a periodic fun... |
| limcrecl 45627 | If ` F ` is a real-valued ... |
| sumnnodd 45628 | A series indexed by ` NN `... |
| lptioo2 45629 | The upper bound of an open... |
| lptioo1 45630 | The lower bound of an open... |
| elprn1 45631 | A member of an unordered p... |
| elprn2 45632 | A member of an unordered p... |
| limcmptdm 45633 | The domain of a maps-to fu... |
| clim2f 45634 | Express the predicate: Th... |
| limcicciooub 45635 | The limit of a function at... |
| ltmod 45636 | A sufficient condition for... |
| islpcn 45637 | A characterization for a l... |
| lptre2pt 45638 | If a set in the real line ... |
| limsupre 45639 | If a sequence is bounded, ... |
| limcresiooub 45640 | The left limit doesn't cha... |
| limcresioolb 45641 | The right limit doesn't ch... |
| limcleqr 45642 | If the left and the right ... |
| lptioo2cn 45643 | The upper bound of an open... |
| lptioo1cn 45644 | The lower bound of an open... |
| neglimc 45645 | Limit of the negative func... |
| addlimc 45646 | Sum of two limits. (Contr... |
| 0ellimcdiv 45647 | If the numerator converges... |
| clim2cf 45648 | Express the predicate ` F ... |
| limclner 45649 | For a limit point, both fr... |
| sublimc 45650 | Subtraction of two limits.... |
| reclimc 45651 | Limit of the reciprocal of... |
| clim0cf 45652 | Express the predicate ` F ... |
| limclr 45653 | For a limit point, both fr... |
| divlimc 45654 | Limit of the quotient of t... |
| expfac 45655 | Factorial grows faster tha... |
| climconstmpt 45656 | A constant sequence conver... |
| climresmpt 45657 | A function restricted to u... |
| climsubmpt 45658 | Limit of the difference of... |
| climsubc2mpt 45659 | Limit of the difference of... |
| climsubc1mpt 45660 | Limit of the difference of... |
| fnlimfv 45661 | The value of the limit fun... |
| climreclf 45662 | The limit of a convergent ... |
| climeldmeq 45663 | Two functions that are eve... |
| climf2 45664 | Express the predicate: Th... |
| fnlimcnv 45665 | The sequence of function v... |
| climeldmeqmpt 45666 | Two functions that are eve... |
| climfveq 45667 | Two functions that are eve... |
| clim2f2 45668 | Express the predicate: Th... |
| climfveqmpt 45669 | Two functions that are eve... |
| climd 45670 | Express the predicate: Th... |
| clim2d 45671 | The limit of complex numbe... |
| fnlimfvre 45672 | The limit function of real... |
| allbutfifvre 45673 | Given a sequence of real-v... |
| climleltrp 45674 | The limit of complex numbe... |
| fnlimfvre2 45675 | The limit function of real... |
| fnlimf 45676 | The limit function of real... |
| fnlimabslt 45677 | A sequence of function val... |
| climfveqf 45678 | Two functions that are eve... |
| climmptf 45679 | Exhibit a function ` G ` w... |
| climfveqmpt3 45680 | Two functions that are eve... |
| climeldmeqf 45681 | Two functions that are eve... |
| climreclmpt 45682 | The limit of B convergent ... |
| limsupref 45683 | If a sequence is bounded, ... |
| limsupbnd1f 45684 | If a sequence is eventuall... |
| climbddf 45685 | A converging sequence of c... |
| climeqf 45686 | Two functions that are eve... |
| climeldmeqmpt3 45687 | Two functions that are eve... |
| limsupcld 45688 | Closure of the superior li... |
| climfv 45689 | The limit of a convergent ... |
| limsupval3 45690 | The superior limit of an i... |
| climfveqmpt2 45691 | Two functions that are eve... |
| limsup0 45692 | The superior limit of the ... |
| climeldmeqmpt2 45693 | Two functions that are eve... |
| limsupresre 45694 | The supremum limit of a fu... |
| climeqmpt 45695 | Two functions that are eve... |
| climfvd 45696 | The limit of a convergent ... |
| limsuplesup 45697 | An upper bound for the sup... |
| limsupresico 45698 | The superior limit doesn't... |
| limsuppnfdlem 45699 | If the restriction of a fu... |
| limsuppnfd 45700 | If the restriction of a fu... |
| limsupresuz 45701 | If the real part of the do... |
| limsupub 45702 | If the limsup is not ` +oo... |
| limsupres 45703 | The superior limit of a re... |
| climinf2lem 45704 | A convergent, nonincreasin... |
| climinf2 45705 | A convergent, nonincreasin... |
| limsupvaluz 45706 | The superior limit, when t... |
| limsupresuz2 45707 | If the domain of a functio... |
| limsuppnflem 45708 | If the restriction of a fu... |
| limsuppnf 45709 | If the restriction of a fu... |
| limsupubuzlem 45710 | If the limsup is not ` +oo... |
| limsupubuz 45711 | For a real-valued function... |
| climinf2mpt 45712 | A bounded below, monotonic... |
| climinfmpt 45713 | A bounded below, monotonic... |
| climinf3 45714 | A convergent, nonincreasin... |
| limsupvaluzmpt 45715 | The superior limit, when t... |
| limsupequzmpt2 45716 | Two functions that are eve... |
| limsupubuzmpt 45717 | If the limsup is not ` +oo... |
| limsupmnflem 45718 | The superior limit of a fu... |
| limsupmnf 45719 | The superior limit of a fu... |
| limsupequzlem 45720 | Two functions that are eve... |
| limsupequz 45721 | Two functions that are eve... |
| limsupre2lem 45722 | Given a function on the ex... |
| limsupre2 45723 | Given a function on the ex... |
| limsupmnfuzlem 45724 | The superior limit of a fu... |
| limsupmnfuz 45725 | The superior limit of a fu... |
| limsupequzmptlem 45726 | Two functions that are eve... |
| limsupequzmpt 45727 | Two functions that are eve... |
| limsupre2mpt 45728 | Given a function on the ex... |
| limsupequzmptf 45729 | Two functions that are eve... |
| limsupre3lem 45730 | Given a function on the ex... |
| limsupre3 45731 | Given a function on the ex... |
| limsupre3mpt 45732 | Given a function on the ex... |
| limsupre3uzlem 45733 | Given a function on the ex... |
| limsupre3uz 45734 | Given a function on the ex... |
| limsupreuz 45735 | Given a function on the re... |
| limsupvaluz2 45736 | The superior limit, when t... |
| limsupreuzmpt 45737 | Given a function on the re... |
| supcnvlimsup 45738 | If a function on a set of ... |
| supcnvlimsupmpt 45739 | If a function on a set of ... |
| 0cnv 45740 | If ` (/) ` is a complex nu... |
| climuzlem 45741 | Express the predicate: Th... |
| climuz 45742 | Express the predicate: Th... |
| lmbr3v 45743 | Express the binary relatio... |
| climisp 45744 | If a sequence converges to... |
| lmbr3 45745 | Express the binary relatio... |
| climrescn 45746 | A sequence converging w.r.... |
| climxrrelem 45747 | If a sequence ranging over... |
| climxrre 45748 | If a sequence ranging over... |
| limsuplt2 45751 | The defining property of t... |
| liminfgord 45752 | Ordering property of the i... |
| limsupvald 45753 | The superior limit of a se... |
| limsupresicompt 45754 | The superior limit doesn't... |
| limsupcli 45755 | Closure of the superior li... |
| liminfgf 45756 | Closure of the inferior li... |
| liminfval 45757 | The inferior limit of a se... |
| climlimsup 45758 | A sequence of real numbers... |
| limsupge 45759 | The defining property of t... |
| liminfgval 45760 | Value of the inferior limi... |
| liminfcl 45761 | Closure of the inferior li... |
| liminfvald 45762 | The inferior limit of a se... |
| liminfval5 45763 | The inferior limit of an i... |
| limsupresxr 45764 | The superior limit of a fu... |
| liminfresxr 45765 | The inferior limit of a fu... |
| liminfval2 45766 | The superior limit, relati... |
| climlimsupcex 45767 | Counterexample for ~ climl... |
| liminfcld 45768 | Closure of the inferior li... |
| liminfresico 45769 | The inferior limit doesn't... |
| limsup10exlem 45770 | The range of the given fun... |
| limsup10ex 45771 | The superior limit of a fu... |
| liminf10ex 45772 | The inferior limit of a fu... |
| liminflelimsuplem 45773 | The superior limit is grea... |
| liminflelimsup 45774 | The superior limit is grea... |
| limsupgtlem 45775 | For any positive real, the... |
| limsupgt 45776 | Given a sequence of real n... |
| liminfresre 45777 | The inferior limit of a fu... |
| liminfresicompt 45778 | The inferior limit doesn't... |
| liminfltlimsupex 45779 | An example where the ` lim... |
| liminfgelimsup 45780 | The inferior limit is grea... |
| liminfvalxr 45781 | Alternate definition of ` ... |
| liminfresuz 45782 | If the real part of the do... |
| liminflelimsupuz 45783 | The superior limit is grea... |
| liminfvalxrmpt 45784 | Alternate definition of ` ... |
| liminfresuz2 45785 | If the domain of a functio... |
| liminfgelimsupuz 45786 | The inferior limit is grea... |
| liminfval4 45787 | Alternate definition of ` ... |
| liminfval3 45788 | Alternate definition of ` ... |
| liminfequzmpt2 45789 | Two functions that are eve... |
| liminfvaluz 45790 | Alternate definition of ` ... |
| liminf0 45791 | The inferior limit of the ... |
| limsupval4 45792 | Alternate definition of ` ... |
| liminfvaluz2 45793 | Alternate definition of ` ... |
| liminfvaluz3 45794 | Alternate definition of ` ... |
| liminflelimsupcex 45795 | A counterexample for ~ lim... |
| limsupvaluz3 45796 | Alternate definition of ` ... |
| liminfvaluz4 45797 | Alternate definition of ` ... |
| limsupvaluz4 45798 | Alternate definition of ` ... |
| climliminflimsupd 45799 | If a sequence of real numb... |
| liminfreuzlem 45800 | Given a function on the re... |
| liminfreuz 45801 | Given a function on the re... |
| liminfltlem 45802 | Given a sequence of real n... |
| liminflt 45803 | Given a sequence of real n... |
| climliminf 45804 | A sequence of real numbers... |
| liminflimsupclim 45805 | A sequence of real numbers... |
| climliminflimsup 45806 | A sequence of real numbers... |
| climliminflimsup2 45807 | A sequence of real numbers... |
| climliminflimsup3 45808 | A sequence of real numbers... |
| climliminflimsup4 45809 | A sequence of real numbers... |
| limsupub2 45810 | A extended real valued fun... |
| limsupubuz2 45811 | A sequence with values in ... |
| xlimpnfxnegmnf 45812 | A sequence converges to ` ... |
| liminflbuz2 45813 | A sequence with values in ... |
| liminfpnfuz 45814 | The inferior limit of a fu... |
| liminflimsupxrre 45815 | A sequence with values in ... |
| xlimrel 45818 | The limit on extended real... |
| xlimres 45819 | A function converges iff i... |
| xlimcl 45820 | The limit of a sequence of... |
| rexlimddv2 45821 | Restricted existential eli... |
| xlimclim 45822 | Given a sequence of reals,... |
| xlimconst 45823 | A constant sequence conver... |
| climxlim 45824 | A converging sequence in t... |
| xlimbr 45825 | Express the binary relatio... |
| fuzxrpmcn 45826 | A function mapping from an... |
| cnrefiisplem 45827 | Lemma for ~ cnrefiisp (som... |
| cnrefiisp 45828 | A non-real, complex number... |
| xlimxrre 45829 | If a sequence ranging over... |
| xlimmnfvlem1 45830 | Lemma for ~ xlimmnfv : the... |
| xlimmnfvlem2 45831 | Lemma for ~ xlimmnf : the ... |
| xlimmnfv 45832 | A function converges to mi... |
| xlimconst2 45833 | A sequence that eventually... |
| xlimpnfvlem1 45834 | Lemma for ~ xlimpnfv : the... |
| xlimpnfvlem2 45835 | Lemma for ~ xlimpnfv : the... |
| xlimpnfv 45836 | A function converges to pl... |
| xlimclim2lem 45837 | Lemma for ~ xlimclim2 . H... |
| xlimclim2 45838 | Given a sequence of extend... |
| xlimmnf 45839 | A function converges to mi... |
| xlimpnf 45840 | A function converges to pl... |
| xlimmnfmpt 45841 | A function converges to pl... |
| xlimpnfmpt 45842 | A function converges to pl... |
| climxlim2lem 45843 | In this lemma for ~ climxl... |
| climxlim2 45844 | A sequence of extended rea... |
| dfxlim2v 45845 | An alternative definition ... |
| dfxlim2 45846 | An alternative definition ... |
| climresd 45847 | A function restricted to u... |
| climresdm 45848 | A real function converges ... |
| dmclimxlim 45849 | A real valued sequence tha... |
| xlimmnflimsup2 45850 | A sequence of extended rea... |
| xlimuni 45851 | An infinite sequence conve... |
| xlimclimdm 45852 | A sequence of extended rea... |
| xlimfun 45853 | The convergence relation o... |
| xlimmnflimsup 45854 | If a sequence of extended ... |
| xlimdm 45855 | Two ways to express that a... |
| xlimpnfxnegmnf2 45856 | A sequence converges to ` ... |
| xlimresdm 45857 | A function converges in th... |
| xlimpnfliminf 45858 | If a sequence of extended ... |
| xlimpnfliminf2 45859 | A sequence of extended rea... |
| xlimliminflimsup 45860 | A sequence of extended rea... |
| xlimlimsupleliminf 45861 | A sequence of extended rea... |
| coseq0 45862 | A complex number whose cos... |
| sinmulcos 45863 | Multiplication formula for... |
| coskpi2 45864 | The cosine of an integer m... |
| cosnegpi 45865 | The cosine of negative ` _... |
| sinaover2ne0 45866 | If ` A ` in ` ( 0 , 2 _pi ... |
| cosknegpi 45867 | The cosine of an integer m... |
| mulcncff 45868 | The multiplication of two ... |
| cncfmptssg 45869 | A continuous complex funct... |
| constcncfg 45870 | A constant function is a c... |
| idcncfg 45871 | The identity function is a... |
| cncfshift 45872 | A periodic continuous func... |
| resincncf 45873 | ` sin ` restricted to real... |
| addccncf2 45874 | Adding a constant is a con... |
| 0cnf 45875 | The empty set is a continu... |
| fsumcncf 45876 | The finite sum of continuo... |
| cncfperiod 45877 | A periodic continuous func... |
| subcncff 45878 | The subtraction of two con... |
| negcncfg 45879 | The opposite of a continuo... |
| cnfdmsn 45880 | A function with a singleto... |
| cncfcompt 45881 | Composition of continuous ... |
| addcncff 45882 | The sum of two continuous ... |
| ioccncflimc 45883 | Limit at the upper bound o... |
| cncfuni 45884 | A complex function on a su... |
| icccncfext 45885 | A continuous function on a... |
| cncficcgt0 45886 | A the absolute value of a ... |
| icocncflimc 45887 | Limit at the lower bound, ... |
| cncfdmsn 45888 | A complex function with a ... |
| divcncff 45889 | The quotient of two contin... |
| cncfshiftioo 45890 | A periodic continuous func... |
| cncfiooicclem1 45891 | A continuous function ` F ... |
| cncfiooicc 45892 | A continuous function ` F ... |
| cncfiooiccre 45893 | A continuous function ` F ... |
| cncfioobdlem 45894 | ` G ` actually extends ` F... |
| cncfioobd 45895 | A continuous function ` F ... |
| jumpncnp 45896 | Jump discontinuity or disc... |
| cxpcncf2 45897 | The complex power function... |
| fprodcncf 45898 | The finite product of cont... |
| add1cncf 45899 | Addition to a constant is ... |
| add2cncf 45900 | Addition to a constant is ... |
| sub1cncfd 45901 | Subtracting a constant is ... |
| sub2cncfd 45902 | Subtraction from a constan... |
| fprodsub2cncf 45903 | ` F ` is continuous. (Con... |
| fprodadd2cncf 45904 | ` F ` is continuous. (Con... |
| fprodsubrecnncnvlem 45905 | The sequence ` S ` of fini... |
| fprodsubrecnncnv 45906 | The sequence ` S ` of fini... |
| fprodaddrecnncnvlem 45907 | The sequence ` S ` of fini... |
| fprodaddrecnncnv 45908 | The sequence ` S ` of fini... |
| dvsinexp 45909 | The derivative of sin^N . ... |
| dvcosre 45910 | The real derivative of the... |
| dvsinax 45911 | Derivative exercise: the d... |
| dvsubf 45912 | The subtraction rule for e... |
| dvmptconst 45913 | Function-builder for deriv... |
| dvcnre 45914 | From complex differentiati... |
| dvmptidg 45915 | Function-builder for deriv... |
| dvresntr 45916 | Function-builder for deriv... |
| fperdvper 45917 | The derivative of a period... |
| dvasinbx 45918 | Derivative exercise: the d... |
| dvresioo 45919 | Restriction of a derivativ... |
| dvdivf 45920 | The quotient rule for ever... |
| dvdivbd 45921 | A sufficient condition for... |
| dvsubcncf 45922 | A sufficient condition for... |
| dvmulcncf 45923 | A sufficient condition for... |
| dvcosax 45924 | Derivative exercise: the d... |
| dvdivcncf 45925 | A sufficient condition for... |
| dvbdfbdioolem1 45926 | Given a function with boun... |
| dvbdfbdioolem2 45927 | A function on an open inte... |
| dvbdfbdioo 45928 | A function on an open inte... |
| ioodvbdlimc1lem1 45929 | If ` F ` has bounded deriv... |
| ioodvbdlimc1lem2 45930 | Limit at the lower bound o... |
| ioodvbdlimc1 45931 | A real function with bound... |
| ioodvbdlimc2lem 45932 | Limit at the upper bound o... |
| ioodvbdlimc2 45933 | A real function with bound... |
| dvdmsscn 45934 | ` X ` is a subset of ` CC ... |
| dvmptmulf 45935 | Function-builder for deriv... |
| dvnmptdivc 45936 | Function-builder for itera... |
| dvdsn1add 45937 | If ` K ` divides ` N ` but... |
| dvxpaek 45938 | Derivative of the polynomi... |
| dvnmptconst 45939 | The ` N ` -th derivative o... |
| dvnxpaek 45940 | The ` n ` -th derivative o... |
| dvnmul 45941 | Function-builder for the `... |
| dvmptfprodlem 45942 | Induction step for ~ dvmpt... |
| dvmptfprod 45943 | Function-builder for deriv... |
| dvnprodlem1 45944 | ` D ` is bijective. (Cont... |
| dvnprodlem2 45945 | Induction step for ~ dvnpr... |
| dvnprodlem3 45946 | The multinomial formula fo... |
| dvnprod 45947 | The multinomial formula fo... |
| itgsin0pilem1 45948 | Calculation of the integra... |
| ibliccsinexp 45949 | sin^n on a closed interval... |
| itgsin0pi 45950 | Calculation of the integra... |
| iblioosinexp 45951 | sin^n on an open integral ... |
| itgsinexplem1 45952 | Integration by parts is ap... |
| itgsinexp 45953 | A recursive formula for th... |
| iblconstmpt 45954 | A constant function is int... |
| itgeq1d 45955 | Equality theorem for an in... |
| mbfres2cn 45956 | Measurability of a piecewi... |
| vol0 45957 | The measure of the empty s... |
| ditgeqiooicc 45958 | A function ` F ` on an ope... |
| volge0 45959 | The volume of a set is alw... |
| cnbdibl 45960 | A continuous bounded funct... |
| snmbl 45961 | A singleton is measurable.... |
| ditgeq3d 45962 | Equality theorem for the d... |
| iblempty 45963 | The empty function is inte... |
| iblsplit 45964 | The union of two integrabl... |
| volsn 45965 | A singleton has 0 Lebesgue... |
| itgvol0 45966 | If the domani is negligibl... |
| itgcoscmulx 45967 | Exercise: the integral of ... |
| iblsplitf 45968 | A version of ~ iblsplit us... |
| ibliooicc 45969 | If a function is integrabl... |
| volioc 45970 | The measure of a left-open... |
| iblspltprt 45971 | If a function is integrabl... |
| itgsincmulx 45972 | Exercise: the integral of ... |
| itgsubsticclem 45973 | lemma for ~ itgsubsticc . ... |
| itgsubsticc 45974 | Integration by u-substitut... |
| itgioocnicc 45975 | The integral of a piecewis... |
| iblcncfioo 45976 | A continuous function ` F ... |
| itgspltprt 45977 | The ` S. ` integral splits... |
| itgiccshift 45978 | The integral of a function... |
| itgperiod 45979 | The integral of a periodic... |
| itgsbtaddcnst 45980 | Integral substitution, add... |
| volico 45981 | The measure of left-closed... |
| sublevolico 45982 | The Lebesgue measure of a ... |
| dmvolss 45983 | Lebesgue measurable sets a... |
| ismbl3 45984 | The predicate " ` A ` is L... |
| volioof 45985 | The function that assigns ... |
| ovolsplit 45986 | The Lebesgue outer measure... |
| fvvolioof 45987 | The function value of the ... |
| volioore 45988 | The measure of an open int... |
| fvvolicof 45989 | The function value of the ... |
| voliooico 45990 | An open interval and a lef... |
| ismbl4 45991 | The predicate " ` A ` is L... |
| volioofmpt 45992 | ` ( ( vol o. (,) ) o. F ) ... |
| volicoff 45993 | ` ( ( vol o. [,) ) o. F ) ... |
| voliooicof 45994 | The Lebesgue measure of op... |
| volicofmpt 45995 | ` ( ( vol o. [,) ) o. F ) ... |
| volicc 45996 | The Lebesgue measure of a ... |
| voliccico 45997 | A closed interval and a le... |
| mbfdmssre 45998 | The domain of a measurable... |
| stoweidlem1 45999 | Lemma for ~ stoweid . Thi... |
| stoweidlem2 46000 | lemma for ~ stoweid : here... |
| stoweidlem3 46001 | Lemma for ~ stoweid : if `... |
| stoweidlem4 46002 | Lemma for ~ stoweid : a cl... |
| stoweidlem5 46003 | There exists a δ as ... |
| stoweidlem6 46004 | Lemma for ~ stoweid : two ... |
| stoweidlem7 46005 | This lemma is used to prov... |
| stoweidlem8 46006 | Lemma for ~ stoweid : two ... |
| stoweidlem9 46007 | Lemma for ~ stoweid : here... |
| stoweidlem10 46008 | Lemma for ~ stoweid . Thi... |
| stoweidlem11 46009 | This lemma is used to prov... |
| stoweidlem12 46010 | Lemma for ~ stoweid . Thi... |
| stoweidlem13 46011 | Lemma for ~ stoweid . Thi... |
| stoweidlem14 46012 | There exists a ` k ` as in... |
| stoweidlem15 46013 | This lemma is used to prov... |
| stoweidlem16 46014 | Lemma for ~ stoweid . The... |
| stoweidlem17 46015 | This lemma proves that the... |
| stoweidlem18 46016 | This theorem proves Lemma ... |
| stoweidlem19 46017 | If a set of real functions... |
| stoweidlem20 46018 | If a set A of real functio... |
| stoweidlem21 46019 | Once the Stone Weierstrass... |
| stoweidlem22 46020 | If a set of real functions... |
| stoweidlem23 46021 | This lemma is used to prov... |
| stoweidlem24 46022 | This lemma proves that for... |
| stoweidlem25 46023 | This lemma proves that for... |
| stoweidlem26 46024 | This lemma is used to prov... |
| stoweidlem27 46025 | This lemma is used to prov... |
| stoweidlem28 46026 | There exists a δ as ... |
| stoweidlem29 46027 | When the hypothesis for th... |
| stoweidlem30 46028 | This lemma is used to prov... |
| stoweidlem31 46029 | This lemma is used to prov... |
| stoweidlem32 46030 | If a set A of real functio... |
| stoweidlem33 46031 | If a set of real functions... |
| stoweidlem34 46032 | This lemma proves that for... |
| stoweidlem35 46033 | This lemma is used to prov... |
| stoweidlem36 46034 | This lemma is used to prov... |
| stoweidlem37 46035 | This lemma is used to prov... |
| stoweidlem38 46036 | This lemma is used to prov... |
| stoweidlem39 46037 | This lemma is used to prov... |
| stoweidlem40 46038 | This lemma proves that q_n... |
| stoweidlem41 46039 | This lemma is used to prov... |
| stoweidlem42 46040 | This lemma is used to prov... |
| stoweidlem43 46041 | This lemma is used to prov... |
| stoweidlem44 46042 | This lemma is used to prov... |
| stoweidlem45 46043 | This lemma proves that, gi... |
| stoweidlem46 46044 | This lemma proves that set... |
| stoweidlem47 46045 | Subtracting a constant fro... |
| stoweidlem48 46046 | This lemma is used to prov... |
| stoweidlem49 46047 | There exists a function q_... |
| stoweidlem50 46048 | This lemma proves that set... |
| stoweidlem51 46049 | There exists a function x ... |
| stoweidlem52 46050 | There exists a neighborhoo... |
| stoweidlem53 46051 | This lemma is used to prov... |
| stoweidlem54 46052 | There exists a function ` ... |
| stoweidlem55 46053 | This lemma proves the exis... |
| stoweidlem56 46054 | This theorem proves Lemma ... |
| stoweidlem57 46055 | There exists a function x ... |
| stoweidlem58 46056 | This theorem proves Lemma ... |
| stoweidlem59 46057 | This lemma proves that the... |
| stoweidlem60 46058 | This lemma proves that the... |
| stoweidlem61 46059 | This lemma proves that the... |
| stoweidlem62 46060 | This theorem proves the St... |
| stoweid 46061 | This theorem proves the St... |
| stowei 46062 | This theorem proves the St... |
| wallispilem1 46063 | ` I ` is monotone: increas... |
| wallispilem2 46064 | A first set of properties ... |
| wallispilem3 46065 | I maps to real values. (C... |
| wallispilem4 46066 | ` F ` maps to explicit exp... |
| wallispilem5 46067 | The sequence ` H ` converg... |
| wallispi 46068 | Wallis' formula for π :... |
| wallispi2lem1 46069 | An intermediate step betwe... |
| wallispi2lem2 46070 | Two expressions are proven... |
| wallispi2 46071 | An alternative version of ... |
| stirlinglem1 46072 | A simple limit of fraction... |
| stirlinglem2 46073 | ` A ` maps to positive rea... |
| stirlinglem3 46074 | Long but simple algebraic ... |
| stirlinglem4 46075 | Algebraic manipulation of ... |
| stirlinglem5 46076 | If ` T ` is between ` 0 ` ... |
| stirlinglem6 46077 | A series that converges to... |
| stirlinglem7 46078 | Algebraic manipulation of ... |
| stirlinglem8 46079 | If ` A ` converges to ` C ... |
| stirlinglem9 46080 | ` ( ( B `` N ) - ( B `` ( ... |
| stirlinglem10 46081 | A bound for any B(N)-B(N +... |
| stirlinglem11 46082 | ` B ` is decreasing. (Con... |
| stirlinglem12 46083 | The sequence ` B ` is boun... |
| stirlinglem13 46084 | ` B ` is decreasing and ha... |
| stirlinglem14 46085 | The sequence ` A ` converg... |
| stirlinglem15 46086 | The Stirling's formula is ... |
| stirling 46087 | Stirling's approximation f... |
| stirlingr 46088 | Stirling's approximation f... |
| dirkerval 46089 | The N_th Dirichlet Kernel.... |
| dirker2re 46090 | The Dirichlet Kernel value... |
| dirkerdenne0 46091 | The Dirichlet Kernel denom... |
| dirkerval2 46092 | The N_th Dirichlet Kernel ... |
| dirkerre 46093 | The Dirichlet Kernel at an... |
| dirkerper 46094 | the Dirichlet Kernel has p... |
| dirkerf 46095 | For any natural number ` N... |
| dirkertrigeqlem1 46096 | Sum of an even number of a... |
| dirkertrigeqlem2 46097 | Trigonomic equality lemma ... |
| dirkertrigeqlem3 46098 | Trigonometric equality lem... |
| dirkertrigeq 46099 | Trigonometric equality for... |
| dirkeritg 46100 | The definite integral of t... |
| dirkercncflem1 46101 | If ` Y ` is a multiple of ... |
| dirkercncflem2 46102 | Lemma used to prove that t... |
| dirkercncflem3 46103 | The Dirichlet Kernel is co... |
| dirkercncflem4 46104 | The Dirichlet Kernel is co... |
| dirkercncf 46105 | For any natural number ` N... |
| fourierdlem1 46106 | A partition interval is a ... |
| fourierdlem2 46107 | Membership in a partition.... |
| fourierdlem3 46108 | Membership in a partition.... |
| fourierdlem4 46109 | ` E ` is a function that m... |
| fourierdlem5 46110 | ` S ` is a function. (Con... |
| fourierdlem6 46111 | ` X ` is in the periodic p... |
| fourierdlem7 46112 | The difference between the... |
| fourierdlem8 46113 | A partition interval is a ... |
| fourierdlem9 46114 | ` H ` is a complex functio... |
| fourierdlem10 46115 | Condition on the bounds of... |
| fourierdlem11 46116 | If there is a partition, t... |
| fourierdlem12 46117 | A point of a partition is ... |
| fourierdlem13 46118 | Value of ` V ` in terms of... |
| fourierdlem14 46119 | Given the partition ` V ` ... |
| fourierdlem15 46120 | The range of the partition... |
| fourierdlem16 46121 | The coefficients of the fo... |
| fourierdlem17 46122 | The defined ` L ` is actua... |
| fourierdlem18 46123 | The function ` S ` is cont... |
| fourierdlem19 46124 | If two elements of ` D ` h... |
| fourierdlem20 46125 | Every interval in the part... |
| fourierdlem21 46126 | The coefficients of the fo... |
| fourierdlem22 46127 | The coefficients of the fo... |
| fourierdlem23 46128 | If ` F ` is continuous and... |
| fourierdlem24 46129 | A sufficient condition for... |
| fourierdlem25 46130 | If ` C ` is not in the ran... |
| fourierdlem26 46131 | Periodic image of a point ... |
| fourierdlem27 46132 | A partition open interval ... |
| fourierdlem28 46133 | Derivative of ` ( F `` ( X... |
| fourierdlem29 46134 | Explicit function value fo... |
| fourierdlem30 46135 | Sum of three small pieces ... |
| fourierdlem31 46136 | If ` A ` is finite and for... |
| fourierdlem32 46137 | Limit of a continuous func... |
| fourierdlem33 46138 | Limit of a continuous func... |
| fourierdlem34 46139 | A partition is one to one.... |
| fourierdlem35 46140 | There is a single point in... |
| fourierdlem36 46141 | ` F ` is an isomorphism. ... |
| fourierdlem37 46142 | ` I ` is a function that m... |
| fourierdlem38 46143 | The function ` F ` is cont... |
| fourierdlem39 46144 | Integration by parts of ... |
| fourierdlem40 46145 | ` H ` is a continuous func... |
| fourierdlem41 46146 | Lemma used to prove that e... |
| fourierdlem42 46147 | The set of points in a mov... |
| fourierdlem43 46148 | ` K ` is a real function. ... |
| fourierdlem44 46149 | A condition for having ` (... |
| fourierdlem46 46150 | The function ` F ` has a l... |
| fourierdlem47 46151 | For ` r ` large enough, th... |
| fourierdlem48 46152 | The given periodic functio... |
| fourierdlem49 46153 | The given periodic functio... |
| fourierdlem50 46154 | Continuity of ` O ` and it... |
| fourierdlem51 46155 | ` X ` is in the periodic p... |
| fourierdlem52 46156 | d16:d17,d18:jca |- ( ph ->... |
| fourierdlem53 46157 | The limit of ` F ( s ) ` a... |
| fourierdlem54 46158 | Given a partition ` Q ` an... |
| fourierdlem55 46159 | ` U ` is a real function. ... |
| fourierdlem56 46160 | Derivative of the ` K ` fu... |
| fourierdlem57 46161 | The derivative of ` O ` . ... |
| fourierdlem58 46162 | The derivative of ` K ` is... |
| fourierdlem59 46163 | The derivative of ` H ` is... |
| fourierdlem60 46164 | Given a differentiable fun... |
| fourierdlem61 46165 | Given a differentiable fun... |
| fourierdlem62 46166 | The function ` K ` is cont... |
| fourierdlem63 46167 | The upper bound of interva... |
| fourierdlem64 46168 | The partition ` V ` is fin... |
| fourierdlem65 46169 | The distance of two adjace... |
| fourierdlem66 46170 | Value of the ` G ` functio... |
| fourierdlem67 46171 | ` G ` is a function. (Con... |
| fourierdlem68 46172 | The derivative of ` O ` is... |
| fourierdlem69 46173 | A piecewise continuous fun... |
| fourierdlem70 46174 | A piecewise continuous fun... |
| fourierdlem71 46175 | A periodic piecewise conti... |
| fourierdlem72 46176 | The derivative of ` O ` is... |
| fourierdlem73 46177 | A version of the Riemann L... |
| fourierdlem74 46178 | Given a piecewise smooth f... |
| fourierdlem75 46179 | Given a piecewise smooth f... |
| fourierdlem76 46180 | Continuity of ` O ` and it... |
| fourierdlem77 46181 | If ` H ` is bounded, then ... |
| fourierdlem78 46182 | ` G ` is continuous when r... |
| fourierdlem79 46183 | ` E ` projects every inter... |
| fourierdlem80 46184 | The derivative of ` O ` is... |
| fourierdlem81 46185 | The integral of a piecewis... |
| fourierdlem82 46186 | Integral by substitution, ... |
| fourierdlem83 46187 | The fourier partial sum fo... |
| fourierdlem84 46188 | If ` F ` is piecewise cont... |
| fourierdlem85 46189 | Limit of the function ` G ... |
| fourierdlem86 46190 | Continuity of ` O ` and it... |
| fourierdlem87 46191 | The integral of ` G ` goes... |
| fourierdlem88 46192 | Given a piecewise continuo... |
| fourierdlem89 46193 | Given a piecewise continuo... |
| fourierdlem90 46194 | Given a piecewise continuo... |
| fourierdlem91 46195 | Given a piecewise continuo... |
| fourierdlem92 46196 | The integral of a piecewis... |
| fourierdlem93 46197 | Integral by substitution (... |
| fourierdlem94 46198 | For a piecewise smooth fun... |
| fourierdlem95 46199 | Algebraic manipulation of ... |
| fourierdlem96 46200 | limit for ` F ` at the low... |
| fourierdlem97 46201 | ` F ` is continuous on the... |
| fourierdlem98 46202 | ` F ` is continuous on the... |
| fourierdlem99 46203 | limit for ` F ` at the upp... |
| fourierdlem100 46204 | A piecewise continuous fun... |
| fourierdlem101 46205 | Integral by substitution f... |
| fourierdlem102 46206 | For a piecewise smooth fun... |
| fourierdlem103 46207 | The half lower part of the... |
| fourierdlem104 46208 | The half upper part of the... |
| fourierdlem105 46209 | A piecewise continuous fun... |
| fourierdlem106 46210 | For a piecewise smooth fun... |
| fourierdlem107 46211 | The integral of a piecewis... |
| fourierdlem108 46212 | The integral of a piecewis... |
| fourierdlem109 46213 | The integral of a piecewis... |
| fourierdlem110 46214 | The integral of a piecewis... |
| fourierdlem111 46215 | The fourier partial sum fo... |
| fourierdlem112 46216 | Here abbreviations (local ... |
| fourierdlem113 46217 | Fourier series convergence... |
| fourierdlem114 46218 | Fourier series convergence... |
| fourierdlem115 46219 | Fourier serier convergence... |
| fourierd 46220 | Fourier series convergence... |
| fourierclimd 46221 | Fourier series convergence... |
| fourierclim 46222 | Fourier series convergence... |
| fourier 46223 | Fourier series convergence... |
| fouriercnp 46224 | If ` F ` is continuous at ... |
| fourier2 46225 | Fourier series convergence... |
| sqwvfoura 46226 | Fourier coefficients for t... |
| sqwvfourb 46227 | Fourier series ` B ` coeff... |
| fourierswlem 46228 | The Fourier series for the... |
| fouriersw 46229 | Fourier series convergence... |
| fouriercn 46230 | If the derivative of ` F `... |
| elaa2lem 46231 | Elementhood in the set of ... |
| elaa2 46232 | Elementhood in the set of ... |
| etransclem1 46233 | ` H ` is a function. (Con... |
| etransclem2 46234 | Derivative of ` G ` . (Co... |
| etransclem3 46235 | The given ` if ` term is a... |
| etransclem4 46236 | ` F ` expressed as a finit... |
| etransclem5 46237 | A change of bound variable... |
| etransclem6 46238 | A change of bound variable... |
| etransclem7 46239 | The given product is an in... |
| etransclem8 46240 | ` F ` is a function. (Con... |
| etransclem9 46241 | If ` K ` divides ` N ` but... |
| etransclem10 46242 | The given ` if ` term is a... |
| etransclem11 46243 | A change of bound variable... |
| etransclem12 46244 | ` C ` applied to ` N ` . ... |
| etransclem13 46245 | ` F ` applied to ` Y ` . ... |
| etransclem14 46246 | Value of the term ` T ` , ... |
| etransclem15 46247 | Value of the term ` T ` , ... |
| etransclem16 46248 | Every element in the range... |
| etransclem17 46249 | The ` N ` -th derivative o... |
| etransclem18 46250 | The given function is inte... |
| etransclem19 46251 | The ` N ` -th derivative o... |
| etransclem20 46252 | ` H ` is smooth. (Contrib... |
| etransclem21 46253 | The ` N ` -th derivative o... |
| etransclem22 46254 | The ` N ` -th derivative o... |
| etransclem23 46255 | This is the claim proof in... |
| etransclem24 46256 | ` P ` divides the I -th de... |
| etransclem25 46257 | ` P ` factorial divides th... |
| etransclem26 46258 | Every term in the sum of t... |
| etransclem27 46259 | The ` N ` -th derivative o... |
| etransclem28 46260 | ` ( P - 1 ) ` factorial di... |
| etransclem29 46261 | The ` N ` -th derivative o... |
| etransclem30 46262 | The ` N ` -th derivative o... |
| etransclem31 46263 | The ` N ` -th derivative o... |
| etransclem32 46264 | This is the proof for the ... |
| etransclem33 46265 | ` F ` is smooth. (Contrib... |
| etransclem34 46266 | The ` N ` -th derivative o... |
| etransclem35 46267 | ` P ` does not divide the ... |
| etransclem36 46268 | The ` N ` -th derivative o... |
| etransclem37 46269 | ` ( P - 1 ) ` factorial di... |
| etransclem38 46270 | ` P ` divides the I -th de... |
| etransclem39 46271 | ` G ` is a function. (Con... |
| etransclem40 46272 | The ` N ` -th derivative o... |
| etransclem41 46273 | ` P ` does not divide the ... |
| etransclem42 46274 | The ` N ` -th derivative o... |
| etransclem43 46275 | ` G ` is a continuous func... |
| etransclem44 46276 | The given finite sum is no... |
| etransclem45 46277 | ` K ` is an integer. (Con... |
| etransclem46 46278 | This is the proof for equa... |
| etransclem47 46279 | ` _e ` is transcendental. ... |
| etransclem48 46280 | ` _e ` is transcendental. ... |
| etransc 46281 | ` _e ` is transcendental. ... |
| rrxtopn 46282 | The topology of the genera... |
| rrxngp 46283 | Generalized Euclidean real... |
| rrxtps 46284 | Generalized Euclidean real... |
| rrxtopnfi 46285 | The topology of the n-dime... |
| rrxtopon 46286 | The topology on generalize... |
| rrxtop 46287 | The topology on generalize... |
| rrndistlt 46288 | Given two points in the sp... |
| rrxtoponfi 46289 | The topology on n-dimensio... |
| rrxunitopnfi 46290 | The base set of the standa... |
| rrxtopn0 46291 | The topology of the zero-d... |
| qndenserrnbllem 46292 | n-dimensional rational num... |
| qndenserrnbl 46293 | n-dimensional rational num... |
| rrxtopn0b 46294 | The topology of the zero-d... |
| qndenserrnopnlem 46295 | n-dimensional rational num... |
| qndenserrnopn 46296 | n-dimensional rational num... |
| qndenserrn 46297 | n-dimensional rational num... |
| rrxsnicc 46298 | A multidimensional singlet... |
| rrnprjdstle 46299 | The distance between two p... |
| rrndsmet 46300 | ` D ` is a metric for the ... |
| rrndsxmet 46301 | ` D ` is an extended metri... |
| ioorrnopnlem 46302 | The a point in an indexed ... |
| ioorrnopn 46303 | The indexed product of ope... |
| ioorrnopnxrlem 46304 | Given a point ` F ` that b... |
| ioorrnopnxr 46305 | The indexed product of ope... |
| issal 46312 | Express the predicate " ` ... |
| pwsal 46313 | The power set of a given s... |
| salunicl 46314 | SAlg sigma-algebra is clos... |
| saluncl 46315 | The union of two sets in a... |
| prsal 46316 | The pair of the empty set ... |
| saldifcl 46317 | The complement of an eleme... |
| 0sal 46318 | The empty set belongs to e... |
| salgenval 46319 | The sigma-algebra generate... |
| saliunclf 46320 | SAlg sigma-algebra is clos... |
| saliuncl 46321 | SAlg sigma-algebra is clos... |
| salincl 46322 | The intersection of two se... |
| saluni 46323 | A set is an element of any... |
| saliinclf 46324 | SAlg sigma-algebra is clos... |
| saliincl 46325 | SAlg sigma-algebra is clos... |
| saldifcl2 46326 | The difference of two elem... |
| intsaluni 46327 | The union of an arbitrary ... |
| intsal 46328 | The arbitrary intersection... |
| salgenn0 46329 | The set used in the defini... |
| salgencl 46330 | ` SalGen ` actually genera... |
| issald 46331 | Sufficient condition to pr... |
| salexct 46332 | An example of nontrivial s... |
| sssalgen 46333 | A set is a subset of the s... |
| salgenss 46334 | The sigma-algebra generate... |
| salgenuni 46335 | The base set of the sigma-... |
| issalgend 46336 | One side of ~ dfsalgen2 . ... |
| salexct2 46337 | An example of a subset tha... |
| unisalgen 46338 | The union of a set belongs... |
| dfsalgen2 46339 | Alternate characterization... |
| salexct3 46340 | An example of a sigma-alge... |
| salgencntex 46341 | This counterexample shows ... |
| salgensscntex 46342 | This counterexample shows ... |
| issalnnd 46343 | Sufficient condition to pr... |
| dmvolsal 46344 | Lebesgue measurable sets f... |
| saldifcld 46345 | The complement of an eleme... |
| saluncld 46346 | The union of two sets in a... |
| salgencld 46347 | ` SalGen ` actually genera... |
| 0sald 46348 | The empty set belongs to e... |
| iooborel 46349 | An open interval is a Bore... |
| salincld 46350 | The intersection of two se... |
| salunid 46351 | A set is an element of any... |
| unisalgen2 46352 | The union of a set belongs... |
| bor1sal 46353 | The Borel sigma-algebra on... |
| iocborel 46354 | A left-open, right-closed ... |
| subsaliuncllem 46355 | A subspace sigma-algebra i... |
| subsaliuncl 46356 | A subspace sigma-algebra i... |
| subsalsal 46357 | A subspace sigma-algebra i... |
| subsaluni 46358 | A set belongs to the subsp... |
| salrestss 46359 | A sigma-algebra restricted... |
| sge0rnre 46362 | When ` sum^ ` is applied t... |
| fge0icoicc 46363 | If ` F ` maps to nonnegati... |
| sge0val 46364 | The value of the sum of no... |
| fge0npnf 46365 | If ` F ` maps to nonnegati... |
| sge0rnn0 46366 | The range used in the defi... |
| sge0vald 46367 | The value of the sum of no... |
| fge0iccico 46368 | A range of nonnegative ext... |
| gsumge0cl 46369 | Closure of group sum, for ... |
| sge0reval 46370 | Value of the sum of nonneg... |
| sge0pnfval 46371 | If a term in the sum of no... |
| fge0iccre 46372 | A range of nonnegative ext... |
| sge0z 46373 | Any nonnegative extended s... |
| sge00 46374 | The sum of nonnegative ext... |
| fsumlesge0 46375 | Every finite subsum of non... |
| sge0revalmpt 46376 | Value of the sum of nonneg... |
| sge0sn 46377 | A sum of a nonnegative ext... |
| sge0tsms 46378 | ` sum^ ` applied to a nonn... |
| sge0cl 46379 | The arbitrary sum of nonne... |
| sge0f1o 46380 | Re-index a nonnegative ext... |
| sge0snmpt 46381 | A sum of a nonnegative ext... |
| sge0ge0 46382 | The sum of nonnegative ext... |
| sge0xrcl 46383 | The arbitrary sum of nonne... |
| sge0repnf 46384 | The of nonnegative extende... |
| sge0fsum 46385 | The arbitrary sum of a fin... |
| sge0rern 46386 | If the sum of nonnegative ... |
| sge0supre 46387 | If the arbitrary sum of no... |
| sge0fsummpt 46388 | The arbitrary sum of a fin... |
| sge0sup 46389 | The arbitrary sum of nonne... |
| sge0less 46390 | A shorter sum of nonnegati... |
| sge0rnbnd 46391 | The range used in the defi... |
| sge0pr 46392 | Sum of a pair of nonnegati... |
| sge0gerp 46393 | The arbitrary sum of nonne... |
| sge0pnffigt 46394 | If the sum of nonnegative ... |
| sge0ssre 46395 | If a sum of nonnegative ex... |
| sge0lefi 46396 | A sum of nonnegative exten... |
| sge0lessmpt 46397 | A shorter sum of nonnegati... |
| sge0ltfirp 46398 | If the sum of nonnegative ... |
| sge0prle 46399 | The sum of a pair of nonne... |
| sge0gerpmpt 46400 | The arbitrary sum of nonne... |
| sge0resrnlem 46401 | The sum of nonnegative ext... |
| sge0resrn 46402 | The sum of nonnegative ext... |
| sge0ssrempt 46403 | If a sum of nonnegative ex... |
| sge0resplit 46404 | ` sum^ ` splits into two p... |
| sge0le 46405 | If all of the terms of sum... |
| sge0ltfirpmpt 46406 | If the extended sum of non... |
| sge0split 46407 | Split a sum of nonnegative... |
| sge0lempt 46408 | If all of the terms of sum... |
| sge0splitmpt 46409 | Split a sum of nonnegative... |
| sge0ss 46410 | Change the index set to a ... |
| sge0iunmptlemfi 46411 | Sum of nonnegative extende... |
| sge0p1 46412 | The addition of the next t... |
| sge0iunmptlemre 46413 | Sum of nonnegative extende... |
| sge0fodjrnlem 46414 | Re-index a nonnegative ext... |
| sge0fodjrn 46415 | Re-index a nonnegative ext... |
| sge0iunmpt 46416 | Sum of nonnegative extende... |
| sge0iun 46417 | Sum of nonnegative extende... |
| sge0nemnf 46418 | The generalized sum of non... |
| sge0rpcpnf 46419 | The sum of an infinite num... |
| sge0rernmpt 46420 | If the sum of nonnegative ... |
| sge0lefimpt 46421 | A sum of nonnegative exten... |
| nn0ssge0 46422 | Nonnegative integers are n... |
| sge0clmpt 46423 | The generalized sum of non... |
| sge0ltfirpmpt2 46424 | If the extended sum of non... |
| sge0isum 46425 | If a series of nonnegative... |
| sge0xrclmpt 46426 | The generalized sum of non... |
| sge0xp 46427 | Combine two generalized su... |
| sge0isummpt 46428 | If a series of nonnegative... |
| sge0ad2en 46429 | The value of the infinite ... |
| sge0isummpt2 46430 | If a series of nonnegative... |
| sge0xaddlem1 46431 | The extended addition of t... |
| sge0xaddlem2 46432 | The extended addition of t... |
| sge0xadd 46433 | The extended addition of t... |
| sge0fsummptf 46434 | The generalized sum of a f... |
| sge0snmptf 46435 | A sum of a nonnegative ext... |
| sge0ge0mpt 46436 | The sum of nonnegative ext... |
| sge0repnfmpt 46437 | The of nonnegative extende... |
| sge0pnffigtmpt 46438 | If the generalized sum of ... |
| sge0splitsn 46439 | Separate out a term in a g... |
| sge0pnffsumgt 46440 | If the sum of nonnegative ... |
| sge0gtfsumgt 46441 | If the generalized sum of ... |
| sge0uzfsumgt 46442 | If a real number is smalle... |
| sge0pnfmpt 46443 | If a term in the sum of no... |
| sge0seq 46444 | A series of nonnegative re... |
| sge0reuz 46445 | Value of the generalized s... |
| sge0reuzb 46446 | Value of the generalized s... |
| ismea 46449 | Express the predicate " ` ... |
| dmmeasal 46450 | The domain of a measure is... |
| meaf 46451 | A measure is a function th... |
| mea0 46452 | The measure of the empty s... |
| nnfoctbdjlem 46453 | There exists a mapping fro... |
| nnfoctbdj 46454 | There exists a mapping fro... |
| meadjuni 46455 | The measure of the disjoin... |
| meacl 46456 | The measure of a set is a ... |
| iundjiunlem 46457 | The sets in the sequence `... |
| iundjiun 46458 | Given a sequence ` E ` of ... |
| meaxrcl 46459 | The measure of a set is an... |
| meadjun 46460 | The measure of the union o... |
| meassle 46461 | The measure of a set is gr... |
| meaunle 46462 | The measure of the union o... |
| meadjiunlem 46463 | The sum of nonnegative ext... |
| meadjiun 46464 | The measure of the disjoin... |
| ismeannd 46465 | Sufficient condition to pr... |
| meaiunlelem 46466 | The measure of the union o... |
| meaiunle 46467 | The measure of the union o... |
| psmeasurelem 46468 | ` M ` applied to a disjoin... |
| psmeasure 46469 | Point supported measure, R... |
| voliunsge0lem 46470 | The Lebesgue measure funct... |
| voliunsge0 46471 | The Lebesgue measure funct... |
| volmea 46472 | The Lebesgue measure on th... |
| meage0 46473 | If the measure of a measur... |
| meadjunre 46474 | The measure of the union o... |
| meassre 46475 | If the measure of a measur... |
| meale0eq0 46476 | A measure that is less tha... |
| meadif 46477 | The measure of the differe... |
| meaiuninclem 46478 | Measures are continuous fr... |
| meaiuninc 46479 | Measures are continuous fr... |
| meaiuninc2 46480 | Measures are continuous fr... |
| meaiunincf 46481 | Measures are continuous fr... |
| meaiuninc3v 46482 | Measures are continuous fr... |
| meaiuninc3 46483 | Measures are continuous fr... |
| meaiininclem 46484 | Measures are continuous fr... |
| meaiininc 46485 | Measures are continuous fr... |
| meaiininc2 46486 | Measures are continuous fr... |
| caragenval 46491 | The sigma-algebra generate... |
| isome 46492 | Express the predicate " ` ... |
| caragenel 46493 | Membership in the Caratheo... |
| omef 46494 | An outer measure is a func... |
| ome0 46495 | The outer measure of the e... |
| omessle 46496 | The outer measure of a set... |
| omedm 46497 | The domain of an outer mea... |
| caragensplit 46498 | If ` E ` is in the set gen... |
| caragenelss 46499 | An element of the Caratheo... |
| carageneld 46500 | Membership in the Caratheo... |
| omecl 46501 | The outer measure of a set... |
| caragenss 46502 | The sigma-algebra generate... |
| omeunile 46503 | The outer measure of the u... |
| caragen0 46504 | The empty set belongs to a... |
| omexrcl 46505 | The outer measure of a set... |
| caragenunidm 46506 | The base set of an outer m... |
| caragensspw 46507 | The sigma-algebra generate... |
| omessre 46508 | If the outer measure of a ... |
| caragenuni 46509 | The base set of the sigma-... |
| caragenuncllem 46510 | The Caratheodory's constru... |
| caragenuncl 46511 | The Caratheodory's constru... |
| caragendifcl 46512 | The Caratheodory's constru... |
| caragenfiiuncl 46513 | The Caratheodory's constru... |
| omeunle 46514 | The outer measure of the u... |
| omeiunle 46515 | The outer measure of the i... |
| omelesplit 46516 | The outer measure of a set... |
| omeiunltfirp 46517 | If the outer measure of a ... |
| omeiunlempt 46518 | The outer measure of the i... |
| carageniuncllem1 46519 | The outer measure of ` A i... |
| carageniuncllem2 46520 | The Caratheodory's constru... |
| carageniuncl 46521 | The Caratheodory's constru... |
| caragenunicl 46522 | The Caratheodory's constru... |
| caragensal 46523 | Caratheodory's method gene... |
| caratheodorylem1 46524 | Lemma used to prove that C... |
| caratheodorylem2 46525 | Caratheodory's constructio... |
| caratheodory 46526 | Caratheodory's constructio... |
| 0ome 46527 | The map that assigns 0 to ... |
| isomenndlem 46528 | ` O ` is sub-additive w.r.... |
| isomennd 46529 | Sufficient condition to pr... |
| caragenel2d 46530 | Membership in the Caratheo... |
| omege0 46531 | If the outer measure of a ... |
| omess0 46532 | If the outer measure of a ... |
| caragencmpl 46533 | A measure built with the C... |
| vonval 46538 | Value of the Lebesgue meas... |
| ovnval 46539 | Value of the Lebesgue oute... |
| elhoi 46540 | Membership in a multidimen... |
| icoresmbl 46541 | A closed-below, open-above... |
| hoissre 46542 | The projection of a half-o... |
| ovnval2 46543 | Value of the Lebesgue oute... |
| volicorecl 46544 | The Lebesgue measure of a ... |
| hoiprodcl 46545 | The pre-measure of half-op... |
| hoicvr 46546 | ` I ` is a countable set o... |
| hoissrrn 46547 | A half-open interval is a ... |
| ovn0val 46548 | The Lebesgue outer measure... |
| ovnn0val 46549 | The value of a (multidimen... |
| ovnval2b 46550 | Value of the Lebesgue oute... |
| volicorescl 46551 | The Lebesgue measure of a ... |
| ovnprodcl 46552 | The product used in the de... |
| hoiprodcl2 46553 | The pre-measure of half-op... |
| hoicvrrex 46554 | Any subset of the multidim... |
| ovnsupge0 46555 | The set used in the defini... |
| ovnlecvr 46556 | Given a subset of multidim... |
| ovnpnfelsup 46557 | ` +oo ` is an element of t... |
| ovnsslelem 46558 | The (multidimensional, non... |
| ovnssle 46559 | The (multidimensional) Leb... |
| ovnlerp 46560 | The Lebesgue outer measure... |
| ovnf 46561 | The Lebesgue outer measure... |
| ovncvrrp 46562 | The Lebesgue outer measure... |
| ovn0lem 46563 | For any finite dimension, ... |
| ovn0 46564 | For any finite dimension, ... |
| ovncl 46565 | The Lebesgue outer measure... |
| ovn02 46566 | For the zero-dimensional s... |
| ovnxrcl 46567 | The Lebesgue outer measure... |
| ovnsubaddlem1 46568 | The Lebesgue outer measure... |
| ovnsubaddlem2 46569 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd 46570 | ` ( voln* `` X ) ` is suba... |
| ovnome 46571 | ` ( voln* `` X ) ` is an o... |
| vonmea 46572 | ` ( voln `` X ) ` is a mea... |
| volicon0 46573 | The measure of a nonempty ... |
| hsphoif 46574 | ` H ` is a function (that ... |
| hoidmvval 46575 | The dimensional volume of ... |
| hoissrrn2 46576 | A half-open interval is a ... |
| hsphoival 46577 | ` H ` is a function (that ... |
| hoiprodcl3 46578 | The pre-measure of half-op... |
| volicore 46579 | The Lebesgue measure of a ... |
| hoidmvcl 46580 | The dimensional volume of ... |
| hoidmv0val 46581 | The dimensional volume of ... |
| hoidmvn0val 46582 | The dimensional volume of ... |
| hsphoidmvle2 46583 | The dimensional volume of ... |
| hsphoidmvle 46584 | The dimensional volume of ... |
| hoidmvval0 46585 | The dimensional volume of ... |
| hoiprodp1 46586 | The dimensional volume of ... |
| sge0hsphoire 46587 | If the generalized sum of ... |
| hoidmvval0b 46588 | The dimensional volume of ... |
| hoidmv1lelem1 46589 | The supremum of ` U ` belo... |
| hoidmv1lelem2 46590 | This is the contradiction ... |
| hoidmv1lelem3 46591 | The dimensional volume of ... |
| hoidmv1le 46592 | The dimensional volume of ... |
| hoidmvlelem1 46593 | The supremum of ` U ` belo... |
| hoidmvlelem2 46594 | This is the contradiction ... |
| hoidmvlelem3 46595 | This is the contradiction ... |
| hoidmvlelem4 46596 | The dimensional volume of ... |
| hoidmvlelem5 46597 | The dimensional volume of ... |
| hoidmvle 46598 | The dimensional volume of ... |
| ovnhoilem1 46599 | The Lebesgue outer measure... |
| ovnhoilem2 46600 | The Lebesgue outer measure... |
| ovnhoi 46601 | The Lebesgue outer measure... |
| dmovn 46602 | The domain of the Lebesgue... |
| hoicoto2 46603 | The half-open interval exp... |
| dmvon 46604 | Lebesgue measurable n-dime... |
| hoi2toco 46605 | The half-open interval exp... |
| hoidifhspval 46606 | ` D ` is a function that r... |
| hspval 46607 | The value of the half-spac... |
| ovnlecvr2 46608 | Given a subset of multidim... |
| ovncvr2 46609 | ` B ` and ` T ` are the le... |
| dmovnsal 46610 | The domain of the Lebesgue... |
| unidmovn 46611 | Base set of the n-dimensio... |
| rrnmbl 46612 | The set of n-dimensional R... |
| hoidifhspval2 46613 | ` D ` is a function that r... |
| hspdifhsp 46614 | A n-dimensional half-open ... |
| unidmvon 46615 | Base set of the n-dimensio... |
| hoidifhspf 46616 | ` D ` is a function that r... |
| hoidifhspval3 46617 | ` D ` is a function that r... |
| hoidifhspdmvle 46618 | The dimensional volume of ... |
| voncmpl 46619 | The Lebesgue measure is co... |
| hoiqssbllem1 46620 | The center of the n-dimens... |
| hoiqssbllem2 46621 | The center of the n-dimens... |
| hoiqssbllem3 46622 | A n-dimensional ball conta... |
| hoiqssbl 46623 | A n-dimensional ball conta... |
| hspmbllem1 46624 | Any half-space of the n-di... |
| hspmbllem2 46625 | Any half-space of the n-di... |
| hspmbllem3 46626 | Any half-space of the n-di... |
| hspmbl 46627 | Any half-space of the n-di... |
| hoimbllem 46628 | Any n-dimensional half-ope... |
| hoimbl 46629 | Any n-dimensional half-ope... |
| opnvonmbllem1 46630 | The half-open interval exp... |
| opnvonmbllem2 46631 | An open subset of the n-di... |
| opnvonmbl 46632 | An open subset of the n-di... |
| opnssborel 46633 | Open sets of a generalized... |
| borelmbl 46634 | All Borel subsets of the n... |
| volicorege0 46635 | The Lebesgue measure of a ... |
| isvonmbl 46636 | The predicate " ` A ` is m... |
| mblvon 46637 | The n-dimensional Lebesgue... |
| vonmblss 46638 | n-dimensional Lebesgue mea... |
| volico2 46639 | The measure of left-closed... |
| vonmblss2 46640 | n-dimensional Lebesgue mea... |
| ovolval2lem 46641 | The value of the Lebesgue ... |
| ovolval2 46642 | The value of the Lebesgue ... |
| ovnsubadd2lem 46643 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd2 46644 | ` ( voln* `` X ) ` is suba... |
| ovolval3 46645 | The value of the Lebesgue ... |
| ovnsplit 46646 | The n-dimensional Lebesgue... |
| ovolval4lem1 46647 | |- ( ( ph /\ n e. A ) -> ... |
| ovolval4lem2 46648 | The value of the Lebesgue ... |
| ovolval4 46649 | The value of the Lebesgue ... |
| ovolval5lem1 46650 | ` |- ( ph -> ( sum^ `` ( n... |
| ovolval5lem2 46651 | ` |- ( ( ph /\ n e. NN ) -... |
| ovolval5lem3 46652 | The value of the Lebesgue ... |
| ovolval5 46653 | The value of the Lebesgue ... |
| ovnovollem1 46654 | if ` F ` is a cover of ` B... |
| ovnovollem2 46655 | if ` I ` is a cover of ` (... |
| ovnovollem3 46656 | The 1-dimensional Lebesgue... |
| ovnovol 46657 | The 1-dimensional Lebesgue... |
| vonvolmbllem 46658 | If a subset ` B ` of real ... |
| vonvolmbl 46659 | A subset of Real numbers i... |
| vonvol 46660 | The 1-dimensional Lebesgue... |
| vonvolmbl2 46661 | A subset ` X ` of the spac... |
| vonvol2 46662 | The 1-dimensional Lebesgue... |
| hoimbl2 46663 | Any n-dimensional half-ope... |
| voncl 46664 | The Lebesgue measure of a ... |
| vonhoi 46665 | The Lebesgue outer measure... |
| vonxrcl 46666 | The Lebesgue measure of a ... |
| ioosshoi 46667 | A n-dimensional open inter... |
| vonn0hoi 46668 | The Lebesgue outer measure... |
| von0val 46669 | The Lebesgue measure (for ... |
| vonhoire 46670 | The Lebesgue measure of a ... |
| iinhoiicclem 46671 | A n-dimensional closed int... |
| iinhoiicc 46672 | A n-dimensional closed int... |
| iunhoiioolem 46673 | A n-dimensional open inter... |
| iunhoiioo 46674 | A n-dimensional open inter... |
| ioovonmbl 46675 | Any n-dimensional open int... |
| iccvonmbllem 46676 | Any n-dimensional closed i... |
| iccvonmbl 46677 | Any n-dimensional closed i... |
| vonioolem1 46678 | The sequence of the measur... |
| vonioolem2 46679 | The n-dimensional Lebesgue... |
| vonioo 46680 | The n-dimensional Lebesgue... |
| vonicclem1 46681 | The sequence of the measur... |
| vonicclem2 46682 | The n-dimensional Lebesgue... |
| vonicc 46683 | The n-dimensional Lebesgue... |
| snvonmbl 46684 | A n-dimensional singleton ... |
| vonn0ioo 46685 | The n-dimensional Lebesgue... |
| vonn0icc 46686 | The n-dimensional Lebesgue... |
| ctvonmbl 46687 | Any n-dimensional countabl... |
| vonn0ioo2 46688 | The n-dimensional Lebesgue... |
| vonsn 46689 | The n-dimensional Lebesgue... |
| vonn0icc2 46690 | The n-dimensional Lebesgue... |
| vonct 46691 | The n-dimensional Lebesgue... |
| vitali2 46692 | There are non-measurable s... |
| pimltmnf2f 46695 | Given a real-valued functi... |
| pimltmnf2 46696 | Given a real-valued functi... |
| preimagelt 46697 | The preimage of a right-op... |
| preimalegt 46698 | The preimage of a left-ope... |
| pimconstlt0 46699 | Given a constant function,... |
| pimconstlt1 46700 | Given a constant function,... |
| pimltpnff 46701 | Given a real-valued functi... |
| pimltpnf 46702 | Given a real-valued functi... |
| pimgtpnf2f 46703 | Given a real-valued functi... |
| pimgtpnf2 46704 | Given a real-valued functi... |
| salpreimagelt 46705 | If all the preimages of le... |
| pimrecltpos 46706 | The preimage of an unbound... |
| salpreimalegt 46707 | If all the preimages of ri... |
| pimiooltgt 46708 | The preimage of an open in... |
| preimaicomnf 46709 | Preimage of an open interv... |
| pimltpnf2f 46710 | Given a real-valued functi... |
| pimltpnf2 46711 | Given a real-valued functi... |
| pimgtmnf2 46712 | Given a real-valued functi... |
| pimdecfgtioc 46713 | Given a nonincreasing func... |
| pimincfltioc 46714 | Given a nondecreasing func... |
| pimdecfgtioo 46715 | Given a nondecreasing func... |
| pimincfltioo 46716 | Given a nondecreasing func... |
| preimaioomnf 46717 | Preimage of an open interv... |
| preimageiingt 46718 | A preimage of a left-close... |
| preimaleiinlt 46719 | A preimage of a left-open,... |
| pimgtmnff 46720 | Given a real-valued functi... |
| pimgtmnf 46721 | Given a real-valued functi... |
| pimrecltneg 46722 | The preimage of an unbound... |
| salpreimagtge 46723 | If all the preimages of le... |
| salpreimaltle 46724 | If all the preimages of ri... |
| issmflem 46725 | The predicate " ` F ` is a... |
| issmf 46726 | The predicate " ` F ` is a... |
| salpreimalelt 46727 | If all the preimages of ri... |
| salpreimagtlt 46728 | If all the preimages of le... |
| smfpreimalt 46729 | Given a function measurabl... |
| smff 46730 | A function measurable w.r.... |
| smfdmss 46731 | The domain of a function m... |
| issmff 46732 | The predicate " ` F ` is a... |
| issmfd 46733 | A sufficient condition for... |
| smfpreimaltf 46734 | Given a function measurabl... |
| issmfdf 46735 | A sufficient condition for... |
| sssmf 46736 | The restriction of a sigma... |
| mbfresmf 46737 | A real-valued measurable f... |
| cnfsmf 46738 | A continuous function is m... |
| incsmflem 46739 | A nondecreasing function i... |
| incsmf 46740 | A real-valued, nondecreasi... |
| smfsssmf 46741 | If a function is measurabl... |
| issmflelem 46742 | The predicate " ` F ` is a... |
| issmfle 46743 | The predicate " ` F ` is a... |
| smfpimltmpt 46744 | Given a function measurabl... |
| smfpimltxr 46745 | Given a function measurabl... |
| issmfdmpt 46746 | A sufficient condition for... |
| smfconst 46747 | Given a sigma-algebra over... |
| sssmfmpt 46748 | The restriction of a sigma... |
| cnfrrnsmf 46749 | A function, continuous fro... |
| smfid 46750 | The identity function is B... |
| bormflebmf 46751 | A Borel measurable functio... |
| smfpreimale 46752 | Given a function measurabl... |
| issmfgtlem 46753 | The predicate " ` F ` is a... |
| issmfgt 46754 | The predicate " ` F ` is a... |
| issmfled 46755 | A sufficient condition for... |
| smfpimltxrmptf 46756 | Given a function measurabl... |
| smfpimltxrmpt 46757 | Given a function measurabl... |
| smfmbfcex 46758 | A constant function, with ... |
| issmfgtd 46759 | A sufficient condition for... |
| smfpreimagt 46760 | Given a function measurabl... |
| smfaddlem1 46761 | Given the sum of two funct... |
| smfaddlem2 46762 | The sum of two sigma-measu... |
| smfadd 46763 | The sum of two sigma-measu... |
| decsmflem 46764 | A nonincreasing function i... |
| decsmf 46765 | A real-valued, nonincreasi... |
| smfpreimagtf 46766 | Given a function measurabl... |
| issmfgelem 46767 | The predicate " ` F ` is a... |
| issmfge 46768 | The predicate " ` F ` is a... |
| smflimlem1 46769 | Lemma for the proof that t... |
| smflimlem2 46770 | Lemma for the proof that t... |
| smflimlem3 46771 | The limit of sigma-measura... |
| smflimlem4 46772 | Lemma for the proof that t... |
| smflimlem5 46773 | Lemma for the proof that t... |
| smflimlem6 46774 | Lemma for the proof that t... |
| smflim 46775 | The limit of sigma-measura... |
| nsssmfmbflem 46776 | The sigma-measurable funct... |
| nsssmfmbf 46777 | The sigma-measurable funct... |
| smfpimgtxr 46778 | Given a function measurabl... |
| smfpimgtmpt 46779 | Given a function measurabl... |
| smfpreimage 46780 | Given a function measurabl... |
| mbfpsssmf 46781 | Real-valued measurable fun... |
| smfpimgtxrmptf 46782 | Given a function measurabl... |
| smfpimgtxrmpt 46783 | Given a function measurabl... |
| smfpimioompt 46784 | Given a function measurabl... |
| smfpimioo 46785 | Given a function measurabl... |
| smfresal 46786 | Given a sigma-measurable f... |
| smfrec 46787 | The reciprocal of a sigma-... |
| smfres 46788 | The restriction of sigma-m... |
| smfmullem1 46789 | The multiplication of two ... |
| smfmullem2 46790 | The multiplication of two ... |
| smfmullem3 46791 | The multiplication of two ... |
| smfmullem4 46792 | The multiplication of two ... |
| smfmul 46793 | The multiplication of two ... |
| smfmulc1 46794 | A sigma-measurable functio... |
| smfdiv 46795 | The fraction of two sigma-... |
| smfpimbor1lem1 46796 | Every open set belongs to ... |
| smfpimbor1lem2 46797 | Given a sigma-measurable f... |
| smfpimbor1 46798 | Given a sigma-measurable f... |
| smf2id 46799 | Twice the identity functio... |
| smfco 46800 | The composition of a Borel... |
| smfneg 46801 | The negative of a sigma-me... |
| smffmptf 46802 | A function measurable w.r.... |
| smffmpt 46803 | A function measurable w.r.... |
| smflim2 46804 | The limit of a sequence of... |
| smfpimcclem 46805 | Lemma for ~ smfpimcc given... |
| smfpimcc 46806 | Given a countable set of s... |
| issmfle2d 46807 | A sufficient condition for... |
| smflimmpt 46808 | The limit of a sequence of... |
| smfsuplem1 46809 | The supremum of a countabl... |
| smfsuplem2 46810 | The supremum of a countabl... |
| smfsuplem3 46811 | The supremum of a countabl... |
| smfsup 46812 | The supremum of a countabl... |
| smfsupmpt 46813 | The supremum of a countabl... |
| smfsupxr 46814 | The supremum of a countabl... |
| smfinflem 46815 | The infimum of a countable... |
| smfinf 46816 | The infimum of a countable... |
| smfinfmpt 46817 | The infimum of a countable... |
| smflimsuplem1 46818 | If ` H ` converges, the ` ... |
| smflimsuplem2 46819 | The superior limit of a se... |
| smflimsuplem3 46820 | The limit of the ` ( H `` ... |
| smflimsuplem4 46821 | If ` H ` converges, the ` ... |
| smflimsuplem5 46822 | ` H ` converges to the sup... |
| smflimsuplem6 46823 | The superior limit of a se... |
| smflimsuplem7 46824 | The superior limit of a se... |
| smflimsuplem8 46825 | The superior limit of a se... |
| smflimsup 46826 | The superior limit of a se... |
| smflimsupmpt 46827 | The superior limit of a se... |
| smfliminflem 46828 | The inferior limit of a co... |
| smfliminf 46829 | The inferior limit of a co... |
| smfliminfmpt 46830 | The inferior limit of a co... |
| adddmmbl 46831 | If two functions have doma... |
| adddmmbl2 46832 | If two functions have doma... |
| muldmmbl 46833 | If two functions have doma... |
| muldmmbl2 46834 | If two functions have doma... |
| smfdmmblpimne 46835 | If a measurable function w... |
| smfdivdmmbl 46836 | If a functions and a sigma... |
| smfpimne 46837 | Given a function measurabl... |
| smfpimne2 46838 | Given a function measurabl... |
| smfdivdmmbl2 46839 | If a functions and a sigma... |
| fsupdm 46840 | The domain of the sup func... |
| fsupdm2 46841 | The domain of the sup func... |
| smfsupdmmbllem 46842 | If a countable set of sigm... |
| smfsupdmmbl 46843 | If a countable set of sigm... |
| finfdm 46844 | The domain of the inf func... |
| finfdm2 46845 | The domain of the inf func... |
| smfinfdmmbllem 46846 | If a countable set of sigm... |
| smfinfdmmbl 46847 | If a countable set of sigm... |
| sigarval 46848 | Define the signed area by ... |
| sigarim 46849 | Signed area takes value in... |
| sigarac 46850 | Signed area is anticommuta... |
| sigaraf 46851 | Signed area is additive by... |
| sigarmf 46852 | Signed area is additive (w... |
| sigaras 46853 | Signed area is additive by... |
| sigarms 46854 | Signed area is additive (w... |
| sigarls 46855 | Signed area is linear by t... |
| sigarid 46856 | Signed area of a flat para... |
| sigarexp 46857 | Expand the signed area for... |
| sigarperm 46858 | Signed area ` ( A - C ) G ... |
| sigardiv 46859 | If signed area between vec... |
| sigarimcd 46860 | Signed area takes value in... |
| sigariz 46861 | If signed area is zero, th... |
| sigarcol 46862 | Given three points ` A ` ,... |
| sharhght 46863 | Let ` A B C ` be a triangl... |
| sigaradd 46864 | Subtracting (double) area ... |
| cevathlem1 46865 | Ceva's theorem first lemma... |
| cevathlem2 46866 | Ceva's theorem second lemm... |
| cevath 46867 | Ceva's theorem. Let ` A B... |
| simpcntrab 46868 | The center of a simple gro... |
| et-ltneverrefl 46869 | Less-than class is never r... |
| et-equeucl 46870 | Alternative proof that equ... |
| et-sqrtnegnre 46871 | The square root of a negat... |
| ormklocald 46872 | If elements of a certain s... |
| ormkglobd 46873 | If all adjacent elements o... |
| natlocalincr 46874 | Global monotonicity on hal... |
| natglobalincr 46875 | Local monotonicity on half... |
| upwordnul 46878 | Empty set is an increasing... |
| upwordisword 46879 | Any increasing sequence is... |
| singoutnword 46880 | Singleton with character o... |
| singoutnupword 46881 | Singleton with character o... |
| upwordsing 46882 | Singleton is an increasing... |
| upwordsseti 46883 | Strictly increasing sequen... |
| tworepnotupword 46884 | Concatenation of identical... |
| upwrdfi 46885 | There is a finite number o... |
| evenwodadd 46886 | If an integer is multiplie... |
| squeezedltsq 46887 | If a real value is squeeze... |
| lambert0 46888 | A value of Lambert W (prod... |
| lamberte 46889 | A value of Lambert W (prod... |
| cjnpoly 46890 | Complex conjugation operat... |
| tannpoly 46891 | The tangent function is no... |
| sinnpoly 46892 | Sine function is not a pol... |
| hirstL-ax3 46893 | The third axiom of a syste... |
| ax3h 46894 | Recover ~ ax-3 from ~ hirs... |
| aibandbiaiffaiffb 46895 | A closed form showing (a i... |
| aibandbiaiaiffb 46896 | A closed form showing (a i... |
| notatnand 46897 | Do not use. Use intnanr i... |
| aistia 46898 | Given a is equivalent to `... |
| aisfina 46899 | Given a is equivalent to `... |
| bothtbothsame 46900 | Given both a, b are equiva... |
| bothfbothsame 46901 | Given both a, b are equiva... |
| aiffbbtat 46902 | Given a is equivalent to b... |
| aisbbisfaisf 46903 | Given a is equivalent to b... |
| axorbtnotaiffb 46904 | Given a is exclusive to b,... |
| aiffnbandciffatnotciffb 46905 | Given a is equivalent to (... |
| axorbciffatcxorb 46906 | Given a is equivalent to (... |
| aibnbna 46907 | Given a implies b, (not b)... |
| aibnbaif 46908 | Given a implies b, not b, ... |
| aiffbtbat 46909 | Given a is equivalent to b... |
| astbstanbst 46910 | Given a is equivalent to T... |
| aistbistaandb 46911 | Given a is equivalent to T... |
| aisbnaxb 46912 | Given a is equivalent to b... |
| atbiffatnnb 46913 | If a implies b, then a imp... |
| bisaiaisb 46914 | Application of bicom1 with... |
| atbiffatnnbalt 46915 | If a implies b, then a imp... |
| abnotbtaxb 46916 | Assuming a, not b, there e... |
| abnotataxb 46917 | Assuming not a, b, there e... |
| conimpf 46918 | Assuming a, not b, and a i... |
| conimpfalt 46919 | Assuming a, not b, and a i... |
| aistbisfiaxb 46920 | Given a is equivalent to T... |
| aisfbistiaxb 46921 | Given a is equivalent to F... |
| aifftbifffaibif 46922 | Given a is equivalent to T... |
| aifftbifffaibifff 46923 | Given a is equivalent to T... |
| atnaiana 46924 | Given a, it is not the cas... |
| ainaiaandna 46925 | Given a, a implies it is n... |
| abcdta 46926 | Given (((a and b) and c) a... |
| abcdtb 46927 | Given (((a and b) and c) a... |
| abcdtc 46928 | Given (((a and b) and c) a... |
| abcdtd 46929 | Given (((a and b) and c) a... |
| abciffcbatnabciffncba 46930 | Operands in a biconditiona... |
| abciffcbatnabciffncbai 46931 | Operands in a biconditiona... |
| nabctnabc 46932 | not ( a -> ( b /\ c ) ) we... |
| jabtaib 46933 | For when pm3.4 lacks a pm3... |
| onenotinotbothi 46934 | From one negated implicati... |
| twonotinotbothi 46935 | From these two negated imp... |
| clifte 46936 | show d is the same as an i... |
| cliftet 46937 | show d is the same as an i... |
| clifteta 46938 | show d is the same as an i... |
| cliftetb 46939 | show d is the same as an i... |
| confun 46940 | Given the hypotheses there... |
| confun2 46941 | Confun simplified to two p... |
| confun3 46942 | Confun's more complex form... |
| confun4 46943 | An attempt at derivative. ... |
| confun5 46944 | An attempt at derivative. ... |
| plcofph 46945 | Given, a,b and a "definiti... |
| pldofph 46946 | Given, a,b c, d, "definiti... |
| plvcofph 46947 | Given, a,b,d, and "definit... |
| plvcofphax 46948 | Given, a,b,d, and "definit... |
| plvofpos 46949 | rh is derivable because ON... |
| mdandyv0 46950 | Given the equivalences set... |
| mdandyv1 46951 | Given the equivalences set... |
| mdandyv2 46952 | Given the equivalences set... |
| mdandyv3 46953 | Given the equivalences set... |
| mdandyv4 46954 | Given the equivalences set... |
| mdandyv5 46955 | Given the equivalences set... |
| mdandyv6 46956 | Given the equivalences set... |
| mdandyv7 46957 | Given the equivalences set... |
| mdandyv8 46958 | Given the equivalences set... |
| mdandyv9 46959 | Given the equivalences set... |
| mdandyv10 46960 | Given the equivalences set... |
| mdandyv11 46961 | Given the equivalences set... |
| mdandyv12 46962 | Given the equivalences set... |
| mdandyv13 46963 | Given the equivalences set... |
| mdandyv14 46964 | Given the equivalences set... |
| mdandyv15 46965 | Given the equivalences set... |
| mdandyvr0 46966 | Given the equivalences set... |
| mdandyvr1 46967 | Given the equivalences set... |
| mdandyvr2 46968 | Given the equivalences set... |
| mdandyvr3 46969 | Given the equivalences set... |
| mdandyvr4 46970 | Given the equivalences set... |
| mdandyvr5 46971 | Given the equivalences set... |
| mdandyvr6 46972 | Given the equivalences set... |
| mdandyvr7 46973 | Given the equivalences set... |
| mdandyvr8 46974 | Given the equivalences set... |
| mdandyvr9 46975 | Given the equivalences set... |
| mdandyvr10 46976 | Given the equivalences set... |
| mdandyvr11 46977 | Given the equivalences set... |
| mdandyvr12 46978 | Given the equivalences set... |
| mdandyvr13 46979 | Given the equivalences set... |
| mdandyvr14 46980 | Given the equivalences set... |
| mdandyvr15 46981 | Given the equivalences set... |
| mdandyvrx0 46982 | Given the exclusivities se... |
| mdandyvrx1 46983 | Given the exclusivities se... |
| mdandyvrx2 46984 | Given the exclusivities se... |
| mdandyvrx3 46985 | Given the exclusivities se... |
| mdandyvrx4 46986 | Given the exclusivities se... |
| mdandyvrx5 46987 | Given the exclusivities se... |
| mdandyvrx6 46988 | Given the exclusivities se... |
| mdandyvrx7 46989 | Given the exclusivities se... |
| mdandyvrx8 46990 | Given the exclusivities se... |
| mdandyvrx9 46991 | Given the exclusivities se... |
| mdandyvrx10 46992 | Given the exclusivities se... |
| mdandyvrx11 46993 | Given the exclusivities se... |
| mdandyvrx12 46994 | Given the exclusivities se... |
| mdandyvrx13 46995 | Given the exclusivities se... |
| mdandyvrx14 46996 | Given the exclusivities se... |
| mdandyvrx15 46997 | Given the exclusivities se... |
| H15NH16TH15IH16 46998 | Given 15 hypotheses and a ... |
| dandysum2p2e4 46999 | CONTRADICTION PROVED AT 1 ... |
| mdandysum2p2e4 47000 | CONTRADICTION PROVED AT 1 ... |
| adh-jarrsc 47001 | Replacement of a nested an... |
| adh-minim 47002 | A single axiom for minimal... |
| adh-minim-ax1-ax2-lem1 47003 | First lemma for the deriva... |
| adh-minim-ax1-ax2-lem2 47004 | Second lemma for the deriv... |
| adh-minim-ax1-ax2-lem3 47005 | Third lemma for the deriva... |
| adh-minim-ax1-ax2-lem4 47006 | Fourth lemma for the deriv... |
| adh-minim-ax1 47007 | Derivation of ~ ax-1 from ... |
| adh-minim-ax2-lem5 47008 | Fifth lemma for the deriva... |
| adh-minim-ax2-lem6 47009 | Sixth lemma for the deriva... |
| adh-minim-ax2c 47010 | Derivation of a commuted f... |
| adh-minim-ax2 47011 | Derivation of ~ ax-2 from ... |
| adh-minim-idALT 47012 | Derivation of ~ id (reflex... |
| adh-minim-pm2.43 47013 | Derivation of ~ pm2.43 Whi... |
| adh-minimp 47014 | Another single axiom for m... |
| adh-minimp-jarr-imim1-ax2c-lem1 47015 | First lemma for the deriva... |
| adh-minimp-jarr-lem2 47016 | Second lemma for the deriv... |
| adh-minimp-jarr-ax2c-lem3 47017 | Third lemma for the deriva... |
| adh-minimp-sylsimp 47018 | Derivation of ~ jarr (also... |
| adh-minimp-ax1 47019 | Derivation of ~ ax-1 from ... |
| adh-minimp-imim1 47020 | Derivation of ~ imim1 ("le... |
| adh-minimp-ax2c 47021 | Derivation of a commuted f... |
| adh-minimp-ax2-lem4 47022 | Fourth lemma for the deriv... |
| adh-minimp-ax2 47023 | Derivation of ~ ax-2 from ... |
| adh-minimp-idALT 47024 | Derivation of ~ id (reflex... |
| adh-minimp-pm2.43 47025 | Derivation of ~ pm2.43 Whi... |
| n0nsn2el 47026 | If a class with one elemen... |
| eusnsn 47027 | There is a unique element ... |
| absnsb 47028 | If the class abstraction `... |
| euabsneu 47029 | Another way to express exi... |
| elprneb 47030 | An element of a proper uno... |
| oppr 47031 | Equality for ordered pairs... |
| opprb 47032 | Equality for unordered pai... |
| or2expropbilem1 47033 | Lemma 1 for ~ or2expropbi ... |
| or2expropbilem2 47034 | Lemma 2 for ~ or2expropbi ... |
| or2expropbi 47035 | If two classes are strictl... |
| eubrv 47036 | If there is a unique set w... |
| eubrdm 47037 | If there is a unique set w... |
| eldmressn 47038 | Element of the domain of a... |
| iota0def 47039 | Example for a defined iota... |
| iota0ndef 47040 | Example for an undefined i... |
| fveqvfvv 47041 | If a function's value at a... |
| fnresfnco 47042 | Composition of two functio... |
| funcoressn 47043 | A composition restricted t... |
| funressnfv 47044 | A restriction to a singlet... |
| funressndmfvrn 47045 | The value of a function ` ... |
| funressnvmo 47046 | A function restricted to a... |
| funressnmo 47047 | A function restricted to a... |
| funressneu 47048 | There is exactly one value... |
| fresfo 47049 | Conditions for a restricti... |
| fsetsniunop 47050 | The class of all functions... |
| fsetabsnop 47051 | The class of all functions... |
| fsetsnf 47052 | The mapping of an element ... |
| fsetsnf1 47053 | The mapping of an element ... |
| fsetsnfo 47054 | The mapping of an element ... |
| fsetsnf1o 47055 | The mapping of an element ... |
| fsetsnprcnex 47056 | The class of all functions... |
| cfsetssfset 47057 | The class of constant func... |
| cfsetsnfsetfv 47058 | The function value of the ... |
| cfsetsnfsetf 47059 | The mapping of the class o... |
| cfsetsnfsetf1 47060 | The mapping of the class o... |
| cfsetsnfsetfo 47061 | The mapping of the class o... |
| cfsetsnfsetf1o 47062 | The mapping of the class o... |
| fsetprcnexALT 47063 | First version of proof for... |
| fcoreslem1 47064 | Lemma 1 for ~ fcores . (C... |
| fcoreslem2 47065 | Lemma 2 for ~ fcores . (C... |
| fcoreslem3 47066 | Lemma 3 for ~ fcores . (C... |
| fcoreslem4 47067 | Lemma 4 for ~ fcores . (C... |
| fcores 47068 | Every composite function `... |
| fcoresf1lem 47069 | Lemma for ~ fcoresf1 . (C... |
| fcoresf1 47070 | If a composition is inject... |
| fcoresf1b 47071 | A composition is injective... |
| fcoresfo 47072 | If a composition is surjec... |
| fcoresfob 47073 | A composition is surjectiv... |
| fcoresf1ob 47074 | A composition is bijective... |
| f1cof1blem 47075 | Lemma for ~ f1cof1b and ~ ... |
| 3f1oss1 47076 | The composition of three b... |
| 3f1oss2 47077 | The composition of three b... |
| f1cof1b 47078 | If the range of ` F ` equa... |
| funfocofob 47079 | If the domain of a functio... |
| fnfocofob 47080 | If the domain of a functio... |
| focofob 47081 | If the domain of a functio... |
| f1ocof1ob 47082 | If the range of ` F ` equa... |
| f1ocof1ob2 47083 | If the range of ` F ` equa... |
| aiotajust 47085 | Soundness justification th... |
| dfaiota2 47087 | Alternate definition of th... |
| reuabaiotaiota 47088 | The iota and the alternate... |
| reuaiotaiota 47089 | The iota and the alternate... |
| aiotaexb 47090 | The alternate iota over a ... |
| aiotavb 47091 | The alternate iota over a ... |
| aiotaint 47092 | This is to ~ df-aiota what... |
| dfaiota3 47093 | Alternate definition of ` ... |
| iotan0aiotaex 47094 | If the iota over a wff ` p... |
| aiotaexaiotaiota 47095 | The alternate iota over a ... |
| aiotaval 47096 | Theorem 8.19 in [Quine] p.... |
| aiota0def 47097 | Example for a defined alte... |
| aiota0ndef 47098 | Example for an undefined a... |
| r19.32 47099 | Theorem 19.32 of [Margaris... |
| rexsb 47100 | An equivalent expression f... |
| rexrsb 47101 | An equivalent expression f... |
| 2rexsb 47102 | An equivalent expression f... |
| 2rexrsb 47103 | An equivalent expression f... |
| cbvral2 47104 | Change bound variables of ... |
| cbvrex2 47105 | Change bound variables of ... |
| ralndv1 47106 | Example for a theorem abou... |
| ralndv2 47107 | Second example for a theor... |
| reuf1odnf 47108 | There is exactly one eleme... |
| reuf1od 47109 | There is exactly one eleme... |
| euoreqb 47110 | There is a set which is eq... |
| 2reu3 47111 | Double restricted existent... |
| 2reu7 47112 | Two equivalent expressions... |
| 2reu8 47113 | Two equivalent expressions... |
| 2reu8i 47114 | Implication of a double re... |
| 2reuimp0 47115 | Implication of a double re... |
| 2reuimp 47116 | Implication of a double re... |
| ralbinrald 47123 | Elemination of a restricte... |
| nvelim 47124 | If a class is the universa... |
| alneu 47125 | If a statement holds for a... |
| eu2ndop1stv 47126 | If there is a unique secon... |
| dfateq12d 47127 | Equality deduction for "de... |
| nfdfat 47128 | Bound-variable hypothesis ... |
| dfdfat2 47129 | Alternate definition of th... |
| fundmdfat 47130 | A function is defined at a... |
| dfatprc 47131 | A function is not defined ... |
| dfatelrn 47132 | The value of a function ` ... |
| dfafv2 47133 | Alternative definition of ... |
| afveq12d 47134 | Equality deduction for fun... |
| afveq1 47135 | Equality theorem for funct... |
| afveq2 47136 | Equality theorem for funct... |
| nfafv 47137 | Bound-variable hypothesis ... |
| csbafv12g 47138 | Move class substitution in... |
| afvfundmfveq 47139 | If a class is a function r... |
| afvnfundmuv 47140 | If a set is not in the dom... |
| ndmafv 47141 | The value of a class outsi... |
| afvvdm 47142 | If the function value of a... |
| nfunsnafv 47143 | If the restriction of a cl... |
| afvvfunressn 47144 | If the function value of a... |
| afvprc 47145 | A function's value at a pr... |
| afvvv 47146 | If a function's value at a... |
| afvpcfv0 47147 | If the value of the altern... |
| afvnufveq 47148 | The value of the alternati... |
| afvvfveq 47149 | The value of the alternati... |
| afv0fv0 47150 | If the value of the altern... |
| afvfvn0fveq 47151 | If the function's value at... |
| afv0nbfvbi 47152 | The function's value at an... |
| afvfv0bi 47153 | The function's value at an... |
| afveu 47154 | The value of a function at... |
| fnbrafvb 47155 | Equivalence of function va... |
| fnopafvb 47156 | Equivalence of function va... |
| funbrafvb 47157 | Equivalence of function va... |
| funopafvb 47158 | Equivalence of function va... |
| funbrafv 47159 | The second argument of a b... |
| funbrafv2b 47160 | Function value in terms of... |
| dfafn5a 47161 | Representation of a functi... |
| dfafn5b 47162 | Representation of a functi... |
| fnrnafv 47163 | The range of a function ex... |
| afvelrnb 47164 | A member of a function's r... |
| afvelrnb0 47165 | A member of a function's r... |
| dfaimafn 47166 | Alternate definition of th... |
| dfaimafn2 47167 | Alternate definition of th... |
| afvelima 47168 | Function value in an image... |
| afvelrn 47169 | A function's value belongs... |
| fnafvelrn 47170 | A function's value belongs... |
| fafvelcdm 47171 | A function's value belongs... |
| ffnafv 47172 | A function maps to a class... |
| afvres 47173 | The value of a restricted ... |
| tz6.12-afv 47174 | Function value. Theorem 6... |
| tz6.12-1-afv 47175 | Function value (Theorem 6.... |
| dmfcoafv 47176 | Domains of a function comp... |
| afvco2 47177 | Value of a function compos... |
| rlimdmafv 47178 | Two ways to express that a... |
| aoveq123d 47179 | Equality deduction for ope... |
| nfaov 47180 | Bound-variable hypothesis ... |
| csbaovg 47181 | Move class substitution in... |
| aovfundmoveq 47182 | If a class is a function r... |
| aovnfundmuv 47183 | If an ordered pair is not ... |
| ndmaov 47184 | The value of an operation ... |
| ndmaovg 47185 | The value of an operation ... |
| aovvdm 47186 | If the operation value of ... |
| nfunsnaov 47187 | If the restriction of a cl... |
| aovvfunressn 47188 | If the operation value of ... |
| aovprc 47189 | The value of an operation ... |
| aovrcl 47190 | Reverse closure for an ope... |
| aovpcov0 47191 | If the alternative value o... |
| aovnuoveq 47192 | The alternative value of t... |
| aovvoveq 47193 | The alternative value of t... |
| aov0ov0 47194 | If the alternative value o... |
| aovovn0oveq 47195 | If the operation's value a... |
| aov0nbovbi 47196 | The operation's value on a... |
| aovov0bi 47197 | The operation's value on a... |
| rspceaov 47198 | A frequently used special ... |
| fnotaovb 47199 | Equivalence of operation v... |
| ffnaov 47200 | An operation maps to a cla... |
| faovcl 47201 | Closure law for an operati... |
| aovmpt4g 47202 | Value of a function given ... |
| aoprssdm 47203 | Domain of closure of an op... |
| ndmaovcl 47204 | The "closure" of an operat... |
| ndmaovrcl 47205 | Reverse closure law, in co... |
| ndmaovcom 47206 | Any operation is commutati... |
| ndmaovass 47207 | Any operation is associati... |
| ndmaovdistr 47208 | Any operation is distribut... |
| dfatafv2iota 47211 | If a function is defined a... |
| ndfatafv2 47212 | The alternate function val... |
| ndfatafv2undef 47213 | The alternate function val... |
| dfatafv2ex 47214 | The alternate function val... |
| afv2ex 47215 | The alternate function val... |
| afv2eq12d 47216 | Equality deduction for fun... |
| afv2eq1 47217 | Equality theorem for funct... |
| afv2eq2 47218 | Equality theorem for funct... |
| nfafv2 47219 | Bound-variable hypothesis ... |
| csbafv212g 47220 | Move class substitution in... |
| fexafv2ex 47221 | The alternate function val... |
| ndfatafv2nrn 47222 | The alternate function val... |
| ndmafv2nrn 47223 | The value of a class outsi... |
| funressndmafv2rn 47224 | The alternate function val... |
| afv2ndefb 47225 | Two ways to say that an al... |
| nfunsnafv2 47226 | If the restriction of a cl... |
| afv2prc 47227 | A function's value at a pr... |
| dfatafv2rnb 47228 | The alternate function val... |
| afv2orxorb 47229 | If a set is in the range o... |
| dmafv2rnb 47230 | The alternate function val... |
| fundmafv2rnb 47231 | The alternate function val... |
| afv2elrn 47232 | An alternate function valu... |
| afv20defat 47233 | If the alternate function ... |
| fnafv2elrn 47234 | An alternate function valu... |
| fafv2elcdm 47235 | An alternate function valu... |
| fafv2elrnb 47236 | An alternate function valu... |
| fcdmvafv2v 47237 | If the codomain of a funct... |
| tz6.12-2-afv2 47238 | Function value when ` F ` ... |
| afv2eu 47239 | The value of a function at... |
| afv2res 47240 | The value of a restricted ... |
| tz6.12-afv2 47241 | Function value (Theorem 6.... |
| tz6.12-1-afv2 47242 | Function value (Theorem 6.... |
| tz6.12c-afv2 47243 | Corollary of Theorem 6.12(... |
| tz6.12i-afv2 47244 | Corollary of Theorem 6.12(... |
| funressnbrafv2 47245 | The second argument of a b... |
| dfatbrafv2b 47246 | Equivalence of function va... |
| dfatopafv2b 47247 | Equivalence of function va... |
| funbrafv2 47248 | The second argument of a b... |
| fnbrafv2b 47249 | Equivalence of function va... |
| fnopafv2b 47250 | Equivalence of function va... |
| funbrafv22b 47251 | Equivalence of function va... |
| funopafv2b 47252 | Equivalence of function va... |
| dfatsnafv2 47253 | Singleton of function valu... |
| dfafv23 47254 | A definition of function v... |
| dfatdmfcoafv2 47255 | Domain of a function compo... |
| dfatcolem 47256 | Lemma for ~ dfatco . (Con... |
| dfatco 47257 | The predicate "defined at"... |
| afv2co2 47258 | Value of a function compos... |
| rlimdmafv2 47259 | Two ways to express that a... |
| dfafv22 47260 | Alternate definition of ` ... |
| afv2ndeffv0 47261 | If the alternate function ... |
| dfatafv2eqfv 47262 | If a function is defined a... |
| afv2rnfveq 47263 | If the alternate function ... |
| afv20fv0 47264 | If the alternate function ... |
| afv2fvn0fveq 47265 | If the function's value at... |
| afv2fv0 47266 | If the function's value at... |
| afv2fv0b 47267 | The function's value at an... |
| afv2fv0xorb 47268 | If a set is in the range o... |
| an4com24 47269 | Rearrangement of 4 conjunc... |
| 3an4ancom24 47270 | Commutative law for a conj... |
| 4an21 47271 | Rearrangement of 4 conjunc... |
| dfnelbr2 47274 | Alternate definition of th... |
| nelbr 47275 | The binary relation of a s... |
| nelbrim 47276 | If a set is related to ano... |
| nelbrnel 47277 | A set is related to anothe... |
| nelbrnelim 47278 | If a set is related to ano... |
| ralralimp 47279 | Selecting one of two alter... |
| otiunsndisjX 47280 | The union of singletons co... |
| fvifeq 47281 | Equality of function value... |
| rnfdmpr 47282 | The range of a one-to-one ... |
| imarnf1pr 47283 | The image of the range of ... |
| funop1 47284 | A function is an ordered p... |
| fun2dmnopgexmpl 47285 | A function with a domain c... |
| opabresex0d 47286 | A collection of ordered pa... |
| opabbrfex0d 47287 | A collection of ordered pa... |
| opabresexd 47288 | A collection of ordered pa... |
| opabbrfexd 47289 | A collection of ordered pa... |
| f1oresf1orab 47290 | Build a bijection by restr... |
| f1oresf1o 47291 | Build a bijection by restr... |
| f1oresf1o2 47292 | Build a bijection by restr... |
| fvmptrab 47293 | Value of a function mappin... |
| fvmptrabdm 47294 | Value of a function mappin... |
| cnambpcma 47295 | ((a-b)+c)-a = c-a holds fo... |
| cnapbmcpd 47296 | ((a+b)-c)+d = ((a+d)+b)-c ... |
| addsubeq0 47297 | The sum of two complex num... |
| leaddsuble 47298 | Addition and subtraction o... |
| 2leaddle2 47299 | If two real numbers are le... |
| ltnltne 47300 | Variant of trichotomy law ... |
| p1lep2 47301 | A real number increasd by ... |
| ltsubsubaddltsub 47302 | If the result of subtracti... |
| zm1nn 47303 | An integer minus 1 is posi... |
| readdcnnred 47304 | The sum of a real number a... |
| resubcnnred 47305 | The difference of a real n... |
| recnmulnred 47306 | The product of a real numb... |
| cndivrenred 47307 | The quotient of an imagina... |
| sqrtnegnre 47308 | The square root of a negat... |
| nn0resubcl 47309 | Closure law for subtractio... |
| zgeltp1eq 47310 | If an integer is between a... |
| 1t10e1p1e11 47311 | 11 is 1 times 10 to the po... |
| deccarry 47312 | Add 1 to a 2 digit number ... |
| eluzge0nn0 47313 | If an integer is greater t... |
| nltle2tri 47314 | Negated extended trichotom... |
| ssfz12 47315 | Subset relationship for fi... |
| elfz2z 47316 | Membership of an integer i... |
| 2elfz3nn0 47317 | If there are two elements ... |
| fz0addcom 47318 | The addition of two member... |
| 2elfz2melfz 47319 | If the sum of two integers... |
| fz0addge0 47320 | The sum of two integers in... |
| elfzlble 47321 | Membership of an integer i... |
| elfzelfzlble 47322 | Membership of an element o... |
| fzopred 47323 | Join a predecessor to the ... |
| fzopredsuc 47324 | Join a predecessor and a s... |
| 1fzopredsuc 47325 | Join 0 and a successor to ... |
| el1fzopredsuc 47326 | An element of an open inte... |
| subsubelfzo0 47327 | Subtracting a difference f... |
| 2ffzoeq 47328 | Two functions over a half-... |
| 2ltceilhalf 47329 | The ceiling of half of an ... |
| ceilhalfgt1 47330 | The ceiling of half of an ... |
| ceilhalfelfzo1 47331 | A positive integer less th... |
| gpgedgvtx1lem 47332 | Lemma for ~ gpgedgvtx1 . ... |
| 2tceilhalfelfzo1 47333 | Two times a positive integ... |
| ceilbi 47334 | A condition equivalent to ... |
| ceilhalf1 47335 | The ceiling of one half is... |
| rehalfge1 47336 | Half of a real number grea... |
| ceilhalfnn 47337 | The ceiling of half of a p... |
| 1elfzo1ceilhalf1 47338 | 1 is in the half-open inte... |
| fldivmod 47339 | Expressing the floor of a ... |
| ceildivmod 47340 | Expressing the ceiling of ... |
| ceil5half3 47341 | The ceiling of half of 5 i... |
| submodaddmod 47342 | Subtraction and addition m... |
| difltmodne 47343 | Two nonnegative integers a... |
| zplusmodne 47344 | A nonnegative integer is n... |
| addmodne 47345 | The sum of a nonnegative i... |
| plusmod5ne 47346 | A nonnegative integer is n... |
| zp1modne 47347 | An integer is not itself p... |
| p1modne 47348 | A nonnegative integer is n... |
| m1modne 47349 | A nonnegative integer is n... |
| minusmod5ne 47350 | A nonnegative integer is n... |
| submodlt 47351 | The difference of an eleme... |
| submodneaddmod 47352 | An integer minus ` B ` is ... |
| m1modnep2mod 47353 | A nonnegative integer minu... |
| minusmodnep2tmod 47354 | A nonnegative integer minu... |
| m1mod0mod1 47355 | An integer decreased by 1 ... |
| elmod2 47356 | An integer modulo 2 is eit... |
| mod0mul 47357 | If an integer is 0 modulo ... |
| modn0mul 47358 | If an integer is not 0 mod... |
| m1modmmod 47359 | An integer decreased by 1 ... |
| difmodm1lt 47360 | The difference between an ... |
| 8mod5e3 47361 | 8 modulo 5 is 3. (Contrib... |
| modmkpkne 47362 | If an integer minus a cons... |
| modmknepk 47363 | A nonnegative integer less... |
| modlt0b 47364 | An integer with an absolut... |
| mod2addne 47365 | The sums of a nonnegative ... |
| modm1nep1 47366 | A nonnegative integer less... |
| modm2nep1 47367 | A nonnegative integer less... |
| modp2nep1 47368 | A nonnegative integer less... |
| modm1nep2 47369 | A nonnegative integer less... |
| modm1nem2 47370 | A nonnegative integer less... |
| modm1p1ne 47371 | If an integer minus one eq... |
| smonoord 47372 | Ordering relation for a st... |
| fsummsndifre 47373 | A finite sum with one of i... |
| fsumsplitsndif 47374 | Separate out a term in a f... |
| fsummmodsndifre 47375 | A finite sum of summands m... |
| fsummmodsnunz 47376 | A finite sum of summands m... |
| setsidel 47377 | The injected slot is an el... |
| setsnidel 47378 | The injected slot is an el... |
| setsv 47379 | The value of the structure... |
| preimafvsnel 47380 | The preimage of a function... |
| preimafvn0 47381 | The preimage of a function... |
| uniimafveqt 47382 | The union of the image of ... |
| uniimaprimaeqfv 47383 | The union of the image of ... |
| setpreimafvex 47384 | The class ` P ` of all pre... |
| elsetpreimafvb 47385 | The characterization of an... |
| elsetpreimafv 47386 | An element of the class ` ... |
| elsetpreimafvssdm 47387 | An element of the class ` ... |
| fvelsetpreimafv 47388 | There is an element in a p... |
| preimafvelsetpreimafv 47389 | The preimage of a function... |
| preimafvsspwdm 47390 | The class ` P ` of all pre... |
| 0nelsetpreimafv 47391 | The empty set is not an el... |
| elsetpreimafvbi 47392 | An element of the preimage... |
| elsetpreimafveqfv 47393 | The elements of the preima... |
| eqfvelsetpreimafv 47394 | If an element of the domai... |
| elsetpreimafvrab 47395 | An element of the preimage... |
| imaelsetpreimafv 47396 | The image of an element of... |
| uniimaelsetpreimafv 47397 | The union of the image of ... |
| elsetpreimafveq 47398 | If two preimages of functi... |
| fundcmpsurinjlem1 47399 | Lemma 1 for ~ fundcmpsurin... |
| fundcmpsurinjlem2 47400 | Lemma 2 for ~ fundcmpsurin... |
| fundcmpsurinjlem3 47401 | Lemma 3 for ~ fundcmpsurin... |
| imasetpreimafvbijlemf 47402 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv 47403 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv1 47404 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemf1 47405 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfo 47406 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbij 47407 | The mapping ` H ` is a bij... |
| fundcmpsurbijinjpreimafv 47408 | Every function ` F : A -->... |
| fundcmpsurinjpreimafv 47409 | Every function ` F : A -->... |
| fundcmpsurinj 47410 | Every function ` F : A -->... |
| fundcmpsurbijinj 47411 | Every function ` F : A -->... |
| fundcmpsurinjimaid 47412 | Every function ` F : A -->... |
| fundcmpsurinjALT 47413 | Alternate proof of ~ fundc... |
| iccpval 47416 | Partition consisting of a ... |
| iccpart 47417 | A special partition. Corr... |
| iccpartimp 47418 | Implications for a class b... |
| iccpartres 47419 | The restriction of a parti... |
| iccpartxr 47420 | If there is a partition, t... |
| iccpartgtprec 47421 | If there is a partition, t... |
| iccpartipre 47422 | If there is a partition, t... |
| iccpartiltu 47423 | If there is a partition, t... |
| iccpartigtl 47424 | If there is a partition, t... |
| iccpartlt 47425 | If there is a partition, t... |
| iccpartltu 47426 | If there is a partition, t... |
| iccpartgtl 47427 | If there is a partition, t... |
| iccpartgt 47428 | If there is a partition, t... |
| iccpartleu 47429 | If there is a partition, t... |
| iccpartgel 47430 | If there is a partition, t... |
| iccpartrn 47431 | If there is a partition, t... |
| iccpartf 47432 | The range of the partition... |
| iccpartel 47433 | If there is a partition, t... |
| iccelpart 47434 | An element of any partitio... |
| iccpartiun 47435 | A half-open interval of ex... |
| icceuelpartlem 47436 | Lemma for ~ icceuelpart . ... |
| icceuelpart 47437 | An element of a partitione... |
| iccpartdisj 47438 | The segments of a partitio... |
| iccpartnel 47439 | A point of a partition is ... |
| fargshiftfv 47440 | If a class is a function, ... |
| fargshiftf 47441 | If a class is a function, ... |
| fargshiftf1 47442 | If a function is 1-1, then... |
| fargshiftfo 47443 | If a function is onto, the... |
| fargshiftfva 47444 | The values of a shifted fu... |
| lswn0 47445 | The last symbol of a not e... |
| nfich1 47448 | The first interchangeable ... |
| nfich2 47449 | The second interchangeable... |
| ichv 47450 | Setvar variables are inter... |
| ichf 47451 | Setvar variables are inter... |
| ichid 47452 | A setvar variable is alway... |
| icht 47453 | A theorem is interchangeab... |
| ichbidv 47454 | Formula building rule for ... |
| ichcircshi 47455 | The setvar variables are i... |
| ichan 47456 | If two setvar variables ar... |
| ichn 47457 | Negation does not affect i... |
| ichim 47458 | Formula building rule for ... |
| dfich2 47459 | Alternate definition of th... |
| ichcom 47460 | The interchangeability of ... |
| ichbi12i 47461 | Equivalence for interchang... |
| icheqid 47462 | In an equality for the sam... |
| icheq 47463 | In an equality of setvar v... |
| ichnfimlem 47464 | Lemma for ~ ichnfim : A s... |
| ichnfim 47465 | If in an interchangeabilit... |
| ichnfb 47466 | If ` x ` and ` y ` are int... |
| ichal 47467 | Move a universal quantifie... |
| ich2al 47468 | Two setvar variables are a... |
| ich2ex 47469 | Two setvar variables are a... |
| ichexmpl1 47470 | Example for interchangeabl... |
| ichexmpl2 47471 | Example for interchangeabl... |
| ich2exprop 47472 | If the setvar variables ar... |
| ichnreuop 47473 | If the setvar variables ar... |
| ichreuopeq 47474 | If the setvar variables ar... |
| sprid 47475 | Two identical representati... |
| elsprel 47476 | An unordered pair is an el... |
| spr0nelg 47477 | The empty set is not an el... |
| sprval 47480 | The set of all unordered p... |
| sprvalpw 47481 | The set of all unordered p... |
| sprssspr 47482 | The set of all unordered p... |
| spr0el 47483 | The empty set is not an un... |
| sprvalpwn0 47484 | The set of all unordered p... |
| sprel 47485 | An element of the set of a... |
| prssspr 47486 | An element of a subset of ... |
| prelspr 47487 | An unordered pair of eleme... |
| prsprel 47488 | The elements of a pair fro... |
| prsssprel 47489 | The elements of a pair fro... |
| sprvalpwle2 47490 | The set of all unordered p... |
| sprsymrelfvlem 47491 | Lemma for ~ sprsymrelf and... |
| sprsymrelf1lem 47492 | Lemma for ~ sprsymrelf1 . ... |
| sprsymrelfolem1 47493 | Lemma 1 for ~ sprsymrelfo ... |
| sprsymrelfolem2 47494 | Lemma 2 for ~ sprsymrelfo ... |
| sprsymrelfv 47495 | The value of the function ... |
| sprsymrelf 47496 | The mapping ` F ` is a fun... |
| sprsymrelf1 47497 | The mapping ` F ` is a one... |
| sprsymrelfo 47498 | The mapping ` F ` is a fun... |
| sprsymrelf1o 47499 | The mapping ` F ` is a bij... |
| sprbisymrel 47500 | There is a bijection betwe... |
| sprsymrelen 47501 | The class ` P ` of subsets... |
| prpair 47502 | Characterization of a prop... |
| prproropf1olem0 47503 | Lemma 0 for ~ prproropf1o ... |
| prproropf1olem1 47504 | Lemma 1 for ~ prproropf1o ... |
| prproropf1olem2 47505 | Lemma 2 for ~ prproropf1o ... |
| prproropf1olem3 47506 | Lemma 3 for ~ prproropf1o ... |
| prproropf1olem4 47507 | Lemma 4 for ~ prproropf1o ... |
| prproropf1o 47508 | There is a bijection betwe... |
| prproropen 47509 | The set of proper pairs an... |
| prproropreud 47510 | There is exactly one order... |
| pairreueq 47511 | Two equivalent representat... |
| paireqne 47512 | Two sets are not equal iff... |
| prprval 47515 | The set of all proper unor... |
| prprvalpw 47516 | The set of all proper unor... |
| prprelb 47517 | An element of the set of a... |
| prprelprb 47518 | A set is an element of the... |
| prprspr2 47519 | The set of all proper unor... |
| prprsprreu 47520 | There is a unique proper u... |
| prprreueq 47521 | There is a unique proper u... |
| sbcpr 47522 | The proper substitution of... |
| reupr 47523 | There is a unique unordere... |
| reuprpr 47524 | There is a unique proper u... |
| poprelb 47525 | Equality for unordered pai... |
| 2exopprim 47526 | The existence of an ordere... |
| reuopreuprim 47527 | There is a unique unordere... |
| fmtno 47530 | The ` N ` th Fermat number... |
| fmtnoge3 47531 | Each Fermat number is grea... |
| fmtnonn 47532 | Each Fermat number is a po... |
| fmtnom1nn 47533 | A Fermat number minus one ... |
| fmtnoodd 47534 | Each Fermat number is odd.... |
| fmtnorn 47535 | A Fermat number is a funct... |
| fmtnof1 47536 | The enumeration of the Fer... |
| fmtnoinf 47537 | The set of Fermat numbers ... |
| fmtnorec1 47538 | The first recurrence relat... |
| sqrtpwpw2p 47539 | The floor of the square ro... |
| fmtnosqrt 47540 | The floor of the square ro... |
| fmtno0 47541 | The ` 0 ` th Fermat number... |
| fmtno1 47542 | The ` 1 ` st Fermat number... |
| fmtnorec2lem 47543 | Lemma for ~ fmtnorec2 (ind... |
| fmtnorec2 47544 | The second recurrence rela... |
| fmtnodvds 47545 | Any Fermat number divides ... |
| goldbachthlem1 47546 | Lemma 1 for ~ goldbachth .... |
| goldbachthlem2 47547 | Lemma 2 for ~ goldbachth .... |
| goldbachth 47548 | Goldbach's theorem: Two d... |
| fmtnorec3 47549 | The third recurrence relat... |
| fmtnorec4 47550 | The fourth recurrence rela... |
| fmtno2 47551 | The ` 2 ` nd Fermat number... |
| fmtno3 47552 | The ` 3 ` rd Fermat number... |
| fmtno4 47553 | The ` 4 ` th Fermat number... |
| fmtno5lem1 47554 | Lemma 1 for ~ fmtno5 . (C... |
| fmtno5lem2 47555 | Lemma 2 for ~ fmtno5 . (C... |
| fmtno5lem3 47556 | Lemma 3 for ~ fmtno5 . (C... |
| fmtno5lem4 47557 | Lemma 4 for ~ fmtno5 . (C... |
| fmtno5 47558 | The ` 5 ` th Fermat number... |
| fmtno0prm 47559 | The ` 0 ` th Fermat number... |
| fmtno1prm 47560 | The ` 1 ` st Fermat number... |
| fmtno2prm 47561 | The ` 2 ` nd Fermat number... |
| 257prm 47562 | 257 is a prime number (the... |
| fmtno3prm 47563 | The ` 3 ` rd Fermat number... |
| odz2prm2pw 47564 | Any power of two is coprim... |
| fmtnoprmfac1lem 47565 | Lemma for ~ fmtnoprmfac1 :... |
| fmtnoprmfac1 47566 | Divisor of Fermat number (... |
| fmtnoprmfac2lem1 47567 | Lemma for ~ fmtnoprmfac2 .... |
| fmtnoprmfac2 47568 | Divisor of Fermat number (... |
| fmtnofac2lem 47569 | Lemma for ~ fmtnofac2 (Ind... |
| fmtnofac2 47570 | Divisor of Fermat number (... |
| fmtnofac1 47571 | Divisor of Fermat number (... |
| fmtno4sqrt 47572 | The floor of the square ro... |
| fmtno4prmfac 47573 | If P was a (prime) factor ... |
| fmtno4prmfac193 47574 | If P was a (prime) factor ... |
| fmtno4nprmfac193 47575 | 193 is not a (prime) facto... |
| fmtno4prm 47576 | The ` 4 `-th Fermat number... |
| 65537prm 47577 | 65537 is a prime number (t... |
| fmtnofz04prm 47578 | The first five Fermat numb... |
| fmtnole4prm 47579 | The first five Fermat numb... |
| fmtno5faclem1 47580 | Lemma 1 for ~ fmtno5fac . ... |
| fmtno5faclem2 47581 | Lemma 2 for ~ fmtno5fac . ... |
| fmtno5faclem3 47582 | Lemma 3 for ~ fmtno5fac . ... |
| fmtno5fac 47583 | The factorization of the `... |
| fmtno5nprm 47584 | The ` 5 ` th Fermat number... |
| prmdvdsfmtnof1lem1 47585 | Lemma 1 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof1lem2 47586 | Lemma 2 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof 47587 | The mapping of a Fermat nu... |
| prmdvdsfmtnof1 47588 | The mapping of a Fermat nu... |
| prminf2 47589 | The set of prime numbers i... |
| 2pwp1prm 47590 | For ` ( ( 2 ^ k ) + 1 ) ` ... |
| 2pwp1prmfmtno 47591 | Every prime number of the ... |
| m2prm 47592 | The second Mersenne number... |
| m3prm 47593 | The third Mersenne number ... |
| flsqrt 47594 | A condition equivalent to ... |
| flsqrt5 47595 | The floor of the square ro... |
| 3ndvds4 47596 | 3 does not divide 4. (Con... |
| 139prmALT 47597 | 139 is a prime number. In... |
| 31prm 47598 | 31 is a prime number. In ... |
| m5prm 47599 | The fifth Mersenne number ... |
| 127prm 47600 | 127 is a prime number. (C... |
| m7prm 47601 | The seventh Mersenne numbe... |
| m11nprm 47602 | The eleventh Mersenne numb... |
| mod42tp1mod8 47603 | If a number is ` 3 ` modul... |
| sfprmdvdsmersenne 47604 | If ` Q ` is a safe prime (... |
| sgprmdvdsmersenne 47605 | If ` P ` is a Sophie Germa... |
| lighneallem1 47606 | Lemma 1 for ~ lighneal . ... |
| lighneallem2 47607 | Lemma 2 for ~ lighneal . ... |
| lighneallem3 47608 | Lemma 3 for ~ lighneal . ... |
| lighneallem4a 47609 | Lemma 1 for ~ lighneallem4... |
| lighneallem4b 47610 | Lemma 2 for ~ lighneallem4... |
| lighneallem4 47611 | Lemma 3 for ~ lighneal . ... |
| lighneal 47612 | If a power of a prime ` P ... |
| modexp2m1d 47613 | The square of an integer w... |
| proththdlem 47614 | Lemma for ~ proththd . (C... |
| proththd 47615 | Proth's theorem (1878). I... |
| 5tcu2e40 47616 | 5 times the cube of 2 is 4... |
| 3exp4mod41 47617 | 3 to the fourth power is -... |
| 41prothprmlem1 47618 | Lemma 1 for ~ 41prothprm .... |
| 41prothprmlem2 47619 | Lemma 2 for ~ 41prothprm .... |
| 41prothprm 47620 | 41 is a _Proth prime_. (C... |
| quad1 47621 | A condition for a quadrati... |
| requad01 47622 | A condition for a quadrati... |
| requad1 47623 | A condition for a quadrati... |
| requad2 47624 | A condition for a quadrati... |
| iseven 47629 | The predicate "is an even ... |
| isodd 47630 | The predicate "is an odd n... |
| evenz 47631 | An even number is an integ... |
| oddz 47632 | An odd number is an intege... |
| evendiv2z 47633 | The result of dividing an ... |
| oddp1div2z 47634 | The result of dividing an ... |
| oddm1div2z 47635 | The result of dividing an ... |
| isodd2 47636 | The predicate "is an odd n... |
| dfodd2 47637 | Alternate definition for o... |
| dfodd6 47638 | Alternate definition for o... |
| dfeven4 47639 | Alternate definition for e... |
| evenm1odd 47640 | The predecessor of an even... |
| evenp1odd 47641 | The successor of an even n... |
| oddp1eveni 47642 | The successor of an odd nu... |
| oddm1eveni 47643 | The predecessor of an odd ... |
| evennodd 47644 | An even number is not an o... |
| oddneven 47645 | An odd number is not an ev... |
| enege 47646 | The negative of an even nu... |
| onego 47647 | The negative of an odd num... |
| m1expevenALTV 47648 | Exponentiation of -1 by an... |
| m1expoddALTV 47649 | Exponentiation of -1 by an... |
| dfeven2 47650 | Alternate definition for e... |
| dfodd3 47651 | Alternate definition for o... |
| iseven2 47652 | The predicate "is an even ... |
| isodd3 47653 | The predicate "is an odd n... |
| 2dvdseven 47654 | 2 divides an even number. ... |
| m2even 47655 | A multiple of 2 is an even... |
| 2ndvdsodd 47656 | 2 does not divide an odd n... |
| 2dvdsoddp1 47657 | 2 divides an odd number in... |
| 2dvdsoddm1 47658 | 2 divides an odd number de... |
| dfeven3 47659 | Alternate definition for e... |
| dfodd4 47660 | Alternate definition for o... |
| dfodd5 47661 | Alternate definition for o... |
| zefldiv2ALTV 47662 | The floor of an even numbe... |
| zofldiv2ALTV 47663 | The floor of an odd number... |
| oddflALTV 47664 | Odd number representation ... |
| iseven5 47665 | The predicate "is an even ... |
| isodd7 47666 | The predicate "is an odd n... |
| dfeven5 47667 | Alternate definition for e... |
| dfodd7 47668 | Alternate definition for o... |
| gcd2odd1 47669 | The greatest common diviso... |
| zneoALTV 47670 | No even integer equals an ... |
| zeoALTV 47671 | An integer is even or odd.... |
| zeo2ALTV 47672 | An integer is even or odd ... |
| nneoALTV 47673 | A positive integer is even... |
| nneoiALTV 47674 | A positive integer is even... |
| odd2np1ALTV 47675 | An integer is odd iff it i... |
| oddm1evenALTV 47676 | An integer is odd iff its ... |
| oddp1evenALTV 47677 | An integer is odd iff its ... |
| oexpnegALTV 47678 | The exponential of the neg... |
| oexpnegnz 47679 | The exponential of the neg... |
| bits0ALTV 47680 | Value of the zeroth bit. ... |
| bits0eALTV 47681 | The zeroth bit of an even ... |
| bits0oALTV 47682 | The zeroth bit of an odd n... |
| divgcdoddALTV 47683 | Either ` A / ( A gcd B ) `... |
| opoeALTV 47684 | The sum of two odds is eve... |
| opeoALTV 47685 | The sum of an odd and an e... |
| omoeALTV 47686 | The difference of two odds... |
| omeoALTV 47687 | The difference of an odd a... |
| oddprmALTV 47688 | A prime not equal to ` 2 `... |
| 0evenALTV 47689 | 0 is an even number. (Con... |
| 0noddALTV 47690 | 0 is not an odd number. (... |
| 1oddALTV 47691 | 1 is an odd number. (Cont... |
| 1nevenALTV 47692 | 1 is not an even number. ... |
| 2evenALTV 47693 | 2 is an even number. (Con... |
| 2noddALTV 47694 | 2 is not an odd number. (... |
| nn0o1gt2ALTV 47695 | An odd nonnegative integer... |
| nnoALTV 47696 | An alternate characterizat... |
| nn0oALTV 47697 | An alternate characterizat... |
| nn0e 47698 | An alternate characterizat... |
| nneven 47699 | An alternate characterizat... |
| nn0onn0exALTV 47700 | For each odd nonnegative i... |
| nn0enn0exALTV 47701 | For each even nonnegative ... |
| nnennexALTV 47702 | For each even positive int... |
| nnpw2evenALTV 47703 | 2 to the power of a positi... |
| epoo 47704 | The sum of an even and an ... |
| emoo 47705 | The difference of an even ... |
| epee 47706 | The sum of two even number... |
| emee 47707 | The difference of two even... |
| evensumeven 47708 | If a summand is even, the ... |
| 3odd 47709 | 3 is an odd number. (Cont... |
| 4even 47710 | 4 is an even number. (Con... |
| 5odd 47711 | 5 is an odd number. (Cont... |
| 6even 47712 | 6 is an even number. (Con... |
| 7odd 47713 | 7 is an odd number. (Cont... |
| 8even 47714 | 8 is an even number. (Con... |
| evenprm2 47715 | A prime number is even iff... |
| oddprmne2 47716 | Every prime number not bei... |
| oddprmuzge3 47717 | A prime number which is od... |
| evenltle 47718 | If an even number is great... |
| odd2prm2 47719 | If an odd number is the su... |
| even3prm2 47720 | If an even number is the s... |
| mogoldbblem 47721 | Lemma for ~ mogoldbb . (C... |
| perfectALTVlem1 47722 | Lemma for ~ perfectALTV . ... |
| perfectALTVlem2 47723 | Lemma for ~ perfectALTV . ... |
| perfectALTV 47724 | The Euclid-Euler theorem, ... |
| fppr 47727 | The set of Fermat pseudopr... |
| fpprmod 47728 | The set of Fermat pseudopr... |
| fpprel 47729 | A Fermat pseudoprime to th... |
| fpprbasnn 47730 | The base of a Fermat pseud... |
| fpprnn 47731 | A Fermat pseudoprime to th... |
| fppr2odd 47732 | A Fermat pseudoprime to th... |
| 11t31e341 47733 | 341 is the product of 11 a... |
| 2exp340mod341 47734 | Eight to the eighth power ... |
| 341fppr2 47735 | 341 is the (smallest) _Pou... |
| 4fppr1 47736 | 4 is the (smallest) Fermat... |
| 8exp8mod9 47737 | Eight to the eighth power ... |
| 9fppr8 47738 | 9 is the (smallest) Fermat... |
| dfwppr 47739 | Alternate definition of a ... |
| fpprwppr 47740 | A Fermat pseudoprime to th... |
| fpprwpprb 47741 | An integer ` X ` which is ... |
| fpprel2 47742 | An alternate definition fo... |
| nfermltl8rev 47743 | Fermat's little theorem wi... |
| nfermltl2rev 47744 | Fermat's little theorem wi... |
| nfermltlrev 47745 | Fermat's little theorem re... |
| isgbe 47752 | The predicate "is an even ... |
| isgbow 47753 | The predicate "is a weak o... |
| isgbo 47754 | The predicate "is an odd G... |
| gbeeven 47755 | An even Goldbach number is... |
| gbowodd 47756 | A weak odd Goldbach number... |
| gbogbow 47757 | A (strong) odd Goldbach nu... |
| gboodd 47758 | An odd Goldbach number is ... |
| gbepos 47759 | Any even Goldbach number i... |
| gbowpos 47760 | Any weak odd Goldbach numb... |
| gbopos 47761 | Any odd Goldbach number is... |
| gbegt5 47762 | Any even Goldbach number i... |
| gbowgt5 47763 | Any weak odd Goldbach numb... |
| gbowge7 47764 | Any weak odd Goldbach numb... |
| gboge9 47765 | Any odd Goldbach number is... |
| gbege6 47766 | Any even Goldbach number i... |
| gbpart6 47767 | The Goldbach partition of ... |
| gbpart7 47768 | The (weak) Goldbach partit... |
| gbpart8 47769 | The Goldbach partition of ... |
| gbpart9 47770 | The (strong) Goldbach part... |
| gbpart11 47771 | The (strong) Goldbach part... |
| 6gbe 47772 | 6 is an even Goldbach numb... |
| 7gbow 47773 | 7 is a weak odd Goldbach n... |
| 8gbe 47774 | 8 is an even Goldbach numb... |
| 9gbo 47775 | 9 is an odd Goldbach numbe... |
| 11gbo 47776 | 11 is an odd Goldbach numb... |
| stgoldbwt 47777 | If the strong ternary Gold... |
| sbgoldbwt 47778 | If the strong binary Goldb... |
| sbgoldbst 47779 | If the strong binary Goldb... |
| sbgoldbaltlem1 47780 | Lemma 1 for ~ sbgoldbalt :... |
| sbgoldbaltlem2 47781 | Lemma 2 for ~ sbgoldbalt :... |
| sbgoldbalt 47782 | An alternate (related to t... |
| sbgoldbb 47783 | If the strong binary Goldb... |
| sgoldbeven3prm 47784 | If the binary Goldbach con... |
| sbgoldbm 47785 | If the strong binary Goldb... |
| mogoldbb 47786 | If the modern version of t... |
| sbgoldbmb 47787 | The strong binary Goldbach... |
| sbgoldbo 47788 | If the strong binary Goldb... |
| nnsum3primes4 47789 | 4 is the sum of at most 3 ... |
| nnsum4primes4 47790 | 4 is the sum of at most 4 ... |
| nnsum3primesprm 47791 | Every prime is "the sum of... |
| nnsum4primesprm 47792 | Every prime is "the sum of... |
| nnsum3primesgbe 47793 | Any even Goldbach number i... |
| nnsum4primesgbe 47794 | Any even Goldbach number i... |
| nnsum3primesle9 47795 | Every integer greater than... |
| nnsum4primesle9 47796 | Every integer greater than... |
| nnsum4primesodd 47797 | If the (weak) ternary Gold... |
| nnsum4primesoddALTV 47798 | If the (strong) ternary Go... |
| evengpop3 47799 | If the (weak) ternary Gold... |
| evengpoap3 47800 | If the (strong) ternary Go... |
| nnsum4primeseven 47801 | If the (weak) ternary Gold... |
| nnsum4primesevenALTV 47802 | If the (strong) ternary Go... |
| wtgoldbnnsum4prm 47803 | If the (weak) ternary Gold... |
| stgoldbnnsum4prm 47804 | If the (strong) ternary Go... |
| bgoldbnnsum3prm 47805 | If the binary Goldbach con... |
| bgoldbtbndlem1 47806 | Lemma 1 for ~ bgoldbtbnd :... |
| bgoldbtbndlem2 47807 | Lemma 2 for ~ bgoldbtbnd .... |
| bgoldbtbndlem3 47808 | Lemma 3 for ~ bgoldbtbnd .... |
| bgoldbtbndlem4 47809 | Lemma 4 for ~ bgoldbtbnd .... |
| bgoldbtbnd 47810 | If the binary Goldbach con... |
| tgoldbachgtALTV 47813 | Variant of Thierry Arnoux'... |
| bgoldbachlt 47814 | The binary Goldbach conjec... |
| tgblthelfgott 47816 | The ternary Goldbach conje... |
| tgoldbachlt 47817 | The ternary Goldbach conje... |
| tgoldbach 47818 | The ternary Goldbach conje... |
| clnbgrprc0 47821 | The closed neighborhood is... |
| clnbgrcl 47822 | If a class ` X ` has at le... |
| clnbgrval 47823 | The closed neighborhood of... |
| dfclnbgr2 47824 | Alternate definition of th... |
| dfclnbgr4 47825 | Alternate definition of th... |
| elclnbgrelnbgr 47826 | An element of the closed n... |
| dfclnbgr3 47827 | Alternate definition of th... |
| clnbgrnvtx0 47828 | If a class ` X ` is not a ... |
| clnbgrel 47829 | Characterization of a memb... |
| clnbgrvtxel 47830 | Every vertex ` K ` is a me... |
| clnbgrisvtx 47831 | Every member ` N ` of the ... |
| clnbgrssvtx 47832 | The closed neighborhood of... |
| clnbgrn0 47833 | The closed neighborhood of... |
| clnbupgr 47834 | The closed neighborhood of... |
| clnbupgrel 47835 | A member of the closed nei... |
| clnbgr0vtx 47836 | In a null graph (with no v... |
| clnbgr0edg 47837 | In an empty graph (with no... |
| clnbgrsym 47838 | In a graph, the closed nei... |
| predgclnbgrel 47839 | If a (not necessarily prop... |
| clnbgredg 47840 | A vertex connected by an e... |
| clnbgrssedg 47841 | The vertices connected by ... |
| edgusgrclnbfin 47842 | The size of the closed nei... |
| clnbusgrfi 47843 | The closed neighborhood of... |
| clnbfiusgrfi 47844 | The closed neighborhood of... |
| clnbgrlevtx 47845 | The size of the closed nei... |
| dfsclnbgr2 47846 | Alternate definition of th... |
| sclnbgrel 47847 | Characterization of a memb... |
| sclnbgrelself 47848 | A vertex ` N ` is a member... |
| sclnbgrisvtx 47849 | Every member ` X ` of the ... |
| dfclnbgr5 47850 | Alternate definition of th... |
| dfnbgr5 47851 | Alternate definition of th... |
| dfnbgrss 47852 | Subset chain for different... |
| dfvopnbgr2 47853 | Alternate definition of th... |
| vopnbgrel 47854 | Characterization of a memb... |
| vopnbgrelself 47855 | A vertex ` N ` is a member... |
| dfclnbgr6 47856 | Alternate definition of th... |
| dfnbgr6 47857 | Alternate definition of th... |
| dfsclnbgr6 47858 | Alternate definition of a ... |
| dfnbgrss2 47859 | Subset chain for different... |
| isisubgr 47862 | The subgraph induced by a ... |
| isubgriedg 47863 | The edges of an induced su... |
| isubgrvtxuhgr 47864 | The subgraph induced by th... |
| isubgredgss 47865 | The edges of an induced su... |
| isubgredg 47866 | An edge of an induced subg... |
| isubgrvtx 47867 | The vertices of an induced... |
| isubgruhgr 47868 | An induced subgraph of a h... |
| isubgrsubgr 47869 | An induced subgraph of a h... |
| isubgrupgr 47870 | An induced subgraph of a p... |
| isubgrumgr 47871 | An induced subgraph of a m... |
| isubgrusgr 47872 | An induced subgraph of a s... |
| isubgr0uhgr 47873 | The subgraph induced by an... |
| grimfn 47879 | The graph isomorphism func... |
| grimdmrel 47880 | The domain of the graph is... |
| isgrim 47882 | An isomorphism of graphs i... |
| grimprop 47883 | Properties of an isomorphi... |
| grimf1o 47884 | An isomorphism of graphs i... |
| grimidvtxedg 47885 | The identity relation rest... |
| grimid 47886 | The identity relation rest... |
| grimuhgr 47887 | If there is a graph isomor... |
| grimcnv 47888 | The converse of a graph is... |
| grimco 47889 | The composition of graph i... |
| uhgrimedgi 47890 | An isomorphism between gra... |
| uhgrimedg 47891 | An isomorphism between gra... |
| uhgrimprop 47892 | An isomorphism between hyp... |
| isuspgrim0lem 47893 | An isomorphism of simple p... |
| isuspgrim0 47894 | An isomorphism of simple p... |
| isuspgrimlem 47895 | Lemma for ~ isuspgrim . (... |
| isuspgrim 47896 | A class is an isomorphism ... |
| upgrimwlklem1 47897 | Lemma 1 for ~ upgrimwlk an... |
| upgrimwlklem2 47898 | Lemma 2 for ~ upgrimwlk . ... |
| upgrimwlklem3 47899 | Lemma 3 for ~ upgrimwlk . ... |
| upgrimwlklem4 47900 | Lemma 4 for ~ upgrimwlk . ... |
| upgrimwlklem5 47901 | Lemma 5 for ~ upgrimwlk . ... |
| upgrimwlk 47902 | Graph isomorphisms between... |
| upgrimwlklen 47903 | Graph isomorphisms between... |
| upgrimtrlslem1 47904 | Lemma 1 for ~ upgrimtrls .... |
| upgrimtrlslem2 47905 | Lemma 2 for ~ upgrimtrls .... |
| upgrimtrls 47906 | Graph isomorphisms between... |
| upgrimpthslem1 47907 | Lemma 1 for ~ upgrimpths .... |
| upgrimpthslem2 47908 | Lemma 2 for ~ upgrimpths .... |
| upgrimpths 47909 | Graph isomorphisms between... |
| upgrimspths 47910 | Graph isomorphisms between... |
| upgrimcycls 47911 | Graph isomorphisms between... |
| brgric 47912 | The relation "is isomorphi... |
| brgrici 47913 | Prove that two graphs are ... |
| gricrcl 47914 | Reverse closure of the "is... |
| dfgric2 47915 | Alternate, explicit defini... |
| gricbri 47916 | Implications of two graphs... |
| gricushgr 47917 | The "is isomorphic to" rel... |
| gricuspgr 47918 | The "is isomorphic to" rel... |
| gricrel 47919 | The "is isomorphic to" rel... |
| gricref 47920 | Graph isomorphism is refle... |
| gricsym 47921 | Graph isomorphism is symme... |
| gricsymb 47922 | Graph isomorphism is symme... |
| grictr 47923 | Graph isomorphism is trans... |
| gricer 47924 | Isomorphism is an equivale... |
| gricen 47925 | Isomorphic graphs have equ... |
| opstrgric 47926 | A graph represented as an ... |
| ushggricedg 47927 | A simple hypergraph (with ... |
| cycldlenngric 47928 | Two simple pseudographs ar... |
| isubgrgrim 47929 | Isomorphic subgraphs induc... |
| uhgrimisgrgriclem 47930 | Lemma for ~ uhgrimisgrgric... |
| uhgrimisgrgric 47931 | For isomorphic hypergraphs... |
| clnbgrisubgrgrim 47932 | Isomorphic subgraphs induc... |
| clnbgrgrimlem 47933 | Lemma for ~ clnbgrgrim : ... |
| clnbgrgrim 47934 | Graph isomorphisms between... |
| grimedg 47935 | Graph isomorphisms map edg... |
| grtriproplem 47938 | Lemma for ~ grtriprop . (... |
| grtri 47939 | The triangles in a graph. ... |
| grtriprop 47940 | The properties of a triang... |
| grtrif1o 47941 | Any bijection onto a trian... |
| isgrtri 47942 | A triangle in a graph. (C... |
| grtrissvtx 47943 | A triangle is a subset of ... |
| grtriclwlk3 47944 | A triangle induces a close... |
| cycl3grtrilem 47945 | Lemma for ~ cycl3grtri . ... |
| cycl3grtri 47946 | The vertices of a cycle of... |
| grtrimap 47947 | Conditions for mapping tri... |
| grimgrtri 47948 | Graph isomorphisms map tri... |
| usgrgrtrirex 47949 | Conditions for a simple gr... |
| stgrfv 47952 | The star graph S_N. (Contr... |
| stgrvtx 47953 | The vertices of the star g... |
| stgriedg 47954 | The indexed edges of the s... |
| stgredg 47955 | The edges of the star grap... |
| stgredgel 47956 | An edge of the star graph ... |
| stgredgiun 47957 | The edges of the star grap... |
| stgrusgra 47958 | The star graph S_N is a si... |
| stgr0 47959 | The star graph S_0 consist... |
| stgr1 47960 | The star graph S_1 consist... |
| stgrvtx0 47961 | The center ("internal node... |
| stgrorder 47962 | The order of a star graph ... |
| stgrnbgr0 47963 | All vertices of a star gra... |
| stgrclnbgr0 47964 | All vertices of a star gra... |
| isubgr3stgrlem1 47965 | Lemma 1 for ~ isubgr3stgr ... |
| isubgr3stgrlem2 47966 | Lemma 2 for ~ isubgr3stgr ... |
| isubgr3stgrlem3 47967 | Lemma 3 for ~ isubgr3stgr ... |
| isubgr3stgrlem4 47968 | Lemma 4 for ~ isubgr3stgr ... |
| isubgr3stgrlem5 47969 | Lemma 5 for ~ isubgr3stgr ... |
| isubgr3stgrlem6 47970 | Lemma 6 for ~ isubgr3stgr ... |
| isubgr3stgrlem7 47971 | Lemma 7 for ~ isubgr3stgr ... |
| isubgr3stgrlem8 47972 | Lemma 8 for ~ isubgr3stgr ... |
| isubgr3stgrlem9 47973 | Lemma 9 for ~ isubgr3stgr ... |
| isubgr3stgr 47974 | If a vertex of a simple gr... |
| grlimfn 47978 | The graph local isomorphis... |
| grlimdmrel 47979 | The domain of the graph lo... |
| isgrlim 47981 | A local isomorphism of gra... |
| isgrlim2 47982 | A local isomorphism of gra... |
| grlimprop 47983 | Properties of a local isom... |
| grlimf1o 47984 | A local isomorphism of gra... |
| grlimprop2 47985 | Properties of a local isom... |
| uhgrimgrlim 47986 | An isomorphism of hypergra... |
| uspgrlimlem1 47987 | Lemma 1 for ~ uspgrlim . ... |
| uspgrlimlem2 47988 | Lemma 2 for ~ uspgrlim . ... |
| uspgrlimlem3 47989 | Lemma 3 for ~ uspgrlim . ... |
| uspgrlimlem4 47990 | Lemma 4 for ~ uspgrlim . ... |
| uspgrlim 47991 | A local isomorphism of sim... |
| usgrlimprop 47992 | Properties of a local isom... |
| grlimgrtrilem1 47993 | Lemma 3 for ~ grlimgrtri .... |
| grlimgrtrilem2 47994 | Lemma 3 for ~ grlimgrtri .... |
| grlimgrtri 47995 | Local isomorphisms between... |
| brgrlic 47996 | The relation "is locally i... |
| brgrilci 47997 | Prove that two graphs are ... |
| grlicrel 47998 | The "is locally isomorphic... |
| grlicrcl 47999 | Reverse closure of the "is... |
| dfgrlic2 48000 | Alternate, explicit defini... |
| grilcbri 48001 | Implications of two graphs... |
| dfgrlic3 48002 | Alternate, explicit defini... |
| grilcbri2 48003 | Implications of two graphs... |
| grlicref 48004 | Graph local isomorphism is... |
| grlicsym 48005 | Graph local isomorphism is... |
| grlicsymb 48006 | Graph local isomorphism is... |
| grlictr 48007 | Graph local isomorphism is... |
| grlicer 48008 | Local isomorphism is an eq... |
| grlicen 48009 | Locally isomorphic graphs ... |
| gricgrlic 48010 | Isomorphic hypergraphs are... |
| clnbgr3stgrgrlic 48011 | If all (closed) neighborho... |
| usgrexmpl1lem 48012 | Lemma for ~ usgrexmpl1 . ... |
| usgrexmpl1 48013 | ` G ` is a simple graph of... |
| usgrexmpl1vtx 48014 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpl1edg 48015 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl1tri 48016 | ` G ` contains a triangle ... |
| usgrexmpl2lem 48017 | Lemma for ~ usgrexmpl2 . ... |
| usgrexmpl2 48018 | ` G ` is a simple graph of... |
| usgrexmpl2vtx 48019 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpl2edg 48020 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl2nblem 48021 | Lemma for ~ usgrexmpl2nb0 ... |
| usgrexmpl2nb0 48022 | The neighborhood of the fi... |
| usgrexmpl2nb1 48023 | The neighborhood of the se... |
| usgrexmpl2nb2 48024 | The neighborhood of the th... |
| usgrexmpl2nb3 48025 | The neighborhood of the fo... |
| usgrexmpl2nb4 48026 | The neighborhood of the fi... |
| usgrexmpl2nb5 48027 | The neighborhood of the si... |
| usgrexmpl2trifr 48028 | ` G ` is triangle-free. (... |
| usgrexmpl12ngric 48029 | The graphs ` H ` and ` G `... |
| usgrexmpl12ngrlic 48030 | The graphs ` H ` and ` G `... |
| gpgov 48033 | The generalized Petersen g... |
| gpgvtx 48034 | The vertices of the genera... |
| gpgiedg 48035 | The indexed edges of the g... |
| gpgedg 48036 | The edges of the generaliz... |
| gpgiedgdmellem 48037 | Lemma for ~ gpgiedgdmel an... |
| gpgvtxel 48038 | A vertex in a generalized ... |
| gpgvtxel2 48039 | The second component of a ... |
| gpgiedgdmel 48040 | An index of edges of the g... |
| gpgedgel 48041 | An edge in a generalized P... |
| gpgprismgriedgdmel 48042 | An index of edges of the g... |
| gpgprismgriedgdmss 48043 | A subset of the index of e... |
| gpgvtx0 48044 | The outside vertices in a ... |
| gpgvtx1 48045 | The inside vertices in a g... |
| opgpgvtx 48046 | A vertex in a generalized ... |
| gpgusgralem 48047 | Lemma for ~ gpgusgra . (C... |
| gpgusgra 48048 | The generalized Petersen g... |
| gpgprismgrusgra 48049 | The generalized Petersen g... |
| gpgorder 48050 | The order of the generaliz... |
| gpg5order 48051 | The order of a generalized... |
| gpgedgvtx0 48052 | The edges starting at an o... |
| gpgedgvtx1 48053 | The edges starting at an i... |
| gpgvtxedg0 48054 | The edges starting at an o... |
| gpgvtxedg1 48055 | The edges starting at an i... |
| gpgedgiov 48056 | The edges of the generaliz... |
| gpgedg2ov 48057 | The edges of the generaliz... |
| gpgedg2iv 48058 | The edges of the generaliz... |
| gpg5nbgrvtx03starlem1 48059 | Lemma 1 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx03starlem2 48060 | Lemma 2 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx03starlem3 48061 | Lemma 3 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx13starlem1 48062 | Lemma 1 for ~ gpg5nbgr3sta... |
| gpg5nbgrvtx13starlem2 48063 | Lemma 2 for ~ gpg5nbgr3sta... |
| gpg5nbgrvtx13starlem3 48064 | Lemma 3 for ~ gpg5nbgr3sta... |
| gpgnbgrvtx0 48065 | The (open) neighborhood of... |
| gpgnbgrvtx1 48066 | The (open) neighborhood of... |
| gpg3nbgrvtx0 48067 | In a generalized Petersen ... |
| gpg3nbgrvtx0ALT 48068 | In a generalized Petersen ... |
| gpg3nbgrvtx1 48069 | In a generalized Petersen ... |
| gpgcubic 48070 | Every generalized Petersen... |
| gpg5nbgrvtx03star 48071 | In a generalized Petersen ... |
| gpg5nbgr3star 48072 | In a generalized Petersen ... |
| gpgvtxdg3 48073 | Every vertex in a generali... |
| gpg3kgrtriexlem1 48074 | Lemma 1 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem2 48075 | Lemma 2 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem3 48076 | Lemma 3 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem4 48077 | Lemma 4 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem5 48078 | Lemma 5 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem6 48079 | Lemma 6 for ~ gpg3kgrtriex... |
| gpg3kgrtriex 48080 | All generalized Petersen g... |
| gpg5gricstgr3 48081 | Each closed neighborhood i... |
| pglem 48082 | Lemma for theorems about P... |
| pgjsgr 48083 | A Petersen graph is a simp... |
| gpg5grlic 48084 | The two generalized Peters... |
| gpgprismgr4cycllem1 48085 | Lemma 1 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem2 48086 | Lemma 2 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem3 48087 | Lemma 3 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem4 48088 | Lemma 4 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem5 48089 | Lemma 5 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem6 48090 | Lemma 6 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem7 48091 | Lemma 7 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem8 48092 | Lemma 8 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem9 48093 | Lemma 9 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem10 48094 | Lemma 10 for ~ gpgprismgr4... |
| gpgprismgr4cycllem11 48095 | Lemma 11 for ~ gpgprismgr4... |
| gpgprismgr4cycl0 48096 | The generalized Petersen g... |
| gpgprismgr4cyclex 48097 | The generalized Petersen g... |
| pgnioedg1 48098 | An inside and an outside v... |
| pgnioedg2 48099 | An inside and an outside v... |
| pgnioedg3 48100 | An inside and an outside v... |
| pgnioedg4 48101 | An inside and an outside v... |
| pgnioedg5 48102 | An inside and an outside v... |
| pgnbgreunbgrlem1 48103 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem1 48104 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem2 48105 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem3 48106 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2 48107 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem3 48108 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem4 48109 | Lemma 4 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem1 48110 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem2 48111 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem3 48112 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5 48113 | Lemma 5 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem6 48114 | Lemma 6 for ~ pgnbgreunbgr... |
| pgnbgreunbgr 48115 | In a Petersen graph, two d... |
| pgn4cyclex 48116 | A cycle in a Petersen grap... |
| pg4cyclnex 48117 | In the Petersen graph G(5,... |
| gpg5ngric 48118 | The two generalized Peters... |
| lgricngricex 48119 | There are two different lo... |
| 1hegrlfgr 48120 | A graph ` G ` with one hyp... |
| upwlksfval 48123 | The set of simple walks (i... |
| isupwlk 48124 | Properties of a pair of fu... |
| isupwlkg 48125 | Generalization of ~ isupwl... |
| upwlkbprop 48126 | Basic properties of a simp... |
| upwlkwlk 48127 | A simple walk is a walk. ... |
| upgrwlkupwlk 48128 | In a pseudograph, a walk i... |
| upgrwlkupwlkb 48129 | In a pseudograph, the defi... |
| upgrisupwlkALT 48130 | Alternate proof of ~ upgri... |
| upgredgssspr 48131 | The set of edges of a pseu... |
| uspgropssxp 48132 | The set ` G ` of "simple p... |
| uspgrsprfv 48133 | The value of the function ... |
| uspgrsprf 48134 | The mapping ` F ` is a fun... |
| uspgrsprf1 48135 | The mapping ` F ` is a one... |
| uspgrsprfo 48136 | The mapping ` F ` is a fun... |
| uspgrsprf1o 48137 | The mapping ` F ` is a bij... |
| uspgrex 48138 | The class ` G ` of all "si... |
| uspgrbispr 48139 | There is a bijection betwe... |
| uspgrspren 48140 | The set ` G ` of the "simp... |
| uspgrymrelen 48141 | The set ` G ` of the "simp... |
| uspgrbisymrel 48142 | There is a bijection betwe... |
| uspgrbisymrelALT 48143 | Alternate proof of ~ uspgr... |
| ovn0dmfun 48144 | If a class operation value... |
| xpsnopab 48145 | A Cartesian product with a... |
| xpiun 48146 | A Cartesian product expres... |
| ovn0ssdmfun 48147 | If a class' operation valu... |
| fnxpdmdm 48148 | The domain of the domain o... |
| cnfldsrngbas 48149 | The base set of a subring ... |
| cnfldsrngadd 48150 | The group addition operati... |
| cnfldsrngmul 48151 | The ring multiplication op... |
| plusfreseq 48152 | If the empty set is not co... |
| mgmplusfreseq 48153 | If the empty set is not co... |
| 0mgm 48154 | A set with an empty base s... |
| opmpoismgm 48155 | A structure with a group a... |
| copissgrp 48156 | A structure with a constan... |
| copisnmnd 48157 | A structure with a constan... |
| 0nodd 48158 | 0 is not an odd integer. ... |
| 1odd 48159 | 1 is an odd integer. (Con... |
| 2nodd 48160 | 2 is not an odd integer. ... |
| oddibas 48161 | Lemma 1 for ~ oddinmgm : ... |
| oddiadd 48162 | Lemma 2 for ~ oddinmgm : ... |
| oddinmgm 48163 | The structure of all odd i... |
| nnsgrpmgm 48164 | The structure of positive ... |
| nnsgrp 48165 | The structure of positive ... |
| nnsgrpnmnd 48166 | The structure of positive ... |
| nn0mnd 48167 | The set of nonnegative int... |
| gsumsplit2f 48168 | Split a group sum into two... |
| gsumdifsndf 48169 | Extract a summand from a f... |
| gsumfsupp 48170 | A group sum of a family ca... |
| iscllaw 48177 | The predicate "is a closed... |
| iscomlaw 48178 | The predicate "is a commut... |
| clcllaw 48179 | Closure of a closed operat... |
| isasslaw 48180 | The predicate "is an assoc... |
| asslawass 48181 | Associativity of an associ... |
| mgmplusgiopALT 48182 | Slot 2 (group operation) o... |
| sgrpplusgaopALT 48183 | Slot 2 (group operation) o... |
| intopval 48190 | The internal (binary) oper... |
| intop 48191 | An internal (binary) opera... |
| clintopval 48192 | The closed (internal binar... |
| assintopval 48193 | The associative (closed in... |
| assintopmap 48194 | The associative (closed in... |
| isclintop 48195 | The predicate "is a closed... |
| clintop 48196 | A closed (internal binary)... |
| assintop 48197 | An associative (closed int... |
| isassintop 48198 | The predicate "is an assoc... |
| clintopcllaw 48199 | The closure law holds for ... |
| assintopcllaw 48200 | The closure low holds for ... |
| assintopasslaw 48201 | The associative low holds ... |
| assintopass 48202 | An associative (closed int... |
| ismgmALT 48211 | The predicate "is a magma"... |
| iscmgmALT 48212 | The predicate "is a commut... |
| issgrpALT 48213 | The predicate "is a semigr... |
| iscsgrpALT 48214 | The predicate "is a commut... |
| mgm2mgm 48215 | Equivalence of the two def... |
| sgrp2sgrp 48216 | Equivalence of the two def... |
| lmod0rng 48217 | If the scalar ring of a mo... |
| nzrneg1ne0 48218 | The additive inverse of th... |
| lidldomn1 48219 | If a (left) ideal (which i... |
| lidlabl 48220 | A (left) ideal of a ring i... |
| lidlrng 48221 | A (left) ideal of a ring i... |
| zlidlring 48222 | The zero (left) ideal of a... |
| uzlidlring 48223 | Only the zero (left) ideal... |
| lidldomnnring 48224 | A (left) ideal of a domain... |
| 0even 48225 | 0 is an even integer. (Co... |
| 1neven 48226 | 1 is not an even integer. ... |
| 2even 48227 | 2 is an even integer. (Co... |
| 2zlidl 48228 | The even integers are a (l... |
| 2zrng 48229 | The ring of integers restr... |
| 2zrngbas 48230 | The base set of R is the s... |
| 2zrngadd 48231 | The group addition operati... |
| 2zrng0 48232 | The additive identity of R... |
| 2zrngamgm 48233 | R is an (additive) magma. ... |
| 2zrngasgrp 48234 | R is an (additive) semigro... |
| 2zrngamnd 48235 | R is an (additive) monoid.... |
| 2zrngacmnd 48236 | R is a commutative (additi... |
| 2zrngagrp 48237 | R is an (additive) group. ... |
| 2zrngaabl 48238 | R is an (additive) abelian... |
| 2zrngmul 48239 | The ring multiplication op... |
| 2zrngmmgm 48240 | R is a (multiplicative) ma... |
| 2zrngmsgrp 48241 | R is a (multiplicative) se... |
| 2zrngALT 48242 | The ring of integers restr... |
| 2zrngnmlid 48243 | R has no multiplicative (l... |
| 2zrngnmrid 48244 | R has no multiplicative (r... |
| 2zrngnmlid2 48245 | R has no multiplicative (l... |
| 2zrngnring 48246 | R is not a unital ring. (... |
| cznrnglem 48247 | Lemma for ~ cznrng : The ... |
| cznabel 48248 | The ring constructed from ... |
| cznrng 48249 | The ring constructed from ... |
| cznnring 48250 | The ring constructed from ... |
| rngcvalALTV 48253 | Value of the category of n... |
| rngcbasALTV 48254 | Set of objects of the cate... |
| rngchomfvalALTV 48255 | Set of arrows of the categ... |
| rngchomALTV 48256 | Set of arrows of the categ... |
| elrngchomALTV 48257 | A morphism of non-unital r... |
| rngccofvalALTV 48258 | Composition in the categor... |
| rngccoALTV 48259 | Composition in the categor... |
| rngccatidALTV 48260 | Lemma for ~ rngccatALTV . ... |
| rngccatALTV 48261 | The category of non-unital... |
| rngcidALTV 48262 | The identity arrow in the ... |
| rngcsectALTV 48263 | A section in the category ... |
| rngcinvALTV 48264 | An inverse in the category... |
| rngcisoALTV 48265 | An isomorphism in the cate... |
| rngchomffvalALTV 48266 | The value of the functiona... |
| rngchomrnghmresALTV 48267 | The value of the functiona... |
| rngcrescrhmALTV 48268 | The category of non-unital... |
| rhmsubcALTVlem1 48269 | Lemma 1 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem2 48270 | Lemma 2 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem3 48271 | Lemma 3 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem4 48272 | Lemma 4 for ~ rhmsubcALTV ... |
| rhmsubcALTV 48273 | According to ~ df-subc , t... |
| rhmsubcALTVcat 48274 | The restriction of the cat... |
| ringcvalALTV 48277 | Value of the category of r... |
| funcringcsetcALTV2lem1 48278 | Lemma 1 for ~ funcringcset... |
| funcringcsetcALTV2lem2 48279 | Lemma 2 for ~ funcringcset... |
| funcringcsetcALTV2lem3 48280 | Lemma 3 for ~ funcringcset... |
| funcringcsetcALTV2lem4 48281 | Lemma 4 for ~ funcringcset... |
| funcringcsetcALTV2lem5 48282 | Lemma 5 for ~ funcringcset... |
| funcringcsetcALTV2lem6 48283 | Lemma 6 for ~ funcringcset... |
| funcringcsetcALTV2lem7 48284 | Lemma 7 for ~ funcringcset... |
| funcringcsetcALTV2lem8 48285 | Lemma 8 for ~ funcringcset... |
| funcringcsetcALTV2lem9 48286 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV2 48287 | The "natural forgetful fun... |
| ringcbasALTV 48288 | Set of objects of the cate... |
| ringchomfvalALTV 48289 | Set of arrows of the categ... |
| ringchomALTV 48290 | Set of arrows of the categ... |
| elringchomALTV 48291 | A morphism of rings is a f... |
| ringccofvalALTV 48292 | Composition in the categor... |
| ringccoALTV 48293 | Composition in the categor... |
| ringccatidALTV 48294 | Lemma for ~ ringccatALTV .... |
| ringccatALTV 48295 | The category of rings is a... |
| ringcidALTV 48296 | The identity arrow in the ... |
| ringcsectALTV 48297 | A section in the category ... |
| ringcinvALTV 48298 | An inverse in the category... |
| ringcisoALTV 48299 | An isomorphism in the cate... |
| ringcbasbasALTV 48300 | An element of the base set... |
| funcringcsetclem1ALTV 48301 | Lemma 1 for ~ funcringcset... |
| funcringcsetclem2ALTV 48302 | Lemma 2 for ~ funcringcset... |
| funcringcsetclem3ALTV 48303 | Lemma 3 for ~ funcringcset... |
| funcringcsetclem4ALTV 48304 | Lemma 4 for ~ funcringcset... |
| funcringcsetclem5ALTV 48305 | Lemma 5 for ~ funcringcset... |
| funcringcsetclem6ALTV 48306 | Lemma 6 for ~ funcringcset... |
| funcringcsetclem7ALTV 48307 | Lemma 7 for ~ funcringcset... |
| funcringcsetclem8ALTV 48308 | Lemma 8 for ~ funcringcset... |
| funcringcsetclem9ALTV 48309 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV 48310 | The "natural forgetful fun... |
| srhmsubcALTVlem1 48311 | Lemma 1 for ~ srhmsubcALTV... |
| srhmsubcALTVlem2 48312 | Lemma 2 for ~ srhmsubcALTV... |
| srhmsubcALTV 48313 | According to ~ df-subc , t... |
| sringcatALTV 48314 | The restriction of the cat... |
| crhmsubcALTV 48315 | According to ~ df-subc , t... |
| cringcatALTV 48316 | The restriction of the cat... |
| drhmsubcALTV 48317 | According to ~ df-subc , t... |
| drngcatALTV 48318 | The restriction of the cat... |
| fldcatALTV 48319 | The restriction of the cat... |
| fldcALTV 48320 | The restriction of the cat... |
| fldhmsubcALTV 48321 | According to ~ df-subc , t... |
| eliunxp2 48322 | Membership in a union of C... |
| mpomptx2 48323 | Express a two-argument fun... |
| cbvmpox2 48324 | Rule to change the bound v... |
| dmmpossx2 48325 | The domain of a mapping is... |
| mpoexxg2 48326 | Existence of an operation ... |
| ovmpordxf 48327 | Value of an operation give... |
| ovmpordx 48328 | Value of an operation give... |
| ovmpox2 48329 | The value of an operation ... |
| fdmdifeqresdif 48330 | The restriction of a condi... |
| ofaddmndmap 48331 | The function operation app... |
| mapsnop 48332 | A singleton of an ordered ... |
| fprmappr 48333 | A function with a domain o... |
| mapprop 48334 | An unordered pair containi... |
| ztprmneprm 48335 | A prime is not an integer ... |
| 2t6m3t4e0 48336 | 2 times 6 minus 3 times 4 ... |
| ssnn0ssfz 48337 | For any finite subset of `... |
| nn0sumltlt 48338 | If the sum of two nonnegat... |
| bcpascm1 48339 | Pascal's rule for the bino... |
| altgsumbc 48340 | The sum of binomial coeffi... |
| altgsumbcALT 48341 | Alternate proof of ~ altgs... |
| zlmodzxzlmod 48342 | The ` ZZ `-module ` ZZ X. ... |
| zlmodzxzel 48343 | An element of the (base se... |
| zlmodzxz0 48344 | The ` 0 ` of the ` ZZ `-mo... |
| zlmodzxzscm 48345 | The scalar multiplication ... |
| zlmodzxzadd 48346 | The addition of the ` ZZ `... |
| zlmodzxzsubm 48347 | The subtraction of the ` Z... |
| zlmodzxzsub 48348 | The subtraction of the ` Z... |
| mgpsumunsn 48349 | Extract a summand/factor f... |
| mgpsumz 48350 | If the group sum for the m... |
| mgpsumn 48351 | If the group sum for the m... |
| exple2lt6 48352 | A nonnegative integer to t... |
| pgrple2abl 48353 | Every symmetric group on a... |
| pgrpgt2nabl 48354 | Every symmetric group on a... |
| invginvrid 48355 | Identity for a multiplicat... |
| rmsupp0 48356 | The support of a mapping o... |
| domnmsuppn0 48357 | The support of a mapping o... |
| rmsuppss 48358 | The support of a mapping o... |
| scmsuppss 48359 | The support of a mapping o... |
| rmsuppfi 48360 | The support of a mapping o... |
| rmfsupp 48361 | A mapping of a multiplicat... |
| scmsuppfi 48362 | The support of a mapping o... |
| scmfsupp 48363 | A mapping of a scalar mult... |
| suppmptcfin 48364 | The support of a mapping w... |
| mptcfsupp 48365 | A mapping with value 0 exc... |
| fsuppmptdmf 48366 | A mapping with a finite do... |
| lmodvsmdi 48367 | Multiple distributive law ... |
| gsumlsscl 48368 | Closure of a group sum in ... |
| assaascl0 48369 | The scalar 0 embedded into... |
| assaascl1 48370 | The scalar 1 embedded into... |
| ply1vr1smo 48371 | The variable in a polynomi... |
| ply1sclrmsm 48372 | The ring multiplication of... |
| coe1id 48373 | Coefficient vector of the ... |
| coe1sclmulval 48374 | The value of the coefficie... |
| ply1mulgsumlem1 48375 | Lemma 1 for ~ ply1mulgsum ... |
| ply1mulgsumlem2 48376 | Lemma 2 for ~ ply1mulgsum ... |
| ply1mulgsumlem3 48377 | Lemma 3 for ~ ply1mulgsum ... |
| ply1mulgsumlem4 48378 | Lemma 4 for ~ ply1mulgsum ... |
| ply1mulgsum 48379 | The product of two polynom... |
| evl1at0 48380 | Polynomial evaluation for ... |
| evl1at1 48381 | Polynomial evaluation for ... |
| linply1 48382 | A term of the form ` x - C... |
| lineval 48383 | A term of the form ` x - C... |
| linevalexample 48384 | The polynomial ` x - 3 ` o... |
| dmatALTval 48389 | The algebra of ` N ` x ` N... |
| dmatALTbas 48390 | The base set of the algebr... |
| dmatALTbasel 48391 | An element of the base set... |
| dmatbas 48392 | The set of all ` N ` x ` N... |
| lincop 48397 | A linear combination as op... |
| lincval 48398 | The value of a linear comb... |
| dflinc2 48399 | Alternative definition of ... |
| lcoop 48400 | A linear combination as op... |
| lcoval 48401 | The value of a linear comb... |
| lincfsuppcl 48402 | A linear combination of ve... |
| linccl 48403 | A linear combination of ve... |
| lincval0 48404 | The value of an empty line... |
| lincvalsng 48405 | The linear combination ove... |
| lincvalsn 48406 | The linear combination ove... |
| lincvalpr 48407 | The linear combination ove... |
| lincval1 48408 | The linear combination ove... |
| lcosn0 48409 | Properties of a linear com... |
| lincvalsc0 48410 | The linear combination whe... |
| lcoc0 48411 | Properties of a linear com... |
| linc0scn0 48412 | If a set contains the zero... |
| lincdifsn 48413 | A vector is a linear combi... |
| linc1 48414 | A vector is a linear combi... |
| lincellss 48415 | A linear combination of a ... |
| lco0 48416 | The set of empty linear co... |
| lcoel0 48417 | The zero vector is always ... |
| lincsum 48418 | The sum of two linear comb... |
| lincscm 48419 | A linear combinations mult... |
| lincsumcl 48420 | The sum of two linear comb... |
| lincscmcl 48421 | The multiplication of a li... |
| lincsumscmcl 48422 | The sum of a linear combin... |
| lincolss 48423 | According to the statement... |
| ellcoellss 48424 | Every linear combination o... |
| lcoss 48425 | A set of vectors of a modu... |
| lspsslco 48426 | Lemma for ~ lspeqlco . (C... |
| lcosslsp 48427 | Lemma for ~ lspeqlco . (C... |
| lspeqlco 48428 | Equivalence of a _span_ of... |
| rellininds 48432 | The class defining the rel... |
| linindsv 48434 | The classes of the module ... |
| islininds 48435 | The property of being a li... |
| linindsi 48436 | The implications of being ... |
| linindslinci 48437 | The implications of being ... |
| islinindfis 48438 | The property of being a li... |
| islinindfiss 48439 | The property of being a li... |
| linindscl 48440 | A linearly independent set... |
| lindepsnlininds 48441 | A linearly dependent subse... |
| islindeps 48442 | The property of being a li... |
| lincext1 48443 | Property 1 of an extension... |
| lincext2 48444 | Property 2 of an extension... |
| lincext3 48445 | Property 3 of an extension... |
| lindslinindsimp1 48446 | Implication 1 for ~ lindsl... |
| lindslinindimp2lem1 48447 | Lemma 1 for ~ lindslininds... |
| lindslinindimp2lem2 48448 | Lemma 2 for ~ lindslininds... |
| lindslinindimp2lem3 48449 | Lemma 3 for ~ lindslininds... |
| lindslinindimp2lem4 48450 | Lemma 4 for ~ lindslininds... |
| lindslinindsimp2lem5 48451 | Lemma 5 for ~ lindslininds... |
| lindslinindsimp2 48452 | Implication 2 for ~ lindsl... |
| lindslininds 48453 | Equivalence of definitions... |
| linds0 48454 | The empty set is always a ... |
| el0ldep 48455 | A set containing the zero ... |
| el0ldepsnzr 48456 | A set containing the zero ... |
| lindsrng01 48457 | Any subset of a module is ... |
| lindszr 48458 | Any subset of a module ove... |
| snlindsntorlem 48459 | Lemma for ~ snlindsntor . ... |
| snlindsntor 48460 | A singleton is linearly in... |
| ldepsprlem 48461 | Lemma for ~ ldepspr . (Co... |
| ldepspr 48462 | If a vector is a scalar mu... |
| lincresunit3lem3 48463 | Lemma 3 for ~ lincresunit3... |
| lincresunitlem1 48464 | Lemma 1 for properties of ... |
| lincresunitlem2 48465 | Lemma for properties of a ... |
| lincresunit1 48466 | Property 1 of a specially ... |
| lincresunit2 48467 | Property 2 of a specially ... |
| lincresunit3lem1 48468 | Lemma 1 for ~ lincresunit3... |
| lincresunit3lem2 48469 | Lemma 2 for ~ lincresunit3... |
| lincresunit3 48470 | Property 3 of a specially ... |
| lincreslvec3 48471 | Property 3 of a specially ... |
| islindeps2 48472 | Conditions for being a lin... |
| islininds2 48473 | Implication of being a lin... |
| isldepslvec2 48474 | Alternative definition of ... |
| lindssnlvec 48475 | A singleton not containing... |
| lmod1lem1 48476 | Lemma 1 for ~ lmod1 . (Co... |
| lmod1lem2 48477 | Lemma 2 for ~ lmod1 . (Co... |
| lmod1lem3 48478 | Lemma 3 for ~ lmod1 . (Co... |
| lmod1lem4 48479 | Lemma 4 for ~ lmod1 . (Co... |
| lmod1lem5 48480 | Lemma 5 for ~ lmod1 . (Co... |
| lmod1 48481 | The (smallest) structure r... |
| lmod1zr 48482 | The (smallest) structure r... |
| lmod1zrnlvec 48483 | There is a (left) module (... |
| lmodn0 48484 | Left modules exist. (Cont... |
| zlmodzxzequa 48485 | Example of an equation wit... |
| zlmodzxznm 48486 | Example of a linearly depe... |
| zlmodzxzldeplem 48487 | A and B are not equal. (C... |
| zlmodzxzequap 48488 | Example of an equation wit... |
| zlmodzxzldeplem1 48489 | Lemma 1 for ~ zlmodzxzldep... |
| zlmodzxzldeplem2 48490 | Lemma 2 for ~ zlmodzxzldep... |
| zlmodzxzldeplem3 48491 | Lemma 3 for ~ zlmodzxzldep... |
| zlmodzxzldeplem4 48492 | Lemma 4 for ~ zlmodzxzldep... |
| zlmodzxzldep 48493 | { A , B } is a linearly de... |
| ldepsnlinclem1 48494 | Lemma 1 for ~ ldepsnlinc .... |
| ldepsnlinclem2 48495 | Lemma 2 for ~ ldepsnlinc .... |
| lvecpsslmod 48496 | The class of all (left) ve... |
| ldepsnlinc 48497 | The reverse implication of... |
| ldepslinc 48498 | For (left) vector spaces, ... |
| suppdm 48499 | If the range of a function... |
| eluz2cnn0n1 48500 | An integer greater than 1 ... |
| divge1b 48501 | The ratio of a real number... |
| divgt1b 48502 | The ratio of a real number... |
| ltsubaddb 48503 | Equivalence for the "less ... |
| ltsubsubb 48504 | Equivalence for the "less ... |
| ltsubadd2b 48505 | Equivalence for the "less ... |
| divsub1dir 48506 | Distribution of division o... |
| expnegico01 48507 | An integer greater than 1 ... |
| elfzolborelfzop1 48508 | An element of a half-open ... |
| pw2m1lepw2m1 48509 | 2 to the power of a positi... |
| zgtp1leeq 48510 | If an integer is between a... |
| flsubz 48511 | An integer can be moved in... |
| nn0onn0ex 48512 | For each odd nonnegative i... |
| nn0enn0ex 48513 | For each even nonnegative ... |
| nnennex 48514 | For each even positive int... |
| nneop 48515 | A positive integer is even... |
| nneom 48516 | A positive integer is even... |
| nn0eo 48517 | A nonnegative integer is e... |
| nnpw2even 48518 | 2 to the power of a positi... |
| zefldiv2 48519 | The floor of an even integ... |
| zofldiv2 48520 | The floor of an odd intege... |
| nn0ofldiv2 48521 | The floor of an odd nonneg... |
| flnn0div2ge 48522 | The floor of a positive in... |
| flnn0ohalf 48523 | The floor of the half of a... |
| logcxp0 48524 | Logarithm of a complex pow... |
| regt1loggt0 48525 | The natural logarithm for ... |
| fdivval 48528 | The quotient of two functi... |
| fdivmpt 48529 | The quotient of two functi... |
| fdivmptf 48530 | The quotient of two functi... |
| refdivmptf 48531 | The quotient of two functi... |
| fdivpm 48532 | The quotient of two functi... |
| refdivpm 48533 | The quotient of two functi... |
| fdivmptfv 48534 | The function value of a qu... |
| refdivmptfv 48535 | The function value of a qu... |
| bigoval 48538 | Set of functions of order ... |
| elbigofrcl 48539 | Reverse closure of the "bi... |
| elbigo 48540 | Properties of a function o... |
| elbigo2 48541 | Properties of a function o... |
| elbigo2r 48542 | Sufficient condition for a... |
| elbigof 48543 | A function of order G(x) i... |
| elbigodm 48544 | The domain of a function o... |
| elbigoimp 48545 | The defining property of a... |
| elbigolo1 48546 | A function (into the posit... |
| rege1logbrege0 48547 | The general logarithm, wit... |
| rege1logbzge0 48548 | The general logarithm, wit... |
| fllogbd 48549 | A real number is between t... |
| relogbmulbexp 48550 | The logarithm of the produ... |
| relogbdivb 48551 | The logarithm of the quoti... |
| logbge0b 48552 | The logarithm of a number ... |
| logblt1b 48553 | The logarithm of a number ... |
| fldivexpfllog2 48554 | The floor of a positive re... |
| nnlog2ge0lt1 48555 | A positive integer is 1 if... |
| logbpw2m1 48556 | The floor of the binary lo... |
| fllog2 48557 | The floor of the binary lo... |
| blenval 48560 | The binary length of an in... |
| blen0 48561 | The binary length of 0. (... |
| blenn0 48562 | The binary length of a "nu... |
| blenre 48563 | The binary length of a pos... |
| blennn 48564 | The binary length of a pos... |
| blennnelnn 48565 | The binary length of a pos... |
| blennn0elnn 48566 | The binary length of a non... |
| blenpw2 48567 | The binary length of a pow... |
| blenpw2m1 48568 | The binary length of a pow... |
| nnpw2blen 48569 | A positive integer is betw... |
| nnpw2blenfzo 48570 | A positive integer is betw... |
| nnpw2blenfzo2 48571 | A positive integer is eith... |
| nnpw2pmod 48572 | Every positive integer can... |
| blen1 48573 | The binary length of 1. (... |
| blen2 48574 | The binary length of 2. (... |
| nnpw2p 48575 | Every positive integer can... |
| nnpw2pb 48576 | A number is a positive int... |
| blen1b 48577 | The binary length of a non... |
| blennnt2 48578 | The binary length of a pos... |
| nnolog2flm1 48579 | The floor of the binary lo... |
| blennn0em1 48580 | The binary length of the h... |
| blennngt2o2 48581 | The binary length of an od... |
| blengt1fldiv2p1 48582 | The binary length of an in... |
| blennn0e2 48583 | The binary length of an ev... |
| digfval 48586 | Operation to obtain the ` ... |
| digval 48587 | The ` K ` th digit of a no... |
| digvalnn0 48588 | The ` K ` th digit of a no... |
| nn0digval 48589 | The ` K ` th digit of a no... |
| dignn0fr 48590 | The digits of the fraction... |
| dignn0ldlem 48591 | Lemma for ~ dignnld . (Co... |
| dignnld 48592 | The leading digits of a po... |
| dig2nn0ld 48593 | The leading digits of a po... |
| dig2nn1st 48594 | The first (relevant) digit... |
| dig0 48595 | All digits of 0 are 0. (C... |
| digexp 48596 | The ` K ` th digit of a po... |
| dig1 48597 | All but one digits of 1 ar... |
| 0dig1 48598 | The ` 0 ` th digit of 1 is... |
| 0dig2pr01 48599 | The integers 0 and 1 corre... |
| dig2nn0 48600 | A digit of a nonnegative i... |
| 0dig2nn0e 48601 | The last bit of an even in... |
| 0dig2nn0o 48602 | The last bit of an odd int... |
| dig2bits 48603 | The ` K ` th digit of a no... |
| dignn0flhalflem1 48604 | Lemma 1 for ~ dignn0flhalf... |
| dignn0flhalflem2 48605 | Lemma 2 for ~ dignn0flhalf... |
| dignn0ehalf 48606 | The digits of the half of ... |
| dignn0flhalf 48607 | The digits of the rounded ... |
| nn0sumshdiglemA 48608 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglemB 48609 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglem1 48610 | Lemma 1 for ~ nn0sumshdig ... |
| nn0sumshdiglem2 48611 | Lemma 2 for ~ nn0sumshdig ... |
| nn0sumshdig 48612 | A nonnegative integer can ... |
| nn0mulfsum 48613 | Trivial algorithm to calcu... |
| nn0mullong 48614 | Standard algorithm (also k... |
| naryfval 48617 | The set of the n-ary (endo... |
| naryfvalixp 48618 | The set of the n-ary (endo... |
| naryfvalel 48619 | An n-ary (endo)function on... |
| naryrcl 48620 | Reverse closure for n-ary ... |
| naryfvalelfv 48621 | The value of an n-ary (end... |
| naryfvalelwrdf 48622 | An n-ary (endo)function on... |
| 0aryfvalel 48623 | A nullary (endo)function o... |
| 0aryfvalelfv 48624 | The value of a nullary (en... |
| 1aryfvalel 48625 | A unary (endo)function on ... |
| fv1arycl 48626 | Closure of a unary (endo)f... |
| 1arympt1 48627 | A unary (endo)function in ... |
| 1arympt1fv 48628 | The value of a unary (endo... |
| 1arymaptfv 48629 | The value of the mapping o... |
| 1arymaptf 48630 | The mapping of unary (endo... |
| 1arymaptf1 48631 | The mapping of unary (endo... |
| 1arymaptfo 48632 | The mapping of unary (endo... |
| 1arymaptf1o 48633 | The mapping of unary (endo... |
| 1aryenef 48634 | The set of unary (endo)fun... |
| 1aryenefmnd 48635 | The set of unary (endo)fun... |
| 2aryfvalel 48636 | A binary (endo)function on... |
| fv2arycl 48637 | Closure of a binary (endo)... |
| 2arympt 48638 | A binary (endo)function in... |
| 2arymptfv 48639 | The value of a binary (end... |
| 2arymaptfv 48640 | The value of the mapping o... |
| 2arymaptf 48641 | The mapping of binary (end... |
| 2arymaptf1 48642 | The mapping of binary (end... |
| 2arymaptfo 48643 | The mapping of binary (end... |
| 2arymaptf1o 48644 | The mapping of binary (end... |
| 2aryenef 48645 | The set of binary (endo)fu... |
| itcoval 48650 | The value of the function ... |
| itcoval0 48651 | A function iterated zero t... |
| itcoval1 48652 | A function iterated once. ... |
| itcoval2 48653 | A function iterated twice.... |
| itcoval3 48654 | A function iterated three ... |
| itcoval0mpt 48655 | A mapping iterated zero ti... |
| itcovalsuc 48656 | The value of the function ... |
| itcovalsucov 48657 | The value of the function ... |
| itcovalendof 48658 | The n-th iterate of an end... |
| itcovalpclem1 48659 | Lemma 1 for ~ itcovalpc : ... |
| itcovalpclem2 48660 | Lemma 2 for ~ itcovalpc : ... |
| itcovalpc 48661 | The value of the function ... |
| itcovalt2lem2lem1 48662 | Lemma 1 for ~ itcovalt2lem... |
| itcovalt2lem2lem2 48663 | Lemma 2 for ~ itcovalt2lem... |
| itcovalt2lem1 48664 | Lemma 1 for ~ itcovalt2 : ... |
| itcovalt2lem2 48665 | Lemma 2 for ~ itcovalt2 : ... |
| itcovalt2 48666 | The value of the function ... |
| ackvalsuc1mpt 48667 | The Ackermann function at ... |
| ackvalsuc1 48668 | The Ackermann function at ... |
| ackval0 48669 | The Ackermann function at ... |
| ackval1 48670 | The Ackermann function at ... |
| ackval2 48671 | The Ackermann function at ... |
| ackval3 48672 | The Ackermann function at ... |
| ackendofnn0 48673 | The Ackermann function at ... |
| ackfnnn0 48674 | The Ackermann function at ... |
| ackval0val 48675 | The Ackermann function at ... |
| ackvalsuc0val 48676 | The Ackermann function at ... |
| ackvalsucsucval 48677 | The Ackermann function at ... |
| ackval0012 48678 | The Ackermann function at ... |
| ackval1012 48679 | The Ackermann function at ... |
| ackval2012 48680 | The Ackermann function at ... |
| ackval3012 48681 | The Ackermann function at ... |
| ackval40 48682 | The Ackermann function at ... |
| ackval41a 48683 | The Ackermann function at ... |
| ackval41 48684 | The Ackermann function at ... |
| ackval42 48685 | The Ackermann function at ... |
| ackval42a 48686 | The Ackermann function at ... |
| ackval50 48687 | The Ackermann function at ... |
| fv1prop 48688 | The function value of unor... |
| fv2prop 48689 | The function value of unor... |
| submuladdmuld 48690 | Transformation of a sum of... |
| affinecomb1 48691 | Combination of two real af... |
| affinecomb2 48692 | Combination of two real af... |
| affineid 48693 | Identity of an affine comb... |
| 1subrec1sub 48694 | Subtract the reciprocal of... |
| resum2sqcl 48695 | The sum of two squares of ... |
| resum2sqgt0 48696 | The sum of the square of a... |
| resum2sqrp 48697 | The sum of the square of a... |
| resum2sqorgt0 48698 | The sum of the square of t... |
| reorelicc 48699 | Membership in and outside ... |
| rrx2pxel 48700 | The x-coordinate of a poin... |
| rrx2pyel 48701 | The y-coordinate of a poin... |
| prelrrx2 48702 | An unordered pair of order... |
| prelrrx2b 48703 | An unordered pair of order... |
| rrx2pnecoorneor 48704 | If two different points ` ... |
| rrx2pnedifcoorneor 48705 | If two different points ` ... |
| rrx2pnedifcoorneorr 48706 | If two different points ` ... |
| rrx2xpref1o 48707 | There is a bijection betwe... |
| rrx2xpreen 48708 | The set of points in the t... |
| rrx2plord 48709 | The lexicographical orderi... |
| rrx2plord1 48710 | The lexicographical orderi... |
| rrx2plord2 48711 | The lexicographical orderi... |
| rrx2plordisom 48712 | The set of points in the t... |
| rrx2plordso 48713 | The lexicographical orderi... |
| ehl2eudisval0 48714 | The Euclidean distance of ... |
| ehl2eudis0lt 48715 | An upper bound of the Eucl... |
| lines 48720 | The lines passing through ... |
| line 48721 | The line passing through t... |
| rrxlines 48722 | Definition of lines passin... |
| rrxline 48723 | The line passing through t... |
| rrxlinesc 48724 | Definition of lines passin... |
| rrxlinec 48725 | The line passing through t... |
| eenglngeehlnmlem1 48726 | Lemma 1 for ~ eenglngeehln... |
| eenglngeehlnmlem2 48727 | Lemma 2 for ~ eenglngeehln... |
| eenglngeehlnm 48728 | The line definition in the... |
| rrx2line 48729 | The line passing through t... |
| rrx2vlinest 48730 | The vertical line passing ... |
| rrx2linest 48731 | The line passing through t... |
| rrx2linesl 48732 | The line passing through t... |
| rrx2linest2 48733 | The line passing through t... |
| elrrx2linest2 48734 | The line passing through t... |
| spheres 48735 | The spheres for given cent... |
| sphere 48736 | A sphere with center ` X `... |
| rrxsphere 48737 | The sphere with center ` M... |
| 2sphere 48738 | The sphere with center ` M... |
| 2sphere0 48739 | The sphere around the orig... |
| line2ylem 48740 | Lemma for ~ line2y . This... |
| line2 48741 | Example for a line ` G ` p... |
| line2xlem 48742 | Lemma for ~ line2x . This... |
| line2x 48743 | Example for a horizontal l... |
| line2y 48744 | Example for a vertical lin... |
| itsclc0lem1 48745 | Lemma for theorems about i... |
| itsclc0lem2 48746 | Lemma for theorems about i... |
| itsclc0lem3 48747 | Lemma for theorems about i... |
| itscnhlc0yqe 48748 | Lemma for ~ itsclc0 . Qua... |
| itschlc0yqe 48749 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqe 48750 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqsollem1 48751 | Lemma 1 for ~ itsclc0yqsol... |
| itsclc0yqsollem2 48752 | Lemma 2 for ~ itsclc0yqsol... |
| itsclc0yqsol 48753 | Lemma for ~ itsclc0 . Sol... |
| itscnhlc0xyqsol 48754 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol1 48755 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol 48756 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsol 48757 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolr 48758 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolb 48759 | Lemma for ~ itsclc0 . Sol... |
| itsclc0 48760 | The intersection points of... |
| itsclc0b 48761 | The intersection points of... |
| itsclinecirc0 48762 | The intersection points of... |
| itsclinecirc0b 48763 | The intersection points of... |
| itsclinecirc0in 48764 | The intersection points of... |
| itsclquadb 48765 | Quadratic equation for the... |
| itsclquadeu 48766 | Quadratic equation for the... |
| 2itscplem1 48767 | Lemma 1 for ~ 2itscp . (C... |
| 2itscplem2 48768 | Lemma 2 for ~ 2itscp . (C... |
| 2itscplem3 48769 | Lemma D for ~ 2itscp . (C... |
| 2itscp 48770 | A condition for a quadrati... |
| itscnhlinecirc02plem1 48771 | Lemma 1 for ~ itscnhlineci... |
| itscnhlinecirc02plem2 48772 | Lemma 2 for ~ itscnhlineci... |
| itscnhlinecirc02plem3 48773 | Lemma 3 for ~ itscnhlineci... |
| itscnhlinecirc02p 48774 | Intersection of a nonhoriz... |
| inlinecirc02plem 48775 | Lemma for ~ inlinecirc02p ... |
| inlinecirc02p 48776 | Intersection of a line wit... |
| inlinecirc02preu 48777 | Intersection of a line wit... |
| pm4.71da 48778 | Deduction converting a bic... |
| logic1 48779 | Distribution of implicatio... |
| logic1a 48780 | Variant of ~ logic1 . (Co... |
| logic2 48781 | Variant of ~ logic1 . (Co... |
| pm5.32dav 48782 | Distribution of implicatio... |
| pm5.32dra 48783 | Reverse distribution of im... |
| exp12bd 48784 | The import-export theorem ... |
| mpbiran3d 48785 | Equivalence with a conjunc... |
| mpbiran4d 48786 | Equivalence with a conjunc... |
| dtrucor3 48787 | An example of how ~ ax-5 w... |
| ralbidb 48788 | Formula-building rule for ... |
| ralbidc 48789 | Formula-building rule for ... |
| r19.41dv 48790 | A complex deduction form o... |
| rmotru 48791 | Two ways of expressing "at... |
| reutru 48792 | Two ways of expressing "ex... |
| reutruALT 48793 | Alternate proof of ~ reutr... |
| reueqbidva 48794 | Formula-building rule for ... |
| reuxfr1dd 48795 | Transfer existential uniqu... |
| ssdisjd 48796 | Subset preserves disjointn... |
| ssdisjdr 48797 | Subset preserves disjointn... |
| disjdifb 48798 | Relative complement is ant... |
| predisj 48799 | Preimages of disjoint sets... |
| vsn 48800 | The singleton of the unive... |
| mosn 48801 | "At most one" element in a... |
| mo0 48802 | "At most one" element in a... |
| mosssn 48803 | "At most one" element in a... |
| mo0sn 48804 | Two ways of expressing "at... |
| mosssn2 48805 | Two ways of expressing "at... |
| unilbss 48806 | Superclass of the greatest... |
| iuneq0 48807 | An indexed union is empty ... |
| iineq0 48808 | An indexed intersection is... |
| iunlub 48809 | The indexed union is the t... |
| iinglb 48810 | The indexed intersection i... |
| iuneqconst2 48811 | Indexed union of identical... |
| iineqconst2 48812 | Indexed intersection of id... |
| inpw 48813 | Two ways of expressing a c... |
| opth1neg 48814 | Two ordered pairs are not ... |
| opth2neg 48815 | Two ordered pairs are not ... |
| brab2dd 48816 | Expressing that two sets a... |
| brab2ddw 48817 | Expressing that two sets a... |
| brab2ddw2 48818 | Expressing that two sets a... |
| iinxp 48819 | Indexed intersection of Ca... |
| intxp 48820 | Intersection of Cartesian ... |
| coxp 48821 | Composition with a Cartesi... |
| cosn 48822 | Composition with an ordere... |
| cosni 48823 | Composition with an ordere... |
| inisegn0a 48824 | The inverse image of a sin... |
| dmrnxp 48825 | A Cartesian product is the... |
| mof0 48826 | There is at most one funct... |
| mof02 48827 | A variant of ~ mof0 . (Co... |
| mof0ALT 48828 | Alternate proof of ~ mof0 ... |
| eufsnlem 48829 | There is exactly one funct... |
| eufsn 48830 | There is exactly one funct... |
| eufsn2 48831 | There is exactly one funct... |
| mofsn 48832 | There is at most one funct... |
| mofsn2 48833 | There is at most one funct... |
| mofsssn 48834 | There is at most one funct... |
| mofmo 48835 | There is at most one funct... |
| mofeu 48836 | The uniqueness of a functi... |
| elfvne0 48837 | If a function value has a ... |
| fdomne0 48838 | A function with non-empty ... |
| f1sn2g 48839 | A function that maps a sin... |
| f102g 48840 | A function that maps the e... |
| f1mo 48841 | A function that maps a set... |
| f002 48842 | A function with an empty c... |
| map0cor 48843 | A function exists iff an e... |
| ffvbr 48844 | Relation with function val... |
| xpco2 48845 | Composition of a Cartesian... |
| ovsng 48846 | The operation value of a s... |
| ovsng2 48847 | The operation value of a s... |
| ovsn 48848 | The operation value of a s... |
| ovsn2 48849 | The operation value of a s... |
| fvconstr 48850 | Two ways of expressing ` A... |
| fvconstrn0 48851 | Two ways of expressing ` A... |
| fvconstr2 48852 | Two ways of expressing ` A... |
| ovmpt4d 48853 | Deduction version of ~ ovm... |
| eqfnovd 48854 | Deduction for equality of ... |
| fonex 48855 | The domain of a surjection... |
| eloprab1st2nd 48856 | Reconstruction of a nested... |
| fmpodg 48857 | Domain and codomain of the... |
| fmpod 48858 | Domain and codomain of the... |
| resinsnlem 48859 | Lemma for ~ resinsnALT . ... |
| resinsn 48860 | Restriction to the interse... |
| resinsnALT 48861 | Restriction to the interse... |
| dftpos5 48862 | Alternate definition of ` ... |
| dftpos6 48863 | Alternate definition of ` ... |
| dmtposss 48864 | The domain of ` tpos F ` i... |
| tposres0 48865 | The transposition of a set... |
| tposresg 48866 | The transposition restrict... |
| tposrescnv 48867 | The transposition restrict... |
| tposres2 48868 | The transposition restrict... |
| tposres3 48869 | The transposition restrict... |
| tposres 48870 | The transposition restrict... |
| tposresxp 48871 | The transposition restrict... |
| tposf1o 48872 | Condition of a bijective t... |
| tposid 48873 | Swap an ordered pair. (Co... |
| tposidres 48874 | Swap an ordered pair. (Co... |
| tposidf1o 48875 | The swap function, or the ... |
| tposideq 48876 | Two ways of expressing the... |
| tposideq2 48877 | Two ways of expressing the... |
| ixpv 48878 | Infinite Cartesian product... |
| fvconst0ci 48879 | A constant function's valu... |
| fvconstdomi 48880 | A constant function's valu... |
| f1omo 48881 | There is at most one eleme... |
| f1omoOLD 48882 | Obsolete version of ~ f1om... |
| f1omoALT 48883 | There is at most one eleme... |
| iccin 48884 | Intersection of two closed... |
| iccdisj2 48885 | If the upper bound of one ... |
| iccdisj 48886 | If the upper bound of one ... |
| slotresfo 48887 | The condition of a structu... |
| mreuniss 48888 | The union of a collection ... |
| clduni 48889 | The union of closed sets i... |
| opncldeqv 48890 | Conditions on open sets ar... |
| opndisj 48891 | Two ways of saying that tw... |
| clddisj 48892 | Two ways of saying that tw... |
| neircl 48893 | Reverse closure of the nei... |
| opnneilem 48894 | Lemma factoring out common... |
| opnneir 48895 | If something is true for a... |
| opnneirv 48896 | A variant of ~ opnneir wit... |
| opnneilv 48897 | The converse of ~ opnneir ... |
| opnneil 48898 | A variant of ~ opnneilv . ... |
| opnneieqv 48899 | The equivalence between ne... |
| opnneieqvv 48900 | The equivalence between ne... |
| restcls2lem 48901 | A closed set in a subspace... |
| restcls2 48902 | A closed set in a subspace... |
| restclsseplem 48903 | Lemma for ~ restclssep . ... |
| restclssep 48904 | Two disjoint closed sets i... |
| cnneiima 48905 | Given a continuous functio... |
| iooii 48906 | Open intervals are open se... |
| icccldii 48907 | Closed intervals are close... |
| i0oii 48908 | ` ( 0 [,) A ) ` is open in... |
| io1ii 48909 | ` ( A (,] 1 ) ` is open in... |
| sepnsepolem1 48910 | Lemma for ~ sepnsepo . (C... |
| sepnsepolem2 48911 | Open neighborhood and neig... |
| sepnsepo 48912 | Open neighborhood and neig... |
| sepdisj 48913 | Separated sets are disjoin... |
| seposep 48914 | If two sets are separated ... |
| sepcsepo 48915 | If two sets are separated ... |
| sepfsepc 48916 | If two sets are separated ... |
| seppsepf 48917 | If two sets are precisely ... |
| seppcld 48918 | If two sets are precisely ... |
| isnrm4 48919 | A topological space is nor... |
| dfnrm2 48920 | A topological space is nor... |
| dfnrm3 48921 | A topological space is nor... |
| iscnrm3lem1 48922 | Lemma for ~ iscnrm3 . Sub... |
| iscnrm3lem2 48923 | Lemma for ~ iscnrm3 provin... |
| iscnrm3lem4 48924 | Lemma for ~ iscnrm3lem5 an... |
| iscnrm3lem5 48925 | Lemma for ~ iscnrm3l . (C... |
| iscnrm3lem6 48926 | Lemma for ~ iscnrm3lem7 . ... |
| iscnrm3lem7 48927 | Lemma for ~ iscnrm3rlem8 a... |
| iscnrm3rlem1 48928 | Lemma for ~ iscnrm3rlem2 .... |
| iscnrm3rlem2 48929 | Lemma for ~ iscnrm3rlem3 .... |
| iscnrm3rlem3 48930 | Lemma for ~ iscnrm3r . Th... |
| iscnrm3rlem4 48931 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem5 48932 | Lemma for ~ iscnrm3rlem6 .... |
| iscnrm3rlem6 48933 | Lemma for ~ iscnrm3rlem7 .... |
| iscnrm3rlem7 48934 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem8 48935 | Lemma for ~ iscnrm3r . Di... |
| iscnrm3r 48936 | Lemma for ~ iscnrm3 . If ... |
| iscnrm3llem1 48937 | Lemma for ~ iscnrm3l . Cl... |
| iscnrm3llem2 48938 | Lemma for ~ iscnrm3l . If... |
| iscnrm3l 48939 | Lemma for ~ iscnrm3 . Giv... |
| iscnrm3 48940 | A completely normal topolo... |
| iscnrm3v 48941 | A topology is completely n... |
| iscnrm4 48942 | A completely normal topolo... |
| isprsd 48943 | Property of being a preord... |
| lubeldm2 48944 | Member of the domain of th... |
| glbeldm2 48945 | Member of the domain of th... |
| lubeldm2d 48946 | Member of the domain of th... |
| glbeldm2d 48947 | Member of the domain of th... |
| lubsscl 48948 | If a subset of ` S ` conta... |
| glbsscl 48949 | If a subset of ` S ` conta... |
| lubprlem 48950 | Lemma for ~ lubprdm and ~ ... |
| lubprdm 48951 | The set of two comparable ... |
| lubpr 48952 | The LUB of the set of two ... |
| glbprlem 48953 | Lemma for ~ glbprdm and ~ ... |
| glbprdm 48954 | The set of two comparable ... |
| glbpr 48955 | The GLB of the set of two ... |
| joindm2 48956 | The join of any two elemen... |
| joindm3 48957 | The join of any two elemen... |
| meetdm2 48958 | The meet of any two elemen... |
| meetdm3 48959 | The meet of any two elemen... |
| posjidm 48960 | Poset join is idempotent. ... |
| posmidm 48961 | Poset meet is idempotent. ... |
| resiposbas 48962 | Construct a poset ( ~ resi... |
| resipos 48963 | A set equipped with an ord... |
| exbaspos 48964 | There exists a poset for a... |
| exbasprs 48965 | There exists a preordered ... |
| basresposfo 48966 | The base function restrict... |
| basresprsfo 48967 | The base function restrict... |
| posnex 48968 | The class of posets is a p... |
| prsnex 48969 | The class of preordered se... |
| toslat 48970 | A toset is a lattice. (Co... |
| isclatd 48971 | The predicate "is a comple... |
| intubeu 48972 | Existential uniqueness of ... |
| unilbeu 48973 | Existential uniqueness of ... |
| ipolublem 48974 | Lemma for ~ ipolubdm and ~... |
| ipolubdm 48975 | The domain of the LUB of t... |
| ipolub 48976 | The LUB of the inclusion p... |
| ipoglblem 48977 | Lemma for ~ ipoglbdm and ~... |
| ipoglbdm 48978 | The domain of the GLB of t... |
| ipoglb 48979 | The GLB of the inclusion p... |
| ipolub0 48980 | The LUB of the empty set i... |
| ipolub00 48981 | The LUB of the empty set i... |
| ipoglb0 48982 | The GLB of the empty set i... |
| mrelatlubALT 48983 | Least upper bounds in a Mo... |
| mrelatglbALT 48984 | Greatest lower bounds in a... |
| mreclat 48985 | A Moore space is a complet... |
| topclat 48986 | A topology is a complete l... |
| toplatglb0 48987 | The empty intersection in ... |
| toplatlub 48988 | Least upper bounds in a to... |
| toplatglb 48989 | Greatest lower bounds in a... |
| toplatjoin 48990 | Joins in a topology are re... |
| toplatmeet 48991 | Meets in a topology are re... |
| topdlat 48992 | A topology is a distributi... |
| elmgpcntrd 48993 | The center of a ring. (Co... |
| asclelbas 48994 | Lifted scalars are in the ... |
| asclelbasALT 48995 | Alternate proof of ~ ascle... |
| asclcntr 48996 | The algebra scalar lifting... |
| asclcom 48997 | Scalars are commutative af... |
| homf0 48998 | The base is empty iff the ... |
| catprslem 48999 | Lemma for ~ catprs . (Con... |
| catprs 49000 | A preorder can be extracte... |
| catprs2 49001 | A category equipped with t... |
| catprsc 49002 | A construction of the preo... |
| catprsc2 49003 | An alternate construction ... |
| endmndlem 49004 | A diagonal hom-set in a ca... |
| oppccatb 49005 | An opposite category is a ... |
| oppcmndclem 49006 | Lemma for ~ oppcmndc . Ev... |
| oppcendc 49007 | The opposite category of a... |
| oppcmndc 49008 | The opposite category of a... |
| idmon 49009 | An identity arrow, or an i... |
| idepi 49010 | An identity arrow, or an i... |
| sectrcl 49011 | Reverse closure for sectio... |
| sectrcl2 49012 | Reverse closure for sectio... |
| invrcl 49013 | Reverse closure for invers... |
| invrcl2 49014 | Reverse closure for invers... |
| isinv2 49015 | The property " ` F ` is an... |
| isisod 49016 | The predicate "is an isomo... |
| upeu2lem 49017 | Lemma for ~ upeu2 . There... |
| sectfn 49018 | The function value of the ... |
| invfn 49019 | The function value of the ... |
| isofnALT 49020 | The function value of the ... |
| isofval2 49021 | Function value of the func... |
| isorcl 49022 | Reverse closure for isomor... |
| isorcl2 49023 | Reverse closure for isomor... |
| isoval2 49024 | The isomorphisms are the d... |
| sectpropdlem 49025 | Lemma for ~ sectpropd . (... |
| sectpropd 49026 | Two structures with the sa... |
| invpropdlem 49027 | Lemma for ~ invpropd . (C... |
| invpropd 49028 | Two structures with the sa... |
| isopropdlem 49029 | Lemma for ~ isopropd . (C... |
| isopropd 49030 | Two structures with the sa... |
| cicfn 49031 | ` ~=c ` is a function on `... |
| cicrcl2 49032 | Isomorphism implies the st... |
| oppccic 49033 | Isomorphic objects are iso... |
| relcic 49034 | The set of isomorphic obje... |
| cicerALT 49035 | Isomorphism is an equivale... |
| cic1st2nd 49036 | Reconstruction of a pair o... |
| cic1st2ndbr 49037 | Rewrite the predicate of i... |
| cicpropdlem 49038 | Lemma for ~ cicpropd . (C... |
| cicpropd 49039 | Two structures with the sa... |
| oppccicb 49040 | Isomorphic objects are iso... |
| oppcciceq 49041 | The opposite category has ... |
| dmdm 49042 | The double domain of a fun... |
| iinfssclem1 49043 | Lemma for ~ iinfssc . (Co... |
| iinfssclem2 49044 | Lemma for ~ iinfssc . (Co... |
| iinfssclem3 49045 | Lemma for ~ iinfssc . (Co... |
| iinfssc 49046 | Indexed intersection of su... |
| iinfsubc 49047 | Indexed intersection of su... |
| iinfprg 49048 | Indexed intersection of fu... |
| infsubc 49049 | The intersection of two su... |
| infsubc2 49050 | The intersection of two su... |
| infsubc2d 49051 | The intersection of two su... |
| discsubclem 49052 | Lemma for ~ discsubc . (C... |
| discsubc 49053 | A discrete category, whose... |
| iinfconstbaslem 49054 | Lemma for ~ iinfconstbas .... |
| iinfconstbas 49055 | The discrete category is t... |
| nelsubclem 49056 | Lemma for ~ nelsubc . (Co... |
| nelsubc 49057 | An empty "hom-set" for non... |
| nelsubc2 49058 | An empty "hom-set" for non... |
| nelsubc3lem 49059 | Lemma for ~ nelsubc3 . (C... |
| nelsubc3 49060 | Remark 4.2(2) of [Adamek] ... |
| ssccatid 49061 | A category ` C ` restricte... |
| resccatlem 49062 | Lemma for ~ resccat . (Co... |
| resccat 49063 | A class ` C ` restricted b... |
| reldmfunc 49064 | The domain of ` Func ` is ... |
| func1st2nd 49065 | Rewrite the functor predic... |
| func1st 49066 | Extract the first member o... |
| func2nd 49067 | Extract the second member ... |
| funcrcl2 49068 | Reverse closure for a func... |
| funcrcl3 49069 | Reverse closure for a func... |
| funcf2lem 49070 | A utility theorem for prov... |
| funcf2lem2 49071 | A utility theorem for prov... |
| 0funcglem 49072 | Lemma for ~ 0funcg . (Con... |
| 0funcg2 49073 | The functor from the empty... |
| 0funcg 49074 | The functor from the empty... |
| 0funclem 49075 | Lemma for ~ 0funcALT . (C... |
| 0func 49076 | The functor from the empty... |
| 0funcALT 49077 | Alternate proof of ~ 0func... |
| func0g 49078 | The source category of a f... |
| func0g2 49079 | The source category of a f... |
| initc 49080 | Sets with empty base are t... |
| cofu1st2nd 49081 | Rewrite the functor compos... |
| rescofuf 49082 | The restriction of functor... |
| cofu1a 49083 | Value of the object part o... |
| cofu2a 49084 | Value of the morphism part... |
| cofucla 49085 | The composition of two fun... |
| funchomf 49086 | Source categories of a fun... |
| idfurcl 49087 | Reverse closure for an ide... |
| idfu1stf1o 49088 | The identity functor/inclu... |
| idfu1stalem 49089 | Lemma for ~ idfu1sta . (C... |
| idfu1sta 49090 | Value of the object part o... |
| idfu1a 49091 | Value of the object part o... |
| idfu2nda 49092 | Value of the morphism part... |
| imasubclem1 49093 | Lemma for ~ imasubc . (Co... |
| imasubclem2 49094 | Lemma for ~ imasubc . (Co... |
| imasubclem3 49095 | Lemma for ~ imasubc . (Co... |
| imaf1homlem 49096 | Lemma for ~ imaf1hom and o... |
| imaf1hom 49097 | The hom-set of an image of... |
| imaidfu2lem 49098 | Lemma for ~ imaidfu2 . (C... |
| imaidfu 49099 | The image of the identity ... |
| imaidfu2 49100 | The image of the identity ... |
| cofid1a 49101 | Express the object part of... |
| cofid2a 49102 | Express the morphism part ... |
| cofid1 49103 | Express the object part of... |
| cofid2 49104 | Express the morphism part ... |
| cofidvala 49105 | The property " ` F ` is a ... |
| cofidf2a 49106 | If " ` F ` is a section of... |
| cofidf1a 49107 | If " ` F ` is a section of... |
| cofidval 49108 | The property " ` <. F , G ... |
| cofidf2 49109 | If " ` F ` is a section of... |
| cofidf1 49110 | If " ` <. F , G >. ` is a ... |
| oppffn 49113 | ` oppFunc ` is a function ... |
| reldmoppf 49114 | The domain of ` oppFunc ` ... |
| oppfvalg 49115 | Value of the opposite func... |
| oppfrcllem 49116 | Lemma for ~ oppfrcl . (Co... |
| oppfrcl 49117 | If an opposite functor of ... |
| oppfrcl2 49118 | If an opposite functor of ... |
| oppfrcl3 49119 | If an opposite functor of ... |
| oppf1st2nd 49120 | Rewrite the opposite funct... |
| 2oppf 49121 | The double opposite functo... |
| eloppf 49122 | The pre-image of a non-emp... |
| eloppf2 49123 | Both components of a pre-i... |
| oppfvallem 49124 | Lemma for ~ oppfval . (Co... |
| oppfval 49125 | Value of the opposite func... |
| oppfval2 49126 | Value of the opposite func... |
| oppfval3 49127 | Value of the opposite func... |
| oppf1 49128 | Value of the object part o... |
| oppf2 49129 | Value of the morphism part... |
| oppfoppc 49130 | The opposite functor is a ... |
| oppfoppc2 49131 | The opposite functor is a ... |
| funcoppc2 49132 | A functor on opposite cate... |
| funcoppc4 49133 | A functor on opposite cate... |
| funcoppc5 49134 | A functor on opposite cate... |
| 2oppffunc 49135 | The opposite functor of an... |
| funcoppc3 49136 | A functor on opposite cate... |
| oppff1 49137 | The operation generating o... |
| oppff1o 49138 | The operation generating o... |
| cofuoppf 49139 | Composition of opposite fu... |
| imasubc 49140 | An image of a full functor... |
| imasubc2 49141 | An image of a full functor... |
| imassc 49142 | An image of a functor sati... |
| imaid 49143 | An image of a functor pres... |
| imaf1co 49144 | An image of a functor whos... |
| imasubc3 49145 | An image of a functor inje... |
| fthcomf 49146 | Source categories of a fai... |
| idfth 49147 | The inclusion functor is a... |
| idemb 49148 | The inclusion functor is a... |
| idsubc 49149 | The source category of an ... |
| idfullsubc 49150 | The source category of an ... |
| cofidfth 49151 | If " ` F ` is a section of... |
| fulloppf 49152 | The opposite functor of a ... |
| fthoppf 49153 | The opposite functor of a ... |
| ffthoppf 49154 | The opposite functor of a ... |
| upciclem1 49155 | Lemma for ~ upcic , ~ upeu... |
| upciclem2 49156 | Lemma for ~ upciclem3 and ... |
| upciclem3 49157 | Lemma for ~ upciclem4 . (... |
| upciclem4 49158 | Lemma for ~ upcic and ~ up... |
| upcic 49159 | A universal property defin... |
| upeu 49160 | A universal property defin... |
| upeu2 49161 | Generate new universal mor... |
| reldmup 49164 | The domain of ` UP ` is a ... |
| upfval 49165 | Function value of the clas... |
| upfval2 49166 | Function value of the clas... |
| upfval3 49167 | Function value of the clas... |
| isuplem 49168 | Lemma for ~ isup and other... |
| isup 49169 | The predicate "is a univer... |
| uppropd 49170 | If two categories have the... |
| reldmup2 49171 | The domain of ` ( D UP E )... |
| relup 49172 | The set of universal pairs... |
| uprcl 49173 | Reverse closure for the cl... |
| up1st2nd 49174 | Rewrite the universal prop... |
| up1st2ndr 49175 | Combine separated parts in... |
| up1st2ndb 49176 | Combine/separate parts in ... |
| up1st2nd2 49177 | Rewrite the universal prop... |
| uprcl2 49178 | Reverse closure for the cl... |
| uprcl3 49179 | Reverse closure for the cl... |
| uprcl4 49180 | Reverse closure for the cl... |
| uprcl5 49181 | Reverse closure for the cl... |
| uobrcl 49182 | Reverse closure for univer... |
| isup2 49183 | The universal property of ... |
| upeu3 49184 | The universal pair ` <. X ... |
| upeu4 49185 | Generate a new universal m... |
| uptposlem 49186 | Lemma for ~ uptpos . (Con... |
| uptpos 49187 | Rewrite the predicate of u... |
| oppcuprcl4 49188 | Reverse closure for the cl... |
| oppcuprcl3 49189 | Reverse closure for the cl... |
| oppcuprcl5 49190 | Reverse closure for the cl... |
| oppcuprcl2 49191 | Reverse closure for the cl... |
| uprcl2a 49192 | Reverse closure for the cl... |
| oppfuprcl 49193 | Reverse closure for the cl... |
| oppfuprcl2 49194 | Reverse closure for the cl... |
| oppcup3lem 49195 | Lemma for ~ oppcup3 . (Co... |
| oppcup 49196 | The universal pair ` <. X ... |
| oppcup2 49197 | The universal property for... |
| oppcup3 49198 | The universal property for... |
| uptrlem1 49199 | Lemma for ~ uptr . (Contr... |
| uptrlem2 49200 | Lemma for ~ uptr . (Contr... |
| uptrlem3 49201 | Lemma for ~ uptr . (Contr... |
| uptr 49202 | Universal property and ful... |
| uptri 49203 | Universal property and ful... |
| uptra 49204 | Universal property and ful... |
| uptrar 49205 | Universal property and ful... |
| uptrai 49206 | Universal property and ful... |
| uobffth 49207 | A fully faithful functor g... |
| uobeqw 49208 | If a full functor (in fact... |
| uobeq 49209 | If a full functor (in fact... |
| uptr2 49210 | Universal property and ful... |
| uptr2a 49211 | Universal property and ful... |
| isnatd 49212 | Property of being a natura... |
| natrcl2 49213 | Reverse closure for a natu... |
| natrcl3 49214 | Reverse closure for a natu... |
| catbas 49215 | The base of the category s... |
| cathomfval 49216 | The hom-sets of the catego... |
| catcofval 49217 | Composition of the categor... |
| natoppf 49218 | A natural transformation i... |
| natoppf2 49219 | A natural transformation i... |
| natoppfb 49220 | A natural transformation i... |
| initoo2 49221 | An initial object is an ob... |
| termoo2 49222 | A terminal object is an ob... |
| zeroo2 49223 | A zero object is an object... |
| oppcinito 49224 | Initial objects are termin... |
| oppctermo 49225 | Terminal objects are initi... |
| oppczeroo 49226 | Zero objects are zero in t... |
| termoeu2 49227 | Terminal objects are essen... |
| initopropdlemlem 49228 | Lemma for ~ initopropdlem ... |
| initopropdlem 49229 | Lemma for ~ initopropd . ... |
| termopropdlem 49230 | Lemma for ~ termopropd . ... |
| zeroopropdlem 49231 | Lemma for ~ zeroopropd . ... |
| initopropd 49232 | Two structures with the sa... |
| termopropd 49233 | Two structures with the sa... |
| zeroopropd 49234 | Two structures with the sa... |
| reldmxpc 49235 | The binary product of cate... |
| reldmxpcALT 49236 | Alternate proof of ~ reldm... |
| elxpcbasex1 49237 | A non-empty base set of th... |
| elxpcbasex1ALT 49238 | Alternate proof of ~ elxpc... |
| elxpcbasex2 49239 | A non-empty base set of th... |
| elxpcbasex2ALT 49240 | Alternate proof of ~ elxpc... |
| xpcfucbas 49241 | The base set of the produc... |
| xpcfuchomfval 49242 | Set of morphisms of the bi... |
| xpcfuchom 49243 | Set of morphisms of the bi... |
| xpcfuchom2 49244 | Value of the set of morphi... |
| xpcfucco2 49245 | Value of composition in th... |
| xpcfuccocl 49246 | The composition of two nat... |
| xpcfucco3 49247 | Value of composition in th... |
| dfswapf2 49250 | Alternate definition of ` ... |
| swapfval 49251 | Value of the swap functor.... |
| swapfelvv 49252 | A swap functor is an order... |
| swapf2fvala 49253 | The morphism part of the s... |
| swapf2fval 49254 | The morphism part of the s... |
| swapf1vala 49255 | The object part of the swa... |
| swapf1val 49256 | The object part of the swa... |
| swapf2fn 49257 | The morphism part of the s... |
| swapf1a 49258 | The object part of the swa... |
| swapf2vala 49259 | The morphism part of the s... |
| swapf2a 49260 | The morphism part of the s... |
| swapf1 49261 | The object part of the swa... |
| swapf2val 49262 | The morphism part of the s... |
| swapf2 49263 | The morphism part of the s... |
| swapf1f1o 49264 | The object part of the swa... |
| swapf2f1o 49265 | The morphism part of the s... |
| swapf2f1oa 49266 | The morphism part of the s... |
| swapf2f1oaALT 49267 | Alternate proof of ~ swapf... |
| swapfid 49268 | Each identity morphism in ... |
| swapfida 49269 | Each identity morphism in ... |
| swapfcoa 49270 | Composition in the source ... |
| swapffunc 49271 | The swap functor is a func... |
| swapfffth 49272 | The swap functor is a full... |
| swapffunca 49273 | The swap functor is a func... |
| swapfiso 49274 | The swap functor is an iso... |
| swapciso 49275 | The product category is ca... |
| oppc1stflem 49276 | A utility theorem for prov... |
| oppc1stf 49277 | The opposite functor of th... |
| oppc2ndf 49278 | The opposite functor of th... |
| 1stfpropd 49279 | If two categories have the... |
| 2ndfpropd 49280 | If two categories have the... |
| diagpropd 49281 | If two categories have the... |
| cofuswapfcl 49282 | The bifunctor pre-composed... |
| cofuswapf1 49283 | The object part of a bifun... |
| cofuswapf2 49284 | The morphism part of a bif... |
| tposcurf1cl 49285 | The partially evaluated tr... |
| tposcurf11 49286 | Value of the double evalua... |
| tposcurf12 49287 | The partially evaluated tr... |
| tposcurf1 49288 | Value of the object part o... |
| tposcurf2 49289 | Value of the transposed cu... |
| tposcurf2val 49290 | Value of a component of th... |
| tposcurf2cl 49291 | The transposed curry funct... |
| tposcurfcl 49292 | The transposed curry funct... |
| diag1 49293 | The constant functor of ` ... |
| diag1a 49294 | The constant functor of ` ... |
| diag1f1lem 49295 | The object part of the dia... |
| diag1f1 49296 | The object part of the dia... |
| diag2f1lem 49297 | Lemma for ~ diag2f1 . The... |
| diag2f1 49298 | If ` B ` is non-empty, the... |
| fucofulem1 49299 | Lemma for proving functor ... |
| fucofulem2 49300 | Lemma for proving functor ... |
| fuco2el 49301 | Equivalence of product fun... |
| fuco2eld 49302 | Equivalence of product fun... |
| fuco2eld2 49303 | Equivalence of product fun... |
| fuco2eld3 49304 | Equivalence of product fun... |
| fucofvalg 49307 | Value of the function givi... |
| fucofval 49308 | Value of the function givi... |
| fucoelvv 49309 | A functor composition bifu... |
| fuco1 49310 | The object part of the fun... |
| fucof1 49311 | The object part of the fun... |
| fuco2 49312 | The morphism part of the f... |
| fucofn2 49313 | The morphism part of the f... |
| fucofvalne 49314 | Value of the function givi... |
| fuco11 49315 | The object part of the fun... |
| fuco11cl 49316 | The object part of the fun... |
| fuco11a 49317 | The object part of the fun... |
| fuco112 49318 | The object part of the fun... |
| fuco111 49319 | The object part of the fun... |
| fuco111x 49320 | The object part of the fun... |
| fuco112x 49321 | The object part of the fun... |
| fuco112xa 49322 | The object part of the fun... |
| fuco11id 49323 | The identity morphism of t... |
| fuco11idx 49324 | The identity morphism of t... |
| fuco21 49325 | The morphism part of the f... |
| fuco11b 49326 | The object part of the fun... |
| fuco11bALT 49327 | Alternate proof of ~ fuco1... |
| fuco22 49328 | The morphism part of the f... |
| fucofn22 49329 | The morphism part of the f... |
| fuco23 49330 | The morphism part of the f... |
| fuco22natlem1 49331 | Lemma for ~ fuco22nat . T... |
| fuco22natlem2 49332 | Lemma for ~ fuco22nat . T... |
| fuco22natlem3 49333 | Combine ~ fuco22natlem2 wi... |
| fuco22natlem 49334 | The composed natural trans... |
| fuco22nat 49335 | The composed natural trans... |
| fucof21 49336 | The morphism part of the f... |
| fucoid 49337 | Each identity morphism in ... |
| fucoid2 49338 | Each identity morphism in ... |
| fuco22a 49339 | The morphism part of the f... |
| fuco23alem 49340 | The naturality property ( ... |
| fuco23a 49341 | The morphism part of the f... |
| fucocolem1 49342 | Lemma for ~ fucoco . Asso... |
| fucocolem2 49343 | Lemma for ~ fucoco . The ... |
| fucocolem3 49344 | Lemma for ~ fucoco . The ... |
| fucocolem4 49345 | Lemma for ~ fucoco . The ... |
| fucoco 49346 | Composition in the source ... |
| fucoco2 49347 | Composition in the source ... |
| fucofunc 49348 | The functor composition bi... |
| fucofunca 49349 | The functor composition bi... |
| fucolid 49350 | Post-compose a natural tra... |
| fucorid 49351 | Pre-composing a natural tr... |
| fucorid2 49352 | Pre-composing a natural tr... |
| postcofval 49353 | Value of the post-composit... |
| postcofcl 49354 | The post-composition funct... |
| precofvallem 49355 | Lemma for ~ precofval to e... |
| precofval 49356 | Value of the pre-compositi... |
| precofvalALT 49357 | Alternate proof of ~ preco... |
| precofval2 49358 | Value of the pre-compositi... |
| precofcl 49359 | The pre-composition functo... |
| precofval3 49360 | Value of the pre-compositi... |
| precoffunc 49361 | The pre-composition functo... |
| reldmprcof 49364 | The domain of ` -o.F ` is ... |
| prcofvalg 49365 | Value of the pre-compositi... |
| prcofvala 49366 | Value of the pre-compositi... |
| prcofval 49367 | Value of the pre-compositi... |
| prcofpropd 49368 | If the categories have the... |
| prcofelvv 49369 | The pre-composition functo... |
| reldmprcof1 49370 | The domain of the object p... |
| reldmprcof2 49371 | The domain of the morphism... |
| prcoftposcurfuco 49372 | The pre-composition functo... |
| prcoftposcurfucoa 49373 | The pre-composition functo... |
| prcoffunc 49374 | The pre-composition functo... |
| prcoffunca 49375 | The pre-composition functo... |
| prcoffunca2 49376 | The pre-composition functo... |
| prcof1 49377 | The object part of the pre... |
| prcof2a 49378 | The morphism part of the p... |
| prcof2 49379 | The morphism part of the p... |
| prcof21a 49380 | The morphism part of the p... |
| prcof22a 49381 | The morphism part of the p... |
| prcofdiag1 49382 | A constant functor pre-com... |
| prcofdiag 49383 | A diagonal functor post-co... |
| catcrcl 49384 | Reverse closure for the ca... |
| catcrcl2 49385 | Reverse closure for the ca... |
| elcatchom 49386 | A morphism of the category... |
| catcsect 49387 | The property " ` F ` is a ... |
| catcinv 49388 | The property " ` F ` is an... |
| catcisoi 49389 | A functor is an isomorphis... |
| uobeq2 49390 | If a full functor (in fact... |
| uobeq3 49391 | An isomorphism between cat... |
| opf11 49392 | The object part of the op ... |
| opf12 49393 | The object part of the op ... |
| opf2fval 49394 | The morphism part of the o... |
| opf2 49395 | The morphism part of the o... |
| fucoppclem 49396 | Lemma for ~ fucoppc . (Co... |
| fucoppcid 49397 | The opposite category of f... |
| fucoppcco 49398 | The opposite category of f... |
| fucoppc 49399 | The isomorphism from the o... |
| fucoppcffth 49400 | A fully faithful functor f... |
| fucoppcfunc 49401 | A functor from the opposit... |
| fucoppccic 49402 | The opposite category of f... |
| oppfdiag1 49403 | A constant functor for opp... |
| oppfdiag1a 49404 | A constant functor for opp... |
| oppfdiag 49405 | A diagonal functor for opp... |
| isthinc 49408 | The predicate "is a thin c... |
| isthinc2 49409 | A thin category is a categ... |
| isthinc3 49410 | A thin category is a categ... |
| thincc 49411 | A thin category is a categ... |
| thinccd 49412 | A thin category is a categ... |
| thincssc 49413 | A thin category is a categ... |
| isthincd2lem1 49414 | Lemma for ~ isthincd2 and ... |
| thincmo2 49415 | Morphisms in the same hom-... |
| thinchom 49416 | A non-empty hom-set of a t... |
| thincmo 49417 | There is at most one morph... |
| thincmoALT 49418 | Alternate proof of ~ thinc... |
| thincmod 49419 | At most one morphism in ea... |
| thincn0eu 49420 | In a thin category, a hom-... |
| thincid 49421 | In a thin category, a morp... |
| thincmon 49422 | In a thin category, all mo... |
| thincepi 49423 | In a thin category, all mo... |
| isthincd2lem2 49424 | Lemma for ~ isthincd2 . (... |
| isthincd 49425 | The predicate "is a thin c... |
| isthincd2 49426 | The predicate " ` C ` is a... |
| oppcthin 49427 | The opposite category of a... |
| oppcthinco 49428 | If the opposite category o... |
| oppcthinendc 49429 | The opposite category of a... |
| oppcthinendcALT 49430 | Alternate proof of ~ oppct... |
| thincpropd 49431 | Two structures with the sa... |
| subthinc 49432 | A subcategory of a thin ca... |
| functhinclem1 49433 | Lemma for ~ functhinc . G... |
| functhinclem2 49434 | Lemma for ~ functhinc . (... |
| functhinclem3 49435 | Lemma for ~ functhinc . T... |
| functhinclem4 49436 | Lemma for ~ functhinc . O... |
| functhinc 49437 | A functor to a thin catego... |
| functhincfun 49438 | A functor to a thin catego... |
| fullthinc 49439 | A functor to a thin catego... |
| fullthinc2 49440 | A full functor to a thin c... |
| thincfth 49441 | A functor from a thin cate... |
| thincciso 49442 | Two thin categories are is... |
| thinccisod 49443 | Two thin categories are is... |
| thincciso2 49444 | Categories isomorphic to a... |
| thincciso3 49445 | Categories isomorphic to a... |
| thincciso4 49446 | Two isomorphic categories ... |
| 0thincg 49447 | Any structure with an empt... |
| 0thinc 49448 | The empty category (see ~ ... |
| indcthing 49449 | An indiscrete category, i.... |
| discthing 49450 | A discrete category, i.e.,... |
| indthinc 49451 | An indiscrete category in ... |
| indthincALT 49452 | An alternate proof of ~ in... |
| prsthinc 49453 | Preordered sets as categor... |
| setcthin 49454 | A category of sets all of ... |
| setc2othin 49455 | The category ` ( SetCat ``... |
| thincsect 49456 | In a thin category, one mo... |
| thincsect2 49457 | In a thin category, ` F ` ... |
| thincinv 49458 | In a thin category, ` F ` ... |
| thinciso 49459 | In a thin category, ` F : ... |
| thinccic 49460 | In a thin category, two ob... |
| istermc 49463 | The predicate "is a termin... |
| istermc2 49464 | The predicate "is a termin... |
| istermc3 49465 | The predicate "is a termin... |
| termcthin 49466 | A terminal category is a t... |
| termcthind 49467 | A terminal category is a t... |
| termccd 49468 | A terminal category is a c... |
| termcbas 49469 | The base of a terminal cat... |
| termco 49470 | The object of a terminal c... |
| termcbas2 49471 | The base of a terminal cat... |
| termcbasmo 49472 | Two objects in a terminal ... |
| termchomn0 49473 | All hom-sets of a terminal... |
| termchommo 49474 | All morphisms of a termina... |
| termcid 49475 | The morphism of a terminal... |
| termcid2 49476 | The morphism of a terminal... |
| termchom 49477 | The hom-set of a terminal ... |
| termchom2 49478 | The hom-set of a terminal ... |
| setcsnterm 49479 | The category of one set, e... |
| setc1oterm 49480 | The category ` ( SetCat ``... |
| setc1obas 49481 | The base of the trivial ca... |
| setc1ohomfval 49482 | Set of morphisms of the tr... |
| setc1ocofval 49483 | Composition in the trivial... |
| setc1oid 49484 | The identity morphism of t... |
| funcsetc1ocl 49485 | The functor to the trivial... |
| funcsetc1o 49486 | Value of the functor to th... |
| isinito2lem 49487 | The predicate "is an initi... |
| isinito2 49488 | The predicate "is an initi... |
| isinito3 49489 | The predicate "is an initi... |
| dfinito4 49490 | An alternate definition of... |
| dftermo4 49491 | An alternate definition of... |
| termcpropd 49492 | Two structures with the sa... |
| oppctermhom 49493 | The opposite category of a... |
| oppctermco 49494 | The opposite category of a... |
| oppcterm 49495 | The opposite category of a... |
| functermclem 49496 | Lemma for ~ functermc . (... |
| functermc 49497 | Functor to a terminal cate... |
| functermc2 49498 | Functor to a terminal cate... |
| functermceu 49499 | There exists a unique func... |
| fulltermc 49500 | A functor to a terminal ca... |
| fulltermc2 49501 | Given a full functor to a ... |
| termcterm 49502 | A terminal category is a t... |
| termcterm2 49503 | A terminal object of the c... |
| termcterm3 49504 | In the category of small c... |
| termcciso 49505 | A category is isomorphic t... |
| termccisoeu 49506 | The isomorphism between te... |
| termc2 49507 | If there exists a unique f... |
| termc 49508 | Alternate definition of ` ... |
| dftermc2 49509 | Alternate definition of ` ... |
| eufunclem 49510 | If there exists a unique f... |
| eufunc 49511 | If there exists a unique f... |
| idfudiag1lem 49512 | Lemma for ~ idfudiag1bas a... |
| idfudiag1bas 49513 | If the identity functor of... |
| idfudiag1 49514 | If the identity functor of... |
| euendfunc 49515 | If there exists a unique e... |
| euendfunc2 49516 | If there exists a unique e... |
| termcarweu 49517 | There exists a unique disj... |
| arweuthinc 49518 | If a structure has a uniqu... |
| arweutermc 49519 | If a structure has a uniqu... |
| dftermc3 49520 | Alternate definition of ` ... |
| termcfuncval 49521 | The value of a functor fro... |
| diag1f1olem 49522 | To any functor from a term... |
| diag1f1o 49523 | The object part of the dia... |
| termcnatval 49524 | Value of natural transform... |
| diag2f1olem 49525 | Lemma for ~ diag2f1o . (C... |
| diag2f1o 49526 | If ` D ` is terminal, the ... |
| diagffth 49527 | The diagonal functor is a ... |
| diagciso 49528 | The diagonal functor is an... |
| diagcic 49529 | Any category ` C ` is isom... |
| funcsn 49530 | The category of one functo... |
| fucterm 49531 | The category of functors t... |
| 0fucterm 49532 | The category of functors f... |
| termfucterm 49533 | All functors between two t... |
| cofuterm 49534 | Post-compose with a functo... |
| uobeqterm 49535 | Universal objects and term... |
| isinito4 49536 | The predicate "is an initi... |
| isinito4a 49537 | The predicate "is an initi... |
| prstcval 49540 | Lemma for ~ prstcnidlem an... |
| prstcnidlem 49541 | Lemma for ~ prstcnid and ~... |
| prstcnid 49542 | Components other than ` Ho... |
| prstcbas 49543 | The base set is unchanged.... |
| prstcleval 49544 | Value of the less-than-or-... |
| prstcle 49545 | Value of the less-than-or-... |
| prstcocval 49546 | Orthocomplementation is un... |
| prstcoc 49547 | Orthocomplementation is un... |
| prstchomval 49548 | Hom-sets of the constructe... |
| prstcprs 49549 | The category is a preorder... |
| prstcthin 49550 | The preordered set is equi... |
| prstchom 49551 | Hom-sets of the constructe... |
| prstchom2 49552 | Hom-sets of the constructe... |
| prstchom2ALT 49553 | Hom-sets of the constructe... |
| oduoppcbas 49554 | The dual of a preordered s... |
| oduoppcciso 49555 | The dual of a preordered s... |
| postcpos 49556 | The converted category is ... |
| postcposALT 49557 | Alternate proof of ~ postc... |
| postc 49558 | The converted category is ... |
| discsntermlem 49559 | A singlegon is an element ... |
| basrestermcfolem 49560 | An element of the class of... |
| discbas 49561 | A discrete category (a cat... |
| discthin 49562 | A discrete category (a cat... |
| discsnterm 49563 | A discrete category (a cat... |
| basrestermcfo 49564 | The base function restrict... |
| termcnex 49565 | The class of all terminal ... |
| mndtcval 49568 | Value of the category buil... |
| mndtcbasval 49569 | The base set of the catego... |
| mndtcbas 49570 | The category built from a ... |
| mndtcob 49571 | Lemma for ~ mndtchom and ~... |
| mndtcbas2 49572 | Two objects in a category ... |
| mndtchom 49573 | The only hom-set of the ca... |
| mndtcco 49574 | The composition of the cat... |
| mndtcco2 49575 | The composition of the cat... |
| mndtccatid 49576 | Lemma for ~ mndtccat and ~... |
| mndtccat 49577 | The function value is a ca... |
| mndtcid 49578 | The identity morphism, or ... |
| oppgoppchom 49579 | The converted opposite mon... |
| oppgoppcco 49580 | The converted opposite mon... |
| oppgoppcid 49581 | The converted opposite mon... |
| grptcmon 49582 | All morphisms in a categor... |
| grptcepi 49583 | All morphisms in a categor... |
| 2arwcatlem1 49584 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem2 49585 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem3 49586 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem4 49587 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem5 49588 | Lemma for ~ 2arwcat . (Co... |
| 2arwcat 49589 | The condition for a struct... |
| incat 49590 | Constructing a category wi... |
| setc1onsubc 49591 | Construct a category with ... |
| cnelsubclem 49592 | Lemma for ~ cnelsubc . (C... |
| cnelsubc 49593 | Remark 4.2(2) of [Adamek] ... |
| lanfn 49598 | ` Lan ` is a function on `... |
| ranfn 49599 | ` Ran ` is a function on `... |
| reldmlan 49600 | The domain of ` Lan ` is a... |
| reldmran 49601 | The domain of ` Ran ` is a... |
| lanfval 49602 | Value of the function gene... |
| ranfval 49603 | Value of the function gene... |
| lanpropd 49604 | If the categories have the... |
| ranpropd 49605 | If the categories have the... |
| reldmlan2 49606 | The domain of ` ( P Lan E ... |
| reldmran2 49607 | The domain of ` ( P Ran E ... |
| lanval 49608 | Value of the set of left K... |
| ranval 49609 | Value of the set of right ... |
| lanrcl 49610 | Reverse closure for left K... |
| ranrcl 49611 | Reverse closure for right ... |
| rellan 49612 | The set of left Kan extens... |
| relran 49613 | The set of right Kan exten... |
| islan 49614 | A left Kan extension is a ... |
| islan2 49615 | A left Kan extension is a ... |
| lanval2 49616 | The set of left Kan extens... |
| isran 49617 | A right Kan extension is a... |
| isran2 49618 | A right Kan extension is a... |
| ranval2 49619 | The set of right Kan exten... |
| ranval3 49620 | The set of right Kan exten... |
| lanrcl2 49621 | Reverse closure for left K... |
| lanrcl3 49622 | Reverse closure for left K... |
| lanrcl4 49623 | The first component of a l... |
| lanrcl5 49624 | The second component of a ... |
| ranrcl2 49625 | Reverse closure for right ... |
| ranrcl3 49626 | Reverse closure for right ... |
| ranrcl4lem 49627 | Lemma for ~ ranrcl4 and ~ ... |
| ranrcl4 49628 | The first component of a r... |
| ranrcl5 49629 | The second component of a ... |
| lanup 49630 | The universal property of ... |
| ranup 49631 | The universal property of ... |
| reldmlmd 49636 | The domain of ` Limit ` is... |
| reldmcmd 49637 | The domain of ` Colimit ` ... |
| lmdfval 49638 | Function value of ` Limit ... |
| cmdfval 49639 | Function value of ` Colimi... |
| lmdrcl 49640 | Reverse closure for a limi... |
| cmdrcl 49641 | Reverse closure for a coli... |
| reldmlmd2 49642 | The domain of ` ( C Limit ... |
| reldmcmd2 49643 | The domain of ` ( C Colimi... |
| lmdfval2 49644 | The set of limits of a dia... |
| cmdfval2 49645 | The set of colimits of a d... |
| lmdpropd 49646 | If the categories have the... |
| cmdpropd 49647 | If the categories have the... |
| rellmd 49648 | The set of limits of a dia... |
| relcmd 49649 | The set of colimits of a d... |
| concl 49650 | A natural transformation f... |
| coccl 49651 | A natural transformation t... |
| concom 49652 | A cone to a diagram commut... |
| coccom 49653 | A co-cone to a diagram com... |
| islmd 49654 | The universal property of ... |
| iscmd 49655 | The universal property of ... |
| lmddu 49656 | The duality of limits and ... |
| cmddu 49657 | The duality of limits and ... |
| initocmd 49658 | Initial objects are the ob... |
| termolmd 49659 | Terminal objects are the o... |
| lmdran 49660 | To each limit of a diagram... |
| cmdlan 49661 | To each colimit of a diagr... |
| nfintd 49662 | Bound-variable hypothesis ... |
| nfiund 49663 | Bound-variable hypothesis ... |
| nfiundg 49664 | Bound-variable hypothesis ... |
| iunord 49665 | The indexed union of a col... |
| iunordi 49666 | The indexed union of a col... |
| spd 49667 | Specialization deduction, ... |
| spcdvw 49668 | A version of ~ spcdv where... |
| tfis2d 49669 | Transfinite Induction Sche... |
| bnd2d 49670 | Deduction form of ~ bnd2 .... |
| dffun3f 49671 | Alternate definition of fu... |
| setrecseq 49674 | Equality theorem for set r... |
| nfsetrecs 49675 | Bound-variable hypothesis ... |
| setrec1lem1 49676 | Lemma for ~ setrec1 . Thi... |
| setrec1lem2 49677 | Lemma for ~ setrec1 . If ... |
| setrec1lem3 49678 | Lemma for ~ setrec1 . If ... |
| setrec1lem4 49679 | Lemma for ~ setrec1 . If ... |
| setrec1 49680 | This is the first of two f... |
| setrec2fun 49681 | This is the second of two ... |
| setrec2lem1 49682 | Lemma for ~ setrec2 . The... |
| setrec2lem2 49683 | Lemma for ~ setrec2 . The... |
| setrec2 49684 | This is the second of two ... |
| setrec2v 49685 | Version of ~ setrec2 with ... |
| setrec2mpt 49686 | Version of ~ setrec2 where... |
| setis 49687 | Version of ~ setrec2 expre... |
| elsetrecslem 49688 | Lemma for ~ elsetrecs . A... |
| elsetrecs 49689 | A set ` A ` is an element ... |
| setrecsss 49690 | The ` setrecs ` operator r... |
| setrecsres 49691 | A recursively generated cl... |
| vsetrec 49692 | Construct ` _V ` using set... |
| 0setrec 49693 | If a function sends the em... |
| onsetreclem1 49694 | Lemma for ~ onsetrec . (C... |
| onsetreclem2 49695 | Lemma for ~ onsetrec . (C... |
| onsetreclem3 49696 | Lemma for ~ onsetrec . (C... |
| onsetrec 49697 | Construct ` On ` using set... |
| elpglem1 49700 | Lemma for ~ elpg . (Contr... |
| elpglem2 49701 | Lemma for ~ elpg . (Contr... |
| elpglem3 49702 | Lemma for ~ elpg . (Contr... |
| elpg 49703 | Membership in the class of... |
| pgindlem 49704 | Lemma for ~ pgind . (Cont... |
| pgindnf 49705 | Version of ~ pgind with ex... |
| pgind 49706 | Induction on partizan game... |
| sbidd 49707 | An identity theorem for su... |
| sbidd-misc 49708 | An identity theorem for su... |
| gte-lte 49713 | Simple relationship betwee... |
| gt-lt 49714 | Simple relationship betwee... |
| gte-lteh 49715 | Relationship between ` <_ ... |
| gt-lth 49716 | Relationship between ` < `... |
| ex-gt 49717 | Simple example of ` > ` , ... |
| ex-gte 49718 | Simple example of ` >_ ` ,... |
| sinhval-named 49725 | Value of the named sinh fu... |
| coshval-named 49726 | Value of the named cosh fu... |
| tanhval-named 49727 | Value of the named tanh fu... |
| sinh-conventional 49728 | Conventional definition of... |
| sinhpcosh 49729 | Prove that ` ( sinh `` A )... |
| secval 49736 | Value of the secant functi... |
| cscval 49737 | Value of the cosecant func... |
| cotval 49738 | Value of the cotangent fun... |
| seccl 49739 | The closure of the secant ... |
| csccl 49740 | The closure of the cosecan... |
| cotcl 49741 | The closure of the cotange... |
| reseccl 49742 | The closure of the secant ... |
| recsccl 49743 | The closure of the cosecan... |
| recotcl 49744 | The closure of the cotange... |
| recsec 49745 | The reciprocal of secant i... |
| reccsc 49746 | The reciprocal of cosecant... |
| reccot 49747 | The reciprocal of cotangen... |
| rectan 49748 | The reciprocal of tangent ... |
| sec0 49749 | The value of the secant fu... |
| onetansqsecsq 49750 | Prove the tangent squared ... |
| cotsqcscsq 49751 | Prove the tangent squared ... |
| ifnmfalse 49752 | If A is not a member of B,... |
| logb2aval 49753 | Define the value of the ` ... |
| mvlraddi 49760 | Move the right term in a s... |
| assraddsubi 49761 | Associate RHS addition-sub... |
| joinlmuladdmuli 49762 | Join AB+CB into (A+C) on L... |
| joinlmulsubmuld 49763 | Join AB-CB into (A-C) on L... |
| joinlmulsubmuli 49764 | Join AB-CB into (A-C) on L... |
| mvlrmuld 49765 | Move the right term in a p... |
| mvlrmuli 49766 | Move the right term in a p... |
| i2linesi 49767 | Solve for the intersection... |
| i2linesd 49768 | Solve for the intersection... |
| alimp-surprise 49769 | Demonstrate that when usin... |
| alimp-no-surprise 49770 | There is no "surprise" in ... |
| empty-surprise 49771 | Demonstrate that when usin... |
| empty-surprise2 49772 | "Prove" that false is true... |
| eximp-surprise 49773 | Show what implication insi... |
| eximp-surprise2 49774 | Show that "there exists" w... |
| alsconv 49779 | There is an equivalence be... |
| alsi1d 49780 | Deduction rule: Given "al... |
| alsi2d 49781 | Deduction rule: Given "al... |
| alsc1d 49782 | Deduction rule: Given "al... |
| alsc2d 49783 | Deduction rule: Given "al... |
| alscn0d 49784 | Deduction rule: Given "al... |
| alsi-no-surprise 49785 | Demonstrate that there is ... |
| 5m4e1 49786 | Prove that 5 - 4 = 1. (Co... |
| 2p2ne5 49787 | Prove that ` 2 + 2 =/= 5 `... |
| resolution 49788 | Resolution rule. This is ... |
| testable 49789 | In classical logic all wff... |
| aacllem 49790 | Lemma for other theorems a... |
| amgmwlem 49791 | Weighted version of ~ amgm... |
| amgmlemALT 49792 | Alternate proof of ~ amgml... |
| amgmw2d 49793 | Weighted arithmetic-geomet... |
| young2d 49794 | Young's inequality for ` n... |
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