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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... |
| pm5.32d 576 | Distribution of implicatio... |
| pm5.32rd 577 | Distribution of implicatio... |
| pm5.32da 578 | Distribution of implicatio... |
| sylan 579 | A syllogism inference. (C... |
| sylanb 580 | A syllogism inference. (C... |
| sylanbr 581 | A syllogism inference. (C... |
| sylanbrc 582 | Syllogism inference. (Con... |
| syl2anc 583 | Syllogism inference combin... |
| syl2anc2 584 | Double syllogism inference... |
| sylancl 585 | Syllogism inference combin... |
| sylancr 586 | Syllogism inference combin... |
| sylancom 587 | Syllogism inference with c... |
| sylanblc 588 | Syllogism inference combin... |
| sylanblrc 589 | Syllogism inference combin... |
| syldan 590 | A syllogism deduction with... |
| sylbida 591 | A syllogism deduction. (C... |
| sylan2 592 | A syllogism inference. (C... |
| sylan2b 593 | A syllogism inference. (C... |
| sylan2br 594 | A syllogism inference. (C... |
| syl2an 595 | A double syllogism inferen... |
| syl2anr 596 | A double syllogism inferen... |
| syl2anb 597 | A double syllogism inferen... |
| syl2anbr 598 | A double syllogism inferen... |
| sylancb 599 | A syllogism inference comb... |
| sylancbr 600 | A syllogism inference comb... |
| syldanl 601 | A syllogism deduction with... |
| syland 602 | A syllogism deduction. (C... |
| sylani 603 | A syllogism inference. (C... |
| sylan2d 604 | A syllogism deduction. (C... |
| sylan2i 605 | A syllogism inference. (C... |
| syl2ani 606 | A syllogism inference. (C... |
| syl2and 607 | A syllogism deduction. (C... |
| anim12d 608 | Conjoin antecedents and co... |
| anim12d1 609 | Variant of ~ anim12d where... |
| anim1d 610 | Add a conjunct to right of... |
| anim2d 611 | Add a conjunct to left of ... |
| anim12i 612 | Conjoin antecedents and co... |
| anim12ci 613 | Variant of ~ anim12i with ... |
| anim1i 614 | Introduce conjunct to both... |
| anim1ci 615 | Introduce conjunct to both... |
| anim2i 616 | Introduce conjunct to both... |
| anim12ii 617 | Conjoin antecedents and co... |
| anim12dan 618 | Conjoin antecedents and co... |
| im2anan9 619 | Deduction joining nested i... |
| im2anan9r 620 | Deduction joining nested i... |
| pm3.45 621 | Theorem *3.45 (Fact) of [W... |
| anbi2i 622 | Introduce a left conjunct ... |
| anbi1i 623 | Introduce a right conjunct... |
| anbi2ci 624 | Variant of ~ anbi2i with c... |
| anbi1ci 625 | Variant of ~ anbi1i with c... |
| bianbi 626 | Exchanging conjunction in ... |
| anbi12i 627 | Conjoin both sides of two ... |
| anbi12ci 628 | Variant of ~ anbi12i with ... |
| anbi2d 629 | Deduction adding a left co... |
| anbi1d 630 | Deduction adding a right c... |
| anbi12d 631 | Deduction joining two equi... |
| anbi1 632 | Introduce a right conjunct... |
| anbi2 633 | Introduce a left conjunct ... |
| anbi1cd 634 | Introduce a proposition as... |
| an2anr 635 | Double commutation in conj... |
| pm4.38 636 | Theorem *4.38 of [Whitehea... |
| bi2anan9 637 | Deduction joining two equi... |
| bi2anan9r 638 | Deduction joining two equi... |
| bi2bian9 639 | Deduction joining two bico... |
| anbiim 640 | Adding biconditional when ... |
| bianass 641 | An inference to merge two ... |
| bianassc 642 | An inference to merge two ... |
| an21 643 | Swap two conjuncts. (Cont... |
| an12 644 | Swap two conjuncts. Note ... |
| an32 645 | A rearrangement of conjunc... |
| an13 646 | A rearrangement of conjunc... |
| an31 647 | A rearrangement of conjunc... |
| an12s 648 | Swap two conjuncts in ante... |
| ancom2s 649 | Inference commuting a nest... |
| an13s 650 | Swap two conjuncts in ante... |
| an32s 651 | Swap two conjuncts in ante... |
| ancom1s 652 | Inference commuting a nest... |
| an31s 653 | Swap two conjuncts in ante... |
| anass1rs 654 | Commutative-associative la... |
| an4 655 | Rearrangement of 4 conjunc... |
| an42 656 | Rearrangement of 4 conjunc... |
| an43 657 | Rearrangement of 4 conjunc... |
| an3 658 | A rearrangement of conjunc... |
| an4s 659 | Inference rearranging 4 co... |
| an42s 660 | Inference rearranging 4 co... |
| anabs1 661 | Absorption into embedded c... |
| anabs5 662 | Absorption into embedded c... |
| anabs7 663 | Absorption into embedded c... |
| anabsan 664 | Absorption of antecedent w... |
| anabss1 665 | Absorption of antecedent i... |
| anabss4 666 | Absorption of antecedent i... |
| anabss5 667 | Absorption of antecedent i... |
| anabsi5 668 | Absorption of antecedent i... |
| anabsi6 669 | Absorption of antecedent i... |
| anabsi7 670 | Absorption of antecedent i... |
| anabsi8 671 | Absorption of antecedent i... |
| anabss7 672 | Absorption of antecedent i... |
| anabsan2 673 | Absorption of antecedent w... |
| anabss3 674 | Absorption of antecedent i... |
| anandi 675 | Distribution of conjunctio... |
| anandir 676 | Distribution of conjunctio... |
| anandis 677 | Inference that undistribut... |
| anandirs 678 | Inference that undistribut... |
| sylanl1 679 | A syllogism inference. (C... |
| sylanl2 680 | A syllogism inference. (C... |
| sylanr1 681 | A syllogism inference. (C... |
| sylanr2 682 | A syllogism inference. (C... |
| syl6an 683 | A syllogism deduction comb... |
| syl2an2r 684 | ~ syl2anr with antecedents... |
| syl2an2 685 | ~ syl2an with antecedents ... |
| mpdan 686 | An inference based on modu... |
| mpancom 687 | An inference based on modu... |
| mpidan 688 | A deduction which "stacks"... |
| mpan 689 | An inference based on modu... |
| mpan2 690 | An inference based on modu... |
| mp2an 691 | An inference based on modu... |
| mp4an 692 | An inference based on modu... |
| mpan2d 693 | A deduction based on modus... |
| mpand 694 | A deduction based on modus... |
| mpani 695 | An inference based on modu... |
| mpan2i 696 | An inference based on modu... |
| mp2ani 697 | An inference based on modu... |
| mp2and 698 | A deduction based on modus... |
| mpanl1 699 | An inference based on modu... |
| mpanl2 700 | An inference based on modu... |
| mpanl12 701 | An inference based on modu... |
| mpanr1 702 | An inference based on modu... |
| mpanr2 703 | An inference based on modu... |
| mpanr12 704 | An inference based on modu... |
| mpanlr1 705 | An inference based on modu... |
| mpbirand 706 | Detach truth from conjunct... |
| mpbiran2d 707 | Detach truth from conjunct... |
| mpbiran 708 | Detach truth from conjunct... |
| mpbiran2 709 | Detach truth from conjunct... |
| mpbir2an 710 | Detach a conjunction of tr... |
| mpbi2and 711 | Detach a conjunction of tr... |
| mpbir2and 712 | Detach a conjunction of tr... |
| adantll 713 | Deduction adding a conjunc... |
| adantlr 714 | Deduction adding a conjunc... |
| adantrl 715 | Deduction adding a conjunc... |
| adantrr 716 | Deduction adding a conjunc... |
| adantlll 717 | Deduction adding a conjunc... |
| adantllr 718 | Deduction adding a conjunc... |
| adantlrl 719 | Deduction adding a conjunc... |
| adantlrr 720 | Deduction adding a conjunc... |
| adantrll 721 | Deduction adding a conjunc... |
| adantrlr 722 | Deduction adding a conjunc... |
| adantrrl 723 | Deduction adding a conjunc... |
| adantrrr 724 | Deduction adding a conjunc... |
| ad2antrr 725 | Deduction adding two conju... |
| ad2antlr 726 | Deduction adding two conju... |
| ad2antrl 727 | Deduction adding two conju... |
| ad2antll 728 | Deduction adding conjuncts... |
| ad3antrrr 729 | Deduction adding three con... |
| ad3antlr 730 | Deduction adding three con... |
| ad4antr 731 | Deduction adding 4 conjunc... |
| ad4antlr 732 | Deduction adding 4 conjunc... |
| ad5antr 733 | Deduction adding 5 conjunc... |
| ad5antlr 734 | Deduction adding 5 conjunc... |
| ad6antr 735 | Deduction adding 6 conjunc... |
| ad6antlr 736 | Deduction adding 6 conjunc... |
| ad7antr 737 | Deduction adding 7 conjunc... |
| ad7antlr 738 | Deduction adding 7 conjunc... |
| ad8antr 739 | Deduction adding 8 conjunc... |
| ad8antlr 740 | Deduction adding 8 conjunc... |
| ad9antr 741 | Deduction adding 9 conjunc... |
| ad9antlr 742 | Deduction adding 9 conjunc... |
| ad10antr 743 | Deduction adding 10 conjun... |
| ad10antlr 744 | Deduction adding 10 conjun... |
| ad2ant2l 745 | Deduction adding two conju... |
| ad2ant2r 746 | Deduction adding two conju... |
| ad2ant2lr 747 | Deduction adding two conju... |
| ad2ant2rl 748 | Deduction adding two conju... |
| adantl3r 749 | Deduction adding 1 conjunc... |
| ad4ant13 750 | Deduction adding conjuncts... |
| ad4ant14 751 | Deduction adding conjuncts... |
| ad4ant23 752 | Deduction adding conjuncts... |
| ad4ant24 753 | Deduction adding conjuncts... |
| adantl4r 754 | Deduction adding 1 conjunc... |
| ad5ant12 755 | Deduction adding conjuncts... |
| ad5ant13 756 | Deduction adding conjuncts... |
| ad5ant14 757 | Deduction adding conjuncts... |
| ad5ant15 758 | Deduction adding conjuncts... |
| ad5ant23 759 | Deduction adding conjuncts... |
| ad5ant24 760 | Deduction adding conjuncts... |
| ad5ant25 761 | Deduction adding conjuncts... |
| adantl5r 762 | Deduction adding 1 conjunc... |
| adantl6r 763 | Deduction adding 1 conjunc... |
| pm3.33 764 | Theorem *3.33 (Syll) of [W... |
| pm3.34 765 | Theorem *3.34 (Syll) of [W... |
| simpll 766 | Simplification of a conjun... |
| simplld 767 | Deduction form of ~ simpll... |
| simplr 768 | Simplification of a conjun... |
| simplrd 769 | Deduction eliminating a do... |
| simprl 770 | Simplification of a conjun... |
| simprld 771 | Deduction eliminating a do... |
| simprr 772 | Simplification of a conjun... |
| simprrd 773 | Deduction form of ~ simprr... |
| simplll 774 | Simplification of a conjun... |
| simpllr 775 | Simplification of a conjun... |
| simplrl 776 | Simplification of a conjun... |
| simplrr 777 | Simplification of a conjun... |
| simprll 778 | Simplification of a conjun... |
| simprlr 779 | Simplification of a conjun... |
| simprrl 780 | Simplification of a conjun... |
| simprrr 781 | Simplification of a conjun... |
| simp-4l 782 | Simplification of a conjun... |
| simp-4r 783 | Simplification of a conjun... |
| simp-5l 784 | Simplification of a conjun... |
| simp-5r 785 | Simplification of a conjun... |
| simp-6l 786 | Simplification of a conjun... |
| simp-6r 787 | Simplification of a conjun... |
| simp-7l 788 | Simplification of a conjun... |
| simp-7r 789 | Simplification of a conjun... |
| simp-8l 790 | Simplification of a conjun... |
| simp-8r 791 | Simplification of a conjun... |
| simp-9l 792 | Simplification of a conjun... |
| simp-9r 793 | Simplification of a conjun... |
| simp-10l 794 | Simplification of a conjun... |
| simp-10r 795 | Simplification of a conjun... |
| simp-11l 796 | Simplification of a conjun... |
| simp-11r 797 | Simplification of a conjun... |
| pm2.01da 798 | Deduction based on reducti... |
| pm2.18da 799 | Deduction based on reducti... |
| impbida 800 | Deduce an equivalence from... |
| pm5.21nd 801 | Eliminate an antecedent im... |
| pm3.35 802 | Conjunctive detachment. T... |
| pm5.74da 803 | Distribution of implicatio... |
| bitr 804 | Theorem *4.22 of [Whitehea... |
| biantr 805 | A transitive law of equiva... |
| pm4.14 806 | Theorem *4.14 of [Whitehea... |
| pm3.37 807 | Theorem *3.37 (Transp) of ... |
| anim12 808 | Conjoin antecedents and co... |
| pm3.4 809 | Conjunction implies implic... |
| exbiri 810 | Inference form of ~ exbir ... |
| pm2.61ian 811 | Elimination of an antecede... |
| pm2.61dan 812 | Elimination of an antecede... |
| pm2.61ddan 813 | Elimination of two anteced... |
| pm2.61dda 814 | Elimination of two anteced... |
| mtand 815 | A modus tollens deduction.... |
| pm2.65da 816 | Deduction for proof by con... |
| condan 817 | Proof by contradiction. (... |
| biadan 818 | An implication is equivale... |
| biadani 819 | Inference associated with ... |
| biadaniALT 820 | Alternate proof of ~ biada... |
| biadanii 821 | Inference associated with ... |
| biadanid 822 | Deduction associated with ... |
| pm5.1 823 | Two propositions are equiv... |
| pm5.21 824 | Two propositions are equiv... |
| pm5.35 825 | Theorem *5.35 of [Whitehea... |
| abai 826 | Introduce one conjunct as ... |
| pm4.45im 827 | Conjunction with implicati... |
| impimprbi 828 | An implication and its rev... |
| nan 829 | Theorem to move a conjunct... |
| pm5.31 830 | Theorem *5.31 of [Whitehea... |
| pm5.31r 831 | Variant of ~ pm5.31 . (Co... |
| pm4.15 832 | Theorem *4.15 of [Whitehea... |
| pm5.36 833 | Theorem *5.36 of [Whitehea... |
| annotanannot 834 | A conjunction with a negat... |
| pm5.33 835 | Theorem *5.33 of [Whitehea... |
| syl12anc 836 | Syllogism combined with co... |
| syl21anc 837 | Syllogism combined with co... |
| syl22anc 838 | Syllogism combined with co... |
| syl1111anc 839 | Four-hypothesis eliminatio... |
| syldbl2 840 | Stacked hypotheseis implie... |
| mpsyl4anc 841 | An elimination deduction. ... |
| pm4.87 842 | Theorem *4.87 of [Whitehea... |
| bimsc1 843 | Removal of conjunct from o... |
| a2and 844 | Deduction distributing a c... |
| animpimp2impd 845 | Deduction deriving nested ... |
| pm4.64 848 | Theorem *4.64 of [Whitehea... |
| pm4.66 849 | Theorem *4.66 of [Whitehea... |
| pm2.53 850 | Theorem *2.53 of [Whitehea... |
| pm2.54 851 | Theorem *2.54 of [Whitehea... |
| imor 852 | Implication in terms of di... |
| imori 853 | Infer disjunction from imp... |
| imorri 854 | Infer implication from dis... |
| pm4.62 855 | Theorem *4.62 of [Whitehea... |
| jaoi 856 | Inference disjoining the a... |
| jao1i 857 | Add a disjunct in the ante... |
| jaod 858 | Deduction disjoining the a... |
| mpjaod 859 | Eliminate a disjunction in... |
| ori 860 | Infer implication from dis... |
| orri 861 | Infer disjunction from imp... |
| orrd 862 | Deduce disjunction from im... |
| ord 863 | Deduce implication from di... |
| orci 864 | Deduction introducing a di... |
| olci 865 | Deduction introducing a di... |
| orc 866 | Introduction of a disjunct... |
| olc 867 | Introduction of a disjunct... |
| pm1.4 868 | Axiom *1.4 of [WhiteheadRu... |
| orcom 869 | Commutative law for disjun... |
| orcomd 870 | Commutation of disjuncts i... |
| orcoms 871 | Commutation of disjuncts i... |
| orcd 872 | Deduction introducing a di... |
| olcd 873 | Deduction introducing a di... |
| orcs 874 | Deduction eliminating disj... |
| olcs 875 | Deduction eliminating disj... |
| olcnd 876 | A lemma for Conjunctive No... |
| orcnd 877 | A lemma for Conjunctive No... |
| mtord 878 | A modus tollens deduction ... |
| pm3.2ni 879 | Infer negated disjunction ... |
| pm2.45 880 | Theorem *2.45 of [Whitehea... |
| pm2.46 881 | Theorem *2.46 of [Whitehea... |
| pm2.47 882 | Theorem *2.47 of [Whitehea... |
| pm2.48 883 | Theorem *2.48 of [Whitehea... |
| pm2.49 884 | Theorem *2.49 of [Whitehea... |
| norbi 885 | If neither of two proposit... |
| nbior 886 | If two propositions are no... |
| orel1 887 | Elimination of disjunction... |
| pm2.25 888 | Theorem *2.25 of [Whitehea... |
| orel2 889 | Elimination of disjunction... |
| pm2.67-2 890 | Slight generalization of T... |
| pm2.67 891 | Theorem *2.67 of [Whitehea... |
| curryax 892 | A non-intuitionistic posit... |
| exmid 893 | Law of excluded middle, al... |
| exmidd 894 | Law of excluded middle in ... |
| pm2.1 895 | Theorem *2.1 of [Whitehead... |
| pm2.13 896 | Theorem *2.13 of [Whitehea... |
| pm2.621 897 | Theorem *2.621 of [Whitehe... |
| pm2.62 898 | Theorem *2.62 of [Whitehea... |
| pm2.68 899 | Theorem *2.68 of [Whitehea... |
| dfor2 900 | Logical 'or' expressed in ... |
| pm2.07 901 | Theorem *2.07 of [Whitehea... |
| pm1.2 902 | Axiom *1.2 of [WhiteheadRu... |
| oridm 903 | Idempotent law for disjunc... |
| pm4.25 904 | Theorem *4.25 of [Whitehea... |
| pm2.4 905 | Theorem *2.4 of [Whitehead... |
| pm2.41 906 | Theorem *2.41 of [Whitehea... |
| orim12i 907 | Disjoin antecedents and co... |
| orim1i 908 | Introduce disjunct to both... |
| orim2i 909 | Introduce disjunct to both... |
| orim12dALT 910 | Alternate proof of ~ orim1... |
| orbi2i 911 | Inference adding a left di... |
| orbi1i 912 | Inference adding a right d... |
| orbi12i 913 | Infer the disjunction of t... |
| orbi2d 914 | Deduction adding a left di... |
| orbi1d 915 | Deduction adding a right d... |
| orbi1 916 | Theorem *4.37 of [Whitehea... |
| orbi12d 917 | Deduction joining two equi... |
| pm1.5 918 | Axiom *1.5 (Assoc) of [Whi... |
| or12 919 | Swap two disjuncts. (Cont... |
| orass 920 | Associative law for disjun... |
| pm2.31 921 | Theorem *2.31 of [Whitehea... |
| pm2.32 922 | Theorem *2.32 of [Whitehea... |
| pm2.3 923 | Theorem *2.3 of [Whitehead... |
| or32 924 | A rearrangement of disjunc... |
| or4 925 | Rearrangement of 4 disjunc... |
| or42 926 | Rearrangement of 4 disjunc... |
| orordi 927 | Distribution of disjunctio... |
| orordir 928 | Distribution of disjunctio... |
| orimdi 929 | Disjunction distributes ov... |
| pm2.76 930 | Theorem *2.76 of [Whitehea... |
| pm2.85 931 | Theorem *2.85 of [Whitehea... |
| pm2.75 932 | Theorem *2.75 of [Whitehea... |
| pm4.78 933 | Implication distributes ov... |
| biort 934 | A disjunction with a true ... |
| biorf 935 | A wff is equivalent to its... |
| biortn 936 | A wff is equivalent to its... |
| biorfi 937 | The dual of ~ biorf is not... |
| biorfri 938 | A wff is equivalent to its... |
| biorfriOLD 939 | Obsolete proof of ~ biorfr... |
| pm2.26 940 | Theorem *2.26 of [Whitehea... |
| pm2.63 941 | Theorem *2.63 of [Whitehea... |
| pm2.64 942 | Theorem *2.64 of [Whitehea... |
| pm2.42 943 | Theorem *2.42 of [Whitehea... |
| pm5.11g 944 | A general instance of Theo... |
| pm5.11 945 | Theorem *5.11 of [Whitehea... |
| pm5.12 946 | Theorem *5.12 of [Whitehea... |
| pm5.14 947 | Theorem *5.14 of [Whitehea... |
| pm5.13 948 | Theorem *5.13 of [Whitehea... |
| pm5.55 949 | Theorem *5.55 of [Whitehea... |
| pm4.72 950 | Implication in terms of bi... |
| imimorb 951 | Simplify an implication be... |
| oibabs 952 | Absorption of disjunction ... |
| orbidi 953 | Disjunction distributes ov... |
| pm5.7 954 | Disjunction distributes ov... |
| jaao 955 | Inference conjoining and d... |
| jaoa 956 | Inference disjoining and c... |
| jaoian 957 | Inference disjoining the a... |
| jaodan 958 | Deduction disjoining the a... |
| mpjaodan 959 | Eliminate a disjunction in... |
| pm3.44 960 | Theorem *3.44 of [Whitehea... |
| jao 961 | Disjunction of antecedents... |
| jaob 962 | Disjunction of antecedents... |
| pm4.77 963 | Theorem *4.77 of [Whitehea... |
| pm3.48 964 | Theorem *3.48 of [Whitehea... |
| orim12d 965 | Disjoin antecedents and co... |
| orim1d 966 | Disjoin antecedents and co... |
| orim2d 967 | Disjoin antecedents and co... |
| orim2 968 | Axiom *1.6 (Sum) of [White... |
| pm2.38 969 | Theorem *2.38 of [Whitehea... |
| pm2.36 970 | Theorem *2.36 of [Whitehea... |
| pm2.37 971 | Theorem *2.37 of [Whitehea... |
| pm2.81 972 | Theorem *2.81 of [Whitehea... |
| pm2.8 973 | Theorem *2.8 of [Whitehead... |
| pm2.73 974 | Theorem *2.73 of [Whitehea... |
| pm2.74 975 | Theorem *2.74 of [Whitehea... |
| pm2.82 976 | Theorem *2.82 of [Whitehea... |
| pm4.39 977 | Theorem *4.39 of [Whitehea... |
| animorl 978 | Conjunction implies disjun... |
| animorr 979 | Conjunction implies disjun... |
| animorlr 980 | Conjunction implies disjun... |
| animorrl 981 | Conjunction implies disjun... |
| ianor 982 | Negated conjunction in ter... |
| anor 983 | Conjunction in terms of di... |
| ioran 984 | Negated disjunction in ter... |
| pm4.52 985 | Theorem *4.52 of [Whitehea... |
| pm4.53 986 | Theorem *4.53 of [Whitehea... |
| pm4.54 987 | Theorem *4.54 of [Whitehea... |
| pm4.55 988 | Theorem *4.55 of [Whitehea... |
| pm4.56 989 | Theorem *4.56 of [Whitehea... |
| oran 990 | Disjunction in terms of co... |
| pm4.57 991 | Theorem *4.57 of [Whitehea... |
| pm3.1 992 | Theorem *3.1 of [Whitehead... |
| pm3.11 993 | Theorem *3.11 of [Whitehea... |
| pm3.12 994 | Theorem *3.12 of [Whitehea... |
| pm3.13 995 | Theorem *3.13 of [Whitehea... |
| pm3.14 996 | Theorem *3.14 of [Whitehea... |
| pm4.44 997 | Theorem *4.44 of [Whitehea... |
| pm4.45 998 | Theorem *4.45 of [Whitehea... |
| orabs 999 | Absorption of redundant in... |
| oranabs 1000 | Absorb a disjunct into a c... |
| pm5.61 1001 | Theorem *5.61 of [Whitehea... |
| pm5.6 1002 | Conjunction in antecedent ... |
| orcanai 1003 | Change disjunction in cons... |
| pm4.79 1004 | Theorem *4.79 of [Whitehea... |
| pm5.53 1005 | Theorem *5.53 of [Whitehea... |
| ordi 1006 | Distributive law for disju... |
| ordir 1007 | Distributive law for disju... |
| andi 1008 | Distributive law for conju... |
| andir 1009 | Distributive law for conju... |
| orddi 1010 | Double distributive law fo... |
| anddi 1011 | Double distributive law fo... |
| pm5.17 1012 | Theorem *5.17 of [Whitehea... |
| pm5.15 1013 | Theorem *5.15 of [Whitehea... |
| pm5.16 1014 | Theorem *5.16 of [Whitehea... |
| xor 1015 | Two ways to express exclus... |
| nbi2 1016 | Two ways to express "exclu... |
| xordi 1017 | Conjunction distributes ov... |
| pm5.54 1018 | Theorem *5.54 of [Whitehea... |
| pm5.62 1019 | Theorem *5.62 of [Whitehea... |
| pm5.63 1020 | Theorem *5.63 of [Whitehea... |
| niabn 1021 | Miscellaneous inference re... |
| ninba 1022 | Miscellaneous inference re... |
| pm4.43 1023 | Theorem *4.43 of [Whitehea... |
| pm4.82 1024 | Theorem *4.82 of [Whitehea... |
| pm4.83 1025 | Theorem *4.83 of [Whitehea... |
| pclem6 1026 | Negation inferred from emb... |
| bigolden 1027 | Dijkstra-Scholten's Golden... |
| pm5.71 1028 | Theorem *5.71 of [Whitehea... |
| pm5.75 1029 | Theorem *5.75 of [Whitehea... |
| ecase2d 1030 | Deduction for elimination ... |
| ecase2dOLD 1031 | Obsolete version of ~ ecas... |
| 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 ... |
| ecase3adOLD 1037 | Obsolete version of ~ ecas... |
| ccase 1038 | Inference for combining ca... |
| ccased 1039 | Deduction for combining ca... |
| ccase2 1040 | Inference for combining ca... |
| 4cases 1041 | Inference eliminating two ... |
| 4casesdan 1042 | Deduction eliminating two ... |
| cases 1043 | Case disjunction according... |
| dedlem0a 1044 | Lemma for an alternate ver... |
| dedlem0b 1045 | Lemma for an alternate ver... |
| dedlema 1046 | Lemma for weak deduction t... |
| dedlemb 1047 | Lemma for weak deduction t... |
| cases2 1048 | Case disjunction according... |
| cases2ALT 1049 | Alternate proof of ~ cases... |
| dfbi3 1050 | An alternate definition of... |
| pm5.24 1051 | Theorem *5.24 of [Whitehea... |
| 4exmid 1052 | The disjunction of the fou... |
| consensus 1053 | The consensus theorem. Th... |
| pm4.42 1054 | Theorem *4.42 of [Whitehea... |
| prlem1 1055 | A specialized lemma for se... |
| prlem2 1056 | A specialized lemma for se... |
| oplem1 1057 | A specialized lemma for se... |
| dn1 1058 | A single axiom for Boolean... |
| bianir 1059 | A closed form of ~ mpbir ,... |
| jaoi2 1060 | Inference removing a negat... |
| jaoi3 1061 | Inference separating a dis... |
| ornld 1062 | Selecting one statement fr... |
| dfifp2 1065 | Alternate definition of th... |
| dfifp3 1066 | Alternate definition of th... |
| dfifp4 1067 | Alternate definition of th... |
| dfifp5 1068 | Alternate definition of th... |
| dfifp6 1069 | Alternate definition of th... |
| dfifp7 1070 | Alternate definition of th... |
| ifpdfbi 1071 | Define the biconditional a... |
| anifp 1072 | The conditional operator i... |
| ifpor 1073 | The conditional operator i... |
| ifpn 1074 | Conditional operator for t... |
| ifptru 1075 | Value of the conditional o... |
| ifpfal 1076 | Value of the conditional o... |
| ifpid 1077 | Value of the conditional o... |
| casesifp 1078 | Version of ~ cases express... |
| ifpbi123d 1079 | Equivalence deduction for ... |
| ifpbi23d 1080 | Equivalence deduction for ... |
| ifpimpda 1081 | Separation of the values o... |
| 1fpid3 1082 | The value of the condition... |
| elimh 1083 | Hypothesis builder for the... |
| dedt 1084 | The weak deduction theorem... |
| con3ALT 1085 | Proof of ~ con3 from its a... |
| 3orass 1090 | Associative law for triple... |
| 3orel1 1091 | Partial elimination of a t... |
| 3orrot 1092 | Rotation law for triple di... |
| 3orcoma 1093 | Commutation law for triple... |
| 3orcomb 1094 | Commutation law for triple... |
| 3anass 1095 | Associative law for triple... |
| 3anan12 1096 | Convert triple conjunction... |
| 3anan32 1097 | Convert triple conjunction... |
| 3ancoma 1098 | Commutation law for triple... |
| 3ancomb 1099 | Commutation law for triple... |
| 3anrot 1100 | Rotation law for triple co... |
| 3anrev 1101 | Reversal law for triple co... |
| anandi3 1102 | Distribution of triple con... |
| anandi3r 1103 | Distribution of triple con... |
| 3anidm 1104 | Idempotent law for conjunc... |
| 3an4anass 1105 | Associative law for four c... |
| 3ioran 1106 | Negated triple disjunction... |
| 3ianor 1107 | Negated triple conjunction... |
| 3anor 1108 | Triple conjunction express... |
| 3oran 1109 | Triple disjunction in term... |
| 3impa 1110 | Importation from double to... |
| 3imp 1111 | Importation inference. (C... |
| 3imp31 1112 | The importation inference ... |
| 3imp231 1113 | Importation inference. (C... |
| 3imp21 1114 | The importation inference ... |
| 3impb 1115 | Importation from double to... |
| 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... |
| 3adantl1 1166 | Deduction adding a conjunc... |
| 3adantl2 1167 | Deduction adding a conjunc... |
| 3adantl3 1168 | Deduction adding a conjunc... |
| 3adantr1 1169 | Deduction adding a conjunc... |
| 3adantr2 1170 | Deduction adding a conjunc... |
| 3adantr3 1171 | Deduction adding a conjunc... |
| ad4ant123 1172 | Deduction adding conjuncts... |
| ad4ant124 1173 | Deduction adding conjuncts... |
| ad4ant134 1174 | Deduction adding conjuncts... |
| ad4ant234 1175 | Deduction adding conjuncts... |
| 3adant1l 1176 | Deduction adding a conjunc... |
| 3adant1r 1177 | Deduction adding a conjunc... |
| 3adant2l 1178 | Deduction adding a conjunc... |
| 3adant2r 1179 | Deduction adding a conjunc... |
| 3adant3l 1180 | Deduction adding a conjunc... |
| 3adant3r 1181 | Deduction adding a conjunc... |
| 3adant3r1 1182 | Deduction adding a conjunc... |
| 3adant3r2 1183 | Deduction adding a conjunc... |
| 3adant3r3 1184 | Deduction adding a conjunc... |
| 3ad2antl1 1185 | Deduction adding conjuncts... |
| 3ad2antl2 1186 | Deduction adding conjuncts... |
| 3ad2antl3 1187 | Deduction adding conjuncts... |
| 3ad2antr1 1188 | Deduction adding conjuncts... |
| 3ad2antr2 1189 | Deduction adding conjuncts... |
| 3ad2antr3 1190 | Deduction adding conjuncts... |
| simpl1 1191 | Simplification of conjunct... |
| simpl2 1192 | Simplification of conjunct... |
| simpl3 1193 | Simplification of conjunct... |
| simpr1 1194 | Simplification of conjunct... |
| simpr2 1195 | Simplification of conjunct... |
| simpr3 1196 | Simplification of conjunct... |
| simp1l 1197 | Simplification of triple c... |
| simp1r 1198 | Simplification of triple c... |
| simp2l 1199 | Simplification of triple c... |
| simp2r 1200 | Simplification of triple c... |
| simp3l 1201 | Simplification of triple c... |
| simp3r 1202 | Simplification of triple c... |
| simp11 1203 | Simplification of doubly t... |
| simp12 1204 | Simplification of doubly t... |
| simp13 1205 | Simplification of doubly t... |
| simp21 1206 | Simplification of doubly t... |
| simp22 1207 | Simplification of doubly t... |
| simp23 1208 | Simplification of doubly t... |
| simp31 1209 | Simplification of doubly t... |
| simp32 1210 | Simplification of doubly t... |
| simp33 1211 | Simplification of doubly t... |
| simpll1 1212 | Simplification of conjunct... |
| simpll2 1213 | Simplification of conjunct... |
| simpll3 1214 | Simplification of conjunct... |
| simplr1 1215 | Simplification of conjunct... |
| simplr2 1216 | Simplification of conjunct... |
| simplr3 1217 | Simplification of conjunct... |
| simprl1 1218 | Simplification of conjunct... |
| simprl2 1219 | Simplification of conjunct... |
| simprl3 1220 | Simplification of conjunct... |
| simprr1 1221 | Simplification of conjunct... |
| simprr2 1222 | Simplification of conjunct... |
| simprr3 1223 | Simplification of conjunct... |
| simpl1l 1224 | Simplification of conjunct... |
| simpl1r 1225 | Simplification of conjunct... |
| simpl2l 1226 | Simplification of conjunct... |
| simpl2r 1227 | Simplification of conjunct... |
| simpl3l 1228 | Simplification of conjunct... |
| simpl3r 1229 | Simplification of conjunct... |
| simpr1l 1230 | Simplification of conjunct... |
| simpr1r 1231 | Simplification of conjunct... |
| simpr2l 1232 | Simplification of conjunct... |
| simpr2r 1233 | Simplification of conjunct... |
| simpr3l 1234 | Simplification of conjunct... |
| simpr3r 1235 | Simplification of conjunct... |
| simp1ll 1236 | Simplification of conjunct... |
| simp1lr 1237 | Simplification of conjunct... |
| simp1rl 1238 | Simplification of conjunct... |
| simp1rr 1239 | Simplification of conjunct... |
| simp2ll 1240 | Simplification of conjunct... |
| simp2lr 1241 | Simplification of conjunct... |
| simp2rl 1242 | Simplification of conjunct... |
| simp2rr 1243 | Simplification of conjunct... |
| simp3ll 1244 | Simplification of conjunct... |
| simp3lr 1245 | Simplification of conjunct... |
| simp3rl 1246 | Simplification of conjunct... |
| simp3rr 1247 | Simplification of conjunct... |
| simpl11 1248 | Simplification of conjunct... |
| simpl12 1249 | Simplification of conjunct... |
| simpl13 1250 | Simplification of conjunct... |
| simpl21 1251 | Simplification of conjunct... |
| simpl22 1252 | Simplification of conjunct... |
| simpl23 1253 | Simplification of conjunct... |
| simpl31 1254 | Simplification of conjunct... |
| simpl32 1255 | Simplification of conjunct... |
| simpl33 1256 | Simplification of conjunct... |
| simpr11 1257 | Simplification of conjunct... |
| simpr12 1258 | Simplification of conjunct... |
| simpr13 1259 | Simplification of conjunct... |
| simpr21 1260 | Simplification of conjunct... |
| simpr22 1261 | Simplification of conjunct... |
| simpr23 1262 | Simplification of conjunct... |
| simpr31 1263 | Simplification of conjunct... |
| simpr32 1264 | Simplification of conjunct... |
| simpr33 1265 | Simplification of conjunct... |
| simp1l1 1266 | Simplification of conjunct... |
| simp1l2 1267 | Simplification of conjunct... |
| simp1l3 1268 | Simplification of conjunct... |
| simp1r1 1269 | Simplification of conjunct... |
| simp1r2 1270 | Simplification of conjunct... |
| simp1r3 1271 | Simplification of conjunct... |
| simp2l1 1272 | Simplification of conjunct... |
| simp2l2 1273 | Simplification of conjunct... |
| simp2l3 1274 | Simplification of conjunct... |
| simp2r1 1275 | Simplification of conjunct... |
| simp2r2 1276 | Simplification of conjunct... |
| simp2r3 1277 | Simplification of conjunct... |
| simp3l1 1278 | Simplification of conjunct... |
| simp3l2 1279 | Simplification of conjunct... |
| simp3l3 1280 | Simplification of conjunct... |
| simp3r1 1281 | Simplification of conjunct... |
| simp3r2 1282 | Simplification of conjunct... |
| simp3r3 1283 | Simplification of conjunct... |
| simp11l 1284 | Simplification of conjunct... |
| simp11r 1285 | Simplification of conjunct... |
| simp12l 1286 | Simplification of conjunct... |
| simp12r 1287 | Simplification of conjunct... |
| simp13l 1288 | Simplification of conjunct... |
| simp13r 1289 | Simplification of conjunct... |
| simp21l 1290 | Simplification of conjunct... |
| simp21r 1291 | Simplification of conjunct... |
| simp22l 1292 | Simplification of conjunct... |
| simp22r 1293 | Simplification of conjunct... |
| simp23l 1294 | Simplification of conjunct... |
| simp23r 1295 | Simplification of conjunct... |
| simp31l 1296 | Simplification of conjunct... |
| simp31r 1297 | Simplification of conjunct... |
| simp32l 1298 | Simplification of conjunct... |
| simp32r 1299 | Simplification of conjunct... |
| simp33l 1300 | Simplification of conjunct... |
| simp33r 1301 | Simplification of conjunct... |
| simp111 1302 | Simplification of conjunct... |
| simp112 1303 | Simplification of conjunct... |
| simp113 1304 | Simplification of conjunct... |
| simp121 1305 | Simplification of conjunct... |
| simp122 1306 | Simplification of conjunct... |
| simp123 1307 | Simplification of conjunct... |
| simp131 1308 | Simplification of conjunct... |
| simp132 1309 | Simplification of conjunct... |
| simp133 1310 | Simplification of conjunct... |
| simp211 1311 | Simplification of conjunct... |
| simp212 1312 | Simplification of conjunct... |
| simp213 1313 | Simplification of conjunct... |
| simp221 1314 | Simplification of conjunct... |
| simp222 1315 | Simplification of conjunct... |
| simp223 1316 | Simplification of conjunct... |
| simp231 1317 | Simplification of conjunct... |
| simp232 1318 | Simplification of conjunct... |
| simp233 1319 | Simplification of conjunct... |
| simp311 1320 | Simplification of conjunct... |
| simp312 1321 | Simplification of conjunct... |
| simp313 1322 | Simplification of conjunct... |
| simp321 1323 | Simplification of conjunct... |
| simp322 1324 | Simplification of conjunct... |
| simp323 1325 | Simplification of conjunct... |
| simp331 1326 | Simplification of conjunct... |
| simp332 1327 | Simplification of conjunct... |
| simp333 1328 | Simplification of conjunct... |
| 3anibar 1329 | Remove a hypothesis from t... |
| 3mix1 1330 | Introduction in triple dis... |
| 3mix2 1331 | Introduction in triple dis... |
| 3mix3 1332 | Introduction in triple dis... |
| 3mix1i 1333 | Introduction in triple dis... |
| 3mix2i 1334 | Introduction in triple dis... |
| 3mix3i 1335 | Introduction in triple dis... |
| 3mix1d 1336 | Deduction introducing trip... |
| 3mix2d 1337 | Deduction introducing trip... |
| 3mix3d 1338 | Deduction introducing trip... |
| 3pm3.2i 1339 | Infer conjunction of premi... |
| pm3.2an3 1340 | Version of ~ pm3.2 for a t... |
| mpbir3an 1341 | Detach a conjunction of tr... |
| mpbir3and 1342 | Detach a conjunction of tr... |
| syl3anbrc 1343 | Syllogism inference. (Con... |
| syl21anbrc 1344 | Syllogism inference. (Con... |
| 3imp3i2an 1345 | An elimination deduction. ... |
| ex3 1346 | Apply ~ ex to a hypothesis... |
| 3imp1 1347 | Importation to left triple... |
| 3impd 1348 | Importation deduction for ... |
| 3imp2 1349 | Importation to right tripl... |
| 3impdi 1350 | Importation inference (und... |
| 3impdir 1351 | Importation inference (und... |
| 3exp1 1352 | Exportation from left trip... |
| 3expd 1353 | Exportation deduction for ... |
| 3exp2 1354 | Exportation from right tri... |
| exp5o 1355 | A triple exportation infer... |
| exp516 1356 | A triple exportation infer... |
| exp520 1357 | A triple exportation infer... |
| 3impexp 1358 | Version of ~ impexp for a ... |
| 3an1rs 1359 | Swap conjuncts. (Contribu... |
| 3anassrs 1360 | Associative law for conjun... |
| ad5ant245 1361 | Deduction adding conjuncts... |
| ad5ant234 1362 | Deduction adding conjuncts... |
| ad5ant235 1363 | Deduction adding conjuncts... |
| ad5ant123 1364 | Deduction adding conjuncts... |
| ad5ant124 1365 | Deduction adding conjuncts... |
| ad5ant125 1366 | Deduction adding conjuncts... |
| ad5ant134 1367 | Deduction adding conjuncts... |
| ad5ant135 1368 | Deduction adding conjuncts... |
| ad5ant145 1369 | Deduction adding conjuncts... |
| ad5ant2345 1370 | Deduction adding conjuncts... |
| syl3anc 1371 | Syllogism combined with co... |
| syl13anc 1372 | Syllogism combined with co... |
| syl31anc 1373 | Syllogism combined with co... |
| syl112anc 1374 | Syllogism combined with co... |
| syl121anc 1375 | Syllogism combined with co... |
| syl211anc 1376 | Syllogism combined with co... |
| syl23anc 1377 | Syllogism combined with co... |
| syl32anc 1378 | Syllogism combined with co... |
| syl122anc 1379 | Syllogism combined with co... |
| syl212anc 1380 | Syllogism combined with co... |
| syl221anc 1381 | Syllogism combined with co... |
| syl113anc 1382 | Syllogism combined with co... |
| syl131anc 1383 | Syllogism combined with co... |
| syl311anc 1384 | Syllogism combined with co... |
| syl33anc 1385 | Syllogism combined with co... |
| syl222anc 1386 | Syllogism combined with co... |
| syl123anc 1387 | Syllogism combined with co... |
| syl132anc 1388 | Syllogism combined with co... |
| syl213anc 1389 | Syllogism combined with co... |
| syl231anc 1390 | Syllogism combined with co... |
| syl312anc 1391 | Syllogism combined with co... |
| syl321anc 1392 | Syllogism combined with co... |
| syl133anc 1393 | Syllogism combined with co... |
| syl313anc 1394 | Syllogism combined with co... |
| syl331anc 1395 | Syllogism combined with co... |
| syl223anc 1396 | Syllogism combined with co... |
| syl232anc 1397 | Syllogism combined with co... |
| syl322anc 1398 | Syllogism combined with co... |
| syl233anc 1399 | Syllogism combined with co... |
| syl323anc 1400 | Syllogism combined with co... |
| syl332anc 1401 | Syllogism combined with co... |
| syl333anc 1402 | A syllogism inference comb... |
| syl3an1b 1403 | A syllogism inference. (C... |
| syl3an2b 1404 | A syllogism inference. (C... |
| syl3an3b 1405 | A syllogism inference. (C... |
| syl3an1br 1406 | A syllogism inference. (C... |
| syl3an2br 1407 | A syllogism inference. (C... |
| syl3an3br 1408 | A syllogism inference. (C... |
| syld3an3 1409 | A syllogism inference. (C... |
| syld3an1 1410 | A syllogism inference. (C... |
| syld3an2 1411 | A syllogism inference. (C... |
| syl3anl1 1412 | A syllogism inference. (C... |
| syl3anl2 1413 | A syllogism inference. (C... |
| syl3anl3 1414 | A syllogism inference. (C... |
| syl3anl 1415 | A triple syllogism inferen... |
| syl3anr1 1416 | A syllogism inference. (C... |
| syl3anr2 1417 | A syllogism inference. (C... |
| syl3anr3 1418 | A syllogism inference. (C... |
| 3anidm12 1419 | Inference from idempotent ... |
| 3anidm13 1420 | Inference from idempotent ... |
| 3anidm23 1421 | Inference from idempotent ... |
| syl2an3an 1422 | ~ syl3an with antecedents ... |
| syl2an23an 1423 | Deduction related to ~ syl... |
| 3ori 1424 | Infer implication from tri... |
| 3jao 1425 | Disjunction of three antec... |
| 3jaob 1426 | Disjunction of three antec... |
| 3jaobOLD 1427 | Obsolete version of ~ 3jao... |
| 3jaoi 1428 | Disjunction of three antec... |
| 3jaod 1429 | Disjunction of three antec... |
| 3jaoian 1430 | Disjunction of three antec... |
| 3jaodan 1431 | Disjunction of three antec... |
| mpjao3dan 1432 | Eliminate a three-way disj... |
| 3jaao 1433 | Inference conjoining and d... |
| syl3an9b 1434 | Nested syllogism inference... |
| 3orbi123d 1435 | Deduction joining 3 equiva... |
| 3anbi123d 1436 | Deduction joining 3 equiva... |
| 3anbi12d 1437 | Deduction conjoining and a... |
| 3anbi13d 1438 | Deduction conjoining and a... |
| 3anbi23d 1439 | Deduction conjoining and a... |
| 3anbi1d 1440 | Deduction adding conjuncts... |
| 3anbi2d 1441 | Deduction adding conjuncts... |
| 3anbi3d 1442 | Deduction adding conjuncts... |
| 3anim123d 1443 | Deduction joining 3 implic... |
| 3orim123d 1444 | Deduction joining 3 implic... |
| an6 1445 | Rearrangement of 6 conjunc... |
| 3an6 1446 | Analogue of ~ an4 for trip... |
| 3or6 1447 | Analogue of ~ or4 for trip... |
| mp3an1 1448 | An inference based on modu... |
| mp3an2 1449 | An inference based on modu... |
| mp3an3 1450 | An inference based on modu... |
| mp3an12 1451 | An inference based on modu... |
| mp3an13 1452 | An inference based on modu... |
| mp3an23 1453 | An inference based on modu... |
| mp3an1i 1454 | An inference based on modu... |
| mp3anl1 1455 | An inference based on modu... |
| mp3anl2 1456 | An inference based on modu... |
| mp3anl3 1457 | An inference based on modu... |
| mp3anr1 1458 | An inference based on modu... |
| mp3anr2 1459 | An inference based on modu... |
| mp3anr3 1460 | An inference based on modu... |
| mp3an 1461 | An inference based on modu... |
| mpd3an3 1462 | An inference based on modu... |
| mpd3an23 1463 | An inference based on modu... |
| mp3and 1464 | A deduction based on modus... |
| mp3an12i 1465 | ~ mp3an with antecedents i... |
| mp3an2i 1466 | ~ mp3an with antecedents i... |
| mp3an3an 1467 | ~ mp3an with antecedents i... |
| mp3an2ani 1468 | An elimination deduction. ... |
| biimp3a 1469 | Infer implication from a l... |
| biimp3ar 1470 | Infer implication from a l... |
| 3anandis 1471 | Inference that undistribut... |
| 3anandirs 1472 | Inference that undistribut... |
| ecase23d 1473 | Deduction for elimination ... |
| 3ecase 1474 | Inference for elimination ... |
| 3bior1fd 1475 | A disjunction is equivalen... |
| 3bior1fand 1476 | A disjunction is equivalen... |
| 3bior2fd 1477 | A wff is equivalent to its... |
| 3biant1d 1478 | A conjunction is equivalen... |
| intn3an1d 1479 | Introduction of a triple c... |
| intn3an2d 1480 | Introduction of a triple c... |
| intn3an3d 1481 | Introduction of a triple c... |
| an3andi 1482 | Distribution of conjunctio... |
| an33rean 1483 | Rearrange a 9-fold conjunc... |
| 3orel2 1484 | Partial elimination of a t... |
| 3orel3 1485 | Partial elimination of a t... |
| 3orel13 1486 | Elimination of two disjunc... |
| 3pm3.2ni 1487 | Triple negated disjunction... |
| nanan 1490 | Conjunction in terms of al... |
| dfnan2 1491 | Alternative denial in term... |
| nanor 1492 | Alternative denial in term... |
| nancom 1493 | Alternative denial is comm... |
| nannan 1494 | Nested alternative denials... |
| nanim 1495 | Implication in terms of al... |
| nannot 1496 | Negation in terms of alter... |
| nanbi 1497 | Biconditional in terms of ... |
| nanbi1 1498 | Introduce a right anti-con... |
| nanbi2 1499 | Introduce a left anti-conj... |
| nanbi12 1500 | Join two logical equivalen... |
| nanbi1i 1501 | Introduce a right anti-con... |
| nanbi2i 1502 | Introduce a left anti-conj... |
| nanbi12i 1503 | Join two logical equivalen... |
| nanbi1d 1504 | Introduce a right anti-con... |
| nanbi2d 1505 | Introduce a left anti-conj... |
| nanbi12d 1506 | Join two logical equivalen... |
| nanass 1507 | A characterization of when... |
| xnor 1510 | Two ways to write XNOR (ex... |
| xorcom 1511 | The connector ` \/_ ` is c... |
| xorass 1512 | The connector ` \/_ ` is a... |
| excxor 1513 | This tautology shows that ... |
| xor2 1514 | Two ways to express "exclu... |
| xoror 1515 | Exclusive disjunction impl... |
| xornan 1516 | Exclusive disjunction impl... |
| xornan2 1517 | XOR implies NAND (written ... |
| xorneg2 1518 | The connector ` \/_ ` is n... |
| xorneg1 1519 | The connector ` \/_ ` is n... |
| xorneg 1520 | The connector ` \/_ ` is u... |
| xorbi12i 1521 | Equality property for excl... |
| xorbi12d 1522 | Equality property for excl... |
| anxordi 1523 | Conjunction distributes ov... |
| xorexmid 1524 | Exclusive-or variant of th... |
| norcom 1527 | The connector ` -\/ ` is c... |
| nornot 1528 | ` -. ` is expressible via ... |
| noran 1529 | ` /\ ` is expressible via ... |
| noror 1530 | ` \/ ` is expressible via ... |
| norasslem1 1531 | This lemma shows the equiv... |
| norasslem2 1532 | This lemma specializes ~ b... |
| norasslem3 1533 | This lemma specializes ~ b... |
| norass 1534 | A characterization of when... |
| trujust 1539 | Soundness justification th... |
| tru 1541 | The truth value ` T. ` is ... |
| dftru2 1542 | An alternate definition of... |
| trut 1543 | A proposition is equivalen... |
| mptru 1544 | Eliminate ` T. ` as an ant... |
| tbtru 1545 | A proposition is equivalen... |
| bitru 1546 | A theorem is equivalent to... |
| trud 1547 | Anything implies ` T. ` . ... |
| truan 1548 | True can be removed from a... |
| fal 1551 | The truth value ` F. ` is ... |
| nbfal 1552 | The negation of a proposit... |
| bifal 1553 | A contradiction is equival... |
| falim 1554 | The truth value ` F. ` imp... |
| falimd 1555 | The truth value ` F. ` imp... |
| dfnot 1556 | Given falsum ` F. ` , we c... |
| inegd 1557 | Negation introduction rule... |
| efald 1558 | Deduction based on reducti... |
| pm2.21fal 1559 | If a wff and its negation ... |
| truimtru 1560 | A ` -> ` identity. (Contr... |
| truimfal 1561 | A ` -> ` identity. (Contr... |
| falimtru 1562 | A ` -> ` identity. (Contr... |
| falimfal 1563 | A ` -> ` identity. (Contr... |
| nottru 1564 | A ` -. ` identity. (Contr... |
| notfal 1565 | A ` -. ` identity. (Contr... |
| trubitru 1566 | A ` <-> ` identity. (Cont... |
| falbitru 1567 | A ` <-> ` identity. (Cont... |
| trubifal 1568 | A ` <-> ` identity. (Cont... |
| falbifal 1569 | A ` <-> ` identity. (Cont... |
| truantru 1570 | A ` /\ ` identity. (Contr... |
| truanfal 1571 | A ` /\ ` identity. (Contr... |
| falantru 1572 | A ` /\ ` identity. (Contr... |
| falanfal 1573 | A ` /\ ` identity. (Contr... |
| truortru 1574 | A ` \/ ` identity. (Contr... |
| truorfal 1575 | A ` \/ ` identity. (Contr... |
| falortru 1576 | A ` \/ ` identity. (Contr... |
| falorfal 1577 | A ` \/ ` identity. (Contr... |
| trunantru 1578 | A ` -/\ ` identity. (Cont... |
| trunanfal 1579 | A ` -/\ ` identity. (Cont... |
| falnantru 1580 | A ` -/\ ` identity. (Cont... |
| falnanfal 1581 | A ` -/\ ` identity. (Cont... |
| truxortru 1582 | A ` \/_ ` identity. (Cont... |
| truxorfal 1583 | A ` \/_ ` identity. (Cont... |
| falxortru 1584 | A ` \/_ ` identity. (Cont... |
| falxorfal 1585 | A ` \/_ ` identity. (Cont... |
| trunortru 1586 | A ` -\/ ` identity. (Cont... |
| trunorfal 1587 | A ` -\/ ` identity. (Cont... |
| falnortru 1588 | A ` -\/ ` identity. (Cont... |
| falnorfal 1589 | A ` -\/ ` identity. (Cont... |
| hadbi123d 1592 | Equality theorem for the a... |
| hadbi123i 1593 | Equality theorem for the a... |
| hadass 1594 | Associative law for the ad... |
| hadbi 1595 | The adder sum is the same ... |
| hadcoma 1596 | Commutative law for the ad... |
| hadcomb 1597 | Commutative law for the ad... |
| hadrot 1598 | Rotation law for the adder... |
| hadnot 1599 | The adder sum distributes ... |
| had1 1600 | If the first input is true... |
| had0 1601 | If the first input is fals... |
| hadifp 1602 | The value of the adder sum... |
| cador 1605 | The adder carry in disjunc... |
| cadan 1606 | The adder carry in conjunc... |
| cadbi123d 1607 | Equality theorem for the a... |
| cadbi123i 1608 | Equality theorem for the a... |
| cadcoma 1609 | Commutative law for the ad... |
| cadcomb 1610 | Commutative law for the ad... |
| cadrot 1611 | Rotation law for the adder... |
| cadnot 1612 | The adder carry distribute... |
| cad11 1613 | If (at least) two inputs a... |
| cad1 1614 | If one input is true, then... |
| cad0 1615 | If one input is false, the... |
| cad0OLD 1616 | Obsolete version of ~ cad0... |
| cadifp 1617 | The value of the carry is,... |
| cadtru 1618 | The adder carry is true as... |
| minimp 1619 | A single axiom for minimal... |
| minimp-syllsimp 1620 | Derivation of Syll-Simp ( ... |
| minimp-ax1 1621 | Derivation of ~ ax-1 from ... |
| minimp-ax2c 1622 | Derivation of a commuted f... |
| minimp-ax2 1623 | Derivation of ~ ax-2 from ... |
| minimp-pm2.43 1624 | Derivation of ~ pm2.43 (al... |
| impsingle 1625 | The shortest single axiom ... |
| impsingle-step4 1626 | Derivation of impsingle-st... |
| impsingle-step8 1627 | Derivation of impsingle-st... |
| impsingle-ax1 1628 | Derivation of impsingle-ax... |
| impsingle-step15 1629 | Derivation of impsingle-st... |
| impsingle-step18 1630 | Derivation of impsingle-st... |
| impsingle-step19 1631 | Derivation of impsingle-st... |
| impsingle-step20 1632 | Derivation of impsingle-st... |
| impsingle-step21 1633 | Derivation of impsingle-st... |
| impsingle-step22 1634 | Derivation of impsingle-st... |
| impsingle-step25 1635 | Derivation of impsingle-st... |
| impsingle-imim1 1636 | Derivation of impsingle-im... |
| impsingle-peirce 1637 | Derivation of impsingle-pe... |
| tarski-bernays-ax2 1638 | Derivation of ~ ax-2 from ... |
| meredith 1639 | Carew Meredith's sole axio... |
| merlem1 1640 | Step 3 of Meredith's proof... |
| merlem2 1641 | Step 4 of Meredith's proof... |
| merlem3 1642 | Step 7 of Meredith's proof... |
| merlem4 1643 | Step 8 of Meredith's proof... |
| merlem5 1644 | Step 11 of Meredith's proo... |
| merlem6 1645 | Step 12 of Meredith's proo... |
| merlem7 1646 | Between steps 14 and 15 of... |
| merlem8 1647 | Step 15 of Meredith's proo... |
| merlem9 1648 | Step 18 of Meredith's proo... |
| merlem10 1649 | Step 19 of Meredith's proo... |
| merlem11 1650 | Step 20 of Meredith's proo... |
| merlem12 1651 | Step 28 of Meredith's proo... |
| merlem13 1652 | Step 35 of Meredith's proo... |
| luk-1 1653 | 1 of 3 axioms for proposit... |
| luk-2 1654 | 2 of 3 axioms for proposit... |
| luk-3 1655 | 3 of 3 axioms for proposit... |
| luklem1 1656 | Used to rederive standard ... |
| luklem2 1657 | Used to rederive standard ... |
| luklem3 1658 | Used to rederive standard ... |
| luklem4 1659 | Used to rederive standard ... |
| luklem5 1660 | Used to rederive standard ... |
| luklem6 1661 | Used to rederive standard ... |
| luklem7 1662 | Used to rederive standard ... |
| luklem8 1663 | Used to rederive standard ... |
| ax1 1664 | Standard propositional axi... |
| ax2 1665 | Standard propositional axi... |
| ax3 1666 | Standard propositional axi... |
| nic-dfim 1667 | This theorem "defines" imp... |
| nic-dfneg 1668 | This theorem "defines" neg... |
| nic-mp 1669 | Derive Nicod's rule of mod... |
| nic-mpALT 1670 | A direct proof of ~ nic-mp... |
| nic-ax 1671 | Nicod's axiom derived from... |
| nic-axALT 1672 | A direct proof of ~ nic-ax... |
| nic-imp 1673 | Inference for ~ nic-mp usi... |
| nic-idlem1 1674 | Lemma for ~ nic-id . (Con... |
| nic-idlem2 1675 | Lemma for ~ nic-id . Infe... |
| nic-id 1676 | Theorem ~ id expressed wit... |
| nic-swap 1677 | The connector ` -/\ ` is s... |
| nic-isw1 1678 | Inference version of ~ nic... |
| nic-isw2 1679 | Inference for swapping nes... |
| nic-iimp1 1680 | Inference version of ~ nic... |
| nic-iimp2 1681 | Inference version of ~ nic... |
| nic-idel 1682 | Inference to remove the tr... |
| nic-ich 1683 | Chained inference. (Contr... |
| nic-idbl 1684 | Double the terms. Since d... |
| nic-bijust 1685 | Biconditional justificatio... |
| nic-bi1 1686 | Inference to extract one s... |
| nic-bi2 1687 | Inference to extract the o... |
| nic-stdmp 1688 | Derive the standard modus ... |
| nic-luk1 1689 | Proof of ~ luk-1 from ~ ni... |
| nic-luk2 1690 | Proof of ~ luk-2 from ~ ni... |
| nic-luk3 1691 | Proof of ~ luk-3 from ~ ni... |
| lukshef-ax1 1692 | This alternative axiom for... |
| lukshefth1 1693 | Lemma for ~ renicax . (Co... |
| lukshefth2 1694 | Lemma for ~ renicax . (Co... |
| renicax 1695 | A rederivation of ~ nic-ax... |
| tbw-bijust 1696 | Justification for ~ tbw-ne... |
| tbw-negdf 1697 | The definition of negation... |
| tbw-ax1 1698 | The first of four axioms i... |
| tbw-ax2 1699 | The second of four axioms ... |
| tbw-ax3 1700 | The third of four axioms i... |
| tbw-ax4 1701 | The fourth of four axioms ... |
| tbwsyl 1702 | Used to rederive the Lukas... |
| tbwlem1 1703 | Used to rederive the Lukas... |
| tbwlem2 1704 | Used to rederive the Lukas... |
| tbwlem3 1705 | Used to rederive the Lukas... |
| tbwlem4 1706 | Used to rederive the Lukas... |
| tbwlem5 1707 | Used to rederive the Lukas... |
| re1luk1 1708 | ~ luk-1 derived from the T... |
| re1luk2 1709 | ~ luk-2 derived from the T... |
| re1luk3 1710 | ~ luk-3 derived from the T... |
| merco1 1711 | A single axiom for proposi... |
| merco1lem1 1712 | Used to rederive the Tarsk... |
| retbwax4 1713 | ~ tbw-ax4 rederived from ~... |
| retbwax2 1714 | ~ tbw-ax2 rederived from ~... |
| merco1lem2 1715 | Used to rederive the Tarsk... |
| merco1lem3 1716 | Used to rederive the Tarsk... |
| merco1lem4 1717 | Used to rederive the Tarsk... |
| merco1lem5 1718 | Used to rederive the Tarsk... |
| merco1lem6 1719 | Used to rederive the Tarsk... |
| merco1lem7 1720 | Used to rederive the Tarsk... |
| retbwax3 1721 | ~ tbw-ax3 rederived from ~... |
| merco1lem8 1722 | Used to rederive the Tarsk... |
| merco1lem9 1723 | Used to rederive the Tarsk... |
| merco1lem10 1724 | Used to rederive the Tarsk... |
| merco1lem11 1725 | Used to rederive the Tarsk... |
| merco1lem12 1726 | Used to rederive the Tarsk... |
| merco1lem13 1727 | Used to rederive the Tarsk... |
| merco1lem14 1728 | Used to rederive the Tarsk... |
| merco1lem15 1729 | Used to rederive the Tarsk... |
| merco1lem16 1730 | Used to rederive the Tarsk... |
| merco1lem17 1731 | Used to rederive the Tarsk... |
| merco1lem18 1732 | Used to rederive the Tarsk... |
| retbwax1 1733 | ~ tbw-ax1 rederived from ~... |
| merco2 1734 | A single axiom for proposi... |
| mercolem1 1735 | Used to rederive the Tarsk... |
| mercolem2 1736 | Used to rederive the Tarsk... |
| mercolem3 1737 | Used to rederive the Tarsk... |
| mercolem4 1738 | Used to rederive the Tarsk... |
| mercolem5 1739 | Used to rederive the Tarsk... |
| mercolem6 1740 | Used to rederive the Tarsk... |
| mercolem7 1741 | Used to rederive the Tarsk... |
| mercolem8 1742 | Used to rederive the Tarsk... |
| re1tbw1 1743 | ~ tbw-ax1 rederived from ~... |
| re1tbw2 1744 | ~ tbw-ax2 rederived from ~... |
| re1tbw3 1745 | ~ tbw-ax3 rederived from ~... |
| re1tbw4 1746 | ~ tbw-ax4 rederived from ~... |
| rb-bijust 1747 | Justification for ~ rb-imd... |
| rb-imdf 1748 | The definition of implicat... |
| anmp 1749 | Modus ponens for ` { \/ , ... |
| rb-ax1 1750 | The first of four axioms i... |
| rb-ax2 1751 | The second of four axioms ... |
| rb-ax3 1752 | The third of four axioms i... |
| rb-ax4 1753 | The fourth of four axioms ... |
| rbsyl 1754 | Used to rederive the Lukas... |
| rblem1 1755 | Used to rederive the Lukas... |
| rblem2 1756 | Used to rederive the Lukas... |
| rblem3 1757 | Used to rederive the Lukas... |
| rblem4 1758 | Used to rederive the Lukas... |
| rblem5 1759 | Used to rederive the Lukas... |
| rblem6 1760 | Used to rederive the Lukas... |
| rblem7 1761 | Used to rederive the Lukas... |
| re1axmp 1762 | ~ ax-mp derived from Russe... |
| re2luk1 1763 | ~ luk-1 derived from Russe... |
| re2luk2 1764 | ~ luk-2 derived from Russe... |
| re2luk3 1765 | ~ luk-3 derived from Russe... |
| mptnan 1766 | Modus ponendo tollens 1, o... |
| mptxor 1767 | Modus ponendo tollens 2, o... |
| mtpor 1768 | Modus tollendo ponens (inc... |
| mtpxor 1769 | Modus tollendo ponens (ori... |
| stoic1a 1770 | Stoic logic Thema 1 (part ... |
| stoic1b 1771 | Stoic logic Thema 1 (part ... |
| stoic2a 1772 | Stoic logic Thema 2 versio... |
| stoic2b 1773 | Stoic logic Thema 2 versio... |
| stoic3 1774 | Stoic logic Thema 3. Stat... |
| stoic4a 1775 | Stoic logic Thema 4 versio... |
| stoic4b 1776 | Stoic logic Thema 4 versio... |
| alnex 1779 | Universal quantification o... |
| eximal 1780 | An equivalence between an ... |
| nf2 1783 | Alternate definition of no... |
| nf3 1784 | Alternate definition of no... |
| nf4 1785 | Alternate definition of no... |
| nfi 1786 | Deduce that ` x ` is not f... |
| nfri 1787 | Consequence of the definit... |
| nfd 1788 | Deduce that ` x ` is not f... |
| nfrd 1789 | Consequence of the definit... |
| nftht 1790 | Closed form of ~ nfth . (... |
| nfntht 1791 | Closed form of ~ nfnth . ... |
| nfntht2 1792 | Closed form of ~ nfnth . ... |
| gen2 1794 | Generalization applied twi... |
| mpg 1795 | Modus ponens combined with... |
| mpgbi 1796 | Modus ponens on biconditio... |
| mpgbir 1797 | Modus ponens on biconditio... |
| nex 1798 | Generalization rule for ne... |
| nfth 1799 | No variable is (effectivel... |
| nfnth 1800 | No variable is (effectivel... |
| hbth 1801 | No variable is (effectivel... |
| nftru 1802 | The true constant has no f... |
| nffal 1803 | The false constant has no ... |
| sptruw 1804 | Version of ~ sp when ` ph ... |
| altru 1805 | For all sets, ` T. ` is tr... |
| alfal 1806 | For all sets, ` -. F. ` is... |
| alim 1808 | Restatement of Axiom ~ ax-... |
| alimi 1809 | Inference quantifying both... |
| 2alimi 1810 | Inference doubly quantifyi... |
| ala1 1811 | Add an antecedent in a uni... |
| al2im 1812 | Closed form of ~ al2imi . ... |
| al2imi 1813 | Inference quantifying ante... |
| alanimi 1814 | Variant of ~ al2imi with c... |
| alimdh 1815 | Deduction form of Theorem ... |
| albi 1816 | Theorem 19.15 of [Margaris... |
| albii 1817 | Inference adding universal... |
| 2albii 1818 | Inference adding two unive... |
| 3albii 1819 | Inference adding three uni... |
| sylgt 1820 | Closed form of ~ sylg . (... |
| sylg 1821 | A syllogism combined with ... |
| alrimih 1822 | Inference form of Theorem ... |
| hbxfrbi 1823 | A utility lemma to transfe... |
| alex 1824 | Universal quantifier in te... |
| exnal 1825 | Existential quantification... |
| 2nalexn 1826 | Part of theorem *11.5 in [... |
| 2exnaln 1827 | Theorem *11.22 in [Whitehe... |
| 2nexaln 1828 | Theorem *11.25 in [Whitehe... |
| alimex 1829 | An equivalence between an ... |
| aleximi 1830 | A variant of ~ al2imi : in... |
| alexbii 1831 | Biconditional form of ~ al... |
| exim 1832 | Theorem 19.22 of [Margaris... |
| eximi 1833 | Inference adding existenti... |
| 2eximi 1834 | Inference adding two exist... |
| eximii 1835 | Inference associated with ... |
| exa1 1836 | Add an antecedent in an ex... |
| 19.38 1837 | Theorem 19.38 of [Margaris... |
| 19.38a 1838 | Under a nonfreeness hypoth... |
| 19.38b 1839 | Under a nonfreeness hypoth... |
| imnang 1840 | Quantified implication in ... |
| alinexa 1841 | A transformation of quanti... |
| exnalimn 1842 | Existential quantification... |
| alexn 1843 | A relationship between two... |
| 2exnexn 1844 | Theorem *11.51 in [Whitehe... |
| exbi 1845 | Theorem 19.18 of [Margaris... |
| exbii 1846 | Inference adding existenti... |
| 2exbii 1847 | Inference adding two exist... |
| 3exbii 1848 | Inference adding three exi... |
| nfbiit 1849 | Equivalence theorem for th... |
| nfbii 1850 | Equality theorem for the n... |
| nfxfr 1851 | A utility lemma to transfe... |
| nfxfrd 1852 | A utility lemma to transfe... |
| nfnbi 1853 | A variable is nonfree in a... |
| nfnbiOLD 1854 | Obsolete version of ~ nfnb... |
| nfnt 1855 | If a variable is nonfree i... |
| nfn 1856 | Inference associated with ... |
| nfnd 1857 | Deduction associated with ... |
| exanali 1858 | A transformation of quanti... |
| 2exanali 1859 | Theorem *11.521 in [Whiteh... |
| exancom 1860 | Commutation of conjunction... |
| exan 1861 | Place a conjunct in the sc... |
| alrimdh 1862 | Deduction form of Theorem ... |
| eximdh 1863 | Deduction from Theorem 19.... |
| nexdh 1864 | Deduction for generalizati... |
| albidh 1865 | Formula-building rule for ... |
| exbidh 1866 | Formula-building rule for ... |
| exsimpl 1867 | Simplification of an exist... |
| exsimpr 1868 | Simplification of an exist... |
| 19.26 1869 | Theorem 19.26 of [Margaris... |
| 19.26-2 1870 | Theorem ~ 19.26 with two q... |
| 19.26-3an 1871 | Theorem ~ 19.26 with tripl... |
| 19.29 1872 | Theorem 19.29 of [Margaris... |
| 19.29r 1873 | Variation of ~ 19.29 . (C... |
| 19.29r2 1874 | Variation of ~ 19.29r with... |
| 19.29x 1875 | Variation of ~ 19.29 with ... |
| 19.35 1876 | Theorem 19.35 of [Margaris... |
| 19.35i 1877 | Inference associated with ... |
| 19.35ri 1878 | Inference associated with ... |
| 19.25 1879 | Theorem 19.25 of [Margaris... |
| 19.30 1880 | Theorem 19.30 of [Margaris... |
| 19.43 1881 | Theorem 19.43 of [Margaris... |
| 19.43OLD 1882 | Obsolete proof of ~ 19.43 ... |
| 19.33 1883 | Theorem 19.33 of [Margaris... |
| 19.33b 1884 | The antecedent provides a ... |
| 19.40 1885 | Theorem 19.40 of [Margaris... |
| 19.40-2 1886 | Theorem *11.42 in [Whitehe... |
| 19.40b 1887 | The antecedent provides a ... |
| albiim 1888 | Split a biconditional and ... |
| 2albiim 1889 | Split a biconditional and ... |
| exintrbi 1890 | Add/remove a conjunct in t... |
| exintr 1891 | Introduce a conjunct in th... |
| alsyl 1892 | Universally quantified and... |
| nfimd 1893 | If in a context ` x ` is n... |
| nfimt 1894 | Closed form of ~ nfim and ... |
| nfim 1895 | If ` x ` is not free in ` ... |
| nfand 1896 | If in a context ` x ` is n... |
| nf3and 1897 | Deduction form of bound-va... |
| nfan 1898 | If ` x ` is not free in ` ... |
| nfnan 1899 | If ` x ` is not free in ` ... |
| nf3an 1900 | If ` x ` is not free in ` ... |
| nfbid 1901 | If in a context ` x ` is n... |
| nfbi 1902 | If ` x ` is not free in ` ... |
| nfor 1903 | If ` x ` is not free in ` ... |
| nf3or 1904 | If ` x ` is not free in ` ... |
| empty 1905 | Two characterizations of t... |
| emptyex 1906 | On the empty domain, any e... |
| emptyal 1907 | On the empty domain, any u... |
| emptynf 1908 | On the empty domain, any v... |
| ax5d 1910 | Version of ~ ax-5 with ant... |
| ax5e 1911 | A rephrasing of ~ ax-5 usi... |
| ax5ea 1912 | If a formula holds for som... |
| nfv 1913 | If ` x ` is not present in... |
| nfvd 1914 | ~ nfv with antecedent. Us... |
| alimdv 1915 | Deduction form of Theorem ... |
| eximdv 1916 | Deduction form of Theorem ... |
| 2alimdv 1917 | Deduction form of Theorem ... |
| 2eximdv 1918 | Deduction form of Theorem ... |
| albidv 1919 | Formula-building rule for ... |
| exbidv 1920 | Formula-building rule for ... |
| nfbidv 1921 | An equality theorem for no... |
| 2albidv 1922 | Formula-building rule for ... |
| 2exbidv 1923 | Formula-building rule for ... |
| 3exbidv 1924 | Formula-building rule for ... |
| 4exbidv 1925 | Formula-building rule for ... |
| alrimiv 1926 | Inference form of Theorem ... |
| alrimivv 1927 | Inference form of Theorem ... |
| alrimdv 1928 | Deduction form of Theorem ... |
| exlimiv 1929 | Inference form of Theorem ... |
| exlimiiv 1930 | Inference (Rule C) associa... |
| exlimivv 1931 | Inference form of Theorem ... |
| exlimdv 1932 | Deduction form of Theorem ... |
| exlimdvv 1933 | Deduction form of Theorem ... |
| exlimddv 1934 | Existential elimination ru... |
| nexdv 1935 | Deduction for generalizati... |
| 2ax5 1936 | Quantification of two vari... |
| stdpc5v 1937 | Version of ~ stdpc5 with a... |
| 19.21v 1938 | Version of ~ 19.21 with a ... |
| 19.32v 1939 | Version of ~ 19.32 with a ... |
| 19.31v 1940 | Version of ~ 19.31 with a ... |
| 19.23v 1941 | Version of ~ 19.23 with a ... |
| 19.23vv 1942 | Theorem ~ 19.23v extended ... |
| pm11.53v 1943 | Version of ~ pm11.53 with ... |
| 19.36imv 1944 | One direction of ~ 19.36v ... |
| 19.36imvOLD 1945 | Obsolete version of ~ 19.3... |
| 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... |
| spvw 1980 | Version of ~ sp when ` x `... |
| 19.3v 1981 | Version of ~ 19.3 with a d... |
| 19.8v 1982 | Version of ~ 19.8a with a ... |
| 19.9v 1983 | Version of ~ 19.9 with a d... |
| 19.39 1984 | Theorem 19.39 of [Margaris... |
| 19.24 1985 | Theorem 19.24 of [Margaris... |
| 19.34 1986 | Theorem 19.34 of [Margaris... |
| 19.36v 1987 | Version of ~ 19.36 with a ... |
| 19.12vvv 1988 | Version of ~ 19.12vv with ... |
| 19.27v 1989 | Version of ~ 19.27 with a ... |
| 19.28v 1990 | Version of ~ 19.28 with a ... |
| 19.37v 1991 | Version of ~ 19.37 with a ... |
| 19.44v 1992 | Version of ~ 19.44 with a ... |
| 19.45v 1993 | Version of ~ 19.45 with a ... |
| spimevw 1994 | Existential introduction, ... |
| spimvw 1995 | A weak form of specializat... |
| spvv 1996 | Specialization, using impl... |
| spfalw 1997 | Version of ~ sp when ` ph ... |
| chvarvv 1998 | Implicit substitution of `... |
| equs4v 1999 | Version of ~ equs4 with a ... |
| alequexv 2000 | Version of ~ equs4v with i... |
| exsbim 2001 | One direction of the equiv... |
| equsv 2002 | If a formula does not cont... |
| equsalvw 2003 | Version of ~ equsalv with ... |
| equsexvw 2004 | Version of ~ equsexv with ... |
| cbvaliw 2005 | Change bound variable. Us... |
| cbvalivw 2006 | Change bound variable. Us... |
| ax7v 2008 | Weakened version of ~ ax-7... |
| ax7v1 2009 | First of two weakened vers... |
| ax7v2 2010 | Second of two weakened ver... |
| equid 2011 | Identity law for equality.... |
| nfequid 2012 | Bound-variable hypothesis ... |
| equcomiv 2013 | Weaker form of ~ equcomi w... |
| ax6evr 2014 | A commuted form of ~ ax6ev... |
| ax7 2015 | Proof of ~ ax-7 from ~ ax7... |
| equcomi 2016 | Commutative law for equali... |
| equcom 2017 | Commutative law for equali... |
| equcomd 2018 | Deduction form of ~ equcom... |
| equcoms 2019 | An inference commuting equ... |
| equtr 2020 | A transitive law for equal... |
| equtrr 2021 | A transitive law for equal... |
| equeuclr 2022 | Commuted version of ~ eque... |
| equeucl 2023 | Equality is a left-Euclide... |
| equequ1 2024 | An equivalence law for equ... |
| equequ2 2025 | An equivalence law for equ... |
| equtr2 2026 | Equality is a left-Euclide... |
| stdpc6 2027 | One of the two equality ax... |
| equvinv 2028 | A variable introduction la... |
| equvinva 2029 | A modified version of the ... |
| equvelv 2030 | A biconditional form of ~ ... |
| ax13b 2031 | An equivalence between two... |
| spfw 2032 | Weak version of ~ sp . Us... |
| spw 2033 | Weak version of the specia... |
| cbvalw 2034 | Change bound variable. Us... |
| cbvalvw 2035 | Change bound variable. Us... |
| cbvexvw 2036 | Change bound variable. Us... |
| cbvaldvaw 2037 | Rule used to change the bo... |
| cbvexdvaw 2038 | Rule used to change the bo... |
| cbval2vw 2039 | Rule used to change bound ... |
| cbvex2vw 2040 | Rule used to change bound ... |
| cbvex4vw 2041 | Rule used to change bound ... |
| alcomimw 2042 | Weak version of ~ ax-11 . ... |
| excomimw 2043 | Weak version of ~ excomim ... |
| alcomw 2044 | Weak version of ~ alcom an... |
| hbn1fw 2045 | Weak version of ~ ax-10 fr... |
| hbn1w 2046 | Weak version of ~ hbn1 . ... |
| hba1w 2047 | Weak version of ~ hba1 . ... |
| hbe1w 2048 | Weak version of ~ hbe1 . ... |
| hbalw 2049 | Weak version of ~ hbal . ... |
| 19.8aw 2050 | If a formula is true, then... |
| exexw 2051 | Existential quantification... |
| spaev 2052 | A special instance of ~ sp... |
| cbvaev 2053 | Change bound variable in a... |
| aevlem0 2054 | Lemma for ~ aevlem . Inst... |
| aevlem 2055 | Lemma for ~ aev and ~ axc1... |
| aeveq 2056 | The antecedent ` A. x x = ... |
| aev 2057 | A "distinctor elimination"... |
| aev2 2058 | A version of ~ aev with tw... |
| hbaev 2059 | All variables are effectiv... |
| naev 2060 | If some set variables can ... |
| naev2 2061 | Generalization of ~ hbnaev... |
| hbnaev 2062 | Any variable is free in ` ... |
| sbjust 2063 | Justification theorem for ... |
| sbt 2066 | A substitution into a theo... |
| sbtru 2067 | The result of substituting... |
| stdpc4 2068 | The specialization axiom o... |
| sbtALT 2069 | Alternate proof of ~ sbt ,... |
| 2stdpc4 2070 | A double specialization us... |
| sbi1 2071 | Distribute substitution ov... |
| spsbim 2072 | Distribute substitution ov... |
| spsbbi 2073 | Biconditional property for... |
| sbimi 2074 | Distribute substitution ov... |
| sb2imi 2075 | Distribute substitution ov... |
| sbbii 2076 | Infer substitution into bo... |
| 2sbbii 2077 | Infer double substitution ... |
| sbimdv 2078 | Deduction substituting bot... |
| sbbidv 2079 | Deduction substituting bot... |
| sban 2080 | Conjunction inside and out... |
| sb3an 2081 | Threefold conjunction insi... |
| spsbe 2082 | Existential generalization... |
| sbequ 2083 | Equality property for subs... |
| sbequi 2084 | An equality theorem for su... |
| sb6 2085 | Alternate definition of su... |
| 2sb6 2086 | Equivalence for double sub... |
| sb1v 2087 | One direction of ~ sb5 , p... |
| sbv 2088 | Substitution for a variabl... |
| sbcom4 2089 | Commutativity law for subs... |
| pm11.07 2090 | Axiom *11.07 in [Whitehead... |
| sbrimvw 2091 | Substitution in an implica... |
| sbbiiev 2092 | An equivalence of substitu... |
| sbievw 2093 | Conversion of implicit sub... |
| sbievwOLD 2094 | Obsolete version of ~ sbie... |
| sbiedvw 2095 | Conversion of implicit sub... |
| 2sbievw 2096 | Conversion of double impli... |
| sbcom3vv 2097 | Substituting ` y ` for ` x... |
| sbievw2 2098 | ~ sbievw applied twice, av... |
| sbco2vv 2099 | A composition law for subs... |
| cbvsbv 2100 | Change the bound variable ... |
| sbco4lem 2101 | Lemma for ~ sbco4 . It re... |
| sbco4 2102 | Two ways of exchanging two... |
| equsb3 2103 | Substitution in an equalit... |
| equsb3r 2104 | Substitution applied to th... |
| equsb1v 2105 | Substitution applied to an... |
| nsb 2106 | Any substitution in an alw... |
| sbn1 2107 | One direction of ~ sbn , u... |
| wel 2109 | Extend wff definition to i... |
| ax8v 2111 | Weakened version of ~ ax-8... |
| ax8v1 2112 | First of two weakened vers... |
| ax8v2 2113 | Second of two weakened ver... |
| ax8 2114 | Proof of ~ ax-8 from ~ ax8... |
| elequ1 2115 | An identity law for the no... |
| elsb1 2116 | Substitution for the first... |
| cleljust 2117 | When the class variables i... |
| ax9v 2119 | Weakened version of ~ ax-9... |
| ax9v1 2120 | First of two weakened vers... |
| ax9v2 2121 | Second of two weakened ver... |
| ax9 2122 | Proof of ~ ax-9 from ~ ax9... |
| elequ2 2123 | An identity law for the no... |
| elequ2g 2124 | A form of ~ elequ2 with a ... |
| elsb2 2125 | Substitution for the secon... |
| elequ12 2126 | An identity law for the no... |
| ru0 2127 | The FOL statement used in ... |
| ax6dgen 2128 | Tarski's system uses the w... |
| ax10w 2129 | Weak version of ~ ax-10 fr... |
| ax11w 2130 | Weak version of ~ ax-11 fr... |
| ax11dgen 2131 | Degenerate instance of ~ a... |
| ax12wlem 2132 | Lemma for weak version of ... |
| ax12w 2133 | Weak version of ~ ax-12 fr... |
| ax12dgen 2134 | Degenerate instance of ~ a... |
| ax12wdemo 2135 | Example of an application ... |
| ax13w 2136 | Weak version (principal in... |
| ax13dgen1 2137 | Degenerate instance of ~ a... |
| ax13dgen2 2138 | Degenerate instance of ~ a... |
| ax13dgen3 2139 | Degenerate instance of ~ a... |
| ax13dgen4 2140 | Degenerate instance of ~ a... |
| hbn1 2142 | Alias for ~ ax-10 to be us... |
| hbe1 2143 | The setvar ` x ` is not fr... |
| hbe1a 2144 | Dual statement of ~ hbe1 .... |
| nf5-1 2145 | One direction of ~ nf5 can... |
| nf5i 2146 | Deduce that ` x ` is not f... |
| nf5dh 2147 | Deduce that ` x ` is not f... |
| nf5dv 2148 | Apply the definition of no... |
| nfnaew 2149 | All variables are effectiv... |
| nfnaewOLD 2150 | Obsolete version of ~ nfna... |
| 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 ... |
| equsexvOLD 2270 | Obsolete version of ~ equs... |
| sbft 2271 | Substitution has no effect... |
| sbf 2272 | Substitution for a variabl... |
| sbf2 2273 | Substitution has no effect... |
| sbh 2274 | Substitution for a variabl... |
| hbs1 2275 | The setvar ` x ` is not fr... |
| nfs1f 2276 | If ` x ` is not free in ` ... |
| sb5 2277 | Alternate definition of su... |
| sb5OLD 2278 | Obsolete version of ~ sb5 ... |
| sb56OLD 2279 | Obsolete version of ~ sbal... |
| equs5av 2280 | A property related to subs... |
| 2sb5 2281 | Equivalence for double sub... |
| sbco4lemOLDOLD 2282 | Obsolete version of ~ sbco... |
| dfsb7 2283 | An alternate definition of... |
| sbn 2284 | Negation inside and outsid... |
| sbex 2285 | Move existential quantifie... |
| nf5 2286 | Alternate definition of ~ ... |
| nf6 2287 | An alternate definition of... |
| nf5d 2288 | Deduce that ` x ` is not f... |
| nf5di 2289 | Since the converse holds b... |
| 19.9h 2290 | A wff may be existentially... |
| 19.21h 2291 | Theorem 19.21 of [Margaris... |
| 19.23h 2292 | Theorem 19.23 of [Margaris... |
| exlimih 2293 | Inference associated with ... |
| exlimdh 2294 | Deduction form of Theorem ... |
| equsalhw 2295 | Version of ~ equsalh with ... |
| equsexhv 2296 | An equivalence related to ... |
| hba1 2297 | The setvar ` x ` is not fr... |
| hbnt 2298 | Closed theorem version of ... |
| hbn 2299 | If ` x ` is not free in ` ... |
| hbnd 2300 | Deduction form of bound-va... |
| hbim1 2301 | A closed form of ~ hbim . ... |
| hbimd 2302 | Deduction form of bound-va... |
| hbim 2303 | If ` x ` is not free in ` ... |
| hban 2304 | If ` x ` is not free in ` ... |
| hb3an 2305 | If ` x ` is not free in ` ... |
| sbi2 2306 | Introduction of implicatio... |
| sbim 2307 | Implication inside and out... |
| sbrim 2308 | Substitution in an implica... |
| sbrimOLD 2309 | Obsolete version of ~ sbri... |
| sblim 2310 | Substitution in an implica... |
| sbor 2311 | Disjunction inside and out... |
| sbbi 2312 | Equivalence inside and out... |
| sblbis 2313 | Introduce left bicondition... |
| sbrbis 2314 | Introduce right biconditio... |
| sbrbif 2315 | Introduce right biconditio... |
| sbnf 2316 | Move nonfree predicate in ... |
| sbnfOLD 2317 | Obsolete version of ~ sbnf... |
| sbiev 2318 | Conversion of implicit sub... |
| sbievOLD 2319 | Obsolete version of ~ sbie... |
| sbiedw 2320 | Conversion of implicit sub... |
| axc7 2321 | Show that the original axi... |
| axc7e 2322 | Abbreviated version of ~ a... |
| modal-b 2323 | The analogue in our predic... |
| 19.9ht 2324 | A closed version of ~ 19.9... |
| axc4 2325 | Show that the original axi... |
| axc4i 2326 | Inference version of ~ axc... |
| nfal 2327 | If ` x ` is not free in ` ... |
| nfex 2328 | If ` x ` is not free in ` ... |
| hbex 2329 | If ` x ` is not free in ` ... |
| nfnf 2330 | If ` x ` is not free in ` ... |
| 19.12 2331 | Theorem 19.12 of [Margaris... |
| nfald 2332 | Deduction form of ~ nfal .... |
| nfexd 2333 | If ` x ` is not free in ` ... |
| nfsbv 2334 | If ` z ` is not free in ` ... |
| nfsbvOLD 2335 | Obsolete version of ~ nfsb... |
| sbco2v 2336 | A composition law for subs... |
| aaan 2337 | Distribute universal quant... |
| aaanOLD 2338 | Obsolete version of ~ aaan... |
| eeor 2339 | Distribute existential qua... |
| eeorOLD 2340 | Obsolete version of ~ eeor... |
| cbv3v 2341 | Rule used to change bound ... |
| cbv1v 2342 | Rule used to change bound ... |
| cbv2w 2343 | Rule used to change bound ... |
| cbvaldw 2344 | Deduction used to change b... |
| cbvexdw 2345 | Deduction used to change b... |
| cbv3hv 2346 | Rule used to change bound ... |
| cbvalv1 2347 | Rule used to change bound ... |
| cbvexv1 2348 | Rule used to change bound ... |
| cbval2v 2349 | Rule used to change bound ... |
| cbvex2v 2350 | Rule used to change bound ... |
| dvelimhw 2351 | Proof of ~ dvelimh without... |
| pm11.53 2352 | Theorem *11.53 in [Whitehe... |
| 19.12vv 2353 | Special case of ~ 19.12 wh... |
| eean 2354 | Distribute existential qua... |
| eeanv 2355 | Distribute a pair of exist... |
| eeeanv 2356 | Distribute three existenti... |
| ee4anv 2357 | Distribute two pairs of ex... |
| sb8v 2358 | Substitution of variable i... |
| sb8f 2359 | Substitution of variable i... |
| sb8fOLD 2360 | Obsolete version of ~ sb8f... |
| sb8ef 2361 | Substitution of variable i... |
| 2sb8ef 2362 | An equivalent expression f... |
| sb6rfv 2363 | Reversed substitution. Ve... |
| sbnf2 2364 | Two ways of expressing " `... |
| exsb 2365 | An equivalent expression f... |
| 2exsb 2366 | An equivalent expression f... |
| sbbib 2367 | Reversal of substitution. ... |
| sbbibvv 2368 | Reversal of substitution. ... |
| cbvsbvf 2369 | Change the bound variable ... |
| cleljustALT 2370 | Alternate proof of ~ clelj... |
| cleljustALT2 2371 | Alternate proof of ~ clelj... |
| equs5aALT 2372 | Alternate proof of ~ equs5... |
| equs5eALT 2373 | Alternate proof of ~ equs5... |
| axc11r 2374 | Same as ~ axc11 but with r... |
| dral1v 2375 | Formula-building lemma for... |
| dral1vOLD 2376 | Obsolete version of ~ dral... |
| drex1v 2377 | Formula-building lemma for... |
| drnf1v 2378 | Formula-building lemma for... |
| drnf1vOLD 2379 | Obsolete version of ~ drnf... |
| ax13v 2381 | A weaker version of ~ ax-1... |
| ax13lem1 2382 | A version of ~ ax13v with ... |
| ax13 2383 | Derive ~ ax-13 from ~ ax13... |
| ax13lem2 2384 | Lemma for ~ nfeqf2 . This... |
| nfeqf2 2385 | An equation between setvar... |
| dveeq2 2386 | Quantifier introduction wh... |
| nfeqf1 2387 | An equation between setvar... |
| dveeq1 2388 | Quantifier introduction wh... |
| nfeqf 2389 | A variable is effectively ... |
| axc9 2390 | Derive set.mm's original ~... |
| ax6e 2391 | At least one individual ex... |
| ax6 2392 | Theorem showing that ~ ax-... |
| axc10 2393 | Show that the original axi... |
| spimt 2394 | Closed theorem form of ~ s... |
| spim 2395 | Specialization, using impl... |
| spimed 2396 | Deduction version of ~ spi... |
| spime 2397 | Existential introduction, ... |
| spimv 2398 | A version of ~ spim with a... |
| spimvALT 2399 | Alternate proof of ~ spimv... |
| spimev 2400 | Distinct-variable version ... |
| spv 2401 | Specialization, using impl... |
| spei 2402 | Inference from existential... |
| chvar 2403 | Implicit substitution of `... |
| chvarv 2404 | Implicit substitution of `... |
| cbv3 2405 | Rule used to change bound ... |
| cbval 2406 | Rule used to change bound ... |
| cbvex 2407 | Rule used to change bound ... |
| cbvalv 2408 | Rule used to change bound ... |
| cbvexv 2409 | Rule used to change bound ... |
| cbv1 2410 | Rule used to change bound ... |
| cbv2 2411 | Rule used to change bound ... |
| cbv3h 2412 | Rule used to change bound ... |
| cbv1h 2413 | Rule used to change bound ... |
| cbv2h 2414 | Rule used to change bound ... |
| cbvald 2415 | Deduction used to change b... |
| cbvexd 2416 | Deduction used to change b... |
| cbvaldva 2417 | Rule used to change the bo... |
| cbvexdva 2418 | Rule used to change the bo... |
| cbval2 2419 | Rule used to change bound ... |
| cbvex2 2420 | Rule used to change bound ... |
| cbval2vv 2421 | Rule used to change bound ... |
| cbvex2vv 2422 | Rule used to change bound ... |
| cbvex4v 2423 | Rule used to change bound ... |
| equs4 2424 | Lemma used in proofs of im... |
| equsal 2425 | An equivalence related to ... |
| equsex 2426 | An equivalence related to ... |
| equsexALT 2427 | Alternate proof of ~ equse... |
| equsalh 2428 | An equivalence related to ... |
| equsexh 2429 | An equivalence related to ... |
| axc15 2430 | Derivation of set.mm's ori... |
| ax12 2431 | Rederivation of Axiom ~ ax... |
| ax12b 2432 | A bidirectional version of... |
| ax13ALT 2433 | Alternate proof of ~ ax13 ... |
| axc11n 2434 | Derive set.mm's original ~... |
| aecom 2435 | Commutation law for identi... |
| aecoms 2436 | A commutation rule for ide... |
| naecoms 2437 | A commutation rule for dis... |
| axc11 2438 | Show that ~ ax-c11 can be ... |
| hbae 2439 | All variables are effectiv... |
| hbnae 2440 | All variables are effectiv... |
| nfae 2441 | All variables are effectiv... |
| nfnae 2442 | All variables are effectiv... |
| hbnaes 2443 | Rule that applies ~ hbnae ... |
| axc16i 2444 | Inference with ~ axc16 as ... |
| axc16nfALT 2445 | Alternate proof of ~ axc16... |
| dral2 2446 | Formula-building lemma for... |
| dral1 2447 | Formula-building lemma for... |
| dral1ALT 2448 | Alternate proof of ~ dral1... |
| drex1 2449 | Formula-building lemma for... |
| drex2 2450 | Formula-building lemma for... |
| drnf1 2451 | Formula-building lemma for... |
| drnf2 2452 | Formula-building lemma for... |
| nfald2 2453 | Variation on ~ nfald which... |
| nfexd2 2454 | Variation on ~ nfexd which... |
| exdistrf 2455 | Distribution of existentia... |
| dvelimf 2456 | Version of ~ dvelimv witho... |
| dvelimdf 2457 | Deduction form of ~ dvelim... |
| dvelimh 2458 | Version of ~ dvelim withou... |
| dvelim 2459 | This theorem can be used t... |
| dvelimv 2460 | Similar to ~ dvelim with f... |
| dvelimnf 2461 | Version of ~ dvelim using ... |
| dveeq2ALT 2462 | Alternate proof of ~ dveeq... |
| equvini 2463 | A variable introduction la... |
| equvel 2464 | A variable elimination law... |
| equs5a 2465 | A property related to subs... |
| equs5e 2466 | A property related to subs... |
| equs45f 2467 | Two ways of expressing sub... |
| equs5 2468 | Lemma used in proofs of su... |
| dveel1 2469 | Quantifier introduction wh... |
| dveel2 2470 | Quantifier introduction wh... |
| axc14 2471 | Axiom ~ ax-c14 is redundan... |
| sb6x 2472 | Equivalence involving subs... |
| sbequ5 2473 | Substitution does not chan... |
| sbequ6 2474 | Substitution does not chan... |
| sb5rf 2475 | Reversed substitution. Us... |
| sb6rf 2476 | Reversed substitution. Fo... |
| ax12vALT 2477 | Alternate proof of ~ ax12v... |
| 2ax6elem 2478 | We can always find values ... |
| 2ax6e 2479 | We can always find values ... |
| 2sb5rf 2480 | Reversed double substituti... |
| 2sb6rf 2481 | Reversed double substituti... |
| sbel2x 2482 | Elimination of double subs... |
| sb4b 2483 | Simplified definition of s... |
| sb3b 2484 | Simplified definition of s... |
| sb3 2485 | One direction of a simplif... |
| sb1 2486 | One direction of a simplif... |
| sb2 2487 | One direction of a simplif... |
| sb4a 2488 | A version of one implicati... |
| dfsb1 2489 | Alternate definition of su... |
| hbsb2 2490 | Bound-variable hypothesis ... |
| nfsb2 2491 | Bound-variable hypothesis ... |
| hbsb2a 2492 | Special case of a bound-va... |
| sb4e 2493 | One direction of a simplif... |
| hbsb2e 2494 | Special case of a bound-va... |
| hbsb3 2495 | If ` y ` is not free in ` ... |
| nfs1 2496 | If ` y ` is not free in ` ... |
| axc16ALT 2497 | Alternate proof of ~ axc16... |
| axc16gALT 2498 | Alternate proof of ~ axc16... |
| equsb1 2499 | Substitution applied to an... |
| equsb2 2500 | Substitution applied to an... |
| dfsb2 2501 | An alternate definition of... |
| dfsb3 2502 | An alternate definition of... |
| drsb1 2503 | Formula-building lemma for... |
| sb2ae 2504 | In the case of two success... |
| sb6f 2505 | Equivalence for substituti... |
| sb5f 2506 | Equivalence for substituti... |
| nfsb4t 2507 | A variable not free in a p... |
| nfsb4 2508 | A variable not free in a p... |
| sbequ8 2509 | Elimination of equality fr... |
| sbie 2510 | Conversion of implicit sub... |
| sbied 2511 | Conversion of implicit sub... |
| sbiedv 2512 | Conversion of implicit sub... |
| 2sbiev 2513 | Conversion of double impli... |
| sbcom3 2514 | Substituting ` y ` for ` x... |
| sbco 2515 | A composition law for subs... |
| sbid2 2516 | An identity law for substi... |
| sbid2v 2517 | An identity law for substi... |
| sbidm 2518 | An idempotent law for subs... |
| sbco2 2519 | A composition law for subs... |
| sbco2d 2520 | A composition law for subs... |
| sbco3 2521 | A composition law for subs... |
| sbcom 2522 | A commutativity law for su... |
| sbtrt 2523 | Partially closed form of ~... |
| sbtr 2524 | A partial converse to ~ sb... |
| sb8 2525 | Substitution of variable i... |
| sb8e 2526 | Substitution of variable i... |
| sb9 2527 | Commutation of quantificat... |
| sb9i 2528 | Commutation of quantificat... |
| sbhb 2529 | Two ways of expressing " `... |
| nfsbd 2530 | Deduction version of ~ nfs... |
| nfsb 2531 | If ` z ` is not free in ` ... |
| hbsb 2532 | If ` z ` is not free in ` ... |
| sb7f 2533 | This version of ~ dfsb7 do... |
| sb7h 2534 | This version of ~ dfsb7 do... |
| sb10f 2535 | Hao Wang's identity axiom ... |
| sbal1 2536 | Check out ~ sbal for a ver... |
| sbal2 2537 | Move quantifier in and out... |
| 2sb8e 2538 | An equivalent expression f... |
| dfmoeu 2539 | An elementary proof of ~ m... |
| dfeumo 2540 | An elementary proof showin... |
| mojust 2542 | Soundness justification th... |
| nexmo 2544 | Nonexistence implies uniqu... |
| exmo 2545 | Any proposition holds for ... |
| moabs 2546 | Absorption of existence co... |
| moim 2547 | The at-most-one quantifier... |
| moimi 2548 | The at-most-one quantifier... |
| moimdv 2549 | The at-most-one quantifier... |
| mobi 2550 | Equivalence theorem for th... |
| mobii 2551 | Formula-building rule for ... |
| mobidv 2552 | Formula-building rule for ... |
| mobid 2553 | Formula-building rule for ... |
| moa1 2554 | If an implication holds fo... |
| moan 2555 | "At most one" is still the... |
| moani 2556 | "At most one" is still tru... |
| moor 2557 | "At most one" is still the... |
| mooran1 2558 | "At most one" imports disj... |
| mooran2 2559 | "At most one" exports disj... |
| nfmo1 2560 | Bound-variable hypothesis ... |
| nfmod2 2561 | Bound-variable hypothesis ... |
| nfmodv 2562 | Bound-variable hypothesis ... |
| nfmov 2563 | Bound-variable hypothesis ... |
| nfmod 2564 | Bound-variable hypothesis ... |
| nfmo 2565 | Bound-variable hypothesis ... |
| mof 2566 | Version of ~ df-mo with di... |
| mo3 2567 | Alternate definition of th... |
| mo 2568 | Equivalent definitions of ... |
| mo4 2569 | At-most-one quantifier exp... |
| mo4f 2570 | At-most-one quantifier exp... |
| eu3v 2573 | An alternate way to expres... |
| eujust 2574 | Soundness justification th... |
| eujustALT 2575 | Alternate proof of ~ eujus... |
| eu6lem 2576 | Lemma of ~ eu6im . A diss... |
| eu6 2577 | Alternate definition of th... |
| eu6im 2578 | One direction of ~ eu6 nee... |
| euf 2579 | Version of ~ eu6 with disj... |
| euex 2580 | Existential uniqueness imp... |
| eumo 2581 | Existential uniqueness imp... |
| eumoi 2582 | Uniqueness inferred from e... |
| exmoeub 2583 | Existence implies that uni... |
| exmoeu 2584 | Existence is equivalent to... |
| moeuex 2585 | Uniqueness implies that ex... |
| moeu 2586 | Uniqueness is equivalent t... |
| eubi 2587 | Equivalence theorem for th... |
| eubii 2588 | Introduce unique existenti... |
| eubidv 2589 | Formula-building rule for ... |
| eubid 2590 | Formula-building rule for ... |
| nfeu1 2591 | Bound-variable hypothesis ... |
| nfeu1ALT 2592 | Alternate proof of ~ nfeu1... |
| nfeud2 2593 | Bound-variable hypothesis ... |
| nfeudw 2594 | Bound-variable hypothesis ... |
| nfeud 2595 | Bound-variable hypothesis ... |
| nfeuw 2596 | Bound-variable hypothesis ... |
| nfeu 2597 | Bound-variable hypothesis ... |
| dfeu 2598 | Rederive ~ df-eu from the ... |
| dfmo 2599 | Rederive ~ df-mo from the ... |
| euequ 2600 | There exists a unique set ... |
| sb8eulem 2601 | Lemma. Factor out the com... |
| sb8euv 2602 | Variable substitution in u... |
| sb8eu 2603 | Variable substitution in u... |
| sb8mo 2604 | Variable substitution for ... |
| cbvmovw 2605 | Change bound variable. Us... |
| cbvmow 2606 | Rule used to change bound ... |
| cbvmo 2607 | Rule used to change bound ... |
| cbveuvw 2608 | Change bound variable. Us... |
| cbveuw 2609 | Version of ~ cbveu with a ... |
| cbveu 2610 | Rule used to change bound ... |
| cbveuALT 2611 | Alternative proof of ~ cbv... |
| eu2 2612 | An alternate way of defini... |
| eu1 2613 | An alternate way to expres... |
| euor 2614 | Introduce a disjunct into ... |
| euorv 2615 | Introduce a disjunct into ... |
| euor2 2616 | Introduce or eliminate a d... |
| sbmo 2617 | Substitution into an at-mo... |
| eu4 2618 | Uniqueness using implicit ... |
| euimmo 2619 | Existential uniqueness imp... |
| euim 2620 | Add unique existential qua... |
| moanimlem 2621 | Factor out the common proo... |
| moanimv 2622 | Introduction of a conjunct... |
| moanim 2623 | Introduction of a conjunct... |
| euan 2624 | Introduction of a conjunct... |
| moanmo 2625 | Nested at-most-one quantif... |
| moaneu 2626 | Nested at-most-one and uni... |
| euanv 2627 | Introduction of a conjunct... |
| mopick 2628 | "At most one" picks a vari... |
| moexexlem 2629 | Factor out the proof skele... |
| 2moexv 2630 | Double quantification with... |
| moexexvw 2631 | "At most one" double quant... |
| 2moswapv 2632 | A condition allowing to sw... |
| 2euswapv 2633 | A condition allowing to sw... |
| 2euexv 2634 | Double quantification with... |
| 2exeuv 2635 | Double existential uniquen... |
| eupick 2636 | Existential uniqueness "pi... |
| eupicka 2637 | Version of ~ eupick with c... |
| eupickb 2638 | Existential uniqueness "pi... |
| eupickbi 2639 | Theorem *14.26 in [Whitehe... |
| mopick2 2640 | "At most one" can show the... |
| moexex 2641 | "At most one" double quant... |
| moexexv 2642 | "At most one" double quant... |
| 2moex 2643 | Double quantification with... |
| 2euex 2644 | Double quantification with... |
| 2eumo 2645 | Nested unique existential ... |
| 2eu2ex 2646 | Double existential uniquen... |
| 2moswap 2647 | A condition allowing to sw... |
| 2euswap 2648 | A condition allowing to sw... |
| 2exeu 2649 | Double existential uniquen... |
| 2mo2 2650 | Two ways of expressing "th... |
| 2mo 2651 | Two ways of expressing "th... |
| 2mos 2652 | Double "there exists at mo... |
| 2mosOLD 2653 | Obsolete version of ~ 2mos... |
| 2eu1 2654 | Double existential uniquen... |
| 2eu1v 2655 | Double existential uniquen... |
| 2eu2 2656 | Double existential uniquen... |
| 2eu3 2657 | Double existential uniquen... |
| 2eu4 2658 | This theorem provides us w... |
| 2eu5 2659 | An alternate definition of... |
| 2eu6 2660 | Two equivalent expressions... |
| 2eu7 2661 | Two equivalent expressions... |
| 2eu8 2662 | Two equivalent expressions... |
| euae 2663 | Two ways to express "exact... |
| exists1 2664 | Two ways to express "exact... |
| exists2 2665 | A condition implying that ... |
| barbara 2666 | "Barbara", one of the fund... |
| celarent 2667 | "Celarent", one of the syl... |
| darii 2668 | "Darii", one of the syllog... |
| dariiALT 2669 | Alternate proof of ~ darii... |
| ferio 2670 | "Ferio" ("Ferioque"), one ... |
| barbarilem 2671 | Lemma for ~ barbari and th... |
| barbari 2672 | "Barbari", one of the syll... |
| barbariALT 2673 | Alternate proof of ~ barba... |
| celaront 2674 | "Celaront", one of the syl... |
| cesare 2675 | "Cesare", one of the syllo... |
| camestres 2676 | "Camestres", one of the sy... |
| festino 2677 | "Festino", one of the syll... |
| festinoALT 2678 | Alternate proof of ~ festi... |
| baroco 2679 | "Baroco", one of the syllo... |
| barocoALT 2680 | Alternate proof of ~ festi... |
| cesaro 2681 | "Cesaro", one of the syllo... |
| camestros 2682 | "Camestros", one of the sy... |
| datisi 2683 | "Datisi", one of the syllo... |
| disamis 2684 | "Disamis", one of the syll... |
| ferison 2685 | "Ferison", one of the syll... |
| bocardo 2686 | "Bocardo", one of the syll... |
| darapti 2687 | "Darapti", one of the syll... |
| daraptiALT 2688 | Alternate proof of ~ darap... |
| felapton 2689 | "Felapton", one of the syl... |
| calemes 2690 | "Calemes", one of the syll... |
| dimatis 2691 | "Dimatis", one of the syll... |
| fresison 2692 | "Fresison", one of the syl... |
| calemos 2693 | "Calemos", one of the syll... |
| fesapo 2694 | "Fesapo", one of the syllo... |
| bamalip 2695 | "Bamalip", one of the syll... |
| axia1 2696 | Left 'and' elimination (in... |
| axia2 2697 | Right 'and' elimination (i... |
| axia3 2698 | 'And' introduction (intuit... |
| axin1 2699 | 'Not' introduction (intuit... |
| axin2 2700 | 'Not' elimination (intuiti... |
| axio 2701 | Definition of 'or' (intuit... |
| axi4 2702 | Specialization (intuitioni... |
| axi5r 2703 | Converse of ~ axc4 (intuit... |
| axial 2704 | The setvar ` x ` is not fr... |
| axie1 2705 | The setvar ` x ` is not fr... |
| axie2 2706 | A key property of existent... |
| axi9 2707 | Axiom of existence (intuit... |
| axi10 2708 | Axiom of Quantifier Substi... |
| axi12 2709 | Axiom of Quantifier Introd... |
| axbnd 2710 | Axiom of Bundling (intuiti... |
| axexte 2712 | The axiom of extensionalit... |
| axextg 2713 | A generalization of the ax... |
| axextb 2714 | A bidirectional version of... |
| axextmo 2715 | There exists at most one s... |
| nulmo 2716 | There exists at most one e... |
| eleq1ab 2719 | Extension (in the sense of... |
| cleljustab 2720 | Extension of ~ cleljust fr... |
| abid 2721 | Simplification of class ab... |
| vexwt 2722 | A standard theorem of pred... |
| vexw 2723 | If ` ph ` is a theorem, th... |
| vextru 2724 | Every setvar is a member o... |
| nfsab1 2725 | Bound-variable hypothesis ... |
| hbab1 2726 | Bound-variable hypothesis ... |
| hbab1OLD 2727 | Obsolete version of ~ hbab... |
| hbab 2728 | Bound-variable hypothesis ... |
| hbabg 2729 | Bound-variable hypothesis ... |
| nfsab 2730 | Bound-variable hypothesis ... |
| nfsabg 2731 | Bound-variable hypothesis ... |
| dfcleq 2733 | The defining characterizat... |
| cvjust 2734 | Every set is a class. Pro... |
| ax9ALT 2735 | Proof of ~ ax-9 from Tarsk... |
| eleq2w2 2736 | A weaker version of ~ eleq... |
| eqriv 2737 | Infer equality of classes ... |
| eqrdv 2738 | Deduce equality of classes... |
| eqrdav 2739 | Deduce equality of classes... |
| eqid 2740 | Law of identity (reflexivi... |
| eqidd 2741 | Class identity law with an... |
| eqeq1d 2742 | Deduction from equality to... |
| eqeq1dALT 2743 | Alternate proof of ~ eqeq1... |
| eqeq1 2744 | Equality implies equivalen... |
| eqeq1i 2745 | Inference from equality to... |
| eqcomd 2746 | Deduction from commutative... |
| eqcom 2747 | Commutative law for class ... |
| eqcoms 2748 | Inference applying commuta... |
| eqcomi 2749 | Inference from commutative... |
| neqcomd 2750 | Commute an inequality. (C... |
| eqeq2d 2751 | Deduction from equality to... |
| eqeq2 2752 | Equality implies equivalen... |
| eqeq2i 2753 | Inference from equality to... |
| eqeqan12d 2754 | A useful inference for sub... |
| eqeqan12rd 2755 | A useful inference for sub... |
| eqeq12d 2756 | A useful inference for sub... |
| eqeq12 2757 | Equality relationship amon... |
| eqeq12i 2758 | A useful inference for sub... |
| eqeq12OLD 2759 | Obsolete version of ~ eqeq... |
| eqeq12dOLD 2760 | Obsolete version of ~ eqeq... |
| eqeqan12dOLD 2761 | Obsolete version of ~ eqeq... |
| eqeqan12dALT 2762 | Alternate proof of ~ eqeqa... |
| eqtr 2763 | Transitive law for class e... |
| eqtr2 2764 | A transitive law for class... |
| eqtr2OLD 2765 | Obsolete version of eqtr2 ... |
| eqtr3 2766 | A transitive law for class... |
| eqtr3OLD 2767 | Obsolete version of ~ eqtr... |
| eqtri 2768 | An equality transitivity i... |
| eqtr2i 2769 | An equality transitivity i... |
| eqtr3i 2770 | An equality transitivity i... |
| eqtr4i 2771 | An equality transitivity i... |
| 3eqtri 2772 | An inference from three ch... |
| 3eqtrri 2773 | An inference from three ch... |
| 3eqtr2i 2774 | An inference from three ch... |
| 3eqtr2ri 2775 | An inference from three ch... |
| 3eqtr3i 2776 | An inference from three ch... |
| 3eqtr3ri 2777 | An inference from three ch... |
| 3eqtr4i 2778 | An inference from three ch... |
| 3eqtr4ri 2779 | An inference from three ch... |
| eqtrd 2780 | An equality transitivity d... |
| eqtr2d 2781 | An equality transitivity d... |
| eqtr3d 2782 | An equality transitivity e... |
| eqtr4d 2783 | An equality transitivity e... |
| 3eqtrd 2784 | A deduction from three cha... |
| 3eqtrrd 2785 | A deduction from three cha... |
| 3eqtr2d 2786 | A deduction from three cha... |
| 3eqtr2rd 2787 | A deduction from three cha... |
| 3eqtr3d 2788 | A deduction from three cha... |
| 3eqtr3rd 2789 | A deduction from three cha... |
| 3eqtr4d 2790 | A deduction from three cha... |
| 3eqtr4rd 2791 | A deduction from three cha... |
| eqtrid 2792 | An equality transitivity d... |
| eqtr2id 2793 | An equality transitivity d... |
| eqtr3id 2794 | An equality transitivity d... |
| eqtr3di 2795 | An equality transitivity d... |
| eqtrdi 2796 | An equality transitivity d... |
| eqtr2di 2797 | An equality transitivity d... |
| eqtr4di 2798 | An equality transitivity d... |
| eqtr4id 2799 | An equality transitivity d... |
| sylan9eq 2800 | An equality transitivity d... |
| sylan9req 2801 | An equality transitivity d... |
| sylan9eqr 2802 | An equality transitivity d... |
| 3eqtr3g 2803 | A chained equality inferen... |
| 3eqtr3a 2804 | A chained equality inferen... |
| 3eqtr4g 2805 | A chained equality inferen... |
| 3eqtr4a 2806 | A chained equality inferen... |
| eq2tri 2807 | A compound transitive infe... |
| iseqsetvlem 2808 | Lemma for ~ iseqsetv-cleq ... |
| iseqsetv-cleq 2809 | Alternate proof of ~ iseqs... |
| abbi 2810 | Equivalent formulas yield ... |
| abbidv 2811 | Equivalent wff's yield equ... |
| abbii 2812 | Equivalent wff's yield equ... |
| abbid 2813 | Equivalent wff's yield equ... |
| abbib 2814 | Equal class abstractions r... |
| cbvabv 2815 | Rule used to change bound ... |
| cbvabw 2816 | Rule used to change bound ... |
| cbvab 2817 | Rule used to change bound ... |
| eqabbw 2818 | Version of ~ eqabb using i... |
| dfclel 2820 | Characterization of the el... |
| elex2 2821 | If a class contains anothe... |
| issettru 2822 | Weak version of ~ isset . ... |
| iseqsetv-clel 2823 | Alternate proof of ~ iseqs... |
| issetlem 2824 | Lemma for ~ elisset and ~ ... |
| elissetv 2825 | An element of a class exis... |
| elisset 2826 | An element of a class exis... |
| eleq1w 2827 | Weaker version of ~ eleq1 ... |
| eleq2w 2828 | Weaker version of ~ eleq2 ... |
| eleq1d 2829 | Deduction from equality to... |
| eleq2d 2830 | Deduction from equality to... |
| eleq2dALT 2831 | Alternate proof of ~ eleq2... |
| eleq1 2832 | Equality implies equivalen... |
| eleq2 2833 | Equality implies equivalen... |
| eleq12 2834 | Equality implies equivalen... |
| eleq1i 2835 | Inference from equality to... |
| eleq2i 2836 | Inference from equality to... |
| eleq12i 2837 | Inference from equality to... |
| eleq12d 2838 | Deduction from equality to... |
| eleq1a 2839 | A transitive-type law rela... |
| eqeltri 2840 | Substitution of equal clas... |
| eqeltrri 2841 | Substitution of equal clas... |
| eleqtri 2842 | Substitution of equal clas... |
| eleqtrri 2843 | Substitution of equal clas... |
| eqeltrd 2844 | Substitution of equal clas... |
| eqeltrrd 2845 | Deduction that substitutes... |
| eleqtrd 2846 | Deduction that substitutes... |
| eleqtrrd 2847 | Deduction that substitutes... |
| eqeltrid 2848 | A membership and equality ... |
| eqeltrrid 2849 | A membership and equality ... |
| eleqtrid 2850 | A membership and equality ... |
| eleqtrrid 2851 | A membership and equality ... |
| eqeltrdi 2852 | A membership and equality ... |
| eqeltrrdi 2853 | A membership and equality ... |
| eleqtrdi 2854 | A membership and equality ... |
| eleqtrrdi 2855 | A membership and equality ... |
| 3eltr3i 2856 | Substitution of equal clas... |
| 3eltr4i 2857 | Substitution of equal clas... |
| 3eltr3d 2858 | Substitution of equal clas... |
| 3eltr4d 2859 | Substitution of equal clas... |
| 3eltr3g 2860 | Substitution of equal clas... |
| 3eltr4g 2861 | Substitution of equal clas... |
| eleq2s 2862 | Substitution of equal clas... |
| eqneltri 2863 | If a class is not an eleme... |
| eqneltrd 2864 | If a class is not an eleme... |
| eqneltrrd 2865 | If a class is not an eleme... |
| neleqtrd 2866 | If a class is not an eleme... |
| neleqtrrd 2867 | If a class is not an eleme... |
| nelneq 2868 | A way of showing two class... |
| nelneq2 2869 | A way of showing two class... |
| eqsb1 2870 | Substitution for the left-... |
| clelsb1 2871 | Substitution for the first... |
| clelsb2 2872 | Substitution for the secon... |
| clelsb2OLD 2873 | Obsolete version of ~ clel... |
| cleqh 2874 | Establish equality between... |
| hbxfreq 2875 | A utility lemma to transfe... |
| hblem 2876 | Change the free variable o... |
| hblemg 2877 | Change the free variable o... |
| eqabdv 2878 | Deduction from a wff to a ... |
| eqabcdv 2879 | Deduction from a wff to a ... |
| eqabi 2880 | Equality of a class variab... |
| abid1 2881 | Every class is equal to a ... |
| abid2 2882 | A simplification of class ... |
| eqab 2883 | One direction of ~ eqabb i... |
| eqabb 2884 | Equality of a class variab... |
| eqabbOLD 2885 | Obsolete version of ~ eqab... |
| eqabcb 2886 | Equality of a class variab... |
| eqabrd 2887 | Equality of a class variab... |
| eqabri 2888 | Equality of a class variab... |
| eqabcri 2889 | Equality of a class variab... |
| clelab 2890 | Membership of a class vari... |
| clabel 2891 | Membership of a class abst... |
| sbab 2892 | The right-hand side of the... |
| nfcjust 2894 | Justification theorem for ... |
| nfci 2896 | Deduce that a class ` A ` ... |
| nfcii 2897 | Deduce that a class ` A ` ... |
| nfcr 2898 | Consequence of the not-fre... |
| nfcrALT 2899 | Alternate version of ~ nfc... |
| nfcri 2900 | Consequence of the not-fre... |
| nfcd 2901 | Deduce that a class ` A ` ... |
| nfcrd 2902 | Consequence of the not-fre... |
| nfcrii 2903 | Consequence of the not-fre... |
| nfceqdf 2904 | An equality theorem for ef... |
| nfceqi 2905 | Equality theorem for class... |
| nfcxfr 2906 | A utility lemma to transfe... |
| nfcxfrd 2907 | A utility lemma to transfe... |
| nfcv 2908 | If ` x ` is disjoint from ... |
| nfcvd 2909 | If ` x ` is disjoint from ... |
| nfab1 2910 | Bound-variable hypothesis ... |
| nfnfc1 2911 | The setvar ` x ` is bound ... |
| clelsb1fw 2912 | Substitution for the first... |
| clelsb1f 2913 | Substitution for the first... |
| nfab 2914 | Bound-variable hypothesis ... |
| nfabg 2915 | Bound-variable hypothesis ... |
| nfaba1 2916 | Bound-variable hypothesis ... |
| nfaba1OLD 2917 | Obsolete version of ~ nfab... |
| nfaba1g 2918 | Bound-variable hypothesis ... |
| nfeqd 2919 | Hypothesis builder for equ... |
| nfeld 2920 | Hypothesis builder for ele... |
| nfnfc 2921 | Hypothesis builder for ` F... |
| nfeq 2922 | Hypothesis builder for equ... |
| nfel 2923 | Hypothesis builder for ele... |
| nfeq1 2924 | Hypothesis builder for equ... |
| nfel1 2925 | Hypothesis builder for ele... |
| nfeq2 2926 | Hypothesis builder for equ... |
| nfel2 2927 | Hypothesis builder for ele... |
| drnfc1 2928 | Formula-building lemma for... |
| drnfc1OLD 2929 | Obsolete version of ~ drnf... |
| drnfc2 2930 | Formula-building lemma for... |
| drnfc2OLD 2931 | Obsolete version of ~ drnf... |
| nfabdw 2932 | Bound-variable hypothesis ... |
| nfabdwOLD 2933 | Obsolete version of ~ nfab... |
| nfabd 2934 | Bound-variable hypothesis ... |
| nfabd2 2935 | Bound-variable hypothesis ... |
| dvelimdc 2936 | Deduction form of ~ dvelim... |
| dvelimc 2937 | Version of ~ dvelim for cl... |
| nfcvf 2938 | If ` x ` and ` y ` are dis... |
| nfcvf2 2939 | If ` x ` and ` y ` are dis... |
| cleqf 2940 | Establish equality between... |
| eqabf 2941 | Equality of a class variab... |
| abid2f 2942 | A simplification of class ... |
| abid2fOLD 2943 | Obsolete version of ~ abid... |
| sbabel 2944 | Theorem to move a substitu... |
| sbabelOLD 2945 | Obsolete version of ~ sbab... |
| neii 2948 | Inference associated with ... |
| neir 2949 | Inference associated with ... |
| nne 2950 | Negation of inequality. (... |
| neneqd 2951 | Deduction eliminating ineq... |
| neneq 2952 | From inequality to non-equ... |
| neqned 2953 | If it is not the case that... |
| neqne 2954 | From non-equality to inequ... |
| neirr 2955 | No class is unequal to its... |
| exmidne 2956 | Excluded middle with equal... |
| eqneqall 2957 | A contradiction concerning... |
| nonconne 2958 | Law of noncontradiction wi... |
| necon3ad 2959 | Contrapositive law deducti... |
| necon3bd 2960 | Contrapositive law deducti... |
| necon2ad 2961 | Contrapositive inference f... |
| necon2bd 2962 | Contrapositive inference f... |
| necon1ad 2963 | Contrapositive deduction f... |
| necon1bd 2964 | Contrapositive deduction f... |
| necon4ad 2965 | Contrapositive inference f... |
| necon4bd 2966 | Contrapositive inference f... |
| necon3d 2967 | Contrapositive law deducti... |
| necon1d 2968 | Contrapositive law deducti... |
| necon2d 2969 | Contrapositive inference f... |
| necon4d 2970 | Contrapositive inference f... |
| necon3ai 2971 | Contrapositive inference f... |
| necon3aiOLD 2972 | Obsolete version of ~ neco... |
| necon3bi 2973 | Contrapositive inference f... |
| necon1ai 2974 | Contrapositive inference f... |
| necon1bi 2975 | Contrapositive inference f... |
| necon2ai 2976 | Contrapositive inference f... |
| necon2bi 2977 | Contrapositive inference f... |
| necon4ai 2978 | Contrapositive inference f... |
| necon3i 2979 | Contrapositive inference f... |
| necon1i 2980 | Contrapositive inference f... |
| necon2i 2981 | Contrapositive inference f... |
| necon4i 2982 | Contrapositive inference f... |
| necon3abid 2983 | Deduction from equality to... |
| necon3bbid 2984 | Deduction from equality to... |
| necon1abid 2985 | Contrapositive deduction f... |
| necon1bbid 2986 | Contrapositive inference f... |
| necon4abid 2987 | Contrapositive law deducti... |
| necon4bbid 2988 | Contrapositive law deducti... |
| necon2abid 2989 | Contrapositive deduction f... |
| necon2bbid 2990 | Contrapositive deduction f... |
| necon3bid 2991 | Deduction from equality to... |
| necon4bid 2992 | Contrapositive law deducti... |
| necon3abii 2993 | Deduction from equality to... |
| necon3bbii 2994 | Deduction from equality to... |
| necon1abii 2995 | Contrapositive inference f... |
| necon1bbii 2996 | Contrapositive inference f... |
| necon2abii 2997 | Contrapositive inference f... |
| necon2bbii 2998 | Contrapositive inference f... |
| necon3bii 2999 | Inference from equality to... |
| necom 3000 | Commutation of inequality.... |
| necomi 3001 | Inference from commutative... |
| necomd 3002 | Deduction from commutative... |
| nesym 3003 | Characterization of inequa... |
| nesymi 3004 | Inference associated with ... |
| nesymir 3005 | Inference associated with ... |
| neeq1d 3006 | Deduction for inequality. ... |
| neeq2d 3007 | Deduction for inequality. ... |
| neeq12d 3008 | Deduction for inequality. ... |
| neeq1 3009 | Equality theorem for inequ... |
| neeq2 3010 | Equality theorem for inequ... |
| neeq1i 3011 | Inference for inequality. ... |
| neeq2i 3012 | Inference for inequality. ... |
| neeq12i 3013 | Inference for inequality. ... |
| eqnetrd 3014 | Substitution of equal clas... |
| eqnetrrd 3015 | Substitution of equal clas... |
| neeqtrd 3016 | Substitution of equal clas... |
| eqnetri 3017 | Substitution of equal clas... |
| eqnetrri 3018 | Substitution of equal clas... |
| neeqtri 3019 | Substitution of equal clas... |
| neeqtrri 3020 | Substitution of equal clas... |
| neeqtrrd 3021 | Substitution of equal clas... |
| eqnetrrid 3022 | A chained equality inferen... |
| 3netr3d 3023 | Substitution of equality i... |
| 3netr4d 3024 | Substitution of equality i... |
| 3netr3g 3025 | Substitution of equality i... |
| 3netr4g 3026 | Substitution of equality i... |
| nebi 3027 | Contraposition law for ine... |
| pm13.18 3028 | Theorem *13.18 in [Whitehe... |
| pm13.181 3029 | Theorem *13.181 in [Whiteh... |
| pm13.181OLD 3030 | Obsolete version of ~ pm13... |
| pm2.61ine 3031 | Inference eliminating an i... |
| pm2.21ddne 3032 | A contradiction implies an... |
| pm2.61ne 3033 | Deduction eliminating an i... |
| pm2.61dne 3034 | Deduction eliminating an i... |
| pm2.61dane 3035 | Deduction eliminating an i... |
| pm2.61da2ne 3036 | Deduction eliminating two ... |
| pm2.61da3ne 3037 | Deduction eliminating thre... |
| pm2.61iine 3038 | Equality version of ~ pm2.... |
| mteqand 3039 | A modus tollens deduction ... |
| neor 3040 | Logical OR with an equalit... |
| neanior 3041 | A De Morgan's law for ineq... |
| ne3anior 3042 | A De Morgan's law for ineq... |
| neorian 3043 | A De Morgan's law for ineq... |
| nemtbir 3044 | An inference from an inequ... |
| nelne1 3045 | Two classes are different ... |
| nelne2 3046 | Two classes are different ... |
| nelelne 3047 | Two classes are different ... |
| neneor 3048 | If two classes are differe... |
| nfne 3049 | Bound-variable hypothesis ... |
| nfned 3050 | Bound-variable hypothesis ... |
| nabbib 3051 | Not equivalent wff's corre... |
| neli 3054 | Inference associated with ... |
| nelir 3055 | Inference associated with ... |
| nelcon3d 3056 | Contrapositive law deducti... |
| neleq12d 3057 | Equality theorem for negat... |
| neleq1 3058 | Equality theorem for negat... |
| neleq2 3059 | Equality theorem for negat... |
| nfnel 3060 | Bound-variable hypothesis ... |
| nfneld 3061 | Bound-variable hypothesis ... |
| nnel 3062 | Negation of negated member... |
| elnelne1 3063 | Two classes are different ... |
| elnelne2 3064 | Two classes are different ... |
| pm2.24nel 3065 | A contradiction concerning... |
| pm2.61danel 3066 | Deduction eliminating an e... |
| rgen 3069 | Generalization rule for re... |
| ralel 3070 | All elements of a class ar... |
| rgenw 3071 | Generalization rule for re... |
| rgen2w 3072 | Generalization rule for re... |
| mprg 3073 | Modus ponens combined with... |
| mprgbir 3074 | Modus ponens on biconditio... |
| raln 3075 | Restricted universally qua... |
| ralnex 3078 | Relationship between restr... |
| dfrex2 3079 | Relationship between restr... |
| nrex 3080 | Inference adding restricte... |
| alral 3081 | Universal quantification i... |
| rexex 3082 | Restricted existence impli... |
| rextru 3083 | Two ways of expressing tha... |
| ralimi2 3084 | Inference quantifying both... |
| reximi2 3085 | Inference quantifying both... |
| ralimia 3086 | Inference quantifying both... |
| reximia 3087 | Inference quantifying both... |
| ralimiaa 3088 | Inference quantifying both... |
| ralimi 3089 | Inference quantifying both... |
| reximi 3090 | Inference quantifying both... |
| ral2imi 3091 | Inference quantifying ante... |
| ralim 3092 | Distribution of restricted... |
| rexim 3093 | Theorem 19.22 of [Margaris... |
| reximiaOLD 3094 | Obsolete version of ~ rexi... |
| ralbii2 3095 | Inference adding different... |
| rexbii2 3096 | Inference adding different... |
| ralbiia 3097 | Inference adding restricte... |
| rexbiia 3098 | Inference adding restricte... |
| ralbii 3099 | Inference adding restricte... |
| rexbii 3100 | Inference adding restricte... |
| ralanid 3101 | Cancellation law for restr... |
| rexanid 3102 | Cancellation law for restr... |
| ralcom3 3103 | A commutation law for rest... |
| ralcom3OLD 3104 | Obsolete version of ~ ralc... |
| dfral2 3105 | Relationship between restr... |
| rexnal 3106 | Relationship between restr... |
| ralinexa 3107 | A transformation of restri... |
| rexanali 3108 | A transformation of restri... |
| ralbi 3109 | Distribute a restricted un... |
| rexbi 3110 | Distribute restricted quan... |
| rexbiOLD 3111 | Obsolete version of ~ rexb... |
| ralrexbid 3112 | Formula-building rule for ... |
| ralrexbidOLD 3113 | Obsolete version of ~ ralr... |
| r19.35 3114 | Restricted quantifier vers... |
| r19.35OLD 3115 | Obsolete version of ~ 19.3... |
| r19.26m 3116 | Version of ~ 19.26 and ~ r... |
| r19.26 3117 | Restricted quantifier vers... |
| r19.26-3 3118 | Version of ~ r19.26 with t... |
| ralbiim 3119 | Split a biconditional and ... |
| r19.29 3120 | Restricted quantifier vers... |
| r19.29OLD 3121 | Obsolete version of ~ r19.... |
| r19.29r 3122 | Restricted quantifier vers... |
| r19.29rOLD 3123 | Obsolete version of ~ r19.... |
| r19.29imd 3124 | Theorem 19.29 of [Margaris... |
| r19.40 3125 | Restricted quantifier vers... |
| r19.30 3126 | Restricted quantifier vers... |
| r19.30OLD 3127 | Obsolete version of ~ 19.3... |
| r19.43 3128 | Restricted quantifier vers... |
| 2ralimi 3129 | Inference quantifying both... |
| 3ralimi 3130 | Inference quantifying both... |
| 4ralimi 3131 | Inference quantifying both... |
| 5ralimi 3132 | Inference quantifying both... |
| 6ralimi 3133 | Inference quantifying both... |
| 2ralbii 3134 | Inference adding two restr... |
| 2rexbii 3135 | Inference adding two restr... |
| 3ralbii 3136 | Inference adding three res... |
| 4ralbii 3137 | Inference adding four rest... |
| 2ralbiim 3138 | Split a biconditional and ... |
| ralnex2 3139 | Relationship between two r... |
| ralnex3 3140 | Relationship between three... |
| rexnal2 3141 | Relationship between two r... |
| rexnal3 3142 | Relationship between three... |
| nrexralim 3143 | Negation of a complex pred... |
| r19.26-2 3144 | Restricted quantifier vers... |
| 2r19.29 3145 | Theorem ~ r19.29 with two ... |
| r19.29d2r 3146 | Theorem 19.29 of [Margaris... |
| r19.29d2rOLD 3147 | Obsolete version of ~ r19.... |
| r2allem 3148 | Lemma factoring out common... |
| r2exlem 3149 | Lemma factoring out common... |
| hbralrimi 3150 | Inference from Theorem 19.... |
| ralrimiv 3151 | Inference from Theorem 19.... |
| ralrimiva 3152 | Inference from Theorem 19.... |
| rexlimiva 3153 | Inference from Theorem 19.... |
| rexlimiv 3154 | Inference from Theorem 19.... |
| nrexdv 3155 | Deduction adding restricte... |
| ralrimivw 3156 | Inference from Theorem 19.... |
| rexlimivw 3157 | Weaker version of ~ rexlim... |
| ralrimdv 3158 | Inference from Theorem 19.... |
| rexlimdv 3159 | Inference from Theorem 19.... |
| ralrimdva 3160 | Inference from Theorem 19.... |
| rexlimdva 3161 | Inference from Theorem 19.... |
| rexlimdvaa 3162 | Inference from Theorem 19.... |
| rexlimdva2 3163 | Inference from Theorem 19.... |
| r19.29an 3164 | A commonly used pattern in... |
| rexlimdv3a 3165 | Inference from Theorem 19.... |
| rexlimdvw 3166 | Inference from Theorem 19.... |
| rexlimddv 3167 | Restricted existential eli... |
| r19.29a 3168 | A commonly used pattern in... |
| ralimdv2 3169 | Inference quantifying both... |
| reximdv2 3170 | Deduction quantifying both... |
| reximdvai 3171 | Deduction quantifying both... |
| reximdvaiOLD 3172 | Obsolete version of ~ rexi... |
| ralimdva 3173 | Deduction quantifying both... |
| reximdva 3174 | Deduction quantifying both... |
| ralimdv 3175 | Deduction quantifying both... |
| reximdv 3176 | Deduction from Theorem 19.... |
| reximddv 3177 | Deduction from Theorem 19.... |
| reximddv3 3178 | Deduction from Theorem 19.... |
| reximssdv 3179 | Derivation of a restricted... |
| ralbidv2 3180 | Formula-building rule for ... |
| rexbidv2 3181 | Formula-building rule for ... |
| ralbidva 3182 | Formula-building rule for ... |
| rexbidva 3183 | Formula-building rule for ... |
| ralbidv 3184 | Formula-building rule for ... |
| rexbidv 3185 | Formula-building rule for ... |
| r19.21v 3186 | Restricted quantifier vers... |
| r19.21vOLD 3187 | Obsolete version of ~ r19.... |
| r19.37v 3188 | Restricted quantifier vers... |
| r19.23v 3189 | Restricted quantifier vers... |
| r19.36v 3190 | Restricted quantifier vers... |
| rexlimivOLD 3191 | Obsolete version of ~ rexl... |
| rexlimivaOLD 3192 | Obsolete version of ~ rexl... |
| rexlimivwOLD 3193 | Obsolete version of ~ rexl... |
| r19.27v 3194 | Restricted quantitifer ver... |
| r19.41v 3195 | Restricted quantifier vers... |
| r19.28v 3196 | Restricted quantifier vers... |
| r19.42v 3197 | Restricted quantifier vers... |
| r19.32v 3198 | Restricted quantifier vers... |
| r19.45v 3199 | Restricted quantifier vers... |
| r19.44v 3200 | One direction of a restric... |
| r2al 3201 | Double restricted universa... |
| r2ex 3202 | Double restricted existent... |
| r3al 3203 | Triple restricted universa... |
| r3ex 3204 | Triple existential quantif... |
| rgen2 3205 | Generalization rule for re... |
| ralrimivv 3206 | Inference from Theorem 19.... |
| rexlimivv 3207 | Inference from Theorem 19.... |
| ralrimivva 3208 | Inference from Theorem 19.... |
| ralrimdvv 3209 | Inference from Theorem 19.... |
| rgen3 3210 | Generalization rule for re... |
| ralrimivvva 3211 | Inference from Theorem 19.... |
| ralimdvva 3212 | Deduction doubly quantifyi... |
| reximdvva 3213 | Deduction doubly quantifyi... |
| ralimdvv 3214 | Deduction doubly quantifyi... |
| ralimd4v 3215 | Deduction quadrupally quan... |
| ralimd6v 3216 | Deduction sextupally quant... |
| ralrimdvva 3217 | Inference from Theorem 19.... |
| rexlimdvv 3218 | Inference from Theorem 19.... |
| rexlimdvva 3219 | Inference from Theorem 19.... |
| rexlimdvvva 3220 | Inference from Theorem 19.... |
| reximddv2 3221 | Double deduction from Theo... |
| r19.29vva 3222 | A commonly used pattern ba... |
| r19.29vvaOLD 3223 | Obsolete version of ~ r19.... |
| 2rexbiia 3224 | Inference adding two restr... |
| 2ralbidva 3225 | Formula-building rule for ... |
| 2rexbidva 3226 | Formula-building rule for ... |
| 2ralbidv 3227 | Formula-building rule for ... |
| 2rexbidv 3228 | Formula-building rule for ... |
| rexralbidv 3229 | Formula-building rule for ... |
| 3ralbidv 3230 | Formula-building rule for ... |
| 4ralbidv 3231 | Formula-building rule for ... |
| 6ralbidv 3232 | Formula-building rule for ... |
| r19.41vv 3233 | Version of ~ r19.41v with ... |
| reeanlem 3234 | Lemma factoring out common... |
| reeanv 3235 | Rearrange restricted exist... |
| 3reeanv 3236 | Rearrange three restricted... |
| 2ralor 3237 | Distribute restricted univ... |
| 2ralorOLD 3238 | Obsolete version of ~ 2ral... |
| risset 3239 | Two ways to say " ` A ` be... |
| nelb 3240 | A definition of ` -. A e. ... |
| nelbOLD 3241 | Obsolete version of ~ nelb... |
| rspw 3242 | Restricted specialization.... |
| cbvralvw 3243 | Change the bound variable ... |
| cbvrexvw 3244 | Change the bound variable ... |
| cbvraldva 3245 | Rule used to change the bo... |
| cbvrexdva 3246 | Rule used to change the bo... |
| cbvral2vw 3247 | Change bound variables of ... |
| cbvrex2vw 3248 | Change bound variables of ... |
| cbvral3vw 3249 | Change bound variables of ... |
| cbvral4vw 3250 | Change bound variables of ... |
| cbvral6vw 3251 | Change bound variables of ... |
| cbvral8vw 3252 | Change bound variables of ... |
| rsp 3253 | Restricted specialization.... |
| rspa 3254 | Restricted specialization.... |
| rspe 3255 | Restricted specialization.... |
| rspec 3256 | Specialization rule for re... |
| r19.21bi 3257 | Inference from Theorem 19.... |
| r19.21be 3258 | Inference from Theorem 19.... |
| r19.21t 3259 | Restricted quantifier vers... |
| r19.21 3260 | Restricted quantifier vers... |
| r19.23t 3261 | Closed theorem form of ~ r... |
| r19.23 3262 | Restricted quantifier vers... |
| ralrimi 3263 | Inference from Theorem 19.... |
| ralrimia 3264 | Inference from Theorem 19.... |
| rexlimi 3265 | Restricted quantifier vers... |
| ralimdaa 3266 | Deduction quantifying both... |
| reximdai 3267 | Deduction from Theorem 19.... |
| r19.37 3268 | Restricted quantifier vers... |
| r19.41 3269 | Restricted quantifier vers... |
| ralrimd 3270 | Inference from Theorem 19.... |
| rexlimd2 3271 | Version of ~ rexlimd with ... |
| rexlimd 3272 | Deduction form of ~ rexlim... |
| r19.29af2 3273 | A commonly used pattern ba... |
| r19.29af 3274 | A commonly used pattern ba... |
| reximd2a 3275 | Deduction quantifying both... |
| ralbida 3276 | Formula-building rule for ... |
| ralbidaOLD 3277 | Obsolete version of ~ ralb... |
| rexbida 3278 | Formula-building rule for ... |
| ralbid 3279 | Formula-building rule for ... |
| rexbid 3280 | Formula-building rule for ... |
| rexbidvALT 3281 | Alternate proof of ~ rexbi... |
| rexbidvaALT 3282 | Alternate proof of ~ rexbi... |
| rsp2 3283 | Restricted specialization,... |
| rsp2e 3284 | Restricted specialization.... |
| rspec2 3285 | Specialization rule for re... |
| rspec3 3286 | Specialization rule for re... |
| r2alf 3287 | Double restricted universa... |
| r2exf 3288 | Double restricted existent... |
| 2ralbida 3289 | Formula-building rule for ... |
| nfra1 3290 | The setvar ` x ` is not fr... |
| nfre1 3291 | The setvar ` x ` is not fr... |
| ralcom4 3292 | Commutation of restricted ... |
| ralcom4OLD 3293 | Obsolete version of ~ ralc... |
| rexcom4 3294 | Commutation of restricted ... |
| ralcom 3295 | Commutation of restricted ... |
| rexcom 3296 | Commutation of restricted ... |
| rexcomOLD 3297 | Obsolete version of ~ rexc... |
| rexcom4a 3298 | Specialized existential co... |
| ralrot3 3299 | Rotate three restricted un... |
| ralcom13 3300 | Swap first and third restr... |
| ralcom13OLD 3301 | Obsolete version of ~ ralc... |
| rexcom13 3302 | Swap first and third restr... |
| rexrot4 3303 | Rotate four restricted exi... |
| 2ex2rexrot 3304 | Rotate two existential qua... |
| nfra2w 3305 | Similar to Lemma 24 of [Mo... |
| nfra2wOLD 3306 | Obsolete version of ~ nfra... |
| hbra1 3307 | The setvar ` x ` is not fr... |
| ralcomf 3308 | Commutation of restricted ... |
| rexcomf 3309 | Commutation of restricted ... |
| cbvralfw 3310 | Rule used to change bound ... |
| cbvrexfw 3311 | Rule used to change bound ... |
| cbvralw 3312 | Rule used to change bound ... |
| cbvrexw 3313 | Rule used to change bound ... |
| hbral 3314 | Bound-variable hypothesis ... |
| nfraldw 3315 | Deduction version of ~ nfr... |
| nfrexdw 3316 | Deduction version of ~ nfr... |
| nfralw 3317 | Bound-variable hypothesis ... |
| nfralwOLD 3318 | Obsolete version of ~ nfra... |
| nfrexw 3319 | Bound-variable hypothesis ... |
| r19.12 3320 | Restricted quantifier vers... |
| r19.12OLD 3321 | Obsolete version of ~ 19.1... |
| reean 3322 | Rearrange restricted exist... |
| cbvralsvw 3323 | Change bound variable by u... |
| cbvrexsvw 3324 | Change bound variable by u... |
| cbvralsvwOLD 3325 | Obsolete version of ~ cbvr... |
| cbvralsvwOLDOLD 3326 | Obsolete version of ~ cbvr... |
| cbvrexsvwOLD 3327 | Obsolete version of ~ cbvr... |
| nfraldwOLD 3328 | Obsolete version of ~ nfra... |
| nfra2wOLDOLD 3329 | Obsolete version of ~ nfra... |
| rexeq 3330 | Equality theorem for restr... |
| raleq 3331 | Equality theorem for restr... |
| raleqi 3332 | Equality inference for res... |
| rexeqi 3333 | Equality inference for res... |
| raleqdv 3334 | Equality deduction for res... |
| rexeqdv 3335 | Equality deduction for res... |
| raleqtrdv 3336 | Substitution of equal clas... |
| rexeqtrdv 3337 | Substitution of equal clas... |
| raleqtrrdv 3338 | Substitution of equal clas... |
| rexeqtrrdv 3339 | Substitution of equal clas... |
| raleqbidva 3340 | Equality deduction for res... |
| rexeqbidva 3341 | Equality deduction for res... |
| raleqbidvv 3342 | Version of ~ raleqbidv wit... |
| raleqbidvvOLD 3343 | Obsolete version of ~ rale... |
| rexeqbidvv 3344 | Version of ~ rexeqbidv wit... |
| rexeqbidvvOLD 3345 | Obsolete version of ~ rexe... |
| raleqbi1dv 3346 | Equality deduction for res... |
| rexeqbi1dv 3347 | Equality deduction for res... |
| raleqOLD 3348 | Obsolete version of ~ rale... |
| rexeqOLD 3349 | Obsolete version of ~ rale... |
| raleleq 3350 | All elements of a class ar... |
| raleleqOLD 3351 | Obsolete version of ~ rale... |
| raleqbii 3352 | Equality deduction for res... |
| rexeqbii 3353 | Equality deduction for res... |
| raleqbidv 3354 | Equality deduction for res... |
| rexeqbidv 3355 | Equality deduction for res... |
| cbvraldva2 3356 | Rule used to change the bo... |
| cbvrexdva2 3357 | Rule used to change the bo... |
| cbvrexdva2OLD 3358 | Obsolete version of ~ cbvr... |
| cbvraldvaOLD 3359 | Obsolete version of ~ cbvr... |
| cbvrexdvaOLD 3360 | Obsolete version of ~ cbvr... |
| raleqf 3361 | Equality theorem for restr... |
| rexeqf 3362 | Equality theorem for restr... |
| rexeqfOLD 3363 | Obsolete version of ~ rexe... |
| raleqbid 3364 | Equality deduction for res... |
| rexeqbid 3365 | Equality deduction for res... |
| sbralie 3366 | Implicit to explicit subst... |
| sbralieALT 3367 | Alternative shorter proof ... |
| cbvralf 3368 | Rule used to change bound ... |
| cbvrexf 3369 | Rule used to change bound ... |
| cbvral 3370 | Rule used to change bound ... |
| cbvrex 3371 | Rule used to change bound ... |
| cbvralv 3372 | Change the bound variable ... |
| cbvrexv 3373 | Change the bound variable ... |
| cbvralsv 3374 | Change bound variable by u... |
| cbvrexsv 3375 | Change bound variable by u... |
| cbvral2v 3376 | Change bound variables of ... |
| cbvrex2v 3377 | Change bound variables of ... |
| cbvral3v 3378 | Change bound variables of ... |
| rgen2a 3379 | Generalization rule for re... |
| nfrald 3380 | Deduction version of ~ nfr... |
| nfrexd 3381 | Deduction version of ~ nfr... |
| nfral 3382 | Bound-variable hypothesis ... |
| nfrex 3383 | Bound-variable hypothesis ... |
| nfra2 3384 | Similar to Lemma 24 of [Mo... |
| ralcom2 3385 | Commutation of restricted ... |
| reu5 3390 | Restricted uniqueness in t... |
| reurmo 3391 | Restricted existential uni... |
| reurex 3392 | Restricted unique existenc... |
| mormo 3393 | Unrestricted "at most one"... |
| rmobiia 3394 | Formula-building rule for ... |
| reubiia 3395 | Formula-building rule for ... |
| rmobii 3396 | Formula-building rule for ... |
| reubii 3397 | Formula-building rule for ... |
| rmoanid 3398 | Cancellation law for restr... |
| reuanid 3399 | Cancellation law for restr... |
| rmoanidOLD 3400 | Obsolete version of ~ rmoa... |
| reuanidOLD 3401 | Obsolete version of ~ reua... |
| 2reu2rex 3402 | Double restricted existent... |
| rmobidva 3403 | Formula-building rule for ... |
| reubidva 3404 | Formula-building rule for ... |
| rmobidv 3405 | Formula-building rule for ... |
| reubidv 3406 | Formula-building rule for ... |
| reueubd 3407 | Restricted existential uni... |
| rmo5 3408 | Restricted "at most one" i... |
| nrexrmo 3409 | Nonexistence implies restr... |
| moel 3410 | "At most one" element in a... |
| cbvrmovw 3411 | Change the bound variable ... |
| cbvreuvw 3412 | Change the bound variable ... |
| moelOLD 3413 | Obsolete version of ~ moel... |
| rmobida 3414 | Formula-building rule for ... |
| reubida 3415 | Formula-building rule for ... |
| rmobidvaOLD 3416 | Obsolete version of ~ rmob... |
| cbvrmow 3417 | Change the bound variable ... |
| cbvreuw 3418 | Change the bound variable ... |
| nfrmo1 3419 | The setvar ` x ` is not fr... |
| nfreu1 3420 | The setvar ` x ` is not fr... |
| nfrmow 3421 | Bound-variable hypothesis ... |
| nfreuw 3422 | Bound-variable hypothesis ... |
| cbvreuwOLD 3423 | Obsolete version of ~ cbvr... |
| cbvreuvwOLD 3424 | Obsolete version of ~ cbvr... |
| rmoeq1 3425 | Equality theorem for restr... |
| reueq1 3426 | Equality theorem for restr... |
| rmoeq1OLD 3427 | Obsolete version of ~ rmoe... |
| reueq1OLD 3428 | Obsolete version of ~ reue... |
| rmoeqd 3429 | Equality deduction for res... |
| reueqd 3430 | Equality deduction for res... |
| rmoeq1f 3431 | Equality theorem for restr... |
| reueq1f 3432 | Equality theorem for restr... |
| nfreuwOLD 3433 | Obsolete version of ~ nfre... |
| nfrmowOLD 3434 | Obsolete version of ~ nfrm... |
| cbvreu 3435 | Change the bound variable ... |
| cbvrmo 3436 | Change the bound variable ... |
| cbvrmov 3437 | Change the bound variable ... |
| cbvreuv 3438 | Change the bound variable ... |
| nfrmod 3439 | Deduction version of ~ nfr... |
| nfreud 3440 | Deduction version of ~ nfr... |
| nfrmo 3441 | Bound-variable hypothesis ... |
| nfreu 3442 | Bound-variable hypothesis ... |
| rabbidva2 3445 | Equivalent wff's yield equ... |
| rabbia2 3446 | Equivalent wff's yield equ... |
| rabbiia 3447 | Equivalent formulas yield ... |
| rabbiiaOLD 3448 | Obsolete version of ~ rabb... |
| rabbii 3449 | Equivalent wff's correspon... |
| rabbidva 3450 | Equivalent wff's yield equ... |
| rabbidv 3451 | Equivalent wff's yield equ... |
| rabbieq 3452 | Equivalent wff's correspon... |
| rabswap 3453 | Swap with a membership rel... |
| cbvrabv 3454 | Rule to change the bound v... |
| rabeqcda 3455 | When ` ps ` is always true... |
| rabeqc 3456 | A restricted class abstrac... |
| rabeqi 3457 | Equality theorem for restr... |
| rabeq 3458 | Equality theorem for restr... |
| rabeqdv 3459 | Equality of restricted cla... |
| rabeqbidva 3460 | Equality of restricted cla... |
| rabeqbidvaOLD 3461 | Obsolete version of ~ rabe... |
| rabeqbidv 3462 | Equality of restricted cla... |
| rabrabi 3463 | Abstract builder restricte... |
| nfrab1 3464 | The abstraction variable i... |
| rabid 3465 | An "identity" law of concr... |
| rabidim1 3466 | Membership in a restricted... |
| reqabi 3467 | Inference from equality of... |
| rabrab 3468 | Abstract builder restricte... |
| rabrabiOLD 3469 | Obsolete version of ~ rabr... |
| rabbida4 3470 | Version of ~ rabbidva2 wit... |
| rabbida 3471 | Equivalent wff's yield equ... |
| rabbid 3472 | Version of ~ rabbidv with ... |
| rabeqd 3473 | Deduction form of ~ rabeq ... |
| rabeqbida 3474 | Version of ~ rabeqbidva wi... |
| rabbi 3475 | Equivalent wff's correspon... |
| rabid2f 3476 | An "identity" law for rest... |
| rabid2im 3477 | One direction of ~ rabid2 ... |
| rabid2 3478 | An "identity" law for rest... |
| rabid2OLD 3479 | Obsolete version of ~ rabi... |
| rabeqf 3480 | Equality theorem for restr... |
| cbvrabw 3481 | Rule to change the bound v... |
| cbvrabwOLD 3482 | Obsolete version of ~ cbvr... |
| nfrabw 3483 | A variable not free in a w... |
| nfrabwOLD 3484 | Obsolete version of ~ nfra... |
| rabbidaOLD 3485 | Obsolete version of ~ rabb... |
| nfrab 3486 | A variable not free in a w... |
| cbvrab 3487 | Rule to change the bound v... |
| vjust 3489 | Justification theorem for ... |
| dfv2 3491 | Alternate definition of th... |
| vex 3492 | All setvar variables are s... |
| elv 3493 | If a proposition is implie... |
| elvd 3494 | If a proposition is implie... |
| el2v 3495 | If a proposition is implie... |
| el3v 3496 | If a proposition is implie... |
| el3v3 3497 | If a proposition is implie... |
| eqv 3498 | The universe contains ever... |
| eqvf 3499 | The universe contains ever... |
| abv 3500 | The class of sets verifyin... |
| abvALT 3501 | Alternate proof of ~ abv ,... |
| isset 3502 | Two ways to express that "... |
| cbvexeqsetf 3503 | The expression ` E. x x = ... |
| issetft 3504 | Closed theorem form of ~ i... |
| issetf 3505 | A version of ~ isset that ... |
| isseti 3506 | A way to say " ` A ` is a ... |
| issetri 3507 | A way to say " ` A ` is a ... |
| eqvisset 3508 | A class equal to a variabl... |
| elex 3509 | If a class is a member of ... |
| elexOLD 3510 | Obsolete version of ~ elex... |
| elexi 3511 | If a class is a member of ... |
| elexd 3512 | If a class is a member of ... |
| elex2OLD 3513 | Obsolete version of ~ elex... |
| elex22 3514 | If two classes each contai... |
| prcnel 3515 | A proper class doesn't bel... |
| ralv 3516 | A universal quantifier res... |
| rexv 3517 | An existential quantifier ... |
| reuv 3518 | A unique existential quant... |
| rmov 3519 | An at-most-one quantifier ... |
| rabab 3520 | A class abstraction restri... |
| rexcom4b 3521 | Specialized existential co... |
| ceqsal1t 3522 | One direction of ~ ceqsalt... |
| ceqsalt 3523 | Closed theorem version of ... |
| ceqsralt 3524 | Restricted quantifier vers... |
| ceqsalg 3525 | A representation of explic... |
| ceqsalgALT 3526 | Alternate proof of ~ ceqsa... |
| ceqsal 3527 | A representation of explic... |
| ceqsalALT 3528 | A representation of explic... |
| ceqsalv 3529 | A representation of explic... |
| ceqsalvOLD 3530 | Obsolete version of ~ ceqs... |
| ceqsralv 3531 | Restricted quantifier vers... |
| ceqsralvOLD 3532 | Obsolete version of ~ ceqs... |
| gencl 3533 | Implicit substitution for ... |
| 2gencl 3534 | Implicit substitution for ... |
| 3gencl 3535 | Implicit substitution for ... |
| cgsexg 3536 | Implicit substitution infe... |
| cgsex2g 3537 | Implicit substitution infe... |
| cgsex4g 3538 | An implicit substitution i... |
| cgsex4gOLD 3539 | Obsolete version of ~ cgse... |
| ceqsex 3540 | Elimination of an existent... |
| ceqsexOLD 3541 | Obsolete version of ~ ceqs... |
| ceqsexv 3542 | Elimination of an existent... |
| ceqsexvOLD 3543 | Obsolete version of ~ ceqs... |
| ceqsexvOLDOLD 3544 | Obsolete version of ~ ceqs... |
| ceqsexv2d 3545 | Elimination of an existent... |
| ceqsexv2dOLD 3546 | Obsolete version of ~ ceqs... |
| ceqsex2 3547 | Elimination of two existen... |
| ceqsex2v 3548 | Elimination of two existen... |
| ceqsex3v 3549 | Elimination of three exist... |
| ceqsex4v 3550 | Elimination of four existe... |
| ceqsex6v 3551 | Elimination of six existen... |
| ceqsex8v 3552 | Elimination of eight exist... |
| gencbvex 3553 | Change of bound variable u... |
| gencbvex2 3554 | Restatement of ~ gencbvex ... |
| gencbval 3555 | Change of bound variable u... |
| sbhypf 3556 | Introduce an explicit subs... |
| sbhypfOLD 3557 | Obsolete version of ~ sbhy... |
| spcimgft 3558 | Closed theorem form of ~ s... |
| spcimgfi1 3559 | A closed version of ~ spci... |
| spcimgfi1OLD 3560 | Obsolete version of ~ spci... |
| spcgft 3561 | A closed version of ~ spcg... |
| spcimgf 3562 | Rule of specialization, us... |
| spcimegf 3563 | Existential specialization... |
| vtoclgft 3564 | Closed theorem form of ~ v... |
| vtocleg 3565 | Implicit substitution of a... |
| vtoclg 3566 | Implicit substitution of a... |
| vtocle 3567 | Implicit substitution of a... |
| vtocleOLD 3568 | Obsolete version of ~ vtoc... |
| vtoclbg 3569 | Implicit substitution of a... |
| vtocl 3570 | Implicit substitution of a... |
| vtoclOLD 3571 | Obsolete version of ~ vtoc... |
| vtocldf 3572 | Implicit substitution of a... |
| vtocld 3573 | Implicit substitution of a... |
| vtocl2d 3574 | Implicit substitution of t... |
| vtoclef 3575 | Implicit substitution of a... |
| vtoclf 3576 | Implicit substitution of a... |
| vtoclfOLD 3577 | Obsolete version of ~ vtoc... |
| vtocl2 3578 | Implicit substitution of c... |
| vtocl3 3579 | Implicit substitution of c... |
| vtoclb 3580 | Implicit substitution of a... |
| vtoclgf 3581 | Implicit substitution of a... |
| vtoclg1f 3582 | Version of ~ vtoclgf with ... |
| vtoclgOLD 3583 | Obsolete version of ~ vtoc... |
| vtocl2gf 3584 | Implicit substitution of a... |
| vtocl3gf 3585 | Implicit substitution of a... |
| vtocl2g 3586 | Implicit substitution of 2... |
| vtocl3g 3587 | Implicit substitution of a... |
| vtoclgaf 3588 | Implicit substitution of a... |
| vtoclga 3589 | Implicit substitution of a... |
| vtocl2ga 3590 | Implicit substitution of 2... |
| vtocl2gaf 3591 | Implicit substitution of 2... |
| vtocl2gafOLD 3592 | Obsolete version of ~ vtoc... |
| vtocl3gaf 3593 | Implicit substitution of 3... |
| vtocl3gafOLD 3594 | Obsolete version of ~ vtoc... |
| vtocl3ga 3595 | Implicit substitution of 3... |
| vtocl3gaOLD 3596 | Obsolete version of ~ vtoc... |
| vtocl3gaOLDOLD 3597 | Obsolete version of ~ vtoc... |
| vtocl4g 3598 | Implicit substitution of 4... |
| vtocl4ga 3599 | Implicit substitution of 4... |
| vtocl4gaOLD 3600 | Obsolete version of ~ vtoc... |
| vtoclegft 3601 | Implicit substitution of a... |
| vtoclegftOLD 3602 | Obsolete version of ~ vtoc... |
| vtoclri 3603 | Implicit substitution of a... |
| spcgf 3604 | Rule of specialization, us... |
| spcegf 3605 | Existential specialization... |
| spcimdv 3606 | Restricted specialization,... |
| spcdv 3607 | Rule of specialization, us... |
| spcimedv 3608 | Restricted existential spe... |
| spcgv 3609 | Rule of specialization, us... |
| spcegv 3610 | Existential specialization... |
| spcedv 3611 | Existential specialization... |
| spc2egv 3612 | Existential specialization... |
| spc2gv 3613 | Specialization with two qu... |
| spc2ed 3614 | Existential specialization... |
| spc2d 3615 | Specialization with 2 quan... |
| spc3egv 3616 | Existential specialization... |
| spc3gv 3617 | Specialization with three ... |
| spcv 3618 | Rule of specialization, us... |
| spcev 3619 | Existential specialization... |
| spc2ev 3620 | Existential specialization... |
| rspct 3621 | A closed version of ~ rspc... |
| rspcdf 3622 | Restricted specialization,... |
| rspc 3623 | Restricted specialization,... |
| rspce 3624 | Restricted existential spe... |
| rspcimdv 3625 | Restricted specialization,... |
| rspcimedv 3626 | Restricted existential spe... |
| rspcdv 3627 | Restricted specialization,... |
| rspcedv 3628 | Restricted existential spe... |
| rspcebdv 3629 | Restricted existential spe... |
| rspcdv2 3630 | Restricted specialization,... |
| rspcv 3631 | Restricted specialization,... |
| rspccv 3632 | Restricted specialization,... |
| rspcva 3633 | Restricted specialization,... |
| rspccva 3634 | Restricted specialization,... |
| rspcev 3635 | Restricted existential spe... |
| rspcdva 3636 | Restricted specialization,... |
| rspcedvd 3637 | Restricted existential spe... |
| rspcedvdw 3638 | Version of ~ rspcedvd wher... |
| rspceb2dv 3639 | Restricted existential spe... |
| rspcime 3640 | Prove a restricted existen... |
| rspceaimv 3641 | Restricted existential spe... |
| rspcedeq1vd 3642 | Restricted existential spe... |
| rspcedeq2vd 3643 | Restricted existential spe... |
| rspc2 3644 | Restricted specialization ... |
| rspc2gv 3645 | Restricted specialization ... |
| rspc2v 3646 | 2-variable restricted spec... |
| rspc2va 3647 | 2-variable restricted spec... |
| rspc2ev 3648 | 2-variable restricted exis... |
| 2rspcedvdw 3649 | Double application of ~ rs... |
| rspc2dv 3650 | 2-variable restricted spec... |
| rspc3v 3651 | 3-variable restricted spec... |
| rspc3ev 3652 | 3-variable restricted exis... |
| 3rspcedvdw 3653 | Triple application of ~ rs... |
| rspc3dv 3654 | 3-variable restricted spec... |
| rspc4v 3655 | 4-variable restricted spec... |
| rspc6v 3656 | 6-variable restricted spec... |
| rspc8v 3657 | 8-variable restricted spec... |
| rspceeqv 3658 | Restricted existential spe... |
| ralxpxfr2d 3659 | Transfer a universal quant... |
| rexraleqim 3660 | Statement following from e... |
| eqvincg 3661 | A variable introduction la... |
| eqvinc 3662 | A variable introduction la... |
| eqvincf 3663 | A variable introduction la... |
| alexeqg 3664 | Two ways to express substi... |
| ceqex 3665 | Equality implies equivalen... |
| ceqsexg 3666 | A representation of explic... |
| ceqsexgv 3667 | Elimination of an existent... |
| ceqsrexv 3668 | Elimination of a restricte... |
| ceqsrexbv 3669 | Elimination of a restricte... |
| ceqsralbv 3670 | Elimination of a restricte... |
| ceqsrex2v 3671 | Elimination of a restricte... |
| clel2g 3672 | Alternate definition of me... |
| clel2 3673 | Alternate definition of me... |
| clel3g 3674 | Alternate definition of me... |
| clel3 3675 | Alternate definition of me... |
| clel4g 3676 | Alternate definition of me... |
| clel4 3677 | Alternate definition of me... |
| clel5 3678 | Alternate definition of cl... |
| pm13.183 3679 | Compare theorem *13.183 in... |
| rr19.3v 3680 | Restricted quantifier vers... |
| rr19.28v 3681 | Restricted quantifier vers... |
| elab6g 3682 | Membership in a class abst... |
| elabd2 3683 | Membership in a class abst... |
| elabd3 3684 | Membership in a class abst... |
| elabgt 3685 | Membership in a class abst... |
| elabgtOLD 3686 | Obsolete version of ~ elab... |
| elabgtOLDOLD 3687 | Obsolete version of ~ elab... |
| elabgf 3688 | Membership in a class abst... |
| elabf 3689 | Membership in a class abst... |
| elabg 3690 | Membership in a class abst... |
| elabgOLD 3691 | Obsolete version of ~ elab... |
| elabgw 3692 | Membership in a class abst... |
| elab2gw 3693 | Membership in a class abst... |
| elab 3694 | Membership in a class abst... |
| elabOLD 3695 | Obsolete version of ~ elab... |
| elab2g 3696 | Membership in a class abst... |
| elabd 3697 | Explicit demonstration the... |
| elab2 3698 | Membership in a class abst... |
| elab4g 3699 | Membership in a class abst... |
| elab3gf 3700 | Membership in a class abst... |
| elab3g 3701 | Membership in a class abst... |
| elab3 3702 | Membership in a class abst... |
| elrabi 3703 | Implication for the member... |
| elrabf 3704 | Membership in a restricted... |
| rabtru 3705 | Abstract builder using the... |
| rabeqcOLD 3706 | Obsolete version of ~ rabe... |
| elrab3t 3707 | Membership in a restricted... |
| elrab 3708 | Membership in a restricted... |
| elrab3 3709 | Membership in a restricted... |
| elrabd 3710 | Membership in a restricted... |
| elrab2 3711 | Membership in a restricted... |
| elrab2w 3712 | Membership in a restricted... |
| ralab 3713 | Universal quantification o... |
| ralabOLD 3714 | Obsolete version of ~ rala... |
| ralrab 3715 | Universal quantification o... |
| rexab 3716 | Existential quantification... |
| rexabOLD 3717 | Obsolete version of ~ rexa... |
| rexrab 3718 | Existential quantification... |
| ralab2 3719 | Universal quantification o... |
| ralrab2 3720 | Universal quantification o... |
| rexab2 3721 | Existential quantification... |
| rexrab2 3722 | Existential quantification... |
| reurab 3723 | Restricted existential uni... |
| abidnf 3724 | Identity used to create cl... |
| dedhb 3725 | A deduction theorem for co... |
| class2seteq 3726 | Writing a set as a class a... |
| nelrdva 3727 | Deduce negative membership... |
| eqeu 3728 | A condition which implies ... |
| moeq 3729 | There exists at most one s... |
| eueq 3730 | A class is a set if and on... |
| eueqi 3731 | There exists a unique set ... |
| eueq2 3732 | Equality has existential u... |
| eueq3 3733 | Equality has existential u... |
| moeq3 3734 | "At most one" property of ... |
| mosub 3735 | "At most one" remains true... |
| mo2icl 3736 | Theorem for inferring "at ... |
| mob2 3737 | Consequence of "at most on... |
| moi2 3738 | Consequence of "at most on... |
| mob 3739 | Equality implied by "at mo... |
| moi 3740 | Equality implied by "at mo... |
| morex 3741 | Derive membership from uni... |
| euxfr2w 3742 | Transfer existential uniqu... |
| euxfrw 3743 | Transfer existential uniqu... |
| euxfr2 3744 | Transfer existential uniqu... |
| euxfr 3745 | Transfer existential uniqu... |
| euind 3746 | Existential uniqueness via... |
| reu2 3747 | A way to express restricte... |
| reu6 3748 | A way to express restricte... |
| reu3 3749 | A way to express restricte... |
| reu6i 3750 | A condition which implies ... |
| eqreu 3751 | A condition which implies ... |
| rmo4 3752 | Restricted "at most one" u... |
| reu4 3753 | Restricted uniqueness usin... |
| reu7 3754 | Restricted uniqueness usin... |
| reu8 3755 | Restricted uniqueness usin... |
| rmo3f 3756 | Restricted "at most one" u... |
| rmo4f 3757 | Restricted "at most one" u... |
| reu2eqd 3758 | Deduce equality from restr... |
| reueq 3759 | Equality has existential u... |
| rmoeq 3760 | Equality's restricted exis... |
| rmoan 3761 | Restricted "at most one" s... |
| rmoim 3762 | Restricted "at most one" i... |
| rmoimia 3763 | Restricted "at most one" i... |
| rmoimi 3764 | Restricted "at most one" i... |
| rmoimi2 3765 | Restricted "at most one" i... |
| 2reu5a 3766 | Double restricted existent... |
| reuimrmo 3767 | Restricted uniqueness impl... |
| 2reuswap 3768 | A condition allowing swap ... |
| 2reuswap2 3769 | A condition allowing swap ... |
| reuxfrd 3770 | Transfer existential uniqu... |
| reuxfr 3771 | Transfer existential uniqu... |
| reuxfr1d 3772 | Transfer existential uniqu... |
| reuxfr1ds 3773 | Transfer existential uniqu... |
| reuxfr1 3774 | Transfer existential uniqu... |
| reuind 3775 | Existential uniqueness via... |
| 2rmorex 3776 | Double restricted quantifi... |
| 2reu5lem1 3777 | Lemma for ~ 2reu5 . Note ... |
| 2reu5lem2 3778 | Lemma for ~ 2reu5 . (Cont... |
| 2reu5lem3 3779 | Lemma for ~ 2reu5 . This ... |
| 2reu5 3780 | Double restricted existent... |
| 2reurmo 3781 | Double restricted quantifi... |
| 2reurex 3782 | Double restricted quantifi... |
| 2rmoswap 3783 | A condition allowing to sw... |
| 2rexreu 3784 | Double restricted existent... |
| cdeqi 3787 | Deduce conditional equalit... |
| cdeqri 3788 | Property of conditional eq... |
| cdeqth 3789 | Deduce conditional equalit... |
| cdeqnot 3790 | Distribute conditional equ... |
| cdeqal 3791 | Distribute conditional equ... |
| cdeqab 3792 | Distribute conditional equ... |
| cdeqal1 3793 | Distribute conditional equ... |
| cdeqab1 3794 | Distribute conditional equ... |
| cdeqim 3795 | Distribute conditional equ... |
| cdeqcv 3796 | Conditional equality for s... |
| cdeqeq 3797 | Distribute conditional equ... |
| cdeqel 3798 | Distribute conditional equ... |
| nfcdeq 3799 | If we have a conditional e... |
| nfccdeq 3800 | Variation of ~ nfcdeq for ... |
| rru 3801 | Relative version of Russel... |
| ru 3802 | Russell's Paradox. Propos... |
| ruOLD 3803 | Obsolete version of ~ ru a... |
| dfsbcq 3806 | Proper substitution of a c... |
| dfsbcq2 3807 | This theorem, which is sim... |
| sbsbc 3808 | Show that ~ df-sb and ~ df... |
| sbceq1d 3809 | Equality theorem for class... |
| sbceq1dd 3810 | Equality theorem for class... |
| sbceqbid 3811 | Equality theorem for class... |
| sbc8g 3812 | This is the closest we can... |
| sbc2or 3813 | The disjunction of two equ... |
| sbcex 3814 | By our definition of prope... |
| sbceq1a 3815 | Equality theorem for class... |
| sbceq2a 3816 | Equality theorem for class... |
| spsbc 3817 | Specialization: if a formu... |
| spsbcd 3818 | Specialization: if a formu... |
| sbcth 3819 | A substitution into a theo... |
| sbcthdv 3820 | Deduction version of ~ sbc... |
| sbcid 3821 | An identity theorem for su... |
| nfsbc1d 3822 | Deduction version of ~ nfs... |
| nfsbc1 3823 | Bound-variable hypothesis ... |
| nfsbc1v 3824 | Bound-variable hypothesis ... |
| nfsbcdw 3825 | Deduction version of ~ nfs... |
| nfsbcw 3826 | Bound-variable hypothesis ... |
| sbccow 3827 | A composition law for clas... |
| nfsbcd 3828 | Deduction version of ~ nfs... |
| nfsbc 3829 | Bound-variable hypothesis ... |
| sbcco 3830 | A composition law for clas... |
| sbcco2 3831 | A composition law for clas... |
| sbc5 3832 | An equivalence for class s... |
| sbc5ALT 3833 | Alternate proof of ~ sbc5 ... |
| sbc6g 3834 | An equivalence for class s... |
| sbc6gOLD 3835 | Obsolete version of ~ sbc6... |
| sbc6 3836 | An equivalence for class s... |
| sbc7 3837 | An equivalence for class s... |
| cbvsbcw 3838 | Change bound variables in ... |
| cbvsbcvw 3839 | Change the bound variable ... |
| cbvsbc 3840 | Change bound variables in ... |
| cbvsbcv 3841 | Change the bound variable ... |
| sbciegft 3842 | Conversion of implicit sub... |
| sbciegftOLD 3843 | Obsolete version of ~ sbci... |
| sbciegf 3844 | Conversion of implicit sub... |
| sbcieg 3845 | Conversion of implicit sub... |
| sbciegOLD 3846 | Obsolete version of ~ sbci... |
| sbcie2g 3847 | Conversion of implicit sub... |
| sbcie 3848 | Conversion of implicit sub... |
| sbciedf 3849 | Conversion of implicit sub... |
| sbcied 3850 | Conversion of implicit sub... |
| sbciedOLD 3851 | Obsolete version of ~ sbci... |
| sbcied2 3852 | Conversion of implicit sub... |
| elrabsf 3853 | Membership in a restricted... |
| eqsbc1 3854 | Substitution for the left-... |
| sbcng 3855 | Move negation in and out o... |
| sbcimg 3856 | Distribution of class subs... |
| sbcan 3857 | Distribution of class subs... |
| sbcor 3858 | Distribution of class subs... |
| sbcbig 3859 | Distribution of class subs... |
| sbcn1 3860 | Move negation in and out o... |
| sbcim1 3861 | Distribution of class subs... |
| sbcim1OLD 3862 | Obsolete version of ~ sbci... |
| sbcbid 3863 | Formula-building deduction... |
| sbcbidv 3864 | Formula-building deduction... |
| sbcbii 3865 | Formula-building inference... |
| sbcbi1 3866 | Distribution of class subs... |
| sbcbi2 3867 | Substituting into equivale... |
| sbcal 3868 | Move universal quantifier ... |
| sbcex2 3869 | Move existential quantifie... |
| sbceqal 3870 | Class version of one impli... |
| sbceqalOLD 3871 | Obsolete version of ~ sbce... |
| sbeqalb 3872 | Theorem *14.121 in [Whiteh... |
| eqsbc2 3873 | Substitution for the right... |
| sbc3an 3874 | Distribution of class subs... |
| sbcel1v 3875 | Class substitution into a ... |
| sbcel2gv 3876 | Class substitution into a ... |
| sbcel21v 3877 | Class substitution into a ... |
| sbcimdv 3878 | Substitution analogue of T... |
| sbcimdvOLD 3879 | Obsolete version of ~ sbci... |
| sbctt 3880 | Substitution for a variabl... |
| sbcgf 3881 | Substitution for a variabl... |
| sbc19.21g 3882 | Substitution for a variabl... |
| sbcg 3883 | Substitution for a variabl... |
| sbcgOLD 3884 | Obsolete version of ~ sbcg... |
| sbcgfi 3885 | Substitution for a variabl... |
| sbc2iegf 3886 | Conversion of implicit sub... |
| sbc2ie 3887 | Conversion of implicit sub... |
| sbc2ieOLD 3888 | Obsolete version of ~ sbc2... |
| sbc2iedv 3889 | Conversion of implicit sub... |
| sbc3ie 3890 | Conversion of implicit sub... |
| sbccomlem 3891 | Lemma for ~ sbccom . (Con... |
| sbccomlemOLD 3892 | Obsolete version of ~ sbcc... |
| sbccom 3893 | Commutative law for double... |
| sbcralt 3894 | Interchange class substitu... |
| sbcrext 3895 | Interchange class substitu... |
| sbcralg 3896 | Interchange class substitu... |
| sbcrex 3897 | Interchange class substitu... |
| sbcreu 3898 | Interchange class substitu... |
| reu8nf 3899 | Restricted uniqueness usin... |
| sbcabel 3900 | Interchange class substitu... |
| rspsbc 3901 | Restricted quantifier vers... |
| rspsbca 3902 | Restricted quantifier vers... |
| rspesbca 3903 | Existence form of ~ rspsbc... |
| spesbc 3904 | Existence form of ~ spsbc ... |
| spesbcd 3905 | form of ~ spsbc . (Contri... |
| sbcth2 3906 | A substitution into a theo... |
| ra4v 3907 | Version of ~ ra4 with a di... |
| ra4 3908 | Restricted quantifier vers... |
| rmo2 3909 | Alternate definition of re... |
| rmo2i 3910 | Condition implying restric... |
| rmo3 3911 | Restricted "at most one" u... |
| rmob 3912 | Consequence of "at most on... |
| rmoi 3913 | Consequence of "at most on... |
| rmob2 3914 | Consequence of "restricted... |
| rmoi2 3915 | Consequence of "restricted... |
| rmoanim 3916 | Introduction of a conjunct... |
| rmoanimALT 3917 | Alternate proof of ~ rmoan... |
| reuan 3918 | Introduction of a conjunct... |
| 2reu1 3919 | Double restricted existent... |
| 2reu2 3920 | Double restricted existent... |
| csb2 3923 | Alternate expression for t... |
| csbeq1 3924 | Analogue of ~ dfsbcq for p... |
| csbeq1d 3925 | Equality deduction for pro... |
| csbeq2 3926 | Substituting into equivale... |
| csbeq2d 3927 | Formula-building deduction... |
| csbeq2dv 3928 | Formula-building deduction... |
| csbeq2i 3929 | Formula-building inference... |
| csbeq12dv 3930 | Formula-building inference... |
| cbvcsbw 3931 | Change bound variables in ... |
| cbvcsb 3932 | Change bound variables in ... |
| cbvcsbv 3933 | Change the bound variable ... |
| csbid 3934 | Analogue of ~ sbid for pro... |
| csbeq1a 3935 | Equality theorem for prope... |
| csbcow 3936 | Composition law for chaine... |
| csbco 3937 | Composition law for chaine... |
| csbtt 3938 | Substitution doesn't affec... |
| csbconstgf 3939 | Substitution doesn't affec... |
| csbconstg 3940 | Substitution doesn't affec... |
| csbconstgOLD 3941 | Obsolete version of ~ csbc... |
| csbgfi 3942 | Substitution for a variabl... |
| csbconstgi 3943 | The proper substitution of... |
| nfcsb1d 3944 | Bound-variable hypothesis ... |
| nfcsb1 3945 | Bound-variable hypothesis ... |
| nfcsb1v 3946 | Bound-variable hypothesis ... |
| nfcsbd 3947 | Deduction version of ~ nfc... |
| nfcsbw 3948 | Bound-variable hypothesis ... |
| nfcsb 3949 | Bound-variable hypothesis ... |
| csbhypf 3950 | Introduce an explicit subs... |
| csbiebt 3951 | Conversion of implicit sub... |
| csbiedf 3952 | Conversion of implicit sub... |
| csbieb 3953 | Bidirectional conversion b... |
| csbiebg 3954 | Bidirectional conversion b... |
| csbiegf 3955 | Conversion of implicit sub... |
| csbief 3956 | Conversion of implicit sub... |
| csbie 3957 | Conversion of implicit sub... |
| csbieOLD 3958 | Obsolete version of ~ csbi... |
| csbied 3959 | Conversion of implicit sub... |
| csbiedOLD 3960 | Obsolete version of ~ csbi... |
| csbied2 3961 | Conversion of implicit sub... |
| csbie2t 3962 | Conversion of implicit sub... |
| csbie2 3963 | Conversion of implicit sub... |
| csbie2g 3964 | Conversion of implicit sub... |
| cbvrabcsfw 3965 | Version of ~ cbvrabcsf wit... |
| cbvralcsf 3966 | A more general version of ... |
| cbvrexcsf 3967 | A more general version of ... |
| cbvreucsf 3968 | A more general version of ... |
| cbvrabcsf 3969 | A more general version of ... |
| cbvralv2 3970 | Rule used to change the bo... |
| cbvrexv2 3971 | Rule used to change the bo... |
| rspc2vd 3972 | Deduction version of 2-var... |
| difjust 3978 | Soundness justification th... |
| unjust 3980 | Soundness justification th... |
| injust 3982 | Soundness justification th... |
| dfin5 3984 | Alternate definition for t... |
| dfdif2 3985 | Alternate definition of cl... |
| eldif 3986 | Expansion of membership in... |
| eldifd 3987 | If a class is in one class... |
| eldifad 3988 | If a class is in the diffe... |
| eldifbd 3989 | If a class is in the diffe... |
| elneeldif 3990 | The elements of a set diff... |
| velcomp 3991 | Characterization of setvar... |
| elin 3992 | Expansion of membership in... |
| dfss2 3994 | Alternate definition of th... |
| dfss 3995 | Variant of subclass defini... |
| dfss3 3997 | Alternate definition of su... |
| dfss6 3998 | Alternate definition of su... |
| dfssf 3999 | Equivalence for subclass r... |
| dfss3f 4000 | Equivalence for subclass r... |
| nfss 4001 | If ` x ` is not free in ` ... |
| ssel 4002 | Membership relationships f... |
| ssel2 4003 | Membership relationships f... |
| sseli 4004 | Membership implication fro... |
| sselii 4005 | Membership inference from ... |
| sselid 4006 | Membership inference from ... |
| sseld 4007 | Membership deduction from ... |
| sselda 4008 | Membership deduction from ... |
| sseldd 4009 | Membership inference from ... |
| ssneld 4010 | If a class is not in anoth... |
| ssneldd 4011 | If an element is not in a ... |
| ssriv 4012 | Inference based on subclas... |
| ssrd 4013 | Deduction based on subclas... |
| ssrdv 4014 | Deduction based on subclas... |
| sstr2 4015 | Transitivity of subclass r... |
| sstr2OLD 4016 | Obsolete version of ~ sstr... |
| sstr 4017 | Transitivity of subclass r... |
| sstri 4018 | Subclass transitivity infe... |
| sstrd 4019 | Subclass transitivity dedu... |
| sstrid 4020 | Subclass transitivity dedu... |
| sstrdi 4021 | Subclass transitivity dedu... |
| sylan9ss 4022 | A subclass transitivity de... |
| sylan9ssr 4023 | A subclass transitivity de... |
| eqss 4024 | The subclass relationship ... |
| eqssi 4025 | Infer equality from two su... |
| eqssd 4026 | Equality deduction from tw... |
| sssseq 4027 | If a class is a subclass o... |
| eqrd 4028 | Deduce equality of classes... |
| eqri 4029 | Infer equality of classes ... |
| eqelssd 4030 | Equality deduction from su... |
| ssid 4031 | Any class is a subclass of... |
| ssidd 4032 | Weakening of ~ ssid . (Co... |
| ssv 4033 | Any class is a subclass of... |
| sseq1 4034 | Equality theorem for subcl... |
| sseq2 4035 | Equality theorem for the s... |
| sseq12 4036 | Equality theorem for the s... |
| sseq1i 4037 | An equality inference for ... |
| sseq2i 4038 | An equality inference for ... |
| sseq12i 4039 | An equality inference for ... |
| sseq1d 4040 | An equality deduction for ... |
| sseq2d 4041 | An equality deduction for ... |
| sseq12d 4042 | An equality deduction for ... |
| eqsstri 4043 | Substitution of equality i... |
| eqsstrri 4044 | Substitution of equality i... |
| sseqtri 4045 | Substitution of equality i... |
| sseqtrri 4046 | Substitution of equality i... |
| eqsstrd 4047 | Substitution of equality i... |
| eqsstrrd 4048 | Substitution of equality i... |
| sseqtrd 4049 | Substitution of equality i... |
| sseqtrrd 4050 | Substitution of equality i... |
| 3sstr3i 4051 | Substitution of equality i... |
| 3sstr4i 4052 | Substitution of equality i... |
| 3sstr3g 4053 | Substitution of equality i... |
| 3sstr4g 4054 | Substitution of equality i... |
| 3sstr3d 4055 | Substitution of equality i... |
| 3sstr4d 4056 | Substitution of equality i... |
| eqsstrid 4057 | A chained subclass and equ... |
| eqsstrrid 4058 | A chained subclass and equ... |
| sseqtrdi 4059 | A chained subclass and equ... |
| sseqtrrdi 4060 | A chained subclass and equ... |
| sseqtrid 4061 | Subclass transitivity dedu... |
| sseqtrrid 4062 | Subclass transitivity dedu... |
| eqsstrdi 4063 | A chained subclass and equ... |
| eqsstrrdi 4064 | A chained subclass and equ... |
| eqimssd 4065 | Equality implies inclusion... |
| eqimsscd 4066 | Equality implies inclusion... |
| eqimss 4067 | Equality implies inclusion... |
| eqimss2 4068 | Equality implies inclusion... |
| eqimssi 4069 | Infer subclass relationshi... |
| eqimss2i 4070 | Infer subclass relationshi... |
| nssne1 4071 | Two classes are different ... |
| nssne2 4072 | Two classes are different ... |
| nss 4073 | Negation of subclass relat... |
| nelss 4074 | Demonstrate by witnesses t... |
| ssrexf 4075 | Restricted existential qua... |
| ssrmof 4076 | "At most one" existential ... |
| ssralv 4077 | Quantification restricted ... |
| ssrexv 4078 | Existential quantification... |
| ss2ralv 4079 | Two quantifications restri... |
| ss2rexv 4080 | Two existential quantifica... |
| ssralvOLD 4081 | Obsolete version of ~ ssra... |
| ssrexvOLD 4082 | Obsolete version of ~ ssre... |
| ralss 4083 | Restricted universal quant... |
| rexss 4084 | Restricted existential qua... |
| ss2ab 4085 | Class abstractions in a su... |
| abss 4086 | Class abstraction in a sub... |
| ssab 4087 | Subclass of a class abstra... |
| ssabral 4088 | The relation for a subclas... |
| ss2abdv 4089 | Deduction of abstraction s... |
| ss2abi 4090 | Inference of abstraction s... |
| abssdv 4091 | Deduction of abstraction s... |
| abssdvOLD 4092 | Obsolete version of ~ abss... |
| abssi 4093 | Inference of abstraction s... |
| ss2rab 4094 | Restricted abstraction cla... |
| rabss 4095 | Restricted class abstracti... |
| ssrab 4096 | Subclass of a restricted c... |
| ssrabdv 4097 | Subclass of a restricted c... |
| rabssdv 4098 | Subclass of a restricted c... |
| ss2rabdv 4099 | Deduction of restricted ab... |
| ss2rabi 4100 | Inference of restricted ab... |
| rabss2 4101 | Subclass law for restricte... |
| ssab2 4102 | Subclass relation for the ... |
| ssrab2 4103 | Subclass relation for a re... |
| rabss3d 4104 | Subclass law for restricte... |
| ssrab3 4105 | Subclass relation for a re... |
| rabssrabd 4106 | Subclass of a restricted c... |
| ssrabeq 4107 | If the restricting class o... |
| rabssab 4108 | A restricted class is a su... |
| eqrrabd 4109 | Deduce equality with a res... |
| uniiunlem 4110 | A subset relationship usef... |
| dfpss2 4111 | Alternate definition of pr... |
| dfpss3 4112 | Alternate definition of pr... |
| psseq1 4113 | Equality theorem for prope... |
| psseq2 4114 | Equality theorem for prope... |
| psseq1i 4115 | An equality inference for ... |
| psseq2i 4116 | An equality inference for ... |
| psseq12i 4117 | An equality inference for ... |
| psseq1d 4118 | An equality deduction for ... |
| psseq2d 4119 | An equality deduction for ... |
| psseq12d 4120 | An equality deduction for ... |
| pssss 4121 | A proper subclass is a sub... |
| pssne 4122 | Two classes in a proper su... |
| pssssd 4123 | Deduce subclass from prope... |
| pssned 4124 | Proper subclasses are uneq... |
| sspss 4125 | Subclass in terms of prope... |
| pssirr 4126 | Proper subclass is irrefle... |
| pssn2lp 4127 | Proper subclass has no 2-c... |
| sspsstri 4128 | Two ways of stating tricho... |
| ssnpss 4129 | Partial trichotomy law for... |
| psstr 4130 | Transitive law for proper ... |
| sspsstr 4131 | Transitive law for subclas... |
| psssstr 4132 | Transitive law for subclas... |
| psstrd 4133 | Proper subclass inclusion ... |
| sspsstrd 4134 | Transitivity involving sub... |
| psssstrd 4135 | Transitivity involving sub... |
| npss 4136 | A class is not a proper su... |
| ssnelpss 4137 | A subclass missing a membe... |
| ssnelpssd 4138 | Subclass inclusion with on... |
| ssexnelpss 4139 | If there is an element of ... |
| dfdif3 4140 | Alternate definition of cl... |
| dfdif3OLD 4141 | Obsolete version of ~ dfdi... |
| difeq1 4142 | Equality theorem for class... |
| difeq2 4143 | Equality theorem for class... |
| difeq12 4144 | Equality theorem for class... |
| difeq1i 4145 | Inference adding differenc... |
| difeq2i 4146 | Inference adding differenc... |
| difeq12i 4147 | Equality inference for cla... |
| difeq1d 4148 | Deduction adding differenc... |
| difeq2d 4149 | Deduction adding differenc... |
| difeq12d 4150 | Equality deduction for cla... |
| difeqri 4151 | Inference from membership ... |
| nfdif 4152 | Bound-variable hypothesis ... |
| nfdifOLD 4153 | Obsolete version of ~ nfdi... |
| eldifi 4154 | Implication of membership ... |
| eldifn 4155 | Implication of membership ... |
| elndif 4156 | A set does not belong to a... |
| neldif 4157 | Implication of membership ... |
| difdif 4158 | Double class difference. ... |
| difss 4159 | Subclass relationship for ... |
| difssd 4160 | A difference of two classe... |
| difss2 4161 | If a class is contained in... |
| difss2d 4162 | If a class is contained in... |
| ssdifss 4163 | Preservation of a subclass... |
| ddif 4164 | Double complement under un... |
| ssconb 4165 | Contraposition law for sub... |
| sscon 4166 | Contraposition law for sub... |
| ssdif 4167 | Difference law for subsets... |
| ssdifd 4168 | If ` A ` is contained in `... |
| sscond 4169 | If ` A ` is contained in `... |
| ssdifssd 4170 | If ` A ` is contained in `... |
| ssdif2d 4171 | If ` A ` is contained in `... |
| raldifb 4172 | Restricted universal quant... |
| rexdifi 4173 | Restricted existential qua... |
| complss 4174 | Complementation reverses i... |
| compleq 4175 | Two classes are equal if a... |
| elun 4176 | Expansion of membership in... |
| elunnel1 4177 | A member of a union that i... |
| elunnel2 4178 | A member of a union that i... |
| uneqri 4179 | Inference from membership ... |
| unidm 4180 | Idempotent law for union o... |
| uncom 4181 | Commutative law for union ... |
| equncom 4182 | If a class equals the unio... |
| equncomi 4183 | Inference form of ~ equnco... |
| uneq1 4184 | Equality theorem for the u... |
| uneq2 4185 | Equality theorem for the u... |
| uneq12 4186 | Equality theorem for the u... |
| uneq1i 4187 | Inference adding union to ... |
| uneq2i 4188 | Inference adding union to ... |
| uneq12i 4189 | Equality inference for the... |
| uneq1d 4190 | Deduction adding union to ... |
| uneq2d 4191 | Deduction adding union to ... |
| uneq12d 4192 | Equality deduction for the... |
| nfun 4193 | Bound-variable hypothesis ... |
| nfunOLD 4194 | Obsolete version of ~ nfun... |
| unass 4195 | Associative law for union ... |
| un12 4196 | A rearrangement of union. ... |
| un23 4197 | A rearrangement of union. ... |
| un4 4198 | A rearrangement of the uni... |
| unundi 4199 | Union distributes over its... |
| unundir 4200 | Union distributes over its... |
| ssun1 4201 | Subclass relationship for ... |
| ssun2 4202 | Subclass relationship for ... |
| ssun3 4203 | Subclass law for union of ... |
| ssun4 4204 | Subclass law for union of ... |
| elun1 4205 | Membership law for union o... |
| elun2 4206 | Membership law for union o... |
| elunant 4207 | A statement is true for ev... |
| unss1 4208 | Subclass law for union of ... |
| ssequn1 4209 | A relationship between sub... |
| unss2 4210 | Subclass law for union of ... |
| unss12 4211 | Subclass law for union of ... |
| ssequn2 4212 | A relationship between sub... |
| unss 4213 | The union of two subclasse... |
| unssi 4214 | An inference showing the u... |
| unssd 4215 | A deduction showing the un... |
| unssad 4216 | If ` ( A u. B ) ` is conta... |
| unssbd 4217 | If ` ( A u. B ) ` is conta... |
| ssun 4218 | A condition that implies i... |
| rexun 4219 | Restricted existential qua... |
| ralunb 4220 | Restricted quantification ... |
| ralun 4221 | Restricted quantification ... |
| elini 4222 | Membership in an intersect... |
| elind 4223 | Deduce membership in an in... |
| elinel1 4224 | Membership in an intersect... |
| elinel2 4225 | Membership in an intersect... |
| elin2 4226 | Membership in a class defi... |
| elin1d 4227 | Elementhood in the first s... |
| elin2d 4228 | Elementhood in the first s... |
| elin3 4229 | Membership in a class defi... |
| incom 4230 | Commutative law for inters... |
| ineqcom 4231 | Two ways of expressing tha... |
| ineqcomi 4232 | Two ways of expressing tha... |
| ineqri 4233 | Inference from membership ... |
| ineq1 4234 | Equality theorem for inter... |
| ineq2 4235 | Equality theorem for inter... |
| ineq12 4236 | Equality theorem for inter... |
| ineq1i 4237 | Equality inference for int... |
| ineq2i 4238 | Equality inference for int... |
| ineq12i 4239 | Equality inference for int... |
| ineq1d 4240 | Equality deduction for int... |
| ineq2d 4241 | Equality deduction for int... |
| ineq12d 4242 | Equality deduction for int... |
| ineqan12d 4243 | Equality deduction for int... |
| sseqin2 4244 | A relationship between sub... |
| nfin 4245 | Bound-variable hypothesis ... |
| nfinOLD 4246 | Obsolete version of ~ nfin... |
| rabbi2dva 4247 | Deduction from a wff to a ... |
| inidm 4248 | Idempotent law for interse... |
| inass 4249 | Associative law for inters... |
| in12 4250 | A rearrangement of interse... |
| in32 4251 | A rearrangement of interse... |
| in13 4252 | A rearrangement of interse... |
| in31 4253 | A rearrangement of interse... |
| inrot 4254 | Rotate the intersection of... |
| in4 4255 | Rearrangement of intersect... |
| inindi 4256 | Intersection distributes o... |
| inindir 4257 | Intersection distributes o... |
| inss1 4258 | The intersection of two cl... |
| inss2 4259 | The intersection of two cl... |
| ssin 4260 | Subclass of intersection. ... |
| ssini 4261 | An inference showing that ... |
| ssind 4262 | A deduction showing that a... |
| ssrin 4263 | Add right intersection to ... |
| sslin 4264 | Add left intersection to s... |
| ssrind 4265 | Add right intersection to ... |
| ss2in 4266 | Intersection of subclasses... |
| ssinss1 4267 | Intersection preserves sub... |
| inss 4268 | Inclusion of an intersecti... |
| rexin 4269 | Restricted existential qua... |
| dfss7 4270 | Alternate definition of su... |
| symdifcom 4273 | Symmetric difference commu... |
| symdifeq1 4274 | Equality theorem for symme... |
| symdifeq2 4275 | Equality theorem for symme... |
| nfsymdif 4276 | Hypothesis builder for sym... |
| elsymdif 4277 | Membership in a symmetric ... |
| dfsymdif4 4278 | Alternate definition of th... |
| elsymdifxor 4279 | Membership in a symmetric ... |
| dfsymdif2 4280 | Alternate definition of th... |
| symdifass 4281 | Symmetric difference is as... |
| difsssymdif 4282 | The symmetric difference c... |
| difsymssdifssd 4283 | If the symmetric differenc... |
| unabs 4284 | Absorption law for union. ... |
| inabs 4285 | Absorption law for interse... |
| nssinpss 4286 | Negation of subclass expre... |
| nsspssun 4287 | Negation of subclass expre... |
| dfss4 4288 | Subclass defined in terms ... |
| dfun2 4289 | An alternate definition of... |
| dfin2 4290 | An alternate definition of... |
| difin 4291 | Difference with intersecti... |
| ssdifim 4292 | Implication of a class dif... |
| ssdifsym 4293 | Symmetric class difference... |
| dfss5 4294 | Alternate definition of su... |
| dfun3 4295 | Union defined in terms of ... |
| dfin3 4296 | Intersection defined in te... |
| dfin4 4297 | Alternate definition of th... |
| invdif 4298 | Intersection with universa... |
| indif 4299 | Intersection with class di... |
| indif2 4300 | Bring an intersection in a... |
| indif1 4301 | Bring an intersection in a... |
| indifcom 4302 | Commutation law for inters... |
| indi 4303 | Distributive law for inter... |
| undi 4304 | Distributive law for union... |
| indir 4305 | Distributive law for inter... |
| undir 4306 | Distributive law for union... |
| unineq 4307 | Infer equality from equali... |
| uneqin 4308 | Equality of union and inte... |
| difundi 4309 | Distributive law for class... |
| difundir 4310 | Distributive law for class... |
| difindi 4311 | Distributive law for class... |
| difindir 4312 | Distributive law for class... |
| indifdi 4313 | Distribute intersection ov... |
| indifdir 4314 | Distribute intersection ov... |
| difdif2 4315 | Class difference by a clas... |
| undm 4316 | De Morgan's law for union.... |
| indm 4317 | De Morgan's law for inters... |
| difun1 4318 | A relationship involving d... |
| undif3 4319 | An equality involving clas... |
| difin2 4320 | Represent a class differen... |
| dif32 4321 | Swap second and third argu... |
| difabs 4322 | Absorption-like law for cl... |
| sscon34b 4323 | Relative complementation r... |
| rcompleq 4324 | Two subclasses are equal i... |
| dfsymdif3 4325 | Alternate definition of th... |
| unabw 4326 | Union of two class abstrac... |
| unab 4327 | Union of two class abstrac... |
| inab 4328 | Intersection of two class ... |
| difab 4329 | Difference of two class ab... |
| abanssl 4330 | A class abstraction with a... |
| abanssr 4331 | A class abstraction with a... |
| notabw 4332 | A class abstraction define... |
| notab 4333 | A class abstraction define... |
| unrab 4334 | Union of two restricted cl... |
| inrab 4335 | Intersection of two restri... |
| inrab2 4336 | Intersection with a restri... |
| difrab 4337 | Difference of two restrict... |
| dfrab3 4338 | Alternate definition of re... |
| dfrab2 4339 | Alternate definition of re... |
| rabdif 4340 | Move difference in and out... |
| notrab 4341 | Complementation of restric... |
| dfrab3ss 4342 | Restricted class abstracti... |
| rabun2 4343 | Abstraction restricted to ... |
| reuun2 4344 | Transfer uniqueness to a s... |
| reuss2 4345 | Transfer uniqueness to a s... |
| reuss 4346 | Transfer uniqueness to a s... |
| reuun1 4347 | Transfer uniqueness to a s... |
| reupick 4348 | Restricted uniqueness "pic... |
| reupick3 4349 | Restricted uniqueness "pic... |
| reupick2 4350 | Restricted uniqueness "pic... |
| euelss 4351 | Transfer uniqueness of an ... |
| dfnul4 4354 | Alternate definition of th... |
| dfnul2 4355 | Alternate definition of th... |
| dfnul3 4356 | Alternate definition of th... |
| dfnul2OLD 4357 | Obsolete version of ~ dfnu... |
| dfnul3OLD 4358 | Obsolete version of ~ dfnu... |
| dfnul4OLD 4359 | Obsolete version of ~ dfnu... |
| noel 4360 | The empty set has no eleme... |
| noelOLD 4361 | Obsolete version of ~ noel... |
| nel02 4362 | The empty set has no eleme... |
| n0i 4363 | If a class has elements, t... |
| ne0i 4364 | If a class has elements, t... |
| ne0d 4365 | Deduction form of ~ ne0i .... |
| n0ii 4366 | If a class has elements, t... |
| ne0ii 4367 | If a class has elements, t... |
| vn0 4368 | The universal class is not... |
| vn0ALT 4369 | Alternate proof of ~ vn0 .... |
| eq0f 4370 | A class is equal to the em... |
| neq0f 4371 | A class is not empty if an... |
| n0f 4372 | A class is nonempty if and... |
| eq0 4373 | A class is equal to the em... |
| eq0ALT 4374 | Alternate proof of ~ eq0 .... |
| neq0 4375 | A class is not empty if an... |
| n0 4376 | A class is nonempty if and... |
| nel0 4377 | From the general negation ... |
| reximdva0 4378 | Restricted existence deduc... |
| rspn0 4379 | Specialization for restric... |
| n0rex 4380 | There is an element in a n... |
| ssn0rex 4381 | There is an element in a c... |
| n0moeu 4382 | A case of equivalence of "... |
| rex0 4383 | Vacuous restricted existen... |
| reu0 4384 | Vacuous restricted uniquen... |
| rmo0 4385 | Vacuous restricted at-most... |
| 0el 4386 | Membership of the empty se... |
| n0el 4387 | Negated membership of the ... |
| eqeuel 4388 | A condition which implies ... |
| ssdif0 4389 | Subclass expressed in term... |
| difn0 4390 | If the difference of two s... |
| pssdifn0 4391 | A proper subclass has a no... |
| pssdif 4392 | A proper subclass has a no... |
| ndisj 4393 | Express that an intersecti... |
| inn0f 4394 | A nonempty intersection. ... |
| inn0 4395 | A nonempty intersection. ... |
| difin0ss 4396 | Difference, intersection, ... |
| inssdif0 4397 | Intersection, subclass, an... |
| difid 4398 | The difference between a c... |
| difidALT 4399 | Alternate proof of ~ difid... |
| dif0 4400 | The difference between a c... |
| ab0w 4401 | The class of sets verifyin... |
| ab0 4402 | The class of sets verifyin... |
| ab0OLD 4403 | Obsolete version of ~ ab0 ... |
| ab0ALT 4404 | Alternate proof of ~ ab0 ,... |
| dfnf5 4405 | Characterization of nonfre... |
| ab0orv 4406 | The class abstraction defi... |
| ab0orvALT 4407 | Alternate proof of ~ ab0or... |
| abn0 4408 | Nonempty class abstraction... |
| rab0 4409 | Any restricted class abstr... |
| rabeq0w 4410 | Condition for a restricted... |
| rabeq0 4411 | Condition for a restricted... |
| rabn0 4412 | Nonempty restricted class ... |
| rabxm 4413 | Law of excluded middle, in... |
| rabnc 4414 | Law of noncontradiction, i... |
| elneldisj 4415 | The set of elements ` s ` ... |
| elnelun 4416 | The union of the set of el... |
| un0 4417 | The union of a class with ... |
| in0 4418 | The intersection of a clas... |
| 0un 4419 | The union of the empty set... |
| 0in 4420 | The intersection of the em... |
| inv1 4421 | The intersection of a clas... |
| unv 4422 | The union of a class with ... |
| 0ss 4423 | The null set is a subset o... |
| ss0b 4424 | Any subset of the empty se... |
| ss0 4425 | Any subset of the empty se... |
| sseq0 4426 | A subclass of an empty cla... |
| ssn0 4427 | A class with a nonempty su... |
| 0dif 4428 | The difference between the... |
| abf 4429 | A class abstraction determ... |
| eq0rdv 4430 | Deduction for equality to ... |
| eq0rdvALT 4431 | Alternate proof of ~ eq0rd... |
| csbprc 4432 | The proper substitution of... |
| csb0 4433 | The proper substitution of... |
| sbcel12 4434 | Distribute proper substitu... |
| sbceqg 4435 | Distribute proper substitu... |
| sbceqi 4436 | Distribution of class subs... |
| sbcnel12g 4437 | Distribute proper substitu... |
| sbcne12 4438 | Distribute proper substitu... |
| sbcel1g 4439 | Move proper substitution i... |
| sbceq1g 4440 | Move proper substitution t... |
| sbcel2 4441 | Move proper substitution i... |
| sbceq2g 4442 | Move proper substitution t... |
| csbcom 4443 | Commutative law for double... |
| sbcnestgfw 4444 | Nest the composition of tw... |
| csbnestgfw 4445 | Nest the composition of tw... |
| sbcnestgw 4446 | Nest the composition of tw... |
| csbnestgw 4447 | Nest the composition of tw... |
| sbcco3gw 4448 | Composition of two substit... |
| sbcnestgf 4449 | Nest the composition of tw... |
| csbnestgf 4450 | Nest the composition of tw... |
| sbcnestg 4451 | Nest the composition of tw... |
| csbnestg 4452 | Nest the composition of tw... |
| sbcco3g 4453 | Composition of two substit... |
| csbco3g 4454 | Composition of two class s... |
| csbnest1g 4455 | Nest the composition of tw... |
| csbidm 4456 | Idempotent law for class s... |
| csbvarg 4457 | The proper substitution of... |
| csbvargi 4458 | The proper substitution of... |
| sbccsb 4459 | Substitution into a wff ex... |
| sbccsb2 4460 | Substitution into a wff ex... |
| rspcsbela 4461 | Special case related to ~ ... |
| sbnfc2 4462 | Two ways of expressing " `... |
| csbab 4463 | Move substitution into a c... |
| csbun 4464 | Distribution of class subs... |
| csbin 4465 | Distribute proper substitu... |
| csbie2df 4466 | Conversion of implicit sub... |
| 2nreu 4467 | If there are two different... |
| un00 4468 | Two classes are empty iff ... |
| vss 4469 | Only the universal class h... |
| 0pss 4470 | The null set is a proper s... |
| npss0 4471 | No set is a proper subset ... |
| pssv 4472 | Any non-universal class is... |
| disj 4473 | Two ways of saying that tw... |
| disjr 4474 | Two ways of saying that tw... |
| disj1 4475 | Two ways of saying that tw... |
| reldisj 4476 | Two ways of saying that tw... |
| disj3 4477 | Two ways of saying that tw... |
| disjne 4478 | Members of disjoint sets a... |
| disjeq0 4479 | Two disjoint sets are equa... |
| disjel 4480 | A set can't belong to both... |
| disj2 4481 | Two ways of saying that tw... |
| disj4 4482 | Two ways of saying that tw... |
| ssdisj 4483 | Intersection with a subcla... |
| disjpss 4484 | A class is a proper subset... |
| undisj1 4485 | The union of disjoint clas... |
| undisj2 4486 | The union of disjoint clas... |
| ssindif0 4487 | Subclass expressed in term... |
| inelcm 4488 | The intersection of classe... |
| minel 4489 | A minimum element of a cla... |
| undif4 4490 | Distribute union over diff... |
| disjssun 4491 | Subset relation for disjoi... |
| vdif0 4492 | Universal class equality i... |
| difrab0eq 4493 | If the difference between ... |
| pssnel 4494 | A proper subclass has a me... |
| disjdif 4495 | A class and its relative c... |
| disjdifr 4496 | A class and its relative c... |
| difin0 4497 | The difference of a class ... |
| unvdif 4498 | The union of a class and i... |
| undif1 4499 | Absorption of difference b... |
| undif2 4500 | Absorption of difference b... |
| undifabs 4501 | Absorption of difference b... |
| inundif 4502 | The intersection and class... |
| disjdif2 4503 | The difference of a class ... |
| difun2 4504 | Absorption of union by dif... |
| undif 4505 | Union of complementary par... |
| undifr 4506 | Union of complementary par... |
| undifrOLD 4507 | Obsolete version of ~ undi... |
| undif5 4508 | An equality involving clas... |
| ssdifin0 4509 | A subset of a difference d... |
| ssdifeq0 4510 | A class is a subclass of i... |
| ssundif 4511 | A condition equivalent to ... |
| difcom 4512 | Swap the arguments of a cl... |
| pssdifcom1 4513 | Two ways to express overla... |
| pssdifcom2 4514 | Two ways to express non-co... |
| difdifdir 4515 | Distributive law for class... |
| uneqdifeq 4516 | Two ways to say that ` A `... |
| raldifeq 4517 | Equality theorem for restr... |
| r19.2z 4518 | Theorem 19.2 of [Margaris]... |
| r19.2zb 4519 | A response to the notion t... |
| r19.3rz 4520 | Restricted quantification ... |
| r19.28z 4521 | Restricted quantifier vers... |
| r19.3rzv 4522 | Restricted quantification ... |
| r19.9rzv 4523 | Restricted quantification ... |
| r19.28zv 4524 | Restricted quantifier vers... |
| r19.37zv 4525 | Restricted quantifier vers... |
| r19.45zv 4526 | Restricted version of Theo... |
| r19.44zv 4527 | Restricted version of Theo... |
| r19.27z 4528 | Restricted quantifier vers... |
| r19.27zv 4529 | Restricted quantifier vers... |
| r19.36zv 4530 | Restricted quantifier vers... |
| ralidmw 4531 | Idempotent law for restric... |
| rzal 4532 | Vacuous quantification is ... |
| rzalALT 4533 | Alternate proof of ~ rzal ... |
| rexn0 4534 | Restricted existential qua... |
| ralidm 4535 | Idempotent law for restric... |
| ral0 4536 | Vacuous universal quantifi... |
| ralf0 4537 | The quantification of a fa... |
| ralnralall 4538 | A contradiction concerning... |
| falseral0 4539 | A false statement can only... |
| raaan 4540 | Rearrange restricted quant... |
| raaanv 4541 | Rearrange restricted quant... |
| sbss 4542 | Set substitution into the ... |
| sbcssg 4543 | Distribute proper substitu... |
| raaan2 4544 | Rearrange restricted quant... |
| 2reu4lem 4545 | Lemma for ~ 2reu4 . (Cont... |
| 2reu4 4546 | Definition of double restr... |
| csbdif 4547 | Distribution of class subs... |
| dfif2 4550 | An alternate definition of... |
| dfif6 4551 | An alternate definition of... |
| ifeq1 4552 | Equality theorem for condi... |
| ifeq2 4553 | Equality theorem for condi... |
| iftrue 4554 | Value of the conditional o... |
| iftruei 4555 | Inference associated with ... |
| iftrued 4556 | Value of the conditional o... |
| iffalse 4557 | Value of the conditional o... |
| iffalsei 4558 | Inference associated with ... |
| iffalsed 4559 | Value of the conditional o... |
| ifnefalse 4560 | When values are unequal, b... |
| ifsb 4561 | Distribute a function over... |
| dfif3 4562 | Alternate definition of th... |
| dfif4 4563 | Alternate definition of th... |
| dfif5 4564 | Alternate definition of th... |
| ifssun 4565 | A conditional class is inc... |
| ifeq12 4566 | Equality theorem for condi... |
| ifeq1d 4567 | Equality deduction for con... |
| ifeq2d 4568 | Equality deduction for con... |
| ifeq12d 4569 | Equality deduction for con... |
| ifbi 4570 | Equivalence theorem for co... |
| ifbid 4571 | Equivalence deduction for ... |
| ifbieq1d 4572 | Equivalence/equality deduc... |
| ifbieq2i 4573 | Equivalence/equality infer... |
| ifbieq2d 4574 | Equivalence/equality deduc... |
| ifbieq12i 4575 | Equivalence deduction for ... |
| ifbieq12d 4576 | Equivalence deduction for ... |
| nfifd 4577 | Deduction form of ~ nfif .... |
| nfif 4578 | Bound-variable hypothesis ... |
| ifeq1da 4579 | Conditional equality. (Co... |
| ifeq2da 4580 | Conditional equality. (Co... |
| ifeq12da 4581 | Equivalence deduction for ... |
| ifbieq12d2 4582 | Equivalence deduction for ... |
| ifclda 4583 | Conditional closure. (Con... |
| ifeqda 4584 | Separation of the values o... |
| elimif 4585 | Elimination of a condition... |
| ifbothda 4586 | A wff ` th ` containing a ... |
| ifboth 4587 | A wff ` th ` containing a ... |
| ifid 4588 | Identical true and false a... |
| eqif 4589 | Expansion of an equality w... |
| ifval 4590 | Another expression of the ... |
| elif 4591 | Membership in a conditiona... |
| ifel 4592 | Membership of a conditiona... |
| ifcl 4593 | Membership (closure) of a ... |
| ifcld 4594 | Membership (closure) of a ... |
| ifcli 4595 | Inference associated with ... |
| ifexd 4596 | Existence of the condition... |
| ifexg 4597 | Existence of the condition... |
| ifex 4598 | Existence of the condition... |
| ifeqor 4599 | The possible values of a c... |
| ifnot 4600 | Negating the first argumen... |
| ifan 4601 | Rewrite a conjunction in a... |
| ifor 4602 | Rewrite a disjunction in a... |
| 2if2 4603 | Resolve two nested conditi... |
| ifcomnan 4604 | Commute the conditions in ... |
| csbif 4605 | Distribute proper substitu... |
| dedth 4606 | Weak deduction theorem tha... |
| dedth2h 4607 | Weak deduction theorem eli... |
| dedth3h 4608 | Weak deduction theorem eli... |
| dedth4h 4609 | Weak deduction theorem eli... |
| dedth2v 4610 | Weak deduction theorem for... |
| dedth3v 4611 | Weak deduction theorem for... |
| dedth4v 4612 | Weak deduction theorem for... |
| elimhyp 4613 | Eliminate a hypothesis con... |
| elimhyp2v 4614 | Eliminate a hypothesis con... |
| elimhyp3v 4615 | Eliminate a hypothesis con... |
| elimhyp4v 4616 | Eliminate a hypothesis con... |
| elimel 4617 | Eliminate a membership hyp... |
| elimdhyp 4618 | Version of ~ elimhyp where... |
| keephyp 4619 | Transform a hypothesis ` p... |
| keephyp2v 4620 | Keep a hypothesis containi... |
| keephyp3v 4621 | Keep a hypothesis containi... |
| pwjust 4623 | Soundness justification th... |
| elpwg 4625 | Membership in a power clas... |
| elpw 4626 | Membership in a power clas... |
| velpw 4627 | Setvar variable membership... |
| elpwd 4628 | Membership in a power clas... |
| elpwi 4629 | Subset relation implied by... |
| elpwb 4630 | Characterization of the el... |
| elpwid 4631 | An element of a power clas... |
| elelpwi 4632 | If ` A ` belongs to a part... |
| sspw 4633 | The powerclass preserves i... |
| sspwi 4634 | The powerclass preserves i... |
| sspwd 4635 | The powerclass preserves i... |
| pweq 4636 | Equality theorem for power... |
| pweqALT 4637 | Alternate proof of ~ pweq ... |
| pweqi 4638 | Equality inference for pow... |
| pweqd 4639 | Equality deduction for pow... |
| pwunss 4640 | The power class of the uni... |
| nfpw 4641 | Bound-variable hypothesis ... |
| pwidg 4642 | A set is an element of its... |
| pwidb 4643 | A class is an element of i... |
| pwid 4644 | A set is a member of its p... |
| pwss 4645 | Subclass relationship for ... |
| pwundif 4646 | Break up the power class o... |
| snjust 4647 | Soundness justification th... |
| sneq 4658 | Equality theorem for singl... |
| sneqi 4659 | Equality inference for sin... |
| sneqd 4660 | Equality deduction for sin... |
| dfsn2 4661 | Alternate definition of si... |
| elsng 4662 | There is exactly one eleme... |
| elsn 4663 | There is exactly one eleme... |
| velsn 4664 | There is only one element ... |
| elsni 4665 | There is at most one eleme... |
| rabsneq 4666 | Equality of class abstract... |
| absn 4667 | Condition for a class abst... |
| dfpr2 4668 | Alternate definition of a ... |
| dfsn2ALT 4669 | Alternate definition of si... |
| elprg 4670 | A member of a pair of clas... |
| elpri 4671 | If a class is an element o... |
| elpr 4672 | A member of a pair of clas... |
| elpr2g 4673 | A member of a pair of sets... |
| elpr2 4674 | A member of a pair of sets... |
| nelpr2 4675 | If a class is not an eleme... |
| nelpr1 4676 | If a class is not an eleme... |
| nelpri 4677 | If an element doesn't matc... |
| prneli 4678 | If an element doesn't matc... |
| nelprd 4679 | If an element doesn't matc... |
| eldifpr 4680 | Membership in a set with t... |
| rexdifpr 4681 | Restricted existential qua... |
| snidg 4682 | A set is a member of its s... |
| snidb 4683 | A class is a set iff it is... |
| snid 4684 | A set is a member of its s... |
| vsnid 4685 | A setvar variable is a mem... |
| elsn2g 4686 | There is exactly one eleme... |
| elsn2 4687 | There is exactly one eleme... |
| nelsn 4688 | If a class is not equal to... |
| rabeqsn 4689 | Conditions for a restricte... |
| rabsssn 4690 | Conditions for a restricte... |
| rabeqsnd 4691 | Conditions for a restricte... |
| ralsnsg 4692 | Substitution expressed in ... |
| rexsns 4693 | Restricted existential qua... |
| rexsngf 4694 | Restricted existential qua... |
| ralsngf 4695 | Restricted universal quant... |
| reusngf 4696 | Restricted existential uni... |
| ralsng 4697 | Substitution expressed in ... |
| rexsng 4698 | Restricted existential qua... |
| reusng 4699 | Restricted existential uni... |
| 2ralsng 4700 | Substitution expressed in ... |
| ralsngOLD 4701 | Obsolete version of ~ rals... |
| rexsngOLD 4702 | Obsolete version of ~ rexs... |
| rexreusng 4703 | Restricted existential uni... |
| exsnrex 4704 | There is a set being the e... |
| ralsn 4705 | Convert a universal quanti... |
| rexsn 4706 | Convert an existential qua... |
| elpwunsn 4707 | Membership in an extension... |
| eqoreldif 4708 | An element of a set is eit... |
| eltpg 4709 | Members of an unordered tr... |
| eldiftp 4710 | Membership in a set with t... |
| eltpi 4711 | A member of an unordered t... |
| eltp 4712 | A member of an unordered t... |
| el7g 4713 | Members of a set with seve... |
| dftp2 4714 | Alternate definition of un... |
| nfpr 4715 | Bound-variable hypothesis ... |
| ifpr 4716 | Membership of a conditiona... |
| ralprgf 4717 | Convert a restricted unive... |
| rexprgf 4718 | Convert a restricted exist... |
| ralprg 4719 | Convert a restricted unive... |
| ralprgOLD 4720 | Obsolete version of ~ ralp... |
| rexprg 4721 | Convert a restricted exist... |
| rexprgOLD 4722 | Obsolete version of ~ rexp... |
| raltpg 4723 | Convert a restricted unive... |
| rextpg 4724 | Convert a restricted exist... |
| ralpr 4725 | Convert a restricted unive... |
| rexpr 4726 | Convert a restricted exist... |
| reuprg0 4727 | Convert a restricted exist... |
| reuprg 4728 | Convert a restricted exist... |
| reurexprg 4729 | Convert a restricted exist... |
| raltp 4730 | Convert a universal quanti... |
| rextp 4731 | Convert an existential qua... |
| nfsn 4732 | Bound-variable hypothesis ... |
| csbsng 4733 | Distribute proper substitu... |
| csbprg 4734 | Distribute proper substitu... |
| elinsn 4735 | If the intersection of two... |
| disjsn 4736 | Intersection with the sing... |
| disjsn2 4737 | Two distinct singletons ar... |
| disjpr2 4738 | Two completely distinct un... |
| disjprsn 4739 | The disjoint intersection ... |
| disjtpsn 4740 | The disjoint intersection ... |
| disjtp2 4741 | Two completely distinct un... |
| snprc 4742 | The singleton of a proper ... |
| snnzb 4743 | A singleton is nonempty if... |
| rmosn 4744 | A restricted at-most-one q... |
| r19.12sn 4745 | Special case of ~ r19.12 w... |
| rabsn 4746 | Condition where a restrict... |
| rabsnifsb 4747 | A restricted class abstrac... |
| rabsnif 4748 | A restricted class abstrac... |
| rabrsn 4749 | A restricted class abstrac... |
| euabsn2 4750 | Another way to express exi... |
| euabsn 4751 | Another way to express exi... |
| reusn 4752 | A way to express restricte... |
| absneu 4753 | Restricted existential uni... |
| rabsneu 4754 | Restricted existential uni... |
| eusn 4755 | Two ways to express " ` A ... |
| rabsnt 4756 | Truth implied by equality ... |
| prcom 4757 | Commutative law for unorde... |
| preq1 4758 | Equality theorem for unord... |
| preq2 4759 | Equality theorem for unord... |
| preq12 4760 | Equality theorem for unord... |
| preq1i 4761 | Equality inference for uno... |
| preq2i 4762 | Equality inference for uno... |
| preq12i 4763 | Equality inference for uno... |
| preq1d 4764 | Equality deduction for uno... |
| preq2d 4765 | Equality deduction for uno... |
| preq12d 4766 | Equality deduction for uno... |
| tpeq1 4767 | Equality theorem for unord... |
| tpeq2 4768 | Equality theorem for unord... |
| tpeq3 4769 | Equality theorem for unord... |
| tpeq1d 4770 | Equality theorem for unord... |
| tpeq2d 4771 | Equality theorem for unord... |
| tpeq3d 4772 | Equality theorem for unord... |
| tpeq123d 4773 | Equality theorem for unord... |
| tprot 4774 | Rotation of the elements o... |
| tpcoma 4775 | Swap 1st and 2nd members o... |
| tpcomb 4776 | Swap 2nd and 3rd members o... |
| tpass 4777 | Split off the first elemen... |
| qdass 4778 | Two ways to write an unord... |
| qdassr 4779 | Two ways to write an unord... |
| tpidm12 4780 | Unordered triple ` { A , A... |
| tpidm13 4781 | Unordered triple ` { A , B... |
| tpidm23 4782 | Unordered triple ` { A , B... |
| tpidm 4783 | Unordered triple ` { A , A... |
| tppreq3 4784 | An unordered triple is an ... |
| prid1g 4785 | An unordered pair contains... |
| prid2g 4786 | An unordered pair contains... |
| prid1 4787 | An unordered pair contains... |
| prid2 4788 | An unordered pair contains... |
| ifpprsnss 4789 | An unordered pair is a sin... |
| prprc1 4790 | A proper class vanishes in... |
| prprc2 4791 | A proper class vanishes in... |
| prprc 4792 | An unordered pair containi... |
| tpid1 4793 | One of the three elements ... |
| tpid1g 4794 | Closed theorem form of ~ t... |
| tpid2 4795 | One of the three elements ... |
| tpid2g 4796 | Closed theorem form of ~ t... |
| tpid3g 4797 | Closed theorem form of ~ t... |
| tpid3 4798 | One of the three elements ... |
| snnzg 4799 | The singleton of a set is ... |
| snn0d 4800 | The singleton of a set is ... |
| snnz 4801 | The singleton of a set is ... |
| prnz 4802 | A pair containing a set is... |
| prnzg 4803 | A pair containing a set is... |
| tpnz 4804 | An unordered triple contai... |
| tpnzd 4805 | An unordered triple contai... |
| raltpd 4806 | Convert a universal quanti... |
| snssb 4807 | Characterization of the in... |
| snssg 4808 | The singleton formed on a ... |
| snssgOLD 4809 | Obsolete version of ~ snss... |
| snss 4810 | The singleton of an elemen... |
| eldifsn 4811 | Membership in a set with a... |
| eldifsnd 4812 | Membership in a set with a... |
| ssdifsn 4813 | Subset of a set with an el... |
| elpwdifsn 4814 | A subset of a set is an el... |
| eldifsni 4815 | Membership in a set with a... |
| eldifsnneq 4816 | An element of a difference... |
| neldifsn 4817 | The class ` A ` is not in ... |
| neldifsnd 4818 | The class ` A ` is not in ... |
| rexdifsn 4819 | Restricted existential qua... |
| raldifsni 4820 | Rearrangement of a propert... |
| raldifsnb 4821 | Restricted universal quant... |
| eldifvsn 4822 | A set is an element of the... |
| difsn 4823 | An element not in a set ca... |
| difprsnss 4824 | Removal of a singleton fro... |
| difprsn1 4825 | Removal of a singleton fro... |
| difprsn2 4826 | Removal of a singleton fro... |
| diftpsn3 4827 | Removal of a singleton fro... |
| difpr 4828 | Removing two elements as p... |
| tpprceq3 4829 | An unordered triple is an ... |
| tppreqb 4830 | An unordered triple is an ... |
| difsnb 4831 | ` ( B \ { A } ) ` equals `... |
| difsnpss 4832 | ` ( B \ { A } ) ` is a pro... |
| snssi 4833 | The singleton of an elemen... |
| snssd 4834 | The singleton of an elemen... |
| difsnid 4835 | If we remove a single elem... |
| eldifeldifsn 4836 | An element of a difference... |
| pw0 4837 | Compute the power set of t... |
| pwpw0 4838 | Compute the power set of t... |
| snsspr1 4839 | A singleton is a subset of... |
| snsspr2 4840 | A singleton is a subset of... |
| snsstp1 4841 | A singleton is a subset of... |
| snsstp2 4842 | A singleton is a subset of... |
| snsstp3 4843 | A singleton is a subset of... |
| prssg 4844 | A pair of elements of a cl... |
| prss 4845 | A pair of elements of a cl... |
| prssi 4846 | A pair of elements of a cl... |
| prssd 4847 | Deduction version of ~ prs... |
| prsspwg 4848 | An unordered pair belongs ... |
| ssprss 4849 | A pair as subset of a pair... |
| ssprsseq 4850 | A proper pair is a subset ... |
| sssn 4851 | The subsets of a singleton... |
| ssunsn2 4852 | The property of being sand... |
| ssunsn 4853 | Possible values for a set ... |
| eqsn 4854 | Two ways to express that a... |
| eqsnd 4855 | Deduce that a set is a sin... |
| eqsndOLD 4856 | Obsolete version of ~ eqsn... |
| issn 4857 | A sufficient condition for... |
| n0snor2el 4858 | A nonempty set is either a... |
| ssunpr 4859 | Possible values for a set ... |
| sspr 4860 | The subsets of a pair. (C... |
| sstp 4861 | The subsets of an unordere... |
| tpss 4862 | An unordered triple of ele... |
| tpssi 4863 | An unordered triple of ele... |
| sneqrg 4864 | Closed form of ~ sneqr . ... |
| sneqr 4865 | If the singletons of two s... |
| snsssn 4866 | If a singleton is a subset... |
| mosneq 4867 | There exists at most one s... |
| sneqbg 4868 | Two singletons of sets are... |
| snsspw 4869 | The singleton of a class i... |
| prsspw 4870 | An unordered pair belongs ... |
| preq1b 4871 | Biconditional equality lem... |
| preq2b 4872 | Biconditional equality lem... |
| preqr1 4873 | Reverse equality lemma for... |
| preqr2 4874 | Reverse equality lemma for... |
| preq12b 4875 | Equality relationship for ... |
| opthpr 4876 | An unordered pair has the ... |
| preqr1g 4877 | Reverse equality lemma for... |
| preq12bg 4878 | Closed form of ~ preq12b .... |
| prneimg 4879 | Two pairs are not equal if... |
| prnebg 4880 | A (proper) pair is not equ... |
| pr1eqbg 4881 | A (proper) pair is equal t... |
| pr1nebg 4882 | A (proper) pair is not equ... |
| preqsnd 4883 | Equivalence for a pair equ... |
| prnesn 4884 | A proper unordered pair is... |
| prneprprc 4885 | A proper unordered pair is... |
| preqsn 4886 | Equivalence for a pair equ... |
| preq12nebg 4887 | Equality relationship for ... |
| prel12g 4888 | Equality of two unordered ... |
| opthprneg 4889 | An unordered pair has the ... |
| elpreqprlem 4890 | Lemma for ~ elpreqpr . (C... |
| elpreqpr 4891 | Equality and membership ru... |
| elpreqprb 4892 | A set is an element of an ... |
| elpr2elpr 4893 | For an element ` A ` of an... |
| dfopif 4894 | Rewrite ~ df-op using ` if... |
| dfopg 4895 | Value of the ordered pair ... |
| dfop 4896 | Value of an ordered pair w... |
| opeq1 4897 | Equality theorem for order... |
| opeq2 4898 | Equality theorem for order... |
| opeq12 4899 | Equality theorem for order... |
| opeq1i 4900 | Equality inference for ord... |
| opeq2i 4901 | Equality inference for ord... |
| opeq12i 4902 | Equality inference for ord... |
| opeq1d 4903 | Equality deduction for ord... |
| opeq2d 4904 | Equality deduction for ord... |
| opeq12d 4905 | Equality deduction for ord... |
| oteq1 4906 | Equality theorem for order... |
| oteq2 4907 | Equality theorem for order... |
| oteq3 4908 | Equality theorem for order... |
| oteq1d 4909 | Equality deduction for ord... |
| oteq2d 4910 | Equality deduction for ord... |
| oteq3d 4911 | Equality deduction for ord... |
| oteq123d 4912 | Equality deduction for ord... |
| nfop 4913 | Bound-variable hypothesis ... |
| nfopd 4914 | Deduction version of bound... |
| csbopg 4915 | Distribution of class subs... |
| opidg 4916 | The ordered pair ` <. A , ... |
| opid 4917 | The ordered pair ` <. A , ... |
| ralunsn 4918 | Restricted quantification ... |
| 2ralunsn 4919 | Double restricted quantifi... |
| opprc 4920 | Expansion of an ordered pa... |
| opprc1 4921 | Expansion of an ordered pa... |
| opprc2 4922 | Expansion of an ordered pa... |
| oprcl 4923 | If an ordered pair has an ... |
| pwsn 4924 | The power set of a singlet... |
| pwpr 4925 | The power set of an unorde... |
| pwtp 4926 | The power set of an unorde... |
| pwpwpw0 4927 | Compute the power set of t... |
| pwv 4928 | The power class of the uni... |
| prproe 4929 | For an element of a proper... |
| 3elpr2eq 4930 | If there are three element... |
| dfuni2 4933 | Alternate definition of cl... |
| eluni 4934 | Membership in class union.... |
| eluni2 4935 | Membership in class union.... |
| elunii 4936 | Membership in class union.... |
| nfunid 4937 | Deduction version of ~ nfu... |
| nfuni 4938 | Bound-variable hypothesis ... |
| uniss 4939 | Subclass relationship for ... |
| unissi 4940 | Subclass relationship for ... |
| unissd 4941 | Subclass relationship for ... |
| unieq 4942 | Equality theorem for class... |
| unieqi 4943 | Inference of equality of t... |
| unieqd 4944 | Deduction of equality of t... |
| eluniab 4945 | Membership in union of a c... |
| elunirab 4946 | Membership in union of a c... |
| uniprg 4947 | The union of a pair is the... |
| unipr 4948 | The union of a pair is the... |
| unisng 4949 | A set equals the union of ... |
| unisn 4950 | A set equals the union of ... |
| unisnv 4951 | A set equals the union of ... |
| unisn3 4952 | Union of a singleton in th... |
| dfnfc2 4953 | An alternative statement o... |
| uniun 4954 | The class union of the uni... |
| uniin 4955 | The class union of the int... |
| ssuni 4956 | Subclass relationship for ... |
| uni0b 4957 | The union of a set is empt... |
| uni0c 4958 | The union of a set is empt... |
| uni0 4959 | The union of the empty set... |
| csbuni 4960 | Distribute proper substitu... |
| elssuni 4961 | An element of a class is a... |
| unissel 4962 | Condition turning a subcla... |
| unissb 4963 | Relationship involving mem... |
| unissbOLD 4964 | Obsolete version of ~ unis... |
| uniss2 4965 | A subclass condition on th... |
| unidif 4966 | If the difference ` A \ B ... |
| ssunieq 4967 | Relationship implying unio... |
| unimax 4968 | Any member of a class is t... |
| pwuni 4969 | A class is a subclass of t... |
| dfint2 4972 | Alternate definition of cl... |
| inteq 4973 | Equality law for intersect... |
| inteqi 4974 | Equality inference for cla... |
| inteqd 4975 | Equality deduction for cla... |
| elint 4976 | Membership in class inters... |
| elint2 4977 | Membership in class inters... |
| elintg 4978 | Membership in class inters... |
| elinti 4979 | Membership in class inters... |
| nfint 4980 | Bound-variable hypothesis ... |
| elintabg 4981 | Two ways of saying a set i... |
| elintab 4982 | Membership in the intersec... |
| elintabOLD 4983 | Obsolete version of ~ elin... |
| elintrab 4984 | Membership in the intersec... |
| elintrabg 4985 | Membership in the intersec... |
| int0 4986 | The intersection of the em... |
| intss1 4987 | An element of a class incl... |
| ssint 4988 | Subclass of a class inters... |
| ssintab 4989 | Subclass of the intersecti... |
| ssintub 4990 | Subclass of the least uppe... |
| ssmin 4991 | Subclass of the minimum va... |
| intmin 4992 | Any member of a class is t... |
| intss 4993 | Intersection of subclasses... |
| intssuni 4994 | The intersection of a none... |
| ssintrab 4995 | Subclass of the intersecti... |
| unissint 4996 | If the union of a class is... |
| intssuni2 4997 | Subclass relationship for ... |
| intminss 4998 | Under subset ordering, the... |
| intmin2 4999 | Any set is the smallest of... |
| intmin3 5000 | Under subset ordering, the... |
| intmin4 5001 | Elimination of a conjunct ... |
| intab 5002 | The intersection of a spec... |
| int0el 5003 | The intersection of a clas... |
| intun 5004 | The class intersection of ... |
| intprg 5005 | The intersection of a pair... |
| intpr 5006 | The intersection of a pair... |
| intsng 5007 | Intersection of a singleto... |
| intsn 5008 | The intersection of a sing... |
| uniintsn 5009 | Two ways to express " ` A ... |
| uniintab 5010 | The union and the intersec... |
| intunsn 5011 | Theorem joining a singleto... |
| rint0 5012 | Relative intersection of a... |
| elrint 5013 | Membership in a restricted... |
| elrint2 5014 | Membership in a restricted... |
| eliun 5019 | Membership in indexed unio... |
| eliin 5020 | Membership in indexed inte... |
| eliuni 5021 | Membership in an indexed u... |
| iuncom 5022 | Commutation of indexed uni... |
| iuncom4 5023 | Commutation of union with ... |
| iunconst 5024 | Indexed union of a constan... |
| iinconst 5025 | Indexed intersection of a ... |
| iuneqconst 5026 | Indexed union of identical... |
| iuniin 5027 | Law combining indexed unio... |
| iinssiun 5028 | An indexed intersection is... |
| iunss1 5029 | Subclass theorem for index... |
| iinss1 5030 | Subclass theorem for index... |
| iuneq1 5031 | Equality theorem for index... |
| iineq1 5032 | Equality theorem for index... |
| ss2iun 5033 | Subclass theorem for index... |
| iuneq2 5034 | Equality theorem for index... |
| iineq2 5035 | Equality theorem for index... |
| iuneq2i 5036 | Equality inference for ind... |
| iineq2i 5037 | Equality inference for ind... |
| iineq2d 5038 | Equality deduction for ind... |
| iuneq2dv 5039 | Equality deduction for ind... |
| iineq2dv 5040 | Equality deduction for ind... |
| iuneq12df 5041 | Equality deduction for ind... |
| iuneq1d 5042 | Equality theorem for index... |
| iuneq12dOLD 5043 | Obsolete version of ~ iune... |
| iuneq12d 5044 | Equality deduction for ind... |
| iuneq2d 5045 | Equality deduction for ind... |
| nfiun 5046 | Bound-variable hypothesis ... |
| nfiin 5047 | Bound-variable hypothesis ... |
| nfiung 5048 | Bound-variable hypothesis ... |
| nfiing 5049 | Bound-variable hypothesis ... |
| nfiu1 5050 | Bound-variable hypothesis ... |
| nfiu1OLD 5051 | Obsolete version of ~ nfiu... |
| nfii1 5052 | Bound-variable hypothesis ... |
| dfiun2g 5053 | Alternate definition of in... |
| dfiun2gOLD 5054 | Obsolete version of ~ dfiu... |
| dfiin2g 5055 | Alternate definition of in... |
| dfiun2 5056 | Alternate definition of in... |
| dfiin2 5057 | Alternate definition of in... |
| dfiunv2 5058 | Define double indexed unio... |
| cbviun 5059 | Rule used to change the bo... |
| cbviin 5060 | Change bound variables in ... |
| cbviung 5061 | Rule used to change the bo... |
| cbviing 5062 | Change bound variables in ... |
| cbviunv 5063 | Rule used to change the bo... |
| cbviinv 5064 | Change bound variables in ... |
| cbviunvg 5065 | Rule used to change the bo... |
| cbviinvg 5066 | Change bound variables in ... |
| iunssf 5067 | Subset theorem for an inde... |
| iunss 5068 | Subset theorem for an inde... |
| ssiun 5069 | Subset implication for an ... |
| ssiun2 5070 | Identity law for subset of... |
| ssiun2s 5071 | Subset relationship for an... |
| iunss2 5072 | A subclass condition on th... |
| iunssd 5073 | Subset theorem for an inde... |
| iunab 5074 | The indexed union of a cla... |
| iunrab 5075 | The indexed union of a res... |
| iunxdif2 5076 | Indexed union with a class... |
| ssiinf 5077 | Subset theorem for an inde... |
| ssiin 5078 | Subset theorem for an inde... |
| iinss 5079 | Subset implication for an ... |
| iinss2 5080 | An indexed intersection is... |
| uniiun 5081 | Class union in terms of in... |
| intiin 5082 | Class intersection in term... |
| iunid 5083 | An indexed union of single... |
| iunidOLD 5084 | Obsolete version of ~ iuni... |
| iun0 5085 | An indexed union of the em... |
| 0iun 5086 | An empty indexed union is ... |
| 0iin 5087 | An empty indexed intersect... |
| viin 5088 | Indexed intersection with ... |
| iunsn 5089 | Indexed union of a singlet... |
| iunn0 5090 | There is a nonempty class ... |
| iinab 5091 | Indexed intersection of a ... |
| iinrab 5092 | Indexed intersection of a ... |
| iinrab2 5093 | Indexed intersection of a ... |
| iunin2 5094 | Indexed union of intersect... |
| iunin1 5095 | Indexed union of intersect... |
| iinun2 5096 | Indexed intersection of un... |
| iundif2 5097 | Indexed union of class dif... |
| iindif1 5098 | Indexed intersection of cl... |
| 2iunin 5099 | Rearrange indexed unions o... |
| iindif2 5100 | Indexed intersection of cl... |
| iinin2 5101 | Indexed intersection of in... |
| iinin1 5102 | Indexed intersection of in... |
| iinvdif 5103 | The indexed intersection o... |
| elriin 5104 | Elementhood in a relative ... |
| riin0 5105 | Relative intersection of a... |
| riinn0 5106 | Relative intersection of a... |
| riinrab 5107 | Relative intersection of a... |
| symdif0 5108 | Symmetric difference with ... |
| symdifv 5109 | The symmetric difference w... |
| symdifid 5110 | The symmetric difference o... |
| iinxsng 5111 | A singleton index picks ou... |
| iinxprg 5112 | Indexed intersection with ... |
| iunxsng 5113 | A singleton index picks ou... |
| iunxsn 5114 | A singleton index picks ou... |
| iunxsngf 5115 | A singleton index picks ou... |
| iunun 5116 | Separate a union in an ind... |
| iunxun 5117 | Separate a union in the in... |
| iunxdif3 5118 | An indexed union where som... |
| iunxprg 5119 | A pair index picks out two... |
| iunxiun 5120 | Separate an indexed union ... |
| iinuni 5121 | A relationship involving u... |
| iununi 5122 | A relationship involving u... |
| sspwuni 5123 | Subclass relationship for ... |
| pwssb 5124 | Two ways to express a coll... |
| elpwpw 5125 | Characterization of the el... |
| pwpwab 5126 | The double power class wri... |
| pwpwssunieq 5127 | The class of sets whose un... |
| elpwuni 5128 | Relationship for power cla... |
| iinpw 5129 | The power class of an inte... |
| iunpwss 5130 | Inclusion of an indexed un... |
| intss2 5131 | A nonempty intersection of... |
| rintn0 5132 | Relative intersection of a... |
| dfdisj2 5135 | Alternate definition for d... |
| disjss2 5136 | If each element of a colle... |
| disjeq2 5137 | Equality theorem for disjo... |
| disjeq2dv 5138 | Equality deduction for dis... |
| disjss1 5139 | A subset of a disjoint col... |
| disjeq1 5140 | Equality theorem for disjo... |
| disjeq1d 5141 | Equality theorem for disjo... |
| disjeq12d 5142 | Equality theorem for disjo... |
| cbvdisj 5143 | Change bound variables in ... |
| cbvdisjv 5144 | Change bound variables in ... |
| nfdisjw 5145 | Bound-variable hypothesis ... |
| nfdisj 5146 | Bound-variable hypothesis ... |
| nfdisj1 5147 | Bound-variable hypothesis ... |
| disjor 5148 | Two ways to say that a col... |
| disjors 5149 | Two ways to say that a col... |
| disji2 5150 | Property of a disjoint col... |
| disji 5151 | Property of a disjoint col... |
| invdisj 5152 | If there is a function ` C... |
| invdisjrab 5153 | The restricted class abstr... |
| disjiun 5154 | A disjoint collection yiel... |
| disjord 5155 | Conditions for a collectio... |
| disjiunb 5156 | Two ways to say that a col... |
| disjiund 5157 | Conditions for a collectio... |
| sndisj 5158 | Any collection of singleto... |
| 0disj 5159 | Any collection of empty se... |
| disjxsn 5160 | A singleton collection is ... |
| disjx0 5161 | An empty collection is dis... |
| disjprg 5162 | A pair collection is disjo... |
| disjxiun 5163 | An indexed union of a disj... |
| disjxun 5164 | The union of two disjoint ... |
| disjss3 5165 | Expand a disjoint collecti... |
| breq 5168 | Equality theorem for binar... |
| breq1 5169 | Equality theorem for a bin... |
| breq2 5170 | Equality theorem for a bin... |
| breq12 5171 | Equality theorem for a bin... |
| breqi 5172 | Equality inference for bin... |
| breq1i 5173 | Equality inference for a b... |
| breq2i 5174 | Equality inference for a b... |
| breq12i 5175 | Equality inference for a b... |
| breq1d 5176 | Equality deduction for a b... |
| breqd 5177 | Equality deduction for a b... |
| breq2d 5178 | Equality deduction for a b... |
| breq12d 5179 | Equality deduction for a b... |
| breq123d 5180 | Equality deduction for a b... |
| breqdi 5181 | Equality deduction for a b... |
| breqan12d 5182 | Equality deduction for a b... |
| breqan12rd 5183 | Equality deduction for a b... |
| eqnbrtrd 5184 | Substitution of equal clas... |
| nbrne1 5185 | Two classes are different ... |
| nbrne2 5186 | Two classes are different ... |
| eqbrtri 5187 | Substitution of equal clas... |
| eqbrtrd 5188 | Substitution of equal clas... |
| eqbrtrri 5189 | Substitution of equal clas... |
| eqbrtrrd 5190 | Substitution of equal clas... |
| breqtri 5191 | Substitution of equal clas... |
| breqtrd 5192 | Substitution of equal clas... |
| breqtrri 5193 | Substitution of equal clas... |
| breqtrrd 5194 | Substitution of equal clas... |
| 3brtr3i 5195 | Substitution of equality i... |
| 3brtr4i 5196 | Substitution of equality i... |
| 3brtr3d 5197 | Substitution of equality i... |
| 3brtr4d 5198 | Substitution of equality i... |
| 3brtr3g 5199 | Substitution of equality i... |
| 3brtr4g 5200 | Substitution of equality i... |
| eqbrtrid 5201 | A chained equality inferen... |
| eqbrtrrid 5202 | A chained equality inferen... |
| breqtrid 5203 | A chained equality inferen... |
| breqtrrid 5204 | A chained equality inferen... |
| eqbrtrdi 5205 | A chained equality inferen... |
| eqbrtrrdi 5206 | A chained equality inferen... |
| breqtrdi 5207 | A chained equality inferen... |
| breqtrrdi 5208 | A chained equality inferen... |
| ssbrd 5209 | Deduction from a subclass ... |
| ssbr 5210 | Implication from a subclas... |
| ssbri 5211 | Inference from a subclass ... |
| nfbrd 5212 | Deduction version of bound... |
| nfbr 5213 | Bound-variable hypothesis ... |
| brab1 5214 | Relationship between a bin... |
| br0 5215 | The empty binary relation ... |
| brne0 5216 | If two sets are in a binar... |
| brun 5217 | The union of two binary re... |
| brin 5218 | The intersection of two re... |
| brdif 5219 | The difference of two bina... |
| sbcbr123 5220 | Move substitution in and o... |
| sbcbr 5221 | Move substitution in and o... |
| sbcbr12g 5222 | Move substitution in and o... |
| sbcbr1g 5223 | Move substitution in and o... |
| sbcbr2g 5224 | Move substitution in and o... |
| brsymdif 5225 | Characterization of the sy... |
| brralrspcev 5226 | Restricted existential spe... |
| brimralrspcev 5227 | Restricted existential spe... |
| opabss 5230 | The collection of ordered ... |
| opabbid 5231 | Equivalent wff's yield equ... |
| opabbidv 5232 | Equivalent wff's yield equ... |
| opabbii 5233 | Equivalent wff's yield equ... |
| nfopabd 5234 | Bound-variable hypothesis ... |
| nfopab 5235 | Bound-variable hypothesis ... |
| nfopab1 5236 | The first abstraction vari... |
| nfopab2 5237 | The second abstraction var... |
| cbvopab 5238 | Rule used to change bound ... |
| cbvopabv 5239 | Rule used to change bound ... |
| cbvopabvOLD 5240 | Obsolete version of ~ cbvo... |
| cbvopab1 5241 | Change first bound variabl... |
| cbvopab1g 5242 | Change first bound variabl... |
| cbvopab2 5243 | Change second bound variab... |
| cbvopab1s 5244 | Change first bound variabl... |
| cbvopab1v 5245 | Rule used to change the fi... |
| cbvopab1vOLD 5246 | Obsolete version of ~ cbvo... |
| cbvopab2v 5247 | Rule used to change the se... |
| unopab 5248 | Union of two ordered pair ... |
| mpteq12da 5251 | An equality inference for ... |
| mpteq12df 5252 | An equality inference for ... |
| mpteq12dfOLD 5253 | Obsolete version of ~ mpte... |
| mpteq12f 5254 | An equality theorem for th... |
| mpteq12dva 5255 | An equality inference for ... |
| mpteq12dvaOLD 5256 | Obsolete version of ~ mpte... |
| mpteq12dv 5257 | An equality inference for ... |
| mpteq12 5258 | An equality theorem for th... |
| mpteq1 5259 | An equality theorem for th... |
| mpteq1OLD 5260 | Obsolete version of ~ mpte... |
| mpteq1d 5261 | An equality theorem for th... |
| mpteq1i 5262 | An equality theorem for th... |
| mpteq1iOLD 5263 | Obsolete version of ~ mpte... |
| mpteq2da 5264 | Slightly more general equa... |
| mpteq2daOLD 5265 | Obsolete version of ~ mpte... |
| mpteq2dva 5266 | Slightly more general equa... |
| mpteq2dvaOLD 5267 | Obsolete version of ~ mpte... |
| mpteq2dv 5268 | An equality inference for ... |
| mpteq2ia 5269 | An equality inference for ... |
| mpteq2iaOLD 5270 | Obsolete version of ~ mpte... |
| mpteq2i 5271 | An equality inference for ... |
| mpteq12i 5272 | An equality inference for ... |
| nfmpt 5273 | Bound-variable hypothesis ... |
| nfmpt1 5274 | Bound-variable hypothesis ... |
| cbvmptf 5275 | Rule to change the bound v... |
| cbvmptfg 5276 | Rule to change the bound v... |
| cbvmpt 5277 | Rule to change the bound v... |
| cbvmptg 5278 | Rule to change the bound v... |
| cbvmptv 5279 | Rule to change the bound v... |
| cbvmptvOLD 5280 | Obsolete version of ~ cbvm... |
| cbvmptvg 5281 | Rule to change the bound v... |
| mptv 5282 | Function with universal do... |
| dftr2 5285 | An alternate way of defini... |
| dftr2c 5286 | Variant of ~ dftr2 with co... |
| dftr5 5287 | An alternate way of defini... |
| dftr5OLD 5288 | Obsolete version of ~ dftr... |
| dftr3 5289 | An alternate way of defini... |
| dftr4 5290 | An alternate way of defini... |
| treq 5291 | Equality theorem for the t... |
| trel 5292 | In a transitive class, the... |
| trel3 5293 | In a transitive class, the... |
| trss 5294 | An element of a transitive... |
| trin 5295 | The intersection of transi... |
| tr0 5296 | The empty set is transitiv... |
| trv 5297 | The universe is transitive... |
| triun 5298 | An indexed union of a clas... |
| truni 5299 | The union of a class of tr... |
| triin 5300 | An indexed intersection of... |
| trint 5301 | The intersection of a clas... |
| trintss 5302 | Any nonempty transitive cl... |
| axrep1 5304 | The version of the Axiom o... |
| axreplem 5305 | Lemma for ~ axrep2 and ~ a... |
| axrep2 5306 | Axiom of Replacement expre... |
| axrep3 5307 | Axiom of Replacement sligh... |
| axrep4 5308 | A more traditional version... |
| axrep5 5309 | Axiom of Replacement (simi... |
| axrep6 5310 | A condensed form of ~ ax-r... |
| axrep6g 5311 | ~ axrep6 in class notation... |
| zfrepclf 5312 | An inference based on the ... |
| zfrep3cl 5313 | An inference based on the ... |
| zfrep4 5314 | A version of Replacement u... |
| axsepgfromrep 5315 | A more general version ~ a... |
| axsep 5316 | Axiom scheme of separation... |
| axsepg 5318 | A more general version of ... |
| zfauscl 5319 | Separation Scheme (Aussond... |
| bm1.3ii 5320 | Convert implication to equ... |
| ax6vsep 5321 | Derive ~ ax6v (a weakened ... |
| axnulALT 5322 | Alternate proof of ~ axnul... |
| axnul 5323 | The Null Set Axiom of ZF s... |
| 0ex 5325 | The Null Set Axiom of ZF s... |
| al0ssb 5326 | The empty set is the uniqu... |
| sseliALT 5327 | Alternate proof of ~ sseli... |
| csbexg 5328 | The existence of proper su... |
| csbex 5329 | The existence of proper su... |
| unisn2 5330 | A version of ~ unisn witho... |
| nalset 5331 | No set contains all sets. ... |
| vnex 5332 | The universal class does n... |
| vprc 5333 | The universal class is not... |
| nvel 5334 | The universal class does n... |
| inex1 5335 | Separation Scheme (Aussond... |
| inex2 5336 | Separation Scheme (Aussond... |
| inex1g 5337 | Closed-form, generalized S... |
| inex2g 5338 | Sufficient condition for a... |
| ssex 5339 | The subset of a set is als... |
| ssexi 5340 | The subset of a set is als... |
| ssexg 5341 | The subset of a set is als... |
| ssexd 5342 | A subclass of a set is a s... |
| abexd 5343 | Conditions for a class abs... |
| abex 5344 | Conditions for a class abs... |
| prcssprc 5345 | The superclass of a proper... |
| sselpwd 5346 | Elementhood to a power set... |
| difexg 5347 | Existence of a difference.... |
| difexi 5348 | Existence of a difference,... |
| difexd 5349 | Existence of a difference.... |
| zfausab 5350 | Separation Scheme (Aussond... |
| elpw2g 5351 | Membership in a power clas... |
| elpw2 5352 | Membership in a power clas... |
| elpwi2 5353 | Membership in a power clas... |
| rabelpw 5354 | A restricted class abstrac... |
| rabexg 5355 | Separation Scheme in terms... |
| rabexgOLD 5356 | Obsolete proof of ~ rabexg... |
| rabex 5357 | Separation Scheme in terms... |
| rabexd 5358 | Separation Scheme in terms... |
| rabex2 5359 | Separation Scheme in terms... |
| rab2ex 5360 | A class abstraction based ... |
| elssabg 5361 | Membership in a class abst... |
| intex 5362 | The intersection of a none... |
| intnex 5363 | If a class intersection is... |
| intexab 5364 | The intersection of a none... |
| intexrab 5365 | The intersection of a none... |
| iinexg 5366 | The existence of a class i... |
| intabs 5367 | Absorption of a redundant ... |
| inuni 5368 | The intersection of a unio... |
| axpweq 5369 | Two equivalent ways to exp... |
| pwnss 5370 | The power set of a set is ... |
| pwne 5371 | No set equals its power se... |
| difelpw 5372 | A difference is an element... |
| class2set 5373 | The class of elements of `... |
| 0elpw 5374 | Every power class contains... |
| pwne0 5375 | A power class is never emp... |
| 0nep0 5376 | The empty set and its powe... |
| 0inp0 5377 | Something cannot be equal ... |
| unidif0 5378 | The removal of the empty s... |
| eqsnuniex 5379 | If a class is equal to the... |
| iin0 5380 | An indexed intersection of... |
| notzfaus 5381 | In the Separation Scheme ~... |
| intv 5382 | The intersection of the un... |
| zfpow 5384 | Axiom of Power Sets expres... |
| axpow2 5385 | A variant of the Axiom of ... |
| axpow3 5386 | A variant of the Axiom of ... |
| elALT2 5387 | Alternate proof of ~ el us... |
| dtruALT2 5388 | Alternate proof of ~ dtru ... |
| dtrucor 5389 | Corollary of ~ dtru . Thi... |
| dtrucor2 5390 | The theorem form of the de... |
| dvdemo1 5391 | Demonstration of a theorem... |
| dvdemo2 5392 | Demonstration of a theorem... |
| nfnid 5393 | A setvar variable is not f... |
| nfcvb 5394 | The "distinctor" expressio... |
| vpwex 5395 | Power set axiom: the power... |
| pwexg 5396 | Power set axiom expressed ... |
| pwexd 5397 | Deduction version of the p... |
| pwex 5398 | Power set axiom expressed ... |
| pwel 5399 | Quantitative version of ~ ... |
| abssexg 5400 | Existence of a class of su... |
| snexALT 5401 | Alternate proof of ~ snex ... |
| p0ex 5402 | The power set of the empty... |
| p0exALT 5403 | Alternate proof of ~ p0ex ... |
| pp0ex 5404 | The power set of the power... |
| ord3ex 5405 | The ordinal number 3 is a ... |
| dtruALT 5406 | Alternate proof of ~ dtru ... |
| axc16b 5407 | This theorem shows that Ax... |
| eunex 5408 | Existential uniqueness imp... |
| eusv1 5409 | Two ways to express single... |
| eusvnf 5410 | Even if ` x ` is free in `... |
| eusvnfb 5411 | Two ways to say that ` A (... |
| eusv2i 5412 | Two ways to express single... |
| eusv2nf 5413 | Two ways to express single... |
| eusv2 5414 | Two ways to express single... |
| reusv1 5415 | Two ways to express single... |
| reusv2lem1 5416 | Lemma for ~ reusv2 . (Con... |
| reusv2lem2 5417 | Lemma for ~ reusv2 . (Con... |
| reusv2lem3 5418 | Lemma for ~ reusv2 . (Con... |
| reusv2lem4 5419 | Lemma for ~ reusv2 . (Con... |
| reusv2lem5 5420 | Lemma for ~ reusv2 . (Con... |
| reusv2 5421 | Two ways to express single... |
| reusv3i 5422 | Two ways of expressing exi... |
| reusv3 5423 | Two ways to express single... |
| eusv4 5424 | Two ways to express single... |
| alxfr 5425 | Transfer universal quantif... |
| ralxfrd 5426 | Transfer universal quantif... |
| rexxfrd 5427 | Transfer existential quant... |
| ralxfr2d 5428 | Transfer universal quantif... |
| rexxfr2d 5429 | Transfer existential quant... |
| ralxfrd2 5430 | Transfer universal quantif... |
| rexxfrd2 5431 | Transfer existence from a ... |
| ralxfr 5432 | Transfer universal quantif... |
| ralxfrALT 5433 | Alternate proof of ~ ralxf... |
| rexxfr 5434 | Transfer existence from a ... |
| rabxfrd 5435 | Membership in a restricted... |
| rabxfr 5436 | Membership in a restricted... |
| reuhypd 5437 | A theorem useful for elimi... |
| reuhyp 5438 | A theorem useful for elimi... |
| zfpair 5439 | The Axiom of Pairing of Ze... |
| axprALT 5440 | Alternate proof of ~ axpr ... |
| axprlem1 5441 | Lemma for ~ axpr . There ... |
| axprlem2 5442 | Lemma for ~ axpr . There ... |
| axprlem3 5443 | Lemma for ~ axpr . Elimin... |
| axprlem4 5444 | Lemma for ~ axpr . The fi... |
| axprlem5 5445 | Lemma for ~ axpr . The se... |
| axpr 5446 | Unabbreviated version of t... |
| zfpair2 5448 | Derive the abbreviated ver... |
| vsnex 5449 | A singleton built on a set... |
| snexg 5450 | A singleton built on a set... |
| snex 5451 | A singleton is a set. The... |
| prex 5452 | The Axiom of Pairing using... |
| exel 5453 | There exist two sets, one ... |
| exexneq 5454 | There exist two different ... |
| exneq 5455 | Given any set (the " ` y `... |
| dtru 5456 | Given any set (the " ` y `... |
| el 5457 | Any set is an element of s... |
| sels 5458 | If a class is a set, then ... |
| selsALT 5459 | Alternate proof of ~ sels ... |
| elALT 5460 | Alternate proof of ~ el , ... |
| dtruOLD 5461 | Obsolete proof of ~ dtru a... |
| snelpwg 5462 | A singleton of a set is a ... |
| snelpwi 5463 | If a set is a member of a ... |
| snelpwiOLD 5464 | Obsolete version of ~ snel... |
| snelpw 5465 | A singleton of a set is a ... |
| prelpw 5466 | An unordered pair of two s... |
| prelpwi 5467 | If two sets are members of... |
| rext 5468 | A theorem similar to exten... |
| sspwb 5469 | The powerclass constructio... |
| unipw 5470 | A class equals the union o... |
| univ 5471 | The union of the universe ... |
| pwtr 5472 | A class is transitive iff ... |
| ssextss 5473 | An extensionality-like pri... |
| ssext 5474 | An extensionality-like pri... |
| nssss 5475 | Negation of subclass relat... |
| pweqb 5476 | Classes are equal if and o... |
| intidg 5477 | The intersection of all se... |
| intidOLD 5478 | Obsolete version of ~ inti... |
| moabex 5479 | "At most one" existence im... |
| rmorabex 5480 | Restricted "at most one" e... |
| euabex 5481 | The abstraction of a wff w... |
| nnullss 5482 | A nonempty class (even if ... |
| exss 5483 | Restricted existence in a ... |
| opex 5484 | An ordered pair of classes... |
| otex 5485 | An ordered triple of class... |
| elopg 5486 | Characterization of the el... |
| elop 5487 | Characterization of the el... |
| opi1 5488 | One of the two elements in... |
| opi2 5489 | One of the two elements of... |
| opeluu 5490 | Each member of an ordered ... |
| op1stb 5491 | Extract the first member o... |
| brv 5492 | Two classes are always in ... |
| opnz 5493 | An ordered pair is nonempt... |
| opnzi 5494 | An ordered pair is nonempt... |
| opth1 5495 | Equality of the first memb... |
| opth 5496 | The ordered pair theorem. ... |
| opthg 5497 | Ordered pair theorem. ` C ... |
| opth1g 5498 | Equality of the first memb... |
| opthg2 5499 | Ordered pair theorem. (Co... |
| opth2 5500 | Ordered pair theorem. (Co... |
| opthneg 5501 | Two ordered pairs are not ... |
| opthne 5502 | Two ordered pairs are not ... |
| otth2 5503 | Ordered triple theorem, wi... |
| otth 5504 | Ordered triple theorem. (... |
| otthg 5505 | Ordered triple theorem, cl... |
| otthne 5506 | Contrapositive of the orde... |
| eqvinop 5507 | A variable introduction la... |
| sbcop1 5508 | The proper substitution of... |
| sbcop 5509 | The proper substitution of... |
| copsexgw 5510 | Version of ~ copsexg with ... |
| copsexg 5511 | Substitution of class ` A ... |
| copsex2t 5512 | Closed theorem form of ~ c... |
| copsex2g 5513 | Implicit substitution infe... |
| copsex4g 5514 | An implicit substitution i... |
| 0nelop 5515 | A property of ordered pair... |
| opwo0id 5516 | An ordered pair is equal t... |
| opeqex 5517 | Equivalence of existence i... |
| oteqex2 5518 | Equivalence of existence i... |
| oteqex 5519 | Equivalence of existence i... |
| opcom 5520 | An ordered pair commutes i... |
| moop2 5521 | "At most one" property of ... |
| opeqsng 5522 | Equivalence for an ordered... |
| opeqsn 5523 | Equivalence for an ordered... |
| opeqpr 5524 | Equivalence for an ordered... |
| snopeqop 5525 | Equivalence for an ordered... |
| propeqop 5526 | Equivalence for an ordered... |
| propssopi 5527 | If a pair of ordered pairs... |
| snopeqopsnid 5528 | Equivalence for an ordered... |
| mosubopt 5529 | "At most one" remains true... |
| mosubop 5530 | "At most one" remains true... |
| euop2 5531 | Transfer existential uniqu... |
| euotd 5532 | Prove existential uniquene... |
| opthwiener 5533 | Justification theorem for ... |
| uniop 5534 | The union of an ordered pa... |
| uniopel 5535 | Ordered pair membership is... |
| opthhausdorff 5536 | Justification theorem for ... |
| opthhausdorff0 5537 | Justification theorem for ... |
| otsndisj 5538 | The singletons consisting ... |
| otiunsndisj 5539 | The union of singletons co... |
| iunopeqop 5540 | Implication of an ordered ... |
| brsnop 5541 | Binary relation for an ord... |
| brtp 5542 | A necessary and sufficient... |
| opabidw 5543 | The law of concretion. Sp... |
| opabid 5544 | The law of concretion. Sp... |
| elopabw 5545 | Membership in a class abst... |
| elopab 5546 | Membership in a class abst... |
| rexopabb 5547 | Restricted existential qua... |
| vopelopabsb 5548 | The law of concretion in t... |
| opelopabsb 5549 | The law of concretion in t... |
| brabsb 5550 | The law of concretion in t... |
| opelopabt 5551 | Closed theorem form of ~ o... |
| opelopabga 5552 | The law of concretion. Th... |
| brabga 5553 | The law of concretion for ... |
| opelopab2a 5554 | Ordered pair membership in... |
| opelopaba 5555 | The law of concretion. Th... |
| braba 5556 | The law of concretion for ... |
| opelopabg 5557 | The law of concretion. Th... |
| brabg 5558 | The law of concretion for ... |
| opelopabgf 5559 | The law of concretion. Th... |
| opelopab2 5560 | Ordered pair membership in... |
| opelopab 5561 | The law of concretion. Th... |
| brab 5562 | The law of concretion for ... |
| opelopabaf 5563 | The law of concretion. Th... |
| opelopabf 5564 | The law of concretion. Th... |
| ssopab2 5565 | Equivalence of ordered pai... |
| ssopab2bw 5566 | Equivalence of ordered pai... |
| eqopab2bw 5567 | Equivalence of ordered pai... |
| ssopab2b 5568 | Equivalence of ordered pai... |
| ssopab2i 5569 | Inference of ordered pair ... |
| ssopab2dv 5570 | Inference of ordered pair ... |
| eqopab2b 5571 | Equivalence of ordered pai... |
| opabn0 5572 | Nonempty ordered pair clas... |
| opab0 5573 | Empty ordered pair class a... |
| csbopab 5574 | Move substitution into a c... |
| csbopabgALT 5575 | Move substitution into a c... |
| csbmpt12 5576 | Move substitution into a m... |
| csbmpt2 5577 | Move substitution into the... |
| iunopab 5578 | Move indexed union inside ... |
| iunopabOLD 5579 | Obsolete version of ~ iuno... |
| elopabr 5580 | Membership in an ordered-p... |
| elopabran 5581 | Membership in an ordered-p... |
| elopabrOLD 5582 | Obsolete version of ~ elop... |
| rbropapd 5583 | Properties of a pair in an... |
| rbropap 5584 | Properties of a pair in a ... |
| 2rbropap 5585 | Properties of a pair in a ... |
| 0nelopab 5586 | The empty set is never an ... |
| 0nelopabOLD 5587 | Obsolete version of ~ 0nel... |
| brabv 5588 | If two classes are in a re... |
| pwin 5589 | The power class of the int... |
| pwssun 5590 | The power class of the uni... |
| pwun 5591 | The power class of the uni... |
| dfid4 5594 | The identity function expr... |
| dfid2 5595 | Alternate definition of th... |
| dfid3 5596 | A stronger version of ~ df... |
| dfid2OLD 5597 | Obsolete version of ~ dfid... |
| epelg 5600 | The membership relation an... |
| epeli 5601 | The membership relation an... |
| epel 5602 | The membership relation an... |
| 0sn0ep 5603 | An example for the members... |
| epn0 5604 | The membership relation is... |
| poss 5609 | Subset theorem for the par... |
| poeq1 5610 | Equality theorem for parti... |
| poeq2 5611 | Equality theorem for parti... |
| poeq12d 5612 | Equality deduction for par... |
| nfpo 5613 | Bound-variable hypothesis ... |
| nfso 5614 | Bound-variable hypothesis ... |
| pocl 5615 | Characteristic properties ... |
| poclOLD 5616 | Obsolete version of ~ pocl... |
| ispod 5617 | Sufficient conditions for ... |
| swopolem 5618 | Perform the substitutions ... |
| swopo 5619 | A strict weak order is a p... |
| poirr 5620 | A partial order is irrefle... |
| potr 5621 | A partial order is a trans... |
| po2nr 5622 | A partial order has no 2-c... |
| po3nr 5623 | A partial order has no 3-c... |
| po2ne 5624 | Two sets related by a part... |
| po0 5625 | Any relation is a partial ... |
| pofun 5626 | The inverse image of a par... |
| sopo 5627 | A strict linear order is a... |
| soss 5628 | Subset theorem for the str... |
| soeq1 5629 | Equality theorem for the s... |
| soeq2 5630 | Equality theorem for the s... |
| soeq12d 5631 | Equality deduction for tot... |
| sonr 5632 | A strict order relation is... |
| sotr 5633 | A strict order relation is... |
| solin 5634 | A strict order relation is... |
| so2nr 5635 | A strict order relation ha... |
| so3nr 5636 | A strict order relation ha... |
| sotric 5637 | A strict order relation sa... |
| sotrieq 5638 | Trichotomy law for strict ... |
| sotrieq2 5639 | Trichotomy law for strict ... |
| soasym 5640 | Asymmetry law for strict o... |
| sotr2 5641 | A transitivity relation. ... |
| issod 5642 | An irreflexive, transitive... |
| issoi 5643 | An irreflexive, transitive... |
| isso2i 5644 | Deduce strict ordering fro... |
| so0 5645 | Any relation is a strict o... |
| somo 5646 | A totally ordered set has ... |
| sotrine 5647 | Trichotomy law for strict ... |
| sotr3 5648 | Transitivity law for stric... |
| dffr6 5655 | Alternate definition of ~ ... |
| frd 5656 | A nonempty subset of an ` ... |
| fri 5657 | A nonempty subset of an ` ... |
| friOLD 5658 | Obsolete version of ~ fri ... |
| seex 5659 | The ` R ` -preimage of an ... |
| exse 5660 | Any relation on a set is s... |
| dffr2 5661 | Alternate definition of we... |
| dffr2ALT 5662 | Alternate proof of ~ dffr2... |
| frc 5663 | Property of well-founded r... |
| frss 5664 | Subset theorem for the wel... |
| sess1 5665 | Subset theorem for the set... |
| sess2 5666 | Subset theorem for the set... |
| freq1 5667 | Equality theorem for the w... |
| freq2 5668 | Equality theorem for the w... |
| freq12d 5669 | Equality deduction for wel... |
| seeq1 5670 | Equality theorem for the s... |
| seeq2 5671 | Equality theorem for the s... |
| seeq12d 5672 | Equality deduction for the... |
| nffr 5673 | Bound-variable hypothesis ... |
| nfse 5674 | Bound-variable hypothesis ... |
| nfwe 5675 | Bound-variable hypothesis ... |
| frirr 5676 | A well-founded relation is... |
| fr2nr 5677 | A well-founded relation ha... |
| fr0 5678 | Any relation is well-found... |
| frminex 5679 | If an element of a well-fo... |
| efrirr 5680 | A well-founded class does ... |
| efrn2lp 5681 | A well-founded class conta... |
| epse 5682 | The membership relation is... |
| tz7.2 5683 | Similar to Theorem 7.2 of ... |
| dfepfr 5684 | An alternate way of saying... |
| epfrc 5685 | A subset of a well-founded... |
| wess 5686 | Subset theorem for the wel... |
| weeq1 5687 | Equality theorem for the w... |
| weeq2 5688 | Equality theorem for the w... |
| weeq12d 5689 | Equality deduction for wel... |
| wefr 5690 | A well-ordering is well-fo... |
| weso 5691 | A well-ordering is a stric... |
| wecmpep 5692 | The elements of a class we... |
| wetrep 5693 | On a class well-ordered by... |
| wefrc 5694 | A nonempty subclass of a c... |
| we0 5695 | Any relation is a well-ord... |
| wereu 5696 | A nonempty subset of an ` ... |
| wereu2 5697 | A nonempty subclass of an ... |
| xpeq1 5714 | Equality theorem for Carte... |
| xpss12 5715 | Subset theorem for Cartesi... |
| xpss 5716 | A Cartesian product is inc... |
| inxpssres 5717 | Intersection with a Cartes... |
| relxp 5718 | A Cartesian product is a r... |
| xpss1 5719 | Subset relation for Cartes... |
| xpss2 5720 | Subset relation for Cartes... |
| xpeq2 5721 | Equality theorem for Carte... |
| elxpi 5722 | Membership in a Cartesian ... |
| elxp 5723 | Membership in a Cartesian ... |
| elxp2 5724 | Membership in a Cartesian ... |
| xpeq12 5725 | Equality theorem for Carte... |
| xpeq1i 5726 | Equality inference for Car... |
| xpeq2i 5727 | Equality inference for Car... |
| xpeq12i 5728 | Equality inference for Car... |
| xpeq1d 5729 | Equality deduction for Car... |
| xpeq2d 5730 | Equality deduction for Car... |
| xpeq12d 5731 | Equality deduction for Car... |
| sqxpeqd 5732 | Equality deduction for a C... |
| nfxp 5733 | Bound-variable hypothesis ... |
| 0nelxp 5734 | The empty set is not a mem... |
| 0nelelxp 5735 | A member of a Cartesian pr... |
| opelxp 5736 | Ordered pair membership in... |
| opelxpi 5737 | Ordered pair membership in... |
| opelxpii 5738 | Ordered pair membership in... |
| opelxpd 5739 | Ordered pair membership in... |
| opelvv 5740 | Ordered pair membership in... |
| opelvvg 5741 | Ordered pair membership in... |
| opelxp1 5742 | The first member of an ord... |
| opelxp2 5743 | The second member of an or... |
| otelxp 5744 | Ordered triple membership ... |
| otelxp1 5745 | The first member of an ord... |
| otel3xp 5746 | An ordered triple is an el... |
| opabssxpd 5747 | An ordered-pair class abst... |
| rabxp 5748 | Class abstraction restrict... |
| brxp 5749 | Binary relation on a Carte... |
| pwvrel 5750 | A set is a binary relation... |
| pwvabrel 5751 | The powerclass of the cart... |
| brrelex12 5752 | Two classes related by a b... |
| brrelex1 5753 | If two classes are related... |
| brrelex2 5754 | If two classes are related... |
| brrelex12i 5755 | Two classes that are relat... |
| brrelex1i 5756 | The first argument of a bi... |
| brrelex2i 5757 | The second argument of a b... |
| nprrel12 5758 | Proper classes are not rel... |
| nprrel 5759 | No proper class is related... |
| 0nelrel0 5760 | A binary relation does not... |
| 0nelrel 5761 | A binary relation does not... |
| fconstmpt 5762 | Representation of a consta... |
| vtoclr 5763 | Variable to class conversi... |
| opthprc 5764 | Justification theorem for ... |
| brel 5765 | Two things in a binary rel... |
| elxp3 5766 | Membership in a Cartesian ... |
| opeliunxp 5767 | Membership in a union of C... |
| xpundi 5768 | Distributive law for Carte... |
| xpundir 5769 | Distributive law for Carte... |
| xpiundi 5770 | Distributive law for Carte... |
| xpiundir 5771 | Distributive law for Carte... |
| iunxpconst 5772 | Membership in a union of C... |
| xpun 5773 | The Cartesian product of t... |
| elvv 5774 | Membership in universal cl... |
| elvvv 5775 | Membership in universal cl... |
| elvvuni 5776 | An ordered pair contains i... |
| brinxp2 5777 | Intersection of binary rel... |
| brinxp 5778 | Intersection of binary rel... |
| opelinxp 5779 | Ordered pair element in an... |
| poinxp 5780 | Intersection of partial or... |
| soinxp 5781 | Intersection of total orde... |
| frinxp 5782 | Intersection of well-found... |
| seinxp 5783 | Intersection of set-like r... |
| weinxp 5784 | Intersection of well-order... |
| posn 5785 | Partial ordering of a sing... |
| sosn 5786 | Strict ordering on a singl... |
| frsn 5787 | Founded relation on a sing... |
| wesn 5788 | Well-ordering of a singlet... |
| elopaelxp 5789 | Membership in an ordered-p... |
| elopaelxpOLD 5790 | Obsolete version of ~ elop... |
| bropaex12 5791 | Two classes related by an ... |
| opabssxp 5792 | An abstraction relation is... |
| brab2a 5793 | The law of concretion for ... |
| optocl 5794 | Implicit substitution of c... |
| 2optocl 5795 | Implicit substitution of c... |
| 3optocl 5796 | Implicit substitution of c... |
| opbrop 5797 | Ordered pair membership in... |
| 0xp 5798 | The Cartesian product with... |
| csbxp 5799 | Distribute proper substitu... |
| releq 5800 | Equality theorem for the r... |
| releqi 5801 | Equality inference for the... |
| releqd 5802 | Equality deduction for the... |
| nfrel 5803 | Bound-variable hypothesis ... |
| sbcrel 5804 | Distribute proper substitu... |
| relss 5805 | Subclass theorem for relat... |
| ssrel 5806 | A subclass relationship de... |
| ssrelOLD 5807 | Obsolete version of ~ ssre... |
| eqrel 5808 | Extensionality principle f... |
| ssrel2 5809 | A subclass relationship de... |
| ssrel3 5810 | Subclass relation in anoth... |
| relssi 5811 | Inference from subclass pr... |
| relssdv 5812 | Deduction from subclass pr... |
| eqrelriv 5813 | Inference from extensional... |
| eqrelriiv 5814 | Inference from extensional... |
| eqbrriv 5815 | Inference from extensional... |
| eqrelrdv 5816 | Deduce equality of relatio... |
| eqbrrdv 5817 | Deduction from extensional... |
| eqbrrdiv 5818 | Deduction from extensional... |
| eqrelrdv2 5819 | A version of ~ eqrelrdv . ... |
| ssrelrel 5820 | A subclass relationship de... |
| eqrelrel 5821 | Extensionality principle f... |
| elrel 5822 | A member of a relation is ... |
| rel0 5823 | The empty set is a relatio... |
| nrelv 5824 | The universal class is not... |
| relsng 5825 | A singleton is a relation ... |
| relsnb 5826 | An at-most-singleton is a ... |
| relsnopg 5827 | A singleton of an ordered ... |
| relsn 5828 | A singleton is a relation ... |
| relsnop 5829 | A singleton of an ordered ... |
| copsex2gb 5830 | Implicit substitution infe... |
| copsex2ga 5831 | Implicit substitution infe... |
| elopaba 5832 | Membership in an ordered-p... |
| xpsspw 5833 | A Cartesian product is inc... |
| unixpss 5834 | The double class union of ... |
| relun 5835 | The union of two relations... |
| relin1 5836 | The intersection with a re... |
| relin2 5837 | The intersection with a re... |
| relinxp 5838 | Intersection with a Cartes... |
| reldif 5839 | A difference cutting down ... |
| reliun 5840 | An indexed union is a rela... |
| reliin 5841 | An indexed intersection is... |
| reluni 5842 | The union of a class is a ... |
| relint 5843 | The intersection of a clas... |
| relopabiv 5844 | A class of ordered pairs i... |
| relopabv 5845 | A class of ordered pairs i... |
| relopabi 5846 | A class of ordered pairs i... |
| relopabiALT 5847 | Alternate proof of ~ relop... |
| relopab 5848 | A class of ordered pairs i... |
| mptrel 5849 | The maps-to notation alway... |
| reli 5850 | The identity relation is a... |
| rele 5851 | The membership relation is... |
| opabid2 5852 | A relation expressed as an... |
| inopab 5853 | Intersection of two ordere... |
| difopab 5854 | Difference of two ordered-... |
| difopabOLD 5855 | Obsolete version of ~ difo... |
| inxp 5856 | Intersection of two Cartes... |
| inxpOLD 5857 | Obsolete version of ~ inxp... |
| xpindi 5858 | Distributive law for Carte... |
| xpindir 5859 | Distributive law for Carte... |
| xpiindi 5860 | Distributive law for Carte... |
| xpriindi 5861 | Distributive law for Carte... |
| eliunxp 5862 | Membership in a union of C... |
| opeliunxp2 5863 | Membership in a union of C... |
| raliunxp 5864 | Write a double restricted ... |
| rexiunxp 5865 | Write a double restricted ... |
| ralxp 5866 | Universal quantification r... |
| rexxp 5867 | Existential quantification... |
| exopxfr 5868 | Transfer ordered-pair exis... |
| exopxfr2 5869 | Transfer ordered-pair exis... |
| djussxp 5870 | Disjoint union is a subset... |
| ralxpf 5871 | Version of ~ ralxp with bo... |
| rexxpf 5872 | Version of ~ rexxp with bo... |
| iunxpf 5873 | Indexed union on a Cartesi... |
| opabbi2dv 5874 | Deduce equality of a relat... |
| relop 5875 | A necessary and sufficient... |
| ideqg 5876 | For sets, the identity rel... |
| ideq 5877 | For sets, the identity rel... |
| ididg 5878 | A set is identical to itse... |
| issetid 5879 | Two ways of expressing set... |
| coss1 5880 | Subclass theorem for compo... |
| coss2 5881 | Subclass theorem for compo... |
| coeq1 5882 | Equality theorem for compo... |
| coeq2 5883 | Equality theorem for compo... |
| coeq1i 5884 | Equality inference for com... |
| coeq2i 5885 | Equality inference for com... |
| coeq1d 5886 | Equality deduction for com... |
| coeq2d 5887 | Equality deduction for com... |
| coeq12i 5888 | Equality inference for com... |
| coeq12d 5889 | Equality deduction for com... |
| nfco 5890 | Bound-variable hypothesis ... |
| brcog 5891 | Ordered pair membership in... |
| opelco2g 5892 | Ordered pair membership in... |
| brcogw 5893 | Ordered pair membership in... |
| eqbrrdva 5894 | Deduction from extensional... |
| brco 5895 | Binary relation on a compo... |
| opelco 5896 | Ordered pair membership in... |
| cnvss 5897 | Subset theorem for convers... |
| cnveq 5898 | Equality theorem for conve... |
| cnveqi 5899 | Equality inference for con... |
| cnveqd 5900 | Equality deduction for con... |
| elcnv 5901 | Membership in a converse r... |
| elcnv2 5902 | Membership in a converse r... |
| nfcnv 5903 | Bound-variable hypothesis ... |
| brcnvg 5904 | The converse of a binary r... |
| opelcnvg 5905 | Ordered-pair membership in... |
| opelcnv 5906 | Ordered-pair membership in... |
| brcnv 5907 | The converse of a binary r... |
| csbcnv 5908 | Move class substitution in... |
| csbcnvgALT 5909 | Move class substitution in... |
| cnvco 5910 | Distributive law of conver... |
| cnvuni 5911 | The converse of a class un... |
| dfdm3 5912 | Alternate definition of do... |
| dfrn2 5913 | Alternate definition of ra... |
| dfrn3 5914 | Alternate definition of ra... |
| elrn2g 5915 | Membership in a range. (C... |
| elrng 5916 | Membership in a range. (C... |
| elrn2 5917 | Membership in a range. (C... |
| elrn 5918 | Membership in a range. (C... |
| ssrelrn 5919 | If a relation is a subset ... |
| dfdm4 5920 | Alternate definition of do... |
| dfdmf 5921 | Definition of domain, usin... |
| csbdm 5922 | Distribute proper substitu... |
| eldmg 5923 | Domain membership. Theore... |
| eldm2g 5924 | Domain membership. Theore... |
| eldm 5925 | Membership in a domain. T... |
| eldm2 5926 | Membership in a domain. T... |
| dmss 5927 | Subset theorem for domain.... |
| dmeq 5928 | Equality theorem for domai... |
| dmeqi 5929 | Equality inference for dom... |
| dmeqd 5930 | Equality deduction for dom... |
| opeldmd 5931 | Membership of first of an ... |
| opeldm 5932 | Membership of first of an ... |
| breldm 5933 | Membership of first of a b... |
| breldmg 5934 | Membership of first of a b... |
| dmun 5935 | The domain of a union is t... |
| dmin 5936 | The domain of an intersect... |
| breldmd 5937 | Membership of first of a b... |
| dmiun 5938 | The domain of an indexed u... |
| dmuni 5939 | The domain of a union. Pa... |
| dmopab 5940 | The domain of a class of o... |
| dmopabelb 5941 | A set is an element of the... |
| dmopab2rex 5942 | The domain of an ordered p... |
| dmopabss 5943 | Upper bound for the domain... |
| dmopab3 5944 | The domain of a restricted... |
| dm0 5945 | The domain of the empty se... |
| dmi 5946 | The domain of the identity... |
| dmv 5947 | The domain of the universe... |
| dmep 5948 | The domain of the membersh... |
| dm0rn0 5949 | An empty domain is equival... |
| rn0 5950 | The range of the empty set... |
| rnep 5951 | The range of the membershi... |
| reldm0 5952 | A relation is empty iff it... |
| dmxp 5953 | The domain of a Cartesian ... |
| dmxpOLD 5954 | Obsolete version of ~ dmxp... |
| dmxpid 5955 | The domain of a Cartesian ... |
| dmxpin 5956 | The domain of the intersec... |
| xpid11 5957 | The Cartesian square is a ... |
| dmcnvcnv 5958 | The domain of the double c... |
| rncnvcnv 5959 | The range of the double co... |
| elreldm 5960 | The first member of an ord... |
| rneq 5961 | Equality theorem for range... |
| rneqi 5962 | Equality inference for ran... |
| rneqd 5963 | Equality deduction for ran... |
| rnss 5964 | Subset theorem for range. ... |
| rnssi 5965 | Subclass inference for ran... |
| brelrng 5966 | The second argument of a b... |
| brelrn 5967 | The second argument of a b... |
| opelrn 5968 | Membership of second membe... |
| releldm 5969 | The first argument of a bi... |
| relelrn 5970 | The second argument of a b... |
| releldmb 5971 | Membership in a domain. (... |
| relelrnb 5972 | Membership in a range. (C... |
| releldmi 5973 | The first argument of a bi... |
| relelrni 5974 | The second argument of a b... |
| dfrnf 5975 | Definition of range, using... |
| nfdm 5976 | Bound-variable hypothesis ... |
| nfrn 5977 | Bound-variable hypothesis ... |
| dmiin 5978 | Domain of an intersection.... |
| rnopab 5979 | The range of a class of or... |
| rnopabss 5980 | Upper bound for the range ... |
| rnopab3 5981 | The range of a restricted ... |
| rnmpt 5982 | The range of a function in... |
| elrnmpt 5983 | The range of a function in... |
| elrnmpt1s 5984 | Elementhood in an image se... |
| elrnmpt1 5985 | Elementhood in an image se... |
| elrnmptg 5986 | Membership in the range of... |
| elrnmpti 5987 | Membership in the range of... |
| elrnmptd 5988 | The range of a function in... |
| elrnmpt1d 5989 | Elementhood in an image se... |
| elrnmptdv 5990 | Elementhood in the range o... |
| elrnmpt2d 5991 | Elementhood in the range o... |
| dfiun3g 5992 | Alternate definition of in... |
| dfiin3g 5993 | Alternate definition of in... |
| dfiun3 5994 | Alternate definition of in... |
| dfiin3 5995 | Alternate definition of in... |
| riinint 5996 | Express a relative indexed... |
| relrn0 5997 | A relation is empty iff it... |
| dmrnssfld 5998 | The domain and range of a ... |
| dmcoss 5999 | Domain of a composition. ... |
| rncoss 6000 | Range of a composition. (... |
| dmcosseq 6001 | Domain of a composition. ... |
| dmcosseqOLD 6002 | Obsolete version of ~ dmco... |
| dmcoeq 6003 | Domain of a composition. ... |
| rncoeq 6004 | Range of a composition. (... |
| reseq1 6005 | Equality theorem for restr... |
| reseq2 6006 | Equality theorem for restr... |
| reseq1i 6007 | Equality inference for res... |
| reseq2i 6008 | Equality inference for res... |
| reseq12i 6009 | Equality inference for res... |
| reseq1d 6010 | Equality deduction for res... |
| reseq2d 6011 | Equality deduction for res... |
| reseq12d 6012 | Equality deduction for res... |
| nfres 6013 | Bound-variable hypothesis ... |
| csbres 6014 | Distribute proper substitu... |
| res0 6015 | A restriction to the empty... |
| dfres3 6016 | Alternate definition of re... |
| opelres 6017 | Ordered pair elementhood i... |
| brres 6018 | Binary relation on a restr... |
| opelresi 6019 | Ordered pair membership in... |
| brresi 6020 | Binary relation on a restr... |
| opres 6021 | Ordered pair membership in... |
| resieq 6022 | A restricted identity rela... |
| opelidres 6023 | ` <. A , A >. ` belongs to... |
| resres 6024 | The restriction of a restr... |
| resundi 6025 | Distributive law for restr... |
| resundir 6026 | Distributive law for restr... |
| resindi 6027 | Class restriction distribu... |
| resindir 6028 | Class restriction distribu... |
| inres 6029 | Move intersection into cla... |
| resdifcom 6030 | Commutative law for restri... |
| resiun1 6031 | Distribution of restrictio... |
| resiun2 6032 | Distribution of restrictio... |
| resss 6033 | A class includes its restr... |
| rescom 6034 | Commutative law for restri... |
| ssres 6035 | Subclass theorem for restr... |
| ssres2 6036 | Subclass theorem for restr... |
| relres 6037 | A restriction is a relatio... |
| resabs1 6038 | Absorption law for restric... |
| resabs1d 6039 | Absorption law for restric... |
| resabs2 6040 | Absorption law for restric... |
| residm 6041 | Idempotent law for restric... |
| dmresss 6042 | The domain of a restrictio... |
| dmres 6043 | The domain of a restrictio... |
| ssdmres 6044 | A domain restricted to a s... |
| dmresexg 6045 | The domain of a restrictio... |
| resima 6046 | A restriction to an image.... |
| resima2 6047 | Image under a restricted c... |
| rnresss 6048 | The range of a restriction... |
| xpssres 6049 | Restriction of a constant ... |
| elinxp 6050 | Membership in an intersect... |
| elres 6051 | Membership in a restrictio... |
| elsnres 6052 | Membership in restriction ... |
| relssres 6053 | Simplification law for res... |
| dmressnsn 6054 | The domain of a restrictio... |
| eldmressnsn 6055 | The element of the domain ... |
| eldmeldmressn 6056 | An element of the domain (... |
| resdm 6057 | A relation restricted to i... |
| resexg 6058 | The restriction of a set i... |
| resexd 6059 | The restriction of a set i... |
| resex 6060 | The restriction of a set i... |
| resindm 6061 | When restricting a relatio... |
| resdmdfsn 6062 | Restricting a relation to ... |
| reldisjun 6063 | Split a relation into two ... |
| relresdm1 6064 | Restriction of a disjoint ... |
| resopab 6065 | Restriction of a class abs... |
| iss 6066 | A subclass of the identity... |
| resopab2 6067 | Restriction of a class abs... |
| resmpt 6068 | Restriction of the mapping... |
| resmpt3 6069 | Unconditional restriction ... |
| resmptf 6070 | Restriction of the mapping... |
| resmptd 6071 | Restriction of the mapping... |
| dfres2 6072 | Alternate definition of th... |
| mptss 6073 | Sufficient condition for i... |
| elimampt 6074 | Membership in the image of... |
| elidinxp 6075 | Characterization of the el... |
| elidinxpid 6076 | Characterization of the el... |
| elrid 6077 | Characterization of the el... |
| idinxpres 6078 | The intersection of the id... |
| idinxpresid 6079 | The intersection of the id... |
| idssxp 6080 | A diagonal set as a subset... |
| opabresid 6081 | The restricted identity re... |
| mptresid 6082 | The restricted identity re... |
| dmresi 6083 | The domain of a restricted... |
| restidsing 6084 | Restriction of the identit... |
| iresn0n0 6085 | The identity function rest... |
| imaeq1 6086 | Equality theorem for image... |
| imaeq2 6087 | Equality theorem for image... |
| imaeq1i 6088 | Equality theorem for image... |
| imaeq2i 6089 | Equality theorem for image... |
| imaeq1d 6090 | Equality theorem for image... |
| imaeq2d 6091 | Equality theorem for image... |
| imaeq12d 6092 | Equality theorem for image... |
| dfima2 6093 | Alternate definition of im... |
| dfima3 6094 | Alternate definition of im... |
| elimag 6095 | Membership in an image. T... |
| elima 6096 | Membership in an image. T... |
| elima2 6097 | Membership in an image. T... |
| elima3 6098 | Membership in an image. T... |
| nfima 6099 | Bound-variable hypothesis ... |
| nfimad 6100 | Deduction version of bound... |
| imadmrn 6101 | The image of the domain of... |
| imassrn 6102 | The image of a class is a ... |
| mptima 6103 | Image of a function in map... |
| mptimass 6104 | Image of a function in map... |
| imai 6105 | Image under the identity r... |
| rnresi 6106 | The range of the restricte... |
| resiima 6107 | The image of a restriction... |
| ima0 6108 | Image of the empty set. T... |
| 0ima 6109 | Image under the empty rela... |
| csbima12 6110 | Move class substitution in... |
| imadisj 6111 | A class whose image under ... |
| imadisjlnd 6112 | Deduction form of one nega... |
| cnvimass 6113 | A preimage under any class... |
| cnvimarndm 6114 | The preimage of the range ... |
| imasng 6115 | The image of a singleton. ... |
| relimasn 6116 | The image of a singleton. ... |
| elrelimasn 6117 | Elementhood in the image o... |
| elimasng1 6118 | Membership in an image of ... |
| elimasn1 6119 | Membership in an image of ... |
| elimasng 6120 | Membership in an image of ... |
| elimasn 6121 | Membership in an image of ... |
| elimasngOLD 6122 | Obsolete version of ~ elim... |
| elimasni 6123 | Membership in an image of ... |
| args 6124 | Two ways to express the cl... |
| elinisegg 6125 | Membership in the inverse ... |
| eliniseg 6126 | Membership in the inverse ... |
| epin 6127 | Any set is equal to its pr... |
| epini 6128 | Any set is equal to its pr... |
| iniseg 6129 | An idiom that signifies an... |
| inisegn0 6130 | Nonemptiness of an initial... |
| dffr3 6131 | Alternate definition of we... |
| dfse2 6132 | Alternate definition of se... |
| imass1 6133 | Subset theorem for image. ... |
| imass2 6134 | Subset theorem for image. ... |
| ndmima 6135 | The image of a singleton o... |
| relcnv 6136 | A converse is a relation. ... |
| relbrcnvg 6137 | When ` R ` is a relation, ... |
| eliniseg2 6138 | Eliminate the class existe... |
| relbrcnv 6139 | When ` R ` is a relation, ... |
| relco 6140 | A composition is a relatio... |
| cotrg 6141 | Two ways of saying that th... |
| cotrgOLD 6142 | Obsolete version of ~ cotr... |
| cotrgOLDOLD 6143 | Obsolete version of ~ cotr... |
| cotr 6144 | Two ways of saying a relat... |
| idrefALT 6145 | Alternate proof of ~ idref... |
| cnvsym 6146 | Two ways of saying a relat... |
| cnvsymOLD 6147 | Obsolete proof of ~ cnvsym... |
| cnvsymOLDOLD 6148 | Obsolete proof of ~ cnvsym... |
| intasym 6149 | Two ways of saying a relat... |
| asymref 6150 | Two ways of saying a relat... |
| asymref2 6151 | Two ways of saying a relat... |
| intirr 6152 | Two ways of saying a relat... |
| brcodir 6153 | Two ways of saying that tw... |
| codir 6154 | Two ways of saying a relat... |
| qfto 6155 | A quantifier-free way of e... |
| xpidtr 6156 | A Cartesian square is a tr... |
| trin2 6157 | The intersection of two tr... |
| poirr2 6158 | A partial order is irrefle... |
| trinxp 6159 | The relation induced by a ... |
| soirri 6160 | A strict order relation is... |
| sotri 6161 | A strict order relation is... |
| son2lpi 6162 | A strict order relation ha... |
| sotri2 6163 | A transitivity relation. ... |
| sotri3 6164 | A transitivity relation. ... |
| poleloe 6165 | Express "less than or equa... |
| poltletr 6166 | Transitive law for general... |
| somin1 6167 | Property of a minimum in a... |
| somincom 6168 | Commutativity of minimum i... |
| somin2 6169 | Property of a minimum in a... |
| soltmin 6170 | Being less than a minimum,... |
| cnvopab 6171 | The converse of a class ab... |
| cnvopabOLD 6172 | Obsolete version of ~ cnvo... |
| mptcnv 6173 | The converse of a mapping ... |
| cnv0 6174 | The converse of the empty ... |
| cnvi 6175 | The converse of the identi... |
| cnvun 6176 | The converse of a union is... |
| cnvdif 6177 | Distributive law for conve... |
| cnvin 6178 | Distributive law for conve... |
| rnun 6179 | Distributive law for range... |
| rnin 6180 | The range of an intersecti... |
| rniun 6181 | The range of an indexed un... |
| rnuni 6182 | The range of a union. Par... |
| imaundi 6183 | Distributive law for image... |
| imaundir 6184 | The image of a union. (Co... |
| cnvimassrndm 6185 | The preimage of a superset... |
| dminss 6186 | An upper bound for interse... |
| imainss 6187 | An upper bound for interse... |
| inimass 6188 | The image of an intersecti... |
| inimasn 6189 | The intersection of the im... |
| cnvxp 6190 | The converse of a Cartesia... |
| xp0 6191 | The Cartesian product with... |
| xpnz 6192 | The Cartesian product of n... |
| xpeq0 6193 | At least one member of an ... |
| xpdisj1 6194 | Cartesian products with di... |
| xpdisj2 6195 | Cartesian products with di... |
| xpsndisj 6196 | Cartesian products with tw... |
| difxp 6197 | Difference of Cartesian pr... |
| difxp1 6198 | Difference law for Cartesi... |
| difxp2 6199 | Difference law for Cartesi... |
| djudisj 6200 | Disjoint unions with disjo... |
| xpdifid 6201 | The set of distinct couple... |
| resdisj 6202 | A double restriction to di... |
| rnxp 6203 | The range of a Cartesian p... |
| dmxpss 6204 | The domain of a Cartesian ... |
| rnxpss 6205 | The range of a Cartesian p... |
| rnxpid 6206 | The range of a Cartesian s... |
| ssxpb 6207 | A Cartesian product subcla... |
| xp11 6208 | The Cartesian product of n... |
| xpcan 6209 | Cancellation law for Carte... |
| xpcan2 6210 | Cancellation law for Carte... |
| ssrnres 6211 | Two ways to express surjec... |
| rninxp 6212 | Two ways to express surjec... |
| dminxp 6213 | Two ways to express totali... |
| imainrect 6214 | Image by a restricted and ... |
| xpima 6215 | Direct image by a Cartesia... |
| xpima1 6216 | Direct image by a Cartesia... |
| xpima2 6217 | Direct image by a Cartesia... |
| xpimasn 6218 | Direct image of a singleto... |
| sossfld 6219 | The base set of a strict o... |
| sofld 6220 | The base set of a nonempty... |
| cnvcnv3 6221 | The set of all ordered pai... |
| dfrel2 6222 | Alternate definition of re... |
| dfrel4v 6223 | A relation can be expresse... |
| dfrel4 6224 | A relation can be expresse... |
| cnvcnv 6225 | The double converse of a c... |
| cnvcnv2 6226 | The double converse of a c... |
| cnvcnvss 6227 | The double converse of a c... |
| cnvrescnv 6228 | Two ways to express the co... |
| cnveqb 6229 | Equality theorem for conve... |
| cnveq0 6230 | A relation empty iff its c... |
| dfrel3 6231 | Alternate definition of re... |
| elid 6232 | Characterization of the el... |
| dmresv 6233 | The domain of a universal ... |
| rnresv 6234 | The range of a universal r... |
| dfrn4 6235 | Range defined in terms of ... |
| csbrn 6236 | Distribute proper substitu... |
| rescnvcnv 6237 | The restriction of the dou... |
| cnvcnvres 6238 | The double converse of the... |
| imacnvcnv 6239 | The image of the double co... |
| dmsnn0 6240 | The domain of a singleton ... |
| rnsnn0 6241 | The range of a singleton i... |
| dmsn0 6242 | The domain of the singleto... |
| cnvsn0 6243 | The converse of the single... |
| dmsn0el 6244 | The domain of a singleton ... |
| relsn2 6245 | A singleton is a relation ... |
| dmsnopg 6246 | The domain of a singleton ... |
| dmsnopss 6247 | The domain of a singleton ... |
| dmpropg 6248 | The domain of an unordered... |
| dmsnop 6249 | The domain of a singleton ... |
| dmprop 6250 | The domain of an unordered... |
| dmtpop 6251 | The domain of an unordered... |
| cnvcnvsn 6252 | Double converse of a singl... |
| dmsnsnsn 6253 | The domain of the singleto... |
| rnsnopg 6254 | The range of a singleton o... |
| rnpropg 6255 | The range of a pair of ord... |
| cnvsng 6256 | Converse of a singleton of... |
| rnsnop 6257 | The range of a singleton o... |
| op1sta 6258 | Extract the first member o... |
| cnvsn 6259 | Converse of a singleton of... |
| op2ndb 6260 | Extract the second member ... |
| op2nda 6261 | Extract the second member ... |
| opswap 6262 | Swap the members of an ord... |
| cnvresima 6263 | An image under the convers... |
| resdm2 6264 | A class restricted to its ... |
| resdmres 6265 | Restriction to the domain ... |
| resresdm 6266 | A restriction by an arbitr... |
| imadmres 6267 | The image of the domain of... |
| resdmss 6268 | Subset relationship for th... |
| resdifdi 6269 | Distributive law for restr... |
| resdifdir 6270 | Distributive law for restr... |
| mptpreima 6271 | The preimage of a function... |
| mptiniseg 6272 | Converse singleton image o... |
| dmmpt 6273 | The domain of the mapping ... |
| dmmptss 6274 | The domain of a mapping is... |
| dmmptg 6275 | The domain of the mapping ... |
| rnmpt0f 6276 | The range of a function in... |
| rnmptn0 6277 | The range of a function in... |
| dfco2 6278 | Alternate definition of a ... |
| dfco2a 6279 | Generalization of ~ dfco2 ... |
| coundi 6280 | Class composition distribu... |
| coundir 6281 | Class composition distribu... |
| cores 6282 | Restricted first member of... |
| resco 6283 | Associative law for the re... |
| imaco 6284 | Image of the composition o... |
| rnco 6285 | The range of the compositi... |
| rnco2 6286 | The range of the compositi... |
| dmco 6287 | The domain of a compositio... |
| coeq0 6288 | A composition of two relat... |
| coiun 6289 | Composition with an indexe... |
| cocnvcnv1 6290 | A composition is not affec... |
| cocnvcnv2 6291 | A composition is not affec... |
| cores2 6292 | Absorption of a reverse (p... |
| co02 6293 | Composition with the empty... |
| co01 6294 | Composition with the empty... |
| coi1 6295 | Composition with the ident... |
| coi2 6296 | Composition with the ident... |
| coires1 6297 | Composition with a restric... |
| coass 6298 | Associative law for class ... |
| relcnvtrg 6299 | General form of ~ relcnvtr... |
| relcnvtr 6300 | A relation is transitive i... |
| relssdmrn 6301 | A relation is included in ... |
| relssdmrnOLD 6302 | Obsolete version of ~ rels... |
| resssxp 6303 | If the ` R ` -image of a c... |
| cnvssrndm 6304 | The converse is a subset o... |
| cossxp 6305 | Composition as a subset of... |
| relrelss 6306 | Two ways to describe the s... |
| unielrel 6307 | The membership relation fo... |
| relfld 6308 | The double union of a rela... |
| relresfld 6309 | Restriction of a relation ... |
| relcoi2 6310 | Composition with the ident... |
| relcoi1 6311 | Composition with the ident... |
| unidmrn 6312 | The double union of the co... |
| relcnvfld 6313 | if ` R ` is a relation, it... |
| dfdm2 6314 | Alternate definition of do... |
| unixp 6315 | The double class union of ... |
| unixp0 6316 | A Cartesian product is emp... |
| unixpid 6317 | Field of a Cartesian squar... |
| ressn 6318 | Restriction of a class to ... |
| cnviin 6319 | The converse of an interse... |
| cnvpo 6320 | The converse of a partial ... |
| cnvso 6321 | The converse of a strict o... |
| xpco 6322 | Composition of two Cartesi... |
| xpcoid 6323 | Composition of two Cartesi... |
| elsnxp 6324 | Membership in a Cartesian ... |
| reu3op 6325 | There is a unique ordered ... |
| reuop 6326 | There is a unique ordered ... |
| opreu2reurex 6327 | There is a unique ordered ... |
| opreu2reu 6328 | If there is a unique order... |
| dfpo2 6329 | Quantifier-free definition... |
| csbcog 6330 | Distribute proper substitu... |
| snres0 6331 | Condition for restriction ... |
| imaindm 6332 | The image is unaffected by... |
| predeq123 6335 | Equality theorem for the p... |
| predeq1 6336 | Equality theorem for the p... |
| predeq2 6337 | Equality theorem for the p... |
| predeq3 6338 | Equality theorem for the p... |
| nfpred 6339 | Bound-variable hypothesis ... |
| csbpredg 6340 | Move class substitution in... |
| predpredss 6341 | If ` A ` is a subset of ` ... |
| predss 6342 | The predecessor class of `... |
| sspred 6343 | Another subset/predecessor... |
| dfpred2 6344 | An alternate definition of... |
| dfpred3 6345 | An alternate definition of... |
| dfpred3g 6346 | An alternate definition of... |
| elpredgg 6347 | Membership in a predecesso... |
| elpredg 6348 | Membership in a predecesso... |
| elpredimg 6349 | Membership in a predecesso... |
| elpredim 6350 | Membership in a predecesso... |
| elpred 6351 | Membership in a predecesso... |
| predexg 6352 | The predecessor class exis... |
| predasetexOLD 6353 | Obsolete form of ~ predexg... |
| dffr4 6354 | Alternate definition of we... |
| predel 6355 | Membership in the predeces... |
| predtrss 6356 | If ` R ` is transitive ove... |
| predpo 6357 | Property of the predecesso... |
| predso 6358 | Property of the predecesso... |
| setlikespec 6359 | If ` R ` is set-like in ` ... |
| predidm 6360 | Idempotent law for the pre... |
| predin 6361 | Intersection law for prede... |
| predun 6362 | Union law for predecessor ... |
| preddif 6363 | Difference law for predece... |
| predep 6364 | The predecessor under the ... |
| trpred 6365 | The class of predecessors ... |
| preddowncl 6366 | A property of classes that... |
| predpoirr 6367 | Given a partial ordering, ... |
| predfrirr 6368 | Given a well-founded relat... |
| pred0 6369 | The predecessor class over... |
| dfse3 6370 | Alternate definition of se... |
| predrelss 6371 | Subset carries from relati... |
| predprc 6372 | The predecessor of a prope... |
| predres 6373 | Predecessor class is unaff... |
| frpomin 6374 | Every nonempty (possibly p... |
| frpomin2 6375 | Every nonempty (possibly p... |
| frpoind 6376 | The principle of well-foun... |
| frpoinsg 6377 | Well-Founded Induction Sch... |
| frpoins2fg 6378 | Well-Founded Induction sch... |
| frpoins2g 6379 | Well-Founded Induction sch... |
| frpoins3g 6380 | Well-Founded Induction sch... |
| tz6.26 6381 | All nonempty subclasses of... |
| tz6.26OLD 6382 | Obsolete proof of ~ tz6.26... |
| tz6.26i 6383 | All nonempty subclasses of... |
| wfi 6384 | The Principle of Well-Orde... |
| wfiOLD 6385 | Obsolete proof of ~ wfi as... |
| wfii 6386 | The Principle of Well-Orde... |
| wfisg 6387 | Well-Ordered Induction Sch... |
| wfisgOLD 6388 | Obsolete version of ~ wfis... |
| wfis 6389 | Well-Ordered Induction Sch... |
| wfis2fg 6390 | Well-Ordered Induction Sch... |
| wfis2fgOLD 6391 | Obsolete version of ~ wfis... |
| wfis2f 6392 | Well-Ordered Induction sch... |
| wfis2g 6393 | Well-Ordered Induction Sch... |
| wfis2 6394 | Well-Ordered Induction sch... |
| wfis3 6395 | Well-Ordered Induction sch... |
| ordeq 6404 | Equality theorem for the o... |
| elong 6405 | An ordinal number is an or... |
| elon 6406 | An ordinal number is an or... |
| eloni 6407 | An ordinal number has the ... |
| elon2 6408 | An ordinal number is an or... |
| limeq 6409 | Equality theorem for the l... |
| ordwe 6410 | Membership well-orders eve... |
| ordtr 6411 | An ordinal class is transi... |
| ordfr 6412 | Membership is well-founded... |
| ordelss 6413 | An element of an ordinal c... |
| trssord 6414 | A transitive subclass of a... |
| ordirr 6415 | No ordinal class is a memb... |
| nordeq 6416 | A member of an ordinal cla... |
| ordn2lp 6417 | An ordinal class cannot be... |
| tz7.5 6418 | A nonempty subclass of an ... |
| ordelord 6419 | An element of an ordinal c... |
| tron 6420 | The class of all ordinal n... |
| ordelon 6421 | An element of an ordinal c... |
| onelon 6422 | An element of an ordinal n... |
| tz7.7 6423 | A transitive class belongs... |
| ordelssne 6424 | For ordinal classes, membe... |
| ordelpss 6425 | For ordinal classes, membe... |
| ordsseleq 6426 | For ordinal classes, inclu... |
| ordin 6427 | The intersection of two or... |
| onin 6428 | The intersection of two or... |
| ordtri3or 6429 | A trichotomy law for ordin... |
| ordtri1 6430 | A trichotomy law for ordin... |
| ontri1 6431 | A trichotomy law for ordin... |
| ordtri2 6432 | A trichotomy law for ordin... |
| ordtri3 6433 | A trichotomy law for ordin... |
| ordtri4 6434 | A trichotomy law for ordin... |
| orddisj 6435 | An ordinal class and its s... |
| onfr 6436 | The ordinal class is well-... |
| onelpss 6437 | Relationship between membe... |
| onsseleq 6438 | Relationship between subse... |
| onelss 6439 | An element of an ordinal n... |
| ordtr1 6440 | Transitive law for ordinal... |
| ordtr2 6441 | Transitive law for ordinal... |
| ordtr3 6442 | Transitive law for ordinal... |
| ontr1 6443 | Transitive law for ordinal... |
| ontr2 6444 | Transitive law for ordinal... |
| onelssex 6445 | Ordinal less than is equiv... |
| ordunidif 6446 | The union of an ordinal st... |
| ordintdif 6447 | If ` B ` is smaller than `... |
| onintss 6448 | If a property is true for ... |
| oneqmini 6449 | A way to show that an ordi... |
| ord0 6450 | The empty set is an ordina... |
| 0elon 6451 | The empty set is an ordina... |
| ord0eln0 6452 | A nonempty ordinal contain... |
| on0eln0 6453 | An ordinal number contains... |
| dflim2 6454 | An alternate definition of... |
| inton 6455 | The intersection of the cl... |
| nlim0 6456 | The empty set is not a lim... |
| limord 6457 | A limit ordinal is ordinal... |
| limuni 6458 | A limit ordinal is its own... |
| limuni2 6459 | The union of a limit ordin... |
| 0ellim 6460 | A limit ordinal contains t... |
| limelon 6461 | A limit ordinal class that... |
| onn0 6462 | The class of all ordinal n... |
| suceq 6463 | Equality of successors. (... |
| elsuci 6464 | Membership in a successor.... |
| elsucg 6465 | Membership in a successor.... |
| elsuc2g 6466 | Variant of membership in a... |
| elsuc 6467 | Membership in a successor.... |
| elsuc2 6468 | Membership in a successor.... |
| nfsuc 6469 | Bound-variable hypothesis ... |
| elelsuc 6470 | Membership in a successor.... |
| sucel 6471 | Membership of a successor ... |
| suc0 6472 | The successor of the empty... |
| sucprc 6473 | A proper class is its own ... |
| unisucs 6474 | The union of the successor... |
| unisucg 6475 | A transitive class is equa... |
| unisuc 6476 | A transitive class is equa... |
| sssucid 6477 | A class is included in its... |
| sucidg 6478 | Part of Proposition 7.23 o... |
| sucid 6479 | A set belongs to its succe... |
| nsuceq0 6480 | No successor is empty. (C... |
| eqelsuc 6481 | A set belongs to the succe... |
| iunsuc 6482 | Inductive definition for t... |
| suctr 6483 | The successor of a transit... |
| trsuc 6484 | A set whose successor belo... |
| trsucss 6485 | A member of the successor ... |
| ordsssuc 6486 | An ordinal is a subset of ... |
| onsssuc 6487 | A subset of an ordinal num... |
| ordsssuc2 6488 | An ordinal subset of an or... |
| onmindif 6489 | When its successor is subt... |
| ordnbtwn 6490 | There is no set between an... |
| onnbtwn 6491 | There is no set between an... |
| sucssel 6492 | A set whose successor is a... |
| orddif 6493 | Ordinal derived from its s... |
| orduniss 6494 | An ordinal class includes ... |
| ordtri2or 6495 | A trichotomy law for ordin... |
| ordtri2or2 6496 | A trichotomy law for ordin... |
| ordtri2or3 6497 | A consequence of total ord... |
| ordelinel 6498 | The intersection of two or... |
| ordssun 6499 | Property of a subclass of ... |
| ordequn 6500 | The maximum (i.e. union) o... |
| ordun 6501 | The maximum (i.e., union) ... |
| onunel 6502 | The union of two ordinals ... |
| ordunisssuc 6503 | A subclass relationship fo... |
| suc11 6504 | The successor operation be... |
| onun2 6505 | The union of two ordinals ... |
| ontr 6506 | An ordinal number is a tra... |
| onunisuc 6507 | An ordinal number is equal... |
| onordi 6508 | An ordinal number is an or... |
| ontrciOLD 6509 | Obsolete version of ~ ontr... |
| onirri 6510 | An ordinal number is not a... |
| oneli 6511 | A member of an ordinal num... |
| onelssi 6512 | A member of an ordinal num... |
| onssneli 6513 | An ordering law for ordina... |
| onssnel2i 6514 | An ordering law for ordina... |
| onelini 6515 | An element of an ordinal n... |
| oneluni 6516 | An ordinal number equals i... |
| onunisuci 6517 | An ordinal number is equal... |
| onsseli 6518 | Subset is equivalent to me... |
| onun2i 6519 | The union of two ordinal n... |
| unizlim 6520 | An ordinal equal to its ow... |
| on0eqel 6521 | An ordinal number either e... |
| snsn0non 6522 | The singleton of the singl... |
| onxpdisj 6523 | Ordinal numbers and ordere... |
| onnev 6524 | The class of ordinal numbe... |
| iotajust 6526 | Soundness justification th... |
| dfiota2 6528 | Alternate definition for d... |
| nfiota1 6529 | Bound-variable hypothesis ... |
| nfiotadw 6530 | Deduction version of ~ nfi... |
| nfiotaw 6531 | Bound-variable hypothesis ... |
| nfiotad 6532 | Deduction version of ~ nfi... |
| nfiota 6533 | Bound-variable hypothesis ... |
| cbviotaw 6534 | Change bound variables in ... |
| cbviotavw 6535 | Change bound variables in ... |
| cbviotavwOLD 6536 | Obsolete version of ~ cbvi... |
| cbviota 6537 | Change bound variables in ... |
| cbviotav 6538 | Change bound variables in ... |
| sb8iota 6539 | Variable substitution in d... |
| iotaeq 6540 | Equality theorem for descr... |
| iotabi 6541 | Equivalence theorem for de... |
| uniabio 6542 | Part of Theorem 8.17 in [Q... |
| iotaval2 6543 | Version of ~ iotaval using... |
| iotauni2 6544 | Version of ~ iotauni using... |
| iotanul2 6545 | Version of ~ iotanul using... |
| iotaval 6546 | Theorem 8.19 in [Quine] p.... |
| iotassuni 6547 | The ` iota ` class is a su... |
| iotaex 6548 | Theorem 8.23 in [Quine] p.... |
| iotavalOLD 6549 | Obsolete version of ~ iota... |
| iotauni 6550 | Equivalence between two di... |
| iotaint 6551 | Equivalence between two di... |
| iota1 6552 | Property of iota. (Contri... |
| iotanul 6553 | Theorem 8.22 in [Quine] p.... |
| iotassuniOLD 6554 | Obsolete version of ~ iota... |
| iotaexOLD 6555 | Obsolete version of ~ iota... |
| iota4 6556 | Theorem *14.22 in [Whitehe... |
| iota4an 6557 | Theorem *14.23 in [Whitehe... |
| iota5 6558 | A method for computing iot... |
| iotabidv 6559 | Formula-building deduction... |
| iotabii 6560 | Formula-building deduction... |
| iotacl 6561 | Membership law for descrip... |
| iota2df 6562 | A condition that allows to... |
| iota2d 6563 | A condition that allows to... |
| iota2 6564 | The unique element such th... |
| iotan0 6565 | Representation of "the uni... |
| sniota 6566 | A class abstraction with a... |
| dfiota4 6567 | The ` iota ` operation usi... |
| csbiota 6568 | Class substitution within ... |
| dffun2 6585 | Alternate definition of a ... |
| dffun2OLD 6586 | Obsolete version of ~ dffu... |
| dffun2OLDOLD 6587 | Obsolete version of ~ dffu... |
| dffun6 6588 | Alternate definition of a ... |
| dffun3 6589 | Alternate definition of fu... |
| dffun3OLD 6590 | Obsolete version of ~ dffu... |
| dffun4 6591 | Alternate definition of a ... |
| dffun5 6592 | Alternate definition of fu... |
| dffun6f 6593 | Definition of function, us... |
| dffun6OLD 6594 | Obsolete version of ~ dffu... |
| funmo 6595 | A function has at most one... |
| funmoOLD 6596 | Obsolete version of ~ funm... |
| funrel 6597 | A function is a relation. ... |
| 0nelfun 6598 | A function does not contai... |
| funss 6599 | Subclass theorem for funct... |
| funeq 6600 | Equality theorem for funct... |
| funeqi 6601 | Equality inference for the... |
| funeqd 6602 | Equality deduction for the... |
| nffun 6603 | Bound-variable hypothesis ... |
| sbcfung 6604 | Distribute proper substitu... |
| funeu 6605 | There is exactly one value... |
| funeu2 6606 | There is exactly one value... |
| dffun7 6607 | Alternate definition of a ... |
| dffun8 6608 | Alternate definition of a ... |
| dffun9 6609 | Alternate definition of a ... |
| funfn 6610 | A class is a function if a... |
| funfnd 6611 | A function is a function o... |
| funi 6612 | The identity relation is a... |
| nfunv 6613 | The universal class is not... |
| funopg 6614 | A Kuratowski ordered pair ... |
| funopab 6615 | A class of ordered pairs i... |
| funopabeq 6616 | A class of ordered pairs o... |
| funopab4 6617 | A class of ordered pairs o... |
| funmpt 6618 | A function in maps-to nota... |
| funmpt2 6619 | Functionality of a class g... |
| funco 6620 | The composition of two fun... |
| funresfunco 6621 | Composition of two functio... |
| funres 6622 | A restriction of a functio... |
| funresd 6623 | A restriction of a functio... |
| funssres 6624 | The restriction of a funct... |
| fun2ssres 6625 | Equality of restrictions o... |
| funun 6626 | The union of functions wit... |
| fununmo 6627 | If the union of classes is... |
| fununfun 6628 | If the union of classes is... |
| fundif 6629 | A function with removed el... |
| funcnvsn 6630 | The converse singleton of ... |
| funsng 6631 | A singleton of an ordered ... |
| fnsng 6632 | Functionality and domain o... |
| funsn 6633 | A singleton of an ordered ... |
| funprg 6634 | A set of two pairs is a fu... |
| funtpg 6635 | A set of three pairs is a ... |
| funpr 6636 | A function with a domain o... |
| funtp 6637 | A function with a domain o... |
| fnsn 6638 | Functionality and domain o... |
| fnprg 6639 | Function with a domain of ... |
| fntpg 6640 | Function with a domain of ... |
| fntp 6641 | A function with a domain o... |
| funcnvpr 6642 | The converse pair of order... |
| funcnvtp 6643 | The converse triple of ord... |
| funcnvqp 6644 | The converse quadruple of ... |
| fun0 6645 | The empty set is a functio... |
| funcnv0 6646 | The converse of the empty ... |
| funcnvcnv 6647 | The double converse of a f... |
| funcnv2 6648 | A simpler equivalence for ... |
| funcnv 6649 | The converse of a class is... |
| funcnv3 6650 | A condition showing a clas... |
| fun2cnv 6651 | The double converse of a c... |
| svrelfun 6652 | A single-valued relation i... |
| fncnv 6653 | Single-rootedness (see ~ f... |
| fun11 6654 | Two ways of stating that `... |
| fununi 6655 | The union of a chain (with... |
| funin 6656 | The intersection with a fu... |
| funres11 6657 | The restriction of a one-t... |
| funcnvres 6658 | The converse of a restrict... |
| cnvresid 6659 | Converse of a restricted i... |
| funcnvres2 6660 | The converse of a restrict... |
| funimacnv 6661 | The image of the preimage ... |
| funimass1 6662 | A kind of contraposition l... |
| funimass2 6663 | A kind of contraposition l... |
| imadif 6664 | The image of a difference ... |
| imain 6665 | The image of an intersecti... |
| funimaexg 6666 | Axiom of Replacement using... |
| funimaexgOLD 6667 | Obsolete version of ~ funi... |
| funimaex 6668 | The image of a set under a... |
| isarep1 6669 | Part of a study of the Axi... |
| isarep1OLD 6670 | Obsolete version of ~ isar... |
| isarep2 6671 | Part of a study of the Axi... |
| fneq1 6672 | Equality theorem for funct... |
| fneq2 6673 | Equality theorem for funct... |
| fneq1d 6674 | Equality deduction for fun... |
| fneq2d 6675 | Equality deduction for fun... |
| fneq12d 6676 | Equality deduction for fun... |
| fneq12 6677 | Equality theorem for funct... |
| fneq1i 6678 | Equality inference for fun... |
| fneq2i 6679 | Equality inference for fun... |
| nffn 6680 | Bound-variable hypothesis ... |
| fnfun 6681 | A function with domain is ... |
| fnfund 6682 | A function with domain is ... |
| fnrel 6683 | A function with domain is ... |
| fndm 6684 | The domain of a function. ... |
| fndmi 6685 | The domain of a function. ... |
| fndmd 6686 | The domain of a function. ... |
| funfni 6687 | Inference to convert a fun... |
| fndmu 6688 | A function has a unique do... |
| fnbr 6689 | The first argument of bina... |
| fnop 6690 | The first argument of an o... |
| fneu 6691 | There is exactly one value... |
| fneu2 6692 | There is exactly one value... |
| fnunres1 6693 | Restriction of a disjoint ... |
| fnunres2 6694 | Restriction of a disjoint ... |
| fnun 6695 | The union of two functions... |
| fnund 6696 | The union of two functions... |
| fnunop 6697 | Extension of a function wi... |
| fncofn 6698 | Composition of a function ... |
| fnco 6699 | Composition of two functio... |
| fncoOLD 6700 | Obsolete version of ~ fnco... |
| fnresdm 6701 | A function does not change... |
| fnresdisj 6702 | A function restricted to a... |
| 2elresin 6703 | Membership in two function... |
| fnssresb 6704 | Restriction of a function ... |
| fnssres 6705 | Restriction of a function ... |
| fnssresd 6706 | Restriction of a function ... |
| fnresin1 6707 | Restriction of a function'... |
| fnresin2 6708 | Restriction of a function'... |
| fnres 6709 | An equivalence for functio... |
| idfn 6710 | The identity relation is a... |
| fnresi 6711 | The restricted identity re... |
| fnima 6712 | The image of a function's ... |
| fn0 6713 | A function with empty doma... |
| fnimadisj 6714 | A class that is disjoint w... |
| fnimaeq0 6715 | Images under a function ne... |
| dfmpt3 6716 | Alternate definition for t... |
| mptfnf 6717 | The maps-to notation defin... |
| fnmptf 6718 | The maps-to notation defin... |
| fnopabg 6719 | Functionality and domain o... |
| fnopab 6720 | Functionality and domain o... |
| mptfng 6721 | The maps-to notation defin... |
| fnmpt 6722 | The maps-to notation defin... |
| fnmptd 6723 | The maps-to notation defin... |
| mpt0 6724 | A mapping operation with e... |
| fnmpti 6725 | Functionality and domain o... |
| dmmpti 6726 | Domain of the mapping oper... |
| dmmptd 6727 | The domain of the mapping ... |
| mptun 6728 | Union of mappings which ar... |
| partfun 6729 | Rewrite a function defined... |
| feq1 6730 | Equality theorem for funct... |
| feq2 6731 | Equality theorem for funct... |
| feq3 6732 | Equality theorem for funct... |
| feq23 6733 | Equality theorem for funct... |
| feq1d 6734 | Equality deduction for fun... |
| feq2d 6735 | Equality deduction for fun... |
| feq3d 6736 | Equality deduction for fun... |
| feq12d 6737 | Equality deduction for fun... |
| feq123d 6738 | Equality deduction for fun... |
| feq123 6739 | Equality theorem for funct... |
| feq1i 6740 | Equality inference for fun... |
| feq2i 6741 | Equality inference for fun... |
| feq12i 6742 | Equality inference for fun... |
| feq23i 6743 | Equality inference for fun... |
| feq23d 6744 | Equality deduction for fun... |
| nff 6745 | Bound-variable hypothesis ... |
| sbcfng 6746 | Distribute proper substitu... |
| sbcfg 6747 | Distribute proper substitu... |
| elimf 6748 | Eliminate a mapping hypoth... |
| ffn 6749 | A mapping is a function wi... |
| ffnd 6750 | A mapping is a function wi... |
| dffn2 6751 | Any function is a mapping ... |
| ffun 6752 | A mapping is a function. ... |
| ffund 6753 | A mapping is a function, d... |
| frel 6754 | A mapping is a relation. ... |
| freld 6755 | A mapping is a relation. ... |
| frn 6756 | The range of a mapping. (... |
| frnd 6757 | Deduction form of ~ frn . ... |
| fdm 6758 | The domain of a mapping. ... |
| fdmd 6759 | Deduction form of ~ fdm . ... |
| fdmi 6760 | Inference associated with ... |
| dffn3 6761 | A function maps to its ran... |
| ffrn 6762 | A function maps to its ran... |
| ffrnb 6763 | Characterization of a func... |
| ffrnbd 6764 | A function maps to its ran... |
| fss 6765 | Expanding the codomain of ... |
| fssd 6766 | Expanding the codomain of ... |
| fssdmd 6767 | Expressing that a class is... |
| fssdm 6768 | Expressing that a class is... |
| fimass 6769 | The image of a class under... |
| fimassd 6770 | The image of a class is a ... |
| fimacnv 6771 | The preimage of the codoma... |
| fcof 6772 | Composition of a function ... |
| fco 6773 | Composition of two functio... |
| fcoOLD 6774 | Obsolete version of ~ fco ... |
| fcod 6775 | Composition of two mapping... |
| fco2 6776 | Functionality of a composi... |
| fssxp 6777 | A mapping is a class of or... |
| funssxp 6778 | Two ways of specifying a p... |
| ffdm 6779 | A mapping is a partial fun... |
| ffdmd 6780 | The domain of a function. ... |
| fdmrn 6781 | A different way to write `... |
| funcofd 6782 | Composition of two functio... |
| fco3OLD 6783 | Obsolete version of ~ func... |
| opelf 6784 | The members of an ordered ... |
| fun 6785 | The union of two functions... |
| fun2 6786 | The union of two functions... |
| fun2d 6787 | The union of functions wit... |
| fnfco 6788 | Composition of two functio... |
| fssres 6789 | Restriction of a function ... |
| fssresd 6790 | Restriction of a function ... |
| fssres2 6791 | Restriction of a restricte... |
| fresin 6792 | An identity for the mappin... |
| resasplit 6793 | If two functions agree on ... |
| fresaun 6794 | The union of two functions... |
| fresaunres2 6795 | From the union of two func... |
| fresaunres1 6796 | From the union of two func... |
| fcoi1 6797 | Composition of a mapping a... |
| fcoi2 6798 | Composition of restricted ... |
| feu 6799 | There is exactly one value... |
| fcnvres 6800 | The converse of a restrict... |
| fimacnvdisj 6801 | The preimage of a class di... |
| fint 6802 | Function into an intersect... |
| fin 6803 | Mapping into an intersecti... |
| f0 6804 | The empty function. (Cont... |
| f00 6805 | A class is a function with... |
| f0bi 6806 | A function with empty doma... |
| f0dom0 6807 | A function is empty iff it... |
| f0rn0 6808 | If there is no element in ... |
| fconst 6809 | A Cartesian product with a... |
| fconstg 6810 | A Cartesian product with a... |
| fnconstg 6811 | A Cartesian product with a... |
| fconst6g 6812 | Constant function with loo... |
| fconst6 6813 | A constant function as a m... |
| f1eq1 6814 | Equality theorem for one-t... |
| f1eq2 6815 | Equality theorem for one-t... |
| f1eq3 6816 | Equality theorem for one-t... |
| nff1 6817 | Bound-variable hypothesis ... |
| dff12 6818 | Alternate definition of a ... |
| f1f 6819 | A one-to-one mapping is a ... |
| f1fn 6820 | A one-to-one mapping is a ... |
| f1fun 6821 | A one-to-one mapping is a ... |
| f1rel 6822 | A one-to-one onto mapping ... |
| f1dm 6823 | The domain of a one-to-one... |
| f1ss 6824 | A function that is one-to-... |
| f1ssr 6825 | A function that is one-to-... |
| f1ssres 6826 | A function that is one-to-... |
| f1resf1 6827 | The restriction of an inje... |
| f1cnvcnv 6828 | Two ways to express that a... |
| f1cof1 6829 | Composition of two one-to-... |
| f1co 6830 | Composition of one-to-one ... |
| f1coOLD 6831 | Obsolete version of ~ f1co... |
| foeq1 6832 | Equality theorem for onto ... |
| foeq2 6833 | Equality theorem for onto ... |
| foeq3 6834 | Equality theorem for onto ... |
| nffo 6835 | Bound-variable hypothesis ... |
| fof 6836 | An onto mapping is a mappi... |
| fofun 6837 | An onto mapping is a funct... |
| fofn 6838 | An onto mapping is a funct... |
| forn 6839 | The codomain of an onto fu... |
| dffo2 6840 | Alternate definition of an... |
| foima 6841 | The image of the domain of... |
| dffn4 6842 | A function maps onto its r... |
| funforn 6843 | A function maps its domain... |
| fodmrnu 6844 | An onto function has uniqu... |
| fimadmfo 6845 | A function is a function o... |
| fores 6846 | Restriction of an onto fun... |
| fimadmfoALT 6847 | Alternate proof of ~ fimad... |
| focnvimacdmdm 6848 | The preimage of the codoma... |
| focofo 6849 | Composition of onto functi... |
| foco 6850 | Composition of onto functi... |
| foconst 6851 | A nonzero constant functio... |
| f1oeq1 6852 | Equality theorem for one-t... |
| f1oeq2 6853 | Equality theorem for one-t... |
| f1oeq3 6854 | Equality theorem for one-t... |
| f1oeq23 6855 | Equality theorem for one-t... |
| f1eq123d 6856 | Equality deduction for one... |
| foeq123d 6857 | Equality deduction for ont... |
| f1oeq123d 6858 | Equality deduction for one... |
| f1oeq1d 6859 | Equality deduction for one... |
| f1oeq2d 6860 | Equality deduction for one... |
| f1oeq3d 6861 | Equality deduction for one... |
| nff1o 6862 | Bound-variable hypothesis ... |
| f1of1 6863 | A one-to-one onto mapping ... |
| f1of 6864 | A one-to-one onto mapping ... |
| f1ofn 6865 | A one-to-one onto mapping ... |
| f1ofun 6866 | A one-to-one onto mapping ... |
| f1orel 6867 | A one-to-one onto mapping ... |
| f1odm 6868 | The domain of a one-to-one... |
| dff1o2 6869 | Alternate definition of on... |
| dff1o3 6870 | Alternate definition of on... |
| f1ofo 6871 | A one-to-one onto function... |
| dff1o4 6872 | Alternate definition of on... |
| dff1o5 6873 | Alternate definition of on... |
| f1orn 6874 | A one-to-one function maps... |
| f1f1orn 6875 | A one-to-one function maps... |
| f1ocnv 6876 | The converse of a one-to-o... |
| f1ocnvb 6877 | A relation is a one-to-one... |
| f1ores 6878 | The restriction of a one-t... |
| f1orescnv 6879 | The converse of a one-to-o... |
| f1imacnv 6880 | Preimage of an image. (Co... |
| foimacnv 6881 | A reverse version of ~ f1i... |
| foun 6882 | The union of two onto func... |
| f1oun 6883 | The union of two one-to-on... |
| f1un 6884 | The union of two one-to-on... |
| resdif 6885 | The restriction of a one-t... |
| resin 6886 | The restriction of a one-t... |
| f1oco 6887 | Composition of one-to-one ... |
| f1cnv 6888 | The converse of an injecti... |
| funcocnv2 6889 | Composition with the conve... |
| fococnv2 6890 | The composition of an onto... |
| f1ococnv2 6891 | The composition of a one-t... |
| f1cocnv2 6892 | Composition of an injectiv... |
| f1ococnv1 6893 | The composition of a one-t... |
| f1cocnv1 6894 | Composition of an injectiv... |
| funcoeqres 6895 | Express a constraint on a ... |
| f1ssf1 6896 | A subset of an injective f... |
| f10 6897 | The empty set maps one-to-... |
| f10d 6898 | The empty set maps one-to-... |
| f1o00 6899 | One-to-one onto mapping of... |
| fo00 6900 | Onto mapping of the empty ... |
| f1o0 6901 | One-to-one onto mapping of... |
| f1oi 6902 | A restriction of the ident... |
| f1ovi 6903 | The identity relation is a... |
| f1osn 6904 | A singleton of an ordered ... |
| f1osng 6905 | A singleton of an ordered ... |
| f1sng 6906 | A singleton of an ordered ... |
| fsnd 6907 | A singleton of an ordered ... |
| f1oprswap 6908 | A two-element swap is a bi... |
| f1oprg 6909 | An unordered pair of order... |
| tz6.12-2 6910 | Function value when ` F ` ... |
| fveu 6911 | The value of a function at... |
| brprcneu 6912 | If ` A ` is a proper class... |
| brprcneuALT 6913 | Alternate proof of ~ brprc... |
| fvprc 6914 | A function's value at a pr... |
| fvprcALT 6915 | Alternate proof of ~ fvprc... |
| rnfvprc 6916 | The range of a function va... |
| fv2 6917 | Alternate definition of fu... |
| dffv3 6918 | A definition of function v... |
| dffv4 6919 | The previous definition of... |
| elfv 6920 | Membership in a function v... |
| fveq1 6921 | Equality theorem for funct... |
| fveq2 6922 | Equality theorem for funct... |
| fveq1i 6923 | Equality inference for fun... |
| fveq1d 6924 | Equality deduction for fun... |
| fveq2i 6925 | Equality inference for fun... |
| fveq2d 6926 | Equality deduction for fun... |
| 2fveq3 6927 | Equality theorem for neste... |
| fveq12i 6928 | Equality deduction for fun... |
| fveq12d 6929 | Equality deduction for fun... |
| fveqeq2d 6930 | Equality deduction for fun... |
| fveqeq2 6931 | Equality deduction for fun... |
| nffv 6932 | Bound-variable hypothesis ... |
| nffvmpt1 6933 | Bound-variable hypothesis ... |
| nffvd 6934 | Deduction version of bound... |
| fvex 6935 | The value of a class exist... |
| fvexi 6936 | The value of a class exist... |
| fvexd 6937 | The value of a class exist... |
| fvif 6938 | Move a conditional outside... |
| iffv 6939 | Move a conditional outside... |
| fv3 6940 | Alternate definition of th... |
| fvres 6941 | The value of a restricted ... |
| fvresd 6942 | The value of a restricted ... |
| funssfv 6943 | The value of a member of t... |
| tz6.12c 6944 | Corollary of Theorem 6.12(... |
| tz6.12-1 6945 | Function value. Theorem 6... |
| tz6.12-1OLD 6946 | Obsolete version of ~ tz6.... |
| tz6.12 6947 | Function value. Theorem 6... |
| tz6.12f 6948 | Function value, using boun... |
| tz6.12cOLD 6949 | Obsolete version of ~ tz6.... |
| tz6.12i 6950 | Corollary of Theorem 6.12(... |
| fvbr0 6951 | Two possibilities for the ... |
| fvrn0 6952 | A function value is a memb... |
| fvn0fvelrn 6953 | If the value of a function... |
| elfvunirn 6954 | A function value is a subs... |
| fvssunirn 6955 | The result of a function v... |
| fvssunirnOLD 6956 | Obsolete version of ~ fvss... |
| ndmfv 6957 | The value of a class outsi... |
| ndmfvrcl 6958 | Reverse closure law for fu... |
| elfvdm 6959 | If a function value has a ... |
| elfvex 6960 | If a function value has a ... |
| elfvexd 6961 | If a function value has a ... |
| eliman0 6962 | A nonempty function value ... |
| nfvres 6963 | The value of a non-member ... |
| nfunsn 6964 | If the restriction of a cl... |
| fvfundmfvn0 6965 | If the "value of a class" ... |
| 0fv 6966 | Function value of the empt... |
| fv2prc 6967 | A function value of a func... |
| elfv2ex 6968 | If a function value of a f... |
| fveqres 6969 | Equal values imply equal v... |
| csbfv12 6970 | Move class substitution in... |
| csbfv2g 6971 | Move class substitution in... |
| csbfv 6972 | Substitution for a functio... |
| funbrfv 6973 | The second argument of a b... |
| funopfv 6974 | The second element in an o... |
| fnbrfvb 6975 | Equivalence of function va... |
| fnopfvb 6976 | Equivalence of function va... |
| funbrfvb 6977 | Equivalence of function va... |
| funopfvb 6978 | Equivalence of function va... |
| fnbrfvb2 6979 | Version of ~ fnbrfvb for f... |
| fdmeu 6980 | There is exactly one codom... |
| funbrfv2b 6981 | Function value in terms of... |
| dffn5 6982 | Representation of a functi... |
| fnrnfv 6983 | The range of a function ex... |
| fvelrnb 6984 | A member of a function's r... |
| foelcdmi 6985 | A member of a surjective f... |
| dfimafn 6986 | Alternate definition of th... |
| dfimafn2 6987 | Alternate definition of th... |
| funimass4 6988 | Membership relation for th... |
| fvelima 6989 | Function value in an image... |
| funimassd 6990 | Sufficient condition for t... |
| fvelimad 6991 | Function value in an image... |
| feqmptd 6992 | Deduction form of ~ dffn5 ... |
| feqresmpt 6993 | Express a restricted funct... |
| feqmptdf 6994 | Deduction form of ~ dffn5f... |
| dffn5f 6995 | Representation of a functi... |
| fvelimab 6996 | Function value in an image... |
| fvelimabd 6997 | Deduction form of ~ fvelim... |
| fimarab 6998 | Expressing the image of a ... |
| unima 6999 | Image of a union. (Contri... |
| fvi 7000 | The value of the identity ... |
| fviss 7001 | The value of the identity ... |
| fniinfv 7002 | The indexed intersection o... |
| fnsnfv 7003 | Singleton of function valu... |
| opabiotafun 7004 | Define a function whose va... |
| opabiotadm 7005 | Define a function whose va... |
| opabiota 7006 | Define a function whose va... |
| fnimapr 7007 | The image of a pair under ... |
| fnimatpd 7008 | The image of an unordered ... |
| ssimaex 7009 | The existence of a subimag... |
| ssimaexg 7010 | The existence of a subimag... |
| funfv 7011 | A simplified expression fo... |
| funfv2 7012 | The value of a function. ... |
| funfv2f 7013 | The value of a function. ... |
| fvun 7014 | Value of the union of two ... |
| fvun1 7015 | The value of a union when ... |
| fvun2 7016 | The value of a union when ... |
| fvun1d 7017 | The value of a union when ... |
| fvun2d 7018 | The value of a union when ... |
| dffv2 7019 | Alternate definition of fu... |
| dmfco 7020 | Domains of a function comp... |
| fvco2 7021 | Value of a function compos... |
| fvco 7022 | Value of a function compos... |
| fvco3 7023 | Value of a function compos... |
| fvco3d 7024 | Value of a function compos... |
| fvco4i 7025 | Conditions for a compositi... |
| fvopab3g 7026 | Value of a function given ... |
| fvopab3ig 7027 | Value of a function given ... |
| brfvopabrbr 7028 | The binary relation of a f... |
| fvmptg 7029 | Value of a function given ... |
| fvmpti 7030 | Value of a function given ... |
| fvmpt 7031 | Value of a function given ... |
| fvmpt2f 7032 | Value of a function given ... |
| fvtresfn 7033 | Functionality of a tuple-r... |
| fvmpts 7034 | Value of a function given ... |
| fvmpt3 7035 | Value of a function given ... |
| fvmpt3i 7036 | Value of a function given ... |
| fvmptdf 7037 | Deduction version of ~ fvm... |
| fvmptd 7038 | Deduction version of ~ fvm... |
| fvmptd2 7039 | Deduction version of ~ fvm... |
| mptrcl 7040 | Reverse closure for a mapp... |
| fvmpt2i 7041 | Value of a function given ... |
| fvmpt2 7042 | Value of a function given ... |
| fvmptss 7043 | If all the values of the m... |
| fvmpt2d 7044 | Deduction version of ~ fvm... |
| fvmptex 7045 | Express a function ` F ` w... |
| fvmptd3f 7046 | Alternate deduction versio... |
| fvmptd2f 7047 | Alternate deduction versio... |
| fvmptdv 7048 | Alternate deduction versio... |
| fvmptdv2 7049 | Alternate deduction versio... |
| mpteqb 7050 | Bidirectional equality the... |
| fvmptt 7051 | Closed theorem form of ~ f... |
| fvmptf 7052 | Value of a function given ... |
| fvmptnf 7053 | The value of a function gi... |
| fvmptd3 7054 | Deduction version of ~ fvm... |
| fvmptd4 7055 | Deduction version of ~ fvm... |
| fvmptn 7056 | This somewhat non-intuitiv... |
| fvmptss2 7057 | A mapping always evaluates... |
| elfvmptrab1w 7058 | Implications for the value... |
| elfvmptrab1 7059 | Implications for the value... |
| elfvmptrab 7060 | Implications for the value... |
| fvopab4ndm 7061 | Value of a function given ... |
| fvmptndm 7062 | Value of a function given ... |
| fvmptrabfv 7063 | Value of a function mappin... |
| fvopab5 7064 | The value of a function th... |
| fvopab6 7065 | Value of a function given ... |
| eqfnfv 7066 | Equality of functions is d... |
| eqfnfv2 7067 | Equality of functions is d... |
| eqfnfv3 7068 | Derive equality of functio... |
| eqfnfvd 7069 | Deduction for equality of ... |
| eqfnfv2f 7070 | Equality of functions is d... |
| eqfunfv 7071 | Equality of functions is d... |
| eqfnun 7072 | Two functions on ` A u. B ... |
| fvreseq0 7073 | Equality of restricted fun... |
| fvreseq1 7074 | Equality of a function res... |
| fvreseq 7075 | Equality of restricted fun... |
| fnmptfvd 7076 | A function with a given do... |
| fndmdif 7077 | Two ways to express the lo... |
| fndmdifcom 7078 | The difference set between... |
| fndmdifeq0 7079 | The difference set of two ... |
| fndmin 7080 | Two ways to express the lo... |
| fneqeql 7081 | Two functions are equal if... |
| fneqeql2 7082 | Two functions are equal if... |
| fnreseql 7083 | Two functions are equal on... |
| chfnrn 7084 | The range of a choice func... |
| funfvop 7085 | Ordered pair with function... |
| funfvbrb 7086 | Two ways to say that ` A `... |
| fvimacnvi 7087 | A member of a preimage is ... |
| fvimacnv 7088 | The argument of a function... |
| funimass3 7089 | A kind of contraposition l... |
| funimass5 7090 | A subclass of a preimage i... |
| funconstss 7091 | Two ways of specifying tha... |
| fvimacnvALT 7092 | Alternate proof of ~ fvima... |
| elpreima 7093 | Membership in the preimage... |
| elpreimad 7094 | Membership in the preimage... |
| fniniseg 7095 | Membership in the preimage... |
| fncnvima2 7096 | Inverse images under funct... |
| fniniseg2 7097 | Inverse point images under... |
| unpreima 7098 | Preimage of a union. (Con... |
| inpreima 7099 | Preimage of an intersectio... |
| difpreima 7100 | Preimage of a difference. ... |
| respreima 7101 | The preimage of a restrict... |
| cnvimainrn 7102 | The preimage of the inters... |
| sspreima 7103 | The preimage of a subset i... |
| iinpreima 7104 | Preimage of an intersectio... |
| intpreima 7105 | Preimage of an intersectio... |
| fimacnvOLD 7106 | Obsolete version of ~ fima... |
| fimacnvinrn 7107 | Taking the converse image ... |
| fimacnvinrn2 7108 | Taking the converse image ... |
| rescnvimafod 7109 | The restriction of a funct... |
| fvn0ssdmfun 7110 | If a class' function value... |
| fnopfv 7111 | Ordered pair with function... |
| fvelrn 7112 | A function's value belongs... |
| nelrnfvne 7113 | A function value cannot be... |
| fveqdmss 7114 | If the empty set is not co... |
| fveqressseq 7115 | If the empty set is not co... |
| fnfvelrn 7116 | A function's value belongs... |
| ffvelcdm 7117 | A function's value belongs... |
| fnfvelrnd 7118 | A function's value belongs... |
| ffvelcdmi 7119 | A function's value belongs... |
| ffvelcdmda 7120 | A function's value belongs... |
| ffvelcdmd 7121 | A function's value belongs... |
| feldmfvelcdm 7122 | A class is an element of t... |
| rexrn 7123 | Restricted existential qua... |
| ralrn 7124 | Restricted universal quant... |
| elrnrexdm 7125 | For any element in the ran... |
| elrnrexdmb 7126 | For any element in the ran... |
| eldmrexrn 7127 | For any element in the dom... |
| eldmrexrnb 7128 | For any element in the dom... |
| fvcofneq 7129 | The values of two function... |
| ralrnmptw 7130 | A restricted quantifier ov... |
| rexrnmptw 7131 | A restricted quantifier ov... |
| ralrnmpt 7132 | A restricted quantifier ov... |
| rexrnmpt 7133 | A restricted quantifier ov... |
| f0cli 7134 | Unconditional closure of a... |
| dff2 7135 | Alternate definition of a ... |
| dff3 7136 | Alternate definition of a ... |
| dff4 7137 | Alternate definition of a ... |
| dffo3 7138 | An onto mapping expressed ... |
| dffo4 7139 | Alternate definition of an... |
| dffo5 7140 | Alternate definition of an... |
| exfo 7141 | A relation equivalent to t... |
| dffo3f 7142 | An onto mapping expressed ... |
| foelrn 7143 | Property of a surjective f... |
| foelrnf 7144 | Property of a surjective f... |
| foco2 7145 | If a composition of two fu... |
| fmpt 7146 | Functionality of the mappi... |
| f1ompt 7147 | Express bijection for a ma... |
| fmpti 7148 | Functionality of the mappi... |
| fvmptelcdm 7149 | The value of a function at... |
| fmptd 7150 | Domain and codomain of the... |
| fmpttd 7151 | Version of ~ fmptd with in... |
| fmpt3d 7152 | Domain and codomain of the... |
| fmptdf 7153 | A version of ~ fmptd using... |
| fompt 7154 | Express being onto for a m... |
| ffnfv 7155 | A function maps to a class... |
| ffnfvf 7156 | A function maps to a class... |
| fnfvrnss 7157 | An upper bound for range d... |
| fcdmssb 7158 | A function is a function i... |
| rnmptss 7159 | The range of an operation ... |
| fmpt2d 7160 | Domain and codomain of the... |
| ffvresb 7161 | A necessary and sufficient... |
| fssrescdmd 7162 | Restriction of a function ... |
| f1oresrab 7163 | Build a bijection between ... |
| f1ossf1o 7164 | Restricting a bijection, w... |
| fmptco 7165 | Composition of two functio... |
| fmptcof 7166 | Version of ~ fmptco where ... |
| fmptcos 7167 | Composition of two functio... |
| cofmpt 7168 | Express composition of a m... |
| fcompt 7169 | Express composition of two... |
| fcoconst 7170 | Composition with a constan... |
| fsn 7171 | A function maps a singleto... |
| fsn2 7172 | A function that maps a sin... |
| fsng 7173 | A function maps a singleto... |
| fsn2g 7174 | A function that maps a sin... |
| xpsng 7175 | The Cartesian product of t... |
| xpprsng 7176 | The Cartesian product of a... |
| xpsn 7177 | The Cartesian product of t... |
| f1o2sn 7178 | A singleton consisting in ... |
| residpr 7179 | Restriction of the identit... |
| dfmpt 7180 | Alternate definition for t... |
| fnasrn 7181 | A function expressed as th... |
| idref 7182 | Two ways to state that a r... |
| funiun 7183 | A function is a union of s... |
| funopsn 7184 | If a function is an ordere... |
| funop 7185 | An ordered pair is a funct... |
| funopdmsn 7186 | The domain of a function w... |
| funsndifnop 7187 | A singleton of an ordered ... |
| funsneqopb 7188 | A singleton of an ordered ... |
| ressnop0 7189 | If ` A ` is not in ` C ` ,... |
| fpr 7190 | A function with a domain o... |
| fprg 7191 | A function with a domain o... |
| ftpg 7192 | A function with a domain o... |
| ftp 7193 | A function with a domain o... |
| fnressn 7194 | A function restricted to a... |
| funressn 7195 | A function restricted to a... |
| fressnfv 7196 | The value of a function re... |
| fvrnressn 7197 | If the value of a function... |
| fvressn 7198 | The value of a function re... |
| fvn0fvelrnOLD 7199 | Obsolete version of ~ fvn0... |
| fvconst 7200 | The value of a constant fu... |
| fnsnr 7201 | If a class belongs to a fu... |
| fnsnb 7202 | A function whose domain is... |
| fmptsn 7203 | Express a singleton functi... |
| fmptsng 7204 | Express a singleton functi... |
| fmptsnd 7205 | Express a singleton functi... |
| fmptap 7206 | Append an additional value... |
| fmptapd 7207 | Append an additional value... |
| fmptpr 7208 | Express a pair function in... |
| fvresi 7209 | The value of a restricted ... |
| fninfp 7210 | Express the class of fixed... |
| fnelfp 7211 | Property of a fixed point ... |
| fndifnfp 7212 | Express the class of non-f... |
| fnelnfp 7213 | Property of a non-fixed po... |
| fnnfpeq0 7214 | A function is the identity... |
| fvunsn 7215 | Remove an ordered pair not... |
| fvsng 7216 | The value of a singleton o... |
| fvsn 7217 | The value of a singleton o... |
| fvsnun1 7218 | The value of a function wi... |
| fvsnun2 7219 | The value of a function wi... |
| fnsnsplit 7220 | Split a function into a si... |
| fsnunf 7221 | Adjoining a point to a fun... |
| fsnunf2 7222 | Adjoining a point to a pun... |
| fsnunfv 7223 | Recover the added point fr... |
| fsnunres 7224 | Recover the original funct... |
| funresdfunsn 7225 | Restricting a function to ... |
| fvpr1g 7226 | The value of a function wi... |
| fvpr2g 7227 | The value of a function wi... |
| fvpr2gOLD 7228 | Obsolete version of ~ fvpr... |
| fvpr1 7229 | The value of a function wi... |
| fvpr1OLD 7230 | Obsolete version of ~ fvpr... |
| fvpr2 7231 | The value of a function wi... |
| fvpr2OLD 7232 | Obsolete version of ~ fvpr... |
| fprb 7233 | A condition for functionho... |
| fvtp1 7234 | The first value of a funct... |
| fvtp2 7235 | The second value of a func... |
| fvtp3 7236 | The third value of a funct... |
| fvtp1g 7237 | The value of a function wi... |
| fvtp2g 7238 | The value of a function wi... |
| fvtp3g 7239 | The value of a function wi... |
| tpres 7240 | An unordered triple of ord... |
| fvconst2g 7241 | The value of a constant fu... |
| fconst2g 7242 | A constant function expres... |
| fvconst2 7243 | The value of a constant fu... |
| fconst2 7244 | A constant function expres... |
| fconst5 7245 | Two ways to express that a... |
| rnmptc 7246 | Range of a constant functi... |
| fnprb 7247 | A function whose domain ha... |
| fntpb 7248 | A function whose domain ha... |
| fnpr2g 7249 | A function whose domain ha... |
| fpr2g 7250 | A function that maps a pai... |
| fconstfv 7251 | A constant function expres... |
| fconst3 7252 | Two ways to express a cons... |
| fconst4 7253 | Two ways to express a cons... |
| resfunexg 7254 | The restriction of a funct... |
| resiexd 7255 | The restriction of the ide... |
| fnex 7256 | If the domain of a functio... |
| fnexd 7257 | If the domain of a functio... |
| funex 7258 | If the domain of a functio... |
| opabex 7259 | Existence of a function ex... |
| mptexg 7260 | If the domain of a functio... |
| mptexgf 7261 | If the domain of a functio... |
| mptex 7262 | If the domain of a functio... |
| mptexd 7263 | If the domain of a functio... |
| mptrabex 7264 | If the domain of a functio... |
| fex 7265 | If the domain of a mapping... |
| fexd 7266 | If the domain of a mapping... |
| mptfvmpt 7267 | A function in maps-to nota... |
| eufnfv 7268 | A function is uniquely det... |
| funfvima 7269 | A function's value in a pr... |
| funfvima2 7270 | A function's value in an i... |
| funfvima2d 7271 | A function's value in a pr... |
| fnfvima 7272 | The function value of an o... |
| fnfvimad 7273 | A function's value belongs... |
| resfvresima 7274 | The value of the function ... |
| funfvima3 7275 | A class including a functi... |
| ralima 7276 | Universal quantification u... |
| rexima 7277 | Existential quantification... |
| reximaOLD 7278 | Obsolete version of ~ rexi... |
| ralimaOLD 7279 | Obsolete version of ~ rali... |
| fvclss 7280 | Upper bound for the class ... |
| elabrex 7281 | Elementhood in an image se... |
| elabrexg 7282 | Elementhood in an image se... |
| abrexco 7283 | Composition of two image m... |
| imaiun 7284 | The image of an indexed un... |
| imauni 7285 | The image of a union is th... |
| fniunfv 7286 | The indexed union of a fun... |
| funiunfv 7287 | The indexed union of a fun... |
| funiunfvf 7288 | The indexed union of a fun... |
| eluniima 7289 | Membership in the union of... |
| elunirn 7290 | Membership in the union of... |
| elunirnALT 7291 | Alternate proof of ~ eluni... |
| elunirn2OLD 7292 | Obsolete version of ~ elfv... |
| fnunirn 7293 | Membership in a union of s... |
| dff13 7294 | A one-to-one function in t... |
| dff13f 7295 | A one-to-one function in t... |
| f1veqaeq 7296 | If the values of a one-to-... |
| f1cofveqaeq 7297 | If the values of a composi... |
| f1cofveqaeqALT 7298 | Alternate proof of ~ f1cof... |
| 2f1fvneq 7299 | If two one-to-one function... |
| f1mpt 7300 | Express injection for a ma... |
| f1fveq 7301 | Equality of function value... |
| f1elima 7302 | Membership in the image of... |
| f1imass 7303 | Taking images under a one-... |
| f1imaeq 7304 | Taking images under a one-... |
| f1imapss 7305 | Taking images under a one-... |
| fpropnf1 7306 | A function, given by an un... |
| f1dom3fv3dif 7307 | The function values for a ... |
| f1dom3el3dif 7308 | The codomain of a 1-1 func... |
| dff14a 7309 | A one-to-one function in t... |
| dff14b 7310 | A one-to-one function in t... |
| f12dfv 7311 | A one-to-one function with... |
| f13dfv 7312 | A one-to-one function with... |
| dff1o6 7313 | A one-to-one onto function... |
| f1ocnvfv1 7314 | The converse value of the ... |
| f1ocnvfv2 7315 | The value of the converse ... |
| f1ocnvfv 7316 | Relationship between the v... |
| f1ocnvfvb 7317 | Relationship between the v... |
| nvof1o 7318 | An involution is a bijecti... |
| nvocnv 7319 | The converse of an involut... |
| f1cdmsn 7320 | If a one-to-one function w... |
| fsnex 7321 | Relate a function with a s... |
| f1prex 7322 | Relate a one-to-one functi... |
| f1ocnvdm 7323 | The value of the converse ... |
| f1ocnvfvrneq 7324 | If the values of a one-to-... |
| fcof1 7325 | An application is injectiv... |
| fcofo 7326 | An application is surjecti... |
| cbvfo 7327 | Change bound variable betw... |
| cbvexfo 7328 | Change bound variable betw... |
| cocan1 7329 | An injection is left-cance... |
| cocan2 7330 | A surjection is right-canc... |
| fcof1oinvd 7331 | Show that a function is th... |
| fcof1od 7332 | A function is bijective if... |
| 2fcoidinvd 7333 | Show that a function is th... |
| fcof1o 7334 | Show that two functions ar... |
| 2fvcoidd 7335 | Show that the composition ... |
| 2fvidf1od 7336 | A function is bijective if... |
| 2fvidinvd 7337 | Show that two functions ar... |
| foeqcnvco 7338 | Condition for function equ... |
| f1eqcocnv 7339 | Condition for function equ... |
| fveqf1o 7340 | Given a bijection ` F ` , ... |
| f1ocoima 7341 | The composition of two bij... |
| nf1const 7342 | A constant function from a... |
| nf1oconst 7343 | A constant function from a... |
| f1ofvswap 7344 | Swapping two values in a b... |
| fvf1pr 7345 | Values of a one-to-one fun... |
| fliftrel 7346 | ` F ` , a function lift, i... |
| fliftel 7347 | Elementhood in the relatio... |
| fliftel1 7348 | Elementhood in the relatio... |
| fliftcnv 7349 | Converse of the relation `... |
| fliftfun 7350 | The function ` F ` is the ... |
| fliftfund 7351 | The function ` F ` is the ... |
| fliftfuns 7352 | The function ` F ` is the ... |
| fliftf 7353 | The domain and range of th... |
| fliftval 7354 | The value of the function ... |
| isoeq1 7355 | Equality theorem for isomo... |
| isoeq2 7356 | Equality theorem for isomo... |
| isoeq3 7357 | Equality theorem for isomo... |
| isoeq4 7358 | Equality theorem for isomo... |
| isoeq5 7359 | Equality theorem for isomo... |
| nfiso 7360 | Bound-variable hypothesis ... |
| isof1o 7361 | An isomorphism is a one-to... |
| isof1oidb 7362 | A function is a bijection ... |
| isof1oopb 7363 | A function is a bijection ... |
| isorel 7364 | An isomorphism connects bi... |
| soisores 7365 | Express the condition of i... |
| soisoi 7366 | Infer isomorphism from one... |
| isoid 7367 | Identity law for isomorphi... |
| isocnv 7368 | Converse law for isomorphi... |
| isocnv2 7369 | Converse law for isomorphi... |
| isocnv3 7370 | Complementation law for is... |
| isores2 7371 | An isomorphism from one we... |
| isores1 7372 | An isomorphism from one we... |
| isores3 7373 | Induced isomorphism on a s... |
| isotr 7374 | Composition (transitive) l... |
| isomin 7375 | Isomorphisms preserve mini... |
| isoini 7376 | Isomorphisms preserve init... |
| isoini2 7377 | Isomorphisms are isomorphi... |
| isofrlem 7378 | Lemma for ~ isofr . (Cont... |
| isoselem 7379 | Lemma for ~ isose . (Cont... |
| isofr 7380 | An isomorphism preserves w... |
| isose 7381 | An isomorphism preserves s... |
| isofr2 7382 | A weak form of ~ isofr tha... |
| isopolem 7383 | Lemma for ~ isopo . (Cont... |
| isopo 7384 | An isomorphism preserves t... |
| isosolem 7385 | Lemma for ~ isoso . (Cont... |
| isoso 7386 | An isomorphism preserves t... |
| isowe 7387 | An isomorphism preserves t... |
| isowe2 7388 | A weak form of ~ isowe tha... |
| f1oiso 7389 | Any one-to-one onto functi... |
| f1oiso2 7390 | Any one-to-one onto functi... |
| f1owe 7391 | Well-ordering of isomorphi... |
| weniso 7392 | A set-like well-ordering h... |
| weisoeq 7393 | Thus, there is at most one... |
| weisoeq2 7394 | Thus, there is at most one... |
| knatar 7395 | The Knaster-Tarski theorem... |
| fvresval 7396 | The value of a restricted ... |
| funeldmb 7397 | If ` (/) ` is not part of ... |
| eqfunresadj 7398 | Law for adjoining an eleme... |
| eqfunressuc 7399 | Law for equality of restri... |
| fnssintima 7400 | Condition for subset of an... |
| imaeqsexvOLD 7401 | Duplicate version of ~ ral... |
| imaeqsalvOLD 7402 | Duplicate version of ~ ral... |
| canth 7403 | No set ` A ` is equinumero... |
| ncanth 7404 | Cantor's theorem fails for... |
| riotaeqdv 7407 | Formula-building deduction... |
| riotabidv 7408 | Formula-building deduction... |
| riotaeqbidv 7409 | Equality deduction for res... |
| riotaex 7410 | Restricted iota is a set. ... |
| riotav 7411 | An iota restricted to the ... |
| riotauni 7412 | Restricted iota in terms o... |
| nfriota1 7413 | The abstraction variable i... |
| nfriotadw 7414 | Deduction version of ~ nfr... |
| cbvriotaw 7415 | Change bound variable in a... |
| cbvriotavw 7416 | Change bound variable in a... |
| cbvriotavwOLD 7417 | Obsolete version of ~ cbvr... |
| nfriotad 7418 | Deduction version of ~ nfr... |
| nfriota 7419 | A variable not free in a w... |
| cbvriota 7420 | Change bound variable in a... |
| cbvriotav 7421 | Change bound variable in a... |
| csbriota 7422 | Interchange class substitu... |
| riotacl2 7423 | Membership law for "the un... |
| riotacl 7424 | Closure of restricted iota... |
| riotasbc 7425 | Substitution law for descr... |
| riotabidva 7426 | Equivalent wff's yield equ... |
| riotabiia 7427 | Equivalent wff's yield equ... |
| riota1 7428 | Property of restricted iot... |
| riota1a 7429 | Property of iota. (Contri... |
| riota2df 7430 | A deduction version of ~ r... |
| riota2f 7431 | This theorem shows a condi... |
| riota2 7432 | This theorem shows a condi... |
| riotaeqimp 7433 | If two restricted iota des... |
| riotaprop 7434 | Properties of a restricted... |
| riota5f 7435 | A method for computing res... |
| riota5 7436 | A method for computing res... |
| riotass2 7437 | Restriction of a unique el... |
| riotass 7438 | Restriction of a unique el... |
| moriotass 7439 | Restriction of a unique el... |
| snriota 7440 | A restricted class abstrac... |
| riotaxfrd 7441 | Change the variable ` x ` ... |
| eusvobj2 7442 | Specify the same property ... |
| eusvobj1 7443 | Specify the same object in... |
| f1ofveu 7444 | There is one domain elemen... |
| f1ocnvfv3 7445 | Value of the converse of a... |
| riotaund 7446 | Restricted iota equals the... |
| riotassuni 7447 | The restricted iota class ... |
| riotaclb 7448 | Bidirectional closure of r... |
| riotarab 7449 | Restricted iota of a restr... |
| oveq 7456 | Equality theorem for opera... |
| oveq1 7457 | Equality theorem for opera... |
| oveq2 7458 | Equality theorem for opera... |
| oveq12 7459 | Equality theorem for opera... |
| oveq1i 7460 | Equality inference for ope... |
| oveq2i 7461 | Equality inference for ope... |
| oveq12i 7462 | Equality inference for ope... |
| oveqi 7463 | Equality inference for ope... |
| oveq123i 7464 | Equality inference for ope... |
| oveq1d 7465 | Equality deduction for ope... |
| oveq2d 7466 | Equality deduction for ope... |
| oveqd 7467 | Equality deduction for ope... |
| oveq12d 7468 | Equality deduction for ope... |
| oveqan12d 7469 | Equality deduction for ope... |
| oveqan12rd 7470 | Equality deduction for ope... |
| oveq123d 7471 | Equality deduction for ope... |
| fvoveq1d 7472 | Equality deduction for nes... |
| fvoveq1 7473 | Equality theorem for neste... |
| ovanraleqv 7474 | Equality theorem for a con... |
| imbrov2fvoveq 7475 | Equality theorem for neste... |
| ovrspc2v 7476 | If an operation value is e... |
| oveqrspc2v 7477 | Restricted specialization ... |
| oveqdr 7478 | Equality of two operations... |
| nfovd 7479 | Deduction version of bound... |
| nfov 7480 | Bound-variable hypothesis ... |
| oprabidw 7481 | The law of concretion. Sp... |
| oprabid 7482 | The law of concretion. Sp... |
| ovex 7483 | The result of an operation... |
| ovexi 7484 | The result of an operation... |
| ovexd 7485 | The result of an operation... |
| ovssunirn 7486 | The result of an operation... |
| 0ov 7487 | Operation value of the emp... |
| ovprc 7488 | The value of an operation ... |
| ovprc1 7489 | The value of an operation ... |
| ovprc2 7490 | The value of an operation ... |
| ovrcl 7491 | Reverse closure for an ope... |
| elfvov1 7492 | Utility theorem: reverse c... |
| elfvov2 7493 | Utility theorem: reverse c... |
| csbov123 7494 | Move class substitution in... |
| csbov 7495 | Move class substitution in... |
| csbov12g 7496 | Move class substitution in... |
| csbov1g 7497 | Move class substitution in... |
| csbov2g 7498 | Move class substitution in... |
| rspceov 7499 | A frequently used special ... |
| elovimad 7500 | Elementhood of the image s... |
| fnbrovb 7501 | Value of a binary operatio... |
| fnotovb 7502 | Equivalence of operation v... |
| opabbrex 7503 | A collection of ordered pa... |
| opabresex2 7504 | Restrictions of a collecti... |
| opabresex2d 7505 | Obsolete version of ~ opab... |
| fvmptopab 7506 | The function value of a ma... |
| fvmptopabOLD 7507 | Obsolete version of ~ fvmp... |
| f1opr 7508 | Condition for an operation... |
| brfvopab 7509 | The classes involved in a ... |
| dfoprab2 7510 | Class abstraction for oper... |
| reloprab 7511 | An operation class abstrac... |
| oprabv 7512 | If a pair and a class are ... |
| nfoprab1 7513 | The abstraction variables ... |
| nfoprab2 7514 | The abstraction variables ... |
| nfoprab3 7515 | The abstraction variables ... |
| nfoprab 7516 | Bound-variable hypothesis ... |
| oprabbid 7517 | Equivalent wff's yield equ... |
| oprabbidv 7518 | Equivalent wff's yield equ... |
| oprabbii 7519 | Equivalent wff's yield equ... |
| ssoprab2 7520 | Equivalence of ordered pai... |
| ssoprab2b 7521 | Equivalence of ordered pai... |
| eqoprab2bw 7522 | Equivalence of ordered pai... |
| eqoprab2b 7523 | Equivalence of ordered pai... |
| mpoeq123 7524 | An equality theorem for th... |
| mpoeq12 7525 | An equality theorem for th... |
| mpoeq123dva 7526 | An equality deduction for ... |
| mpoeq123dv 7527 | An equality deduction for ... |
| mpoeq123i 7528 | An equality inference for ... |
| mpoeq3dva 7529 | Slightly more general equa... |
| mpoeq3ia 7530 | An equality inference for ... |
| mpoeq3dv 7531 | An equality deduction for ... |
| nfmpo1 7532 | Bound-variable hypothesis ... |
| nfmpo2 7533 | Bound-variable hypothesis ... |
| nfmpo 7534 | Bound-variable hypothesis ... |
| 0mpo0 7535 | A mapping operation with e... |
| mpo0v 7536 | A mapping operation with e... |
| mpo0 7537 | A mapping operation with e... |
| oprab4 7538 | Two ways to state the doma... |
| cbvoprab1 7539 | Rule used to change first ... |
| cbvoprab2 7540 | Change the second bound va... |
| cbvoprab12 7541 | Rule used to change first ... |
| cbvoprab12v 7542 | Rule used to change first ... |
| cbvoprab3 7543 | Rule used to change the th... |
| cbvoprab3v 7544 | Rule used to change the th... |
| cbvmpox 7545 | Rule to change the bound v... |
| cbvmpo 7546 | Rule to change the bound v... |
| cbvmpov 7547 | Rule to change the bound v... |
| elimdelov 7548 | Eliminate a hypothesis whi... |
| brif1 7549 | Move a relation inside and... |
| ovif 7550 | Move a conditional outside... |
| ovif2 7551 | Move a conditional outside... |
| ovif12 7552 | Move a conditional outside... |
| ifov 7553 | Move a conditional outside... |
| dmoprab 7554 | The domain of an operation... |
| dmoprabss 7555 | The domain of an operation... |
| rnoprab 7556 | The range of an operation ... |
| rnoprab2 7557 | The range of a restricted ... |
| reldmoprab 7558 | The domain of an operation... |
| oprabss 7559 | Structure of an operation ... |
| eloprabga 7560 | The law of concretion for ... |
| eloprabgaOLD 7561 | Obsolete version of ~ elop... |
| eloprabg 7562 | The law of concretion for ... |
| ssoprab2i 7563 | Inference of operation cla... |
| mpov 7564 | Operation with universal d... |
| mpomptx 7565 | Express a two-argument fun... |
| mpompt 7566 | Express a two-argument fun... |
| mpodifsnif 7567 | A mapping with two argumen... |
| mposnif 7568 | A mapping with two argumen... |
| fconstmpo 7569 | Representation of a consta... |
| resoprab 7570 | Restriction of an operatio... |
| resoprab2 7571 | Restriction of an operator... |
| resmpo 7572 | Restriction of the mapping... |
| funoprabg 7573 | "At most one" is a suffici... |
| funoprab 7574 | "At most one" is a suffici... |
| fnoprabg 7575 | Functionality and domain o... |
| mpofun 7576 | The maps-to notation for a... |
| fnoprab 7577 | Functionality and domain o... |
| ffnov 7578 | An operation maps to a cla... |
| fovcld 7579 | Closure law for an operati... |
| fovcl 7580 | Closure law for an operati... |
| eqfnov 7581 | Equality of two operations... |
| eqfnov2 7582 | Two operators with the sam... |
| fnov 7583 | Representation of a functi... |
| mpo2eqb 7584 | Bidirectional equality the... |
| rnmpo 7585 | The range of an operation ... |
| reldmmpo 7586 | The domain of an operation... |
| elrnmpog 7587 | Membership in the range of... |
| elrnmpo 7588 | Membership in the range of... |
| elimampo 7589 | Membership in the image of... |
| elrnmpores 7590 | Membership in the range of... |
| ralrnmpo 7591 | A restricted quantifier ov... |
| rexrnmpo 7592 | A restricted quantifier ov... |
| ovid 7593 | The value of an operation ... |
| ovidig 7594 | The value of an operation ... |
| ovidi 7595 | The value of an operation ... |
| ov 7596 | The value of an operation ... |
| ovigg 7597 | The value of an operation ... |
| ovig 7598 | The value of an operation ... |
| ovmpt4g 7599 | Value of a function given ... |
| ovmpos 7600 | Value of a function given ... |
| ov2gf 7601 | The value of an operation ... |
| ovmpodxf 7602 | Value of an operation give... |
| ovmpodx 7603 | Value of an operation give... |
| ovmpod 7604 | Value of an operation give... |
| ovmpox 7605 | The value of an operation ... |
| ovmpoga 7606 | Value of an operation give... |
| ovmpoa 7607 | Value of an operation give... |
| ovmpodf 7608 | Alternate deduction versio... |
| ovmpodv 7609 | Alternate deduction versio... |
| ovmpodv2 7610 | Alternate deduction versio... |
| ovmpog 7611 | Value of an operation give... |
| ovmpo 7612 | Value of an operation give... |
| ovmpot 7613 | The value of an operation ... |
| fvmpopr2d 7614 | Value of an operation give... |
| ov3 7615 | The value of an operation ... |
| ov6g 7616 | The value of an operation ... |
| ovg 7617 | The value of an operation ... |
| ovres 7618 | The value of a restricted ... |
| ovresd 7619 | Lemma for converting metri... |
| oprres 7620 | The restriction of an oper... |
| oprssov 7621 | The value of a member of t... |
| fovcdm 7622 | An operation's value belon... |
| fovcdmda 7623 | An operation's value belon... |
| fovcdmd 7624 | An operation's value belon... |
| fnrnov 7625 | The range of an operation ... |
| foov 7626 | An onto mapping of an oper... |
| fnovrn 7627 | An operation's value belon... |
| ovelrn 7628 | A member of an operation's... |
| funimassov 7629 | Membership relation for th... |
| ovelimab 7630 | Operation value in an imag... |
| ovima0 7631 | An operation value is a me... |
| ovconst2 7632 | The value of a constant op... |
| oprssdm 7633 | Domain of closure of an op... |
| nssdmovg 7634 | The value of an operation ... |
| ndmovg 7635 | The value of an operation ... |
| ndmov 7636 | The value of an operation ... |
| ndmovcl 7637 | The closure of an operatio... |
| ndmovrcl 7638 | Reverse closure law, when ... |
| ndmovcom 7639 | Any operation is commutati... |
| ndmovass 7640 | Any operation is associati... |
| ndmovdistr 7641 | Any operation is distribut... |
| ndmovord 7642 | Elimination of redundant a... |
| ndmovordi 7643 | Elimination of redundant a... |
| caovclg 7644 | Convert an operation closu... |
| caovcld 7645 | Convert an operation closu... |
| caovcl 7646 | Convert an operation closu... |
| caovcomg 7647 | Convert an operation commu... |
| caovcomd 7648 | Convert an operation commu... |
| caovcom 7649 | Convert an operation commu... |
| caovassg 7650 | Convert an operation assoc... |
| caovassd 7651 | Convert an operation assoc... |
| caovass 7652 | Convert an operation assoc... |
| caovcang 7653 | Convert an operation cance... |
| caovcand 7654 | Convert an operation cance... |
| caovcanrd 7655 | Commute the arguments of a... |
| caovcan 7656 | Convert an operation cance... |
| caovordig 7657 | Convert an operation order... |
| caovordid 7658 | Convert an operation order... |
| caovordg 7659 | Convert an operation order... |
| caovordd 7660 | Convert an operation order... |
| caovord2d 7661 | Operation ordering law wit... |
| caovord3d 7662 | Ordering law. (Contribute... |
| caovord 7663 | Convert an operation order... |
| caovord2 7664 | Operation ordering law wit... |
| caovord3 7665 | Ordering law. (Contribute... |
| caovdig 7666 | Convert an operation distr... |
| caovdid 7667 | Convert an operation distr... |
| caovdir2d 7668 | Convert an operation distr... |
| caovdirg 7669 | Convert an operation rever... |
| caovdird 7670 | Convert an operation distr... |
| caovdi 7671 | Convert an operation distr... |
| caov32d 7672 | Rearrange arguments in a c... |
| caov12d 7673 | Rearrange arguments in a c... |
| caov31d 7674 | Rearrange arguments in a c... |
| caov13d 7675 | Rearrange arguments in a c... |
| caov4d 7676 | Rearrange arguments in a c... |
| caov411d 7677 | Rearrange arguments in a c... |
| caov42d 7678 | Rearrange arguments in a c... |
| caov32 7679 | Rearrange arguments in a c... |
| caov12 7680 | Rearrange arguments in a c... |
| caov31 7681 | Rearrange arguments in a c... |
| caov13 7682 | Rearrange arguments in a c... |
| caov4 7683 | Rearrange arguments in a c... |
| caov411 7684 | Rearrange arguments in a c... |
| caov42 7685 | Rearrange arguments in a c... |
| caovdir 7686 | Reverse distributive law. ... |
| caovdilem 7687 | Lemma used by real number ... |
| caovlem2 7688 | Lemma used in real number ... |
| caovmo 7689 | Uniqueness of inverse elem... |
| imaeqexov 7690 | Substitute an operation va... |
| imaeqalov 7691 | Substitute an operation va... |
| mpondm0 7692 | The value of an operation ... |
| elmpocl 7693 | If a two-parameter class i... |
| elmpocl1 7694 | If a two-parameter class i... |
| elmpocl2 7695 | If a two-parameter class i... |
| elovmpod 7696 | Utility lemma for two-para... |
| elovmpo 7697 | Utility lemma for two-para... |
| elovmporab 7698 | Implications for the value... |
| elovmporab1w 7699 | Implications for the value... |
| elovmporab1 7700 | Implications for the value... |
| 2mpo0 7701 | If the operation value of ... |
| relmptopab 7702 | Any function to sets of or... |
| f1ocnvd 7703 | Describe an implicit one-t... |
| f1od 7704 | Describe an implicit one-t... |
| f1ocnv2d 7705 | Describe an implicit one-t... |
| f1o2d 7706 | Describe an implicit one-t... |
| f1opw2 7707 | A one-to-one mapping induc... |
| f1opw 7708 | A one-to-one mapping induc... |
| elovmpt3imp 7709 | If the value of a function... |
| ovmpt3rab1 7710 | The value of an operation ... |
| ovmpt3rabdm 7711 | If the value of a function... |
| elovmpt3rab1 7712 | Implications for the value... |
| elovmpt3rab 7713 | Implications for the value... |
| ofeqd 7718 | Equality theorem for funct... |
| ofeq 7719 | Equality theorem for funct... |
| ofreq 7720 | Equality theorem for funct... |
| ofexg 7721 | A function operation restr... |
| nfof 7722 | Hypothesis builder for fun... |
| nfofr 7723 | Hypothesis builder for fun... |
| ofrfvalg 7724 | Value of a relation applie... |
| offval 7725 | Value of an operation appl... |
| ofrfval 7726 | Value of a relation applie... |
| ofval 7727 | Evaluate a function operat... |
| ofrval 7728 | Exhibit a function relatio... |
| offn 7729 | The function operation pro... |
| offun 7730 | The function operation pro... |
| offval2f 7731 | The function operation exp... |
| ofmresval 7732 | Value of a restriction of ... |
| fnfvof 7733 | Function value of a pointw... |
| off 7734 | The function operation pro... |
| ofres 7735 | Restrict the operands of a... |
| offval2 7736 | The function operation exp... |
| ofrfval2 7737 | The function relation acti... |
| ofmpteq 7738 | Value of a pointwise opera... |
| coof 7739 | The composition of a _homo... |
| ofco 7740 | The composition of a funct... |
| offveq 7741 | Convert an identity of the... |
| offveqb 7742 | Equivalent expressions for... |
| ofc1 7743 | Left operation by a consta... |
| ofc2 7744 | Right operation by a const... |
| ofc12 7745 | Function operation on two ... |
| caofref 7746 | Transfer a reflexive law t... |
| caofinvl 7747 | Transfer a left inverse la... |
| caofid0l 7748 | Transfer a left identity l... |
| caofid0r 7749 | Transfer a right identity ... |
| caofid1 7750 | Transfer a right absorptio... |
| caofid2 7751 | Transfer a right absorptio... |
| caofcom 7752 | Transfer a commutative law... |
| caofrss 7753 | Transfer a relation subset... |
| caofass 7754 | Transfer an associative la... |
| caoftrn 7755 | Transfer a transitivity la... |
| caofdi 7756 | Transfer a distributive la... |
| caofdir 7757 | Transfer a reverse distrib... |
| caonncan 7758 | Transfer ~ nncan -shaped l... |
| relrpss 7761 | The proper subset relation... |
| brrpssg 7762 | The proper subset relation... |
| brrpss 7763 | The proper subset relation... |
| porpss 7764 | Every class is partially o... |
| sorpss 7765 | Express strict ordering un... |
| sorpssi 7766 | Property of a chain of set... |
| sorpssun 7767 | A chain of sets is closed ... |
| sorpssin 7768 | A chain of sets is closed ... |
| sorpssuni 7769 | In a chain of sets, a maxi... |
| sorpssint 7770 | In a chain of sets, a mini... |
| sorpsscmpl 7771 | The componentwise compleme... |
| zfun 7773 | Axiom of Union expressed w... |
| axun2 7774 | A variant of the Axiom of ... |
| uniex2 7775 | The Axiom of Union using t... |
| vuniex 7776 | The union of a setvar is a... |
| uniexg 7777 | The ZF Axiom of Union in c... |
| uniex 7778 | The Axiom of Union in clas... |
| uniexd 7779 | Deduction version of the Z... |
| unexg 7780 | The union of two sets is a... |
| unex 7781 | The union of two sets is a... |
| unexOLD 7782 | Obsolete proof of ~ unex a... |
| tpex 7783 | An unordered triple of cla... |
| unexb 7784 | Existence of union is equi... |
| unexbOLD 7785 | Obsolete proof of ~ unexb ... |
| unexgOLD 7786 | Obsolete proof of ~ unexg ... |
| xpexg 7787 | The Cartesian product of t... |
| xpexd 7788 | The Cartesian product of t... |
| 3xpexg 7789 | The Cartesian product of t... |
| xpex 7790 | The Cartesian product of t... |
| unexd 7791 | The union of two sets is a... |
| sqxpexg 7792 | The Cartesian square of a ... |
| abnexg 7793 | Sufficient condition for a... |
| abnex 7794 | Sufficient condition for a... |
| snnex 7795 | The class of all singleton... |
| pwnex 7796 | The class of all power set... |
| difex2 7797 | If the subtrahend of a cla... |
| difsnexi 7798 | If the difference of a cla... |
| uniuni 7799 | Expression for double unio... |
| uniexr 7800 | Converse of the Axiom of U... |
| uniexb 7801 | The Axiom of Union and its... |
| pwexr 7802 | Converse of the Axiom of P... |
| pwexb 7803 | The Axiom of Power Sets an... |
| elpwpwel 7804 | A class belongs to a doubl... |
| eldifpw 7805 | Membership in a power clas... |
| elpwun 7806 | Membership in the power cl... |
| pwuncl 7807 | Power classes are closed u... |
| iunpw 7808 | An indexed union of a powe... |
| fr3nr 7809 | A well-founded relation ha... |
| epne3 7810 | A well-founded class conta... |
| dfwe2 7811 | Alternate definition of we... |
| epweon 7812 | The membership relation we... |
| epweonALT 7813 | Alternate proof of ~ epweo... |
| ordon 7814 | The class of all ordinal n... |
| onprc 7815 | No set contains all ordina... |
| ssorduni 7816 | The union of a class of or... |
| ssonuni 7817 | The union of a set of ordi... |
| ssonunii 7818 | The union of a set of ordi... |
| ordeleqon 7819 | A way to express the ordin... |
| ordsson 7820 | Any ordinal class is a sub... |
| dford5 7821 | A class is ordinal iff it ... |
| onss 7822 | An ordinal number is a sub... |
| predon 7823 | The predecessor of an ordi... |
| predonOLD 7824 | Obsolete version of ~ pred... |
| ssonprc 7825 | Two ways of saying a class... |
| onuni 7826 | The union of an ordinal nu... |
| orduni 7827 | The union of an ordinal cl... |
| onint 7828 | The intersection (infimum)... |
| onint0 7829 | The intersection of a clas... |
| onssmin 7830 | A nonempty class of ordina... |
| onminesb 7831 | If a property is true for ... |
| onminsb 7832 | If a property is true for ... |
| oninton 7833 | The intersection of a none... |
| onintrab 7834 | The intersection of a clas... |
| onintrab2 7835 | An existence condition equ... |
| onnmin 7836 | No member of a set of ordi... |
| onnminsb 7837 | An ordinal number smaller ... |
| oneqmin 7838 | A way to show that an ordi... |
| uniordint 7839 | The union of a set of ordi... |
| onminex 7840 | If a wff is true for an or... |
| sucon 7841 | The class of all ordinal n... |
| sucexb 7842 | A successor exists iff its... |
| sucexg 7843 | The successor of a set is ... |
| sucex 7844 | The successor of a set is ... |
| onmindif2 7845 | The minimum of a class of ... |
| ordsuci 7846 | The successor of an ordina... |
| sucexeloni 7847 | If the successor of an ord... |
| sucexeloniOLD 7848 | Obsolete version of ~ suce... |
| onsuc 7849 | The successor of an ordina... |
| suceloniOLD 7850 | Obsolete version of ~ onsu... |
| ordsuc 7851 | A class is ordinal if and ... |
| ordsucOLD 7852 | Obsolete version of ~ ords... |
| ordpwsuc 7853 | The collection of ordinals... |
| onpwsuc 7854 | The collection of ordinal ... |
| onsucb 7855 | A class is an ordinal numb... |
| ordsucss 7856 | The successor of an elemen... |
| onpsssuc 7857 | An ordinal number is a pro... |
| ordelsuc 7858 | A set belongs to an ordina... |
| onsucmin 7859 | The successor of an ordina... |
| ordsucelsuc 7860 | Membership is inherited by... |
| ordsucsssuc 7861 | The subclass relationship ... |
| ordsucuniel 7862 | Given an element ` A ` of ... |
| ordsucun 7863 | The successor of the maxim... |
| ordunpr 7864 | The maximum of two ordinal... |
| ordunel 7865 | The maximum of two ordinal... |
| onsucuni 7866 | A class of ordinal numbers... |
| ordsucuni 7867 | An ordinal class is a subc... |
| orduniorsuc 7868 | An ordinal class is either... |
| unon 7869 | The class of all ordinal n... |
| ordunisuc 7870 | An ordinal class is equal ... |
| orduniss2 7871 | The union of the ordinal s... |
| onsucuni2 7872 | A successor ordinal is the... |
| 0elsuc 7873 | The successor of an ordina... |
| limon 7874 | The class of ordinal numbe... |
| onuniorsuc 7875 | An ordinal number is eithe... |
| onssi 7876 | An ordinal number is a sub... |
| onsuci 7877 | The successor of an ordina... |
| onuniorsuciOLD 7878 | Obsolete version of ~ onun... |
| onuninsuci 7879 | An ordinal is equal to its... |
| onsucssi 7880 | A set belongs to an ordina... |
| nlimsucg 7881 | A successor is not a limit... |
| orduninsuc 7882 | An ordinal class is equal ... |
| ordunisuc2 7883 | An ordinal equal to its un... |
| ordzsl 7884 | An ordinal is zero, a succ... |
| onzsl 7885 | An ordinal number is zero,... |
| dflim3 7886 | An alternate definition of... |
| dflim4 7887 | An alternate definition of... |
| limsuc 7888 | The successor of a member ... |
| limsssuc 7889 | A class includes a limit o... |
| nlimon 7890 | Two ways to express the cl... |
| limuni3 7891 | The union of a nonempty cl... |
| tfi 7892 | The Principle of Transfini... |
| tfisg 7893 | A closed form of ~ tfis . ... |
| tfis 7894 | Transfinite Induction Sche... |
| tfis2f 7895 | Transfinite Induction Sche... |
| tfis2 7896 | Transfinite Induction Sche... |
| tfis3 7897 | Transfinite Induction Sche... |
| tfisi 7898 | A transfinite induction sc... |
| tfinds 7899 | Principle of Transfinite I... |
| tfindsg 7900 | Transfinite Induction (inf... |
| tfindsg2 7901 | Transfinite Induction (inf... |
| tfindes 7902 | Transfinite Induction with... |
| tfinds2 7903 | Transfinite Induction (inf... |
| tfinds3 7904 | Principle of Transfinite I... |
| dfom2 7907 | An alternate definition of... |
| elom 7908 | Membership in omega. The ... |
| omsson 7909 | Omega is a subset of ` On ... |
| limomss 7910 | The class of natural numbe... |
| nnon 7911 | A natural number is an ord... |
| nnoni 7912 | A natural number is an ord... |
| nnord 7913 | A natural number is ordina... |
| trom 7914 | The class of finite ordina... |
| ordom 7915 | The class of finite ordina... |
| elnn 7916 | A member of a natural numb... |
| omon 7917 | The class of natural numbe... |
| omelon2 7918 | Omega is an ordinal number... |
| nnlim 7919 | A natural number is not a ... |
| omssnlim 7920 | The class of natural numbe... |
| limom 7921 | Omega is a limit ordinal. ... |
| peano2b 7922 | A class belongs to omega i... |
| nnsuc 7923 | A nonzero natural number i... |
| omsucne 7924 | A natural number is not th... |
| ssnlim 7925 | An ordinal subclass of non... |
| omsinds 7926 | Strong (or "total") induct... |
| omsindsOLD 7927 | Obsolete version of ~ omsi... |
| omun 7928 | The union of two finite or... |
| peano1 7929 | Zero is a natural number. ... |
| peano1OLD 7930 | Obsolete version of ~ pean... |
| peano2 7931 | The successor of any natur... |
| peano3 7932 | The successor of any natur... |
| peano4 7933 | Two natural numbers are eq... |
| peano5 7934 | The induction postulate: a... |
| peano5OLD 7935 | Obsolete version of ~ pean... |
| nn0suc 7936 | A natural number is either... |
| find 7937 | The Principle of Finite In... |
| finds 7938 | Principle of Finite Induct... |
| findsg 7939 | Principle of Finite Induct... |
| finds2 7940 | Principle of Finite Induct... |
| finds1 7941 | Principle of Finite Induct... |
| findes 7942 | Finite induction with expl... |
| dmexg 7943 | The domain of a set is a s... |
| rnexg 7944 | The range of a set is a se... |
| dmexd 7945 | The domain of a set is a s... |
| fndmexd 7946 | If a function is a set, it... |
| dmfex 7947 | If a mapping is a set, its... |
| fndmexb 7948 | The domain of a function i... |
| fdmexb 7949 | The domain of a function i... |
| dmfexALT 7950 | Alternate proof of ~ dmfex... |
| dmex 7951 | The domain of a set is a s... |
| rnex 7952 | The range of a set is a se... |
| iprc 7953 | The identity function is a... |
| resiexg 7954 | The existence of a restric... |
| imaexg 7955 | The image of a set is a se... |
| imaex 7956 | The image of a set is a se... |
| rnexd 7957 | The range of a set is a se... |
| imaexd 7958 | The image of a set is a se... |
| exse2 7959 | Any set relation is set-li... |
| xpexr 7960 | If a Cartesian product is ... |
| xpexr2 7961 | If a nonempty Cartesian pr... |
| xpexcnv 7962 | A condition where the conv... |
| soex 7963 | If the relation in a stric... |
| elxp4 7964 | Membership in a Cartesian ... |
| elxp5 7965 | Membership in a Cartesian ... |
| cnvexg 7966 | The converse of a set is a... |
| cnvex 7967 | The converse of a set is a... |
| relcnvexb 7968 | A relation is a set iff it... |
| f1oexrnex 7969 | If the range of a 1-1 onto... |
| f1oexbi 7970 | There is a one-to-one onto... |
| coexg 7971 | The composition of two set... |
| coex 7972 | The composition of two set... |
| coexd 7973 | The composition of two set... |
| funcnvuni 7974 | The union of a chain (with... |
| fun11uni 7975 | The union of a chain (with... |
| fex2 7976 | A function with bounded do... |
| fabexd 7977 | Existence of a set of func... |
| fabexg 7978 | Existence of a set of func... |
| fabexgOLD 7979 | Obsolete version of ~ fabe... |
| fabex 7980 | Existence of a set of func... |
| mapex 7981 | The class of all functions... |
| f1oabexg 7982 | The class of all 1-1-onto ... |
| f1oabexgOLD 7983 | Obsolete version of ~ f1oa... |
| fiunlem 7984 | Lemma for ~ fiun and ~ f1i... |
| fiun 7985 | The union of a chain (with... |
| f1iun 7986 | The union of a chain (with... |
| fviunfun 7987 | The function value of an i... |
| ffoss 7988 | Relationship between a map... |
| f11o 7989 | Relationship between one-t... |
| resfunexgALT 7990 | Alternate proof of ~ resfu... |
| cofunexg 7991 | Existence of a composition... |
| cofunex2g 7992 | Existence of a composition... |
| fnexALT 7993 | Alternate proof of ~ fnex ... |
| funexw 7994 | Weak version of ~ funex th... |
| mptexw 7995 | Weak version of ~ mptex th... |
| funrnex 7996 | If the domain of a functio... |
| zfrep6 7997 | A version of the Axiom of ... |
| focdmex 7998 | If the domain of an onto f... |
| f1dmex 7999 | If the codomain of a one-t... |
| f1ovv 8000 | The codomain/range of a 1-... |
| fvclex 8001 | Existence of the class of ... |
| fvresex 8002 | Existence of the class of ... |
| abrexexg 8003 | Existence of a class abstr... |
| abrexexgOLD 8004 | Obsolete version of ~ abre... |
| abrexex 8005 | Existence of a class abstr... |
| iunexg 8006 | The existence of an indexe... |
| abrexex2g 8007 | Existence of an existentia... |
| opabex3d 8008 | Existence of an ordered pa... |
| opabex3rd 8009 | Existence of an ordered pa... |
| opabex3 8010 | Existence of an ordered pa... |
| iunex 8011 | The existence of an indexe... |
| abrexex2 8012 | Existence of an existentia... |
| abexssex 8013 | Existence of a class abstr... |
| abexex 8014 | A condition where a class ... |
| f1oweALT 8015 | Alternate proof of ~ f1owe... |
| wemoiso 8016 | Thus, there is at most one... |
| wemoiso2 8017 | Thus, there is at most one... |
| oprabexd 8018 | Existence of an operator a... |
| oprabex 8019 | Existence of an operation ... |
| oprabex3 8020 | Existence of an operation ... |
| oprabrexex2 8021 | Existence of an existentia... |
| ab2rexex 8022 | Existence of a class abstr... |
| ab2rexex2 8023 | Existence of an existentia... |
| xpexgALT 8024 | Alternate proof of ~ xpexg... |
| offval3 8025 | General value of ` ( F oF ... |
| offres 8026 | Pointwise combination comm... |
| ofmres 8027 | Equivalent expressions for... |
| ofmresex 8028 | Existence of a restriction... |
| mptcnfimad 8029 | The converse of a mapping ... |
| 1stval 8034 | The value of the function ... |
| 2ndval 8035 | The value of the function ... |
| 1stnpr 8036 | Value of the first-member ... |
| 2ndnpr 8037 | Value of the second-member... |
| 1st0 8038 | The value of the first-mem... |
| 2nd0 8039 | The value of the second-me... |
| op1st 8040 | Extract the first member o... |
| op2nd 8041 | Extract the second member ... |
| op1std 8042 | Extract the first member o... |
| op2ndd 8043 | Extract the second member ... |
| op1stg 8044 | Extract the first member o... |
| op2ndg 8045 | Extract the second member ... |
| ot1stg 8046 | Extract the first member o... |
| ot2ndg 8047 | Extract the second member ... |
| ot3rdg 8048 | Extract the third member o... |
| 1stval2 8049 | Alternate value of the fun... |
| 2ndval2 8050 | Alternate value of the fun... |
| oteqimp 8051 | The components of an order... |
| fo1st 8052 | The ` 1st ` function maps ... |
| fo2nd 8053 | The ` 2nd ` function maps ... |
| br1steqg 8054 | Uniqueness condition for t... |
| br2ndeqg 8055 | Uniqueness condition for t... |
| f1stres 8056 | Mapping of a restriction o... |
| f2ndres 8057 | Mapping of a restriction o... |
| fo1stres 8058 | Onto mapping of a restrict... |
| fo2ndres 8059 | Onto mapping of a restrict... |
| 1st2val 8060 | Value of an alternate defi... |
| 2nd2val 8061 | Value of an alternate defi... |
| 1stcof 8062 | Composition of the first m... |
| 2ndcof 8063 | Composition of the second ... |
| xp1st 8064 | Location of the first elem... |
| xp2nd 8065 | Location of the second ele... |
| elxp6 8066 | Membership in a Cartesian ... |
| elxp7 8067 | Membership in a Cartesian ... |
| eqopi 8068 | Equality with an ordered p... |
| xp2 8069 | Representation of Cartesia... |
| unielxp 8070 | The membership relation fo... |
| 1st2nd2 8071 | Reconstruction of a member... |
| 1st2ndb 8072 | Reconstruction of an order... |
| xpopth 8073 | An ordered pair theorem fo... |
| eqop 8074 | Two ways to express equali... |
| eqop2 8075 | Two ways to express equali... |
| op1steq 8076 | Two ways of expressing tha... |
| opreuopreu 8077 | There is a unique ordered ... |
| el2xptp 8078 | A member of a nested Carte... |
| el2xptp0 8079 | A member of a nested Carte... |
| el2xpss 8080 | Version of ~ elrel for tri... |
| 2nd1st 8081 | Swap the members of an ord... |
| 1st2nd 8082 | Reconstruction of a member... |
| 1stdm 8083 | The first ordered pair com... |
| 2ndrn 8084 | The second ordered pair co... |
| 1st2ndbr 8085 | Express an element of a re... |
| releldm2 8086 | Two ways of expressing mem... |
| reldm 8087 | An expression for the doma... |
| releldmdifi 8088 | One way of expressing memb... |
| funfv1st2nd 8089 | The function value for the... |
| funelss 8090 | If the first component of ... |
| funeldmdif 8091 | Two ways of expressing mem... |
| sbcopeq1a 8092 | Equality theorem for subst... |
| csbopeq1a 8093 | Equality theorem for subst... |
| sbcoteq1a 8094 | Equality theorem for subst... |
| dfopab2 8095 | A way to define an ordered... |
| dfoprab3s 8096 | A way to define an operati... |
| dfoprab3 8097 | Operation class abstractio... |
| dfoprab4 8098 | Operation class abstractio... |
| dfoprab4f 8099 | Operation class abstractio... |
| opabex2 8100 | Condition for an operation... |
| opabn1stprc 8101 | An ordered-pair class abst... |
| opiota 8102 | The property of a uniquely... |
| cnvoprab 8103 | The converse of a class ab... |
| dfxp3 8104 | Define the Cartesian produ... |
| elopabi 8105 | A consequence of membershi... |
| eloprabi 8106 | A consequence of membershi... |
| mpomptsx 8107 | Express a two-argument fun... |
| mpompts 8108 | Express a two-argument fun... |
| dmmpossx 8109 | The domain of a mapping is... |
| fmpox 8110 | Functionality, domain and ... |
| fmpo 8111 | Functionality, domain and ... |
| fnmpo 8112 | Functionality and domain o... |
| fnmpoi 8113 | Functionality and domain o... |
| dmmpo 8114 | Domain of a class given by... |
| ovmpoelrn 8115 | An operation's value belon... |
| dmmpoga 8116 | Domain of an operation giv... |
| dmmpog 8117 | Domain of an operation giv... |
| mpoexxg 8118 | Existence of an operation ... |
| mpoexg 8119 | Existence of an operation ... |
| mpoexga 8120 | If the domain of an operat... |
| mpoexw 8121 | Weak version of ~ mpoex th... |
| mpoex 8122 | If the domain of an operat... |
| mptmpoopabbrd 8123 | The operation value of a f... |
| mptmpoopabbrdOLD 8124 | Obsolete version of ~ mptm... |
| mptmpoopabovd 8125 | The operation value of a f... |
| mptmpoopabbrdOLDOLD 8126 | Obsolete version of ~ mptm... |
| mptmpoopabovdOLD 8127 | Obsolete version of ~ mptm... |
| el2mpocsbcl 8128 | If the operation value of ... |
| el2mpocl 8129 | If the operation value of ... |
| fnmpoovd 8130 | A function with a Cartesia... |
| offval22 8131 | The function operation exp... |
| brovpreldm 8132 | If a binary relation holds... |
| bropopvvv 8133 | If a binary relation holds... |
| bropfvvvvlem 8134 | Lemma for ~ bropfvvvv . (... |
| bropfvvvv 8135 | If a binary relation holds... |
| ovmptss 8136 | If all the values of the m... |
| relmpoopab 8137 | Any function to sets of or... |
| fmpoco 8138 | Composition of two functio... |
| oprabco 8139 | Composition of a function ... |
| oprab2co 8140 | Composition of operator ab... |
| df1st2 8141 | An alternate possible defi... |
| df2nd2 8142 | An alternate possible defi... |
| 1stconst 8143 | The mapping of a restricti... |
| 2ndconst 8144 | The mapping of a restricti... |
| dfmpo 8145 | Alternate definition for t... |
| mposn 8146 | An operation (in maps-to n... |
| curry1 8147 | Composition with ` ``' ( 2... |
| curry1val 8148 | The value of a curried fun... |
| curry1f 8149 | Functionality of a curried... |
| curry2 8150 | Composition with ` ``' ( 1... |
| curry2f 8151 | Functionality of a curried... |
| curry2val 8152 | The value of a curried fun... |
| cnvf1olem 8153 | Lemma for ~ cnvf1o . (Con... |
| cnvf1o 8154 | Describe a function that m... |
| fparlem1 8155 | Lemma for ~ fpar . (Contr... |
| fparlem2 8156 | Lemma for ~ fpar . (Contr... |
| fparlem3 8157 | Lemma for ~ fpar . (Contr... |
| fparlem4 8158 | Lemma for ~ fpar . (Contr... |
| fpar 8159 | Merge two functions in par... |
| fsplit 8160 | A function that can be use... |
| fsplitfpar 8161 | Merge two functions with a... |
| offsplitfpar 8162 | Express the function opera... |
| f2ndf 8163 | The ` 2nd ` (second compon... |
| fo2ndf 8164 | The ` 2nd ` (second compon... |
| f1o2ndf1 8165 | The ` 2nd ` (second compon... |
| opco1 8166 | Value of an operation prec... |
| opco2 8167 | Value of an operation prec... |
| opco1i 8168 | Inference form of ~ opco1 ... |
| frxp 8169 | A lexicographical ordering... |
| xporderlem 8170 | Lemma for lexicographical ... |
| poxp 8171 | A lexicographical ordering... |
| soxp 8172 | A lexicographical ordering... |
| wexp 8173 | A lexicographical ordering... |
| fnwelem 8174 | Lemma for ~ fnwe . (Contr... |
| fnwe 8175 | A variant on lexicographic... |
| fnse 8176 | Condition for the well-ord... |
| fvproj 8177 | Value of a function on ord... |
| fimaproj 8178 | Image of a cartesian produ... |
| ralxpes 8179 | A version of ~ ralxp with ... |
| ralxp3f 8180 | Restricted for all over a ... |
| ralxp3 8181 | Restricted for all over a ... |
| ralxp3es 8182 | Restricted for-all over a ... |
| frpoins3xpg 8183 | Special case of founded pa... |
| frpoins3xp3g 8184 | Special case of founded pa... |
| xpord2lem 8185 | Lemma for Cartesian produc... |
| poxp2 8186 | Another way of partially o... |
| frxp2 8187 | Another way of giving a we... |
| xpord2pred 8188 | Calculate the predecessor ... |
| sexp2 8189 | Condition for the relation... |
| xpord2indlem 8190 | Induction over the Cartesi... |
| xpord2ind 8191 | Induction over the Cartesi... |
| xpord3lem 8192 | Lemma for triple ordering.... |
| poxp3 8193 | Triple Cartesian product p... |
| frxp3 8194 | Give well-foundedness over... |
| xpord3pred 8195 | Calculate the predecsessor... |
| sexp3 8196 | Show that the triple order... |
| xpord3inddlem 8197 | Induction over the triple ... |
| xpord3indd 8198 | Induction over the triple ... |
| xpord3ind 8199 | Induction over the triple ... |
| orderseqlem 8200 | Lemma for ~ poseq and ~ so... |
| poseq 8201 | A partial ordering of ordi... |
| soseq 8202 | A linear ordering of ordin... |
| suppval 8205 | The value of the operation... |
| supp0prc 8206 | The support of a class is ... |
| suppvalbr 8207 | The value of the operation... |
| supp0 8208 | The support of the empty s... |
| suppval1 8209 | The value of the operation... |
| suppvalfng 8210 | The value of the operation... |
| suppvalfn 8211 | The value of the operation... |
| elsuppfng 8212 | An element of the support ... |
| elsuppfn 8213 | An element of the support ... |
| fvdifsupp 8214 | Function value is zero out... |
| cnvimadfsn 8215 | The support of functions "... |
| suppimacnvss 8216 | The support of functions "... |
| suppimacnv 8217 | Support sets of functions ... |
| fsuppeq 8218 | Two ways of writing the su... |
| fsuppeqg 8219 | Version of ~ fsuppeq avoid... |
| suppssdm 8220 | The support of a function ... |
| suppsnop 8221 | The support of a singleton... |
| snopsuppss 8222 | The support of a singleton... |
| fvn0elsupp 8223 | If the function value for ... |
| fvn0elsuppb 8224 | The function value for a g... |
| rexsupp 8225 | Existential quantification... |
| ressuppss 8226 | The support of the restric... |
| suppun 8227 | The support of a class/fun... |
| ressuppssdif 8228 | The support of the restric... |
| mptsuppdifd 8229 | The support of a function ... |
| mptsuppd 8230 | The support of a function ... |
| extmptsuppeq 8231 | The support of an extended... |
| suppfnss 8232 | The support of a function ... |
| funsssuppss 8233 | The support of a function ... |
| fnsuppres 8234 | Two ways to express restri... |
| fnsuppeq0 8235 | The support of a function ... |
| fczsupp0 8236 | The support of a constant ... |
| suppss 8237 | Show that the support of a... |
| suppssr 8238 | A function is zero outside... |
| suppssrg 8239 | A function is zero outside... |
| suppssov1 8240 | Formula building theorem f... |
| suppssov2 8241 | Formula building theorem f... |
| suppssof1 8242 | Formula building theorem f... |
| suppss2 8243 | Show that the support of a... |
| suppsssn 8244 | Show that the support of a... |
| suppssfv 8245 | Formula building theorem f... |
| suppofssd 8246 | Condition for the support ... |
| suppofss1d 8247 | Condition for the support ... |
| suppofss2d 8248 | Condition for the support ... |
| suppco 8249 | The support of the composi... |
| suppcoss 8250 | The support of the composi... |
| supp0cosupp0 8251 | The support of the composi... |
| imacosupp 8252 | The image of the support o... |
| opeliunxp2f 8253 | Membership in a union of C... |
| mpoxeldm 8254 | If there is an element of ... |
| mpoxneldm 8255 | If the first argument of a... |
| mpoxopn0yelv 8256 | If there is an element of ... |
| mpoxopynvov0g 8257 | If the second argument of ... |
| mpoxopxnop0 8258 | If the first argument of a... |
| mpoxopx0ov0 8259 | If the first argument of a... |
| mpoxopxprcov0 8260 | If the components of the f... |
| mpoxopynvov0 8261 | If the second argument of ... |
| mpoxopoveq 8262 | Value of an operation give... |
| mpoxopovel 8263 | Element of the value of an... |
| mpoxopoveqd 8264 | Value of an operation give... |
| brovex 8265 | A binary relation of the v... |
| brovmpoex 8266 | A binary relation of the v... |
| sprmpod 8267 | The extension of a binary ... |
| tposss 8270 | Subset theorem for transpo... |
| tposeq 8271 | Equality theorem for trans... |
| tposeqd 8272 | Equality theorem for trans... |
| tposssxp 8273 | The transposition is a sub... |
| reltpos 8274 | The transposition is a rel... |
| brtpos2 8275 | Value of the transposition... |
| brtpos0 8276 | The behavior of ` tpos ` w... |
| reldmtpos 8277 | Necessary and sufficient c... |
| brtpos 8278 | The transposition swaps ar... |
| ottpos 8279 | The transposition swaps th... |
| relbrtpos 8280 | The transposition swaps ar... |
| dmtpos 8281 | The domain of ` tpos F ` w... |
| rntpos 8282 | The range of ` tpos F ` wh... |
| tposexg 8283 | The transposition of a set... |
| ovtpos 8284 | The transposition swaps th... |
| tposfun 8285 | The transposition of a fun... |
| dftpos2 8286 | Alternate definition of ` ... |
| dftpos3 8287 | Alternate definition of ` ... |
| dftpos4 8288 | Alternate definition of ` ... |
| tpostpos 8289 | Value of the double transp... |
| tpostpos2 8290 | Value of the double transp... |
| tposfn2 8291 | The domain of a transposit... |
| tposfo2 8292 | Condition for a surjective... |
| tposf2 8293 | The domain and codomain of... |
| tposf12 8294 | Condition for an injective... |
| tposf1o2 8295 | Condition of a bijective t... |
| tposfo 8296 | The domain and codomain/ra... |
| tposf 8297 | The domain and codomain of... |
| tposfn 8298 | Functionality of a transpo... |
| tpos0 8299 | Transposition of the empty... |
| tposco 8300 | Transposition of a composi... |
| tpossym 8301 | Two ways to say a function... |
| tposeqi 8302 | Equality theorem for trans... |
| tposex 8303 | A transposition is a set. ... |
| nftpos 8304 | Hypothesis builder for tra... |
| tposoprab 8305 | Transposition of a class o... |
| tposmpo 8306 | Transposition of a two-arg... |
| tposconst 8307 | The transposition of a con... |
| mpocurryd 8312 | The currying of an operati... |
| mpocurryvald 8313 | The value of a curried ope... |
| fvmpocurryd 8314 | The value of the value of ... |
| pwuninel2 8317 | Proof of ~ pwuninel under ... |
| pwuninel 8318 | The powerclass of the unio... |
| undefval 8319 | Value of the undefined val... |
| undefnel2 8320 | The undefined value genera... |
| undefnel 8321 | The undefined value genera... |
| undefne0 8322 | The undefined value genera... |
| frecseq123 8325 | Equality theorem for the w... |
| nffrecs 8326 | Bound-variable hypothesis ... |
| csbfrecsg 8327 | Move class substitution in... |
| fpr3g 8328 | Functions defined by well-... |
| frrlem1 8329 | Lemma for well-founded rec... |
| frrlem2 8330 | Lemma for well-founded rec... |
| frrlem3 8331 | Lemma for well-founded rec... |
| frrlem4 8332 | Lemma for well-founded rec... |
| frrlem5 8333 | Lemma for well-founded rec... |
| frrlem6 8334 | Lemma for well-founded rec... |
| frrlem7 8335 | Lemma for well-founded rec... |
| frrlem8 8336 | Lemma for well-founded rec... |
| frrlem9 8337 | Lemma for well-founded rec... |
| frrlem10 8338 | Lemma for well-founded rec... |
| frrlem11 8339 | Lemma for well-founded rec... |
| frrlem12 8340 | Lemma for well-founded rec... |
| frrlem13 8341 | Lemma for well-founded rec... |
| frrlem14 8342 | Lemma for well-founded rec... |
| fprlem1 8343 | Lemma for well-founded rec... |
| fprlem2 8344 | Lemma for well-founded rec... |
| fpr2a 8345 | Weak version of ~ fpr2 whi... |
| fpr1 8346 | Law of well-founded recurs... |
| fpr2 8347 | Law of well-founded recurs... |
| fpr3 8348 | Law of well-founded recurs... |
| frrrel 8349 | Show without using the axi... |
| frrdmss 8350 | Show without using the axi... |
| frrdmcl 8351 | Show without using the axi... |
| fprfung 8352 | A "function" defined by we... |
| fprresex 8353 | The restriction of a funct... |
| dfwrecsOLD 8356 | Obsolete definition of the... |
| wrecseq123 8357 | General equality theorem f... |
| wrecseq123OLD 8358 | Obsolete version of ~ wrec... |
| nfwrecs 8359 | Bound-variable hypothesis ... |
| nfwrecsOLD 8360 | Obsolete proof of ~ nfwrec... |
| wrecseq1 8361 | Equality theorem for the w... |
| wrecseq2 8362 | Equality theorem for the w... |
| wrecseq3 8363 | Equality theorem for the w... |
| csbwrecsg 8364 | Move class substitution in... |
| wfr3g 8365 | Functions defined by well-... |
| wfrlem1OLD 8366 | Lemma for well-ordered rec... |
| wfrlem2OLD 8367 | Lemma for well-ordered rec... |
| wfrlem3OLD 8368 | Lemma for well-ordered rec... |
| wfrlem3OLDa 8369 | Lemma for well-ordered rec... |
| wfrlem4OLD 8370 | Lemma for well-ordered rec... |
| wfrlem5OLD 8371 | Lemma for well-ordered rec... |
| wfrrelOLD 8372 | Obsolete proof of ~ wfrrel... |
| wfrdmssOLD 8373 | Obsolete proof of ~ wfrdms... |
| wfrlem8OLD 8374 | Lemma for well-ordered rec... |
| wfrdmclOLD 8375 | Obsolete version of ~ wfrd... |
| wfrlem10OLD 8376 | Lemma for well-ordered rec... |
| wfrfunOLD 8377 | Obsolete proof of ~ wfrfun... |
| wfrlem12OLD 8378 | Lemma for well-ordered rec... |
| wfrlem13OLD 8379 | Lemma for well-ordered rec... |
| wfrlem14OLD 8380 | Lemma for well-ordered rec... |
| wfrlem15OLD 8381 | Lemma for well-ordered rec... |
| wfrlem16OLD 8382 | Lemma for well-ordered rec... |
| wfrlem17OLD 8383 | Without using ~ ax-rep , s... |
| wfr2aOLD 8384 | Obsolete version of ~ wfr2... |
| wfr1OLD 8385 | Obsolete version of ~ wfr1... |
| wfr2OLD 8386 | Obsolete version of ~ wfr2... |
| wfrrel 8387 | The well-ordered recursion... |
| wfrdmss 8388 | The domain of the well-ord... |
| wfrdmcl 8389 | The predecessor class of a... |
| wfrfun 8390 | The "function" generated b... |
| wfrresex 8391 | Show without using the axi... |
| wfr2a 8392 | A weak version of ~ wfr2 w... |
| wfr1 8393 | The Principle of Well-Orde... |
| wfr2 8394 | The Principle of Well-Orde... |
| wfr3 8395 | The principle of Well-Orde... |
| wfr3OLD 8396 | Obsolete form of ~ wfr3 as... |
| iunon 8397 | The indexed union of a set... |
| iinon 8398 | The nonempty indexed inter... |
| onfununi 8399 | A property of functions on... |
| onovuni 8400 | A variant of ~ onfununi fo... |
| onoviun 8401 | A variant of ~ onovuni wit... |
| onnseq 8402 | There are no length ` _om ... |
| dfsmo2 8405 | Alternate definition of a ... |
| issmo 8406 | Conditions for which ` A `... |
| issmo2 8407 | Alternate definition of a ... |
| smoeq 8408 | Equality theorem for stric... |
| smodm 8409 | The domain of a strictly m... |
| smores 8410 | A strictly monotone functi... |
| smores3 8411 | A strictly monotone functi... |
| smores2 8412 | A strictly monotone ordina... |
| smodm2 8413 | The domain of a strictly m... |
| smofvon2 8414 | The function values of a s... |
| iordsmo 8415 | The identity relation rest... |
| smo0 8416 | The null set is a strictly... |
| smofvon 8417 | If ` B ` is a strictly mon... |
| smoel 8418 | If ` x ` is less than ` y ... |
| smoiun 8419 | The value of a strictly mo... |
| smoiso 8420 | If ` F ` is an isomorphism... |
| smoel2 8421 | A strictly monotone ordina... |
| smo11 8422 | A strictly monotone ordina... |
| smoord 8423 | A strictly monotone ordina... |
| smoword 8424 | A strictly monotone ordina... |
| smogt 8425 | A strictly monotone ordina... |
| smocdmdom 8426 | The codomain of a strictly... |
| smoiso2 8427 | The strictly monotone ordi... |
| dfrecs3 8430 | The old definition of tran... |
| dfrecs3OLD 8431 | Obsolete version of ~ dfre... |
| recseq 8432 | Equality theorem for ` rec... |
| nfrecs 8433 | Bound-variable hypothesis ... |
| tfrlem1 8434 | A technical lemma for tran... |
| tfrlem3a 8435 | Lemma for transfinite recu... |
| tfrlem3 8436 | Lemma for transfinite recu... |
| tfrlem4 8437 | Lemma for transfinite recu... |
| tfrlem5 8438 | Lemma for transfinite recu... |
| recsfval 8439 | Lemma for transfinite recu... |
| tfrlem6 8440 | Lemma for transfinite recu... |
| tfrlem7 8441 | Lemma for transfinite recu... |
| tfrlem8 8442 | Lemma for transfinite recu... |
| tfrlem9 8443 | Lemma for transfinite recu... |
| tfrlem9a 8444 | Lemma for transfinite recu... |
| tfrlem10 8445 | Lemma for transfinite recu... |
| tfrlem11 8446 | Lemma for transfinite recu... |
| tfrlem12 8447 | Lemma for transfinite recu... |
| tfrlem13 8448 | Lemma for transfinite recu... |
| tfrlem14 8449 | Lemma for transfinite recu... |
| tfrlem15 8450 | Lemma for transfinite recu... |
| tfrlem16 8451 | Lemma for finite recursion... |
| tfr1a 8452 | A weak version of ~ tfr1 w... |
| tfr2a 8453 | A weak version of ~ tfr2 w... |
| tfr2b 8454 | Without assuming ~ ax-rep ... |
| tfr1 8455 | Principle of Transfinite R... |
| tfr2 8456 | Principle of Transfinite R... |
| tfr3 8457 | Principle of Transfinite R... |
| tfr1ALT 8458 | Alternate proof of ~ tfr1 ... |
| tfr2ALT 8459 | Alternate proof of ~ tfr2 ... |
| tfr3ALT 8460 | Alternate proof of ~ tfr3 ... |
| recsfnon 8461 | Strong transfinite recursi... |
| recsval 8462 | Strong transfinite recursi... |
| tz7.44lem1 8463 | The ordered pair abstracti... |
| tz7.44-1 8464 | The value of ` F ` at ` (/... |
| tz7.44-2 8465 | The value of ` F ` at a su... |
| tz7.44-3 8466 | The value of ` F ` at a li... |
| rdgeq1 8469 | Equality theorem for the r... |
| rdgeq2 8470 | Equality theorem for the r... |
| rdgeq12 8471 | Equality theorem for the r... |
| nfrdg 8472 | Bound-variable hypothesis ... |
| rdglem1 8473 | Lemma used with the recurs... |
| rdgfun 8474 | The recursive definition g... |
| rdgdmlim 8475 | The domain of the recursiv... |
| rdgfnon 8476 | The recursive definition g... |
| rdgvalg 8477 | Value of the recursive def... |
| rdgval 8478 | Value of the recursive def... |
| rdg0 8479 | The initial value of the r... |
| rdgseg 8480 | The initial segments of th... |
| rdgsucg 8481 | The value of the recursive... |
| rdgsuc 8482 | The value of the recursive... |
| rdglimg 8483 | The value of the recursive... |
| rdglim 8484 | The value of the recursive... |
| rdg0g 8485 | The initial value of the r... |
| rdgsucmptf 8486 | The value of the recursive... |
| rdgsucmptnf 8487 | The value of the recursive... |
| rdgsucmpt2 8488 | This version of ~ rdgsucmp... |
| rdgsucmpt 8489 | The value of the recursive... |
| rdglim2 8490 | The value of the recursive... |
| rdglim2a 8491 | The value of the recursive... |
| rdg0n 8492 | If ` A ` is a proper class... |
| frfnom 8493 | The function generated by ... |
| fr0g 8494 | The initial value resultin... |
| frsuc 8495 | The successor value result... |
| frsucmpt 8496 | The successor value result... |
| frsucmptn 8497 | The value of the finite re... |
| frsucmpt2 8498 | The successor value result... |
| tz7.48lem 8499 | A way of showing an ordina... |
| tz7.48-2 8500 | Proposition 7.48(2) of [Ta... |
| tz7.48-1 8501 | Proposition 7.48(1) of [Ta... |
| tz7.48-3 8502 | Proposition 7.48(3) of [Ta... |
| tz7.49 8503 | Proposition 7.49 of [Takeu... |
| tz7.49c 8504 | Corollary of Proposition 7... |
| seqomlem0 8507 | Lemma for ` seqom ` . Cha... |
| seqomlem1 8508 | Lemma for ` seqom ` . The... |
| seqomlem2 8509 | Lemma for ` seqom ` . (Co... |
| seqomlem3 8510 | Lemma for ` seqom ` . (Co... |
| seqomlem4 8511 | Lemma for ` seqom ` . (Co... |
| seqomeq12 8512 | Equality theorem for ` seq... |
| fnseqom 8513 | An index-aware recursive d... |
| seqom0g 8514 | Value of an index-aware re... |
| seqomsuc 8515 | Value of an index-aware re... |
| omsucelsucb 8516 | Membership is inherited by... |
| df1o2 8531 | Expanded value of the ordi... |
| df2o3 8532 | Expanded value of the ordi... |
| df2o2 8533 | Expanded value of the ordi... |
| 1oex 8534 | Ordinal 1 is a set. (Cont... |
| 2oex 8535 | ` 2o ` is a set. (Contrib... |
| 1on 8536 | Ordinal 1 is an ordinal nu... |
| 1onOLD 8537 | Obsolete version of ~ 1on ... |
| 2on 8538 | Ordinal 2 is an ordinal nu... |
| 2onOLD 8539 | Obsolete version of ~ 2on ... |
| 2on0 8540 | Ordinal two is not zero. ... |
| ord3 8541 | Ordinal 3 is an ordinal cl... |
| 3on 8542 | Ordinal 3 is an ordinal nu... |
| 4on 8543 | Ordinal 4 is an ordinal nu... |
| 1oexOLD 8544 | Obsolete version of ~ 1oex... |
| 2oexOLD 8545 | Obsolete version of ~ 2oex... |
| 1n0 8546 | Ordinal one is not equal t... |
| nlim1 8547 | 1 is not a limit ordinal. ... |
| nlim2 8548 | 2 is not a limit ordinal. ... |
| xp01disj 8549 | Cartesian products with th... |
| xp01disjl 8550 | Cartesian products with th... |
| ordgt0ge1 8551 | Two ways to express that a... |
| ordge1n0 8552 | An ordinal greater than or... |
| el1o 8553 | Membership in ordinal one.... |
| ord1eln01 8554 | An ordinal that is not 0 o... |
| ord2eln012 8555 | An ordinal that is not 0, ... |
| 1ellim 8556 | A limit ordinal contains 1... |
| 2ellim 8557 | A limit ordinal contains 2... |
| dif1o 8558 | Two ways to say that ` A `... |
| ondif1 8559 | Two ways to say that ` A `... |
| ondif2 8560 | Two ways to say that ` A `... |
| 2oconcl 8561 | Closure of the pair swappi... |
| 0lt1o 8562 | Ordinal zero is less than ... |
| dif20el 8563 | An ordinal greater than on... |
| 0we1 8564 | The empty set is a well-or... |
| brwitnlem 8565 | Lemma for relations which ... |
| fnoa 8566 | Functionality and domain o... |
| fnom 8567 | Functionality and domain o... |
| fnoe 8568 | Functionality and domain o... |
| oav 8569 | Value of ordinal addition.... |
| omv 8570 | Value of ordinal multiplic... |
| oe0lem 8571 | A helper lemma for ~ oe0 a... |
| oev 8572 | Value of ordinal exponenti... |
| oevn0 8573 | Value of ordinal exponenti... |
| oa0 8574 | Addition with zero. Propo... |
| om0 8575 | Ordinal multiplication wit... |
| oe0m 8576 | Value of zero raised to an... |
| om0x 8577 | Ordinal multiplication wit... |
| oe0m0 8578 | Ordinal exponentiation wit... |
| oe0m1 8579 | Ordinal exponentiation wit... |
| oe0 8580 | Ordinal exponentiation wit... |
| oev2 8581 | Alternate value of ordinal... |
| oasuc 8582 | Addition with successor. ... |
| oesuclem 8583 | Lemma for ~ oesuc . (Cont... |
| omsuc 8584 | Multiplication with succes... |
| oesuc 8585 | Ordinal exponentiation wit... |
| onasuc 8586 | Addition with successor. ... |
| onmsuc 8587 | Multiplication with succes... |
| onesuc 8588 | Exponentiation with a succ... |
| oa1suc 8589 | Addition with 1 is same as... |
| oalim 8590 | Ordinal addition with a li... |
| omlim 8591 | Ordinal multiplication wit... |
| oelim 8592 | Ordinal exponentiation wit... |
| oacl 8593 | Closure law for ordinal ad... |
| omcl 8594 | Closure law for ordinal mu... |
| oecl 8595 | Closure law for ordinal ex... |
| oa0r 8596 | Ordinal addition with zero... |
| om0r 8597 | Ordinal multiplication wit... |
| o1p1e2 8598 | 1 + 1 = 2 for ordinal numb... |
| o2p2e4 8599 | 2 + 2 = 4 for ordinal numb... |
| om1 8600 | Ordinal multiplication wit... |
| om1r 8601 | Ordinal multiplication wit... |
| oe1 8602 | Ordinal exponentiation wit... |
| oe1m 8603 | Ordinal exponentiation wit... |
| oaordi 8604 | Ordering property of ordin... |
| oaord 8605 | Ordering property of ordin... |
| oacan 8606 | Left cancellation law for ... |
| oaword 8607 | Weak ordering property of ... |
| oawordri 8608 | Weak ordering property of ... |
| oaord1 8609 | An ordinal is less than it... |
| oaword1 8610 | An ordinal is less than or... |
| oaword2 8611 | An ordinal is less than or... |
| oawordeulem 8612 | Lemma for ~ oawordex . (C... |
| oawordeu 8613 | Existence theorem for weak... |
| oawordexr 8614 | Existence theorem for weak... |
| oawordex 8615 | Existence theorem for weak... |
| oaordex 8616 | Existence theorem for orde... |
| oa00 8617 | An ordinal sum is zero iff... |
| oalimcl 8618 | The ordinal sum with a lim... |
| oaass 8619 | Ordinal addition is associ... |
| oarec 8620 | Recursive definition of or... |
| oaf1o 8621 | Left addition by a constan... |
| oacomf1olem 8622 | Lemma for ~ oacomf1o . (C... |
| oacomf1o 8623 | Define a bijection from ` ... |
| omordi 8624 | Ordering property of ordin... |
| omord2 8625 | Ordering property of ordin... |
| omord 8626 | Ordering property of ordin... |
| omcan 8627 | Left cancellation law for ... |
| omword 8628 | Weak ordering property of ... |
| omwordi 8629 | Weak ordering property of ... |
| omwordri 8630 | Weak ordering property of ... |
| omword1 8631 | An ordinal is less than or... |
| omword2 8632 | An ordinal is less than or... |
| om00 8633 | The product of two ordinal... |
| om00el 8634 | The product of two nonzero... |
| omordlim 8635 | Ordering involving the pro... |
| omlimcl 8636 | The product of any nonzero... |
| odi 8637 | Distributive law for ordin... |
| omass 8638 | Multiplication of ordinal ... |
| oneo 8639 | If an ordinal number is ev... |
| omeulem1 8640 | Lemma for ~ omeu : existen... |
| omeulem2 8641 | Lemma for ~ omeu : uniquen... |
| omopth2 8642 | An ordered pair-like theor... |
| omeu 8643 | The division algorithm for... |
| oen0 8644 | Ordinal exponentiation wit... |
| oeordi 8645 | Ordering law for ordinal e... |
| oeord 8646 | Ordering property of ordin... |
| oecan 8647 | Left cancellation law for ... |
| oeword 8648 | Weak ordering property of ... |
| oewordi 8649 | Weak ordering property of ... |
| oewordri 8650 | Weak ordering property of ... |
| oeworde 8651 | Ordinal exponentiation com... |
| oeordsuc 8652 | Ordering property of ordin... |
| oelim2 8653 | Ordinal exponentiation wit... |
| oeoalem 8654 | Lemma for ~ oeoa . (Contr... |
| oeoa 8655 | Sum of exponents law for o... |
| oeoelem 8656 | Lemma for ~ oeoe . (Contr... |
| oeoe 8657 | Product of exponents law f... |
| oelimcl 8658 | The ordinal exponential wi... |
| oeeulem 8659 | Lemma for ~ oeeu . (Contr... |
| oeeui 8660 | The division algorithm for... |
| oeeu 8661 | The division algorithm for... |
| nna0 8662 | Addition with zero. Theor... |
| nnm0 8663 | Multiplication with zero. ... |
| nnasuc 8664 | Addition with successor. ... |
| nnmsuc 8665 | Multiplication with succes... |
| nnesuc 8666 | Exponentiation with a succ... |
| nna0r 8667 | Addition to zero. Remark ... |
| nnm0r 8668 | Multiplication with zero. ... |
| nnacl 8669 | Closure of addition of nat... |
| nnmcl 8670 | Closure of multiplication ... |
| nnecl 8671 | Closure of exponentiation ... |
| nnacli 8672 | ` _om ` is closed under ad... |
| nnmcli 8673 | ` _om ` is closed under mu... |
| nnarcl 8674 | Reverse closure law for ad... |
| nnacom 8675 | Addition of natural number... |
| nnaordi 8676 | Ordering property of addit... |
| nnaord 8677 | Ordering property of addit... |
| nnaordr 8678 | Ordering property of addit... |
| nnawordi 8679 | Adding to both sides of an... |
| nnaass 8680 | Addition of natural number... |
| nndi 8681 | Distributive law for natur... |
| nnmass 8682 | Multiplication of natural ... |
| nnmsucr 8683 | Multiplication with succes... |
| nnmcom 8684 | Multiplication of natural ... |
| nnaword 8685 | Weak ordering property of ... |
| nnacan 8686 | Cancellation law for addit... |
| nnaword1 8687 | Weak ordering property of ... |
| nnaword2 8688 | Weak ordering property of ... |
| nnmordi 8689 | Ordering property of multi... |
| nnmord 8690 | Ordering property of multi... |
| nnmword 8691 | Weak ordering property of ... |
| nnmcan 8692 | Cancellation law for multi... |
| nnmwordi 8693 | Weak ordering property of ... |
| nnmwordri 8694 | Weak ordering property of ... |
| nnawordex 8695 | Equivalence for weak order... |
| nnaordex 8696 | Equivalence for ordering. ... |
| nnaordex2 8697 | Equivalence for ordering. ... |
| 1onn 8698 | The ordinal 1 is a natural... |
| 1onnALT 8699 | Shorter proof of ~ 1onn us... |
| 2onn 8700 | The ordinal 2 is a natural... |
| 2onnALT 8701 | Shorter proof of ~ 2onn us... |
| 3onn 8702 | The ordinal 3 is a natural... |
| 4onn 8703 | The ordinal 4 is a natural... |
| 1one2o 8704 | Ordinal one is not ordinal... |
| oaabslem 8705 | Lemma for ~ oaabs . (Cont... |
| oaabs 8706 | Ordinal addition absorbs a... |
| oaabs2 8707 | The absorption law ~ oaabs... |
| omabslem 8708 | Lemma for ~ omabs . (Cont... |
| omabs 8709 | Ordinal multiplication is ... |
| nnm1 8710 | Multiply an element of ` _... |
| nnm2 8711 | Multiply an element of ` _... |
| nn2m 8712 | Multiply an element of ` _... |
| nnneo 8713 | If a natural number is eve... |
| nneob 8714 | A natural number is even i... |
| omsmolem 8715 | Lemma for ~ omsmo . (Cont... |
| omsmo 8716 | A strictly monotonic ordin... |
| omopthlem1 8717 | Lemma for ~ omopthi . (Co... |
| omopthlem2 8718 | Lemma for ~ omopthi . (Co... |
| omopthi 8719 | An ordered pair theorem fo... |
| omopth 8720 | An ordered pair theorem fo... |
| nnasmo 8721 | There is at most one left ... |
| eldifsucnn 8722 | Condition for membership i... |
| on2recsfn 8725 | Show that double recursion... |
| on2recsov 8726 | Calculate the value of the... |
| on2ind 8727 | Double induction over ordi... |
| on3ind 8728 | Triple induction over ordi... |
| coflton 8729 | Cofinality theorem for ord... |
| cofon1 8730 | Cofinality theorem for ord... |
| cofon2 8731 | Cofinality theorem for ord... |
| cofonr 8732 | Inverse cofinality law for... |
| naddfn 8733 | Natural addition is a func... |
| naddcllem 8734 | Lemma for ordinal addition... |
| naddcl 8735 | Closure law for natural ad... |
| naddov 8736 | The value of natural addit... |
| naddov2 8737 | Alternate expression for n... |
| naddov3 8738 | Alternate expression for n... |
| naddf 8739 | Function statement for nat... |
| naddcom 8740 | Natural addition commutes.... |
| naddrid 8741 | Ordinal zero is the additi... |
| naddlid 8742 | Ordinal zero is the additi... |
| naddssim 8743 | Ordinal less-than-or-equal... |
| naddelim 8744 | Ordinal less-than is prese... |
| naddel1 8745 | Ordinal less-than is not a... |
| naddel2 8746 | Ordinal less-than is not a... |
| naddss1 8747 | Ordinal less-than-or-equal... |
| naddss2 8748 | Ordinal less-than-or-equal... |
| naddword1 8749 | Weak-ordering principle fo... |
| naddword2 8750 | Weak-ordering principle fo... |
| naddunif 8751 | Uniformity theorem for nat... |
| naddasslem1 8752 | Lemma for ~ naddass . Exp... |
| naddasslem2 8753 | Lemma for ~ naddass . Exp... |
| naddass 8754 | Natural ordinal addition i... |
| nadd32 8755 | Commutative/associative la... |
| nadd4 8756 | Rearragement of terms in a... |
| nadd42 8757 | Rearragement of terms in a... |
| naddel12 8758 | Natural addition to both s... |
| naddsuc2 8759 | Natural addition with succ... |
| naddoa 8760 | Natural addition of a natu... |
| omnaddcl 8761 | The naturals are closed un... |
| dfer2 8766 | Alternate definition of eq... |
| dfec2 8768 | Alternate definition of ` ... |
| ecexg 8769 | An equivalence class modul... |
| ecexr 8770 | A nonempty equivalence cla... |
| ereq1 8772 | Equality theorem for equiv... |
| ereq2 8773 | Equality theorem for equiv... |
| errel 8774 | An equivalence relation is... |
| erdm 8775 | The domain of an equivalen... |
| ercl 8776 | Elementhood in the field o... |
| ersym 8777 | An equivalence relation is... |
| ercl2 8778 | Elementhood in the field o... |
| ersymb 8779 | An equivalence relation is... |
| ertr 8780 | An equivalence relation is... |
| ertrd 8781 | A transitivity relation fo... |
| ertr2d 8782 | A transitivity relation fo... |
| ertr3d 8783 | A transitivity relation fo... |
| ertr4d 8784 | A transitivity relation fo... |
| erref 8785 | An equivalence relation is... |
| ercnv 8786 | The converse of an equival... |
| errn 8787 | The range and domain of an... |
| erssxp 8788 | An equivalence relation is... |
| erex 8789 | An equivalence relation is... |
| erexb 8790 | An equivalence relation is... |
| iserd 8791 | A reflexive, symmetric, tr... |
| iseri 8792 | A reflexive, symmetric, tr... |
| iseriALT 8793 | Alternate proof of ~ iseri... |
| brinxper 8794 | Conditions for a reflexive... |
| brdifun 8795 | Evaluate the incomparabili... |
| swoer 8796 | Incomparability under a st... |
| swoord1 8797 | The incomparability equiva... |
| swoord2 8798 | The incomparability equiva... |
| swoso 8799 | If the incomparability rel... |
| eqerlem 8800 | Lemma for ~ eqer . (Contr... |
| eqer 8801 | Equivalence relation invol... |
| ider 8802 | The identity relation is a... |
| 0er 8803 | The empty set is an equiva... |
| eceq1 8804 | Equality theorem for equiv... |
| eceq1d 8805 | Equality theorem for equiv... |
| eceq2 8806 | Equality theorem for equiv... |
| eceq2i 8807 | Equality theorem for the `... |
| eceq2d 8808 | Equality theorem for the `... |
| elecg 8809 | Membership in an equivalen... |
| ecref 8810 | All elements are in their ... |
| elec 8811 | Membership in an equivalen... |
| relelec 8812 | Membership in an equivalen... |
| ecss 8813 | An equivalence class is a ... |
| ecdmn0 8814 | A representative of a none... |
| ereldm 8815 | Equality of equivalence cl... |
| erth 8816 | Basic property of equivale... |
| erth2 8817 | Basic property of equivale... |
| erthi 8818 | Basic property of equivale... |
| erdisj 8819 | Equivalence classes do not... |
| ecidsn 8820 | An equivalence class modul... |
| qseq1 8821 | Equality theorem for quoti... |
| qseq2 8822 | Equality theorem for quoti... |
| qseq2i 8823 | Equality theorem for quoti... |
| qseq1d 8824 | Equality theorem for quoti... |
| qseq2d 8825 | Equality theorem for quoti... |
| qseq12 8826 | Equality theorem for quoti... |
| 0qs 8827 | Quotient set with the empt... |
| elqsg 8828 | Closed form of ~ elqs . (... |
| elqs 8829 | Membership in a quotient s... |
| elqsi 8830 | Membership in a quotient s... |
| elqsecl 8831 | Membership in a quotient s... |
| ecelqsg 8832 | Membership of an equivalen... |
| ecelqsi 8833 | Membership of an equivalen... |
| ecopqsi 8834 | "Closure" law for equivale... |
| qsexg 8835 | A quotient set exists. (C... |
| qsex 8836 | A quotient set exists. (C... |
| uniqs 8837 | The union of a quotient se... |
| qsss 8838 | A quotient set is a set of... |
| uniqs2 8839 | The union of a quotient se... |
| snec 8840 | The singleton of an equiva... |
| ecqs 8841 | Equivalence class in terms... |
| ecid 8842 | A set is equal to its cose... |
| qsid 8843 | A set is equal to its quot... |
| ectocld 8844 | Implicit substitution of c... |
| ectocl 8845 | Implicit substitution of c... |
| elqsn0 8846 | A quotient set does not co... |
| ecelqsdm 8847 | Membership of an equivalen... |
| xpider 8848 | A Cartesian square is an e... |
| iiner 8849 | The intersection of a none... |
| riiner 8850 | The relative intersection ... |
| erinxp 8851 | A restricted equivalence r... |
| ecinxp 8852 | Restrict the relation in a... |
| qsinxp 8853 | Restrict the equivalence r... |
| qsdisj 8854 | Members of a quotient set ... |
| qsdisj2 8855 | A quotient set is a disjoi... |
| qsel 8856 | If an element of a quotien... |
| uniinqs 8857 | Class union distributes ov... |
| qliftlem 8858 | Lemma for theorems about a... |
| qliftrel 8859 | ` F ` , a function lift, i... |
| qliftel 8860 | Elementhood in the relatio... |
| qliftel1 8861 | Elementhood in the relatio... |
| qliftfun 8862 | The function ` F ` is the ... |
| qliftfund 8863 | The function ` F ` is the ... |
| qliftfuns 8864 | The function ` F ` is the ... |
| qliftf 8865 | The domain and codomain of... |
| qliftval 8866 | The value of the function ... |
| ecoptocl 8867 | Implicit substitution of c... |
| 2ecoptocl 8868 | Implicit substitution of c... |
| 3ecoptocl 8869 | Implicit substitution of c... |
| brecop 8870 | Binary relation on a quoti... |
| brecop2 8871 | Binary relation on a quoti... |
| eroveu 8872 | Lemma for ~ erov and ~ ero... |
| erovlem 8873 | Lemma for ~ erov and ~ ero... |
| erov 8874 | The value of an operation ... |
| eroprf 8875 | Functionality of an operat... |
| erov2 8876 | The value of an operation ... |
| eroprf2 8877 | Functionality of an operat... |
| ecopoveq 8878 | This is the first of sever... |
| ecopovsym 8879 | Assuming the operation ` F... |
| ecopovtrn 8880 | Assuming that operation ` ... |
| ecopover 8881 | Assuming that operation ` ... |
| eceqoveq 8882 | Equality of equivalence re... |
| ecovcom 8883 | Lemma used to transfer a c... |
| ecovass 8884 | Lemma used to transfer an ... |
| ecovdi 8885 | Lemma used to transfer a d... |
| mapprc 8890 | When ` A ` is a proper cla... |
| pmex 8891 | The class of all partial f... |
| mapexOLD 8892 | Obsolete version of ~ mape... |
| fnmap 8893 | Set exponentiation has a u... |
| fnpm 8894 | Partial function exponenti... |
| reldmmap 8895 | Set exponentiation is a we... |
| mapvalg 8896 | The value of set exponenti... |
| pmvalg 8897 | The value of the partial m... |
| mapval 8898 | The value of set exponenti... |
| elmapg 8899 | Membership relation for se... |
| elmapd 8900 | Deduction form of ~ elmapg... |
| elmapdd 8901 | Deduction associated with ... |
| mapdm0 8902 | The empty set is the only ... |
| elpmg 8903 | The predicate "is a partia... |
| elpm2g 8904 | The predicate "is a partia... |
| elpm2r 8905 | Sufficient condition for b... |
| elpmi 8906 | A partial function is a fu... |
| pmfun 8907 | A partial function is a fu... |
| elmapex 8908 | Eliminate antecedent for m... |
| elmapi 8909 | A mapping is a function, f... |
| mapfset 8910 | If ` B ` is a set, the val... |
| mapssfset 8911 | The value of the set expon... |
| mapfoss 8912 | The value of the set expon... |
| fsetsspwxp 8913 | The class of all functions... |
| fset0 8914 | The set of functions from ... |
| fsetdmprc0 8915 | The set of functions with ... |
| fsetex 8916 | The set of functions betwe... |
| f1setex 8917 | The set of injections betw... |
| fosetex 8918 | The set of surjections bet... |
| f1osetex 8919 | The set of bijections betw... |
| fsetfcdm 8920 | The class of functions wit... |
| fsetfocdm 8921 | The class of functions wit... |
| fsetprcnex 8922 | The class of all functions... |
| fsetcdmex 8923 | The class of all functions... |
| fsetexb 8924 | The class of all functions... |
| elmapfn 8925 | A mapping is a function wi... |
| elmapfun 8926 | A mapping is always a func... |
| elmapssres 8927 | A restricted mapping is a ... |
| fpmg 8928 | A total function is a part... |
| pmss12g 8929 | Subset relation for the se... |
| pmresg 8930 | Elementhood of a restricte... |
| elmap 8931 | Membership relation for se... |
| mapval2 8932 | Alternate expression for t... |
| elpm 8933 | The predicate "is a partia... |
| elpm2 8934 | The predicate "is a partia... |
| fpm 8935 | A total function is a part... |
| mapsspm 8936 | Set exponentiation is a su... |
| pmsspw 8937 | Partial maps are a subset ... |
| mapsspw 8938 | Set exponentiation is a su... |
| mapfvd 8939 | The value of a function th... |
| elmapresaun 8940 | ~ fresaun transposed to ma... |
| fvmptmap 8941 | Special case of ~ fvmpt fo... |
| map0e 8942 | Set exponentiation with an... |
| map0b 8943 | Set exponentiation with an... |
| map0g 8944 | Set exponentiation is empt... |
| 0map0sn0 8945 | The set of mappings of the... |
| mapsnd 8946 | The value of set exponenti... |
| map0 8947 | Set exponentiation is empt... |
| mapsn 8948 | The value of set exponenti... |
| mapss 8949 | Subset inheritance for set... |
| fdiagfn 8950 | Functionality of the diago... |
| fvdiagfn 8951 | Functionality of the diago... |
| mapsnconst 8952 | Every singleton map is a c... |
| mapsncnv 8953 | Expression for the inverse... |
| mapsnf1o2 8954 | Explicit bijection between... |
| mapsnf1o3 8955 | Explicit bijection in the ... |
| ralxpmap 8956 | Quantification over functi... |
| dfixp 8959 | Eliminate the expression `... |
| ixpsnval 8960 | The value of an infinite C... |
| elixp2 8961 | Membership in an infinite ... |
| fvixp 8962 | Projection of a factor of ... |
| ixpfn 8963 | A nuple is a function. (C... |
| elixp 8964 | Membership in an infinite ... |
| elixpconst 8965 | Membership in an infinite ... |
| ixpconstg 8966 | Infinite Cartesian product... |
| ixpconst 8967 | Infinite Cartesian product... |
| ixpeq1 8968 | Equality theorem for infin... |
| ixpeq1d 8969 | Equality theorem for infin... |
| ss2ixp 8970 | Subclass theorem for infin... |
| ixpeq2 8971 | Equality theorem for infin... |
| ixpeq2dva 8972 | Equality theorem for infin... |
| ixpeq2dv 8973 | Equality theorem for infin... |
| cbvixp 8974 | Change bound variable in a... |
| cbvixpv 8975 | Change bound variable in a... |
| nfixpw 8976 | Bound-variable hypothesis ... |
| nfixp 8977 | Bound-variable hypothesis ... |
| nfixp1 8978 | The index variable in an i... |
| ixpprc 8979 | A cartesian product of pro... |
| ixpf 8980 | A member of an infinite Ca... |
| uniixp 8981 | The union of an infinite C... |
| ixpexg 8982 | The existence of an infini... |
| ixpin 8983 | The intersection of two in... |
| ixpiin 8984 | The indexed intersection o... |
| ixpint 8985 | The intersection of a coll... |
| ixp0x 8986 | An infinite Cartesian prod... |
| ixpssmap2g 8987 | An infinite Cartesian prod... |
| ixpssmapg 8988 | An infinite Cartesian prod... |
| 0elixp 8989 | Membership of the empty se... |
| ixpn0 8990 | The infinite Cartesian pro... |
| ixp0 8991 | The infinite Cartesian pro... |
| ixpssmap 8992 | An infinite Cartesian prod... |
| resixp 8993 | Restriction of an element ... |
| undifixp 8994 | Union of two projections o... |
| mptelixpg 8995 | Condition for an explicit ... |
| resixpfo 8996 | Restriction of elements of... |
| elixpsn 8997 | Membership in a class of s... |
| ixpsnf1o 8998 | A bijection between a clas... |
| mapsnf1o 8999 | A bijection between a set ... |
| boxriin 9000 | A rectangular subset of a ... |
| boxcutc 9001 | The relative complement of... |
| relen 9010 | Equinumerosity is a relati... |
| reldom 9011 | Dominance is a relation. ... |
| relsdom 9012 | Strict dominance is a rela... |
| encv 9013 | If two classes are equinum... |
| breng 9014 | Equinumerosity relation. ... |
| bren 9015 | Equinumerosity relation. ... |
| brenOLD 9016 | Obsolete version of ~ bren... |
| brdom2g 9017 | Dominance relation. This ... |
| brdomg 9018 | Dominance relation. (Cont... |
| brdomgOLD 9019 | Obsolete version of ~ brdo... |
| brdomi 9020 | Dominance relation. (Cont... |
| brdomiOLD 9021 | Obsolete version of ~ brdo... |
| brdom 9022 | Dominance relation. (Cont... |
| domen 9023 | Dominance in terms of equi... |
| domeng 9024 | Dominance in terms of equi... |
| ctex 9025 | A countable set is a set. ... |
| f1oen4g 9026 | The domain and range of a ... |
| f1dom4g 9027 | The domain of a one-to-one... |
| f1oen3g 9028 | The domain and range of a ... |
| f1dom3g 9029 | The domain of a one-to-one... |
| f1oen2g 9030 | The domain and range of a ... |
| f1dom2g 9031 | The domain of a one-to-one... |
| f1dom2gOLD 9032 | Obsolete version of ~ f1do... |
| f1oeng 9033 | The domain and range of a ... |
| f1domg 9034 | The domain of a one-to-one... |
| f1oen 9035 | The domain and range of a ... |
| f1dom 9036 | The domain of a one-to-one... |
| brsdom 9037 | Strict dominance relation,... |
| isfi 9038 | Express " ` A ` is finite"... |
| enssdom 9039 | Equinumerosity implies dom... |
| dfdom2 9040 | Alternate definition of do... |
| endom 9041 | Equinumerosity implies dom... |
| sdomdom 9042 | Strict dominance implies d... |
| sdomnen 9043 | Strict dominance implies n... |
| brdom2 9044 | Dominance in terms of stri... |
| bren2 9045 | Equinumerosity expressed i... |
| enrefg 9046 | Equinumerosity is reflexiv... |
| enref 9047 | Equinumerosity is reflexiv... |
| eqeng 9048 | Equality implies equinumer... |
| domrefg 9049 | Dominance is reflexive. (... |
| en2d 9050 | Equinumerosity inference f... |
| en3d 9051 | Equinumerosity inference f... |
| en2i 9052 | Equinumerosity inference f... |
| en3i 9053 | Equinumerosity inference f... |
| dom2lem 9054 | A mapping (first hypothesi... |
| dom2d 9055 | A mapping (first hypothesi... |
| dom3d 9056 | A mapping (first hypothesi... |
| dom2 9057 | A mapping (first hypothesi... |
| dom3 9058 | A mapping (first hypothesi... |
| idssen 9059 | Equality implies equinumer... |
| domssl 9060 | If ` A ` is a subset of ` ... |
| domssr 9061 | If ` C ` is a superset of ... |
| ssdomg 9062 | A set dominates its subset... |
| ener 9063 | Equinumerosity is an equiv... |
| ensymb 9064 | Symmetry of equinumerosity... |
| ensym 9065 | Symmetry of equinumerosity... |
| ensymi 9066 | Symmetry of equinumerosity... |
| ensymd 9067 | Symmetry of equinumerosity... |
| entr 9068 | Transitivity of equinumero... |
| domtr 9069 | Transitivity of dominance ... |
| entri 9070 | A chained equinumerosity i... |
| entr2i 9071 | A chained equinumerosity i... |
| entr3i 9072 | A chained equinumerosity i... |
| entr4i 9073 | A chained equinumerosity i... |
| endomtr 9074 | Transitivity of equinumero... |
| domentr 9075 | Transitivity of dominance ... |
| f1imaeng 9076 | If a function is one-to-on... |
| f1imaen2g 9077 | If a function is one-to-on... |
| f1imaen3g 9078 | If a set function is one-t... |
| f1imaen 9079 | If a function is one-to-on... |
| en0 9080 | The empty set is equinumer... |
| en0OLD 9081 | Obsolete version of ~ en0 ... |
| en0ALT 9082 | Shorter proof of ~ en0 , d... |
| en0r 9083 | The empty set is equinumer... |
| ensn1 9084 | A singleton is equinumerou... |
| ensn1OLD 9085 | Obsolete version of ~ ensn... |
| ensn1g 9086 | A singleton is equinumerou... |
| enpr1g 9087 | ` { A , A } ` has only one... |
| en1 9088 | A set is equinumerous to o... |
| en1OLD 9089 | Obsolete version of ~ en1 ... |
| en1b 9090 | A set is equinumerous to o... |
| en1bOLD 9091 | Obsolete version of ~ en1b... |
| reuen1 9092 | Two ways to express "exact... |
| euen1 9093 | Two ways to express "exact... |
| euen1b 9094 | Two ways to express " ` A ... |
| en1uniel 9095 | A singleton contains its s... |
| en1unielOLD 9096 | Obsolete version of ~ en1u... |
| 2dom 9097 | A set that dominates ordin... |
| fundmen 9098 | A function is equinumerous... |
| fundmeng 9099 | A function is equinumerous... |
| cnven 9100 | A relational set is equinu... |
| cnvct 9101 | If a set is countable, so ... |
| fndmeng 9102 | A function is equinumerate... |
| mapsnend 9103 | Set exponentiation to a si... |
| mapsnen 9104 | Set exponentiation to a si... |
| snmapen 9105 | Set exponentiation: a sing... |
| snmapen1 9106 | Set exponentiation: a sing... |
| map1 9107 | Set exponentiation: ordina... |
| en2sn 9108 | Two singletons are equinum... |
| en2snOLD 9109 | Obsolete version of ~ en2s... |
| 0fi 9110 | The empty set is finite. ... |
| snfi 9111 | A singleton is finite. (C... |
| snfiOLD 9112 | Obsolete version of ~ snfi... |
| fiprc 9113 | The class of finite sets i... |
| unen 9114 | Equinumerosity of union of... |
| enrefnn 9115 | Equinumerosity is reflexiv... |
| en2prd 9116 | Two unordered pairs are eq... |
| enpr2d 9117 | A pair with distinct eleme... |
| enpr2dOLD 9118 | Obsolete version of ~ enpr... |
| ssct 9119 | Any subset of a countable ... |
| ssctOLD 9120 | Obsolete version of ~ ssct... |
| difsnen 9121 | All decrements of a set ar... |
| domdifsn 9122 | Dominance over a set with ... |
| xpsnen 9123 | A set is equinumerous to i... |
| xpsneng 9124 | A set is equinumerous to i... |
| xp1en 9125 | One times a cardinal numbe... |
| endisj 9126 | Any two sets are equinumer... |
| undom 9127 | Dominance law for union. ... |
| undomOLD 9128 | Obsolete version of ~ undo... |
| xpcomf1o 9129 | The canonical bijection fr... |
| xpcomco 9130 | Composition with the bijec... |
| xpcomen 9131 | Commutative law for equinu... |
| xpcomeng 9132 | Commutative law for equinu... |
| xpsnen2g 9133 | A set is equinumerous to i... |
| xpassen 9134 | Associative law for equinu... |
| xpdom2 9135 | Dominance law for Cartesia... |
| xpdom2g 9136 | Dominance law for Cartesia... |
| xpdom1g 9137 | Dominance law for Cartesia... |
| xpdom3 9138 | A set is dominated by its ... |
| xpdom1 9139 | Dominance law for Cartesia... |
| domunsncan 9140 | A singleton cancellation l... |
| omxpenlem 9141 | Lemma for ~ omxpen . (Con... |
| omxpen 9142 | The cardinal and ordinal p... |
| omf1o 9143 | Construct an explicit bije... |
| pw2f1olem 9144 | Lemma for ~ pw2f1o . (Con... |
| pw2f1o 9145 | The power set of a set is ... |
| pw2eng 9146 | The power set of a set is ... |
| pw2en 9147 | The power set of a set is ... |
| fopwdom 9148 | Covering implies injection... |
| enfixsn 9149 | Given two equipollent sets... |
| sucdom2OLD 9150 | Obsolete version of ~ sucd... |
| sbthlem1 9151 | Lemma for ~ sbth . (Contr... |
| sbthlem2 9152 | Lemma for ~ sbth . (Contr... |
| sbthlem3 9153 | Lemma for ~ sbth . (Contr... |
| sbthlem4 9154 | Lemma for ~ sbth . (Contr... |
| sbthlem5 9155 | Lemma for ~ sbth . (Contr... |
| sbthlem6 9156 | Lemma for ~ sbth . (Contr... |
| sbthlem7 9157 | Lemma for ~ sbth . (Contr... |
| sbthlem8 9158 | Lemma for ~ sbth . (Contr... |
| sbthlem9 9159 | Lemma for ~ sbth . (Contr... |
| sbthlem10 9160 | Lemma for ~ sbth . (Contr... |
| sbth 9161 | Schroeder-Bernstein Theore... |
| sbthb 9162 | Schroeder-Bernstein Theore... |
| sbthcl 9163 | Schroeder-Bernstein Theore... |
| dfsdom2 9164 | Alternate definition of st... |
| brsdom2 9165 | Alternate definition of st... |
| sdomnsym 9166 | Strict dominance is asymme... |
| domnsym 9167 | Theorem 22(i) of [Suppes] ... |
| 0domg 9168 | Any set dominates the empt... |
| 0domgOLD 9169 | Obsolete version of ~ 0dom... |
| dom0 9170 | A set dominated by the emp... |
| dom0OLD 9171 | Obsolete version of ~ dom0... |
| 0sdomg 9172 | A set strictly dominates t... |
| 0sdomgOLD 9173 | Obsolete version of ~ 0sdo... |
| 0dom 9174 | Any set dominates the empt... |
| 0sdom 9175 | A set strictly dominates t... |
| sdom0 9176 | The empty set does not str... |
| sdom0OLD 9177 | Obsolete version of ~ sdom... |
| sdomdomtr 9178 | Transitivity of strict dom... |
| sdomentr 9179 | Transitivity of strict dom... |
| domsdomtr 9180 | Transitivity of dominance ... |
| ensdomtr 9181 | Transitivity of equinumero... |
| sdomirr 9182 | Strict dominance is irrefl... |
| sdomtr 9183 | Strict dominance is transi... |
| sdomn2lp 9184 | Strict dominance has no 2-... |
| enen1 9185 | Equality-like theorem for ... |
| enen2 9186 | Equality-like theorem for ... |
| domen1 9187 | Equality-like theorem for ... |
| domen2 9188 | Equality-like theorem for ... |
| sdomen1 9189 | Equality-like theorem for ... |
| sdomen2 9190 | Equality-like theorem for ... |
| domtriord 9191 | Dominance is trichotomous ... |
| sdomel 9192 | For ordinals, strict domin... |
| sdomdif 9193 | The difference of a set fr... |
| onsdominel 9194 | An ordinal with more eleme... |
| domunsn 9195 | Dominance over a set with ... |
| fodomr 9196 | There exists a mapping fro... |
| pwdom 9197 | Injection of sets implies ... |
| canth2 9198 | Cantor's Theorem. No set ... |
| canth2g 9199 | Cantor's theorem with the ... |
| 2pwuninel 9200 | The power set of the power... |
| 2pwne 9201 | No set equals the power se... |
| disjen 9202 | A stronger form of ~ pwuni... |
| disjenex 9203 | Existence version of ~ dis... |
| domss2 9204 | A corollary of ~ disjenex ... |
| domssex2 9205 | A corollary of ~ disjenex ... |
| domssex 9206 | Weakening of ~ domssex2 to... |
| xpf1o 9207 | Construct a bijection on a... |
| xpen 9208 | Equinumerosity law for Car... |
| mapen 9209 | Two set exponentiations ar... |
| mapdom1 9210 | Order-preserving property ... |
| mapxpen 9211 | Equinumerosity law for dou... |
| xpmapenlem 9212 | Lemma for ~ xpmapen . (Co... |
| xpmapen 9213 | Equinumerosity law for set... |
| mapunen 9214 | Equinumerosity law for set... |
| map2xp 9215 | A cardinal power with expo... |
| mapdom2 9216 | Order-preserving property ... |
| mapdom3 9217 | Set exponentiation dominat... |
| pwen 9218 | If two sets are equinumero... |
| ssenen 9219 | Equinumerosity of equinume... |
| limenpsi 9220 | A limit ordinal is equinum... |
| limensuci 9221 | A limit ordinal is equinum... |
| limensuc 9222 | A limit ordinal is equinum... |
| infensuc 9223 | Any infinite ordinal is eq... |
| dif1enlem 9224 | Lemma for ~ rexdif1en and ... |
| dif1enlemOLD 9225 | Obsolete version of ~ dif1... |
| rexdif1en 9226 | If a set is equinumerous t... |
| rexdif1enOLD 9227 | Obsolete version of ~ rexd... |
| dif1en 9228 | If a set ` A ` is equinume... |
| dif1ennn 9229 | If a set ` A ` is equinume... |
| dif1enOLD 9230 | Obsolete version of ~ dif1... |
| findcard 9231 | Schema for induction on th... |
| findcard2 9232 | Schema for induction on th... |
| findcard2s 9233 | Variation of ~ findcard2 r... |
| findcard2d 9234 | Deduction version of ~ fin... |
| nnfi 9235 | Natural numbers are finite... |
| pssnn 9236 | A proper subset of a natur... |
| ssnnfi 9237 | A subset of a natural numb... |
| ssnnfiOLD 9238 | Obsolete version of ~ ssnn... |
| 0finOLD 9239 | Obsolete version of ~ 0fi ... |
| unfi 9240 | The union of two finite se... |
| unfid 9241 | The union of two finite se... |
| ssfi 9242 | A subset of a finite set i... |
| ssfiALT 9243 | Shorter proof of ~ ssfi us... |
| diffi 9244 | If ` A ` is finite, ` ( A ... |
| cnvfi 9245 | If a set is finite, its co... |
| fnfi 9246 | A version of ~ fnex for fi... |
| f1oenfi 9247 | If the domain of a one-to-... |
| f1oenfirn 9248 | If the range of a one-to-o... |
| f1domfi 9249 | If the codomain of a one-t... |
| f1domfi2 9250 | If the domain of a one-to-... |
| enreffi 9251 | Equinumerosity is reflexiv... |
| ensymfib 9252 | Symmetry of equinumerosity... |
| entrfil 9253 | Transitivity of equinumero... |
| enfii 9254 | A set equinumerous to a fi... |
| enfi 9255 | Equinumerous sets have the... |
| enfiALT 9256 | Shorter proof of ~ enfi us... |
| domfi 9257 | A set dominated by a finit... |
| entrfi 9258 | Transitivity of equinumero... |
| entrfir 9259 | Transitivity of equinumero... |
| domtrfil 9260 | Transitivity of dominance ... |
| domtrfi 9261 | Transitivity of dominance ... |
| domtrfir 9262 | Transitivity of dominance ... |
| f1imaenfi 9263 | If a function is one-to-on... |
| ssdomfi 9264 | A finite set dominates its... |
| ssdomfi2 9265 | A set dominates its finite... |
| sbthfilem 9266 | Lemma for ~ sbthfi . (Con... |
| sbthfi 9267 | Schroeder-Bernstein Theore... |
| domnsymfi 9268 | If a set dominates a finit... |
| sdomdomtrfi 9269 | Transitivity of strict dom... |
| domsdomtrfi 9270 | Transitivity of dominance ... |
| sucdom2 9271 | Strict dominance of a set ... |
| phplem1 9272 | Lemma for Pigeonhole Princ... |
| phplem2 9273 | Lemma for Pigeonhole Princ... |
| nneneq 9274 | Two equinumerous natural n... |
| php 9275 | Pigeonhole Principle. A n... |
| php2 9276 | Corollary of Pigeonhole Pr... |
| php3 9277 | Corollary of Pigeonhole Pr... |
| php4 9278 | Corollary of the Pigeonhol... |
| php5 9279 | Corollary of the Pigeonhol... |
| phpeqd 9280 | Corollary of the Pigeonhol... |
| nndomog 9281 | Cardinal ordering agrees w... |
| phplem1OLD 9282 | Obsolete lemma for ~ php a... |
| phplem2OLD 9283 | Obsolete lemma for ~ php a... |
| phplem3OLD 9284 | Obsolete version of ~ phpl... |
| phplem4OLD 9285 | Obsolete version of ~ phpl... |
| nneneqOLD 9286 | Obsolete version of ~ nnen... |
| phpOLD 9287 | Obsolete version of ~ php ... |
| php2OLD 9288 | Obsolete version of ~ php2... |
| php3OLD 9289 | Obsolete version of ~ php3... |
| phpeqdOLD 9290 | Obsolete version of ~ phpe... |
| nndomogOLD 9291 | Obsolete version of ~ nndo... |
| snnen2oOLD 9292 | Obsolete version of ~ snne... |
| onomeneq 9293 | An ordinal number equinume... |
| onomeneqOLD 9294 | Obsolete version of ~ onom... |
| onfin 9295 | An ordinal number is finit... |
| onfin2 9296 | A set is a natural number ... |
| nnfiOLD 9297 | Obsolete version of ~ nnfi... |
| nndomo 9298 | Cardinal ordering agrees w... |
| nnsdomo 9299 | Cardinal ordering agrees w... |
| sucdom 9300 | Strict dominance of a set ... |
| sucdomOLD 9301 | Obsolete version of ~ sucd... |
| snnen2o 9302 | A singleton ` { A } ` is n... |
| 0sdom1dom 9303 | Strict dominance over 0 is... |
| 0sdom1domALT 9304 | Alternate proof of ~ 0sdom... |
| 1sdom2 9305 | Ordinal 1 is strictly domi... |
| 1sdom2ALT 9306 | Alternate proof of ~ 1sdom... |
| sdom1 9307 | A set has less than one me... |
| sdom1OLD 9308 | Obsolete version of ~ sdom... |
| modom 9309 | Two ways to express "at mo... |
| modom2 9310 | Two ways to express "at mo... |
| rex2dom 9311 | A set that has at least 2 ... |
| 1sdom2dom 9312 | Strict dominance over 1 is... |
| 1sdom 9313 | A set that strictly domina... |
| 1sdomOLD 9314 | Obsolete version of ~ 1sdo... |
| unxpdomlem1 9315 | Lemma for ~ unxpdom . (Tr... |
| unxpdomlem2 9316 | Lemma for ~ unxpdom . (Co... |
| unxpdomlem3 9317 | Lemma for ~ unxpdom . (Co... |
| unxpdom 9318 | Cartesian product dominate... |
| unxpdom2 9319 | Corollary of ~ unxpdom . ... |
| sucxpdom 9320 | Cartesian product dominate... |
| pssinf 9321 | A set equinumerous to a pr... |
| fisseneq 9322 | A finite set is equal to i... |
| ominf 9323 | The set of natural numbers... |
| ominfOLD 9324 | Obsolete version of ~ omin... |
| isinf 9325 | Any set that is not finite... |
| isinfOLD 9326 | Obsolete version of ~ isin... |
| fineqvlem 9327 | Lemma for ~ fineqv . (Con... |
| fineqv 9328 | If the Axiom of Infinity i... |
| enfiiOLD 9329 | Obsolete version of ~ enfi... |
| xpfir 9330 | The components of a nonemp... |
| ssfid 9331 | A subset of a finite set i... |
| infi 9332 | The intersection of two se... |
| rabfi 9333 | A restricted class built f... |
| finresfin 9334 | The restriction of a finit... |
| f1finf1o 9335 | Any injection from one fin... |
| f1finf1oOLD 9336 | Obsolete version of ~ f1fi... |
| nfielex 9337 | If a class is not finite, ... |
| en1eqsn 9338 | A set with one element is ... |
| en1eqsnOLD 9339 | Obsolete version of ~ en1e... |
| en1eqsnbi 9340 | A set containing an elemen... |
| dif1ennnALT 9341 | Alternate proof of ~ dif1e... |
| enp1ilem 9342 | Lemma for uses of ~ enp1i ... |
| enp1i 9343 | Proof induction for ~ en2 ... |
| enp1iOLD 9344 | Obsolete version of ~ enp1... |
| en2 9345 | A set equinumerous to ordi... |
| en3 9346 | A set equinumerous to ordi... |
| en4 9347 | A set equinumerous to ordi... |
| findcard3 9348 | Schema for strong inductio... |
| findcard3OLD 9349 | Obsolete version of ~ find... |
| ac6sfi 9350 | A version of ~ ac6s for fi... |
| frfi 9351 | A partial order is well-fo... |
| fimax2g 9352 | A finite set has a maximum... |
| fimaxg 9353 | A finite set has a maximum... |
| fisupg 9354 | Lemma showing existence an... |
| wofi 9355 | A total order on a finite ... |
| ordunifi 9356 | The maximum of a finite co... |
| nnunifi 9357 | The union (supremum) of a ... |
| unblem1 9358 | Lemma for ~ unbnn . After... |
| unblem2 9359 | Lemma for ~ unbnn . The v... |
| unblem3 9360 | Lemma for ~ unbnn . The v... |
| unblem4 9361 | Lemma for ~ unbnn . The f... |
| unbnn 9362 | Any unbounded subset of na... |
| unbnn2 9363 | Version of ~ unbnn that do... |
| isfinite2 9364 | Any set strictly dominated... |
| nnsdomg 9365 | Omega strictly dominates a... |
| nnsdomgOLD 9366 | Obsolete version of ~ nnsd... |
| isfiniteg 9367 | A set is finite iff it is ... |
| infsdomnn 9368 | An infinite set strictly d... |
| infsdomnnOLD 9369 | Obsolete version of ~ infs... |
| infn0 9370 | An infinite set is not emp... |
| infn0ALT 9371 | Shorter proof of ~ infn0 u... |
| fin2inf 9372 | This (useless) theorem, wh... |
| unfilem1 9373 | Lemma for proving that the... |
| unfilem2 9374 | Lemma for proving that the... |
| unfilem3 9375 | Lemma for proving that the... |
| unfir 9376 | If a union is finite, the ... |
| unfib 9377 | A union is finite if and o... |
| unfi2 9378 | The union of two finite se... |
| difinf 9379 | An infinite set ` A ` minu... |
| fodomfi 9380 | An onto function implies d... |
| fofi 9381 | If an onto function has a ... |
| f1fi 9382 | If a 1-to-1 function has a... |
| imafi 9383 | Images of finite sets are ... |
| imafiOLD 9384 | Obsolete version of ~ imaf... |
| pwfir 9385 | If the power set of a set ... |
| pwfilem 9386 | Lemma for ~ pwfi . (Contr... |
| pwfi 9387 | The power set of a finite ... |
| xpfi 9388 | The Cartesian product of t... |
| xpfiOLD 9389 | Obsolete version of ~ xpfi... |
| 3xpfi 9390 | The Cartesian product of t... |
| domunfican 9391 | A finite set union cancell... |
| infcntss 9392 | Every infinite set has a d... |
| prfi 9393 | An unordered pair is finit... |
| prfiALT 9394 | Shorter proof of ~ prfi us... |
| tpfi 9395 | An unordered triple is fin... |
| fiint 9396 | Equivalent ways of stating... |
| fiintOLD 9397 | Obsolete version of ~ fiin... |
| fodomfir 9398 | There exists a mapping fro... |
| fodomfib 9399 | Equivalence of an onto map... |
| fodomfiOLD 9400 | Obsolete version of ~ fodo... |
| fodomfibOLD 9401 | Obsolete version of ~ fodo... |
| fofinf1o 9402 | Any surjection from one fi... |
| rneqdmfinf1o 9403 | Any function from a finite... |
| fidomdm 9404 | Any finite set dominates i... |
| dmfi 9405 | The domain of a finite set... |
| fundmfibi 9406 | A function is finite if an... |
| resfnfinfin 9407 | The restriction of a funct... |
| residfi 9408 | A restricted identity func... |
| cnvfiALT 9409 | Shorter proof of ~ cnvfi u... |
| rnfi 9410 | The range of a finite set ... |
| f1dmvrnfibi 9411 | A one-to-one function whos... |
| f1vrnfibi 9412 | A one-to-one function whic... |
| iunfi 9413 | The finite union of finite... |
| unifi 9414 | The finite union of finite... |
| unifi2 9415 | The finite union of finite... |
| infssuni 9416 | If an infinite set ` A ` i... |
| unirnffid 9417 | The union of the range of ... |
| pwfilemOLD 9418 | Obsolete version of ~ pwfi... |
| pwfiOLD 9419 | Obsolete version of ~ pwfi... |
| mapfi 9420 | Set exponentiation of fini... |
| ixpfi 9421 | A Cartesian product of fin... |
| ixpfi2 9422 | A Cartesian product of fin... |
| mptfi 9423 | A finite mapping set is fi... |
| abrexfi 9424 | An image set from a finite... |
| cnvimamptfin 9425 | A preimage of a mapping wi... |
| elfpw 9426 | Membership in a class of f... |
| unifpw 9427 | A set is the union of its ... |
| f1opwfi 9428 | A one-to-one mapping induc... |
| fissuni 9429 | A finite subset of a union... |
| fipreima 9430 | Given a finite subset ` A ... |
| finsschain 9431 | A finite subset of the uni... |
| indexfi 9432 | If for every element of a ... |
| relfsupp 9435 | The property of a function... |
| relprcnfsupp 9436 | A proper class is never fi... |
| isfsupp 9437 | The property of a class to... |
| isfsuppd 9438 | Deduction form of ~ isfsup... |
| funisfsupp 9439 | The property of a function... |
| fsuppimp 9440 | Implications of a class be... |
| fsuppimpd 9441 | A finitely supported funct... |
| fsuppfund 9442 | A finitely supported funct... |
| fisuppfi 9443 | A function on a finite set... |
| fidmfisupp 9444 | A function with a finite d... |
| fdmfisuppfi 9445 | The support of a function ... |
| fdmfifsupp 9446 | A function with a finite d... |
| fsuppmptdm 9447 | A mapping with a finite do... |
| fndmfisuppfi 9448 | The support of a function ... |
| fndmfifsupp 9449 | A function with a finite d... |
| suppeqfsuppbi 9450 | If two functions have the ... |
| suppssfifsupp 9451 | If the support of a functi... |
| fsuppsssupp 9452 | If the support of a functi... |
| fsuppsssuppgd 9453 | If the support of a functi... |
| fsuppss 9454 | A subset of a finitely sup... |
| fsuppssov1 9455 | Formula building theorem f... |
| fsuppxpfi 9456 | The cartesian product of t... |
| fczfsuppd 9457 | A constant function with v... |
| fsuppun 9458 | The union of two finitely ... |
| fsuppunfi 9459 | The union of the support o... |
| fsuppunbi 9460 | If the union of two classe... |
| 0fsupp 9461 | The empty set is a finitel... |
| snopfsupp 9462 | A singleton containing an ... |
| funsnfsupp 9463 | Finite support for a funct... |
| fsuppres 9464 | The restriction of a finit... |
| fmptssfisupp 9465 | The restriction of a mappi... |
| ressuppfi 9466 | If the support of the rest... |
| resfsupp 9467 | If the restriction of a fu... |
| resfifsupp 9468 | The restriction of a funct... |
| ffsuppbi 9469 | Two ways of saying that a ... |
| fsuppmptif 9470 | A function mapping an argu... |
| sniffsupp 9471 | A function mapping all but... |
| fsuppcolem 9472 | Lemma for ~ fsuppco . For... |
| fsuppco 9473 | The composition of a 1-1 f... |
| fsuppco2 9474 | The composition of a funct... |
| fsuppcor 9475 | The composition of a funct... |
| mapfienlem1 9476 | Lemma 1 for ~ mapfien . (... |
| mapfienlem2 9477 | Lemma 2 for ~ mapfien . (... |
| mapfienlem3 9478 | Lemma 3 for ~ mapfien . (... |
| mapfien 9479 | A bijection of the base se... |
| mapfien2 9480 | Equinumerousity relation f... |
| fival 9483 | The set of all the finite ... |
| elfi 9484 | Specific properties of an ... |
| elfi2 9485 | The empty intersection nee... |
| elfir 9486 | Sufficient condition for a... |
| intrnfi 9487 | Sufficient condition for t... |
| iinfi 9488 | An indexed intersection of... |
| inelfi 9489 | The intersection of two se... |
| ssfii 9490 | Any element of a set ` A `... |
| fi0 9491 | The set of finite intersec... |
| fieq0 9492 | A set is empty iff the cla... |
| fiin 9493 | The elements of ` ( fi `` ... |
| dffi2 9494 | The set of finite intersec... |
| fiss 9495 | Subset relationship for fu... |
| inficl 9496 | A set which is closed unde... |
| fipwuni 9497 | The set of finite intersec... |
| fisn 9498 | A singleton is closed unde... |
| fiuni 9499 | The union of the finite in... |
| fipwss 9500 | If a set is a family of su... |
| elfiun 9501 | A finite intersection of e... |
| dffi3 9502 | The set of finite intersec... |
| fifo 9503 | Describe a surjection from... |
| marypha1lem 9504 | Core induction for Philip ... |
| marypha1 9505 | (Philip) Hall's marriage t... |
| marypha2lem1 9506 | Lemma for ~ marypha2 . Pr... |
| marypha2lem2 9507 | Lemma for ~ marypha2 . Pr... |
| marypha2lem3 9508 | Lemma for ~ marypha2 . Pr... |
| marypha2lem4 9509 | Lemma for ~ marypha2 . Pr... |
| marypha2 9510 | Version of ~ marypha1 usin... |
| dfsup2 9515 | Quantifier-free definition... |
| supeq1 9516 | Equality theorem for supre... |
| supeq1d 9517 | Equality deduction for sup... |
| supeq1i 9518 | Equality inference for sup... |
| supeq2 9519 | Equality theorem for supre... |
| supeq3 9520 | Equality theorem for supre... |
| supeq123d 9521 | Equality deduction for sup... |
| nfsup 9522 | Hypothesis builder for sup... |
| supmo 9523 | Any class ` B ` has at mos... |
| supexd 9524 | A supremum is a set. (Con... |
| supeu 9525 | A supremum is unique. Sim... |
| supval2 9526 | Alternate expression for t... |
| eqsup 9527 | Sufficient condition for a... |
| eqsupd 9528 | Sufficient condition for a... |
| supcl 9529 | A supremum belongs to its ... |
| supub 9530 | A supremum is an upper bou... |
| suplub 9531 | A supremum is the least up... |
| suplub2 9532 | Bidirectional form of ~ su... |
| supnub 9533 | An upper bound is not less... |
| supex 9534 | A supremum is a set. (Con... |
| sup00 9535 | The supremum under an empt... |
| sup0riota 9536 | The supremum of an empty s... |
| sup0 9537 | The supremum of an empty s... |
| supmax 9538 | The greatest element of a ... |
| fisup2g 9539 | A finite set satisfies the... |
| fisupcl 9540 | A nonempty finite set cont... |
| supgtoreq 9541 | The supremum of a finite s... |
| suppr 9542 | The supremum of a pair. (... |
| supsn 9543 | The supremum of a singleto... |
| supisolem 9544 | Lemma for ~ supiso . (Con... |
| supisoex 9545 | Lemma for ~ supiso . (Con... |
| supiso 9546 | Image of a supremum under ... |
| infeq1 9547 | Equality theorem for infim... |
| infeq1d 9548 | Equality deduction for inf... |
| infeq1i 9549 | Equality inference for inf... |
| infeq2 9550 | Equality theorem for infim... |
| infeq3 9551 | Equality theorem for infim... |
| infeq123d 9552 | Equality deduction for inf... |
| nfinf 9553 | Hypothesis builder for inf... |
| infexd 9554 | An infimum is a set. (Con... |
| eqinf 9555 | Sufficient condition for a... |
| eqinfd 9556 | Sufficient condition for a... |
| infval 9557 | Alternate expression for t... |
| infcllem 9558 | Lemma for ~ infcl , ~ infl... |
| infcl 9559 | An infimum belongs to its ... |
| inflb 9560 | An infimum is a lower boun... |
| infglb 9561 | An infimum is the greatest... |
| infglbb 9562 | Bidirectional form of ~ in... |
| infnlb 9563 | A lower bound is not great... |
| infex 9564 | An infimum is a set. (Con... |
| infmin 9565 | The smallest element of a ... |
| infmo 9566 | Any class ` B ` has at mos... |
| infeu 9567 | An infimum is unique. (Co... |
| fimin2g 9568 | A finite set has a minimum... |
| fiming 9569 | A finite set has a minimum... |
| fiinfg 9570 | Lemma showing existence an... |
| fiinf2g 9571 | A finite set satisfies the... |
| fiinfcl 9572 | A nonempty finite set cont... |
| infltoreq 9573 | The infimum of a finite se... |
| infpr 9574 | The infimum of a pair. (C... |
| infsupprpr 9575 | The infimum of a proper pa... |
| infsn 9576 | The infimum of a singleton... |
| inf00 9577 | The infimum regarding an e... |
| infempty 9578 | The infimum of an empty se... |
| infiso 9579 | Image of an infimum under ... |
| dfoi 9582 | Rewrite ~ df-oi with abbre... |
| oieq1 9583 | Equality theorem for ordin... |
| oieq2 9584 | Equality theorem for ordin... |
| nfoi 9585 | Hypothesis builder for ord... |
| ordiso2 9586 | Generalize ~ ordiso to pro... |
| ordiso 9587 | Order-isomorphic ordinal n... |
| ordtypecbv 9588 | Lemma for ~ ordtype . (Co... |
| ordtypelem1 9589 | Lemma for ~ ordtype . (Co... |
| ordtypelem2 9590 | Lemma for ~ ordtype . (Co... |
| ordtypelem3 9591 | Lemma for ~ ordtype . (Co... |
| ordtypelem4 9592 | Lemma for ~ ordtype . (Co... |
| ordtypelem5 9593 | Lemma for ~ ordtype . (Co... |
| ordtypelem6 9594 | Lemma for ~ ordtype . (Co... |
| ordtypelem7 9595 | Lemma for ~ ordtype . ` ra... |
| ordtypelem8 9596 | Lemma for ~ ordtype . (Co... |
| ordtypelem9 9597 | Lemma for ~ ordtype . Eit... |
| ordtypelem10 9598 | Lemma for ~ ordtype . Usi... |
| oi0 9599 | Definition of the ordinal ... |
| oicl 9600 | The order type of the well... |
| oif 9601 | The order isomorphism of t... |
| oiiso2 9602 | The order isomorphism of t... |
| ordtype 9603 | For any set-like well-orde... |
| oiiniseg 9604 | ` ran F ` is an initial se... |
| ordtype2 9605 | For any set-like well-orde... |
| oiexg 9606 | The order isomorphism on a... |
| oion 9607 | The order type of the well... |
| oiiso 9608 | The order isomorphism of t... |
| oien 9609 | The order type of a well-o... |
| oieu 9610 | Uniqueness of the unique o... |
| oismo 9611 | When ` A ` is a subclass o... |
| oiid 9612 | The order type of an ordin... |
| hartogslem1 9613 | Lemma for ~ hartogs . (Co... |
| hartogslem2 9614 | Lemma for ~ hartogs . (Co... |
| hartogs 9615 | The class of ordinals domi... |
| wofib 9616 | The only sets which are we... |
| wemaplem1 9617 | Value of the lexicographic... |
| wemaplem2 9618 | Lemma for ~ wemapso . Tra... |
| wemaplem3 9619 | Lemma for ~ wemapso . Tra... |
| wemappo 9620 | Construct lexicographic or... |
| wemapsolem 9621 | Lemma for ~ wemapso . (Co... |
| wemapso 9622 | Construct lexicographic or... |
| wemapso2lem 9623 | Lemma for ~ wemapso2 . (C... |
| wemapso2 9624 | An alternative to having a... |
| card2on 9625 | The alternate definition o... |
| card2inf 9626 | The alternate definition o... |
| harf 9629 | Functionality of the Harto... |
| harcl 9630 | Values of the Hartogs func... |
| harval 9631 | Function value of the Hart... |
| elharval 9632 | The Hartogs number of a se... |
| harndom 9633 | The Hartogs number of a se... |
| harword 9634 | Weak ordering property of ... |
| relwdom 9637 | Weak dominance is a relati... |
| brwdom 9638 | Property of weak dominance... |
| brwdomi 9639 | Property of weak dominance... |
| brwdomn0 9640 | Weak dominance over nonemp... |
| 0wdom 9641 | Any set weakly dominates t... |
| fowdom 9642 | An onto function implies w... |
| wdomref 9643 | Reflexivity of weak domina... |
| brwdom2 9644 | Alternate characterization... |
| domwdom 9645 | Weak dominance is implied ... |
| wdomtr 9646 | Transitivity of weak domin... |
| wdomen1 9647 | Equality-like theorem for ... |
| wdomen2 9648 | Equality-like theorem for ... |
| wdompwdom 9649 | Weak dominance strengthens... |
| canthwdom 9650 | Cantor's Theorem, stated u... |
| wdom2d 9651 | Deduce weak dominance from... |
| wdomd 9652 | Deduce weak dominance from... |
| brwdom3 9653 | Condition for weak dominan... |
| brwdom3i 9654 | Weak dominance implies exi... |
| unwdomg 9655 | Weak dominance of a (disjo... |
| xpwdomg 9656 | Weak dominance of a Cartes... |
| wdomima2g 9657 | A set is weakly dominant o... |
| wdomimag 9658 | A set is weakly dominant o... |
| unxpwdom2 9659 | Lemma for ~ unxpwdom . (C... |
| unxpwdom 9660 | If a Cartesian product is ... |
| ixpiunwdom 9661 | Describe an onto function ... |
| harwdom 9662 | The value of the Hartogs f... |
| axreg2 9664 | Axiom of Regularity expres... |
| zfregcl 9665 | The Axiom of Regularity wi... |
| zfreg 9666 | The Axiom of Regularity us... |
| elirrv 9667 | The membership relation is... |
| elirr 9668 | No class is a member of it... |
| elneq 9669 | A class is not equal to an... |
| nelaneq 9670 | A class is not an element ... |
| epinid0 9671 | The membership relation an... |
| sucprcreg 9672 | A class is equal to its su... |
| ruv 9673 | The Russell class is equal... |
| ruALT 9674 | Alternate proof of ~ ru , ... |
| disjcsn 9675 | A class is disjoint from i... |
| zfregfr 9676 | The membership relation is... |
| en2lp 9677 | No class has 2-cycle membe... |
| elnanel 9678 | Two classes are not elemen... |
| cnvepnep 9679 | The membership (epsilon) r... |
| epnsym 9680 | The membership (epsilon) r... |
| elnotel 9681 | A class cannot be an eleme... |
| elnel 9682 | A class cannot be an eleme... |
| en3lplem1 9683 | Lemma for ~ en3lp . (Cont... |
| en3lplem2 9684 | Lemma for ~ en3lp . (Cont... |
| en3lp 9685 | No class has 3-cycle membe... |
| preleqg 9686 | Equality of two unordered ... |
| preleq 9687 | Equality of two unordered ... |
| preleqALT 9688 | Alternate proof of ~ prele... |
| opthreg 9689 | Theorem for alternate repr... |
| suc11reg 9690 | The successor operation be... |
| dford2 9691 | Assuming ~ ax-reg , an ord... |
| inf0 9692 | Existence of ` _om ` impli... |
| inf1 9693 | Variation of Axiom of Infi... |
| inf2 9694 | Variation of Axiom of Infi... |
| inf3lema 9695 | Lemma for our Axiom of Inf... |
| inf3lemb 9696 | Lemma for our Axiom of Inf... |
| inf3lemc 9697 | Lemma for our Axiom of Inf... |
| inf3lemd 9698 | Lemma for our Axiom of Inf... |
| inf3lem1 9699 | Lemma for our Axiom of Inf... |
| inf3lem2 9700 | Lemma for our Axiom of Inf... |
| inf3lem3 9701 | Lemma for our Axiom of Inf... |
| inf3lem4 9702 | Lemma for our Axiom of Inf... |
| inf3lem5 9703 | Lemma for our Axiom of Inf... |
| inf3lem6 9704 | Lemma for our Axiom of Inf... |
| inf3lem7 9705 | Lemma for our Axiom of Inf... |
| inf3 9706 | Our Axiom of Infinity ~ ax... |
| infeq5i 9707 | Half of ~ infeq5 . (Contr... |
| infeq5 9708 | The statement "there exist... |
| zfinf 9710 | Axiom of Infinity expresse... |
| axinf2 9711 | A standard version of Axio... |
| zfinf2 9713 | A standard version of the ... |
| omex 9714 | The existence of omega (th... |
| axinf 9715 | The first version of the A... |
| inf5 9716 | The statement "there exist... |
| omelon 9717 | Omega is an ordinal number... |
| dfom3 9718 | The class of natural numbe... |
| elom3 9719 | A simplification of ~ elom... |
| dfom4 9720 | A simplification of ~ df-o... |
| dfom5 9721 | ` _om ` is the smallest li... |
| oancom 9722 | Ordinal addition is not co... |
| isfinite 9723 | A set is finite iff it is ... |
| fict 9724 | A finite set is countable ... |
| nnsdom 9725 | A natural number is strict... |
| omenps 9726 | Omega is equinumerous to a... |
| omensuc 9727 | The set of natural numbers... |
| infdifsn 9728 | Removing a singleton from ... |
| infdiffi 9729 | Removing a finite set from... |
| unbnn3 9730 | Any unbounded subset of na... |
| noinfep 9731 | Using the Axiom of Regular... |
| cantnffval 9734 | The value of the Cantor no... |
| cantnfdm 9735 | The domain of the Cantor n... |
| cantnfvalf 9736 | Lemma for ~ cantnf . The ... |
| cantnfs 9737 | Elementhood in the set of ... |
| cantnfcl 9738 | Basic properties of the or... |
| cantnfval 9739 | The value of the Cantor no... |
| cantnfval2 9740 | Alternate expression for t... |
| cantnfsuc 9741 | The value of the recursive... |
| cantnfle 9742 | A lower bound on the ` CNF... |
| cantnflt 9743 | An upper bound on the part... |
| cantnflt2 9744 | An upper bound on the ` CN... |
| cantnff 9745 | The ` CNF ` function is a ... |
| cantnf0 9746 | The value of the zero func... |
| cantnfrescl 9747 | A function is finitely sup... |
| cantnfres 9748 | The ` CNF ` function respe... |
| cantnfp1lem1 9749 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem2 9750 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem3 9751 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1 9752 | If ` F ` is created by add... |
| oemapso 9753 | The relation ` T ` is a st... |
| oemapval 9754 | Value of the relation ` T ... |
| oemapvali 9755 | If ` F < G ` , then there ... |
| cantnflem1a 9756 | Lemma for ~ cantnf . (Con... |
| cantnflem1b 9757 | Lemma for ~ cantnf . (Con... |
| cantnflem1c 9758 | Lemma for ~ cantnf . (Con... |
| cantnflem1d 9759 | Lemma for ~ cantnf . (Con... |
| cantnflem1 9760 | Lemma for ~ cantnf . This... |
| cantnflem2 9761 | Lemma for ~ cantnf . (Con... |
| cantnflem3 9762 | Lemma for ~ cantnf . Here... |
| cantnflem4 9763 | Lemma for ~ cantnf . Comp... |
| cantnf 9764 | The Cantor Normal Form the... |
| oemapwe 9765 | The lexicographic order on... |
| cantnffval2 9766 | An alternate definition of... |
| cantnff1o 9767 | Simplify the isomorphism o... |
| wemapwe 9768 | Construct lexicographic or... |
| oef1o 9769 | A bijection of the base se... |
| cnfcomlem 9770 | Lemma for ~ cnfcom . (Con... |
| cnfcom 9771 | Any ordinal ` B ` is equin... |
| cnfcom2lem 9772 | Lemma for ~ cnfcom2 . (Co... |
| cnfcom2 9773 | Any nonzero ordinal ` B ` ... |
| cnfcom3lem 9774 | Lemma for ~ cnfcom3 . (Co... |
| cnfcom3 9775 | Any infinite ordinal ` B `... |
| cnfcom3clem 9776 | Lemma for ~ cnfcom3c . (C... |
| cnfcom3c 9777 | Wrap the construction of ~... |
| ttrcleq 9780 | Equality theorem for trans... |
| nfttrcld 9781 | Bound variable hypothesis ... |
| nfttrcl 9782 | Bound variable hypothesis ... |
| relttrcl 9783 | The transitive closure of ... |
| brttrcl 9784 | Characterization of elemen... |
| brttrcl2 9785 | Characterization of elemen... |
| ssttrcl 9786 | If ` R ` is a relation, th... |
| ttrcltr 9787 | The transitive closure of ... |
| ttrclresv 9788 | The transitive closure of ... |
| ttrclco 9789 | Composition law for the tr... |
| cottrcl 9790 | Composition law for the tr... |
| ttrclss 9791 | If ` R ` is a subclass of ... |
| dmttrcl 9792 | The domain of a transitive... |
| rnttrcl 9793 | The range of a transitive ... |
| ttrclexg 9794 | If ` R ` is a set, then so... |
| dfttrcl2 9795 | When ` R ` is a set and a ... |
| ttrclselem1 9796 | Lemma for ~ ttrclse . Sho... |
| ttrclselem2 9797 | Lemma for ~ ttrclse . Sho... |
| ttrclse 9798 | If ` R ` is set-like over ... |
| trcl 9799 | For any set ` A ` , show t... |
| tz9.1 9800 | Every set has a transitive... |
| tz9.1c 9801 | Alternate expression for t... |
| epfrs 9802 | The strong form of the Axi... |
| zfregs 9803 | The strong form of the Axi... |
| zfregs2 9804 | Alternate strong form of t... |
| setind 9805 | Set (epsilon) induction. ... |
| setind2 9806 | Set (epsilon) induction, s... |
| tcvalg 9809 | Value of the transitive cl... |
| tcid 9810 | Defining property of the t... |
| tctr 9811 | Defining property of the t... |
| tcmin 9812 | Defining property of the t... |
| tc2 9813 | A variant of the definitio... |
| tcsni 9814 | The transitive closure of ... |
| tcss 9815 | The transitive closure fun... |
| tcel 9816 | The transitive closure fun... |
| tcidm 9817 | The transitive closure fun... |
| tc0 9818 | The transitive closure of ... |
| tc00 9819 | The transitive closure is ... |
| frmin 9820 | Every (possibly proper) su... |
| frind 9821 | A subclass of a well-found... |
| frinsg 9822 | Well-Founded Induction Sch... |
| frins 9823 | Well-Founded Induction Sch... |
| frins2f 9824 | Well-Founded Induction sch... |
| frins2 9825 | Well-Founded Induction sch... |
| frins3 9826 | Well-Founded Induction sch... |
| frr3g 9827 | Functions defined by well-... |
| frrlem15 9828 | Lemma for general well-fou... |
| frrlem16 9829 | Lemma for general well-fou... |
| frr1 9830 | Law of general well-founde... |
| frr2 9831 | Law of general well-founde... |
| frr3 9832 | Law of general well-founde... |
| r1funlim 9837 | The cumulative hierarchy o... |
| r1fnon 9838 | The cumulative hierarchy o... |
| r10 9839 | Value of the cumulative hi... |
| r1sucg 9840 | Value of the cumulative hi... |
| r1suc 9841 | Value of the cumulative hi... |
| r1limg 9842 | Value of the cumulative hi... |
| r1lim 9843 | Value of the cumulative hi... |
| r1fin 9844 | The first ` _om ` levels o... |
| r1sdom 9845 | Each stage in the cumulati... |
| r111 9846 | The cumulative hierarchy i... |
| r1tr 9847 | The cumulative hierarchy o... |
| r1tr2 9848 | The union of a cumulative ... |
| r1ordg 9849 | Ordering relation for the ... |
| r1ord3g 9850 | Ordering relation for the ... |
| r1ord 9851 | Ordering relation for the ... |
| r1ord2 9852 | Ordering relation for the ... |
| r1ord3 9853 | Ordering relation for the ... |
| r1sssuc 9854 | The value of the cumulativ... |
| r1pwss 9855 | Each set of the cumulative... |
| r1sscl 9856 | Each set of the cumulative... |
| r1val1 9857 | The value of the cumulativ... |
| tz9.12lem1 9858 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem2 9859 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem3 9860 | Lemma for ~ tz9.12 . (Con... |
| tz9.12 9861 | A set is well-founded if a... |
| tz9.13 9862 | Every set is well-founded,... |
| tz9.13g 9863 | Every set is well-founded,... |
| rankwflemb 9864 | Two ways of saying a set i... |
| rankf 9865 | The domain and codomain of... |
| rankon 9866 | The rank of a set is an or... |
| r1elwf 9867 | Any member of the cumulati... |
| rankvalb 9868 | Value of the rank function... |
| rankr1ai 9869 | One direction of ~ rankr1a... |
| rankvaln 9870 | Value of the rank function... |
| rankidb 9871 | Identity law for the rank ... |
| rankdmr1 9872 | A rank is a member of the ... |
| rankr1ag 9873 | A version of ~ rankr1a tha... |
| rankr1bg 9874 | A relationship between ran... |
| r1rankidb 9875 | Any set is a subset of the... |
| r1elssi 9876 | The range of the ` R1 ` fu... |
| r1elss 9877 | The range of the ` R1 ` fu... |
| pwwf 9878 | A power set is well-founde... |
| sswf 9879 | A subset of a well-founded... |
| snwf 9880 | A singleton is well-founde... |
| unwf 9881 | A binary union is well-fou... |
| prwf 9882 | An unordered pair is well-... |
| opwf 9883 | An ordered pair is well-fo... |
| unir1 9884 | The cumulative hierarchy o... |
| jech9.3 9885 | Every set belongs to some ... |
| rankwflem 9886 | Every set is well-founded,... |
| rankval 9887 | Value of the rank function... |
| rankvalg 9888 | Value of the rank function... |
| rankval2 9889 | Value of an alternate defi... |
| uniwf 9890 | A union is well-founded if... |
| rankr1clem 9891 | Lemma for ~ rankr1c . (Co... |
| rankr1c 9892 | A relationship between the... |
| rankidn 9893 | A relationship between the... |
| rankpwi 9894 | The rank of a power set. ... |
| rankelb 9895 | The membership relation is... |
| wfelirr 9896 | A well-founded set is not ... |
| rankval3b 9897 | The value of the rank func... |
| ranksnb 9898 | The rank of a singleton. ... |
| rankonidlem 9899 | Lemma for ~ rankonid . (C... |
| rankonid 9900 | The rank of an ordinal num... |
| onwf 9901 | The ordinals are all well-... |
| onssr1 9902 | Initial segments of the or... |
| rankr1g 9903 | A relationship between the... |
| rankid 9904 | Identity law for the rank ... |
| rankr1 9905 | A relationship between the... |
| ssrankr1 9906 | A relationship between an ... |
| rankr1a 9907 | A relationship between ran... |
| r1val2 9908 | The value of the cumulativ... |
| r1val3 9909 | The value of the cumulativ... |
| rankel 9910 | The membership relation is... |
| rankval3 9911 | The value of the rank func... |
| bndrank 9912 | Any class whose elements h... |
| unbndrank 9913 | The elements of a proper c... |
| rankpw 9914 | The rank of a power set. ... |
| ranklim 9915 | The rank of a set belongs ... |
| r1pw 9916 | A stronger property of ` R... |
| r1pwALT 9917 | Alternate shorter proof of... |
| r1pwcl 9918 | The cumulative hierarchy o... |
| rankssb 9919 | The subset relation is inh... |
| rankss 9920 | The subset relation is inh... |
| rankunb 9921 | The rank of the union of t... |
| rankprb 9922 | The rank of an unordered p... |
| rankopb 9923 | The rank of an ordered pai... |
| rankuni2b 9924 | The value of the rank func... |
| ranksn 9925 | The rank of a singleton. ... |
| rankuni2 9926 | The rank of a union. Part... |
| rankun 9927 | The rank of the union of t... |
| rankpr 9928 | The rank of an unordered p... |
| rankop 9929 | The rank of an ordered pai... |
| r1rankid 9930 | Any set is a subset of the... |
| rankeq0b 9931 | A set is empty iff its ran... |
| rankeq0 9932 | A set is empty iff its ran... |
| rankr1id 9933 | The rank of the hierarchy ... |
| rankuni 9934 | The rank of a union. Part... |
| rankr1b 9935 | A relationship between ran... |
| ranksuc 9936 | The rank of a successor. ... |
| rankuniss 9937 | Upper bound of the rank of... |
| rankval4 9938 | The rank of a set is the s... |
| rankbnd 9939 | The rank of a set is bound... |
| rankbnd2 9940 | The rank of a set is bound... |
| rankc1 9941 | A relationship that can be... |
| rankc2 9942 | A relationship that can be... |
| rankelun 9943 | Rank membership is inherit... |
| rankelpr 9944 | Rank membership is inherit... |
| rankelop 9945 | Rank membership is inherit... |
| rankxpl 9946 | A lower bound on the rank ... |
| rankxpu 9947 | An upper bound on the rank... |
| rankfu 9948 | An upper bound on the rank... |
| rankmapu 9949 | An upper bound on the rank... |
| rankxplim 9950 | The rank of a Cartesian pr... |
| rankxplim2 9951 | If the rank of a Cartesian... |
| rankxplim3 9952 | The rank of a Cartesian pr... |
| rankxpsuc 9953 | The rank of a Cartesian pr... |
| tcwf 9954 | The transitive closure fun... |
| tcrank 9955 | This theorem expresses two... |
| scottex 9956 | Scott's trick collects all... |
| scott0 9957 | Scott's trick collects all... |
| scottexs 9958 | Theorem scheme version of ... |
| scott0s 9959 | Theorem scheme version of ... |
| cplem1 9960 | Lemma for the Collection P... |
| cplem2 9961 | Lemma for the Collection P... |
| cp 9962 | Collection Principle. Thi... |
| bnd 9963 | A very strong generalizati... |
| bnd2 9964 | A variant of the Boundedne... |
| kardex 9965 | The collection of all sets... |
| karden 9966 | If we allow the Axiom of R... |
| htalem 9967 | Lemma for defining an emul... |
| hta 9968 | A ZFC emulation of Hilbert... |
| djueq12 9975 | Equality theorem for disjo... |
| djueq1 9976 | Equality theorem for disjo... |
| djueq2 9977 | Equality theorem for disjo... |
| nfdju 9978 | Bound-variable hypothesis ... |
| djuex 9979 | The disjoint union of sets... |
| djuexb 9980 | The disjoint union of two ... |
| djulcl 9981 | Left closure of disjoint u... |
| djurcl 9982 | Right closure of disjoint ... |
| djulf1o 9983 | The left injection functio... |
| djurf1o 9984 | The right injection functi... |
| inlresf 9985 | The left injection restric... |
| inlresf1 9986 | The left injection restric... |
| inrresf 9987 | The right injection restri... |
| inrresf1 9988 | The right injection restri... |
| djuin 9989 | The images of any classes ... |
| djur 9990 | A member of a disjoint uni... |
| djuss 9991 | A disjoint union is a subc... |
| djuunxp 9992 | The union of a disjoint un... |
| djuexALT 9993 | Alternate proof of ~ djuex... |
| eldju1st 9994 | The first component of an ... |
| eldju2ndl 9995 | The second component of an... |
| eldju2ndr 9996 | The second component of an... |
| djuun 9997 | The disjoint union of two ... |
| 1stinl 9998 | The first component of the... |
| 2ndinl 9999 | The second component of th... |
| 1stinr 10000 | The first component of the... |
| 2ndinr 10001 | The second component of th... |
| updjudhf 10002 | The mapping of an element ... |
| updjudhcoinlf 10003 | The composition of the map... |
| updjudhcoinrg 10004 | The composition of the map... |
| updjud 10005 | Universal property of the ... |
| cardf2 10014 | The cardinality function i... |
| cardon 10015 | The cardinal number of a s... |
| isnum2 10016 | A way to express well-orde... |
| isnumi 10017 | A set equinumerous to an o... |
| ennum 10018 | Equinumerous sets are equi... |
| finnum 10019 | Every finite set is numera... |
| onenon 10020 | Every ordinal number is nu... |
| tskwe 10021 | A Tarski set is well-order... |
| xpnum 10022 | The cartesian product of n... |
| cardval3 10023 | An alternate definition of... |
| cardid2 10024 | Any numerable set is equin... |
| isnum3 10025 | A set is numerable iff it ... |
| oncardval 10026 | The value of the cardinal ... |
| oncardid 10027 | Any ordinal number is equi... |
| cardonle 10028 | The cardinal of an ordinal... |
| card0 10029 | The cardinality of the emp... |
| cardidm 10030 | The cardinality function i... |
| oncard 10031 | A set is a cardinal number... |
| ficardom 10032 | The cardinal number of a f... |
| ficardid 10033 | A finite set is equinumero... |
| cardnn 10034 | The cardinality of a natur... |
| cardnueq0 10035 | The empty set is the only ... |
| cardne 10036 | No member of a cardinal nu... |
| carden2a 10037 | If two sets have equal non... |
| carden2b 10038 | If two sets are equinumero... |
| card1 10039 | A set has cardinality one ... |
| cardsn 10040 | A singleton has cardinalit... |
| carddomi2 10041 | Two sets have the dominanc... |
| sdomsdomcardi 10042 | A set strictly dominates i... |
| cardlim 10043 | An infinite cardinal is a ... |
| cardsdomelir 10044 | A cardinal strictly domina... |
| cardsdomel 10045 | A cardinal strictly domina... |
| iscard 10046 | Two ways to express the pr... |
| iscard2 10047 | Two ways to express the pr... |
| carddom2 10048 | Two numerable sets have th... |
| harcard 10049 | The class of ordinal numbe... |
| cardprclem 10050 | Lemma for ~ cardprc . (Co... |
| cardprc 10051 | The class of all cardinal ... |
| carduni 10052 | The union of a set of card... |
| cardiun 10053 | The indexed union of a set... |
| cardennn 10054 | If ` A ` is equinumerous t... |
| cardsucinf 10055 | The cardinality of the suc... |
| cardsucnn 10056 | The cardinality of the suc... |
| cardom 10057 | The set of natural numbers... |
| carden2 10058 | Two numerable sets are equ... |
| cardsdom2 10059 | A numerable set is strictl... |
| domtri2 10060 | Trichotomy of dominance fo... |
| nnsdomel 10061 | Strict dominance and eleme... |
| cardval2 10062 | An alternate version of th... |
| isinffi 10063 | An infinite set contains s... |
| fidomtri 10064 | Trichotomy of dominance wi... |
| fidomtri2 10065 | Trichotomy of dominance wi... |
| harsdom 10066 | The Hartogs number of a we... |
| onsdom 10067 | Any well-orderable set is ... |
| harval2 10068 | An alternate expression fo... |
| harsucnn 10069 | The next cardinal after a ... |
| cardmin2 10070 | The smallest ordinal that ... |
| pm54.43lem 10071 | In Theorem *54.43 of [Whit... |
| pm54.43 10072 | Theorem *54.43 of [Whitehe... |
| enpr2 10073 | An unordered pair with dis... |
| pr2nelemOLD 10074 | Obsolete version of ~ enpr... |
| pr2ne 10075 | If an unordered pair has t... |
| pr2neOLD 10076 | Obsolete version of ~ pr2n... |
| prdom2 10077 | An unordered pair has at m... |
| en2eqpr 10078 | Building a set with two el... |
| en2eleq 10079 | Express a set of pair card... |
| en2other2 10080 | Taking the other element t... |
| dif1card 10081 | The cardinality of a nonem... |
| leweon 10082 | Lexicographical order is a... |
| r0weon 10083 | A set-like well-ordering o... |
| infxpenlem 10084 | Lemma for ~ infxpen . (Co... |
| infxpen 10085 | Every infinite ordinal is ... |
| xpomen 10086 | The Cartesian product of o... |
| xpct 10087 | The cartesian product of t... |
| infxpidm2 10088 | Every infinite well-ordera... |
| infxpenc 10089 | A canonical version of ~ i... |
| infxpenc2lem1 10090 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem2 10091 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem3 10092 | Lemma for ~ infxpenc2 . (... |
| infxpenc2 10093 | Existence form of ~ infxpe... |
| iunmapdisj 10094 | The union ` U_ n e. C ( A ... |
| fseqenlem1 10095 | Lemma for ~ fseqen . (Con... |
| fseqenlem2 10096 | Lemma for ~ fseqen . (Con... |
| fseqdom 10097 | One half of ~ fseqen . (C... |
| fseqen 10098 | A set that is equinumerous... |
| infpwfidom 10099 | The collection of finite s... |
| dfac8alem 10100 | Lemma for ~ dfac8a . If t... |
| dfac8a 10101 | Numeration theorem: every ... |
| dfac8b 10102 | The well-ordering theorem:... |
| dfac8clem 10103 | Lemma for ~ dfac8c . (Con... |
| dfac8c 10104 | If the union of a set is w... |
| ac10ct 10105 | A proof of the well-orderi... |
| ween 10106 | A set is numerable iff it ... |
| ac5num 10107 | A version of ~ ac5b with t... |
| ondomen 10108 | If a set is dominated by a... |
| numdom 10109 | A set dominated by a numer... |
| ssnum 10110 | A subset of a numerable se... |
| onssnum 10111 | All subsets of the ordinal... |
| indcardi 10112 | Indirect strong induction ... |
| acnrcl 10113 | Reverse closure for the ch... |
| acneq 10114 | Equality theorem for the c... |
| isacn 10115 | The property of being a ch... |
| acni 10116 | The property of being a ch... |
| acni2 10117 | The property of being a ch... |
| acni3 10118 | The property of being a ch... |
| acnlem 10119 | Construct a mapping satisf... |
| numacn 10120 | A well-orderable set has c... |
| finacn 10121 | Every set has finite choic... |
| acndom 10122 | A set with long choice seq... |
| acnnum 10123 | A set ` X ` which has choi... |
| acnen 10124 | The class of choice sets o... |
| acndom2 10125 | A set smaller than one wit... |
| acnen2 10126 | The class of sets with cho... |
| fodomacn 10127 | A version of ~ fodom that ... |
| fodomnum 10128 | A version of ~ fodom that ... |
| fonum 10129 | A surjection maps numerabl... |
| numwdom 10130 | A surjection maps numerabl... |
| fodomfi2 10131 | Onto functions define domi... |
| wdomfil 10132 | Weak dominance agrees with... |
| infpwfien 10133 | Any infinite well-orderabl... |
| inffien 10134 | The set of finite intersec... |
| wdomnumr 10135 | Weak dominance agrees with... |
| alephfnon 10136 | The aleph function is a fu... |
| aleph0 10137 | The first infinite cardina... |
| alephlim 10138 | Value of the aleph functio... |
| alephsuc 10139 | Value of the aleph functio... |
| alephon 10140 | An aleph is an ordinal num... |
| alephcard 10141 | Every aleph is a cardinal ... |
| alephnbtwn 10142 | No cardinal can be sandwic... |
| alephnbtwn2 10143 | No set has equinumerosity ... |
| alephordilem1 10144 | Lemma for ~ alephordi . (... |
| alephordi 10145 | Strict ordering property o... |
| alephord 10146 | Ordering property of the a... |
| alephord2 10147 | Ordering property of the a... |
| alephord2i 10148 | Ordering property of the a... |
| alephord3 10149 | Ordering property of the a... |
| alephsucdom 10150 | A set dominated by an alep... |
| alephsuc2 10151 | An alternate representatio... |
| alephdom 10152 | Relationship between inclu... |
| alephgeom 10153 | Every aleph is greater tha... |
| alephislim 10154 | Every aleph is a limit ord... |
| aleph11 10155 | The aleph function is one-... |
| alephf1 10156 | The aleph function is a on... |
| alephsdom 10157 | If an ordinal is smaller t... |
| alephdom2 10158 | A dominated initial ordina... |
| alephle 10159 | The argument of the aleph ... |
| cardaleph 10160 | Given any transfinite card... |
| cardalephex 10161 | Every transfinite cardinal... |
| infenaleph 10162 | An infinite numerable set ... |
| isinfcard 10163 | Two ways to express the pr... |
| iscard3 10164 | Two ways to express the pr... |
| cardnum 10165 | Two ways to express the cl... |
| alephinit 10166 | An infinite initial ordina... |
| carduniima 10167 | The union of the image of ... |
| cardinfima 10168 | If a mapping to cardinals ... |
| alephiso 10169 | Aleph is an order isomorph... |
| alephprc 10170 | The class of all transfini... |
| alephsson 10171 | The class of transfinite c... |
| unialeph 10172 | The union of the class of ... |
| alephsmo 10173 | The aleph function is stri... |
| alephf1ALT 10174 | Alternate proof of ~ aleph... |
| alephfplem1 10175 | Lemma for ~ alephfp . (Co... |
| alephfplem2 10176 | Lemma for ~ alephfp . (Co... |
| alephfplem3 10177 | Lemma for ~ alephfp . (Co... |
| alephfplem4 10178 | Lemma for ~ alephfp . (Co... |
| alephfp 10179 | The aleph function has a f... |
| alephfp2 10180 | The aleph function has at ... |
| alephval3 10181 | An alternate way to expres... |
| alephsucpw2 10182 | The power set of an aleph ... |
| mappwen 10183 | Power rule for cardinal ar... |
| finnisoeu 10184 | A finite totally ordered s... |
| iunfictbso 10185 | Countability of a countabl... |
| aceq1 10188 | Equivalence of two version... |
| aceq0 10189 | Equivalence of two version... |
| aceq2 10190 | Equivalence of two version... |
| aceq3lem 10191 | Lemma for ~ dfac3 . (Cont... |
| dfac3 10192 | Equivalence of two version... |
| dfac4 10193 | Equivalence of two version... |
| dfac5lem1 10194 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem2 10195 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem3 10196 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4 10197 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem5 10198 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4OLD 10199 | Obsolete version of ~ dfac... |
| dfac5 10200 | Equivalence of two version... |
| dfac2a 10201 | Our Axiom of Choice (in th... |
| dfac2b 10202 | Axiom of Choice (first for... |
| dfac2 10203 | Axiom of Choice (first for... |
| dfac7 10204 | Equivalence of the Axiom o... |
| dfac0 10205 | Equivalence of two version... |
| dfac1 10206 | Equivalence of two version... |
| dfac8 10207 | A proof of the equivalency... |
| dfac9 10208 | Equivalence of the axiom o... |
| dfac10 10209 | Axiom of Choice equivalent... |
| dfac10c 10210 | Axiom of Choice equivalent... |
| dfac10b 10211 | Axiom of Choice equivalent... |
| acacni 10212 | A choice equivalent: every... |
| dfacacn 10213 | A choice equivalent: every... |
| dfac13 10214 | The axiom of choice holds ... |
| dfac12lem1 10215 | Lemma for ~ dfac12 . (Con... |
| dfac12lem2 10216 | Lemma for ~ dfac12 . (Con... |
| dfac12lem3 10217 | Lemma for ~ dfac12 . (Con... |
| dfac12r 10218 | The axiom of choice holds ... |
| dfac12k 10219 | Equivalence of ~ dfac12 an... |
| dfac12a 10220 | The axiom of choice holds ... |
| dfac12 10221 | The axiom of choice holds ... |
| kmlem1 10222 | Lemma for 5-quantifier AC ... |
| kmlem2 10223 | Lemma for 5-quantifier AC ... |
| kmlem3 10224 | Lemma for 5-quantifier AC ... |
| kmlem4 10225 | Lemma for 5-quantifier AC ... |
| kmlem5 10226 | Lemma for 5-quantifier AC ... |
| kmlem6 10227 | Lemma for 5-quantifier AC ... |
| kmlem7 10228 | Lemma for 5-quantifier AC ... |
| kmlem8 10229 | Lemma for 5-quantifier AC ... |
| kmlem9 10230 | Lemma for 5-quantifier AC ... |
| kmlem10 10231 | Lemma for 5-quantifier AC ... |
| kmlem11 10232 | Lemma for 5-quantifier AC ... |
| kmlem12 10233 | Lemma for 5-quantifier AC ... |
| kmlem13 10234 | Lemma for 5-quantifier AC ... |
| kmlem14 10235 | Lemma for 5-quantifier AC ... |
| kmlem15 10236 | Lemma for 5-quantifier AC ... |
| kmlem16 10237 | Lemma for 5-quantifier AC ... |
| dfackm 10238 | Equivalence of the Axiom o... |
| undjudom 10239 | Cardinal addition dominate... |
| endjudisj 10240 | Equinumerosity of a disjoi... |
| djuen 10241 | Disjoint unions of equinum... |
| djuenun 10242 | Disjoint union is equinume... |
| dju1en 10243 | Cardinal addition with car... |
| dju1dif 10244 | Adding and subtracting one... |
| dju1p1e2 10245 | 1+1=2 for cardinal number ... |
| dju1p1e2ALT 10246 | Alternate proof of ~ dju1p... |
| dju0en 10247 | Cardinal addition with car... |
| xp2dju 10248 | Two times a cardinal numbe... |
| djucomen 10249 | Commutative law for cardin... |
| djuassen 10250 | Associative law for cardin... |
| xpdjuen 10251 | Cardinal multiplication di... |
| mapdjuen 10252 | Sum of exponents law for c... |
| pwdjuen 10253 | Sum of exponents law for c... |
| djudom1 10254 | Ordering law for cardinal ... |
| djudom2 10255 | Ordering law for cardinal ... |
| djudoml 10256 | A set is dominated by its ... |
| djuxpdom 10257 | Cartesian product dominate... |
| djufi 10258 | The disjoint union of two ... |
| cdainflem 10259 | Any partition of omega int... |
| djuinf 10260 | A set is infinite iff the ... |
| infdju1 10261 | An infinite set is equinum... |
| pwdju1 10262 | The sum of a powerset with... |
| pwdjuidm 10263 | If the natural numbers inj... |
| djulepw 10264 | If ` A ` is idempotent und... |
| onadju 10265 | The cardinal and ordinal s... |
| cardadju 10266 | The cardinal sum is equinu... |
| djunum 10267 | The disjoint union of two ... |
| unnum 10268 | The union of two numerable... |
| nnadju 10269 | The cardinal and ordinal s... |
| nnadjuALT 10270 | Shorter proof of ~ nnadju ... |
| ficardadju 10271 | The disjoint union of fini... |
| ficardun 10272 | The cardinality of the uni... |
| ficardun2 10273 | The cardinality of the uni... |
| pwsdompw 10274 | Lemma for ~ domtriom . Th... |
| unctb 10275 | The union of two countable... |
| infdjuabs 10276 | Absorption law for additio... |
| infunabs 10277 | An infinite set is equinum... |
| infdju 10278 | The sum of two cardinal nu... |
| infdif 10279 | The cardinality of an infi... |
| infdif2 10280 | Cardinality ordering for a... |
| infxpdom 10281 | Dominance law for multipli... |
| infxpabs 10282 | Absorption law for multipl... |
| infunsdom1 10283 | The union of two sets that... |
| infunsdom 10284 | The union of two sets that... |
| infxp 10285 | Absorption law for multipl... |
| pwdjudom 10286 | A property of dominance ov... |
| infpss 10287 | Every infinite set has an ... |
| infmap2 10288 | An exponentiation law for ... |
| ackbij2lem1 10289 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem1 10290 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem2 10291 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem3 10292 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem4 10293 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem5 10294 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem6 10295 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem7 10296 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem8 10297 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem9 10298 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem10 10299 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem11 10300 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem12 10301 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem13 10302 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem14 10303 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem15 10304 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem16 10305 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem17 10306 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem18 10307 | Lemma for ~ ackbij1 . (Co... |
| ackbij1 10308 | The Ackermann bijection, p... |
| ackbij1b 10309 | The Ackermann bijection, p... |
| ackbij2lem2 10310 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem3 10311 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem4 10312 | Lemma for ~ ackbij2 . (Co... |
| ackbij2 10313 | The Ackermann bijection, p... |
| r1om 10314 | The set of hereditarily fi... |
| fictb 10315 | A set is countable iff its... |
| cflem 10316 | A lemma used to simplify c... |
| cflemOLD 10317 | Obsolete version of ~ cfle... |
| cfval 10318 | Value of the cofinality fu... |
| cff 10319 | Cofinality is a function o... |
| cfub 10320 | An upper bound on cofinali... |
| cflm 10321 | Value of the cofinality fu... |
| cf0 10322 | Value of the cofinality fu... |
| cardcf 10323 | Cofinality is a cardinal n... |
| cflecard 10324 | Cofinality is bounded by t... |
| cfle 10325 | Cofinality is bounded by i... |
| cfon 10326 | The cofinality of any set ... |
| cfeq0 10327 | Only the ordinal zero has ... |
| cfsuc 10328 | Value of the cofinality fu... |
| cff1 10329 | There is always a map from... |
| cfflb 10330 | If there is a cofinal map ... |
| cfval2 10331 | Another expression for the... |
| coflim 10332 | A simpler expression for t... |
| cflim3 10333 | Another expression for the... |
| cflim2 10334 | The cofinality function is... |
| cfom 10335 | Value of the cofinality fu... |
| cfss 10336 | There is a cofinal subset ... |
| cfslb 10337 | Any cofinal subset of ` A ... |
| cfslbn 10338 | Any subset of ` A ` smalle... |
| cfslb2n 10339 | Any small collection of sm... |
| cofsmo 10340 | Any cofinal map implies th... |
| cfsmolem 10341 | Lemma for ~ cfsmo . (Cont... |
| cfsmo 10342 | The map in ~ cff1 can be a... |
| cfcoflem 10343 | Lemma for ~ cfcof , showin... |
| coftr 10344 | If there is a cofinal map ... |
| cfcof 10345 | If there is a cofinal map ... |
| cfidm 10346 | The cofinality function is... |
| alephsing 10347 | The cofinality of a limit ... |
| sornom 10348 | The range of a single-step... |
| isfin1a 10363 | Definition of a Ia-finite ... |
| fin1ai 10364 | Property of a Ia-finite se... |
| isfin2 10365 | Definition of a II-finite ... |
| fin2i 10366 | Property of a II-finite se... |
| isfin3 10367 | Definition of a III-finite... |
| isfin4 10368 | Definition of a IV-finite ... |
| fin4i 10369 | Infer that a set is IV-inf... |
| isfin5 10370 | Definition of a V-finite s... |
| isfin6 10371 | Definition of a VI-finite ... |
| isfin7 10372 | Definition of a VII-finite... |
| sdom2en01 10373 | A set with less than two e... |
| infpssrlem1 10374 | Lemma for ~ infpssr . (Co... |
| infpssrlem2 10375 | Lemma for ~ infpssr . (Co... |
| infpssrlem3 10376 | Lemma for ~ infpssr . (Co... |
| infpssrlem4 10377 | Lemma for ~ infpssr . (Co... |
| infpssrlem5 10378 | Lemma for ~ infpssr . (Co... |
| infpssr 10379 | Dedekind infinity implies ... |
| fin4en1 10380 | Dedekind finite is a cardi... |
| ssfin4 10381 | Dedekind finite sets have ... |
| domfin4 10382 | A set dominated by a Dedek... |
| ominf4 10383 | ` _om ` is Dedekind infini... |
| infpssALT 10384 | Alternate proof of ~ infps... |
| isfin4-2 10385 | Alternate definition of IV... |
| isfin4p1 10386 | Alternate definition of IV... |
| fin23lem7 10387 | Lemma for ~ isfin2-2 . Th... |
| fin23lem11 10388 | Lemma for ~ isfin2-2 . (C... |
| fin2i2 10389 | A II-finite set contains m... |
| isfin2-2 10390 | ` Fin2 ` expressed in term... |
| ssfin2 10391 | A subset of a II-finite se... |
| enfin2i 10392 | II-finiteness is a cardina... |
| fin23lem24 10393 | Lemma for ~ fin23 . In a ... |
| fincssdom 10394 | In a chain of finite sets,... |
| fin23lem25 10395 | Lemma for ~ fin23 . In a ... |
| fin23lem26 10396 | Lemma for ~ fin23lem22 . ... |
| fin23lem23 10397 | Lemma for ~ fin23lem22 . ... |
| fin23lem22 10398 | Lemma for ~ fin23 but coul... |
| fin23lem27 10399 | The mapping constructed in... |
| isfin3ds 10400 | Property of a III-finite s... |
| ssfin3ds 10401 | A subset of a III-finite s... |
| fin23lem12 10402 | The beginning of the proof... |
| fin23lem13 10403 | Lemma for ~ fin23 . Each ... |
| fin23lem14 10404 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem15 10405 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem16 10406 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem19 10407 | Lemma for ~ fin23 . The f... |
| fin23lem20 10408 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem17 10409 | Lemma for ~ fin23 . By ? ... |
| fin23lem21 10410 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem28 10411 | Lemma for ~ fin23 . The r... |
| fin23lem29 10412 | Lemma for ~ fin23 . The r... |
| fin23lem30 10413 | Lemma for ~ fin23 . The r... |
| fin23lem31 10414 | Lemma for ~ fin23 . The r... |
| fin23lem32 10415 | Lemma for ~ fin23 . Wrap ... |
| fin23lem33 10416 | Lemma for ~ fin23 . Disch... |
| fin23lem34 10417 | Lemma for ~ fin23 . Estab... |
| fin23lem35 10418 | Lemma for ~ fin23 . Stric... |
| fin23lem36 10419 | Lemma for ~ fin23 . Weak ... |
| fin23lem38 10420 | Lemma for ~ fin23 . The c... |
| fin23lem39 10421 | Lemma for ~ fin23 . Thus,... |
| fin23lem40 10422 | Lemma for ~ fin23 . ` Fin2... |
| fin23lem41 10423 | Lemma for ~ fin23 . A set... |
| isf32lem1 10424 | Lemma for ~ isfin3-2 . De... |
| isf32lem2 10425 | Lemma for ~ isfin3-2 . No... |
| isf32lem3 10426 | Lemma for ~ isfin3-2 . Be... |
| isf32lem4 10427 | Lemma for ~ isfin3-2 . Be... |
| isf32lem5 10428 | Lemma for ~ isfin3-2 . Th... |
| isf32lem6 10429 | Lemma for ~ isfin3-2 . Ea... |
| isf32lem7 10430 | Lemma for ~ isfin3-2 . Di... |
| isf32lem8 10431 | Lemma for ~ isfin3-2 . K ... |
| isf32lem9 10432 | Lemma for ~ isfin3-2 . Co... |
| isf32lem10 10433 | Lemma for isfin3-2 . Writ... |
| isf32lem11 10434 | Lemma for ~ isfin3-2 . Re... |
| isf32lem12 10435 | Lemma for ~ isfin3-2 . (C... |
| isfin32i 10436 | One half of ~ isfin3-2 . ... |
| isf33lem 10437 | Lemma for ~ isfin3-3 . (C... |
| isfin3-2 10438 | Weakly Dedekind-infinite s... |
| isfin3-3 10439 | Weakly Dedekind-infinite s... |
| fin33i 10440 | Inference from ~ isfin3-3 ... |
| compsscnvlem 10441 | Lemma for ~ compsscnv . (... |
| compsscnv 10442 | Complementation on a power... |
| isf34lem1 10443 | Lemma for ~ isfin3-4 . (C... |
| isf34lem2 10444 | Lemma for ~ isfin3-4 . (C... |
| compssiso 10445 | Complementation is an anti... |
| isf34lem3 10446 | Lemma for ~ isfin3-4 . (C... |
| compss 10447 | Express image under of the... |
| isf34lem4 10448 | Lemma for ~ isfin3-4 . (C... |
| isf34lem5 10449 | Lemma for ~ isfin3-4 . (C... |
| isf34lem7 10450 | Lemma for ~ isfin3-4 . (C... |
| isf34lem6 10451 | Lemma for ~ isfin3-4 . (C... |
| fin34i 10452 | Inference from ~ isfin3-4 ... |
| isfin3-4 10453 | Weakly Dedekind-infinite s... |
| fin11a 10454 | Every I-finite set is Ia-f... |
| enfin1ai 10455 | Ia-finiteness is a cardina... |
| isfin1-2 10456 | A set is finite in the usu... |
| isfin1-3 10457 | A set is I-finite iff ever... |
| isfin1-4 10458 | A set is I-finite iff ever... |
| dffin1-5 10459 | Compact quantifier-free ve... |
| fin23 10460 | Every II-finite set (every... |
| fin34 10461 | Every III-finite set is IV... |
| isfin5-2 10462 | Alternate definition of V-... |
| fin45 10463 | Every IV-finite set is V-f... |
| fin56 10464 | Every V-finite set is VI-f... |
| fin17 10465 | Every I-finite set is VII-... |
| fin67 10466 | Every VI-finite set is VII... |
| isfin7-2 10467 | A set is VII-finite iff it... |
| fin71num 10468 | A well-orderable set is VI... |
| dffin7-2 10469 | Class form of ~ isfin7-2 .... |
| dfacfin7 10470 | Axiom of Choice equivalent... |
| fin1a2lem1 10471 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem2 10472 | Lemma for ~ fin1a2 . The ... |
| fin1a2lem3 10473 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem4 10474 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem5 10475 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem6 10476 | Lemma for ~ fin1a2 . Esta... |
| fin1a2lem7 10477 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem8 10478 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem9 10479 | Lemma for ~ fin1a2 . In a... |
| fin1a2lem10 10480 | Lemma for ~ fin1a2 . A no... |
| fin1a2lem11 10481 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem12 10482 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem13 10483 | Lemma for ~ fin1a2 . (Con... |
| fin12 10484 | Weak theorem which skips I... |
| fin1a2s 10485 | An II-infinite set can hav... |
| fin1a2 10486 | Every Ia-finite set is II-... |
| itunifval 10487 | Function value of iterated... |
| itunifn 10488 | Functionality of the itera... |
| ituni0 10489 | A zero-fold iterated union... |
| itunisuc 10490 | Successor iterated union. ... |
| itunitc1 10491 | Each union iterate is a me... |
| itunitc 10492 | The union of all union ite... |
| ituniiun 10493 | Unwrap an iterated union f... |
| hsmexlem7 10494 | Lemma for ~ hsmex . Prope... |
| hsmexlem8 10495 | Lemma for ~ hsmex . Prope... |
| hsmexlem9 10496 | Lemma for ~ hsmex . Prope... |
| hsmexlem1 10497 | Lemma for ~ hsmex . Bound... |
| hsmexlem2 10498 | Lemma for ~ hsmex . Bound... |
| hsmexlem3 10499 | Lemma for ~ hsmex . Clear... |
| hsmexlem4 10500 | Lemma for ~ hsmex . The c... |
| hsmexlem5 10501 | Lemma for ~ hsmex . Combi... |
| hsmexlem6 10502 | Lemma for ~ hsmex . (Cont... |
| hsmex 10503 | The collection of heredita... |
| hsmex2 10504 | The set of hereditary size... |
| hsmex3 10505 | The set of hereditary size... |
| axcc2lem 10507 | Lemma for ~ axcc2 . (Cont... |
| axcc2 10508 | A possibly more useful ver... |
| axcc3 10509 | A possibly more useful ver... |
| axcc4 10510 | A version of ~ axcc3 that ... |
| acncc 10511 | An ~ ax-cc equivalent: eve... |
| axcc4dom 10512 | Relax the constraint on ~ ... |
| domtriomlem 10513 | Lemma for ~ domtriom . (C... |
| domtriom 10514 | Trichotomy of equinumerosi... |
| fin41 10515 | Under countable choice, th... |
| dominf 10516 | A nonempty set that is a s... |
| dcomex 10518 | The Axiom of Dependent Cho... |
| axdc2lem 10519 | Lemma for ~ axdc2 . We co... |
| axdc2 10520 | An apparent strengthening ... |
| axdc3lem 10521 | The class ` S ` of finite ... |
| axdc3lem2 10522 | Lemma for ~ axdc3 . We ha... |
| axdc3lem3 10523 | Simple substitution lemma ... |
| axdc3lem4 10524 | Lemma for ~ axdc3 . We ha... |
| axdc3 10525 | Dependent Choice. Axiom D... |
| axdc4lem 10526 | Lemma for ~ axdc4 . (Cont... |
| axdc4 10527 | A more general version of ... |
| axcclem 10528 | Lemma for ~ axcc . (Contr... |
| axcc 10529 | Although CC can be proven ... |
| zfac 10531 | Axiom of Choice expressed ... |
| ac2 10532 | Axiom of Choice equivalent... |
| ac3 10533 | Axiom of Choice using abbr... |
| axac3 10535 | This theorem asserts that ... |
| ackm 10536 | A remarkable equivalent to... |
| axac2 10537 | Derive ~ ax-ac2 from ~ ax-... |
| axac 10538 | Derive ~ ax-ac from ~ ax-a... |
| axaci 10539 | Apply a choice equivalent.... |
| cardeqv 10540 | All sets are well-orderabl... |
| numth3 10541 | All sets are well-orderabl... |
| numth2 10542 | Numeration theorem: any se... |
| numth 10543 | Numeration theorem: every ... |
| ac7 10544 | An Axiom of Choice equival... |
| ac7g 10545 | An Axiom of Choice equival... |
| ac4 10546 | Equivalent of Axiom of Cho... |
| ac4c 10547 | Equivalent of Axiom of Cho... |
| ac5 10548 | An Axiom of Choice equival... |
| ac5b 10549 | Equivalent of Axiom of Cho... |
| ac6num 10550 | A version of ~ ac6 which t... |
| ac6 10551 | Equivalent of Axiom of Cho... |
| ac6c4 10552 | Equivalent of Axiom of Cho... |
| ac6c5 10553 | Equivalent of Axiom of Cho... |
| ac9 10554 | An Axiom of Choice equival... |
| ac6s 10555 | Equivalent of Axiom of Cho... |
| ac6n 10556 | Equivalent of Axiom of Cho... |
| ac6s2 10557 | Generalization of the Axio... |
| ac6s3 10558 | Generalization of the Axio... |
| ac6sg 10559 | ~ ac6s with sethood as ant... |
| ac6sf 10560 | Version of ~ ac6 with boun... |
| ac6s4 10561 | Generalization of the Axio... |
| ac6s5 10562 | Generalization of the Axio... |
| ac8 10563 | An Axiom of Choice equival... |
| ac9s 10564 | An Axiom of Choice equival... |
| numthcor 10565 | Any set is strictly domina... |
| weth 10566 | Well-ordering theorem: any... |
| zorn2lem1 10567 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem2 10568 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem3 10569 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem4 10570 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem5 10571 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem6 10572 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem7 10573 | Lemma for ~ zorn2 . (Cont... |
| zorn2g 10574 | Zorn's Lemma of [Monk1] p.... |
| zorng 10575 | Zorn's Lemma. If the unio... |
| zornn0g 10576 | Variant of Zorn's lemma ~ ... |
| zorn2 10577 | Zorn's Lemma of [Monk1] p.... |
| zorn 10578 | Zorn's Lemma. If the unio... |
| zornn0 10579 | Variant of Zorn's lemma ~ ... |
| ttukeylem1 10580 | Lemma for ~ ttukey . Expa... |
| ttukeylem2 10581 | Lemma for ~ ttukey . A pr... |
| ttukeylem3 10582 | Lemma for ~ ttukey . (Con... |
| ttukeylem4 10583 | Lemma for ~ ttukey . (Con... |
| ttukeylem5 10584 | Lemma for ~ ttukey . The ... |
| ttukeylem6 10585 | Lemma for ~ ttukey . (Con... |
| ttukeylem7 10586 | Lemma for ~ ttukey . (Con... |
| ttukey2g 10587 | The Teichmüller-Tukey... |
| ttukeyg 10588 | The Teichmüller-Tukey... |
| ttukey 10589 | The Teichmüller-Tukey... |
| axdclem 10590 | Lemma for ~ axdc . (Contr... |
| axdclem2 10591 | Lemma for ~ axdc . Using ... |
| axdc 10592 | This theorem derives ~ ax-... |
| fodomg 10593 | An onto function implies d... |
| fodom 10594 | An onto function implies d... |
| dmct 10595 | The domain of a countable ... |
| rnct 10596 | The range of a countable s... |
| fodomb 10597 | Equivalence of an onto map... |
| wdomac 10598 | When assuming AC, weak and... |
| brdom3 10599 | Equivalence to a dominance... |
| brdom5 10600 | An equivalence to a domina... |
| brdom4 10601 | An equivalence to a domina... |
| brdom7disj 10602 | An equivalence to a domina... |
| brdom6disj 10603 | An equivalence to a domina... |
| fin71ac 10604 | Once we allow AC, the "str... |
| imadomg 10605 | An image of a function und... |
| fimact 10606 | The image by a function of... |
| fnrndomg 10607 | The range of a function is... |
| fnct 10608 | If the domain of a functio... |
| mptct 10609 | A countable mapping set is... |
| iunfo 10610 | Existence of an onto funct... |
| iundom2g 10611 | An upper bound for the car... |
| iundomg 10612 | An upper bound for the car... |
| iundom 10613 | An upper bound for the car... |
| unidom 10614 | An upper bound for the car... |
| uniimadom 10615 | An upper bound for the car... |
| uniimadomf 10616 | An upper bound for the car... |
| cardval 10617 | The value of the cardinal ... |
| cardid 10618 | Any set is equinumerous to... |
| cardidg 10619 | Any set is equinumerous to... |
| cardidd 10620 | Any set is equinumerous to... |
| cardf 10621 | The cardinality function i... |
| carden 10622 | Two sets are equinumerous ... |
| cardeq0 10623 | Only the empty set has car... |
| unsnen 10624 | Equinumerosity of a set wi... |
| carddom 10625 | Two sets have the dominanc... |
| cardsdom 10626 | Two sets have the strict d... |
| domtri 10627 | Trichotomy law for dominan... |
| entric 10628 | Trichotomy of equinumerosi... |
| entri2 10629 | Trichotomy of dominance an... |
| entri3 10630 | Trichotomy of dominance. ... |
| sdomsdomcard 10631 | A set strictly dominates i... |
| canth3 10632 | Cantor's theorem in terms ... |
| infxpidm 10633 | Every infinite class is eq... |
| ondomon 10634 | The class of ordinals domi... |
| cardmin 10635 | The smallest ordinal that ... |
| ficard 10636 | A set is finite iff its ca... |
| infinf 10637 | Equivalence between two in... |
| unirnfdomd 10638 | The union of the range of ... |
| konigthlem 10639 | Lemma for ~ konigth . (Co... |
| konigth 10640 | Konig's Theorem. If ` m (... |
| alephsucpw 10641 | The power set of an aleph ... |
| aleph1 10642 | The set exponentiation of ... |
| alephval2 10643 | An alternate way to expres... |
| dominfac 10644 | A nonempty set that is a s... |
| iunctb 10645 | The countable union of cou... |
| unictb 10646 | The countable union of cou... |
| infmap 10647 | An exponentiation law for ... |
| alephadd 10648 | The sum of two alephs is t... |
| alephmul 10649 | The product of two alephs ... |
| alephexp1 10650 | An exponentiation law for ... |
| alephsuc3 10651 | An alternate representatio... |
| alephexp2 10652 | An expression equinumerous... |
| alephreg 10653 | A successor aleph is regul... |
| pwcfsdom 10654 | A corollary of Konig's The... |
| cfpwsdom 10655 | A corollary of Konig's The... |
| alephom 10656 | From ~ canth2 , we know th... |
| smobeth 10657 | The beth function is stric... |
| nd1 10658 | A lemma for proving condit... |
| nd2 10659 | A lemma for proving condit... |
| nd3 10660 | A lemma for proving condit... |
| nd4 10661 | A lemma for proving condit... |
| axextnd 10662 | A version of the Axiom of ... |
| axrepndlem1 10663 | Lemma for the Axiom of Rep... |
| axrepndlem2 10664 | Lemma for the Axiom of Rep... |
| axrepnd 10665 | A version of the Axiom of ... |
| axunndlem1 10666 | Lemma for the Axiom of Uni... |
| axunnd 10667 | A version of the Axiom of ... |
| axpowndlem1 10668 | Lemma for the Axiom of Pow... |
| axpowndlem2 10669 | Lemma for the Axiom of Pow... |
| axpowndlem3 10670 | Lemma for the Axiom of Pow... |
| axpowndlem4 10671 | Lemma for the Axiom of Pow... |
| axpownd 10672 | A version of the Axiom of ... |
| axregndlem1 10673 | Lemma for the Axiom of Reg... |
| axregndlem2 10674 | Lemma for the Axiom of Reg... |
| axregnd 10675 | A version of the Axiom of ... |
| axinfndlem1 10676 | Lemma for the Axiom of Inf... |
| axinfnd 10677 | A version of the Axiom of ... |
| axacndlem1 10678 | Lemma for the Axiom of Cho... |
| axacndlem2 10679 | Lemma for the Axiom of Cho... |
| axacndlem3 10680 | Lemma for the Axiom of Cho... |
| axacndlem4 10681 | Lemma for the Axiom of Cho... |
| axacndlem5 10682 | Lemma for the Axiom of Cho... |
| axacnd 10683 | A version of the Axiom of ... |
| zfcndext 10684 | Axiom of Extensionality ~ ... |
| zfcndrep 10685 | Axiom of Replacement ~ ax-... |
| zfcndun 10686 | Axiom of Union ~ ax-un , r... |
| zfcndpow 10687 | Axiom of Power Sets ~ ax-p... |
| zfcndreg 10688 | Axiom of Regularity ~ ax-r... |
| zfcndinf 10689 | Axiom of Infinity ~ ax-inf... |
| zfcndac 10690 | Axiom of Choice ~ ax-ac , ... |
| elgch 10693 | Elementhood in the collect... |
| fingch 10694 | A finite set is a GCH-set.... |
| gchi 10695 | The only GCH-sets which ha... |
| gchen1 10696 | If ` A <_ B < ~P A ` , and... |
| gchen2 10697 | If ` A < B <_ ~P A ` , and... |
| gchor 10698 | If ` A <_ B <_ ~P A ` , an... |
| engch 10699 | The property of being a GC... |
| gchdomtri 10700 | Under certain conditions, ... |
| fpwwe2cbv 10701 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem1 10702 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem2 10703 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem3 10704 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem4 10705 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem5 10706 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem6 10707 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem7 10708 | Lemma for ~ fpwwe2 . Show... |
| fpwwe2lem8 10709 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem9 10710 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem10 10711 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem11 10712 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem12 10713 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2 10714 | Given any function ` F ` f... |
| fpwwecbv 10715 | Lemma for ~ fpwwe . (Cont... |
| fpwwelem 10716 | Lemma for ~ fpwwe . (Cont... |
| fpwwe 10717 | Given any function ` F ` f... |
| canth4 10718 | An "effective" form of Can... |
| canthnumlem 10719 | Lemma for ~ canthnum . (C... |
| canthnum 10720 | The set of well-orderable ... |
| canthwelem 10721 | Lemma for ~ canthwe . (Co... |
| canthwe 10722 | The set of well-orders of ... |
| canthp1lem1 10723 | Lemma for ~ canthp1 . (Co... |
| canthp1lem2 10724 | Lemma for ~ canthp1 . (Co... |
| canthp1 10725 | A slightly stronger form o... |
| finngch 10726 | The exclusion of finite se... |
| gchdju1 10727 | An infinite GCH-set is ide... |
| gchinf 10728 | An infinite GCH-set is Ded... |
| pwfseqlem1 10729 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem2 10730 | Lemma for ~ pwfseq . (Con... |
| pwfseqlem3 10731 | Lemma for ~ pwfseq . Usin... |
| pwfseqlem4a 10732 | Lemma for ~ pwfseqlem4 . ... |
| pwfseqlem4 10733 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem5 10734 | Lemma for ~ pwfseq . Alth... |
| pwfseq 10735 | The powerset of a Dedekind... |
| pwxpndom2 10736 | The powerset of a Dedekind... |
| pwxpndom 10737 | The powerset of a Dedekind... |
| pwdjundom 10738 | The powerset of a Dedekind... |
| gchdjuidm 10739 | An infinite GCH-set is ide... |
| gchxpidm 10740 | An infinite GCH-set is ide... |
| gchpwdom 10741 | A relationship between dom... |
| gchaleph 10742 | If ` ( aleph `` A ) ` is a... |
| gchaleph2 10743 | If ` ( aleph `` A ) ` and ... |
| hargch 10744 | If ` A + ~~ ~P A ` , then ... |
| alephgch 10745 | If ` ( aleph `` suc A ) ` ... |
| gch2 10746 | It is sufficient to requir... |
| gch3 10747 | An equivalent formulation ... |
| gch-kn 10748 | The equivalence of two ver... |
| gchaclem 10749 | Lemma for ~ gchac (obsolet... |
| gchhar 10750 | A "local" form of ~ gchac ... |
| gchacg 10751 | A "local" form of ~ gchac ... |
| gchac 10752 | The Generalized Continuum ... |
| elwina 10757 | Conditions of weak inacces... |
| elina 10758 | Conditions of strong inacc... |
| winaon 10759 | A weakly inaccessible card... |
| inawinalem 10760 | Lemma for ~ inawina . (Co... |
| inawina 10761 | Every strongly inaccessibl... |
| omina 10762 | ` _om ` is a strongly inac... |
| winacard 10763 | A weakly inaccessible card... |
| winainflem 10764 | A weakly inaccessible card... |
| winainf 10765 | A weakly inaccessible card... |
| winalim 10766 | A weakly inaccessible card... |
| winalim2 10767 | A nontrivial weakly inacce... |
| winafp 10768 | A nontrivial weakly inacce... |
| winafpi 10769 | This theorem, which states... |
| gchina 10770 | Assuming the GCH, weakly a... |
| iswun 10775 | Properties of a weak unive... |
| wuntr 10776 | A weak universe is transit... |
| wununi 10777 | A weak universe is closed ... |
| wunpw 10778 | A weak universe is closed ... |
| wunelss 10779 | The elements of a weak uni... |
| wunpr 10780 | A weak universe is closed ... |
| wunun 10781 | A weak universe is closed ... |
| wuntp 10782 | A weak universe is closed ... |
| wunss 10783 | A weak universe is closed ... |
| wunin 10784 | A weak universe is closed ... |
| wundif 10785 | A weak universe is closed ... |
| wunint 10786 | A weak universe is closed ... |
| wunsn 10787 | A weak universe is closed ... |
| wunsuc 10788 | A weak universe is closed ... |
| wun0 10789 | A weak universe contains t... |
| wunr1om 10790 | A weak universe is infinit... |
| wunom 10791 | A weak universe contains a... |
| wunfi 10792 | A weak universe contains a... |
| wunop 10793 | A weak universe is closed ... |
| wunot 10794 | A weak universe is closed ... |
| wunxp 10795 | A weak universe is closed ... |
| wunpm 10796 | A weak universe is closed ... |
| wunmap 10797 | A weak universe is closed ... |
| wunf 10798 | A weak universe is closed ... |
| wundm 10799 | A weak universe is closed ... |
| wunrn 10800 | A weak universe is closed ... |
| wuncnv 10801 | A weak universe is closed ... |
| wunres 10802 | A weak universe is closed ... |
| wunfv 10803 | A weak universe is closed ... |
| wunco 10804 | A weak universe is closed ... |
| wuntpos 10805 | A weak universe is closed ... |
| intwun 10806 | The intersection of a coll... |
| r1limwun 10807 | Each limit stage in the cu... |
| r1wunlim 10808 | The weak universes in the ... |
| wunex2 10809 | Construct a weak universe ... |
| wunex 10810 | Construct a weak universe ... |
| uniwun 10811 | Every set is contained in ... |
| wunex3 10812 | Construct a weak universe ... |
| wuncval 10813 | Value of the weak universe... |
| wuncid 10814 | The weak universe closure ... |
| wunccl 10815 | The weak universe closure ... |
| wuncss 10816 | The weak universe closure ... |
| wuncidm 10817 | The weak universe closure ... |
| wuncval2 10818 | Our earlier expression for... |
| eltskg 10821 | Properties of a Tarski cla... |
| eltsk2g 10822 | Properties of a Tarski cla... |
| tskpwss 10823 | First axiom of a Tarski cl... |
| tskpw 10824 | Second axiom of a Tarski c... |
| tsken 10825 | Third axiom of a Tarski cl... |
| 0tsk 10826 | The empty set is a (transi... |
| tsksdom 10827 | An element of a Tarski cla... |
| tskssel 10828 | A part of a Tarski class s... |
| tskss 10829 | The subsets of an element ... |
| tskin 10830 | The intersection of two el... |
| tsksn 10831 | A singleton of an element ... |
| tsktrss 10832 | A transitive element of a ... |
| tsksuc 10833 | If an element of a Tarski ... |
| tsk0 10834 | A nonempty Tarski class co... |
| tsk1 10835 | One is an element of a non... |
| tsk2 10836 | Two is an element of a non... |
| 2domtsk 10837 | If a Tarski class is not e... |
| tskr1om 10838 | A nonempty Tarski class is... |
| tskr1om2 10839 | A nonempty Tarski class co... |
| tskinf 10840 | A nonempty Tarski class is... |
| tskpr 10841 | If ` A ` and ` B ` are mem... |
| tskop 10842 | If ` A ` and ` B ` are mem... |
| tskxpss 10843 | A Cartesian product of two... |
| tskwe2 10844 | A Tarski class is well-ord... |
| inttsk 10845 | The intersection of a coll... |
| inar1 10846 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omALT 10847 | Alternate proof of ~ r1om ... |
| rankcf 10848 | Any set must be at least a... |
| inatsk 10849 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omtsk 10850 | The set of hereditarily fi... |
| tskord 10851 | A Tarski class contains al... |
| tskcard 10852 | An even more direct relati... |
| r1tskina 10853 | There is a direct relation... |
| tskuni 10854 | The union of an element of... |
| tskwun 10855 | A nonempty transitive Tars... |
| tskint 10856 | The intersection of an ele... |
| tskun 10857 | The union of two elements ... |
| tskxp 10858 | The Cartesian product of t... |
| tskmap 10859 | Set exponentiation is an e... |
| tskurn 10860 | A transitive Tarski class ... |
| elgrug 10863 | Properties of a Grothendie... |
| grutr 10864 | A Grothendieck universe is... |
| gruelss 10865 | A Grothendieck universe is... |
| grupw 10866 | A Grothendieck universe co... |
| gruss 10867 | Any subset of an element o... |
| grupr 10868 | A Grothendieck universe co... |
| gruurn 10869 | A Grothendieck universe co... |
| gruiun 10870 | If ` B ( x ) ` is a family... |
| gruuni 10871 | A Grothendieck universe co... |
| grurn 10872 | A Grothendieck universe co... |
| gruima 10873 | A Grothendieck universe co... |
| gruel 10874 | Any element of an element ... |
| grusn 10875 | A Grothendieck universe co... |
| gruop 10876 | A Grothendieck universe co... |
| gruun 10877 | A Grothendieck universe co... |
| gruxp 10878 | A Grothendieck universe co... |
| grumap 10879 | A Grothendieck universe co... |
| gruixp 10880 | A Grothendieck universe co... |
| gruiin 10881 | A Grothendieck universe co... |
| gruf 10882 | A Grothendieck universe co... |
| gruen 10883 | A Grothendieck universe co... |
| gruwun 10884 | A nonempty Grothendieck un... |
| intgru 10885 | The intersection of a fami... |
| ingru 10886 | The intersection of a univ... |
| wfgru 10887 | The wellfounded part of a ... |
| grudomon 10888 | Each ordinal that is compa... |
| gruina 10889 | If a Grothendieck universe... |
| grur1a 10890 | A characterization of Grot... |
| grur1 10891 | A characterization of Grot... |
| grutsk1 10892 | Grothendieck universes are... |
| grutsk 10893 | Grothendieck universes are... |
| axgroth5 10895 | The Tarski-Grothendieck ax... |
| axgroth2 10896 | Alternate version of the T... |
| grothpw 10897 | Derive the Axiom of Power ... |
| grothpwex 10898 | Derive the Axiom of Power ... |
| axgroth6 10899 | The Tarski-Grothendieck ax... |
| grothomex 10900 | The Tarski-Grothendieck Ax... |
| grothac 10901 | The Tarski-Grothendieck Ax... |
| axgroth3 10902 | Alternate version of the T... |
| axgroth4 10903 | Alternate version of the T... |
| grothprimlem 10904 | Lemma for ~ grothprim . E... |
| grothprim 10905 | The Tarski-Grothendieck Ax... |
| grothtsk 10906 | The Tarski-Grothendieck Ax... |
| inaprc 10907 | An equivalent to the Tarsk... |
| tskmval 10910 | Value of our tarski map. ... |
| tskmid 10911 | The set ` A ` is an elemen... |
| tskmcl 10912 | A Tarski class that contai... |
| sstskm 10913 | Being a part of ` ( tarski... |
| eltskm 10914 | Belonging to ` ( tarskiMap... |
| elni 10947 | Membership in the class of... |
| elni2 10948 | Membership in the class of... |
| pinn 10949 | A positive integer is a na... |
| pion 10950 | A positive integer is an o... |
| piord 10951 | A positive integer is ordi... |
| niex 10952 | The class of positive inte... |
| 0npi 10953 | The empty set is not a pos... |
| 1pi 10954 | Ordinal 'one' is a positiv... |
| addpiord 10955 | Positive integer addition ... |
| mulpiord 10956 | Positive integer multiplic... |
| mulidpi 10957 | 1 is an identity element f... |
| ltpiord 10958 | Positive integer 'less tha... |
| ltsopi 10959 | Positive integer 'less tha... |
| ltrelpi 10960 | Positive integer 'less tha... |
| dmaddpi 10961 | Domain of addition on posi... |
| dmmulpi 10962 | Domain of multiplication o... |
| addclpi 10963 | Closure of addition of pos... |
| mulclpi 10964 | Closure of multiplication ... |
| addcompi 10965 | Addition of positive integ... |
| addasspi 10966 | Addition of positive integ... |
| mulcompi 10967 | Multiplication of positive... |
| mulasspi 10968 | Multiplication of positive... |
| distrpi 10969 | Multiplication of positive... |
| addcanpi 10970 | Addition cancellation law ... |
| mulcanpi 10971 | Multiplication cancellatio... |
| addnidpi 10972 | There is no identity eleme... |
| ltexpi 10973 | Ordering on positive integ... |
| ltapi 10974 | Ordering property of addit... |
| ltmpi 10975 | Ordering property of multi... |
| 1lt2pi 10976 | One is less than two (one ... |
| nlt1pi 10977 | No positive integer is les... |
| indpi 10978 | Principle of Finite Induct... |
| enqbreq 10990 | Equivalence relation for p... |
| enqbreq2 10991 | Equivalence relation for p... |
| enqer 10992 | The equivalence relation f... |
| enqex 10993 | The equivalence relation f... |
| nqex 10994 | The class of positive frac... |
| 0nnq 10995 | The empty set is not a pos... |
| elpqn 10996 | Each positive fraction is ... |
| ltrelnq 10997 | Positive fraction 'less th... |
| pinq 10998 | The representatives of pos... |
| 1nq 10999 | The positive fraction 'one... |
| nqereu 11000 | There is a unique element ... |
| nqerf 11001 | Corollary of ~ nqereu : th... |
| nqercl 11002 | Corollary of ~ nqereu : cl... |
| nqerrel 11003 | Any member of ` ( N. X. N.... |
| nqerid 11004 | Corollary of ~ nqereu : th... |
| enqeq 11005 | Corollary of ~ nqereu : if... |
| nqereq 11006 | The function ` /Q ` acts a... |
| addpipq2 11007 | Addition of positive fract... |
| addpipq 11008 | Addition of positive fract... |
| addpqnq 11009 | Addition of positive fract... |
| mulpipq2 11010 | Multiplication of positive... |
| mulpipq 11011 | Multiplication of positive... |
| mulpqnq 11012 | Multiplication of positive... |
| ordpipq 11013 | Ordering of positive fract... |
| ordpinq 11014 | Ordering of positive fract... |
| addpqf 11015 | Closure of addition on pos... |
| addclnq 11016 | Closure of addition on pos... |
| mulpqf 11017 | Closure of multiplication ... |
| mulclnq 11018 | Closure of multiplication ... |
| addnqf 11019 | Domain of addition on posi... |
| mulnqf 11020 | Domain of multiplication o... |
| addcompq 11021 | Addition of positive fract... |
| addcomnq 11022 | Addition of positive fract... |
| mulcompq 11023 | Multiplication of positive... |
| mulcomnq 11024 | Multiplication of positive... |
| adderpqlem 11025 | Lemma for ~ adderpq . (Co... |
| mulerpqlem 11026 | Lemma for ~ mulerpq . (Co... |
| adderpq 11027 | Addition is compatible wit... |
| mulerpq 11028 | Multiplication is compatib... |
| addassnq 11029 | Addition of positive fract... |
| mulassnq 11030 | Multiplication of positive... |
| mulcanenq 11031 | Lemma for distributive law... |
| distrnq 11032 | Multiplication of positive... |
| 1nqenq 11033 | The equivalence class of r... |
| mulidnq 11034 | Multiplication identity el... |
| recmulnq 11035 | Relationship between recip... |
| recidnq 11036 | A positive fraction times ... |
| recclnq 11037 | Closure law for positive f... |
| recrecnq 11038 | Reciprocal of reciprocal o... |
| dmrecnq 11039 | Domain of reciprocal on po... |
| ltsonq 11040 | 'Less than' is a strict or... |
| lterpq 11041 | Compatibility of ordering ... |
| ltanq 11042 | Ordering property of addit... |
| ltmnq 11043 | Ordering property of multi... |
| 1lt2nq 11044 | One is less than two (one ... |
| ltaddnq 11045 | The sum of two fractions i... |
| ltexnq 11046 | Ordering on positive fract... |
| halfnq 11047 | One-half of any positive f... |
| nsmallnq 11048 | The is no smallest positiv... |
| ltbtwnnq 11049 | There exists a number betw... |
| ltrnq 11050 | Ordering property of recip... |
| archnq 11051 | For any fraction, there is... |
| npex 11057 | The class of positive real... |
| elnp 11058 | Membership in positive rea... |
| elnpi 11059 | Membership in positive rea... |
| prn0 11060 | A positive real is not emp... |
| prpssnq 11061 | A positive real is a subse... |
| elprnq 11062 | A positive real is a set o... |
| 0npr 11063 | The empty set is not a pos... |
| prcdnq 11064 | A positive real is closed ... |
| prub 11065 | A positive fraction not in... |
| prnmax 11066 | A positive real has no lar... |
| npomex 11067 | A simplifying observation,... |
| prnmadd 11068 | A positive real has no lar... |
| ltrelpr 11069 | Positive real 'less than' ... |
| genpv 11070 | Value of general operation... |
| genpelv 11071 | Membership in value of gen... |
| genpprecl 11072 | Pre-closure law for genera... |
| genpdm 11073 | Domain of general operatio... |
| genpn0 11074 | The result of an operation... |
| genpss 11075 | The result of an operation... |
| genpnnp 11076 | The result of an operation... |
| genpcd 11077 | Downward closure of an ope... |
| genpnmax 11078 | An operation on positive r... |
| genpcl 11079 | Closure of an operation on... |
| genpass 11080 | Associativity of an operat... |
| plpv 11081 | Value of addition on posit... |
| mpv 11082 | Value of multiplication on... |
| dmplp 11083 | Domain of addition on posi... |
| dmmp 11084 | Domain of multiplication o... |
| nqpr 11085 | The canonical embedding of... |
| 1pr 11086 | The positive real number '... |
| addclprlem1 11087 | Lemma to prove downward cl... |
| addclprlem2 11088 | Lemma to prove downward cl... |
| addclpr 11089 | Closure of addition on pos... |
| mulclprlem 11090 | Lemma to prove downward cl... |
| mulclpr 11091 | Closure of multiplication ... |
| addcompr 11092 | Addition of positive reals... |
| addasspr 11093 | Addition of positive reals... |
| mulcompr 11094 | Multiplication of positive... |
| mulasspr 11095 | Multiplication of positive... |
| distrlem1pr 11096 | Lemma for distributive law... |
| distrlem4pr 11097 | Lemma for distributive law... |
| distrlem5pr 11098 | Lemma for distributive law... |
| distrpr 11099 | Multiplication of positive... |
| 1idpr 11100 | 1 is an identity element f... |
| ltprord 11101 | Positive real 'less than' ... |
| psslinpr 11102 | Proper subset is a linear ... |
| ltsopr 11103 | Positive real 'less than' ... |
| prlem934 11104 | Lemma 9-3.4 of [Gleason] p... |
| ltaddpr 11105 | The sum of two positive re... |
| ltaddpr2 11106 | The sum of two positive re... |
| ltexprlem1 11107 | Lemma for Proposition 9-3.... |
| ltexprlem2 11108 | Lemma for Proposition 9-3.... |
| ltexprlem3 11109 | Lemma for Proposition 9-3.... |
| ltexprlem4 11110 | Lemma for Proposition 9-3.... |
| ltexprlem5 11111 | Lemma for Proposition 9-3.... |
| ltexprlem6 11112 | Lemma for Proposition 9-3.... |
| ltexprlem7 11113 | Lemma for Proposition 9-3.... |
| ltexpri 11114 | Proposition 9-3.5(iv) of [... |
| ltaprlem 11115 | Lemma for Proposition 9-3.... |
| ltapr 11116 | Ordering property of addit... |
| addcanpr 11117 | Addition cancellation law ... |
| prlem936 11118 | Lemma 9-3.6 of [Gleason] p... |
| reclem2pr 11119 | Lemma for Proposition 9-3.... |
| reclem3pr 11120 | Lemma for Proposition 9-3.... |
| reclem4pr 11121 | Lemma for Proposition 9-3.... |
| recexpr 11122 | The reciprocal of a positi... |
| suplem1pr 11123 | The union of a nonempty, b... |
| suplem2pr 11124 | The union of a set of posi... |
| supexpr 11125 | The union of a nonempty, b... |
| enrer 11134 | The equivalence relation f... |
| nrex1 11135 | The class of signed reals ... |
| enrbreq 11136 | Equivalence relation for s... |
| enreceq 11137 | Equivalence class equality... |
| enrex 11138 | The equivalence relation f... |
| ltrelsr 11139 | Signed real 'less than' is... |
| addcmpblnr 11140 | Lemma showing compatibilit... |
| mulcmpblnrlem 11141 | Lemma used in lemma showin... |
| mulcmpblnr 11142 | Lemma showing compatibilit... |
| prsrlem1 11143 | Decomposing signed reals i... |
| addsrmo 11144 | There is at most one resul... |
| mulsrmo 11145 | There is at most one resul... |
| addsrpr 11146 | Addition of signed reals i... |
| mulsrpr 11147 | Multiplication of signed r... |
| ltsrpr 11148 | Ordering of signed reals i... |
| gt0srpr 11149 | Greater than zero in terms... |
| 0nsr 11150 | The empty set is not a sig... |
| 0r 11151 | The constant ` 0R ` is a s... |
| 1sr 11152 | The constant ` 1R ` is a s... |
| m1r 11153 | The constant ` -1R ` is a ... |
| addclsr 11154 | Closure of addition on sig... |
| mulclsr 11155 | Closure of multiplication ... |
| dmaddsr 11156 | Domain of addition on sign... |
| dmmulsr 11157 | Domain of multiplication o... |
| addcomsr 11158 | Addition of signed reals i... |
| addasssr 11159 | Addition of signed reals i... |
| mulcomsr 11160 | Multiplication of signed r... |
| mulasssr 11161 | Multiplication of signed r... |
| distrsr 11162 | Multiplication of signed r... |
| m1p1sr 11163 | Minus one plus one is zero... |
| m1m1sr 11164 | Minus one times minus one ... |
| ltsosr 11165 | Signed real 'less than' is... |
| 0lt1sr 11166 | 0 is less than 1 for signe... |
| 1ne0sr 11167 | 1 and 0 are distinct for s... |
| 0idsr 11168 | The signed real number 0 i... |
| 1idsr 11169 | 1 is an identity element f... |
| 00sr 11170 | A signed real times 0 is 0... |
| ltasr 11171 | Ordering property of addit... |
| pn0sr 11172 | A signed real plus its neg... |
| negexsr 11173 | Existence of negative sign... |
| recexsrlem 11174 | The reciprocal of a positi... |
| addgt0sr 11175 | The sum of two positive si... |
| mulgt0sr 11176 | The product of two positiv... |
| sqgt0sr 11177 | The square of a nonzero si... |
| recexsr 11178 | The reciprocal of a nonzer... |
| mappsrpr 11179 | Mapping from positive sign... |
| ltpsrpr 11180 | Mapping of order from posi... |
| map2psrpr 11181 | Equivalence for positive s... |
| supsrlem 11182 | Lemma for supremum theorem... |
| supsr 11183 | A nonempty, bounded set of... |
| opelcn 11200 | Ordered pair membership in... |
| opelreal 11201 | Ordered pair membership in... |
| elreal 11202 | Membership in class of rea... |
| elreal2 11203 | Ordered pair membership in... |
| 0ncn 11204 | The empty set is not a com... |
| ltrelre 11205 | 'Less than' is a relation ... |
| addcnsr 11206 | Addition of complex number... |
| mulcnsr 11207 | Multiplication of complex ... |
| eqresr 11208 | Equality of real numbers i... |
| addresr 11209 | Addition of real numbers i... |
| mulresr 11210 | Multiplication of real num... |
| ltresr 11211 | Ordering of real subset of... |
| ltresr2 11212 | Ordering of real subset of... |
| dfcnqs 11213 | Technical trick to permit ... |
| addcnsrec 11214 | Technical trick to permit ... |
| mulcnsrec 11215 | Technical trick to permit ... |
| axaddf 11216 | Addition is an operation o... |
| axmulf 11217 | Multiplication is an opera... |
| axcnex 11218 | The complex numbers form a... |
| axresscn 11219 | The real numbers are a sub... |
| ax1cn 11220 | 1 is a complex number. Ax... |
| axicn 11221 | ` _i ` is a complex number... |
| axaddcl 11222 | Closure law for addition o... |
| axaddrcl 11223 | Closure law for addition i... |
| axmulcl 11224 | Closure law for multiplica... |
| axmulrcl 11225 | Closure law for multiplica... |
| axmulcom 11226 | Multiplication of complex ... |
| axaddass 11227 | Addition of complex number... |
| axmulass 11228 | Multiplication of complex ... |
| axdistr 11229 | Distributive law for compl... |
| axi2m1 11230 | i-squared equals -1 (expre... |
| ax1ne0 11231 | 1 and 0 are distinct. Axi... |
| ax1rid 11232 | ` 1 ` is an identity eleme... |
| axrnegex 11233 | Existence of negative of r... |
| axrrecex 11234 | Existence of reciprocal of... |
| axcnre 11235 | A complex number can be ex... |
| axpre-lttri 11236 | Ordering on reals satisfie... |
| axpre-lttrn 11237 | Ordering on reals is trans... |
| axpre-ltadd 11238 | Ordering property of addit... |
| axpre-mulgt0 11239 | The product of two positiv... |
| axpre-sup 11240 | A nonempty, bounded-above ... |
| wuncn 11241 | A weak universe containing... |
| cnex 11267 | Alias for ~ ax-cnex . See... |
| addcl 11268 | Alias for ~ ax-addcl , for... |
| readdcl 11269 | Alias for ~ ax-addrcl , fo... |
| mulcl 11270 | Alias for ~ ax-mulcl , for... |
| remulcl 11271 | Alias for ~ ax-mulrcl , fo... |
| mulcom 11272 | Alias for ~ ax-mulcom , fo... |
| addass 11273 | Alias for ~ ax-addass , fo... |
| mulass 11274 | Alias for ~ ax-mulass , fo... |
| adddi 11275 | Alias for ~ ax-distr , for... |
| recn 11276 | A real number is a complex... |
| reex 11277 | The real numbers form a se... |
| reelprrecn 11278 | Reals are a subset of the ... |
| cnelprrecn 11279 | Complex numbers are a subs... |
| mpoaddf 11280 | Addition is an operation o... |
| mpomulf 11281 | Multiplication is an opera... |
| elimne0 11282 | Hypothesis for weak deduct... |
| adddir 11283 | Distributive law for compl... |
| 0cn 11284 | Zero is a complex number. ... |
| 0cnd 11285 | Zero is a complex number, ... |
| c0ex 11286 | Zero is a set. (Contribut... |
| 1cnd 11287 | One is a complex number, d... |
| 1ex 11288 | One is a set. (Contribute... |
| cnre 11289 | Alias for ~ ax-cnre , for ... |
| mulrid 11290 | The number 1 is an identit... |
| mullid 11291 | Identity law for multiplic... |
| 1re 11292 | The number 1 is real. Thi... |
| 1red 11293 | The number 1 is real, dedu... |
| 0re 11294 | The number 0 is real. Rem... |
| 0red 11295 | The number 0 is real, dedu... |
| mulridi 11296 | Identity law for multiplic... |
| mullidi 11297 | Identity law for multiplic... |
| addcli 11298 | Closure law for addition. ... |
| mulcli 11299 | Closure law for multiplica... |
| mulcomi 11300 | Commutative law for multip... |
| mulcomli 11301 | Commutative law for multip... |
| addassi 11302 | Associative law for additi... |
| mulassi 11303 | Associative law for multip... |
| adddii 11304 | Distributive law (left-dis... |
| adddiri 11305 | Distributive law (right-di... |
| recni 11306 | A real number is a complex... |
| readdcli 11307 | Closure law for addition o... |
| remulcli 11308 | Closure law for multiplica... |
| mulridd 11309 | Identity law for multiplic... |
| mullidd 11310 | Identity law for multiplic... |
| addcld 11311 | Closure law for addition. ... |
| mulcld 11312 | Closure law for multiplica... |
| mulcomd 11313 | Commutative law for multip... |
| addassd 11314 | Associative law for additi... |
| mulassd 11315 | Associative law for multip... |
| adddid 11316 | Distributive law (left-dis... |
| adddird 11317 | Distributive law (right-di... |
| adddirp1d 11318 | Distributive law, plus 1 v... |
| joinlmuladdmuld 11319 | Join AB+CB into (A+C) on L... |
| recnd 11320 | Deduction from real number... |
| readdcld 11321 | Closure law for addition o... |
| remulcld 11322 | Closure law for multiplica... |
| pnfnre 11333 | Plus infinity is not a rea... |
| pnfnre2 11334 | Plus infinity is not a rea... |
| mnfnre 11335 | Minus infinity is not a re... |
| ressxr 11336 | The standard reals are a s... |
| rexpssxrxp 11337 | The Cartesian product of s... |
| rexr 11338 | A standard real is an exte... |
| 0xr 11339 | Zero is an extended real. ... |
| renepnf 11340 | No (finite) real equals pl... |
| renemnf 11341 | No real equals minus infin... |
| rexrd 11342 | A standard real is an exte... |
| renepnfd 11343 | No (finite) real equals pl... |
| renemnfd 11344 | No real equals minus infin... |
| pnfex 11345 | Plus infinity exists. (Co... |
| pnfxr 11346 | Plus infinity belongs to t... |
| pnfnemnf 11347 | Plus and minus infinity ar... |
| mnfnepnf 11348 | Minus and plus infinity ar... |
| mnfxr 11349 | Minus infinity belongs to ... |
| rexri 11350 | A standard real is an exte... |
| 1xr 11351 | ` 1 ` is an extended real ... |
| renfdisj 11352 | The reals and the infiniti... |
| ltrelxr 11353 | "Less than" is a relation ... |
| ltrel 11354 | "Less than" is a relation.... |
| lerelxr 11355 | "Less than or equal to" is... |
| lerel 11356 | "Less than or equal to" is... |
| xrlenlt 11357 | "Less than or equal to" ex... |
| xrlenltd 11358 | "Less than or equal to" ex... |
| xrltnle 11359 | "Less than" expressed in t... |
| xrnltled 11360 | "Not less than" implies "l... |
| ssxr 11361 | The three (non-exclusive) ... |
| ltxrlt 11362 | The standard less-than ` <... |
| axlttri 11363 | Ordering on reals satisfie... |
| axlttrn 11364 | Ordering on reals is trans... |
| axltadd 11365 | Ordering property of addit... |
| axmulgt0 11366 | The product of two positiv... |
| axsup 11367 | A nonempty, bounded-above ... |
| lttr 11368 | Alias for ~ axlttrn , for ... |
| mulgt0 11369 | The product of two positiv... |
| lenlt 11370 | 'Less than or equal to' ex... |
| ltnle 11371 | 'Less than' expressed in t... |
| ltso 11372 | 'Less than' is a strict or... |
| gtso 11373 | 'Greater than' is a strict... |
| lttri2 11374 | Consequence of trichotomy.... |
| lttri3 11375 | Trichotomy law for 'less t... |
| lttri4 11376 | Trichotomy law for 'less t... |
| letri3 11377 | Trichotomy law. (Contribu... |
| leloe 11378 | 'Less than or equal to' ex... |
| eqlelt 11379 | Equality in terms of 'less... |
| ltle 11380 | 'Less than' implies 'less ... |
| leltne 11381 | 'Less than or equal to' im... |
| lelttr 11382 | Transitive law. (Contribu... |
| leltletr 11383 | Transitive law, weaker for... |
| ltletr 11384 | Transitive law. (Contribu... |
| ltleletr 11385 | Transitive law, weaker for... |
| letr 11386 | Transitive law. (Contribu... |
| ltnr 11387 | 'Less than' is irreflexive... |
| leid 11388 | 'Less than or equal to' is... |
| ltne 11389 | 'Less than' implies not eq... |
| ltnsym 11390 | 'Less than' is not symmetr... |
| ltnsym2 11391 | 'Less than' is antisymmetr... |
| letric 11392 | Trichotomy law. (Contribu... |
| ltlen 11393 | 'Less than' expressed in t... |
| eqle 11394 | Equality implies 'less tha... |
| eqled 11395 | Equality implies 'less tha... |
| ltadd2 11396 | Addition to both sides of ... |
| ne0gt0 11397 | A nonzero nonnegative numb... |
| lecasei 11398 | Ordering elimination by ca... |
| lelttric 11399 | Trichotomy law. (Contribu... |
| ltlecasei 11400 | Ordering elimination by ca... |
| ltnri 11401 | 'Less than' is irreflexive... |
| eqlei 11402 | Equality implies 'less tha... |
| eqlei2 11403 | Equality implies 'less tha... |
| gtneii 11404 | 'Less than' implies not eq... |
| ltneii 11405 | 'Greater than' implies not... |
| lttri2i 11406 | Consequence of trichotomy.... |
| lttri3i 11407 | Consequence of trichotomy.... |
| letri3i 11408 | Consequence of trichotomy.... |
| leloei 11409 | 'Less than or equal to' in... |
| ltleni 11410 | 'Less than' expressed in t... |
| ltnsymi 11411 | 'Less than' is not symmetr... |
| lenlti 11412 | 'Less than or equal to' in... |
| ltnlei 11413 | 'Less than' in terms of 'l... |
| ltlei 11414 | 'Less than' implies 'less ... |
| ltleii 11415 | 'Less than' implies 'less ... |
| ltnei 11416 | 'Less than' implies not eq... |
| letrii 11417 | Trichotomy law for 'less t... |
| lttri 11418 | 'Less than' is transitive.... |
| lelttri 11419 | 'Less than or equal to', '... |
| ltletri 11420 | 'Less than', 'less than or... |
| letri 11421 | 'Less than or equal to' is... |
| le2tri3i 11422 | Extended trichotomy law fo... |
| ltadd2i 11423 | Addition to both sides of ... |
| mulgt0i 11424 | The product of two positiv... |
| mulgt0ii 11425 | The product of two positiv... |
| ltnrd 11426 | 'Less than' is irreflexive... |
| gtned 11427 | 'Less than' implies not eq... |
| ltned 11428 | 'Greater than' implies not... |
| ne0gt0d 11429 | A nonzero nonnegative numb... |
| lttrid 11430 | Ordering on reals satisfie... |
| lttri2d 11431 | Consequence of trichotomy.... |
| lttri3d 11432 | Consequence of trichotomy.... |
| lttri4d 11433 | Trichotomy law for 'less t... |
| letri3d 11434 | Consequence of trichotomy.... |
| leloed 11435 | 'Less than or equal to' in... |
| eqleltd 11436 | Equality in terms of 'less... |
| ltlend 11437 | 'Less than' expressed in t... |
| lenltd 11438 | 'Less than or equal to' in... |
| ltnled 11439 | 'Less than' in terms of 'l... |
| ltled 11440 | 'Less than' implies 'less ... |
| ltnsymd 11441 | 'Less than' implies 'less ... |
| nltled 11442 | 'Not less than ' implies '... |
| lensymd 11443 | 'Less than or equal to' im... |
| letrid 11444 | Trichotomy law for 'less t... |
| leltned 11445 | 'Less than or equal to' im... |
| leneltd 11446 | 'Less than or equal to' an... |
| mulgt0d 11447 | The product of two positiv... |
| ltadd2d 11448 | Addition to both sides of ... |
| letrd 11449 | Transitive law deduction f... |
| lelttrd 11450 | Transitive law deduction f... |
| ltadd2dd 11451 | Addition to both sides of ... |
| ltletrd 11452 | Transitive law deduction f... |
| lttrd 11453 | Transitive law deduction f... |
| lelttrdi 11454 | If a number is less than a... |
| dedekind 11455 | The Dedekind cut theorem. ... |
| dedekindle 11456 | The Dedekind cut theorem, ... |
| mul12 11457 | Commutative/associative la... |
| mul32 11458 | Commutative/associative la... |
| mul31 11459 | Commutative/associative la... |
| mul4 11460 | Rearrangement of 4 factors... |
| mul4r 11461 | Rearrangement of 4 factors... |
| muladd11 11462 | A simple product of sums e... |
| 1p1times 11463 | Two times a number. (Cont... |
| peano2cn 11464 | A theorem for complex numb... |
| peano2re 11465 | A theorem for reals analog... |
| readdcan 11466 | Cancellation law for addit... |
| 00id 11467 | ` 0 ` is its own additive ... |
| mul02lem1 11468 | Lemma for ~ mul02 . If an... |
| mul02lem2 11469 | Lemma for ~ mul02 . Zero ... |
| mul02 11470 | Multiplication by ` 0 ` . ... |
| mul01 11471 | Multiplication by ` 0 ` . ... |
| addrid 11472 | ` 0 ` is an additive ident... |
| cnegex 11473 | Existence of the negative ... |
| cnegex2 11474 | Existence of a left invers... |
| addlid 11475 | ` 0 ` is a left identity f... |
| addcan 11476 | Cancellation law for addit... |
| addcan2 11477 | Cancellation law for addit... |
| addcom 11478 | Addition commutes. This u... |
| addridi 11479 | ` 0 ` is an additive ident... |
| addlidi 11480 | ` 0 ` is a left identity f... |
| mul02i 11481 | Multiplication by 0. Theo... |
| mul01i 11482 | Multiplication by ` 0 ` . ... |
| addcomi 11483 | Addition commutes. Based ... |
| addcomli 11484 | Addition commutes. (Contr... |
| addcani 11485 | Cancellation law for addit... |
| addcan2i 11486 | Cancellation law for addit... |
| mul12i 11487 | Commutative/associative la... |
| mul32i 11488 | Commutative/associative la... |
| mul4i 11489 | Rearrangement of 4 factors... |
| mul02d 11490 | Multiplication by 0. Theo... |
| mul01d 11491 | Multiplication by ` 0 ` . ... |
| addridd 11492 | ` 0 ` is an additive ident... |
| addlidd 11493 | ` 0 ` is a left identity f... |
| addcomd 11494 | Addition commutes. Based ... |
| addcand 11495 | Cancellation law for addit... |
| addcan2d 11496 | Cancellation law for addit... |
| addcanad 11497 | Cancelling a term on the l... |
| addcan2ad 11498 | Cancelling a term on the r... |
| addneintrd 11499 | Introducing a term on the ... |
| addneintr2d 11500 | Introducing a term on the ... |
| mul12d 11501 | Commutative/associative la... |
| mul32d 11502 | Commutative/associative la... |
| mul31d 11503 | Commutative/associative la... |
| mul4d 11504 | Rearrangement of 4 factors... |
| muladd11r 11505 | A simple product of sums e... |
| comraddd 11506 | Commute RHS addition, in d... |
| ltaddneg 11507 | Adding a negative number t... |
| ltaddnegr 11508 | Adding a negative number t... |
| add12 11509 | Commutative/associative la... |
| add32 11510 | Commutative/associative la... |
| add32r 11511 | Commutative/associative la... |
| add4 11512 | Rearrangement of 4 terms i... |
| add42 11513 | Rearrangement of 4 terms i... |
| add12i 11514 | Commutative/associative la... |
| add32i 11515 | Commutative/associative la... |
| add4i 11516 | Rearrangement of 4 terms i... |
| add42i 11517 | Rearrangement of 4 terms i... |
| add12d 11518 | Commutative/associative la... |
| add32d 11519 | Commutative/associative la... |
| add4d 11520 | Rearrangement of 4 terms i... |
| add42d 11521 | Rearrangement of 4 terms i... |
| 0cnALT 11526 | Alternate proof of ~ 0cn w... |
| 0cnALT2 11527 | Alternate proof of ~ 0cnAL... |
| negeu 11528 | Existential uniqueness of ... |
| subval 11529 | Value of subtraction, whic... |
| negeq 11530 | Equality theorem for negat... |
| negeqi 11531 | Equality inference for neg... |
| negeqd 11532 | Equality deduction for neg... |
| nfnegd 11533 | Deduction version of ~ nfn... |
| nfneg 11534 | Bound-variable hypothesis ... |
| csbnegg 11535 | Move class substitution in... |
| negex 11536 | A negative is a set. (Con... |
| subcl 11537 | Closure law for subtractio... |
| negcl 11538 | Closure law for negative. ... |
| negicn 11539 | ` -u _i ` is a complex num... |
| subf 11540 | Subtraction is an operatio... |
| subadd 11541 | Relationship between subtr... |
| subadd2 11542 | Relationship between subtr... |
| subsub23 11543 | Swap subtrahend and result... |
| pncan 11544 | Cancellation law for subtr... |
| pncan2 11545 | Cancellation law for subtr... |
| pncan3 11546 | Subtraction and addition o... |
| npcan 11547 | Cancellation law for subtr... |
| addsubass 11548 | Associative-type law for a... |
| addsub 11549 | Law for addition and subtr... |
| subadd23 11550 | Commutative/associative la... |
| addsub12 11551 | Commutative/associative la... |
| 2addsub 11552 | Law for subtraction and ad... |
| addsubeq4 11553 | Relation between sums and ... |
| pncan3oi 11554 | Subtraction and addition o... |
| mvrraddi 11555 | Move the right term in a s... |
| mvlladdi 11556 | Move the left term in a su... |
| subid 11557 | Subtraction of a number fr... |
| subid1 11558 | Identity law for subtracti... |
| npncan 11559 | Cancellation law for subtr... |
| nppcan 11560 | Cancellation law for subtr... |
| nnpcan 11561 | Cancellation law for subtr... |
| nppcan3 11562 | Cancellation law for subtr... |
| subcan2 11563 | Cancellation law for subtr... |
| subeq0 11564 | If the difference between ... |
| npncan2 11565 | Cancellation law for subtr... |
| subsub2 11566 | Law for double subtraction... |
| nncan 11567 | Cancellation law for subtr... |
| subsub 11568 | Law for double subtraction... |
| nppcan2 11569 | Cancellation law for subtr... |
| subsub3 11570 | Law for double subtraction... |
| subsub4 11571 | Law for double subtraction... |
| sub32 11572 | Swap the second and third ... |
| nnncan 11573 | Cancellation law for subtr... |
| nnncan1 11574 | Cancellation law for subtr... |
| nnncan2 11575 | Cancellation law for subtr... |
| npncan3 11576 | Cancellation law for subtr... |
| pnpcan 11577 | Cancellation law for mixed... |
| pnpcan2 11578 | Cancellation law for mixed... |
| pnncan 11579 | Cancellation law for mixed... |
| ppncan 11580 | Cancellation law for mixed... |
| addsub4 11581 | Rearrangement of 4 terms i... |
| subadd4 11582 | Rearrangement of 4 terms i... |
| sub4 11583 | Rearrangement of 4 terms i... |
| neg0 11584 | Minus 0 equals 0. (Contri... |
| negid 11585 | Addition of a number and i... |
| negsub 11586 | Relationship between subtr... |
| subneg 11587 | Relationship between subtr... |
| negneg 11588 | A number is equal to the n... |
| neg11 11589 | Negative is one-to-one. (... |
| negcon1 11590 | Negative contraposition la... |
| negcon2 11591 | Negative contraposition la... |
| negeq0 11592 | A number is zero iff its n... |
| subcan 11593 | Cancellation law for subtr... |
| negsubdi 11594 | Distribution of negative o... |
| negdi 11595 | Distribution of negative o... |
| negdi2 11596 | Distribution of negative o... |
| negsubdi2 11597 | Distribution of negative o... |
| neg2sub 11598 | Relationship between subtr... |
| renegcli 11599 | Closure law for negative o... |
| resubcli 11600 | Closure law for subtractio... |
| renegcl 11601 | Closure law for negative o... |
| resubcl 11602 | Closure law for subtractio... |
| negreb 11603 | The negative of a real is ... |
| peano2cnm 11604 | "Reverse" second Peano pos... |
| peano2rem 11605 | "Reverse" second Peano pos... |
| negcli 11606 | Closure law for negative. ... |
| negidi 11607 | Addition of a number and i... |
| negnegi 11608 | A number is equal to the n... |
| subidi 11609 | Subtraction of a number fr... |
| subid1i 11610 | Identity law for subtracti... |
| negne0bi 11611 | A number is nonzero iff it... |
| negrebi 11612 | The negative of a real is ... |
| negne0i 11613 | The negative of a nonzero ... |
| subcli 11614 | Closure law for subtractio... |
| pncan3i 11615 | Subtraction and addition o... |
| negsubi 11616 | Relationship between subtr... |
| subnegi 11617 | Relationship between subtr... |
| subeq0i 11618 | If the difference between ... |
| neg11i 11619 | Negative is one-to-one. (... |
| negcon1i 11620 | Negative contraposition la... |
| negcon2i 11621 | Negative contraposition la... |
| negdii 11622 | Distribution of negative o... |
| negsubdii 11623 | Distribution of negative o... |
| negsubdi2i 11624 | Distribution of negative o... |
| subaddi 11625 | Relationship between subtr... |
| subadd2i 11626 | Relationship between subtr... |
| subaddrii 11627 | Relationship between subtr... |
| subsub23i 11628 | Swap subtrahend and result... |
| addsubassi 11629 | Associative-type law for s... |
| addsubi 11630 | Law for subtraction and ad... |
| subcani 11631 | Cancellation law for subtr... |
| subcan2i 11632 | Cancellation law for subtr... |
| pnncani 11633 | Cancellation law for mixed... |
| addsub4i 11634 | Rearrangement of 4 terms i... |
| 0reALT 11635 | Alternate proof of ~ 0re .... |
| negcld 11636 | Closure law for negative. ... |
| subidd 11637 | Subtraction of a number fr... |
| subid1d 11638 | Identity law for subtracti... |
| negidd 11639 | Addition of a number and i... |
| negnegd 11640 | A number is equal to the n... |
| negeq0d 11641 | A number is zero iff its n... |
| negne0bd 11642 | A number is nonzero iff it... |
| negcon1d 11643 | Contraposition law for una... |
| negcon1ad 11644 | Contraposition law for una... |
| neg11ad 11645 | The negatives of two compl... |
| negned 11646 | If two complex numbers are... |
| negne0d 11647 | The negative of a nonzero ... |
| negrebd 11648 | The negative of a real is ... |
| subcld 11649 | Closure law for subtractio... |
| pncand 11650 | Cancellation law for subtr... |
| pncan2d 11651 | Cancellation law for subtr... |
| pncan3d 11652 | Subtraction and addition o... |
| npcand 11653 | Cancellation law for subtr... |
| nncand 11654 | Cancellation law for subtr... |
| negsubd 11655 | Relationship between subtr... |
| subnegd 11656 | Relationship between subtr... |
| subeq0d 11657 | If the difference between ... |
| subne0d 11658 | Two unequal numbers have n... |
| subeq0ad 11659 | The difference of two comp... |
| subne0ad 11660 | If the difference of two c... |
| neg11d 11661 | If the difference between ... |
| negdid 11662 | Distribution of negative o... |
| negdi2d 11663 | Distribution of negative o... |
| negsubdid 11664 | Distribution of negative o... |
| negsubdi2d 11665 | Distribution of negative o... |
| neg2subd 11666 | Relationship between subtr... |
| subaddd 11667 | Relationship between subtr... |
| subadd2d 11668 | Relationship between subtr... |
| addsubassd 11669 | Associative-type law for s... |
| addsubd 11670 | Law for subtraction and ad... |
| subadd23d 11671 | Commutative/associative la... |
| addsub12d 11672 | Commutative/associative la... |
| npncand 11673 | Cancellation law for subtr... |
| nppcand 11674 | Cancellation law for subtr... |
| nppcan2d 11675 | Cancellation law for subtr... |
| nppcan3d 11676 | Cancellation law for subtr... |
| subsubd 11677 | Law for double subtraction... |
| subsub2d 11678 | Law for double subtraction... |
| subsub3d 11679 | Law for double subtraction... |
| subsub4d 11680 | Law for double subtraction... |
| sub32d 11681 | Swap the second and third ... |
| nnncand 11682 | Cancellation law for subtr... |
| nnncan1d 11683 | Cancellation law for subtr... |
| nnncan2d 11684 | Cancellation law for subtr... |
| npncan3d 11685 | Cancellation law for subtr... |
| pnpcand 11686 | Cancellation law for mixed... |
| pnpcan2d 11687 | Cancellation law for mixed... |
| pnncand 11688 | Cancellation law for mixed... |
| ppncand 11689 | Cancellation law for mixed... |
| subcand 11690 | Cancellation law for subtr... |
| subcan2d 11691 | Cancellation law for subtr... |
| subcanad 11692 | Cancellation law for subtr... |
| subneintrd 11693 | Introducing subtraction on... |
| subcan2ad 11694 | Cancellation law for subtr... |
| subneintr2d 11695 | Introducing subtraction on... |
| addsub4d 11696 | Rearrangement of 4 terms i... |
| subadd4d 11697 | Rearrangement of 4 terms i... |
| sub4d 11698 | Rearrangement of 4 terms i... |
| 2addsubd 11699 | Law for subtraction and ad... |
| addsubeq4d 11700 | Relation between sums and ... |
| subeqxfrd 11701 | Transfer two terms of a su... |
| mvlraddd 11702 | Move the right term in a s... |
| mvlladdd 11703 | Move the left term in a su... |
| mvrraddd 11704 | Move the right term in a s... |
| mvrladdd 11705 | Move the left term in a su... |
| assraddsubd 11706 | Associate RHS addition-sub... |
| subaddeqd 11707 | Transfer two terms of a su... |
| addlsub 11708 | Left-subtraction: Subtrac... |
| addrsub 11709 | Right-subtraction: Subtra... |
| subexsub 11710 | A subtraction law: Exchan... |
| addid0 11711 | If adding a number to a an... |
| addn0nid 11712 | Adding a nonzero number to... |
| pnpncand 11713 | Addition/subtraction cance... |
| subeqrev 11714 | Reverse the order of subtr... |
| addeq0 11715 | Two complex numbers add up... |
| pncan1 11716 | Cancellation law for addit... |
| npcan1 11717 | Cancellation law for subtr... |
| subeq0bd 11718 | If two complex numbers are... |
| renegcld 11719 | Closure law for negative o... |
| resubcld 11720 | Closure law for subtractio... |
| negn0 11721 | The image under negation o... |
| negf1o 11722 | Negation is an isomorphism... |
| kcnktkm1cn 11723 | k times k minus 1 is a com... |
| muladd 11724 | Product of two sums. (Con... |
| subdi 11725 | Distribution of multiplica... |
| subdir 11726 | Distribution of multiplica... |
| ine0 11727 | The imaginary unit ` _i ` ... |
| mulneg1 11728 | Product with negative is n... |
| mulneg2 11729 | The product with a negativ... |
| mulneg12 11730 | Swap the negative sign in ... |
| mul2neg 11731 | Product of two negatives. ... |
| submul2 11732 | Convert a subtraction to a... |
| mulm1 11733 | Product with minus one is ... |
| addneg1mul 11734 | Addition with product with... |
| mulsub 11735 | Product of two differences... |
| mulsub2 11736 | Swap the order of subtract... |
| mulm1i 11737 | Product with minus one is ... |
| mulneg1i 11738 | Product with negative is n... |
| mulneg2i 11739 | Product with negative is n... |
| mul2negi 11740 | Product of two negatives. ... |
| subdii 11741 | Distribution of multiplica... |
| subdiri 11742 | Distribution of multiplica... |
| muladdi 11743 | Product of two sums. (Con... |
| mulm1d 11744 | Product with minus one is ... |
| mulneg1d 11745 | Product with negative is n... |
| mulneg2d 11746 | Product with negative is n... |
| mul2negd 11747 | Product of two negatives. ... |
| subdid 11748 | Distribution of multiplica... |
| subdird 11749 | Distribution of multiplica... |
| muladdd 11750 | Product of two sums. (Con... |
| mulsubd 11751 | Product of two differences... |
| muls1d 11752 | Multiplication by one minu... |
| mulsubfacd 11753 | Multiplication followed by... |
| addmulsub 11754 | The product of a sum and a... |
| subaddmulsub 11755 | The difference with a prod... |
| mulsubaddmulsub 11756 | A special difference of a ... |
| gt0ne0 11757 | Positive implies nonzero. ... |
| lt0ne0 11758 | A number which is less tha... |
| ltadd1 11759 | Addition to both sides of ... |
| leadd1 11760 | Addition to both sides of ... |
| leadd2 11761 | Addition to both sides of ... |
| ltsubadd 11762 | 'Less than' relationship b... |
| ltsubadd2 11763 | 'Less than' relationship b... |
| lesubadd 11764 | 'Less than or equal to' re... |
| lesubadd2 11765 | 'Less than or equal to' re... |
| ltaddsub 11766 | 'Less than' relationship b... |
| ltaddsub2 11767 | 'Less than' relationship b... |
| leaddsub 11768 | 'Less than or equal to' re... |
| leaddsub2 11769 | 'Less than or equal to' re... |
| suble 11770 | Swap subtrahends in an ine... |
| lesub 11771 | Swap subtrahends in an ine... |
| ltsub23 11772 | 'Less than' relationship b... |
| ltsub13 11773 | 'Less than' relationship b... |
| le2add 11774 | Adding both sides of two '... |
| ltleadd 11775 | Adding both sides of two o... |
| leltadd 11776 | Adding both sides of two o... |
| lt2add 11777 | Adding both sides of two '... |
| addgt0 11778 | The sum of 2 positive numb... |
| addgegt0 11779 | The sum of nonnegative and... |
| addgtge0 11780 | The sum of nonnegative and... |
| addge0 11781 | The sum of 2 nonnegative n... |
| ltaddpos 11782 | Adding a positive number t... |
| ltaddpos2 11783 | Adding a positive number t... |
| ltsubpos 11784 | Subtracting a positive num... |
| posdif 11785 | Comparison of two numbers ... |
| lesub1 11786 | Subtraction from both side... |
| lesub2 11787 | Subtraction of both sides ... |
| ltsub1 11788 | Subtraction from both side... |
| ltsub2 11789 | Subtraction of both sides ... |
| lt2sub 11790 | Subtracting both sides of ... |
| le2sub 11791 | Subtracting both sides of ... |
| ltneg 11792 | Negative of both sides of ... |
| ltnegcon1 11793 | Contraposition of negative... |
| ltnegcon2 11794 | Contraposition of negative... |
| leneg 11795 | Negative of both sides of ... |
| lenegcon1 11796 | Contraposition of negative... |
| lenegcon2 11797 | Contraposition of negative... |
| lt0neg1 11798 | Comparison of a number and... |
| lt0neg2 11799 | Comparison of a number and... |
| le0neg1 11800 | Comparison of a number and... |
| le0neg2 11801 | Comparison of a number and... |
| addge01 11802 | A number is less than or e... |
| addge02 11803 | A number is less than or e... |
| add20 11804 | Two nonnegative numbers ar... |
| subge0 11805 | Nonnegative subtraction. ... |
| suble0 11806 | Nonpositive subtraction. ... |
| leaddle0 11807 | The sum of a real number a... |
| subge02 11808 | Nonnegative subtraction. ... |
| lesub0 11809 | Lemma to show a nonnegativ... |
| mulge0 11810 | The product of two nonnega... |
| mullt0 11811 | The product of two negativ... |
| msqgt0 11812 | A nonzero square is positi... |
| msqge0 11813 | A square is nonnegative. ... |
| 0lt1 11814 | 0 is less than 1. Theorem... |
| 0le1 11815 | 0 is less than or equal to... |
| relin01 11816 | An interval law for less t... |
| ltordlem 11817 | Lemma for ~ ltord1 . (Con... |
| ltord1 11818 | Infer an ordering relation... |
| leord1 11819 | Infer an ordering relation... |
| eqord1 11820 | A strictly increasing real... |
| ltord2 11821 | Infer an ordering relation... |
| leord2 11822 | Infer an ordering relation... |
| eqord2 11823 | A strictly decreasing real... |
| wloglei 11824 | Form of ~ wlogle where bot... |
| wlogle 11825 | If the predicate ` ch ( x ... |
| leidi 11826 | 'Less than or equal to' is... |
| gt0ne0i 11827 | Positive means nonzero (us... |
| gt0ne0ii 11828 | Positive implies nonzero. ... |
| msqgt0i 11829 | A nonzero square is positi... |
| msqge0i 11830 | A square is nonnegative. ... |
| addgt0i 11831 | Addition of 2 positive num... |
| addge0i 11832 | Addition of 2 nonnegative ... |
| addgegt0i 11833 | Addition of nonnegative an... |
| addgt0ii 11834 | Addition of 2 positive num... |
| add20i 11835 | Two nonnegative numbers ar... |
| ltnegi 11836 | Negative of both sides of ... |
| lenegi 11837 | Negative of both sides of ... |
| ltnegcon2i 11838 | Contraposition of negative... |
| mulge0i 11839 | The product of two nonnega... |
| lesub0i 11840 | Lemma to show a nonnegativ... |
| ltaddposi 11841 | Adding a positive number t... |
| posdifi 11842 | Comparison of two numbers ... |
| ltnegcon1i 11843 | Contraposition of negative... |
| lenegcon1i 11844 | Contraposition of negative... |
| subge0i 11845 | Nonnegative subtraction. ... |
| ltadd1i 11846 | Addition to both sides of ... |
| leadd1i 11847 | Addition to both sides of ... |
| leadd2i 11848 | Addition to both sides of ... |
| ltsubaddi 11849 | 'Less than' relationship b... |
| lesubaddi 11850 | 'Less than or equal to' re... |
| ltsubadd2i 11851 | 'Less than' relationship b... |
| lesubadd2i 11852 | 'Less than or equal to' re... |
| ltaddsubi 11853 | 'Less than' relationship b... |
| lt2addi 11854 | Adding both side of two in... |
| le2addi 11855 | Adding both side of two in... |
| gt0ne0d 11856 | Positive implies nonzero. ... |
| lt0ne0d 11857 | Something less than zero i... |
| leidd 11858 | 'Less than or equal to' is... |
| msqgt0d 11859 | A nonzero square is positi... |
| msqge0d 11860 | A square is nonnegative. ... |
| lt0neg1d 11861 | Comparison of a number and... |
| lt0neg2d 11862 | Comparison of a number and... |
| le0neg1d 11863 | Comparison of a number and... |
| le0neg2d 11864 | Comparison of a number and... |
| addgegt0d 11865 | Addition of nonnegative an... |
| addgtge0d 11866 | Addition of positive and n... |
| addgt0d 11867 | Addition of 2 positive num... |
| addge0d 11868 | Addition of 2 nonnegative ... |
| mulge0d 11869 | The product of two nonnega... |
| ltnegd 11870 | Negative of both sides of ... |
| lenegd 11871 | Negative of both sides of ... |
| ltnegcon1d 11872 | Contraposition of negative... |
| ltnegcon2d 11873 | Contraposition of negative... |
| lenegcon1d 11874 | Contraposition of negative... |
| lenegcon2d 11875 | Contraposition of negative... |
| ltaddposd 11876 | Adding a positive number t... |
| ltaddpos2d 11877 | Adding a positive number t... |
| ltsubposd 11878 | Subtracting a positive num... |
| posdifd 11879 | Comparison of two numbers ... |
| addge01d 11880 | A number is less than or e... |
| addge02d 11881 | A number is less than or e... |
| subge0d 11882 | Nonnegative subtraction. ... |
| suble0d 11883 | Nonpositive subtraction. ... |
| subge02d 11884 | Nonnegative subtraction. ... |
| ltadd1d 11885 | Addition to both sides of ... |
| leadd1d 11886 | Addition to both sides of ... |
| leadd2d 11887 | Addition to both sides of ... |
| ltsubaddd 11888 | 'Less than' relationship b... |
| lesubaddd 11889 | 'Less than or equal to' re... |
| ltsubadd2d 11890 | 'Less than' relationship b... |
| lesubadd2d 11891 | 'Less than or equal to' re... |
| ltaddsubd 11892 | 'Less than' relationship b... |
| ltaddsub2d 11893 | 'Less than' relationship b... |
| leaddsub2d 11894 | 'Less than or equal to' re... |
| subled 11895 | Swap subtrahends in an ine... |
| lesubd 11896 | Swap subtrahends in an ine... |
| ltsub23d 11897 | 'Less than' relationship b... |
| ltsub13d 11898 | 'Less than' relationship b... |
| lesub1d 11899 | Subtraction from both side... |
| lesub2d 11900 | Subtraction of both sides ... |
| ltsub1d 11901 | Subtraction from both side... |
| ltsub2d 11902 | Subtraction of both sides ... |
| ltadd1dd 11903 | Addition to both sides of ... |
| ltsub1dd 11904 | Subtraction from both side... |
| ltsub2dd 11905 | Subtraction of both sides ... |
| leadd1dd 11906 | Addition to both sides of ... |
| leadd2dd 11907 | Addition to both sides of ... |
| lesub1dd 11908 | Subtraction from both side... |
| lesub2dd 11909 | Subtraction of both sides ... |
| lesub3d 11910 | The result of subtracting ... |
| le2addd 11911 | Adding both side of two in... |
| le2subd 11912 | Subtracting both sides of ... |
| ltleaddd 11913 | Adding both sides of two o... |
| leltaddd 11914 | Adding both sides of two o... |
| lt2addd 11915 | Adding both side of two in... |
| lt2subd 11916 | Subtracting both sides of ... |
| possumd 11917 | Condition for a positive s... |
| sublt0d 11918 | When a subtraction gives a... |
| ltaddsublt 11919 | Addition and subtraction o... |
| 1le1 11920 | One is less than or equal ... |
| ixi 11921 | ` _i ` times itself is min... |
| recextlem1 11922 | Lemma for ~ recex . (Cont... |
| recextlem2 11923 | Lemma for ~ recex . (Cont... |
| recex 11924 | Existence of reciprocal of... |
| mulcand 11925 | Cancellation law for multi... |
| mulcan2d 11926 | Cancellation law for multi... |
| mulcanad 11927 | Cancellation of a nonzero ... |
| mulcan2ad 11928 | Cancellation of a nonzero ... |
| mulcan 11929 | Cancellation law for multi... |
| mulcan2 11930 | Cancellation law for multi... |
| mulcani 11931 | Cancellation law for multi... |
| mul0or 11932 | If a product is zero, one ... |
| mulne0b 11933 | The product of two nonzero... |
| mulne0 11934 | The product of two nonzero... |
| mulne0i 11935 | The product of two nonzero... |
| muleqadd 11936 | Property of numbers whose ... |
| receu 11937 | Existential uniqueness of ... |
| mulnzcnf 11938 | Multiplication maps nonzer... |
| msq0i 11939 | A number is zero iff its s... |
| mul0ori 11940 | If a product is zero, one ... |
| msq0d 11941 | A number is zero iff its s... |
| mul0ord 11942 | If a product is zero, one ... |
| mulne0bd 11943 | The product of two nonzero... |
| mulne0d 11944 | The product of two nonzero... |
| mulcan1g 11945 | A generalized form of the ... |
| mulcan2g 11946 | A generalized form of the ... |
| mulne0bad 11947 | A factor of a nonzero comp... |
| mulne0bbd 11948 | A factor of a nonzero comp... |
| 1div0 11951 | You can't divide by zero, ... |
| 1div0OLD 11952 | Obsolete version of ~ 1div... |
| divval 11953 | Value of division: if ` A ... |
| divmul 11954 | Relationship between divis... |
| divmul2 11955 | Relationship between divis... |
| divmul3 11956 | Relationship between divis... |
| divcl 11957 | Closure law for division. ... |
| reccl 11958 | Closure law for reciprocal... |
| divcan2 11959 | A cancellation law for div... |
| divcan1 11960 | A cancellation law for div... |
| diveq0 11961 | A ratio is zero iff the nu... |
| divne0b 11962 | The ratio of nonzero numbe... |
| divne0 11963 | The ratio of nonzero numbe... |
| recne0 11964 | The reciprocal of a nonzer... |
| recid 11965 | Multiplication of a number... |
| recid2 11966 | Multiplication of a number... |
| divrec 11967 | Relationship between divis... |
| divrec2 11968 | Relationship between divis... |
| divass 11969 | An associative law for div... |
| div23 11970 | A commutative/associative ... |
| div32 11971 | A commutative/associative ... |
| div13 11972 | A commutative/associative ... |
| div12 11973 | A commutative/associative ... |
| divmulass 11974 | An associative law for div... |
| divmulasscom 11975 | An associative/commutative... |
| divdir 11976 | Distribution of division o... |
| divcan3 11977 | A cancellation law for div... |
| divcan4 11978 | A cancellation law for div... |
| div11 11979 | One-to-one relationship fo... |
| div11OLD 11980 | Obsolete version of ~ div1... |
| diveq1 11981 | Equality in terms of unit ... |
| divid 11982 | A number divided by itself... |
| dividOLD 11983 | Obsolete version of ~ divi... |
| div0 11984 | Division into zero is zero... |
| div0OLD 11985 | Obsolete version of ~ div0... |
| div1 11986 | A number divided by 1 is i... |
| 1div1e1 11987 | 1 divided by 1 is 1. (Con... |
| divneg 11988 | Move negative sign inside ... |
| muldivdir 11989 | Distribution of division o... |
| divsubdir 11990 | Distribution of division o... |
| subdivcomb1 11991 | Bring a term in a subtract... |
| subdivcomb2 11992 | Bring a term in a subtract... |
| recrec 11993 | A number is equal to the r... |
| rec11 11994 | Reciprocal is one-to-one. ... |
| rec11r 11995 | Mutual reciprocals. (Cont... |
| divmuldiv 11996 | Multiplication of two rati... |
| divdivdiv 11997 | Division of two ratios. T... |
| divcan5 11998 | Cancellation of common fac... |
| divmul13 11999 | Swap the denominators in t... |
| divmul24 12000 | Swap the numerators in the... |
| divmuleq 12001 | Cross-multiply in an equal... |
| recdiv 12002 | The reciprocal of a ratio.... |
| divcan6 12003 | Cancellation of inverted f... |
| divdiv32 12004 | Swap denominators in a div... |
| divcan7 12005 | Cancel equal divisors in a... |
| dmdcan 12006 | Cancellation law for divis... |
| divdiv1 12007 | Division into a fraction. ... |
| divdiv2 12008 | Division by a fraction. (... |
| recdiv2 12009 | Division into a reciprocal... |
| ddcan 12010 | Cancellation in a double d... |
| divadddiv 12011 | Addition of two ratios. T... |
| divsubdiv 12012 | Subtraction of two ratios.... |
| conjmul 12013 | Two numbers whose reciproc... |
| rereccl 12014 | Closure law for reciprocal... |
| redivcl 12015 | Closure law for division o... |
| eqneg 12016 | A number equal to its nega... |
| eqnegd 12017 | A complex number equals it... |
| eqnegad 12018 | If a complex number equals... |
| div2neg 12019 | Quotient of two negatives.... |
| divneg2 12020 | Move negative sign inside ... |
| recclzi 12021 | Closure law for reciprocal... |
| recne0zi 12022 | The reciprocal of a nonzer... |
| recidzi 12023 | Multiplication of a number... |
| div1i 12024 | A number divided by 1 is i... |
| eqnegi 12025 | A number equal to its nega... |
| reccli 12026 | Closure law for reciprocal... |
| recidi 12027 | Multiplication of a number... |
| recreci 12028 | A number is equal to the r... |
| dividi 12029 | A number divided by itself... |
| div0i 12030 | Division into zero is zero... |
| divclzi 12031 | Closure law for division. ... |
| divcan1zi 12032 | A cancellation law for div... |
| divcan2zi 12033 | A cancellation law for div... |
| divreczi 12034 | Relationship between divis... |
| divcan3zi 12035 | A cancellation law for div... |
| divcan4zi 12036 | A cancellation law for div... |
| rec11i 12037 | Reciprocal is one-to-one. ... |
| divcli 12038 | Closure law for division. ... |
| divcan2i 12039 | A cancellation law for div... |
| divcan1i 12040 | A cancellation law for div... |
| divreci 12041 | Relationship between divis... |
| divcan3i 12042 | A cancellation law for div... |
| divcan4i 12043 | A cancellation law for div... |
| divne0i 12044 | The ratio of nonzero numbe... |
| rec11ii 12045 | Reciprocal is one-to-one. ... |
| divasszi 12046 | An associative law for div... |
| divmulzi 12047 | Relationship between divis... |
| divdirzi 12048 | Distribution of division o... |
| divdiv23zi 12049 | Swap denominators in a div... |
| divmuli 12050 | Relationship between divis... |
| divdiv32i 12051 | Swap denominators in a div... |
| divassi 12052 | An associative law for div... |
| divdiri 12053 | Distribution of division o... |
| div23i 12054 | A commutative/associative ... |
| div11i 12055 | One-to-one relationship fo... |
| divmuldivi 12056 | Multiplication of two rati... |
| divmul13i 12057 | Swap denominators of two r... |
| divadddivi 12058 | Addition of two ratios. T... |
| divdivdivi 12059 | Division of two ratios. T... |
| rerecclzi 12060 | Closure law for reciprocal... |
| rereccli 12061 | Closure law for reciprocal... |
| redivclzi 12062 | Closure law for division o... |
| redivcli 12063 | Closure law for division o... |
| div1d 12064 | A number divided by 1 is i... |
| reccld 12065 | Closure law for reciprocal... |
| recne0d 12066 | The reciprocal of a nonzer... |
| recidd 12067 | Multiplication of a number... |
| recid2d 12068 | Multiplication of a number... |
| recrecd 12069 | A number is equal to the r... |
| dividd 12070 | A number divided by itself... |
| div0d 12071 | Division into zero is zero... |
| divcld 12072 | Closure law for division. ... |
| divcan1d 12073 | A cancellation law for div... |
| divcan2d 12074 | A cancellation law for div... |
| divrecd 12075 | Relationship between divis... |
| divrec2d 12076 | Relationship between divis... |
| divcan3d 12077 | A cancellation law for div... |
| divcan4d 12078 | A cancellation law for div... |
| diveq0d 12079 | A ratio is zero iff the nu... |
| diveq1d 12080 | Equality in terms of unit ... |
| diveq1ad 12081 | The quotient of two comple... |
| diveq0ad 12082 | A fraction of complex numb... |
| divne1d 12083 | If two complex numbers are... |
| divne0bd 12084 | A ratio is zero iff the nu... |
| divnegd 12085 | Move negative sign inside ... |
| divneg2d 12086 | Move negative sign inside ... |
| div2negd 12087 | Quotient of two negatives.... |
| divne0d 12088 | The ratio of nonzero numbe... |
| recdivd 12089 | The reciprocal of a ratio.... |
| recdiv2d 12090 | Division into a reciprocal... |
| divcan6d 12091 | Cancellation of inverted f... |
| ddcand 12092 | Cancellation in a double d... |
| rec11d 12093 | Reciprocal is one-to-one. ... |
| divmuld 12094 | Relationship between divis... |
| div32d 12095 | A commutative/associative ... |
| div13d 12096 | A commutative/associative ... |
| divdiv32d 12097 | Swap denominators in a div... |
| divcan5d 12098 | Cancellation of common fac... |
| divcan5rd 12099 | Cancellation of common fac... |
| divcan7d 12100 | Cancel equal divisors in a... |
| dmdcand 12101 | Cancellation law for divis... |
| dmdcan2d 12102 | Cancellation law for divis... |
| divdiv1d 12103 | Division into a fraction. ... |
| divdiv2d 12104 | Division by a fraction. (... |
| divmul2d 12105 | Relationship between divis... |
| divmul3d 12106 | Relationship between divis... |
| divassd 12107 | An associative law for div... |
| div12d 12108 | A commutative/associative ... |
| div23d 12109 | A commutative/associative ... |
| divdird 12110 | Distribution of division o... |
| divsubdird 12111 | Distribution of division o... |
| div11d 12112 | One-to-one relationship fo... |
| divmuldivd 12113 | Multiplication of two rati... |
| divmul13d 12114 | Swap denominators of two r... |
| divmul24d 12115 | Swap the numerators in the... |
| divadddivd 12116 | Addition of two ratios. T... |
| divsubdivd 12117 | Subtraction of two ratios.... |
| divmuleqd 12118 | Cross-multiply in an equal... |
| divdivdivd 12119 | Division of two ratios. T... |
| diveq1bd 12120 | If two complex numbers are... |
| div2sub 12121 | Swap the order of subtract... |
| div2subd 12122 | Swap subtrahend and minuen... |
| rereccld 12123 | Closure law for reciprocal... |
| redivcld 12124 | Closure law for division o... |
| subrec 12125 | Subtraction of reciprocals... |
| subreci 12126 | Subtraction of reciprocals... |
| subrecd 12127 | Subtraction of reciprocals... |
| mvllmuld 12128 | Move the left term in a pr... |
| mvllmuli 12129 | Move the left term in a pr... |
| ldiv 12130 | Left-division. (Contribut... |
| rdiv 12131 | Right-division. (Contribu... |
| mdiv 12132 | A division law. (Contribu... |
| lineq 12133 | Solution of a (scalar) lin... |
| elimgt0 12134 | Hypothesis for weak deduct... |
| elimge0 12135 | Hypothesis for weak deduct... |
| ltp1 12136 | A number is less than itse... |
| lep1 12137 | A number is less than or e... |
| ltm1 12138 | A number minus 1 is less t... |
| lem1 12139 | A number minus 1 is less t... |
| letrp1 12140 | A transitive property of '... |
| p1le 12141 | A transitive property of p... |
| recgt0 12142 | The reciprocal of a positi... |
| prodgt0 12143 | Infer that a multiplicand ... |
| prodgt02 12144 | Infer that a multiplier is... |
| ltmul1a 12145 | Lemma for ~ ltmul1 . Mult... |
| ltmul1 12146 | Multiplication of both sid... |
| ltmul2 12147 | Multiplication of both sid... |
| lemul1 12148 | Multiplication of both sid... |
| lemul2 12149 | Multiplication of both sid... |
| lemul1a 12150 | Multiplication of both sid... |
| lemul2a 12151 | Multiplication of both sid... |
| ltmul12a 12152 | Comparison of product of t... |
| lemul12b 12153 | Comparison of product of t... |
| lemul12a 12154 | Comparison of product of t... |
| mulgt1OLD 12155 | Obsolete version of ~ mulg... |
| ltmulgt11 12156 | Multiplication by a number... |
| ltmulgt12 12157 | Multiplication by a number... |
| mulgt1 12158 | The product of two numbers... |
| lemulge11 12159 | Multiplication by a number... |
| lemulge12 12160 | Multiplication by a number... |
| ltdiv1 12161 | Division of both sides of ... |
| lediv1 12162 | Division of both sides of ... |
| gt0div 12163 | Division of a positive num... |
| ge0div 12164 | Division of a nonnegative ... |
| divgt0 12165 | The ratio of two positive ... |
| divge0 12166 | The ratio of nonnegative a... |
| mulge0b 12167 | A condition for multiplica... |
| mulle0b 12168 | A condition for multiplica... |
| mulsuble0b 12169 | A condition for multiplica... |
| ltmuldiv 12170 | 'Less than' relationship b... |
| ltmuldiv2 12171 | 'Less than' relationship b... |
| ltdivmul 12172 | 'Less than' relationship b... |
| ledivmul 12173 | 'Less than or equal to' re... |
| ltdivmul2 12174 | 'Less than' relationship b... |
| lt2mul2div 12175 | 'Less than' relationship b... |
| ledivmul2 12176 | 'Less than or equal to' re... |
| lemuldiv 12177 | 'Less than or equal' relat... |
| lemuldiv2 12178 | 'Less than or equal' relat... |
| ltrec 12179 | The reciprocal of both sid... |
| lerec 12180 | The reciprocal of both sid... |
| lt2msq1 12181 | Lemma for ~ lt2msq . (Con... |
| lt2msq 12182 | Two nonnegative numbers co... |
| ltdiv2 12183 | Division of a positive num... |
| ltrec1 12184 | Reciprocal swap in a 'less... |
| lerec2 12185 | Reciprocal swap in a 'less... |
| ledivdiv 12186 | Invert ratios of positive ... |
| lediv2 12187 | Division of a positive num... |
| ltdiv23 12188 | Swap denominator with othe... |
| lediv23 12189 | Swap denominator with othe... |
| lediv12a 12190 | Comparison of ratio of two... |
| lediv2a 12191 | Division of both sides of ... |
| reclt1 12192 | The reciprocal of a positi... |
| recgt1 12193 | The reciprocal of a positi... |
| recgt1i 12194 | The reciprocal of a number... |
| recp1lt1 12195 | Construct a number less th... |
| recreclt 12196 | Given a positive number ` ... |
| le2msq 12197 | The square function on non... |
| msq11 12198 | The square of a nonnegativ... |
| ledivp1 12199 | "Less than or equal to" an... |
| squeeze0 12200 | If a nonnegative number is... |
| ltp1i 12201 | A number is less than itse... |
| recgt0i 12202 | The reciprocal of a positi... |
| recgt0ii 12203 | The reciprocal of a positi... |
| prodgt0i 12204 | Infer that a multiplicand ... |
| divgt0i 12205 | The ratio of two positive ... |
| divge0i 12206 | The ratio of nonnegative a... |
| ltreci 12207 | The reciprocal of both sid... |
| lereci 12208 | The reciprocal of both sid... |
| lt2msqi 12209 | The square function on non... |
| le2msqi 12210 | The square function on non... |
| msq11i 12211 | The square of a nonnegativ... |
| divgt0i2i 12212 | The ratio of two positive ... |
| ltrecii 12213 | The reciprocal of both sid... |
| divgt0ii 12214 | The ratio of two positive ... |
| ltmul1i 12215 | Multiplication of both sid... |
| ltdiv1i 12216 | Division of both sides of ... |
| ltmuldivi 12217 | 'Less than' relationship b... |
| ltmul2i 12218 | Multiplication of both sid... |
| lemul1i 12219 | Multiplication of both sid... |
| lemul2i 12220 | Multiplication of both sid... |
| ltdiv23i 12221 | Swap denominator with othe... |
| ledivp1i 12222 | "Less than or equal to" an... |
| ltdivp1i 12223 | Less-than and division rel... |
| ltdiv23ii 12224 | Swap denominator with othe... |
| ltmul1ii 12225 | Multiplication of both sid... |
| ltdiv1ii 12226 | Division of both sides of ... |
| ltp1d 12227 | A number is less than itse... |
| lep1d 12228 | A number is less than or e... |
| ltm1d 12229 | A number minus 1 is less t... |
| lem1d 12230 | A number minus 1 is less t... |
| recgt0d 12231 | The reciprocal of a positi... |
| divgt0d 12232 | The ratio of two positive ... |
| mulgt1d 12233 | The product of two numbers... |
| lemulge11d 12234 | Multiplication by a number... |
| lemulge12d 12235 | Multiplication by a number... |
| lemul1ad 12236 | Multiplication of both sid... |
| lemul2ad 12237 | Multiplication of both sid... |
| ltmul12ad 12238 | Comparison of product of t... |
| lemul12ad 12239 | Comparison of product of t... |
| lemul12bd 12240 | Comparison of product of t... |
| fimaxre 12241 | A finite set of real numbe... |
| fimaxre2 12242 | A nonempty finite set of r... |
| fimaxre3 12243 | A nonempty finite set of r... |
| fiminre 12244 | A nonempty finite set of r... |
| fiminre2 12245 | A nonempty finite set of r... |
| negfi 12246 | The negation of a finite s... |
| lbreu 12247 | If a set of reals contains... |
| lbcl 12248 | If a set of reals contains... |
| lble 12249 | If a set of reals contains... |
| lbinf 12250 | If a set of reals contains... |
| lbinfcl 12251 | If a set of reals contains... |
| lbinfle 12252 | If a set of reals contains... |
| sup2 12253 | A nonempty, bounded-above ... |
| sup3 12254 | A version of the completen... |
| infm3lem 12255 | Lemma for ~ infm3 . (Cont... |
| infm3 12256 | The completeness axiom for... |
| suprcl 12257 | Closure of supremum of a n... |
| suprub 12258 | A member of a nonempty bou... |
| suprubd 12259 | Natural deduction form of ... |
| suprcld 12260 | Natural deduction form of ... |
| suprlub 12261 | The supremum of a nonempty... |
| suprnub 12262 | An upper bound is not less... |
| suprleub 12263 | The supremum of a nonempty... |
| supaddc 12264 | The supremum function dist... |
| supadd 12265 | The supremum function dist... |
| supmul1 12266 | The supremum function dist... |
| supmullem1 12267 | Lemma for ~ supmul . (Con... |
| supmullem2 12268 | Lemma for ~ supmul . (Con... |
| supmul 12269 | The supremum function dist... |
| sup3ii 12270 | A version of the completen... |
| suprclii 12271 | Closure of supremum of a n... |
| suprubii 12272 | A member of a nonempty bou... |
| suprlubii 12273 | The supremum of a nonempty... |
| suprnubii 12274 | An upper bound is not less... |
| suprleubii 12275 | The supremum of a nonempty... |
| riotaneg 12276 | The negative of the unique... |
| negiso 12277 | Negation is an order anti-... |
| dfinfre 12278 | The infimum of a set of re... |
| infrecl 12279 | Closure of infimum of a no... |
| infrenegsup 12280 | The infimum of a set of re... |
| infregelb 12281 | Any lower bound of a nonem... |
| infrelb 12282 | If a nonempty set of real ... |
| infrefilb 12283 | The infimum of a finite se... |
| supfirege 12284 | The supremum of a finite s... |
| inelr 12285 | The imaginary unit ` _i ` ... |
| rimul 12286 | A real number times the im... |
| cru 12287 | The representation of comp... |
| crne0 12288 | The real representation of... |
| creur 12289 | The real part of a complex... |
| creui 12290 | The imaginary part of a co... |
| cju 12291 | The complex conjugate of a... |
| ofsubeq0 12292 | Function analogue of ~ sub... |
| ofnegsub 12293 | Function analogue of ~ neg... |
| ofsubge0 12294 | Function analogue of ~ sub... |
| nnexALT 12297 | Alternate proof of ~ nnex ... |
| peano5nni 12298 | Peano's inductive postulat... |
| nnssre 12299 | The positive integers are ... |
| nnsscn 12300 | The positive integers are ... |
| nnex 12301 | The set of positive intege... |
| nnre 12302 | A positive integer is a re... |
| nncn 12303 | A positive integer is a co... |
| nnrei 12304 | A positive integer is a re... |
| nncni 12305 | A positive integer is a co... |
| 1nn 12306 | Peano postulate: 1 is a po... |
| peano2nn 12307 | Peano postulate: a success... |
| dfnn2 12308 | Alternate definition of th... |
| dfnn3 12309 | Alternate definition of th... |
| nnred 12310 | A positive integer is a re... |
| nncnd 12311 | A positive integer is a co... |
| peano2nnd 12312 | Peano postulate: a success... |
| nnind 12313 | Principle of Mathematical ... |
| nnindALT 12314 | Principle of Mathematical ... |
| nnindd 12315 | Principle of Mathematical ... |
| nn1m1nn 12316 | Every positive integer is ... |
| nn1suc 12317 | If a statement holds for 1... |
| nnaddcl 12318 | Closure of addition of pos... |
| nnmulcl 12319 | Closure of multiplication ... |
| nnmulcli 12320 | Closure of multiplication ... |
| nnmtmip 12321 | "Minus times minus is plus... |
| nn2ge 12322 | There exists a positive in... |
| nnge1 12323 | A positive integer is one ... |
| nngt1ne1 12324 | A positive integer is grea... |
| nnle1eq1 12325 | A positive integer is less... |
| nngt0 12326 | A positive integer is posi... |
| nnnlt1 12327 | A positive integer is not ... |
| nnnle0 12328 | A positive integer is not ... |
| nnne0 12329 | A positive integer is nonz... |
| nnneneg 12330 | No positive integer is equ... |
| 0nnn 12331 | Zero is not a positive int... |
| 0nnnALT 12332 | Alternate proof of ~ 0nnn ... |
| nnne0ALT 12333 | Alternate version of ~ nnn... |
| nngt0i 12334 | A positive integer is posi... |
| nnne0i 12335 | A positive integer is nonz... |
| nndivre 12336 | The quotient of a real and... |
| nnrecre 12337 | The reciprocal of a positi... |
| nnrecgt0 12338 | The reciprocal of a positi... |
| nnsub 12339 | Subtraction of positive in... |
| nnsubi 12340 | Subtraction of positive in... |
| nndiv 12341 | Two ways to express " ` A ... |
| nndivtr 12342 | Transitive property of div... |
| nnge1d 12343 | A positive integer is one ... |
| nngt0d 12344 | A positive integer is posi... |
| nnne0d 12345 | A positive integer is nonz... |
| nnrecred 12346 | The reciprocal of a positi... |
| nnaddcld 12347 | Closure of addition of pos... |
| nnmulcld 12348 | Closure of multiplication ... |
| nndivred 12349 | A positive integer is one ... |
| 0ne1 12366 | Zero is different from one... |
| 1m1e0 12367 | One minus one equals zero.... |
| 2nn 12368 | 2 is a positive integer. ... |
| 2re 12369 | The number 2 is real. (Co... |
| 2cn 12370 | The number 2 is a complex ... |
| 2cnALT 12371 | Alternate proof of ~ 2cn .... |
| 2ex 12372 | The number 2 is a set. (C... |
| 2cnd 12373 | The number 2 is a complex ... |
| 3nn 12374 | 3 is a positive integer. ... |
| 3re 12375 | The number 3 is real. (Co... |
| 3cn 12376 | The number 3 is a complex ... |
| 3ex 12377 | The number 3 is a set. (C... |
| 4nn 12378 | 4 is a positive integer. ... |
| 4re 12379 | The number 4 is real. (Co... |
| 4cn 12380 | The number 4 is a complex ... |
| 5nn 12381 | 5 is a positive integer. ... |
| 5re 12382 | The number 5 is real. (Co... |
| 5cn 12383 | The number 5 is a complex ... |
| 6nn 12384 | 6 is a positive integer. ... |
| 6re 12385 | The number 6 is real. (Co... |
| 6cn 12386 | The number 6 is a complex ... |
| 7nn 12387 | 7 is a positive integer. ... |
| 7re 12388 | The number 7 is real. (Co... |
| 7cn 12389 | The number 7 is a complex ... |
| 8nn 12390 | 8 is a positive integer. ... |
| 8re 12391 | The number 8 is real. (Co... |
| 8cn 12392 | The number 8 is a complex ... |
| 9nn 12393 | 9 is a positive integer. ... |
| 9re 12394 | The number 9 is real. (Co... |
| 9cn 12395 | The number 9 is a complex ... |
| 0le0 12396 | Zero is nonnegative. (Con... |
| 0le2 12397 | The number 0 is less than ... |
| 2pos 12398 | The number 2 is positive. ... |
| 2ne0 12399 | The number 2 is nonzero. ... |
| 3pos 12400 | The number 3 is positive. ... |
| 3ne0 12401 | The number 3 is nonzero. ... |
| 4pos 12402 | The number 4 is positive. ... |
| 4ne0 12403 | The number 4 is nonzero. ... |
| 5pos 12404 | The number 5 is positive. ... |
| 6pos 12405 | The number 6 is positive. ... |
| 7pos 12406 | The number 7 is positive. ... |
| 8pos 12407 | The number 8 is positive. ... |
| 9pos 12408 | The number 9 is positive. ... |
| neg1cn 12409 | -1 is a complex number. (... |
| neg1rr 12410 | -1 is a real number. (Con... |
| neg1ne0 12411 | -1 is nonzero. (Contribut... |
| neg1lt0 12412 | -1 is less than 0. (Contr... |
| negneg1e1 12413 | ` -u -u 1 ` is 1. (Contri... |
| 1pneg1e0 12414 | ` 1 + -u 1 ` is 0. (Contr... |
| 0m0e0 12415 | 0 minus 0 equals 0. (Cont... |
| 1m0e1 12416 | 1 - 0 = 1. (Contributed b... |
| 0p1e1 12417 | 0 + 1 = 1. (Contributed b... |
| fv0p1e1 12418 | Function value at ` N + 1 ... |
| 1p0e1 12419 | 1 + 0 = 1. (Contributed b... |
| 1p1e2 12420 | 1 + 1 = 2. (Contributed b... |
| 2m1e1 12421 | 2 - 1 = 1. The result is ... |
| 1e2m1 12422 | 1 = 2 - 1. (Contributed b... |
| 3m1e2 12423 | 3 - 1 = 2. (Contributed b... |
| 4m1e3 12424 | 4 - 1 = 3. (Contributed b... |
| 5m1e4 12425 | 5 - 1 = 4. (Contributed b... |
| 6m1e5 12426 | 6 - 1 = 5. (Contributed b... |
| 7m1e6 12427 | 7 - 1 = 6. (Contributed b... |
| 8m1e7 12428 | 8 - 1 = 7. (Contributed b... |
| 9m1e8 12429 | 9 - 1 = 8. (Contributed b... |
| 2p2e4 12430 | Two plus two equals four. ... |
| 2times 12431 | Two times a number. (Cont... |
| times2 12432 | A number times 2. (Contri... |
| 2timesi 12433 | Two times a number. (Cont... |
| times2i 12434 | A number times 2. (Contri... |
| 2txmxeqx 12435 | Two times a complex number... |
| 2div2e1 12436 | 2 divided by 2 is 1. (Con... |
| 2p1e3 12437 | 2 + 1 = 3. (Contributed b... |
| 1p2e3 12438 | 1 + 2 = 3. For a shorter ... |
| 1p2e3ALT 12439 | Alternate proof of ~ 1p2e3... |
| 3p1e4 12440 | 3 + 1 = 4. (Contributed b... |
| 4p1e5 12441 | 4 + 1 = 5. (Contributed b... |
| 5p1e6 12442 | 5 + 1 = 6. (Contributed b... |
| 6p1e7 12443 | 6 + 1 = 7. (Contributed b... |
| 7p1e8 12444 | 7 + 1 = 8. (Contributed b... |
| 8p1e9 12445 | 8 + 1 = 9. (Contributed b... |
| 3p2e5 12446 | 3 + 2 = 5. (Contributed b... |
| 3p3e6 12447 | 3 + 3 = 6. (Contributed b... |
| 4p2e6 12448 | 4 + 2 = 6. (Contributed b... |
| 4p3e7 12449 | 4 + 3 = 7. (Contributed b... |
| 4p4e8 12450 | 4 + 4 = 8. (Contributed b... |
| 5p2e7 12451 | 5 + 2 = 7. (Contributed b... |
| 5p3e8 12452 | 5 + 3 = 8. (Contributed b... |
| 5p4e9 12453 | 5 + 4 = 9. (Contributed b... |
| 6p2e8 12454 | 6 + 2 = 8. (Contributed b... |
| 6p3e9 12455 | 6 + 3 = 9. (Contributed b... |
| 7p2e9 12456 | 7 + 2 = 9. (Contributed b... |
| 1t1e1 12457 | 1 times 1 equals 1. (Cont... |
| 2t1e2 12458 | 2 times 1 equals 2. (Cont... |
| 2t2e4 12459 | 2 times 2 equals 4. (Cont... |
| 3t1e3 12460 | 3 times 1 equals 3. (Cont... |
| 3t2e6 12461 | 3 times 2 equals 6. (Cont... |
| 3t3e9 12462 | 3 times 3 equals 9. (Cont... |
| 4t2e8 12463 | 4 times 2 equals 8. (Cont... |
| 2t0e0 12464 | 2 times 0 equals 0. (Cont... |
| 4d2e2 12465 | One half of four is two. ... |
| 1lt2 12466 | 1 is less than 2. (Contri... |
| 2lt3 12467 | 2 is less than 3. (Contri... |
| 1lt3 12468 | 1 is less than 3. (Contri... |
| 3lt4 12469 | 3 is less than 4. (Contri... |
| 2lt4 12470 | 2 is less than 4. (Contri... |
| 1lt4 12471 | 1 is less than 4. (Contri... |
| 4lt5 12472 | 4 is less than 5. (Contri... |
| 3lt5 12473 | 3 is less than 5. (Contri... |
| 2lt5 12474 | 2 is less than 5. (Contri... |
| 1lt5 12475 | 1 is less than 5. (Contri... |
| 5lt6 12476 | 5 is less than 6. (Contri... |
| 4lt6 12477 | 4 is less than 6. (Contri... |
| 3lt6 12478 | 3 is less than 6. (Contri... |
| 2lt6 12479 | 2 is less than 6. (Contri... |
| 1lt6 12480 | 1 is less than 6. (Contri... |
| 6lt7 12481 | 6 is less than 7. (Contri... |
| 5lt7 12482 | 5 is less than 7. (Contri... |
| 4lt7 12483 | 4 is less than 7. (Contri... |
| 3lt7 12484 | 3 is less than 7. (Contri... |
| 2lt7 12485 | 2 is less than 7. (Contri... |
| 1lt7 12486 | 1 is less than 7. (Contri... |
| 7lt8 12487 | 7 is less than 8. (Contri... |
| 6lt8 12488 | 6 is less than 8. (Contri... |
| 5lt8 12489 | 5 is less than 8. (Contri... |
| 4lt8 12490 | 4 is less than 8. (Contri... |
| 3lt8 12491 | 3 is less than 8. (Contri... |
| 2lt8 12492 | 2 is less than 8. (Contri... |
| 1lt8 12493 | 1 is less than 8. (Contri... |
| 8lt9 12494 | 8 is less than 9. (Contri... |
| 7lt9 12495 | 7 is less than 9. (Contri... |
| 6lt9 12496 | 6 is less than 9. (Contri... |
| 5lt9 12497 | 5 is less than 9. (Contri... |
| 4lt9 12498 | 4 is less than 9. (Contri... |
| 3lt9 12499 | 3 is less than 9. (Contri... |
| 2lt9 12500 | 2 is less than 9. (Contri... |
| 1lt9 12501 | 1 is less than 9. (Contri... |
| 0ne2 12502 | 0 is not equal to 2. (Con... |
| 1ne2 12503 | 1 is not equal to 2. (Con... |
| 1le2 12504 | 1 is less than or equal to... |
| 2cnne0 12505 | 2 is a nonzero complex num... |
| 2rene0 12506 | 2 is a nonzero real number... |
| 1le3 12507 | 1 is less than or equal to... |
| neg1mulneg1e1 12508 | ` -u 1 x. -u 1 ` is 1. (C... |
| halfre 12509 | One-half is real. (Contri... |
| halfcn 12510 | One-half is a complex numb... |
| halfgt0 12511 | One-half is greater than z... |
| halfge0 12512 | One-half is not negative. ... |
| halflt1 12513 | One-half is less than one.... |
| 1mhlfehlf 12514 | Prove that 1 - 1/2 = 1/2. ... |
| 8th4div3 12515 | An eighth of four thirds i... |
| halfpm6th 12516 | One half plus or minus one... |
| it0e0 12517 | i times 0 equals 0. (Cont... |
| 2mulicn 12518 | ` ( 2 x. _i ) e. CC ` . (... |
| 2muline0 12519 | ` ( 2 x. _i ) =/= 0 ` . (... |
| halfcl 12520 | Closure of half of a numbe... |
| rehalfcl 12521 | Real closure of half. (Co... |
| half0 12522 | Half of a number is zero i... |
| 2halves 12523 | Two halves make a whole. ... |
| halfpos2 12524 | A number is positive iff i... |
| halfpos 12525 | A positive number is great... |
| halfnneg2 12526 | A number is nonnegative if... |
| halfaddsubcl 12527 | Closure of half-sum and ha... |
| halfaddsub 12528 | Sum and difference of half... |
| subhalfhalf 12529 | Subtracting the half of a ... |
| lt2halves 12530 | A sum is less than the who... |
| addltmul 12531 | Sum is less than product f... |
| nominpos 12532 | There is no smallest posit... |
| avglt1 12533 | Ordering property for aver... |
| avglt2 12534 | Ordering property for aver... |
| avgle1 12535 | Ordering property for aver... |
| avgle2 12536 | Ordering property for aver... |
| avgle 12537 | The average of two numbers... |
| 2timesd 12538 | Two times a number. (Cont... |
| times2d 12539 | A number times 2. (Contri... |
| halfcld 12540 | Closure of half of a numbe... |
| 2halvesd 12541 | Two halves make a whole. ... |
| rehalfcld 12542 | Real closure of half. (Co... |
| lt2halvesd 12543 | A sum is less than the who... |
| rehalfcli 12544 | Half a real number is real... |
| lt2addmuld 12545 | If two real numbers are le... |
| add1p1 12546 | Adding two times 1 to a nu... |
| sub1m1 12547 | Subtracting two times 1 fr... |
| cnm2m1cnm3 12548 | Subtracting 2 and afterwar... |
| xp1d2m1eqxm1d2 12549 | A complex number increased... |
| div4p1lem1div2 12550 | An integer greater than 5,... |
| nnunb 12551 | The set of positive intege... |
| arch 12552 | Archimedean property of re... |
| nnrecl 12553 | There exists a positive in... |
| bndndx 12554 | A bounded real sequence ` ... |
| elnn0 12557 | Nonnegative integers expre... |
| nnssnn0 12558 | Positive naturals are a su... |
| nn0ssre 12559 | Nonnegative integers are a... |
| nn0sscn 12560 | Nonnegative integers are a... |
| nn0ex 12561 | The set of nonnegative int... |
| nnnn0 12562 | A positive integer is a no... |
| nnnn0i 12563 | A positive integer is a no... |
| nn0re 12564 | A nonnegative integer is a... |
| nn0cn 12565 | A nonnegative integer is a... |
| nn0rei 12566 | A nonnegative integer is a... |
| nn0cni 12567 | A nonnegative integer is a... |
| dfn2 12568 | The set of positive intege... |
| elnnne0 12569 | The positive integer prope... |
| 0nn0 12570 | 0 is a nonnegative integer... |
| 1nn0 12571 | 1 is a nonnegative integer... |
| 2nn0 12572 | 2 is a nonnegative integer... |
| 3nn0 12573 | 3 is a nonnegative integer... |
| 4nn0 12574 | 4 is a nonnegative integer... |
| 5nn0 12575 | 5 is a nonnegative integer... |
| 6nn0 12576 | 6 is a nonnegative integer... |
| 7nn0 12577 | 7 is a nonnegative integer... |
| 8nn0 12578 | 8 is a nonnegative integer... |
| 9nn0 12579 | 9 is a nonnegative integer... |
| nn0ge0 12580 | A nonnegative integer is g... |
| nn0nlt0 12581 | A nonnegative integer is n... |
| nn0ge0i 12582 | Nonnegative integers are n... |
| nn0le0eq0 12583 | A nonnegative integer is l... |
| nn0p1gt0 12584 | A nonnegative integer incr... |
| nnnn0addcl 12585 | A positive integer plus a ... |
| nn0nnaddcl 12586 | A nonnegative integer plus... |
| 0mnnnnn0 12587 | The result of subtracting ... |
| un0addcl 12588 | If ` S ` is closed under a... |
| un0mulcl 12589 | If ` S ` is closed under m... |
| nn0addcl 12590 | Closure of addition of non... |
| nn0mulcl 12591 | Closure of multiplication ... |
| nn0addcli 12592 | Closure of addition of non... |
| nn0mulcli 12593 | Closure of multiplication ... |
| nn0p1nn 12594 | A nonnegative integer plus... |
| peano2nn0 12595 | Second Peano postulate for... |
| nnm1nn0 12596 | A positive integer minus 1... |
| elnn0nn 12597 | The nonnegative integer pr... |
| elnnnn0 12598 | The positive integer prope... |
| elnnnn0b 12599 | The positive integer prope... |
| elnnnn0c 12600 | The positive integer prope... |
| nn0addge1 12601 | A number is less than or e... |
| nn0addge2 12602 | A number is less than or e... |
| nn0addge1i 12603 | A number is less than or e... |
| nn0addge2i 12604 | A number is less than or e... |
| nn0sub 12605 | Subtraction of nonnegative... |
| ltsubnn0 12606 | Subtracting a nonnegative ... |
| nn0negleid 12607 | A nonnegative integer is g... |
| difgtsumgt 12608 | If the difference of a rea... |
| nn0le2xi 12609 | A nonnegative integer is l... |
| nn0lele2xi 12610 | 'Less than or equal to' im... |
| fcdmnn0supp 12611 | Two ways to write the supp... |
| fcdmnn0fsupp 12612 | A function into ` NN0 ` is... |
| fcdmnn0suppg 12613 | Version of ~ fcdmnn0supp a... |
| fcdmnn0fsuppg 12614 | Version of ~ fcdmnn0fsupp ... |
| nnnn0d 12615 | A positive integer is a no... |
| nn0red 12616 | A nonnegative integer is a... |
| nn0cnd 12617 | A nonnegative integer is a... |
| nn0ge0d 12618 | A nonnegative integer is g... |
| nn0addcld 12619 | Closure of addition of non... |
| nn0mulcld 12620 | Closure of multiplication ... |
| nn0readdcl 12621 | Closure law for addition o... |
| nn0n0n1ge2 12622 | A nonnegative integer whic... |
| nn0n0n1ge2b 12623 | A nonnegative integer is n... |
| nn0ge2m1nn 12624 | If a nonnegative integer i... |
| nn0ge2m1nn0 12625 | If a nonnegative integer i... |
| nn0nndivcl 12626 | Closure law for dividing o... |
| elxnn0 12629 | An extended nonnegative in... |
| nn0ssxnn0 12630 | The standard nonnegative i... |
| nn0xnn0 12631 | A standard nonnegative int... |
| xnn0xr 12632 | An extended nonnegative in... |
| 0xnn0 12633 | Zero is an extended nonneg... |
| pnf0xnn0 12634 | Positive infinity is an ex... |
| nn0nepnf 12635 | No standard nonnegative in... |
| nn0xnn0d 12636 | A standard nonnegative int... |
| nn0nepnfd 12637 | No standard nonnegative in... |
| xnn0nemnf 12638 | No extended nonnegative in... |
| xnn0xrnemnf 12639 | The extended nonnegative i... |
| xnn0nnn0pnf 12640 | An extended nonnegative in... |
| elz 12643 | Membership in the set of i... |
| nnnegz 12644 | The negative of a positive... |
| zre 12645 | An integer is a real. (Co... |
| zcn 12646 | An integer is a complex nu... |
| zrei 12647 | An integer is a real numbe... |
| zssre 12648 | The integers are a subset ... |
| zsscn 12649 | The integers are a subset ... |
| zex 12650 | The set of integers exists... |
| elnnz 12651 | Positive integer property ... |
| 0z 12652 | Zero is an integer. (Cont... |
| 0zd 12653 | Zero is an integer, deduct... |
| elnn0z 12654 | Nonnegative integer proper... |
| elznn0nn 12655 | Integer property expressed... |
| elznn0 12656 | Integer property expressed... |
| elznn 12657 | Integer property expressed... |
| zle0orge1 12658 | There is no integer in the... |
| elz2 12659 | Membership in the set of i... |
| dfz2 12660 | Alternative definition of ... |
| zexALT 12661 | Alternate proof of ~ zex .... |
| nnz 12662 | A positive integer is an i... |
| nnssz 12663 | Positive integers are a su... |
| nn0ssz 12664 | Nonnegative integers are a... |
| nnzOLD 12665 | Obsolete version of ~ nnz ... |
| nn0z 12666 | A nonnegative integer is a... |
| nn0zd 12667 | A nonnegative integer is a... |
| nnzd 12668 | A positive integer is an i... |
| nnzi 12669 | A positive integer is an i... |
| nn0zi 12670 | A nonnegative integer is a... |
| elnnz1 12671 | Positive integer property ... |
| znnnlt1 12672 | An integer is not a positi... |
| nnzrab 12673 | Positive integers expresse... |
| nn0zrab 12674 | Nonnegative integers expre... |
| 1z 12675 | One is an integer. (Contr... |
| 1zzd 12676 | One is an integer, deducti... |
| 2z 12677 | 2 is an integer. (Contrib... |
| 3z 12678 | 3 is an integer. (Contrib... |
| 4z 12679 | 4 is an integer. (Contrib... |
| znegcl 12680 | Closure law for negative i... |
| neg1z 12681 | -1 is an integer. (Contri... |
| znegclb 12682 | A complex number is an int... |
| nn0negz 12683 | The negative of a nonnegat... |
| nn0negzi 12684 | The negative of a nonnegat... |
| zaddcl 12685 | Closure of addition of int... |
| peano2z 12686 | Second Peano postulate gen... |
| zsubcl 12687 | Closure of subtraction of ... |
| peano2zm 12688 | "Reverse" second Peano pos... |
| zletr 12689 | Transitive law of ordering... |
| zrevaddcl 12690 | Reverse closure law for ad... |
| znnsub 12691 | The positive difference of... |
| znn0sub 12692 | The nonnegative difference... |
| nzadd 12693 | The sum of a real number n... |
| zmulcl 12694 | Closure of multiplication ... |
| zltp1le 12695 | Integer ordering relation.... |
| zleltp1 12696 | Integer ordering relation.... |
| zlem1lt 12697 | Integer ordering relation.... |
| zltlem1 12698 | Integer ordering relation.... |
| zgt0ge1 12699 | An integer greater than ` ... |
| nnleltp1 12700 | Positive integer ordering ... |
| nnltp1le 12701 | Positive integer ordering ... |
| nnaddm1cl 12702 | Closure of addition of pos... |
| nn0ltp1le 12703 | Nonnegative integer orderi... |
| nn0leltp1 12704 | Nonnegative integer orderi... |
| nn0ltlem1 12705 | Nonnegative integer orderi... |
| nn0sub2 12706 | Subtraction of nonnegative... |
| nn0lt10b 12707 | A nonnegative integer less... |
| nn0lt2 12708 | A nonnegative integer less... |
| nn0le2is012 12709 | A nonnegative integer whic... |
| nn0lem1lt 12710 | Nonnegative integer orderi... |
| nnlem1lt 12711 | Positive integer ordering ... |
| nnltlem1 12712 | Positive integer ordering ... |
| nnm1ge0 12713 | A positive integer decreas... |
| nn0ge0div 12714 | Division of a nonnegative ... |
| zdiv 12715 | Two ways to express " ` M ... |
| zdivadd 12716 | Property of divisibility: ... |
| zdivmul 12717 | Property of divisibility: ... |
| zextle 12718 | An extensionality-like pro... |
| zextlt 12719 | An extensionality-like pro... |
| recnz 12720 | The reciprocal of a number... |
| btwnnz 12721 | A number between an intege... |
| gtndiv 12722 | A larger number does not d... |
| halfnz 12723 | One-half is not an integer... |
| 3halfnz 12724 | Three halves is not an int... |
| suprzcl 12725 | The supremum of a bounded-... |
| prime 12726 | Two ways to express " ` A ... |
| msqznn 12727 | The square of a nonzero in... |
| zneo 12728 | No even integer equals an ... |
| nneo 12729 | A positive integer is even... |
| nneoi 12730 | A positive integer is even... |
| zeo 12731 | An integer is even or odd.... |
| zeo2 12732 | An integer is even or odd ... |
| peano2uz2 12733 | Second Peano postulate for... |
| peano5uzi 12734 | Peano's inductive postulat... |
| peano5uzti 12735 | Peano's inductive postulat... |
| dfuzi 12736 | An expression for the uppe... |
| uzind 12737 | Induction on the upper int... |
| uzind2 12738 | Induction on the upper int... |
| uzind3 12739 | Induction on the upper int... |
| nn0ind 12740 | Principle of Mathematical ... |
| nn0indALT 12741 | Principle of Mathematical ... |
| nn0indd 12742 | Principle of Mathematical ... |
| fzind 12743 | Induction on the integers ... |
| fnn0ind 12744 | Induction on the integers ... |
| nn0ind-raph 12745 | Principle of Mathematical ... |
| zindd 12746 | Principle of Mathematical ... |
| fzindd 12747 | Induction on the integers ... |
| btwnz 12748 | Any real number can be san... |
| zred 12749 | An integer is a real numbe... |
| zcnd 12750 | An integer is a complex nu... |
| znegcld 12751 | Closure law for negative i... |
| peano2zd 12752 | Deduction from second Pean... |
| zaddcld 12753 | Closure of addition of int... |
| zsubcld 12754 | Closure of subtraction of ... |
| zmulcld 12755 | Closure of multiplication ... |
| znnn0nn 12756 | The negative of a negative... |
| zadd2cl 12757 | Increasing an integer by 2... |
| zriotaneg 12758 | The negative of the unique... |
| suprfinzcl 12759 | The supremum of a nonempty... |
| 9p1e10 12762 | 9 + 1 = 10. (Contributed ... |
| dfdec10 12763 | Version of the definition ... |
| decex 12764 | A decimal number is a set.... |
| deceq1 12765 | Equality theorem for the d... |
| deceq2 12766 | Equality theorem for the d... |
| deceq1i 12767 | Equality theorem for the d... |
| deceq2i 12768 | Equality theorem for the d... |
| deceq12i 12769 | Equality theorem for the d... |
| numnncl 12770 | Closure for a numeral (wit... |
| num0u 12771 | Add a zero in the units pl... |
| num0h 12772 | Add a zero in the higher p... |
| numcl 12773 | Closure for a decimal inte... |
| numsuc 12774 | The successor of a decimal... |
| deccl 12775 | Closure for a numeral. (C... |
| 10nn 12776 | 10 is a positive integer. ... |
| 10pos 12777 | The number 10 is positive.... |
| 10nn0 12778 | 10 is a nonnegative intege... |
| 10re 12779 | The number 10 is real. (C... |
| decnncl 12780 | Closure for a numeral. (C... |
| dec0u 12781 | Add a zero in the units pl... |
| dec0h 12782 | Add a zero in the higher p... |
| numnncl2 12783 | Closure for a decimal inte... |
| decnncl2 12784 | Closure for a decimal inte... |
| numlt 12785 | Comparing two decimal inte... |
| numltc 12786 | Comparing two decimal inte... |
| le9lt10 12787 | A "decimal digit" (i.e. a ... |
| declt 12788 | Comparing two decimal inte... |
| decltc 12789 | Comparing two decimal inte... |
| declth 12790 | Comparing two decimal inte... |
| decsuc 12791 | The successor of a decimal... |
| 3declth 12792 | Comparing two decimal inte... |
| 3decltc 12793 | Comparing two decimal inte... |
| decle 12794 | Comparing two decimal inte... |
| decleh 12795 | Comparing two decimal inte... |
| declei 12796 | Comparing a digit to a dec... |
| numlti 12797 | Comparing a digit to a dec... |
| declti 12798 | Comparing a digit to a dec... |
| decltdi 12799 | Comparing a digit to a dec... |
| numsucc 12800 | The successor of a decimal... |
| decsucc 12801 | The successor of a decimal... |
| 1e0p1 12802 | The successor of zero. (C... |
| dec10p 12803 | Ten plus an integer. (Con... |
| numma 12804 | Perform a multiply-add of ... |
| nummac 12805 | Perform a multiply-add of ... |
| numma2c 12806 | Perform a multiply-add of ... |
| numadd 12807 | Add two decimal integers `... |
| numaddc 12808 | Add two decimal integers `... |
| nummul1c 12809 | The product of a decimal i... |
| nummul2c 12810 | The product of a decimal i... |
| decma 12811 | Perform a multiply-add of ... |
| decmac 12812 | Perform a multiply-add of ... |
| decma2c 12813 | Perform a multiply-add of ... |
| decadd 12814 | Add two numerals ` M ` and... |
| decaddc 12815 | Add two numerals ` M ` and... |
| decaddc2 12816 | Add two numerals ` M ` and... |
| decrmanc 12817 | Perform a multiply-add of ... |
| decrmac 12818 | Perform a multiply-add of ... |
| decaddm10 12819 | The sum of two multiples o... |
| decaddi 12820 | Add two numerals ` M ` and... |
| decaddci 12821 | Add two numerals ` M ` and... |
| decaddci2 12822 | Add two numerals ` M ` and... |
| decsubi 12823 | Difference between a numer... |
| decmul1 12824 | The product of a numeral w... |
| decmul1c 12825 | The product of a numeral w... |
| decmul2c 12826 | The product of a numeral w... |
| decmulnc 12827 | The product of a numeral w... |
| 11multnc 12828 | The product of 11 (as nume... |
| decmul10add 12829 | A multiplication of a numb... |
| 6p5lem 12830 | Lemma for ~ 6p5e11 and rel... |
| 5p5e10 12831 | 5 + 5 = 10. (Contributed ... |
| 6p4e10 12832 | 6 + 4 = 10. (Contributed ... |
| 6p5e11 12833 | 6 + 5 = 11. (Contributed ... |
| 6p6e12 12834 | 6 + 6 = 12. (Contributed ... |
| 7p3e10 12835 | 7 + 3 = 10. (Contributed ... |
| 7p4e11 12836 | 7 + 4 = 11. (Contributed ... |
| 7p5e12 12837 | 7 + 5 = 12. (Contributed ... |
| 7p6e13 12838 | 7 + 6 = 13. (Contributed ... |
| 7p7e14 12839 | 7 + 7 = 14. (Contributed ... |
| 8p2e10 12840 | 8 + 2 = 10. (Contributed ... |
| 8p3e11 12841 | 8 + 3 = 11. (Contributed ... |
| 8p4e12 12842 | 8 + 4 = 12. (Contributed ... |
| 8p5e13 12843 | 8 + 5 = 13. (Contributed ... |
| 8p6e14 12844 | 8 + 6 = 14. (Contributed ... |
| 8p7e15 12845 | 8 + 7 = 15. (Contributed ... |
| 8p8e16 12846 | 8 + 8 = 16. (Contributed ... |
| 9p2e11 12847 | 9 + 2 = 11. (Contributed ... |
| 9p3e12 12848 | 9 + 3 = 12. (Contributed ... |
| 9p4e13 12849 | 9 + 4 = 13. (Contributed ... |
| 9p5e14 12850 | 9 + 5 = 14. (Contributed ... |
| 9p6e15 12851 | 9 + 6 = 15. (Contributed ... |
| 9p7e16 12852 | 9 + 7 = 16. (Contributed ... |
| 9p8e17 12853 | 9 + 8 = 17. (Contributed ... |
| 9p9e18 12854 | 9 + 9 = 18. (Contributed ... |
| 10p10e20 12855 | 10 + 10 = 20. (Contribute... |
| 10m1e9 12856 | 10 - 1 = 9. (Contributed ... |
| 4t3lem 12857 | Lemma for ~ 4t3e12 and rel... |
| 4t3e12 12858 | 4 times 3 equals 12. (Con... |
| 4t4e16 12859 | 4 times 4 equals 16. (Con... |
| 5t2e10 12860 | 5 times 2 equals 10. (Con... |
| 5t3e15 12861 | 5 times 3 equals 15. (Con... |
| 5t4e20 12862 | 5 times 4 equals 20. (Con... |
| 5t5e25 12863 | 5 times 5 equals 25. (Con... |
| 6t2e12 12864 | 6 times 2 equals 12. (Con... |
| 6t3e18 12865 | 6 times 3 equals 18. (Con... |
| 6t4e24 12866 | 6 times 4 equals 24. (Con... |
| 6t5e30 12867 | 6 times 5 equals 30. (Con... |
| 6t6e36 12868 | 6 times 6 equals 36. (Con... |
| 7t2e14 12869 | 7 times 2 equals 14. (Con... |
| 7t3e21 12870 | 7 times 3 equals 21. (Con... |
| 7t4e28 12871 | 7 times 4 equals 28. (Con... |
| 7t5e35 12872 | 7 times 5 equals 35. (Con... |
| 7t6e42 12873 | 7 times 6 equals 42. (Con... |
| 7t7e49 12874 | 7 times 7 equals 49. (Con... |
| 8t2e16 12875 | 8 times 2 equals 16. (Con... |
| 8t3e24 12876 | 8 times 3 equals 24. (Con... |
| 8t4e32 12877 | 8 times 4 equals 32. (Con... |
| 8t5e40 12878 | 8 times 5 equals 40. (Con... |
| 8t6e48 12879 | 8 times 6 equals 48. (Con... |
| 8t7e56 12880 | 8 times 7 equals 56. (Con... |
| 8t8e64 12881 | 8 times 8 equals 64. (Con... |
| 9t2e18 12882 | 9 times 2 equals 18. (Con... |
| 9t3e27 12883 | 9 times 3 equals 27. (Con... |
| 9t4e36 12884 | 9 times 4 equals 36. (Con... |
| 9t5e45 12885 | 9 times 5 equals 45. (Con... |
| 9t6e54 12886 | 9 times 6 equals 54. (Con... |
| 9t7e63 12887 | 9 times 7 equals 63. (Con... |
| 9t8e72 12888 | 9 times 8 equals 72. (Con... |
| 9t9e81 12889 | 9 times 9 equals 81. (Con... |
| 9t11e99 12890 | 9 times 11 equals 99. (Co... |
| 9lt10 12891 | 9 is less than 10. (Contr... |
| 8lt10 12892 | 8 is less than 10. (Contr... |
| 7lt10 12893 | 7 is less than 10. (Contr... |
| 6lt10 12894 | 6 is less than 10. (Contr... |
| 5lt10 12895 | 5 is less than 10. (Contr... |
| 4lt10 12896 | 4 is less than 10. (Contr... |
| 3lt10 12897 | 3 is less than 10. (Contr... |
| 2lt10 12898 | 2 is less than 10. (Contr... |
| 1lt10 12899 | 1 is less than 10. (Contr... |
| decbin0 12900 | Decompose base 4 into base... |
| decbin2 12901 | Decompose base 4 into base... |
| decbin3 12902 | Decompose base 4 into base... |
| halfthird 12903 | Half minus a third. (Cont... |
| 5recm6rec 12904 | One fifth minus one sixth.... |
| uzval 12907 | The value of the upper int... |
| uzf 12908 | The domain and codomain of... |
| eluz1 12909 | Membership in the upper se... |
| eluzel2 12910 | Implication of membership ... |
| eluz2 12911 | Membership in an upper set... |
| eluzmn 12912 | Membership in an earlier u... |
| eluz1i 12913 | Membership in an upper set... |
| eluzuzle 12914 | An integer in an upper set... |
| eluzelz 12915 | A member of an upper set o... |
| eluzelre 12916 | A member of an upper set o... |
| eluzelcn 12917 | A member of an upper set o... |
| eluzle 12918 | Implication of membership ... |
| eluz 12919 | Membership in an upper set... |
| uzid 12920 | Membership of the least me... |
| uzidd 12921 | Membership of the least me... |
| uzn0 12922 | The upper integers are all... |
| uztrn 12923 | Transitive law for sets of... |
| uztrn2 12924 | Transitive law for sets of... |
| uzneg 12925 | Contraposition law for upp... |
| uzssz 12926 | An upper set of integers i... |
| uzssre 12927 | An upper set of integers i... |
| uzss 12928 | Subset relationship for tw... |
| uztric 12929 | Totality of the ordering r... |
| uz11 12930 | The upper integers functio... |
| eluzp1m1 12931 | Membership in the next upp... |
| eluzp1l 12932 | Strict ordering implied by... |
| eluzp1p1 12933 | Membership in the next upp... |
| eluzadd 12934 | Membership in a later uppe... |
| eluzsub 12935 | Membership in an earlier u... |
| eluzaddi 12936 | Membership in a later uppe... |
| eluzaddiOLD 12937 | Obsolete version of ~ eluz... |
| eluzsubi 12938 | Membership in an earlier u... |
| eluzsubiOLD 12939 | Obsolete version of ~ eluz... |
| eluzaddOLD 12940 | Obsolete version of ~ eluz... |
| eluzsubOLD 12941 | Obsolete version of ~ eluz... |
| subeluzsub 12942 | Membership of a difference... |
| uzm1 12943 | Choices for an element of ... |
| uznn0sub 12944 | The nonnegative difference... |
| uzin 12945 | Intersection of two upper ... |
| uzp1 12946 | Choices for an element of ... |
| nn0uz 12947 | Nonnegative integers expre... |
| nnuz 12948 | Positive integers expresse... |
| elnnuz 12949 | A positive integer express... |
| elnn0uz 12950 | A nonnegative integer expr... |
| eluz2nn 12951 | An integer greater than or... |
| eluz4eluz2 12952 | An integer greater than or... |
| eluz4nn 12953 | An integer greater than or... |
| eluzge2nn0 12954 | If an integer is greater t... |
| eluz2n0 12955 | An integer greater than or... |
| uzuzle23 12956 | An integer in the upper se... |
| eluzge3nn 12957 | If an integer is greater t... |
| uz3m2nn 12958 | An integer greater than or... |
| 1eluzge0 12959 | 1 is an integer greater th... |
| 2eluzge0 12960 | 2 is an integer greater th... |
| 2eluzge1 12961 | 2 is an integer greater th... |
| uznnssnn 12962 | The upper integers startin... |
| raluz 12963 | Restricted universal quant... |
| raluz2 12964 | Restricted universal quant... |
| rexuz 12965 | Restricted existential qua... |
| rexuz2 12966 | Restricted existential qua... |
| 2rexuz 12967 | Double existential quantif... |
| peano2uz 12968 | Second Peano postulate for... |
| peano2uzs 12969 | Second Peano postulate for... |
| peano2uzr 12970 | Reversed second Peano axio... |
| uzaddcl 12971 | Addition closure law for a... |
| nn0pzuz 12972 | The sum of a nonnegative i... |
| uzind4 12973 | Induction on the upper set... |
| uzind4ALT 12974 | Induction on the upper set... |
| uzind4s 12975 | Induction on the upper set... |
| uzind4s2 12976 | Induction on the upper set... |
| uzind4i 12977 | Induction on the upper int... |
| uzwo 12978 | Well-ordering principle: a... |
| uzwo2 12979 | Well-ordering principle: a... |
| nnwo 12980 | Well-ordering principle: a... |
| nnwof 12981 | Well-ordering principle: a... |
| nnwos 12982 | Well-ordering principle: a... |
| indstr 12983 | Strong Mathematical Induct... |
| eluznn0 12984 | Membership in a nonnegativ... |
| eluznn 12985 | Membership in a positive u... |
| eluz2b1 12986 | Two ways to say "an intege... |
| eluz2gt1 12987 | An integer greater than or... |
| eluz2b2 12988 | Two ways to say "an intege... |
| eluz2b3 12989 | Two ways to say "an intege... |
| uz2m1nn 12990 | One less than an integer g... |
| 1nuz2 12991 | 1 is not in ` ( ZZ>= `` 2 ... |
| elnn1uz2 12992 | A positive integer is eith... |
| uz2mulcl 12993 | Closure of multiplication ... |
| indstr2 12994 | Strong Mathematical Induct... |
| uzinfi 12995 | Extract the lower bound of... |
| nninf 12996 | The infimum of the set of ... |
| nn0inf 12997 | The infimum of the set of ... |
| infssuzle 12998 | The infimum of a subset of... |
| infssuzcl 12999 | The infimum of a subset of... |
| ublbneg 13000 | The image under negation o... |
| eqreznegel 13001 | Two ways to express the im... |
| supminf 13002 | The supremum of a bounded-... |
| lbzbi 13003 | If a set of reals is bound... |
| zsupss 13004 | Any nonempty bounded subse... |
| suprzcl2 13005 | The supremum of a bounded-... |
| suprzub 13006 | The supremum of a bounded-... |
| uzsupss 13007 | Any bounded subset of an u... |
| nn01to3 13008 | A (nonnegative) integer be... |
| nn0ge2m1nnALT 13009 | Alternate proof of ~ nn0ge... |
| uzwo3 13010 | Well-ordering principle: a... |
| zmin 13011 | There is a unique smallest... |
| zmax 13012 | There is a unique largest ... |
| zbtwnre 13013 | There is a unique integer ... |
| rebtwnz 13014 | There is a unique greatest... |
| elq 13017 | Membership in the set of r... |
| qmulz 13018 | If ` A ` is rational, then... |
| znq 13019 | The ratio of an integer an... |
| qre 13020 | A rational number is a rea... |
| zq 13021 | An integer is a rational n... |
| qred 13022 | A rational number is a rea... |
| zssq 13023 | The integers are a subset ... |
| nn0ssq 13024 | The nonnegative integers a... |
| nnssq 13025 | The positive integers are ... |
| qssre 13026 | The rationals are a subset... |
| qsscn 13027 | The rationals are a subset... |
| qex 13028 | The set of rational number... |
| nnq 13029 | A positive integer is rati... |
| qcn 13030 | A rational number is a com... |
| qexALT 13031 | Alternate proof of ~ qex .... |
| qaddcl 13032 | Closure of addition of rat... |
| qnegcl 13033 | Closure law for the negati... |
| qmulcl 13034 | Closure of multiplication ... |
| qsubcl 13035 | Closure of subtraction of ... |
| qreccl 13036 | Closure of reciprocal of r... |
| qdivcl 13037 | Closure of division of rat... |
| qrevaddcl 13038 | Reverse closure law for ad... |
| nnrecq 13039 | The reciprocal of a positi... |
| irradd 13040 | The sum of an irrational n... |
| irrmul 13041 | The product of an irration... |
| elpq 13042 | A positive rational is the... |
| elpqb 13043 | A class is a positive rati... |
| rpnnen1lem2 13044 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem1 13045 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem3 13046 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem4 13047 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem5 13048 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem6 13049 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1 13050 | One half of ~ rpnnen , whe... |
| reexALT 13051 | Alternate proof of ~ reex ... |
| cnref1o 13052 | There is a natural one-to-... |
| cnexALT 13053 | The set of complex numbers... |
| xrex 13054 | The set of extended reals ... |
| mpoaddex 13055 | The addition operation is ... |
| addex 13056 | The addition operation is ... |
| mpomulex 13057 | The multiplication operati... |
| mulex 13058 | The multiplication operati... |
| elrp 13061 | Membership in the set of p... |
| elrpii 13062 | Membership in the set of p... |
| 1rp 13063 | 1 is a positive real. (Co... |
| 2rp 13064 | 2 is a positive real. (Co... |
| 3rp 13065 | 3 is a positive real. (Co... |
| rpssre 13066 | The positive reals are a s... |
| rpre 13067 | A positive real is a real.... |
| rpxr 13068 | A positive real is an exte... |
| rpcn 13069 | A positive real is a compl... |
| nnrp 13070 | A positive integer is a po... |
| rpgt0 13071 | A positive real is greater... |
| rpge0 13072 | A positive real is greater... |
| rpregt0 13073 | A positive real is a posit... |
| rprege0 13074 | A positive real is a nonne... |
| rpne0 13075 | A positive real is nonzero... |
| rprene0 13076 | A positive real is a nonze... |
| rpcnne0 13077 | A positive real is a nonze... |
| rpcndif0 13078 | A positive real number is ... |
| ralrp 13079 | Quantification over positi... |
| rexrp 13080 | Quantification over positi... |
| rpaddcl 13081 | Closure law for addition o... |
| rpmulcl 13082 | Closure law for multiplica... |
| rpmtmip 13083 | "Minus times minus is plus... |
| rpdivcl 13084 | Closure law for division o... |
| rpreccl 13085 | Closure law for reciprocat... |
| rphalfcl 13086 | Closure law for half of a ... |
| rpgecl 13087 | A number greater than or e... |
| rphalflt 13088 | Half of a positive real is... |
| rerpdivcl 13089 | Closure law for division o... |
| ge0p1rp 13090 | A nonnegative number plus ... |
| rpneg 13091 | Either a nonzero real or i... |
| negelrp 13092 | Elementhood of a negation ... |
| negelrpd 13093 | The negation of a negative... |
| 0nrp 13094 | Zero is not a positive rea... |
| ltsubrp 13095 | Subtracting a positive rea... |
| ltaddrp 13096 | Adding a positive number t... |
| difrp 13097 | Two ways to say one number... |
| elrpd 13098 | Membership in the set of p... |
| nnrpd 13099 | A positive integer is a po... |
| zgt1rpn0n1 13100 | An integer greater than 1 ... |
| rpred 13101 | A positive real is a real.... |
| rpxrd 13102 | A positive real is an exte... |
| rpcnd 13103 | A positive real is a compl... |
| rpgt0d 13104 | A positive real is greater... |
| rpge0d 13105 | A positive real is greater... |
| rpne0d 13106 | A positive real is nonzero... |
| rpregt0d 13107 | A positive real is real an... |
| rprege0d 13108 | A positive real is real an... |
| rprene0d 13109 | A positive real is a nonze... |
| rpcnne0d 13110 | A positive real is a nonze... |
| rpreccld 13111 | Closure law for reciprocat... |
| rprecred 13112 | Closure law for reciprocat... |
| rphalfcld 13113 | Closure law for half of a ... |
| reclt1d 13114 | The reciprocal of a positi... |
| recgt1d 13115 | The reciprocal of a positi... |
| rpaddcld 13116 | Closure law for addition o... |
| rpmulcld 13117 | Closure law for multiplica... |
| rpdivcld 13118 | Closure law for division o... |
| ltrecd 13119 | The reciprocal of both sid... |
| lerecd 13120 | The reciprocal of both sid... |
| ltrec1d 13121 | Reciprocal swap in a 'less... |
| lerec2d 13122 | Reciprocal swap in a 'less... |
| lediv2ad 13123 | Division of both sides of ... |
| ltdiv2d 13124 | Division of a positive num... |
| lediv2d 13125 | Division of a positive num... |
| ledivdivd 13126 | Invert ratios of positive ... |
| divge1 13127 | The ratio of a number over... |
| divlt1lt 13128 | A real number divided by a... |
| divle1le 13129 | A real number divided by a... |
| ledivge1le 13130 | If a number is less than o... |
| ge0p1rpd 13131 | A nonnegative number plus ... |
| rerpdivcld 13132 | Closure law for division o... |
| ltsubrpd 13133 | Subtracting a positive rea... |
| ltaddrpd 13134 | Adding a positive number t... |
| ltaddrp2d 13135 | Adding a positive number t... |
| ltmulgt11d 13136 | Multiplication by a number... |
| ltmulgt12d 13137 | Multiplication by a number... |
| gt0divd 13138 | Division of a positive num... |
| ge0divd 13139 | Division of a nonnegative ... |
| rpgecld 13140 | A number greater than or e... |
| divge0d 13141 | The ratio of nonnegative a... |
| ltmul1d 13142 | The ratio of nonnegative a... |
| ltmul2d 13143 | Multiplication of both sid... |
| lemul1d 13144 | Multiplication of both sid... |
| lemul2d 13145 | Multiplication of both sid... |
| ltdiv1d 13146 | Division of both sides of ... |
| lediv1d 13147 | Division of both sides of ... |
| ltmuldivd 13148 | 'Less than' relationship b... |
| ltmuldiv2d 13149 | 'Less than' relationship b... |
| lemuldivd 13150 | 'Less than or equal to' re... |
| lemuldiv2d 13151 | 'Less than or equal to' re... |
| ltdivmuld 13152 | 'Less than' relationship b... |
| ltdivmul2d 13153 | 'Less than' relationship b... |
| ledivmuld 13154 | 'Less than or equal to' re... |
| ledivmul2d 13155 | 'Less than or equal to' re... |
| ltmul1dd 13156 | The ratio of nonnegative a... |
| ltmul2dd 13157 | Multiplication of both sid... |
| ltdiv1dd 13158 | Division of both sides of ... |
| lediv1dd 13159 | Division of both sides of ... |
| lediv12ad 13160 | Comparison of ratio of two... |
| mul2lt0rlt0 13161 | If the result of a multipl... |
| mul2lt0rgt0 13162 | If the result of a multipl... |
| mul2lt0llt0 13163 | If the result of a multipl... |
| mul2lt0lgt0 13164 | If the result of a multipl... |
| mul2lt0bi 13165 | If the result of a multipl... |
| prodge0rd 13166 | Infer that a multiplicand ... |
| prodge0ld 13167 | Infer that a multiplier is... |
| ltdiv23d 13168 | Swap denominator with othe... |
| lediv23d 13169 | Swap denominator with othe... |
| lt2mul2divd 13170 | The ratio of nonnegative a... |
| nnledivrp 13171 | Division of a positive int... |
| nn0ledivnn 13172 | Division of a nonnegative ... |
| addlelt 13173 | If the sum of a real numbe... |
| ltxr 13180 | The 'less than' binary rel... |
| elxr 13181 | Membership in the set of e... |
| xrnemnf 13182 | An extended real other tha... |
| xrnepnf 13183 | An extended real other tha... |
| xrltnr 13184 | The extended real 'less th... |
| ltpnf 13185 | Any (finite) real is less ... |
| ltpnfd 13186 | Any (finite) real is less ... |
| 0ltpnf 13187 | Zero is less than plus inf... |
| mnflt 13188 | Minus infinity is less tha... |
| mnfltd 13189 | Minus infinity is less tha... |
| mnflt0 13190 | Minus infinity is less tha... |
| mnfltpnf 13191 | Minus infinity is less tha... |
| mnfltxr 13192 | Minus infinity is less tha... |
| pnfnlt 13193 | No extended real is greate... |
| nltmnf 13194 | No extended real is less t... |
| pnfge 13195 | Plus infinity is an upper ... |
| xnn0n0n1ge2b 13196 | An extended nonnegative in... |
| 0lepnf 13197 | 0 less than or equal to po... |
| xnn0ge0 13198 | An extended nonnegative in... |
| mnfle 13199 | Minus infinity is less tha... |
| mnfled 13200 | Minus infinity is less tha... |
| xrltnsym 13201 | Ordering on the extended r... |
| xrltnsym2 13202 | 'Less than' is antisymmetr... |
| xrlttri 13203 | Ordering on the extended r... |
| xrlttr 13204 | Ordering on the extended r... |
| xrltso 13205 | 'Less than' is a strict or... |
| xrlttri2 13206 | Trichotomy law for 'less t... |
| xrlttri3 13207 | Trichotomy law for 'less t... |
| xrleloe 13208 | 'Less than or equal' expre... |
| xrleltne 13209 | 'Less than or equal to' im... |
| xrltlen 13210 | 'Less than' expressed in t... |
| dfle2 13211 | Alternative definition of ... |
| dflt2 13212 | Alternative definition of ... |
| xrltle 13213 | 'Less than' implies 'less ... |
| xrltled 13214 | 'Less than' implies 'less ... |
| xrleid 13215 | 'Less than or equal to' is... |
| xrleidd 13216 | 'Less than or equal to' is... |
| xrletri 13217 | Trichotomy law for extende... |
| xrletri3 13218 | Trichotomy law for extende... |
| xrletrid 13219 | Trichotomy law for extende... |
| xrlelttr 13220 | Transitive law for orderin... |
| xrltletr 13221 | Transitive law for orderin... |
| xrletr 13222 | Transitive law for orderin... |
| xrlttrd 13223 | Transitive law for orderin... |
| xrlelttrd 13224 | Transitive law for orderin... |
| xrltletrd 13225 | Transitive law for orderin... |
| xrletrd 13226 | Transitive law for orderin... |
| xrltne 13227 | 'Less than' implies not eq... |
| nltpnft 13228 | An extended real is not le... |
| xgepnf 13229 | An extended real which is ... |
| ngtmnft 13230 | An extended real is not gr... |
| xlemnf 13231 | An extended real which is ... |
| xrrebnd 13232 | An extended real is real i... |
| xrre 13233 | A way of proving that an e... |
| xrre2 13234 | An extended real between t... |
| xrre3 13235 | A way of proving that an e... |
| ge0gtmnf 13236 | A nonnegative extended rea... |
| ge0nemnf 13237 | A nonnegative extended rea... |
| xrrege0 13238 | A nonnegative extended rea... |
| xrmax1 13239 | An extended real is less t... |
| xrmax2 13240 | An extended real is less t... |
| xrmin1 13241 | The minimum of two extende... |
| xrmin2 13242 | The minimum of two extende... |
| xrmaxeq 13243 | The maximum of two extende... |
| xrmineq 13244 | The minimum of two extende... |
| xrmaxlt 13245 | Two ways of saying the max... |
| xrltmin 13246 | Two ways of saying an exte... |
| xrmaxle 13247 | Two ways of saying the max... |
| xrlemin 13248 | Two ways of saying a numbe... |
| max1 13249 | A number is less than or e... |
| max1ALT 13250 | A number is less than or e... |
| max2 13251 | A number is less than or e... |
| 2resupmax 13252 | The supremum of two real n... |
| min1 13253 | The minimum of two numbers... |
| min2 13254 | The minimum of two numbers... |
| maxle 13255 | Two ways of saying the max... |
| lemin 13256 | Two ways of saying a numbe... |
| maxlt 13257 | Two ways of saying the max... |
| ltmin 13258 | Two ways of saying a numbe... |
| lemaxle 13259 | A real number which is les... |
| max0sub 13260 | Decompose a real number in... |
| ifle 13261 | An if statement transforms... |
| z2ge 13262 | There exists an integer gr... |
| qbtwnre 13263 | The rational numbers are d... |
| qbtwnxr 13264 | The rational numbers are d... |
| qsqueeze 13265 | If a nonnegative real is l... |
| qextltlem 13266 | Lemma for ~ qextlt and qex... |
| qextlt 13267 | An extensionality-like pro... |
| qextle 13268 | An extensionality-like pro... |
| xralrple 13269 | Show that ` A ` is less th... |
| alrple 13270 | Show that ` A ` is less th... |
| xnegeq 13271 | Equality of two extended n... |
| xnegex 13272 | A negative extended real e... |
| xnegpnf 13273 | Minus ` +oo ` . Remark of... |
| xnegmnf 13274 | Minus ` -oo ` . Remark of... |
| rexneg 13275 | Minus a real number. Rema... |
| xneg0 13276 | The negative of zero. (Co... |
| xnegcl 13277 | Closure of extended real n... |
| xnegneg 13278 | Extended real version of ~... |
| xneg11 13279 | Extended real version of ~... |
| xltnegi 13280 | Forward direction of ~ xlt... |
| xltneg 13281 | Extended real version of ~... |
| xleneg 13282 | Extended real version of ~... |
| xlt0neg1 13283 | Extended real version of ~... |
| xlt0neg2 13284 | Extended real version of ~... |
| xle0neg1 13285 | Extended real version of ~... |
| xle0neg2 13286 | Extended real version of ~... |
| xaddval 13287 | Value of the extended real... |
| xaddf 13288 | The extended real addition... |
| xmulval 13289 | Value of the extended real... |
| xaddpnf1 13290 | Addition of positive infin... |
| xaddpnf2 13291 | Addition of positive infin... |
| xaddmnf1 13292 | Addition of negative infin... |
| xaddmnf2 13293 | Addition of negative infin... |
| pnfaddmnf 13294 | Addition of positive and n... |
| mnfaddpnf 13295 | Addition of negative and p... |
| rexadd 13296 | The extended real addition... |
| rexsub 13297 | Extended real subtraction ... |
| rexaddd 13298 | The extended real addition... |
| xnn0xaddcl 13299 | The extended nonnegative i... |
| xaddnemnf 13300 | Closure of extended real a... |
| xaddnepnf 13301 | Closure of extended real a... |
| xnegid 13302 | Extended real version of ~... |
| xaddcl 13303 | The extended real addition... |
| xaddcom 13304 | The extended real addition... |
| xaddrid 13305 | Extended real version of ~... |
| xaddlid 13306 | Extended real version of ~... |
| xaddridd 13307 | ` 0 ` is a right identity ... |
| xnn0lem1lt 13308 | Extended nonnegative integ... |
| xnn0lenn0nn0 13309 | An extended nonnegative in... |
| xnn0le2is012 13310 | An extended nonnegative in... |
| xnn0xadd0 13311 | The sum of two extended no... |
| xnegdi 13312 | Extended real version of ~... |
| xaddass 13313 | Associativity of extended ... |
| xaddass2 13314 | Associativity of extended ... |
| xpncan 13315 | Extended real version of ~... |
| xnpcan 13316 | Extended real version of ~... |
| xleadd1a 13317 | Extended real version of ~... |
| xleadd2a 13318 | Commuted form of ~ xleadd1... |
| xleadd1 13319 | Weakened version of ~ xlea... |
| xltadd1 13320 | Extended real version of ~... |
| xltadd2 13321 | Extended real version of ~... |
| xaddge0 13322 | The sum of nonnegative ext... |
| xle2add 13323 | Extended real version of ~... |
| xlt2add 13324 | Extended real version of ~... |
| xsubge0 13325 | Extended real version of ~... |
| xposdif 13326 | Extended real version of ~... |
| xlesubadd 13327 | Under certain conditions, ... |
| xmullem 13328 | Lemma for ~ rexmul . (Con... |
| xmullem2 13329 | Lemma for ~ xmulneg1 . (C... |
| xmulcom 13330 | Extended real multiplicati... |
| xmul01 13331 | Extended real version of ~... |
| xmul02 13332 | Extended real version of ~... |
| xmulneg1 13333 | Extended real version of ~... |
| xmulneg2 13334 | Extended real version of ~... |
| rexmul 13335 | The extended real multipli... |
| xmulf 13336 | The extended real multipli... |
| xmulcl 13337 | Closure of extended real m... |
| xmulpnf1 13338 | Multiplication by plus inf... |
| xmulpnf2 13339 | Multiplication by plus inf... |
| xmulmnf1 13340 | Multiplication by minus in... |
| xmulmnf2 13341 | Multiplication by minus in... |
| xmulpnf1n 13342 | Multiplication by plus inf... |
| xmulrid 13343 | Extended real version of ~... |
| xmullid 13344 | Extended real version of ~... |
| xmulm1 13345 | Extended real version of ~... |
| xmulasslem2 13346 | Lemma for ~ xmulass . (Co... |
| xmulgt0 13347 | Extended real version of ~... |
| xmulge0 13348 | Extended real version of ~... |
| xmulasslem 13349 | Lemma for ~ xmulass . (Co... |
| xmulasslem3 13350 | Lemma for ~ xmulass . (Co... |
| xmulass 13351 | Associativity of the exten... |
| xlemul1a 13352 | Extended real version of ~... |
| xlemul2a 13353 | Extended real version of ~... |
| xlemul1 13354 | Extended real version of ~... |
| xlemul2 13355 | Extended real version of ~... |
| xltmul1 13356 | Extended real version of ~... |
| xltmul2 13357 | Extended real version of ~... |
| xadddilem 13358 | Lemma for ~ xadddi . (Con... |
| xadddi 13359 | Distributive property for ... |
| xadddir 13360 | Commuted version of ~ xadd... |
| xadddi2 13361 | The assumption that the mu... |
| xadddi2r 13362 | Commuted version of ~ xadd... |
| x2times 13363 | Extended real version of ~... |
| xnegcld 13364 | Closure of extended real n... |
| xaddcld 13365 | The extended real addition... |
| xmulcld 13366 | Closure of extended real m... |
| xadd4d 13367 | Rearrangement of 4 terms i... |
| xnn0add4d 13368 | Rearrangement of 4 terms i... |
| xrsupexmnf 13369 | Adding minus infinity to a... |
| xrinfmexpnf 13370 | Adding plus infinity to a ... |
| xrsupsslem 13371 | Lemma for ~ xrsupss . (Co... |
| xrinfmsslem 13372 | Lemma for ~ xrinfmss . (C... |
| xrsupss 13373 | Any subset of extended rea... |
| xrinfmss 13374 | Any subset of extended rea... |
| xrinfmss2 13375 | Any subset of extended rea... |
| xrub 13376 | By quantifying only over r... |
| supxr 13377 | The supremum of a set of e... |
| supxr2 13378 | The supremum of a set of e... |
| supxrcl 13379 | The supremum of an arbitra... |
| supxrun 13380 | The supremum of the union ... |
| supxrmnf 13381 | Adding minus infinity to a... |
| supxrpnf 13382 | The supremum of a set of e... |
| supxrunb1 13383 | The supremum of an unbound... |
| supxrunb2 13384 | The supremum of an unbound... |
| supxrbnd1 13385 | The supremum of a bounded-... |
| supxrbnd2 13386 | The supremum of a bounded-... |
| xrsup0 13387 | The supremum of an empty s... |
| supxrub 13388 | A member of a set of exten... |
| supxrlub 13389 | The supremum of a set of e... |
| supxrleub 13390 | The supremum of a set of e... |
| supxrre 13391 | The real and extended real... |
| supxrbnd 13392 | The supremum of a bounded-... |
| supxrgtmnf 13393 | The supremum of a nonempty... |
| supxrre1 13394 | The supremum of a nonempty... |
| supxrre2 13395 | The supremum of a nonempty... |
| supxrss 13396 | Smaller sets of extended r... |
| infxrcl 13397 | The infimum of an arbitrar... |
| infxrlb 13398 | A member of a set of exten... |
| infxrgelb 13399 | The infimum of a set of ex... |
| infxrre 13400 | The real and extended real... |
| infxrmnf 13401 | The infinimum of a set of ... |
| xrinf0 13402 | The infimum of the empty s... |
| infxrss 13403 | Larger sets of extended re... |
| reltre 13404 | For all real numbers there... |
| rpltrp 13405 | For all positive real numb... |
| reltxrnmnf 13406 | For all extended real numb... |
| infmremnf 13407 | The infimum of the reals i... |
| infmrp1 13408 | The infimum of the positiv... |
| ixxval 13417 | Value of the interval func... |
| elixx1 13418 | Membership in an interval ... |
| ixxf 13419 | The set of intervals of ex... |
| ixxex 13420 | The set of intervals of ex... |
| ixxssxr 13421 | The set of intervals of ex... |
| elixx3g 13422 | Membership in a set of ope... |
| ixxssixx 13423 | An interval is a subset of... |
| ixxdisj 13424 | Split an interval into dis... |
| ixxun 13425 | Split an interval into two... |
| ixxin 13426 | Intersection of two interv... |
| ixxss1 13427 | Subset relationship for in... |
| ixxss2 13428 | Subset relationship for in... |
| ixxss12 13429 | Subset relationship for in... |
| ixxub 13430 | Extract the upper bound of... |
| ixxlb 13431 | Extract the lower bound of... |
| iooex 13432 | The set of open intervals ... |
| iooval 13433 | Value of the open interval... |
| ioo0 13434 | An empty open interval of ... |
| ioon0 13435 | An open interval of extend... |
| ndmioo 13436 | The open interval function... |
| iooid 13437 | An open interval with iden... |
| elioo3g 13438 | Membership in a set of ope... |
| elioore 13439 | A member of an open interv... |
| lbioo 13440 | An open interval does not ... |
| ubioo 13441 | An open interval does not ... |
| iooval2 13442 | Value of the open interval... |
| iooin 13443 | Intersection of two open i... |
| iooss1 13444 | Subset relationship for op... |
| iooss2 13445 | Subset relationship for op... |
| iocval 13446 | Value of the open-below, c... |
| icoval 13447 | Value of the closed-below,... |
| iccval 13448 | Value of the closed interv... |
| elioo1 13449 | Membership in an open inte... |
| elioo2 13450 | Membership in an open inte... |
| elioc1 13451 | Membership in an open-belo... |
| elico1 13452 | Membership in a closed-bel... |
| elicc1 13453 | Membership in a closed int... |
| iccid 13454 | A closed interval with ide... |
| ico0 13455 | An empty open interval of ... |
| ioc0 13456 | An empty open interval of ... |
| icc0 13457 | An empty closed interval o... |
| dfrp2 13458 | Alternate definition of th... |
| elicod 13459 | Membership in a left-close... |
| icogelb 13460 | An element of a left-close... |
| elicore 13461 | A member of a left-closed ... |
| ubioc1 13462 | The upper bound belongs to... |
| lbico1 13463 | The lower bound belongs to... |
| iccleub 13464 | An element of a closed int... |
| iccgelb 13465 | An element of a closed int... |
| elioo5 13466 | Membership in an open inte... |
| eliooxr 13467 | A nonempty open interval s... |
| eliooord 13468 | Ordering implied by a memb... |
| elioo4g 13469 | Membership in an open inte... |
| ioossre 13470 | An open interval is a set ... |
| ioosscn 13471 | An open interval is a set ... |
| elioc2 13472 | Membership in an open-belo... |
| elico2 13473 | Membership in a closed-bel... |
| elicc2 13474 | Membership in a closed rea... |
| elicc2i 13475 | Inference for membership i... |
| elicc4 13476 | Membership in a closed rea... |
| iccss 13477 | Condition for a closed int... |
| iccssioo 13478 | Condition for a closed int... |
| icossico 13479 | Condition for a closed-bel... |
| iccss2 13480 | Condition for a closed int... |
| iccssico 13481 | Condition for a closed int... |
| iccssioo2 13482 | Condition for a closed int... |
| iccssico2 13483 | Condition for a closed int... |
| ioomax 13484 | The open interval from min... |
| iccmax 13485 | The closed interval from m... |
| ioopos 13486 | The set of positive reals ... |
| ioorp 13487 | The set of positive reals ... |
| iooshf 13488 | Shift the arguments of the... |
| iocssre 13489 | A closed-above interval wi... |
| icossre 13490 | A closed-below interval wi... |
| iccssre 13491 | A closed real interval is ... |
| iccssxr 13492 | A closed interval is a set... |
| iocssxr 13493 | An open-below, closed-abov... |
| icossxr 13494 | A closed-below, open-above... |
| ioossicc 13495 | An open interval is a subs... |
| iccssred 13496 | A closed real interval is ... |
| eliccxr 13497 | A member of a closed inter... |
| icossicc 13498 | A closed-below, open-above... |
| iocssicc 13499 | A closed-above, open-below... |
| ioossico 13500 | An open interval is a subs... |
| iocssioo 13501 | Condition for a closed int... |
| icossioo 13502 | Condition for a closed int... |
| ioossioo 13503 | Condition for an open inte... |
| iccsupr 13504 | A nonempty subset of a clo... |
| elioopnf 13505 | Membership in an unbounded... |
| elioomnf 13506 | Membership in an unbounded... |
| elicopnf 13507 | Membership in a closed unb... |
| repos 13508 | Two ways of saying that a ... |
| ioof 13509 | The set of open intervals ... |
| iccf 13510 | The set of closed interval... |
| unirnioo 13511 | The union of the range of ... |
| dfioo2 13512 | Alternate definition of th... |
| ioorebas 13513 | Open intervals are element... |
| xrge0neqmnf 13514 | A nonnegative extended rea... |
| xrge0nre 13515 | An extended real which is ... |
| elrege0 13516 | The predicate "is a nonneg... |
| nn0rp0 13517 | A nonnegative integer is a... |
| rge0ssre 13518 | Nonnegative real numbers a... |
| elxrge0 13519 | Elementhood in the set of ... |
| 0e0icopnf 13520 | 0 is a member of ` ( 0 [,)... |
| 0e0iccpnf 13521 | 0 is a member of ` ( 0 [,]... |
| ge0addcl 13522 | The nonnegative reals are ... |
| ge0mulcl 13523 | The nonnegative reals are ... |
| ge0xaddcl 13524 | The nonnegative reals are ... |
| ge0xmulcl 13525 | The nonnegative extended r... |
| lbicc2 13526 | The lower bound of a close... |
| ubicc2 13527 | The upper bound of a close... |
| elicc01 13528 | Membership in the closed r... |
| elunitrn 13529 | The closed unit interval i... |
| elunitcn 13530 | The closed unit interval i... |
| 0elunit 13531 | Zero is an element of the ... |
| 1elunit 13532 | One is an element of the c... |
| iooneg 13533 | Membership in a negated op... |
| iccneg 13534 | Membership in a negated cl... |
| icoshft 13535 | A shifted real is a member... |
| icoshftf1o 13536 | Shifting a closed-below, o... |
| icoun 13537 | The union of two adjacent ... |
| icodisj 13538 | Adjacent left-closed right... |
| ioounsn 13539 | The union of an open inter... |
| snunioo 13540 | The closure of one end of ... |
| snunico 13541 | The closure of the open en... |
| snunioc 13542 | The closure of the open en... |
| prunioo 13543 | The closure of an open rea... |
| ioodisj 13544 | If the upper bound of one ... |
| ioojoin 13545 | Join two open intervals to... |
| difreicc 13546 | The class difference of ` ... |
| iccsplit 13547 | Split a closed interval in... |
| iccshftr 13548 | Membership in a shifted in... |
| iccshftri 13549 | Membership in a shifted in... |
| iccshftl 13550 | Membership in a shifted in... |
| iccshftli 13551 | Membership in a shifted in... |
| iccdil 13552 | Membership in a dilated in... |
| iccdili 13553 | Membership in a dilated in... |
| icccntr 13554 | Membership in a contracted... |
| icccntri 13555 | Membership in a contracted... |
| divelunit 13556 | A condition for a ratio to... |
| lincmb01cmp 13557 | A linear combination of tw... |
| iccf1o 13558 | Describe a bijection from ... |
| iccen 13559 | Any nontrivial closed inte... |
| xov1plusxeqvd 13560 | A complex number ` X ` is ... |
| unitssre 13561 | ` ( 0 [,] 1 ) ` is a subse... |
| unitsscn 13562 | The closed unit interval i... |
| supicc 13563 | Supremum of a bounded set ... |
| supiccub 13564 | The supremum of a bounded ... |
| supicclub 13565 | The supremum of a bounded ... |
| supicclub2 13566 | The supremum of a bounded ... |
| zltaddlt1le 13567 | The sum of an integer and ... |
| xnn0xrge0 13568 | An extended nonnegative in... |
| fzval 13571 | The value of a finite set ... |
| fzval2 13572 | An alternative way of expr... |
| fzf 13573 | Establish the domain and c... |
| elfz1 13574 | Membership in a finite set... |
| elfz 13575 | Membership in a finite set... |
| elfz2 13576 | Membership in a finite set... |
| elfzd 13577 | Membership in a finite set... |
| elfz5 13578 | Membership in a finite set... |
| elfz4 13579 | Membership in a finite set... |
| elfzuzb 13580 | Membership in a finite set... |
| eluzfz 13581 | Membership in a finite set... |
| elfzuz 13582 | A member of a finite set o... |
| elfzuz3 13583 | Membership in a finite set... |
| elfzel2 13584 | Membership in a finite set... |
| elfzel1 13585 | Membership in a finite set... |
| elfzelz 13586 | A member of a finite set o... |
| elfzelzd 13587 | A member of a finite set o... |
| fzssz 13588 | A finite sequence of integ... |
| elfzle1 13589 | A member of a finite set o... |
| elfzle2 13590 | A member of a finite set o... |
| elfzuz2 13591 | Implication of membership ... |
| elfzle3 13592 | Membership in a finite set... |
| eluzfz1 13593 | Membership in a finite set... |
| eluzfz2 13594 | Membership in a finite set... |
| eluzfz2b 13595 | Membership in a finite set... |
| elfz3 13596 | Membership in a finite set... |
| elfz1eq 13597 | Membership in a finite set... |
| elfzubelfz 13598 | If there is a member in a ... |
| peano2fzr 13599 | A Peano-postulate-like the... |
| fzn0 13600 | Properties of a finite int... |
| fz0 13601 | A finite set of sequential... |
| fzn 13602 | A finite set of sequential... |
| fzen 13603 | A shifted finite set of se... |
| fz1n 13604 | A 1-based finite set of se... |
| 0nelfz1 13605 | 0 is not an element of a f... |
| 0fz1 13606 | Two ways to say a finite 1... |
| fz10 13607 | There are no integers betw... |
| uzsubsubfz 13608 | Membership of an integer g... |
| uzsubsubfz1 13609 | Membership of an integer g... |
| ige3m2fz 13610 | Membership of an integer g... |
| fzsplit2 13611 | Split a finite interval of... |
| fzsplit 13612 | Split a finite interval of... |
| fzdisj 13613 | Condition for two finite i... |
| fz01en 13614 | 0-based and 1-based finite... |
| elfznn 13615 | A member of a finite set o... |
| elfz1end 13616 | A nonempty finite range of... |
| fz1ssnn 13617 | A finite set of positive i... |
| fznn0sub 13618 | Subtraction closure for a ... |
| fzmmmeqm 13619 | Subtracting the difference... |
| fzaddel 13620 | Membership of a sum in a f... |
| fzadd2 13621 | Membership of a sum in a f... |
| fzsubel 13622 | Membership of a difference... |
| fzopth 13623 | A finite set of sequential... |
| fzass4 13624 | Two ways to express a nond... |
| fzss1 13625 | Subset relationship for fi... |
| fzss2 13626 | Subset relationship for fi... |
| fzssuz 13627 | A finite set of sequential... |
| fzsn 13628 | A finite interval of integ... |
| fzssp1 13629 | Subset relationship for fi... |
| fzssnn 13630 | Finite sets of sequential ... |
| ssfzunsnext 13631 | A subset of a finite seque... |
| ssfzunsn 13632 | A subset of a finite seque... |
| fzsuc 13633 | Join a successor to the en... |
| fzpred 13634 | Join a predecessor to the ... |
| fzpreddisj 13635 | A finite set of sequential... |
| elfzp1 13636 | Append an element to a fin... |
| fzp1ss 13637 | Subset relationship for fi... |
| fzelp1 13638 | Membership in a set of seq... |
| fzp1elp1 13639 | Add one to an element of a... |
| fznatpl1 13640 | Shift membership in a fini... |
| fzpr 13641 | A finite interval of integ... |
| fztp 13642 | A finite interval of integ... |
| fz12pr 13643 | An integer range between 1... |
| fzsuc2 13644 | Join a successor to the en... |
| fzp1disj 13645 | ` ( M ... ( N + 1 ) ) ` is... |
| fzdifsuc 13646 | Remove a successor from th... |
| fzprval 13647 | Two ways of defining the f... |
| fztpval 13648 | Two ways of defining the f... |
| fzrev 13649 | Reversal of start and end ... |
| fzrev2 13650 | Reversal of start and end ... |
| fzrev2i 13651 | Reversal of start and end ... |
| fzrev3 13652 | The "complement" of a memb... |
| fzrev3i 13653 | The "complement" of a memb... |
| fznn 13654 | Finite set of sequential i... |
| elfz1b 13655 | Membership in a 1-based fi... |
| elfz1uz 13656 | Membership in a 1-based fi... |
| elfzm11 13657 | Membership in a finite set... |
| uzsplit 13658 | Express an upper integer s... |
| uzdisj 13659 | The first ` N ` elements o... |
| fseq1p1m1 13660 | Add/remove an item to/from... |
| fseq1m1p1 13661 | Add/remove an item to/from... |
| fz1sbc 13662 | Quantification over a one-... |
| elfzp1b 13663 | An integer is a member of ... |
| elfzm1b 13664 | An integer is a member of ... |
| elfzp12 13665 | Options for membership in ... |
| fzm1 13666 | Choices for an element of ... |
| fzneuz 13667 | No finite set of sequentia... |
| fznuz 13668 | Disjointness of the upper ... |
| uznfz 13669 | Disjointness of the upper ... |
| fzp1nel 13670 | One plus the upper bound o... |
| fzrevral 13671 | Reversal of scanning order... |
| fzrevral2 13672 | Reversal of scanning order... |
| fzrevral3 13673 | Reversal of scanning order... |
| fzshftral 13674 | Shift the scanning order i... |
| ige2m1fz1 13675 | Membership of an integer g... |
| ige2m1fz 13676 | Membership in a 0-based fi... |
| elfz2nn0 13677 | Membership in a finite set... |
| fznn0 13678 | Characterization of a fini... |
| elfznn0 13679 | A member of a finite set o... |
| elfz3nn0 13680 | The upper bound of a nonem... |
| fz0ssnn0 13681 | Finite sets of sequential ... |
| fz1ssfz0 13682 | Subset relationship for fi... |
| 0elfz 13683 | 0 is an element of a finit... |
| nn0fz0 13684 | A nonnegative integer is a... |
| elfz0add 13685 | An element of a finite set... |
| fz0sn 13686 | An integer range from 0 to... |
| fz0tp 13687 | An integer range from 0 to... |
| fz0to3un2pr 13688 | An integer range from 0 to... |
| fz0to4untppr 13689 | An integer range from 0 to... |
| fz0to5un2tp 13690 | An integer range from 0 to... |
| elfz0ubfz0 13691 | An element of a finite set... |
| elfz0fzfz0 13692 | A member of a finite set o... |
| fz0fzelfz0 13693 | If a member of a finite se... |
| fznn0sub2 13694 | Subtraction closure for a ... |
| uzsubfz0 13695 | Membership of an integer g... |
| fz0fzdiffz0 13696 | The difference of an integ... |
| elfzmlbm 13697 | Subtracting the lower boun... |
| elfzmlbp 13698 | Subtracting the lower boun... |
| fzctr 13699 | Lemma for theorems about t... |
| difelfzle 13700 | The difference of two inte... |
| difelfznle 13701 | The difference of two inte... |
| nn0split 13702 | Express the set of nonnega... |
| nn0disj 13703 | The first ` N + 1 ` elemen... |
| fz0sn0fz1 13704 | A finite set of sequential... |
| fvffz0 13705 | The function value of a fu... |
| 1fv 13706 | A function on a singleton.... |
| 4fvwrd4 13707 | The first four function va... |
| 2ffzeq 13708 | Two functions over 0-based... |
| preduz 13709 | The value of the predecess... |
| prednn 13710 | The value of the predecess... |
| prednn0 13711 | The value of the predecess... |
| predfz 13712 | Calculate the predecessor ... |
| fzof 13715 | Functionality of the half-... |
| elfzoel1 13716 | Reverse closure for half-o... |
| elfzoel2 13717 | Reverse closure for half-o... |
| elfzoelz 13718 | Reverse closure for half-o... |
| fzoval 13719 | Value of the half-open int... |
| elfzo 13720 | Membership in a half-open ... |
| elfzo2 13721 | Membership in a half-open ... |
| elfzouz 13722 | Membership in a half-open ... |
| nelfzo 13723 | An integer not being a mem... |
| fzolb 13724 | The left endpoint of a hal... |
| fzolb2 13725 | The left endpoint of a hal... |
| elfzole1 13726 | A member in a half-open in... |
| elfzolt2 13727 | A member in a half-open in... |
| elfzolt3 13728 | Membership in a half-open ... |
| elfzolt2b 13729 | A member in a half-open in... |
| elfzolt3b 13730 | Membership in a half-open ... |
| elfzop1le2 13731 | A member in a half-open in... |
| fzonel 13732 | A half-open range does not... |
| elfzouz2 13733 | The upper bound of a half-... |
| elfzofz 13734 | A half-open range is conta... |
| elfzo3 13735 | Express membership in a ha... |
| fzon0 13736 | A half-open integer interv... |
| fzossfz 13737 | A half-open range is conta... |
| fzossz 13738 | A half-open integer interv... |
| fzon 13739 | A half-open set of sequent... |
| fzo0n 13740 | A half-open range of nonne... |
| fzonlt0 13741 | A half-open integer range ... |
| fzo0 13742 | Half-open sets with equal ... |
| fzonnsub 13743 | If ` K < N ` then ` N - K ... |
| fzonnsub2 13744 | If ` M < N ` then ` N - M ... |
| fzoss1 13745 | Subset relationship for ha... |
| fzoss2 13746 | Subset relationship for ha... |
| fzossrbm1 13747 | Subset of a half-open rang... |
| fzo0ss1 13748 | Subset relationship for ha... |
| fzossnn0 13749 | A half-open integer range ... |
| fzospliti 13750 | One direction of splitting... |
| fzosplit 13751 | Split a half-open integer ... |
| fzodisj 13752 | Abutting half-open integer... |
| fzouzsplit 13753 | Split an upper integer set... |
| fzouzdisj 13754 | A half-open integer range ... |
| fzoun 13755 | A half-open integer range ... |
| fzodisjsn 13756 | A half-open integer range ... |
| prinfzo0 13757 | The intersection of a half... |
| lbfzo0 13758 | An integer is strictly gre... |
| elfzo0 13759 | Membership in a half-open ... |
| elfzo0z 13760 | Membership in a half-open ... |
| nn0p1elfzo 13761 | A nonnegative integer incr... |
| elfzo0le 13762 | A member in a half-open ra... |
| elfzonn0 13763 | A member of a half-open ra... |
| fzonmapblen 13764 | The result of subtracting ... |
| fzofzim 13765 | If a nonnegative integer i... |
| fz1fzo0m1 13766 | Translation of one between... |
| fzossnn 13767 | Half-open integer ranges s... |
| elfzo1 13768 | Membership in a half-open ... |
| fzo1fzo0n0 13769 | An integer between 1 and a... |
| fzo0n0 13770 | A half-open integer range ... |
| fzoaddel 13771 | Translate membership in a ... |
| fzo0addel 13772 | Translate membership in a ... |
| fzo0addelr 13773 | Translate membership in a ... |
| fzoaddel2 13774 | Translate membership in a ... |
| elfzoext 13775 | Membership of an integer i... |
| elincfzoext 13776 | Membership of an increased... |
| fzosubel 13777 | Translate membership in a ... |
| fzosubel2 13778 | Membership in a translated... |
| fzosubel3 13779 | Membership in a translated... |
| eluzgtdifelfzo 13780 | Membership of the differen... |
| ige2m2fzo 13781 | Membership of an integer g... |
| fzocatel 13782 | Translate membership in a ... |
| ubmelfzo 13783 | If an integer in a 1-based... |
| elfzodifsumelfzo 13784 | If an integer is in a half... |
| elfzom1elp1fzo 13785 | Membership of an integer i... |
| elfzom1elfzo 13786 | Membership in a half-open ... |
| fzval3 13787 | Expressing a closed intege... |
| fz0add1fz1 13788 | Translate membership in a ... |
| fzosn 13789 | Expressing a singleton as ... |
| elfzomin 13790 | Membership of an integer i... |
| zpnn0elfzo 13791 | Membership of an integer i... |
| zpnn0elfzo1 13792 | Membership of an integer i... |
| fzosplitsnm1 13793 | Removing a singleton from ... |
| elfzonlteqm1 13794 | If an element of a half-op... |
| fzonn0p1 13795 | A nonnegative integer is e... |
| fzossfzop1 13796 | A half-open range of nonne... |
| fzonn0p1p1 13797 | If a nonnegative integer i... |
| elfzom1p1elfzo 13798 | Increasing an element of a... |
| fzo0ssnn0 13799 | Half-open integer ranges s... |
| fzo01 13800 | Expressing the singleton o... |
| fzo12sn 13801 | A 1-based half-open intege... |
| fzo13pr 13802 | A 1-based half-open intege... |
| fzo0to2pr 13803 | A half-open integer range ... |
| fzo0to3tp 13804 | A half-open integer range ... |
| fzo0to42pr 13805 | A half-open integer range ... |
| fzo1to4tp 13806 | A half-open integer range ... |
| fzo0sn0fzo1 13807 | A half-open range of nonne... |
| elfzo0l 13808 | A member of a half-open ra... |
| fzoend 13809 | The endpoint of a half-ope... |
| fzo0end 13810 | The endpoint of a zero-bas... |
| ssfzo12 13811 | Subset relationship for ha... |
| ssfzoulel 13812 | If a half-open integer ran... |
| ssfzo12bi 13813 | Subset relationship for ha... |
| fzoopth 13814 | A half-open integer range ... |
| ubmelm1fzo 13815 | The result of subtracting ... |
| fzofzp1 13816 | If a point is in a half-op... |
| fzofzp1b 13817 | If a point is in a half-op... |
| elfzom1b 13818 | An integer is a member of ... |
| elfzom1elp1fzo1 13819 | Membership of a nonnegativ... |
| elfzo1elm1fzo0 13820 | Membership of a positive i... |
| elfzonelfzo 13821 | If an element of a half-op... |
| fzonfzoufzol 13822 | If an element of a half-op... |
| elfzomelpfzo 13823 | An integer increased by an... |
| elfznelfzo 13824 | A value in a finite set of... |
| elfznelfzob 13825 | A value in a finite set of... |
| peano2fzor 13826 | A Peano-postulate-like the... |
| fzosplitsn 13827 | Extending a half-open rang... |
| fzosplitpr 13828 | Extending a half-open inte... |
| fzosplitprm1 13829 | Extending a half-open inte... |
| fzosplitsni 13830 | Membership in a half-open ... |
| fzisfzounsn 13831 | A finite interval of integ... |
| elfzr 13832 | A member of a finite inter... |
| elfzlmr 13833 | A member of a finite inter... |
| elfz0lmr 13834 | A member of a finite inter... |
| fzostep1 13835 | Two possibilities for a nu... |
| fzoshftral 13836 | Shift the scanning order i... |
| fzind2 13837 | Induction on the integers ... |
| fvinim0ffz 13838 | The function values for th... |
| injresinjlem 13839 | Lemma for ~ injresinj . (... |
| injresinj 13840 | A function whose restricti... |
| subfzo0 13841 | The difference between two... |
| fvf1tp 13842 | Values of a one-to-one fun... |
| flval 13847 | Value of the floor (greate... |
| flcl 13848 | The floor (greatest intege... |
| reflcl 13849 | The floor (greatest intege... |
| fllelt 13850 | A basic property of the fl... |
| flcld 13851 | The floor (greatest intege... |
| flle 13852 | A basic property of the fl... |
| flltp1 13853 | A basic property of the fl... |
| fllep1 13854 | A basic property of the fl... |
| fraclt1 13855 | The fractional part of a r... |
| fracle1 13856 | The fractional part of a r... |
| fracge0 13857 | The fractional part of a r... |
| flge 13858 | The floor function value i... |
| fllt 13859 | The floor function value i... |
| flflp1 13860 | Move floor function betwee... |
| flid 13861 | An integer is its own floo... |
| flidm 13862 | The floor function is idem... |
| flidz 13863 | A real number equals its f... |
| flltnz 13864 | The floor of a non-integer... |
| flwordi 13865 | Ordering relation for the ... |
| flword2 13866 | Ordering relation for the ... |
| flval2 13867 | An alternate way to define... |
| flval3 13868 | An alternate way to define... |
| flbi 13869 | A condition equivalent to ... |
| flbi2 13870 | A condition equivalent to ... |
| adddivflid 13871 | The floor of a sum of an i... |
| ico01fl0 13872 | The floor of a real number... |
| flge0nn0 13873 | The floor of a number grea... |
| flge1nn 13874 | The floor of a number grea... |
| fldivnn0 13875 | The floor function of a di... |
| refldivcl 13876 | The floor function of a di... |
| divfl0 13877 | The floor of a fraction is... |
| fladdz 13878 | An integer can be moved in... |
| flzadd 13879 | An integer can be moved in... |
| flmulnn0 13880 | Move a nonnegative integer... |
| btwnzge0 13881 | A real bounded between an ... |
| 2tnp1ge0ge0 13882 | Two times an integer plus ... |
| flhalf 13883 | Ordering relation for the ... |
| fldivle 13884 | The floor function of a di... |
| fldivnn0le 13885 | The floor function of a di... |
| flltdivnn0lt 13886 | The floor function of a di... |
| ltdifltdiv 13887 | If the dividend of a divis... |
| fldiv4p1lem1div2 13888 | The floor of an integer eq... |
| fldiv4lem1div2uz2 13889 | The floor of an integer gr... |
| fldiv4lem1div2 13890 | The floor of a positive in... |
| ceilval 13891 | The value of the ceiling f... |
| dfceil2 13892 | Alternative definition of ... |
| ceilval2 13893 | The value of the ceiling f... |
| ceicl 13894 | The ceiling function retur... |
| ceilcl 13895 | Closure of the ceiling fun... |
| ceilcld 13896 | Closure of the ceiling fun... |
| ceige 13897 | The ceiling of a real numb... |
| ceilge 13898 | The ceiling of a real numb... |
| ceilged 13899 | The ceiling of a real numb... |
| ceim1l 13900 | One less than the ceiling ... |
| ceilm1lt 13901 | One less than the ceiling ... |
| ceile 13902 | The ceiling of a real numb... |
| ceille 13903 | The ceiling of a real numb... |
| ceilid 13904 | An integer is its own ceil... |
| ceilidz 13905 | A real number equals its c... |
| flleceil 13906 | The floor of a real number... |
| fleqceilz 13907 | A real number is an intege... |
| quoremz 13908 | Quotient and remainder of ... |
| quoremnn0 13909 | Quotient and remainder of ... |
| quoremnn0ALT 13910 | Alternate proof of ~ quore... |
| intfrac2 13911 | Decompose a real into inte... |
| intfracq 13912 | Decompose a rational numbe... |
| fldiv 13913 | Cancellation of the embedd... |
| fldiv2 13914 | Cancellation of an embedde... |
| fznnfl 13915 | Finite set of sequential i... |
| uzsup 13916 | An upper set of integers i... |
| ioopnfsup 13917 | An upper set of reals is u... |
| icopnfsup 13918 | An upper set of reals is u... |
| rpsup 13919 | The positive reals are unb... |
| resup 13920 | The real numbers are unbou... |
| xrsup 13921 | The extended real numbers ... |
| modval 13924 | The value of the modulo op... |
| modvalr 13925 | The value of the modulo op... |
| modcl 13926 | Closure law for the modulo... |
| flpmodeq 13927 | Partition of a division in... |
| modcld 13928 | Closure law for the modulo... |
| mod0 13929 | ` A mod B ` is zero iff ` ... |
| mulmod0 13930 | The product of an integer ... |
| negmod0 13931 | ` A ` is divisible by ` B ... |
| modge0 13932 | The modulo operation is no... |
| modlt 13933 | The modulo operation is le... |
| modelico 13934 | Modular reduction produces... |
| moddiffl 13935 | Value of the modulo operat... |
| moddifz 13936 | The modulo operation diffe... |
| modfrac 13937 | The fractional part of a n... |
| flmod 13938 | The floor function express... |
| intfrac 13939 | Break a number into its in... |
| zmod10 13940 | An integer modulo 1 is 0. ... |
| zmod1congr 13941 | Two arbitrary integers are... |
| modmulnn 13942 | Move a positive integer in... |
| modvalp1 13943 | The value of the modulo op... |
| zmodcl 13944 | Closure law for the modulo... |
| zmodcld 13945 | Closure law for the modulo... |
| zmodfz 13946 | An integer mod ` B ` lies ... |
| zmodfzo 13947 | An integer mod ` B ` lies ... |
| zmodfzp1 13948 | An integer mod ` B ` lies ... |
| modid 13949 | Identity law for modulo. ... |
| modid0 13950 | A positive real number mod... |
| modid2 13951 | Identity law for modulo. ... |
| zmodid2 13952 | Identity law for modulo re... |
| zmodidfzo 13953 | Identity law for modulo re... |
| zmodidfzoimp 13954 | Identity law for modulo re... |
| 0mod 13955 | Special case: 0 modulo a p... |
| 1mod 13956 | Special case: 1 modulo a r... |
| modabs 13957 | Absorption law for modulo.... |
| modabs2 13958 | Absorption law for modulo.... |
| modcyc 13959 | The modulo operation is pe... |
| modcyc2 13960 | The modulo operation is pe... |
| modadd1 13961 | Addition property of the m... |
| modaddabs 13962 | Absorption law for modulo.... |
| modaddmod 13963 | The sum of a real number m... |
| muladdmodid 13964 | The sum of a positive real... |
| mulp1mod1 13965 | The product of an integer ... |
| modmuladd 13966 | Decomposition of an intege... |
| modmuladdim 13967 | Implication of a decomposi... |
| modmuladdnn0 13968 | Implication of a decomposi... |
| negmod 13969 | The negation of a number m... |
| m1modnnsub1 13970 | Minus one modulo a positiv... |
| m1modge3gt1 13971 | Minus one modulo an intege... |
| addmodid 13972 | The sum of a positive inte... |
| addmodidr 13973 | The sum of a positive inte... |
| modadd2mod 13974 | The sum of a real number m... |
| modm1p1mod0 13975 | If a real number modulo a ... |
| modltm1p1mod 13976 | If a real number modulo a ... |
| modmul1 13977 | Multiplication property of... |
| modmul12d 13978 | Multiplication property of... |
| modnegd 13979 | Negation property of the m... |
| modadd12d 13980 | Additive property of the m... |
| modsub12d 13981 | Subtraction property of th... |
| modsubmod 13982 | The difference of a real n... |
| modsubmodmod 13983 | The difference of a real n... |
| 2txmodxeq0 13984 | Two times a positive real ... |
| 2submod 13985 | If a real number is betwee... |
| modifeq2int 13986 | If a nonnegative integer i... |
| modaddmodup 13987 | The sum of an integer modu... |
| modaddmodlo 13988 | The sum of an integer modu... |
| modmulmod 13989 | The product of a real numb... |
| modmulmodr 13990 | The product of an integer ... |
| modaddmulmod 13991 | The sum of a real number a... |
| moddi 13992 | Distribute multiplication ... |
| modsubdir 13993 | Distribute the modulo oper... |
| modeqmodmin 13994 | A real number equals the d... |
| modirr 13995 | A number modulo an irratio... |
| modfzo0difsn 13996 | For a number within a half... |
| modsumfzodifsn 13997 | The sum of a number within... |
| modlteq 13998 | Two nonnegative integers l... |
| addmodlteq 13999 | Two nonnegative integers l... |
| om2uz0i 14000 | The mapping ` G ` is a one... |
| om2uzsuci 14001 | The value of ` G ` (see ~ ... |
| om2uzuzi 14002 | The value ` G ` (see ~ om2... |
| om2uzlti 14003 | Less-than relation for ` G... |
| om2uzlt2i 14004 | The mapping ` G ` (see ~ o... |
| om2uzrani 14005 | Range of ` G ` (see ~ om2u... |
| om2uzf1oi 14006 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzisoi 14007 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzoi 14008 | An alternative definition ... |
| om2uzrdg 14009 | A helper lemma for the val... |
| uzrdglem 14010 | A helper lemma for the val... |
| uzrdgfni 14011 | The recursive definition g... |
| uzrdg0i 14012 | Initial value of a recursi... |
| uzrdgsuci 14013 | Successor value of a recur... |
| ltweuz 14014 | ` < ` is a well-founded re... |
| ltwenn 14015 | Less than well-orders the ... |
| ltwefz 14016 | Less than well-orders a se... |
| uzenom 14017 | An upper integer set is de... |
| uzinf 14018 | An upper integer set is in... |
| nnnfi 14019 | The set of positive intege... |
| uzrdgxfr 14020 | Transfer the value of the ... |
| fzennn 14021 | The cardinality of a finit... |
| fzen2 14022 | The cardinality of a finit... |
| cardfz 14023 | The cardinality of a finit... |
| hashgf1o 14024 | ` G ` maps ` _om ` one-to-... |
| fzfi 14025 | A finite interval of integ... |
| fzfid 14026 | Commonly used special case... |
| fzofi 14027 | Half-open integer sets are... |
| fsequb 14028 | The values of a finite rea... |
| fsequb2 14029 | The values of a finite rea... |
| fseqsupcl 14030 | The values of a finite rea... |
| fseqsupubi 14031 | The values of a finite rea... |
| nn0ennn 14032 | The nonnegative integers a... |
| nnenom 14033 | The set of positive intege... |
| nnct 14034 | ` NN ` is countable. (Con... |
| uzindi 14035 | Indirect strong induction ... |
| axdc4uzlem 14036 | Lemma for ~ axdc4uz . (Co... |
| axdc4uz 14037 | A version of ~ axdc4 that ... |
| ssnn0fi 14038 | A subset of the nonnegativ... |
| rabssnn0fi 14039 | A subset of the nonnegativ... |
| uzsinds 14040 | Strong (or "total") induct... |
| nnsinds 14041 | Strong (or "total") induct... |
| nn0sinds 14042 | Strong (or "total") induct... |
| fsuppmapnn0fiublem 14043 | Lemma for ~ fsuppmapnn0fiu... |
| fsuppmapnn0fiub 14044 | If all functions of a fini... |
| fsuppmapnn0fiubex 14045 | If all functions of a fini... |
| fsuppmapnn0fiub0 14046 | If all functions of a fini... |
| suppssfz 14047 | Condition for a function o... |
| fsuppmapnn0ub 14048 | If a function over the non... |
| fsuppmapnn0fz 14049 | If a function over the non... |
| mptnn0fsupp 14050 | A mapping from the nonnega... |
| mptnn0fsuppd 14051 | A mapping from the nonnega... |
| mptnn0fsuppr 14052 | A finitely supported mappi... |
| f13idfv 14053 | A one-to-one function with... |
| seqex 14056 | Existence of the sequence ... |
| seqeq1 14057 | Equality theorem for the s... |
| seqeq2 14058 | Equality theorem for the s... |
| seqeq3 14059 | Equality theorem for the s... |
| seqeq1d 14060 | Equality deduction for the... |
| seqeq2d 14061 | Equality deduction for the... |
| seqeq3d 14062 | Equality deduction for the... |
| seqeq123d 14063 | Equality deduction for the... |
| nfseq 14064 | Hypothesis builder for the... |
| seqval 14065 | Value of the sequence buil... |
| seqfn 14066 | The sequence builder funct... |
| seq1 14067 | Value of the sequence buil... |
| seq1i 14068 | Value of the sequence buil... |
| seqp1 14069 | Value of the sequence buil... |
| seqexw 14070 | Weak version of ~ seqex th... |
| seqp1d 14071 | Value of the sequence buil... |
| seqm1 14072 | Value of the sequence buil... |
| seqcl2 14073 | Closure properties of the ... |
| seqf2 14074 | Range of the recursive seq... |
| seqcl 14075 | Closure properties of the ... |
| seqf 14076 | Range of the recursive seq... |
| seqfveq2 14077 | Equality of sequences. (C... |
| seqfeq2 14078 | Equality of sequences. (C... |
| seqfveq 14079 | Equality of sequences. (C... |
| seqfeq 14080 | Equality of sequences. (C... |
| seqshft2 14081 | Shifting the index set of ... |
| seqres 14082 | Restricting its characteri... |
| serf 14083 | An infinite series of comp... |
| serfre 14084 | An infinite series of real... |
| monoord 14085 | Ordering relation for a mo... |
| monoord2 14086 | Ordering relation for a mo... |
| sermono 14087 | The partial sums in an inf... |
| seqsplit 14088 | Split a sequence into two ... |
| seq1p 14089 | Removing the first term fr... |
| seqcaopr3 14090 | Lemma for ~ seqcaopr2 . (... |
| seqcaopr2 14091 | The sum of two infinite se... |
| seqcaopr 14092 | The sum of two infinite se... |
| seqf1olem2a 14093 | Lemma for ~ seqf1o . (Con... |
| seqf1olem1 14094 | Lemma for ~ seqf1o . (Con... |
| seqf1olem2 14095 | Lemma for ~ seqf1o . (Con... |
| seqf1o 14096 | Rearrange a sum via an arb... |
| seradd 14097 | The sum of two infinite se... |
| sersub 14098 | The difference of two infi... |
| seqid3 14099 | A sequence that consists e... |
| seqid 14100 | Discarding the first few t... |
| seqid2 14101 | The last few partial sums ... |
| seqhomo 14102 | Apply a homomorphism to a ... |
| seqz 14103 | If the operation ` .+ ` ha... |
| seqfeq4 14104 | Equality of series under d... |
| seqfeq3 14105 | Equality of series under d... |
| seqdistr 14106 | The distributive property ... |
| ser0 14107 | The value of the partial s... |
| ser0f 14108 | A zero-valued infinite ser... |
| serge0 14109 | A finite sum of nonnegativ... |
| serle 14110 | Comparison of partial sums... |
| ser1const 14111 | Value of the partial serie... |
| seqof 14112 | Distribute function operat... |
| seqof2 14113 | Distribute function operat... |
| expval 14116 | Value of exponentiation to... |
| expnnval 14117 | Value of exponentiation to... |
| exp0 14118 | Value of a complex number ... |
| 0exp0e1 14119 | The zeroth power of zero e... |
| exp1 14120 | Value of a complex number ... |
| expp1 14121 | Value of a complex number ... |
| expneg 14122 | Value of a complex number ... |
| expneg2 14123 | Value of a complex number ... |
| expn1 14124 | A complex number raised to... |
| expcllem 14125 | Lemma for proving nonnegat... |
| expcl2lem 14126 | Lemma for proving integer ... |
| nnexpcl 14127 | Closure of exponentiation ... |
| nn0expcl 14128 | Closure of exponentiation ... |
| zexpcl 14129 | Closure of exponentiation ... |
| qexpcl 14130 | Closure of exponentiation ... |
| reexpcl 14131 | Closure of exponentiation ... |
| expcl 14132 | Closure law for nonnegativ... |
| rpexpcl 14133 | Closure law for integer ex... |
| qexpclz 14134 | Closure of integer exponen... |
| reexpclz 14135 | Closure of integer exponen... |
| expclzlem 14136 | Lemma for ~ expclz . (Con... |
| expclz 14137 | Closure law for integer ex... |
| m1expcl2 14138 | Closure of integer exponen... |
| m1expcl 14139 | Closure of exponentiation ... |
| zexpcld 14140 | Closure of exponentiation ... |
| nn0expcli 14141 | Closure of exponentiation ... |
| nn0sqcl 14142 | The square of a nonnegativ... |
| expm1t 14143 | Exponentiation in terms of... |
| 1exp 14144 | Value of 1 raised to an in... |
| expeq0 14145 | A positive integer power i... |
| expne0 14146 | A positive integer power i... |
| expne0i 14147 | An integer power is nonzer... |
| expgt0 14148 | A positive real raised to ... |
| expnegz 14149 | Value of a nonzero complex... |
| 0exp 14150 | Value of zero raised to a ... |
| expge0 14151 | A nonnegative real raised ... |
| expge1 14152 | A real greater than or equ... |
| expgt1 14153 | A real greater than 1 rais... |
| mulexp 14154 | Nonnegative integer expone... |
| mulexpz 14155 | Integer exponentiation of ... |
| exprec 14156 | Integer exponentiation of ... |
| expadd 14157 | Sum of exponents law for n... |
| expaddzlem 14158 | Lemma for ~ expaddz . (Co... |
| expaddz 14159 | Sum of exponents law for i... |
| expmul 14160 | Product of exponents law f... |
| expmulz 14161 | Product of exponents law f... |
| m1expeven 14162 | Exponentiation of negative... |
| expsub 14163 | Exponent subtraction law f... |
| expp1z 14164 | Value of a nonzero complex... |
| expm1 14165 | Value of a nonzero complex... |
| expdiv 14166 | Nonnegative integer expone... |
| sqval 14167 | Value of the square of a c... |
| sqneg 14168 | The square of the negative... |
| sqsubswap 14169 | Swap the order of subtract... |
| sqcl 14170 | Closure of square. (Contr... |
| sqmul 14171 | Distribution of squaring o... |
| sqeq0 14172 | A complex number is zero i... |
| sqdiv 14173 | Distribution of squaring o... |
| sqdivid 14174 | The square of a nonzero co... |
| sqne0 14175 | A complex number is nonzer... |
| resqcl 14176 | Closure of squaring in rea... |
| resqcld 14177 | Closure of squaring in rea... |
| sqgt0 14178 | The square of a nonzero re... |
| sqn0rp 14179 | The square of a nonzero re... |
| nnsqcl 14180 | The positive naturals are ... |
| zsqcl 14181 | Integers are closed under ... |
| qsqcl 14182 | The square of a rational i... |
| sq11 14183 | The square function is one... |
| nn0sq11 14184 | The square function is one... |
| lt2sq 14185 | The square function is inc... |
| le2sq 14186 | The square function is non... |
| le2sq2 14187 | The square function is non... |
| sqge0 14188 | The square of a real is no... |
| sqge0d 14189 | The square of a real is no... |
| zsqcl2 14190 | The square of an integer i... |
| 0expd 14191 | Value of zero raised to a ... |
| exp0d 14192 | Value of a complex number ... |
| exp1d 14193 | Value of a complex number ... |
| expeq0d 14194 | If a positive integer powe... |
| sqvald 14195 | Value of square. Inferenc... |
| sqcld 14196 | Closure of square. (Contr... |
| sqeq0d 14197 | A number is zero iff its s... |
| expcld 14198 | Closure law for nonnegativ... |
| expp1d 14199 | Value of a complex number ... |
| expaddd 14200 | Sum of exponents law for n... |
| expmuld 14201 | Product of exponents law f... |
| sqrecd 14202 | Square of reciprocal is re... |
| expclzd 14203 | Closure law for integer ex... |
| expne0d 14204 | A nonnegative integer powe... |
| expnegd 14205 | Value of a nonzero complex... |
| exprecd 14206 | An integer power of a reci... |
| expp1zd 14207 | Value of a nonzero complex... |
| expm1d 14208 | Value of a nonzero complex... |
| expsubd 14209 | Exponent subtraction law f... |
| sqmuld 14210 | Distribution of squaring o... |
| sqdivd 14211 | Distribution of squaring o... |
| expdivd 14212 | Nonnegative integer expone... |
| mulexpd 14213 | Nonnegative integer expone... |
| znsqcld 14214 | The square of a nonzero in... |
| reexpcld 14215 | Closure of exponentiation ... |
| expge0d 14216 | A nonnegative real raised ... |
| expge1d 14217 | A real greater than or equ... |
| ltexp2a 14218 | Exponent ordering relation... |
| expmordi 14219 | Base ordering relationship... |
| rpexpmord 14220 | Base ordering relationship... |
| expcan 14221 | Cancellation law for integ... |
| ltexp2 14222 | Strict ordering law for ex... |
| leexp2 14223 | Ordering law for exponenti... |
| leexp2a 14224 | Weak ordering relationship... |
| ltexp2r 14225 | The integer powers of a fi... |
| leexp2r 14226 | Weak ordering relationship... |
| leexp1a 14227 | Weak base ordering relatio... |
| exple1 14228 | A real between 0 and 1 inc... |
| expubnd 14229 | An upper bound on ` A ^ N ... |
| sumsqeq0 14230 | The sum of two squres of r... |
| sqvali 14231 | Value of square. Inferenc... |
| sqcli 14232 | Closure of square. (Contr... |
| sqeq0i 14233 | A complex number is zero i... |
| sqrecii 14234 | The square of a reciprocal... |
| sqmuli 14235 | Distribution of squaring o... |
| sqdivi 14236 | Distribution of squaring o... |
| resqcli 14237 | Closure of square in reals... |
| sqgt0i 14238 | The square of a nonzero re... |
| sqge0i 14239 | The square of a real is no... |
| lt2sqi 14240 | The square function on non... |
| le2sqi 14241 | The square function on non... |
| sq11i 14242 | The square function is one... |
| sq0 14243 | The square of 0 is 0. (Co... |
| sq0i 14244 | If a number is zero, then ... |
| sq0id 14245 | If a number is zero, then ... |
| sq1 14246 | The square of 1 is 1. (Co... |
| neg1sqe1 14247 | The square of ` -u 1 ` is ... |
| sq2 14248 | The square of 2 is 4. (Co... |
| sq3 14249 | The square of 3 is 9. (Co... |
| sq4e2t8 14250 | The square of 4 is 2 times... |
| cu2 14251 | The cube of 2 is 8. (Cont... |
| irec 14252 | The reciprocal of ` _i ` .... |
| i2 14253 | ` _i ` squared. (Contribu... |
| i3 14254 | ` _i ` cubed. (Contribute... |
| i4 14255 | ` _i ` to the fourth power... |
| nnlesq 14256 | A positive integer is less... |
| zzlesq 14257 | An integer is less than or... |
| iexpcyc 14258 | Taking ` _i ` to the ` K `... |
| expnass 14259 | A counterexample showing t... |
| sqlecan 14260 | Cancel one factor of a squ... |
| subsq 14261 | Factor the difference of t... |
| subsq2 14262 | Express the difference of ... |
| binom2i 14263 | The square of a binomial. ... |
| subsqi 14264 | Factor the difference of t... |
| sqeqori 14265 | The squares of two complex... |
| subsq0i 14266 | The two solutions to the d... |
| sqeqor 14267 | The squares of two complex... |
| binom2 14268 | The square of a binomial. ... |
| binom2d 14269 | Deduction form of ~ binom2... |
| binom21 14270 | Special case of ~ binom2 w... |
| binom2sub 14271 | Expand the square of a sub... |
| binom2sub1 14272 | Special case of ~ binom2su... |
| binom2subi 14273 | Expand the square of a sub... |
| mulbinom2 14274 | The square of a binomial w... |
| binom3 14275 | The cube of a binomial. (... |
| sq01 14276 | If a complex number equals... |
| zesq 14277 | An integer is even iff its... |
| nnesq 14278 | A positive integer is even... |
| crreczi 14279 | Reciprocal of a complex nu... |
| bernneq 14280 | Bernoulli's inequality, du... |
| bernneq2 14281 | Variation of Bernoulli's i... |
| bernneq3 14282 | A corollary of ~ bernneq .... |
| expnbnd 14283 | Exponentiation with a base... |
| expnlbnd 14284 | The reciprocal of exponent... |
| expnlbnd2 14285 | The reciprocal of exponent... |
| expmulnbnd 14286 | Exponentiation with a base... |
| digit2 14287 | Two ways to express the ` ... |
| digit1 14288 | Two ways to express the ` ... |
| modexp 14289 | Exponentiation property of... |
| discr1 14290 | A nonnegative quadratic fo... |
| discr 14291 | If a quadratic polynomial ... |
| expnngt1 14292 | If an integer power with a... |
| expnngt1b 14293 | An integer power with an i... |
| sqoddm1div8 14294 | A squared odd number minus... |
| nnsqcld 14295 | The naturals are closed un... |
| nnexpcld 14296 | Closure of exponentiation ... |
| nn0expcld 14297 | Closure of exponentiation ... |
| rpexpcld 14298 | Closure law for exponentia... |
| ltexp2rd 14299 | The power of a positive nu... |
| reexpclzd 14300 | Closure of exponentiation ... |
| sqgt0d 14301 | The square of a nonzero re... |
| ltexp2d 14302 | Ordering relationship for ... |
| leexp2d 14303 | Ordering law for exponenti... |
| expcand 14304 | Ordering relationship for ... |
| leexp2ad 14305 | Ordering relationship for ... |
| leexp2rd 14306 | Ordering relationship for ... |
| lt2sqd 14307 | The square function on non... |
| le2sqd 14308 | The square function on non... |
| sq11d 14309 | The square function is one... |
| ltexp1d 14310 | Elevating to a positive po... |
| ltexp1dd 14311 | Raising both sides of 'les... |
| exp11nnd 14312 | The function elevating non... |
| mulsubdivbinom2 14313 | The square of a binomial w... |
| muldivbinom2 14314 | The square of a binomial w... |
| sq10 14315 | The square of 10 is 100. ... |
| sq10e99m1 14316 | The square of 10 is 99 plu... |
| 3dec 14317 | A "decimal constructor" wh... |
| nn0le2msqi 14318 | The square function on non... |
| nn0opthlem1 14319 | A rather pretty lemma for ... |
| nn0opthlem2 14320 | Lemma for ~ nn0opthi . (C... |
| nn0opthi 14321 | An ordered pair theorem fo... |
| nn0opth2i 14322 | An ordered pair theorem fo... |
| nn0opth2 14323 | An ordered pair theorem fo... |
| facnn 14326 | Value of the factorial fun... |
| fac0 14327 | The factorial of 0. (Cont... |
| fac1 14328 | The factorial of 1. (Cont... |
| facp1 14329 | The factorial of a success... |
| fac2 14330 | The factorial of 2. (Cont... |
| fac3 14331 | The factorial of 3. (Cont... |
| fac4 14332 | The factorial of 4. (Cont... |
| facnn2 14333 | Value of the factorial fun... |
| faccl 14334 | Closure of the factorial f... |
| faccld 14335 | Closure of the factorial f... |
| facmapnn 14336 | The factorial function res... |
| facne0 14337 | The factorial function is ... |
| facdiv 14338 | A positive integer divides... |
| facndiv 14339 | No positive integer (great... |
| facwordi 14340 | Ordering property of facto... |
| faclbnd 14341 | A lower bound for the fact... |
| faclbnd2 14342 | A lower bound for the fact... |
| faclbnd3 14343 | A lower bound for the fact... |
| faclbnd4lem1 14344 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4lem2 14345 | Lemma for ~ faclbnd4 . Us... |
| faclbnd4lem3 14346 | Lemma for ~ faclbnd4 . Th... |
| faclbnd4lem4 14347 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4 14348 | Variant of ~ faclbnd5 prov... |
| faclbnd5 14349 | The factorial function gro... |
| faclbnd6 14350 | Geometric lower bound for ... |
| facubnd 14351 | An upper bound for the fac... |
| facavg 14352 | The product of two factori... |
| bcval 14355 | Value of the binomial coef... |
| bcval2 14356 | Value of the binomial coef... |
| bcval3 14357 | Value of the binomial coef... |
| bcval4 14358 | Value of the binomial coef... |
| bcrpcl 14359 | Closure of the binomial co... |
| bccmpl 14360 | "Complementing" its second... |
| bcn0 14361 | ` N ` choose 0 is 1. Rema... |
| bc0k 14362 | The binomial coefficient "... |
| bcnn 14363 | ` N ` choose ` N ` is 1. ... |
| bcn1 14364 | Binomial coefficient: ` N ... |
| bcnp1n 14365 | Binomial coefficient: ` N ... |
| bcm1k 14366 | The proportion of one bino... |
| bcp1n 14367 | The proportion of one bino... |
| bcp1nk 14368 | The proportion of one bino... |
| bcval5 14369 | Write out the top and bott... |
| bcn2 14370 | Binomial coefficient: ` N ... |
| bcp1m1 14371 | Compute the binomial coeff... |
| bcpasc 14372 | Pascal's rule for the bino... |
| bccl 14373 | A binomial coefficient, in... |
| bccl2 14374 | A binomial coefficient, in... |
| bcn2m1 14375 | Compute the binomial coeff... |
| bcn2p1 14376 | Compute the binomial coeff... |
| permnn 14377 | The number of permutations... |
| bcnm1 14378 | The binomial coefficient o... |
| 4bc3eq4 14379 | The value of four choose t... |
| 4bc2eq6 14380 | The value of four choose t... |
| hashkf 14383 | The finite part of the siz... |
| hashgval 14384 | The value of the ` # ` fun... |
| hashginv 14385 | The converse of ` G ` maps... |
| hashinf 14386 | The value of the ` # ` fun... |
| hashbnd 14387 | If ` A ` has size bounded ... |
| hashfxnn0 14388 | The size function is a fun... |
| hashf 14389 | The size function maps all... |
| hashxnn0 14390 | The value of the hash func... |
| hashresfn 14391 | Restriction of the domain ... |
| dmhashres 14392 | Restriction of the domain ... |
| hashnn0pnf 14393 | The value of the hash func... |
| hashnnn0genn0 14394 | If the size of a set is no... |
| hashnemnf 14395 | The size of a set is never... |
| hashv01gt1 14396 | The size of a set is eithe... |
| hashfz1 14397 | The set ` ( 1 ... N ) ` ha... |
| hashen 14398 | Two finite sets have the s... |
| hasheni 14399 | Equinumerous sets have the... |
| hasheqf1o 14400 | The size of two finite set... |
| fiinfnf1o 14401 | There is no bijection betw... |
| hasheqf1oi 14402 | The size of two sets is eq... |
| hashf1rn 14403 | The size of a finite set w... |
| hasheqf1od 14404 | The size of two sets is eq... |
| fz1eqb 14405 | Two possibly-empty 1-based... |
| hashcard 14406 | The size function of the c... |
| hashcl 14407 | Closure of the ` # ` funct... |
| hashxrcl 14408 | Extended real closure of t... |
| hashclb 14409 | Reverse closure of the ` #... |
| nfile 14410 | The size of any infinite s... |
| hashvnfin 14411 | A set of finite size is a ... |
| hashnfinnn0 14412 | The size of an infinite se... |
| isfinite4 14413 | A finite set is equinumero... |
| hasheq0 14414 | Two ways of saying a set i... |
| hashneq0 14415 | Two ways of saying a set i... |
| hashgt0n0 14416 | If the size of a set is gr... |
| hashnncl 14417 | Positive natural closure o... |
| hash0 14418 | The empty set has size zer... |
| hashelne0d 14419 | A set with an element has ... |
| hashsng 14420 | The size of a singleton. ... |
| hashen1 14421 | A set has size 1 if and on... |
| hash1elsn 14422 | A set of size 1 with a kno... |
| hashrabrsn 14423 | The size of a restricted c... |
| hashrabsn01 14424 | The size of a restricted c... |
| hashrabsn1 14425 | If the size of a restricte... |
| hashfn 14426 | A function is equinumerous... |
| fseq1hash 14427 | The value of the size func... |
| hashgadd 14428 | ` G ` maps ordinal additio... |
| hashgval2 14429 | A short expression for the... |
| hashdom 14430 | Dominance relation for the... |
| hashdomi 14431 | Non-strict order relation ... |
| hashsdom 14432 | Strict dominance relation ... |
| hashun 14433 | The size of the union of d... |
| hashun2 14434 | The size of the union of f... |
| hashun3 14435 | The size of the union of f... |
| hashinfxadd 14436 | The extended real addition... |
| hashunx 14437 | The size of the union of d... |
| hashge0 14438 | The cardinality of a set i... |
| hashgt0 14439 | The cardinality of a nonem... |
| hashge1 14440 | The cardinality of a nonem... |
| 1elfz0hash 14441 | 1 is an element of the fin... |
| hashnn0n0nn 14442 | If a nonnegative integer i... |
| hashunsng 14443 | The size of the union of a... |
| hashunsngx 14444 | The size of the union of a... |
| hashunsnggt 14445 | The size of a set is great... |
| hashprg 14446 | The size of an unordered p... |
| elprchashprn2 14447 | If one element of an unord... |
| hashprb 14448 | The size of an unordered p... |
| hashprdifel 14449 | The elements of an unorder... |
| prhash2ex 14450 | There is (at least) one se... |
| hashle00 14451 | If the size of a set is le... |
| hashgt0elex 14452 | If the size of a set is gr... |
| hashgt0elexb 14453 | The size of a set is great... |
| hashp1i 14454 | Size of a finite ordinal. ... |
| hash1 14455 | Size of a finite ordinal. ... |
| hash2 14456 | Size of a finite ordinal. ... |
| hash3 14457 | Size of a finite ordinal. ... |
| hash4 14458 | Size of a finite ordinal. ... |
| pr0hash2ex 14459 | There is (at least) one se... |
| hashss 14460 | The size of a subset is le... |
| prsshashgt1 14461 | The size of a superset of ... |
| hashin 14462 | The size of the intersecti... |
| hashssdif 14463 | The size of the difference... |
| hashdif 14464 | The size of the difference... |
| hashdifsn 14465 | The size of the difference... |
| hashdifpr 14466 | The size of the difference... |
| hashsn01 14467 | The size of a singleton is... |
| hashsnle1 14468 | The size of a singleton is... |
| hashsnlei 14469 | Get an upper bound on a co... |
| hash1snb 14470 | The size of a set is 1 if ... |
| euhash1 14471 | The size of a set is 1 in ... |
| hash1n0 14472 | If the size of a set is 1 ... |
| hashgt12el 14473 | In a set with more than on... |
| hashgt12el2 14474 | In a set with more than on... |
| hashgt23el 14475 | A set with more than two e... |
| hashunlei 14476 | Get an upper bound on a co... |
| hashsslei 14477 | Get an upper bound on a co... |
| hashfz 14478 | Value of the numeric cardi... |
| fzsdom2 14479 | Condition for finite range... |
| hashfzo 14480 | Cardinality of a half-open... |
| hashfzo0 14481 | Cardinality of a half-open... |
| hashfzp1 14482 | Value of the numeric cardi... |
| hashfz0 14483 | Value of the numeric cardi... |
| hashxplem 14484 | Lemma for ~ hashxp . (Con... |
| hashxp 14485 | The size of the Cartesian ... |
| hashmap 14486 | The size of the set expone... |
| hashpw 14487 | The size of the power set ... |
| hashfun 14488 | A finite set is a function... |
| hashres 14489 | The number of elements of ... |
| hashreshashfun 14490 | The number of elements of ... |
| hashimarn 14491 | The size of the image of a... |
| hashimarni 14492 | If the size of the image o... |
| hashfundm 14493 | The size of a set function... |
| hashf1dmrn 14494 | The size of the domain of ... |
| hashf1dmcdm 14495 | The size of the domain of ... |
| resunimafz0 14496 | TODO-AV: Revise using ` F... |
| fnfz0hash 14497 | The size of a function on ... |
| ffz0hash 14498 | The size of a function on ... |
| fnfz0hashnn0 14499 | The size of a function on ... |
| ffzo0hash 14500 | The size of a function on ... |
| fnfzo0hash 14501 | The size of a function on ... |
| fnfzo0hashnn0 14502 | The value of the size func... |
| hashbclem 14503 | Lemma for ~ hashbc : induc... |
| hashbc 14504 | The binomial coefficient c... |
| hashfacen 14505 | The number of bijections b... |
| hashf1lem1 14506 | Lemma for ~ hashf1 . (Con... |
| hashf1lem2 14507 | Lemma for ~ hashf1 . (Con... |
| hashf1 14508 | The permutation number ` |... |
| hashfac 14509 | A factorial counts the num... |
| leiso 14510 | Two ways to write a strict... |
| leisorel 14511 | Version of ~ isorel for st... |
| fz1isolem 14512 | Lemma for ~ fz1iso . (Con... |
| fz1iso 14513 | Any finite ordered set has... |
| ishashinf 14514 | Any set that is not finite... |
| seqcoll 14515 | The function ` F ` contain... |
| seqcoll2 14516 | The function ` F ` contain... |
| phphashd 14517 | Corollary of the Pigeonhol... |
| phphashrd 14518 | Corollary of the Pigeonhol... |
| hashprlei 14519 | An unordered pair has at m... |
| hash2pr 14520 | A set of size two is an un... |
| hash2prde 14521 | A set of size two is an un... |
| hash2exprb 14522 | A set of size two is an un... |
| hash2prb 14523 | A set of size two is a pro... |
| prprrab 14524 | The set of proper pairs of... |
| nehash2 14525 | The cardinality of a set w... |
| hash2prd 14526 | A set of size two is an un... |
| hash2pwpr 14527 | If the size of a subset of... |
| hashle2pr 14528 | A nonempty set of size les... |
| hashle2prv 14529 | A nonempty subset of a pow... |
| pr2pwpr 14530 | The set of subsets of a pa... |
| hashge2el2dif 14531 | A set with size at least 2... |
| hashge2el2difr 14532 | A set with at least 2 diff... |
| hashge2el2difb 14533 | A set has size at least 2 ... |
| hashdmpropge2 14534 | The size of the domain of ... |
| hashtplei 14535 | An unordered triple has at... |
| hashtpg 14536 | The size of an unordered t... |
| hash7g 14537 | The size of an unordered s... |
| hashge3el3dif 14538 | A set with size at least 3... |
| elss2prb 14539 | An element of the set of s... |
| hash2sspr 14540 | A subset of size two is an... |
| exprelprel 14541 | If there is an element of ... |
| hash3tr 14542 | A set of size three is an ... |
| hash1to3 14543 | If the size of a set is be... |
| hash3tpde 14544 | A set of size three is an ... |
| hash3tpexb 14545 | A set of size three is an ... |
| hash3tpb 14546 | A set of size three is a p... |
| tpf1ofv0 14547 | The value of a one-to-one ... |
| tpf1ofv1 14548 | The value of a one-to-one ... |
| tpf1ofv2 14549 | The value of a one-to-one ... |
| tpf 14550 | A function into a (proper)... |
| tpfo 14551 | A function onto a (proper)... |
| tpf1o 14552 | A bijection onto a (proper... |
| fundmge2nop0 14553 | A function with a domain c... |
| fundmge2nop 14554 | A function with a domain c... |
| fun2dmnop0 14555 | A function with a domain c... |
| fun2dmnop 14556 | A function with a domain c... |
| hashdifsnp1 14557 | If the size of a set is a ... |
| fi1uzind 14558 | Properties of an ordered p... |
| brfi1uzind 14559 | Properties of a binary rel... |
| brfi1ind 14560 | Properties of a binary rel... |
| brfi1indALT 14561 | Alternate proof of ~ brfi1... |
| opfi1uzind 14562 | Properties of an ordered p... |
| opfi1ind 14563 | Properties of an ordered p... |
| iswrd 14566 | Property of being a word o... |
| wrdval 14567 | Value of the set of words ... |
| iswrdi 14568 | A zero-based sequence is a... |
| wrdf 14569 | A word is a zero-based seq... |
| iswrdb 14570 | A word over an alphabet is... |
| wrddm 14571 | The indices of a word (i.e... |
| sswrd 14572 | The set of words respects ... |
| snopiswrd 14573 | A singleton of an ordered ... |
| wrdexg 14574 | The set of words over a se... |
| wrdexb 14575 | The set of words over a se... |
| wrdexi 14576 | The set of words over a se... |
| wrdsymbcl 14577 | A symbol within a word ove... |
| wrdfn 14578 | A word is a function with ... |
| wrdv 14579 | A word over an alphabet is... |
| wrdlndm 14580 | The length of a word is no... |
| iswrdsymb 14581 | An arbitrary word is a wor... |
| wrdfin 14582 | A word is a finite set. (... |
| lencl 14583 | The length of a word is a ... |
| lennncl 14584 | The length of a nonempty w... |
| wrdffz 14585 | A word is a function from ... |
| wrdeq 14586 | Equality theorem for the s... |
| wrdeqi 14587 | Equality theorem for the s... |
| iswrddm0 14588 | A function with empty doma... |
| wrd0 14589 | The empty set is a word (t... |
| 0wrd0 14590 | The empty word is the only... |
| ffz0iswrd 14591 | A sequence with zero-based... |
| wrdsymb 14592 | A word is a word over the ... |
| nfwrd 14593 | Hypothesis builder for ` W... |
| csbwrdg 14594 | Class substitution for the... |
| wrdnval 14595 | Words of a fixed length ar... |
| wrdmap 14596 | Words as a mapping. (Cont... |
| hashwrdn 14597 | If there is only a finite ... |
| wrdnfi 14598 | If there is only a finite ... |
| wrdsymb0 14599 | A symbol at a position "ou... |
| wrdlenge1n0 14600 | A word with length at leas... |
| len0nnbi 14601 | The length of a word is a ... |
| wrdlenge2n0 14602 | A word with length at leas... |
| wrdsymb1 14603 | The first symbol of a none... |
| wrdlen1 14604 | A word of length 1 starts ... |
| fstwrdne 14605 | The first symbol of a none... |
| fstwrdne0 14606 | The first symbol of a none... |
| eqwrd 14607 | Two words are equal iff th... |
| elovmpowrd 14608 | Implications for the value... |
| elovmptnn0wrd 14609 | Implications for the value... |
| wrdred1 14610 | A word truncated by a symb... |
| wrdred1hash 14611 | The length of a word trunc... |
| lsw 14614 | Extract the last symbol of... |
| lsw0 14615 | The last symbol of an empt... |
| lsw0g 14616 | The last symbol of an empt... |
| lsw1 14617 | The last symbol of a word ... |
| lswcl 14618 | Closure of the last symbol... |
| lswlgt0cl 14619 | The last symbol of a nonem... |
| ccatfn 14622 | The concatenation operator... |
| ccatfval 14623 | Value of the concatenation... |
| ccatcl 14624 | The concatenation of two w... |
| ccatlen 14625 | The length of a concatenat... |
| ccat0 14626 | The concatenation of two w... |
| ccatval1 14627 | Value of a symbol in the l... |
| ccatval2 14628 | Value of a symbol in the r... |
| ccatval3 14629 | Value of a symbol in the r... |
| elfzelfzccat 14630 | An element of a finite set... |
| ccatvalfn 14631 | The concatenation of two w... |
| ccatsymb 14632 | The symbol at a given posi... |
| ccatfv0 14633 | The first symbol of a conc... |
| ccatval1lsw 14634 | The last symbol of the lef... |
| ccatval21sw 14635 | The first symbol of the ri... |
| ccatlid 14636 | Concatenation of a word by... |
| ccatrid 14637 | Concatenation of a word by... |
| ccatass 14638 | Associative law for concat... |
| ccatrn 14639 | The range of a concatenate... |
| ccatidid 14640 | Concatenation of the empty... |
| lswccatn0lsw 14641 | The last symbol of a word ... |
| lswccat0lsw 14642 | The last symbol of a word ... |
| ccatalpha 14643 | A concatenation of two arb... |
| ccatrcl1 14644 | Reverse closure of a conca... |
| ids1 14647 | Identity function protecti... |
| s1val 14648 | Value of a singleton word.... |
| s1rn 14649 | The range of a singleton w... |
| s1eq 14650 | Equality theorem for a sin... |
| s1eqd 14651 | Equality theorem for a sin... |
| s1cl 14652 | A singleton word is a word... |
| s1cld 14653 | A singleton word is a word... |
| s1prc 14654 | Value of a singleton word ... |
| s1cli 14655 | A singleton word is a word... |
| s1len 14656 | Length of a singleton word... |
| s1nz 14657 | A singleton word is not th... |
| s1dm 14658 | The domain of a singleton ... |
| s1dmALT 14659 | Alternate version of ~ s1d... |
| s1fv 14660 | Sole symbol of a singleton... |
| lsws1 14661 | The last symbol of a singl... |
| eqs1 14662 | A word of length 1 is a si... |
| wrdl1exs1 14663 | A word of length 1 is a si... |
| wrdl1s1 14664 | A word of length 1 is a si... |
| s111 14665 | The singleton word functio... |
| ccatws1cl 14666 | The concatenation of a wor... |
| ccatws1clv 14667 | The concatenation of a wor... |
| ccat2s1cl 14668 | The concatenation of two s... |
| ccats1alpha 14669 | A concatenation of a word ... |
| ccatws1len 14670 | The length of the concaten... |
| ccatws1lenp1b 14671 | The length of a word is ` ... |
| wrdlenccats1lenm1 14672 | The length of a word is th... |
| ccat2s1len 14673 | The length of the concaten... |
| ccatw2s1cl 14674 | The concatenation of a wor... |
| ccatw2s1len 14675 | The length of the concaten... |
| ccats1val1 14676 | Value of a symbol in the l... |
| ccats1val2 14677 | Value of the symbol concat... |
| ccat1st1st 14678 | The first symbol of a word... |
| ccat2s1p1 14679 | Extract the first of two c... |
| ccat2s1p2 14680 | Extract the second of two ... |
| ccatw2s1ass 14681 | Associative law for a conc... |
| ccatws1n0 14682 | The concatenation of a wor... |
| ccatws1ls 14683 | The last symbol of the con... |
| lswccats1 14684 | The last symbol of a word ... |
| lswccats1fst 14685 | The last symbol of a nonem... |
| ccatw2s1p1 14686 | Extract the symbol of the ... |
| ccatw2s1p2 14687 | Extract the second of two ... |
| ccat2s1fvw 14688 | Extract a symbol of a word... |
| ccat2s1fst 14689 | The first symbol of the co... |
| swrdnznd 14692 | The value of a subword ope... |
| swrdval 14693 | Value of a subword. (Cont... |
| swrd00 14694 | A zero length substring. ... |
| swrdcl 14695 | Closure of the subword ext... |
| swrdval2 14696 | Value of the subword extra... |
| swrdlen 14697 | Length of an extracted sub... |
| swrdfv 14698 | A symbol in an extracted s... |
| swrdfv0 14699 | The first symbol in an ext... |
| swrdf 14700 | A subword of a word is a f... |
| swrdvalfn 14701 | Value of the subword extra... |
| swrdrn 14702 | The range of a subword of ... |
| swrdlend 14703 | The value of the subword e... |
| swrdnd 14704 | The value of the subword e... |
| swrdnd2 14705 | Value of the subword extra... |
| swrdnnn0nd 14706 | The value of a subword ope... |
| swrdnd0 14707 | The value of a subword ope... |
| swrd0 14708 | A subword of an empty set ... |
| swrdrlen 14709 | Length of a right-anchored... |
| swrdlen2 14710 | Length of an extracted sub... |
| swrdfv2 14711 | A symbol in an extracted s... |
| swrdwrdsymb 14712 | A subword is a word over t... |
| swrdsb0eq 14713 | Two subwords with the same... |
| swrdsbslen 14714 | Two subwords with the same... |
| swrdspsleq 14715 | Two words have a common su... |
| swrds1 14716 | Extract a single symbol fr... |
| swrdlsw 14717 | Extract the last single sy... |
| ccatswrd 14718 | Joining two adjacent subwo... |
| swrdccat2 14719 | Recover the right half of ... |
| pfxnndmnd 14722 | The value of a prefix oper... |
| pfxval 14723 | Value of a prefix operatio... |
| pfx00 14724 | The zero length prefix is ... |
| pfx0 14725 | A prefix of an empty set i... |
| pfxval0 14726 | Value of a prefix operatio... |
| pfxcl 14727 | Closure of the prefix extr... |
| pfxmpt 14728 | Value of the prefix extrac... |
| pfxres 14729 | Value of the subword extra... |
| pfxf 14730 | A prefix of a word is a fu... |
| pfxfn 14731 | Value of the prefix extrac... |
| pfxfv 14732 | A symbol in a prefix of a ... |
| pfxlen 14733 | Length of a prefix. (Cont... |
| pfxid 14734 | A word is a prefix of itse... |
| pfxrn 14735 | The range of a prefix of a... |
| pfxn0 14736 | A prefix consisting of at ... |
| pfxnd 14737 | The value of a prefix oper... |
| pfxnd0 14738 | The value of a prefix oper... |
| pfxwrdsymb 14739 | A prefix of a word is a wo... |
| addlenrevpfx 14740 | The sum of the lengths of ... |
| addlenpfx 14741 | The sum of the lengths of ... |
| pfxfv0 14742 | The first symbol of a pref... |
| pfxtrcfv 14743 | A symbol in a word truncat... |
| pfxtrcfv0 14744 | The first symbol in a word... |
| pfxfvlsw 14745 | The last symbol in a nonem... |
| pfxeq 14746 | The prefixes of two words ... |
| pfxtrcfvl 14747 | The last symbol in a word ... |
| pfxsuffeqwrdeq 14748 | Two words are equal if and... |
| pfxsuff1eqwrdeq 14749 | Two (nonempty) words are e... |
| disjwrdpfx 14750 | Sets of words are disjoint... |
| ccatpfx 14751 | Concatenating a prefix wit... |
| pfxccat1 14752 | Recover the left half of a... |
| pfx1 14753 | The prefix of length one o... |
| swrdswrdlem 14754 | Lemma for ~ swrdswrd . (C... |
| swrdswrd 14755 | A subword of a subword is ... |
| pfxswrd 14756 | A prefix of a subword is a... |
| swrdpfx 14757 | A subword of a prefix is a... |
| pfxpfx 14758 | A prefix of a prefix is a ... |
| pfxpfxid 14759 | A prefix of a prefix with ... |
| pfxcctswrd 14760 | The concatenation of the p... |
| lenpfxcctswrd 14761 | The length of the concaten... |
| lenrevpfxcctswrd 14762 | The length of the concaten... |
| pfxlswccat 14763 | Reconstruct a nonempty wor... |
| ccats1pfxeq 14764 | The last symbol of a word ... |
| ccats1pfxeqrex 14765 | There exists a symbol such... |
| ccatopth 14766 | An ~ opth -like theorem fo... |
| ccatopth2 14767 | An ~ opth -like theorem fo... |
| ccatlcan 14768 | Concatenation of words is ... |
| ccatrcan 14769 | Concatenation of words is ... |
| wrdeqs1cat 14770 | Decompose a nonempty word ... |
| cats1un 14771 | Express a word with an ext... |
| wrdind 14772 | Perform induction over the... |
| wrd2ind 14773 | Perform induction over the... |
| swrdccatfn 14774 | The subword of a concatena... |
| swrdccatin1 14775 | The subword of a concatena... |
| pfxccatin12lem4 14776 | Lemma 4 for ~ pfxccatin12 ... |
| pfxccatin12lem2a 14777 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem1 14778 | Lemma 1 for ~ pfxccatin12 ... |
| swrdccatin2 14779 | The subword of a concatena... |
| pfxccatin12lem2c 14780 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem2 14781 | Lemma 2 for ~ pfxccatin12 ... |
| pfxccatin12lem3 14782 | Lemma 3 for ~ pfxccatin12 ... |
| pfxccatin12 14783 | The subword of a concatena... |
| pfxccat3 14784 | The subword of a concatena... |
| swrdccat 14785 | The subword of a concatena... |
| pfxccatpfx1 14786 | A prefix of a concatenatio... |
| pfxccatpfx2 14787 | A prefix of a concatenatio... |
| pfxccat3a 14788 | A prefix of a concatenatio... |
| swrdccat3blem 14789 | Lemma for ~ swrdccat3b . ... |
| swrdccat3b 14790 | A suffix of a concatenatio... |
| pfxccatid 14791 | A prefix of a concatenatio... |
| ccats1pfxeqbi 14792 | A word is a prefix of a wo... |
| swrdccatin1d 14793 | The subword of a concatena... |
| swrdccatin2d 14794 | The subword of a concatena... |
| pfxccatin12d 14795 | The subword of a concatena... |
| reuccatpfxs1lem 14796 | Lemma for ~ reuccatpfxs1 .... |
| reuccatpfxs1 14797 | There is a unique word hav... |
| reuccatpfxs1v 14798 | There is a unique word hav... |
| splval 14801 | Value of the substring rep... |
| splcl 14802 | Closure of the substring r... |
| splid 14803 | Splicing a subword for the... |
| spllen 14804 | The length of a splice. (... |
| splfv1 14805 | Symbols to the left of a s... |
| splfv2a 14806 | Symbols within the replace... |
| splval2 14807 | Value of a splice, assumin... |
| revval 14810 | Value of the word reversin... |
| revcl 14811 | The reverse of a word is a... |
| revlen 14812 | The reverse of a word has ... |
| revfv 14813 | Reverse of a word at a poi... |
| rev0 14814 | The empty word is its own ... |
| revs1 14815 | Singleton words are their ... |
| revccat 14816 | Antiautomorphic property o... |
| revrev 14817 | Reversal is an involution ... |
| reps 14820 | Construct a function mappi... |
| repsundef 14821 | A function mapping a half-... |
| repsconst 14822 | Construct a function mappi... |
| repsf 14823 | The constructed function m... |
| repswsymb 14824 | The symbols of a "repeated... |
| repsw 14825 | A function mapping a half-... |
| repswlen 14826 | The length of a "repeated ... |
| repsw0 14827 | The "repeated symbol word"... |
| repsdf2 14828 | Alternative definition of ... |
| repswsymball 14829 | All the symbols of a "repe... |
| repswsymballbi 14830 | A word is a "repeated symb... |
| repswfsts 14831 | The first symbol of a none... |
| repswlsw 14832 | The last symbol of a nonem... |
| repsw1 14833 | The "repeated symbol word"... |
| repswswrd 14834 | A subword of a "repeated s... |
| repswpfx 14835 | A prefix of a repeated sym... |
| repswccat 14836 | The concatenation of two "... |
| repswrevw 14837 | The reverse of a "repeated... |
| cshfn 14840 | Perform a cyclical shift f... |
| cshword 14841 | Perform a cyclical shift f... |
| cshnz 14842 | A cyclical shift is the em... |
| 0csh0 14843 | Cyclically shifting an emp... |
| cshw0 14844 | A word cyclically shifted ... |
| cshwmodn 14845 | Cyclically shifting a word... |
| cshwsublen 14846 | Cyclically shifting a word... |
| cshwn 14847 | A word cyclically shifted ... |
| cshwcl 14848 | A cyclically shifted word ... |
| cshwlen 14849 | The length of a cyclically... |
| cshwf 14850 | A cyclically shifted word ... |
| cshwfn 14851 | A cyclically shifted word ... |
| cshwrn 14852 | The range of a cyclically ... |
| cshwidxmod 14853 | The symbol at a given inde... |
| cshwidxmodr 14854 | The symbol at a given inde... |
| cshwidx0mod 14855 | The symbol at index 0 of a... |
| cshwidx0 14856 | The symbol at index 0 of a... |
| cshwidxm1 14857 | The symbol at index ((n-N)... |
| cshwidxm 14858 | The symbol at index (n-N) ... |
| cshwidxn 14859 | The symbol at index (n-1) ... |
| cshf1 14860 | Cyclically shifting a word... |
| cshinj 14861 | If a word is injectiv (reg... |
| repswcshw 14862 | A cyclically shifted "repe... |
| 2cshw 14863 | Cyclically shifting a word... |
| 2cshwid 14864 | Cyclically shifting a word... |
| lswcshw 14865 | The last symbol of a word ... |
| 2cshwcom 14866 | Cyclically shifting a word... |
| cshwleneq 14867 | If the results of cyclical... |
| 3cshw 14868 | Cyclically shifting a word... |
| cshweqdif2 14869 | If cyclically shifting two... |
| cshweqdifid 14870 | If cyclically shifting a w... |
| cshweqrep 14871 | If cyclically shifting a w... |
| cshw1 14872 | If cyclically shifting a w... |
| cshw1repsw 14873 | If cyclically shifting a w... |
| cshwsexa 14874 | The class of (different!) ... |
| cshwsexaOLD 14875 | Obsolete version of ~ cshw... |
| 2cshwcshw 14876 | If a word is a cyclically ... |
| scshwfzeqfzo 14877 | For a nonempty word the se... |
| cshwcshid 14878 | A cyclically shifted word ... |
| cshwcsh2id 14879 | A cyclically shifted word ... |
| cshimadifsn 14880 | The image of a cyclically ... |
| cshimadifsn0 14881 | The image of a cyclically ... |
| wrdco 14882 | Mapping a word by a functi... |
| lenco 14883 | Length of a mapped word is... |
| s1co 14884 | Mapping of a singleton wor... |
| revco 14885 | Mapping of words (i.e., a ... |
| ccatco 14886 | Mapping of words commutes ... |
| cshco 14887 | Mapping of words commutes ... |
| swrdco 14888 | Mapping of words commutes ... |
| pfxco 14889 | Mapping of words commutes ... |
| lswco 14890 | Mapping of (nonempty) word... |
| repsco 14891 | Mapping of words commutes ... |
| cats1cld 14906 | Closure of concatenation w... |
| cats1co 14907 | Closure of concatenation w... |
| cats1cli 14908 | Closure of concatenation w... |
| cats1fvn 14909 | The last symbol of a conca... |
| cats1fv 14910 | A symbol other than the la... |
| cats1len 14911 | The length of concatenatio... |
| cats1cat 14912 | Closure of concatenation w... |
| cats2cat 14913 | Closure of concatenation o... |
| s2eqd 14914 | Equality theorem for a dou... |
| s3eqd 14915 | Equality theorem for a len... |
| s4eqd 14916 | Equality theorem for a len... |
| s5eqd 14917 | Equality theorem for a len... |
| s6eqd 14918 | Equality theorem for a len... |
| s7eqd 14919 | Equality theorem for a len... |
| s8eqd 14920 | Equality theorem for a len... |
| s3eq2 14921 | Equality theorem for a len... |
| s2cld 14922 | A doubleton word is a word... |
| s3cld 14923 | A length 3 string is a wor... |
| s4cld 14924 | A length 4 string is a wor... |
| s5cld 14925 | A length 5 string is a wor... |
| s6cld 14926 | A length 6 string is a wor... |
| s7cld 14927 | A length 7 string is a wor... |
| s8cld 14928 | A length 7 string is a wor... |
| s2cl 14929 | A doubleton word is a word... |
| s3cl 14930 | A length 3 string is a wor... |
| s2cli 14931 | A doubleton word is a word... |
| s3cli 14932 | A length 3 string is a wor... |
| s4cli 14933 | A length 4 string is a wor... |
| s5cli 14934 | A length 5 string is a wor... |
| s6cli 14935 | A length 6 string is a wor... |
| s7cli 14936 | A length 7 string is a wor... |
| s8cli 14937 | A length 8 string is a wor... |
| s2fv0 14938 | Extract the first symbol f... |
| s2fv1 14939 | Extract the second symbol ... |
| s2len 14940 | The length of a doubleton ... |
| s2dm 14941 | The domain of a doubleton ... |
| s3fv0 14942 | Extract the first symbol f... |
| s3fv1 14943 | Extract the second symbol ... |
| s3fv2 14944 | Extract the third symbol f... |
| s3len 14945 | The length of a length 3 s... |
| s4fv0 14946 | Extract the first symbol f... |
| s4fv1 14947 | Extract the second symbol ... |
| s4fv2 14948 | Extract the third symbol f... |
| s4fv3 14949 | Extract the fourth symbol ... |
| s4len 14950 | The length of a length 4 s... |
| s5len 14951 | The length of a length 5 s... |
| s6len 14952 | The length of a length 6 s... |
| s7len 14953 | The length of a length 7 s... |
| s8len 14954 | The length of a length 8 s... |
| lsws2 14955 | The last symbol of a doubl... |
| lsws3 14956 | The last symbol of a 3 let... |
| lsws4 14957 | The last symbol of a 4 let... |
| s2prop 14958 | A length 2 word is an unor... |
| s2dmALT 14959 | Alternate version of ~ s2d... |
| s3tpop 14960 | A length 3 word is an unor... |
| s4prop 14961 | A length 4 word is a union... |
| s3fn 14962 | A length 3 word is a funct... |
| funcnvs1 14963 | The converse of a singleto... |
| funcnvs2 14964 | The converse of a length 2... |
| funcnvs3 14965 | The converse of a length 3... |
| funcnvs4 14966 | The converse of a length 4... |
| s2f1o 14967 | A length 2 word with mutua... |
| f1oun2prg 14968 | A union of unordered pairs... |
| s4f1o 14969 | A length 4 word with mutua... |
| s4dom 14970 | The domain of a length 4 w... |
| s2co 14971 | Mapping a doubleton word b... |
| s3co 14972 | Mapping a length 3 string ... |
| s0s1 14973 | Concatenation of fixed len... |
| s1s2 14974 | Concatenation of fixed len... |
| s1s3 14975 | Concatenation of fixed len... |
| s1s4 14976 | Concatenation of fixed len... |
| s1s5 14977 | Concatenation of fixed len... |
| s1s6 14978 | Concatenation of fixed len... |
| s1s7 14979 | Concatenation of fixed len... |
| s2s2 14980 | Concatenation of fixed len... |
| s4s2 14981 | Concatenation of fixed len... |
| s4s3 14982 | Concatenation of fixed len... |
| s4s4 14983 | Concatenation of fixed len... |
| s3s4 14984 | Concatenation of fixed len... |
| s2s5 14985 | Concatenation of fixed len... |
| s5s2 14986 | Concatenation of fixed len... |
| s2eq2s1eq 14987 | Two length 2 words are equ... |
| s2eq2seq 14988 | Two length 2 words are equ... |
| s3eqs2s1eq 14989 | Two length 3 words are equ... |
| s3eq3seq 14990 | Two length 3 words are equ... |
| swrds2 14991 | Extract two adjacent symbo... |
| swrds2m 14992 | Extract two adjacent symbo... |
| wrdlen2i 14993 | Implications of a word of ... |
| wrd2pr2op 14994 | A word of length two repre... |
| wrdlen2 14995 | A word of length two. (Co... |
| wrdlen2s2 14996 | A word of length two as do... |
| wrdl2exs2 14997 | A word of length two is a ... |
| pfx2 14998 | A prefix of length two. (... |
| wrd3tpop 14999 | A word of length three rep... |
| wrdlen3s3 15000 | A word of length three as ... |
| repsw2 15001 | The "repeated symbol word"... |
| repsw3 15002 | The "repeated symbol word"... |
| swrd2lsw 15003 | Extract the last two symbo... |
| 2swrd2eqwrdeq 15004 | Two words of length at lea... |
| ccatw2s1ccatws2 15005 | The concatenation of a wor... |
| ccat2s1fvwALT 15006 | Alternate proof of ~ ccat2... |
| wwlktovf 15007 | Lemma 1 for ~ wrd2f1tovbij... |
| wwlktovf1 15008 | Lemma 2 for ~ wrd2f1tovbij... |
| wwlktovfo 15009 | Lemma 3 for ~ wrd2f1tovbij... |
| wwlktovf1o 15010 | Lemma 4 for ~ wrd2f1tovbij... |
| wrd2f1tovbij 15011 | There is a bijection betwe... |
| eqwrds3 15012 | A word is equal with a len... |
| wrdl3s3 15013 | A word of length 3 is a le... |
| s2rn 15014 | Range of a length 2 string... |
| s3rn 15015 | Range of a length 3 string... |
| s7rn 15016 | Range of a length 7 string... |
| s7f1o 15017 | A length 7 word with mutua... |
| s3sndisj 15018 | The singletons consisting ... |
| s3iunsndisj 15019 | The union of singletons co... |
| ofccat 15020 | Letterwise operations on w... |
| ofs1 15021 | Letterwise operations on a... |
| ofs2 15022 | Letterwise operations on a... |
| coss12d 15023 | Subset deduction for compo... |
| trrelssd 15024 | The composition of subclas... |
| xpcogend 15025 | The most interesting case ... |
| xpcoidgend 15026 | If two classes are not dis... |
| cotr2g 15027 | Two ways of saying that th... |
| cotr2 15028 | Two ways of saying a relat... |
| cotr3 15029 | Two ways of saying a relat... |
| coemptyd 15030 | Deduction about compositio... |
| xptrrel 15031 | The cross product is alway... |
| 0trrel 15032 | The empty class is a trans... |
| cleq1lem 15033 | Equality implies bijection... |
| cleq1 15034 | Equality of relations impl... |
| clsslem 15035 | The closure of a subclass ... |
| trcleq1 15040 | Equality of relations impl... |
| trclsslem 15041 | The transitive closure (as... |
| trcleq2lem 15042 | Equality implies bijection... |
| cvbtrcl 15043 | Change of bound variable i... |
| trcleq12lem 15044 | Equality implies bijection... |
| trclexlem 15045 | Existence of relation impl... |
| trclublem 15046 | If a relation exists then ... |
| trclubi 15047 | The Cartesian product of t... |
| trclubgi 15048 | The union with the Cartesi... |
| trclub 15049 | The Cartesian product of t... |
| trclubg 15050 | The union with the Cartesi... |
| trclfv 15051 | The transitive closure of ... |
| brintclab 15052 | Two ways to express a bina... |
| brtrclfv 15053 | Two ways of expressing the... |
| brcnvtrclfv 15054 | Two ways of expressing the... |
| brtrclfvcnv 15055 | Two ways of expressing the... |
| brcnvtrclfvcnv 15056 | Two ways of expressing the... |
| trclfvss 15057 | The transitive closure (as... |
| trclfvub 15058 | The transitive closure of ... |
| trclfvlb 15059 | The transitive closure of ... |
| trclfvcotr 15060 | The transitive closure of ... |
| trclfvlb2 15061 | The transitive closure of ... |
| trclfvlb3 15062 | The transitive closure of ... |
| cotrtrclfv 15063 | The transitive closure of ... |
| trclidm 15064 | The transitive closure of ... |
| trclun 15065 | Transitive closure of a un... |
| trclfvg 15066 | The value of the transitiv... |
| trclfvcotrg 15067 | The value of the transitiv... |
| reltrclfv 15068 | The transitive closure of ... |
| dmtrclfv 15069 | The domain of the transiti... |
| reldmrelexp 15072 | The domain of the repeated... |
| relexp0g 15073 | A relation composed zero t... |
| relexp0 15074 | A relation composed zero t... |
| relexp0d 15075 | A relation composed zero t... |
| relexpsucnnr 15076 | A reduction for relation e... |
| relexp1g 15077 | A relation composed once i... |
| dfid5 15078 | Identity relation is equal... |
| dfid6 15079 | Identity relation expresse... |
| relexp1d 15080 | A relation composed once i... |
| relexpsucnnl 15081 | A reduction for relation e... |
| relexpsucl 15082 | A reduction for relation e... |
| relexpsucr 15083 | A reduction for relation e... |
| relexpsucrd 15084 | A reduction for relation e... |
| relexpsucld 15085 | A reduction for relation e... |
| relexpcnv 15086 | Commutation of converse an... |
| relexpcnvd 15087 | Commutation of converse an... |
| relexp0rel 15088 | The exponentiation of a cl... |
| relexprelg 15089 | The exponentiation of a cl... |
| relexprel 15090 | The exponentiation of a re... |
| relexpreld 15091 | The exponentiation of a re... |
| relexpnndm 15092 | The domain of an exponenti... |
| relexpdmg 15093 | The domain of an exponenti... |
| relexpdm 15094 | The domain of an exponenti... |
| relexpdmd 15095 | The domain of an exponenti... |
| relexpnnrn 15096 | The range of an exponentia... |
| relexprng 15097 | The range of an exponentia... |
| relexprn 15098 | The range of an exponentia... |
| relexprnd 15099 | The range of an exponentia... |
| relexpfld 15100 | The field of an exponentia... |
| relexpfldd 15101 | The field of an exponentia... |
| relexpaddnn 15102 | Relation composition becom... |
| relexpuzrel 15103 | The exponentiation of a cl... |
| relexpaddg 15104 | Relation composition becom... |
| relexpaddd 15105 | Relation composition becom... |
| rtrclreclem1 15108 | The reflexive, transitive ... |
| dfrtrclrec2 15109 | If two elements are connec... |
| rtrclreclem2 15110 | The reflexive, transitive ... |
| rtrclreclem3 15111 | The reflexive, transitive ... |
| rtrclreclem4 15112 | The reflexive, transitive ... |
| dfrtrcl2 15113 | The two definitions ` t* `... |
| relexpindlem 15114 | Principle of transitive in... |
| relexpind 15115 | Principle of transitive in... |
| rtrclind 15116 | Principle of transitive in... |
| shftlem 15119 | Two ways to write a shifte... |
| shftuz 15120 | A shift of the upper integ... |
| shftfval 15121 | The value of the sequence ... |
| shftdm 15122 | Domain of a relation shift... |
| shftfib 15123 | Value of a fiber of the re... |
| shftfn 15124 | Functionality and domain o... |
| shftval 15125 | Value of a sequence shifte... |
| shftval2 15126 | Value of a sequence shifte... |
| shftval3 15127 | Value of a sequence shifte... |
| shftval4 15128 | Value of a sequence shifte... |
| shftval5 15129 | Value of a shifted sequenc... |
| shftf 15130 | Functionality of a shifted... |
| 2shfti 15131 | Composite shift operations... |
| shftidt2 15132 | Identity law for the shift... |
| shftidt 15133 | Identity law for the shift... |
| shftcan1 15134 | Cancellation law for the s... |
| shftcan2 15135 | Cancellation law for the s... |
| seqshft 15136 | Shifting the index set of ... |
| sgnval 15139 | Value of the signum functi... |
| sgn0 15140 | The signum of 0 is 0. (Co... |
| sgnp 15141 | The signum of a positive e... |
| sgnrrp 15142 | The signum of a positive r... |
| sgn1 15143 | The signum of 1 is 1. (Co... |
| sgnpnf 15144 | The signum of ` +oo ` is 1... |
| sgnn 15145 | The signum of a negative e... |
| sgnmnf 15146 | The signum of ` -oo ` is -... |
| cjval 15153 | The value of the conjugate... |
| cjth 15154 | The defining property of t... |
| cjf 15155 | Domain and codomain of the... |
| cjcl 15156 | The conjugate of a complex... |
| reval 15157 | The value of the real part... |
| imval 15158 | The value of the imaginary... |
| imre 15159 | The imaginary part of a co... |
| reim 15160 | The real part of a complex... |
| recl 15161 | The real part of a complex... |
| imcl 15162 | The imaginary part of a co... |
| ref 15163 | Domain and codomain of the... |
| imf 15164 | Domain and codomain of the... |
| crre 15165 | The real part of a complex... |
| crim 15166 | The real part of a complex... |
| replim 15167 | Reconstruct a complex numb... |
| remim 15168 | Value of the conjugate of ... |
| reim0 15169 | The imaginary part of a re... |
| reim0b 15170 | A number is real iff its i... |
| rereb 15171 | A number is real iff it eq... |
| mulre 15172 | A product with a nonzero r... |
| rere 15173 | A real number equals its r... |
| cjreb 15174 | A number is real iff it eq... |
| recj 15175 | Real part of a complex con... |
| reneg 15176 | Real part of negative. (C... |
| readd 15177 | Real part distributes over... |
| resub 15178 | Real part distributes over... |
| remullem 15179 | Lemma for ~ remul , ~ immu... |
| remul 15180 | Real part of a product. (... |
| remul2 15181 | Real part of a product. (... |
| rediv 15182 | Real part of a division. ... |
| imcj 15183 | Imaginary part of a comple... |
| imneg 15184 | The imaginary part of a ne... |
| imadd 15185 | Imaginary part distributes... |
| imsub 15186 | Imaginary part distributes... |
| immul 15187 | Imaginary part of a produc... |
| immul2 15188 | Imaginary part of a produc... |
| imdiv 15189 | Imaginary part of a divisi... |
| cjre 15190 | A real number equals its c... |
| cjcj 15191 | The conjugate of the conju... |
| cjadd 15192 | Complex conjugate distribu... |
| cjmul 15193 | Complex conjugate distribu... |
| ipcnval 15194 | Standard inner product on ... |
| cjmulrcl 15195 | A complex number times its... |
| cjmulval 15196 | A complex number times its... |
| cjmulge0 15197 | A complex number times its... |
| cjneg 15198 | Complex conjugate of negat... |
| addcj 15199 | A number plus its conjugat... |
| cjsub 15200 | Complex conjugate distribu... |
| cjexp 15201 | Complex conjugate of posit... |
| imval2 15202 | The imaginary part of a nu... |
| re0 15203 | The real part of zero. (C... |
| im0 15204 | The imaginary part of zero... |
| re1 15205 | The real part of one. (Co... |
| im1 15206 | The imaginary part of one.... |
| rei 15207 | The real part of ` _i ` . ... |
| imi 15208 | The imaginary part of ` _i... |
| cj0 15209 | The conjugate of zero. (C... |
| cji 15210 | The complex conjugate of t... |
| cjreim 15211 | The conjugate of a represe... |
| cjreim2 15212 | The conjugate of the repre... |
| cj11 15213 | Complex conjugate is a one... |
| cjne0 15214 | A number is nonzero iff it... |
| cjdiv 15215 | Complex conjugate distribu... |
| cnrecnv 15216 | The inverse to the canonic... |
| sqeqd 15217 | A deduction for showing tw... |
| recli 15218 | The real part of a complex... |
| imcli 15219 | The imaginary part of a co... |
| cjcli 15220 | Closure law for complex co... |
| replimi 15221 | Construct a complex number... |
| cjcji 15222 | The conjugate of the conju... |
| reim0bi 15223 | A number is real iff its i... |
| rerebi 15224 | A real number equals its r... |
| cjrebi 15225 | A number is real iff it eq... |
| recji 15226 | Real part of a complex con... |
| imcji 15227 | Imaginary part of a comple... |
| cjmulrcli 15228 | A complex number times its... |
| cjmulvali 15229 | A complex number times its... |
| cjmulge0i 15230 | A complex number times its... |
| renegi 15231 | Real part of negative. (C... |
| imnegi 15232 | Imaginary part of negative... |
| cjnegi 15233 | Complex conjugate of negat... |
| addcji 15234 | A number plus its conjugat... |
| readdi 15235 | Real part distributes over... |
| imaddi 15236 | Imaginary part distributes... |
| remuli 15237 | Real part of a product. (... |
| immuli 15238 | Imaginary part of a produc... |
| cjaddi 15239 | Complex conjugate distribu... |
| cjmuli 15240 | Complex conjugate distribu... |
| ipcni 15241 | Standard inner product on ... |
| cjdivi 15242 | Complex conjugate distribu... |
| crrei 15243 | The real part of a complex... |
| crimi 15244 | The imaginary part of a co... |
| recld 15245 | The real part of a complex... |
| imcld 15246 | The imaginary part of a co... |
| cjcld 15247 | Closure law for complex co... |
| replimd 15248 | Construct a complex number... |
| remimd 15249 | Value of the conjugate of ... |
| cjcjd 15250 | The conjugate of the conju... |
| reim0bd 15251 | A number is real iff its i... |
| rerebd 15252 | A real number equals its r... |
| cjrebd 15253 | A number is real iff it eq... |
| cjne0d 15254 | A number is nonzero iff it... |
| recjd 15255 | Real part of a complex con... |
| imcjd 15256 | Imaginary part of a comple... |
| cjmulrcld 15257 | A complex number times its... |
| cjmulvald 15258 | A complex number times its... |
| cjmulge0d 15259 | A complex number times its... |
| renegd 15260 | Real part of negative. (C... |
| imnegd 15261 | Imaginary part of negative... |
| cjnegd 15262 | Complex conjugate of negat... |
| addcjd 15263 | A number plus its conjugat... |
| cjexpd 15264 | Complex conjugate of posit... |
| readdd 15265 | Real part distributes over... |
| imaddd 15266 | Imaginary part distributes... |
| resubd 15267 | Real part distributes over... |
| imsubd 15268 | Imaginary part distributes... |
| remuld 15269 | Real part of a product. (... |
| immuld 15270 | Imaginary part of a produc... |
| cjaddd 15271 | Complex conjugate distribu... |
| cjmuld 15272 | Complex conjugate distribu... |
| ipcnd 15273 | Standard inner product on ... |
| cjdivd 15274 | Complex conjugate distribu... |
| rered 15275 | A real number equals its r... |
| reim0d 15276 | The imaginary part of a re... |
| cjred 15277 | A real number equals its c... |
| remul2d 15278 | Real part of a product. (... |
| immul2d 15279 | Imaginary part of a produc... |
| redivd 15280 | Real part of a division. ... |
| imdivd 15281 | Imaginary part of a divisi... |
| crred 15282 | The real part of a complex... |
| crimd 15283 | The imaginary part of a co... |
| sqrtval 15288 | Value of square root funct... |
| absval 15289 | The absolute value (modulu... |
| rennim 15290 | A real number does not lie... |
| cnpart 15291 | The specification of restr... |
| sqrt0 15292 | The square root of zero is... |
| 01sqrexlem1 15293 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem2 15294 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem3 15295 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem4 15296 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem5 15297 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem6 15298 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem7 15299 | Lemma for ~ 01sqrex . (Co... |
| 01sqrex 15300 | Existence of a square root... |
| resqrex 15301 | Existence of a square root... |
| sqrmo 15302 | Uniqueness for the square ... |
| resqreu 15303 | Existence and uniqueness f... |
| resqrtcl 15304 | Closure of the square root... |
| resqrtthlem 15305 | Lemma for ~ resqrtth . (C... |
| resqrtth 15306 | Square root theorem over t... |
| remsqsqrt 15307 | Square of square root. (C... |
| sqrtge0 15308 | The square root function i... |
| sqrtgt0 15309 | The square root function i... |
| sqrtmul 15310 | Square root distributes ov... |
| sqrtle 15311 | Square root is monotonic. ... |
| sqrtlt 15312 | Square root is strictly mo... |
| sqrt11 15313 | The square root function i... |
| sqrt00 15314 | A square root is zero iff ... |
| rpsqrtcl 15315 | The square root of a posit... |
| sqrtdiv 15316 | Square root distributes ov... |
| sqrtneglem 15317 | The square root of a negat... |
| sqrtneg 15318 | The square root of a negat... |
| sqrtsq2 15319 | Relationship between squar... |
| sqrtsq 15320 | Square root of square. (C... |
| sqrtmsq 15321 | Square root of square. (C... |
| sqrt1 15322 | The square root of 1 is 1.... |
| sqrt4 15323 | The square root of 4 is 2.... |
| sqrt9 15324 | The square root of 9 is 3.... |
| sqrt2gt1lt2 15325 | The square root of 2 is bo... |
| sqrtm1 15326 | The imaginary unit is the ... |
| nn0sqeq1 15327 | A natural number with squa... |
| absneg 15328 | Absolute value of the nega... |
| abscl 15329 | Real closure of absolute v... |
| abscj 15330 | The absolute value of a nu... |
| absvalsq 15331 | Square of value of absolut... |
| absvalsq2 15332 | Square of value of absolut... |
| sqabsadd 15333 | Square of absolute value o... |
| sqabssub 15334 | Square of absolute value o... |
| absval2 15335 | Value of absolute value fu... |
| abs0 15336 | The absolute value of 0. ... |
| absi 15337 | The absolute value of the ... |
| absge0 15338 | Absolute value is nonnegat... |
| absrpcl 15339 | The absolute value of a no... |
| abs00 15340 | The absolute value of a nu... |
| abs00ad 15341 | A complex number is zero i... |
| abs00bd 15342 | If a complex number is zer... |
| absreimsq 15343 | Square of the absolute val... |
| absreim 15344 | Absolute value of a number... |
| absmul 15345 | Absolute value distributes... |
| absdiv 15346 | Absolute value distributes... |
| absid 15347 | A nonnegative number is it... |
| abs1 15348 | The absolute value of one ... |
| absnid 15349 | For a negative number, its... |
| leabs 15350 | A real number is less than... |
| absor 15351 | The absolute value of a re... |
| absre 15352 | Absolute value of a real n... |
| absresq 15353 | Square of the absolute val... |
| absmod0 15354 | ` A ` is divisible by ` B ... |
| absexp 15355 | Absolute value of positive... |
| absexpz 15356 | Absolute value of integer ... |
| abssq 15357 | Square can be moved in and... |
| sqabs 15358 | The squares of two reals a... |
| absrele 15359 | The absolute value of a co... |
| absimle 15360 | The absolute value of a co... |
| max0add 15361 | The sum of the positive an... |
| absz 15362 | A real number is an intege... |
| nn0abscl 15363 | The absolute value of an i... |
| zabscl 15364 | The absolute value of an i... |
| abslt 15365 | Absolute value and 'less t... |
| absle 15366 | Absolute value and 'less t... |
| abssubne0 15367 | If the absolute value of a... |
| absdiflt 15368 | The absolute value of a di... |
| absdifle 15369 | The absolute value of a di... |
| elicc4abs 15370 | Membership in a symmetric ... |
| lenegsq 15371 | Comparison to a nonnegativ... |
| releabs 15372 | The real part of a number ... |
| recval 15373 | Reciprocal expressed with ... |
| absidm 15374 | The absolute value functio... |
| absgt0 15375 | The absolute value of a no... |
| nnabscl 15376 | The absolute value of a no... |
| abssub 15377 | Swapping order of subtract... |
| abssubge0 15378 | Absolute value of a nonneg... |
| abssuble0 15379 | Absolute value of a nonpos... |
| absmax 15380 | The maximum of two numbers... |
| abstri 15381 | Triangle inequality for ab... |
| abs3dif 15382 | Absolute value of differen... |
| abs2dif 15383 | Difference of absolute val... |
| abs2dif2 15384 | Difference of absolute val... |
| abs2difabs 15385 | Absolute value of differen... |
| abs1m 15386 | For any complex number, th... |
| recan 15387 | Cancellation law involving... |
| absf 15388 | Mapping domain and codomai... |
| abs3lem 15389 | Lemma involving absolute v... |
| abslem2 15390 | Lemma involving absolute v... |
| rddif 15391 | The difference between a r... |
| absrdbnd 15392 | Bound on the absolute valu... |
| fzomaxdiflem 15393 | Lemma for ~ fzomaxdif . (... |
| fzomaxdif 15394 | A bound on the separation ... |
| uzin2 15395 | The upper integers are clo... |
| rexanuz 15396 | Combine two different uppe... |
| rexanre 15397 | Combine two different uppe... |
| rexfiuz 15398 | Combine finitely many diff... |
| rexuz3 15399 | Restrict the base of the u... |
| rexanuz2 15400 | Combine two different uppe... |
| r19.29uz 15401 | A version of ~ 19.29 for u... |
| r19.2uz 15402 | A version of ~ r19.2z for ... |
| rexuzre 15403 | Convert an upper real quan... |
| rexico 15404 | Restrict the base of an up... |
| cau3lem 15405 | Lemma for ~ cau3 . (Contr... |
| cau3 15406 | Convert between three-quan... |
| cau4 15407 | Change the base of a Cauch... |
| caubnd2 15408 | A Cauchy sequence of compl... |
| caubnd 15409 | A Cauchy sequence of compl... |
| sqreulem 15410 | Lemma for ~ sqreu : write ... |
| sqreu 15411 | Existence and uniqueness f... |
| sqrtcl 15412 | Closure of the square root... |
| sqrtthlem 15413 | Lemma for ~ sqrtth . (Con... |
| sqrtf 15414 | Mapping domain and codomai... |
| sqrtth 15415 | Square root theorem over t... |
| sqrtrege0 15416 | The square root function m... |
| eqsqrtor 15417 | Solve an equation containi... |
| eqsqrtd 15418 | A deduction for showing th... |
| eqsqrt2d 15419 | A deduction for showing th... |
| amgm2 15420 | Arithmetic-geometric mean ... |
| sqrtthi 15421 | Square root theorem. Theo... |
| sqrtcli 15422 | The square root of a nonne... |
| sqrtgt0i 15423 | The square root of a posit... |
| sqrtmsqi 15424 | Square root of square. (C... |
| sqrtsqi 15425 | Square root of square. (C... |
| sqsqrti 15426 | Square of square root. (C... |
| sqrtge0i 15427 | The square root of a nonne... |
| absidi 15428 | A nonnegative number is it... |
| absnidi 15429 | A negative number is the n... |
| leabsi 15430 | A real number is less than... |
| absori 15431 | The absolute value of a re... |
| absrei 15432 | Absolute value of a real n... |
| sqrtpclii 15433 | The square root of a posit... |
| sqrtgt0ii 15434 | The square root of a posit... |
| sqrt11i 15435 | The square root function i... |
| sqrtmuli 15436 | Square root distributes ov... |
| sqrtmulii 15437 | Square root distributes ov... |
| sqrtmsq2i 15438 | Relationship between squar... |
| sqrtlei 15439 | Square root is monotonic. ... |
| sqrtlti 15440 | Square root is strictly mo... |
| abslti 15441 | Absolute value and 'less t... |
| abslei 15442 | Absolute value and 'less t... |
| cnsqrt00 15443 | A square root of a complex... |
| absvalsqi 15444 | Square of value of absolut... |
| absvalsq2i 15445 | Square of value of absolut... |
| abscli 15446 | Real closure of absolute v... |
| absge0i 15447 | Absolute value is nonnegat... |
| absval2i 15448 | Value of absolute value fu... |
| abs00i 15449 | The absolute value of a nu... |
| absgt0i 15450 | The absolute value of a no... |
| absnegi 15451 | Absolute value of negative... |
| abscji 15452 | The absolute value of a nu... |
| releabsi 15453 | The real part of a number ... |
| abssubi 15454 | Swapping order of subtract... |
| absmuli 15455 | Absolute value distributes... |
| sqabsaddi 15456 | Square of absolute value o... |
| sqabssubi 15457 | Square of absolute value o... |
| absdivzi 15458 | Absolute value distributes... |
| abstrii 15459 | Triangle inequality for ab... |
| abs3difi 15460 | Absolute value of differen... |
| abs3lemi 15461 | Lemma involving absolute v... |
| rpsqrtcld 15462 | The square root of a posit... |
| sqrtgt0d 15463 | The square root of a posit... |
| absnidd 15464 | A negative number is the n... |
| leabsd 15465 | A real number is less than... |
| absord 15466 | The absolute value of a re... |
| absred 15467 | Absolute value of a real n... |
| resqrtcld 15468 | The square root of a nonne... |
| sqrtmsqd 15469 | Square root of square. (C... |
| sqrtsqd 15470 | Square root of square. (C... |
| sqrtge0d 15471 | The square root of a nonne... |
| sqrtnegd 15472 | The square root of a negat... |
| absidd 15473 | A nonnegative number is it... |
| sqrtdivd 15474 | Square root distributes ov... |
| sqrtmuld 15475 | Square root distributes ov... |
| sqrtsq2d 15476 | Relationship between squar... |
| sqrtled 15477 | Square root is monotonic. ... |
| sqrtltd 15478 | Square root is strictly mo... |
| sqr11d 15479 | The square root function i... |
| absltd 15480 | Absolute value and 'less t... |
| absled 15481 | Absolute value and 'less t... |
| abssubge0d 15482 | Absolute value of a nonneg... |
| abssuble0d 15483 | Absolute value of a nonpos... |
| absdifltd 15484 | The absolute value of a di... |
| absdifled 15485 | The absolute value of a di... |
| icodiamlt 15486 | Two elements in a half-ope... |
| abscld 15487 | Real closure of absolute v... |
| sqrtcld 15488 | Closure of the square root... |
| sqrtrege0d 15489 | The real part of the squar... |
| sqsqrtd 15490 | Square root theorem. Theo... |
| msqsqrtd 15491 | Square root theorem. Theo... |
| sqr00d 15492 | A square root is zero iff ... |
| absvalsqd 15493 | Square of value of absolut... |
| absvalsq2d 15494 | Square of value of absolut... |
| absge0d 15495 | Absolute value is nonnegat... |
| absval2d 15496 | Value of absolute value fu... |
| abs00d 15497 | The absolute value of a nu... |
| absne0d 15498 | The absolute value of a nu... |
| absrpcld 15499 | The absolute value of a no... |
| absnegd 15500 | Absolute value of negative... |
| abscjd 15501 | The absolute value of a nu... |
| releabsd 15502 | The real part of a number ... |
| absexpd 15503 | Absolute value of positive... |
| abssubd 15504 | Swapping order of subtract... |
| absmuld 15505 | Absolute value distributes... |
| absdivd 15506 | Absolute value distributes... |
| abstrid 15507 | Triangle inequality for ab... |
| abs2difd 15508 | Difference of absolute val... |
| abs2dif2d 15509 | Difference of absolute val... |
| abs2difabsd 15510 | Absolute value of differen... |
| abs3difd 15511 | Absolute value of differen... |
| abs3lemd 15512 | Lemma involving absolute v... |
| reusq0 15513 | A complex number is the sq... |
| bhmafibid1cn 15514 | The Brahmagupta-Fibonacci ... |
| bhmafibid2cn 15515 | The Brahmagupta-Fibonacci ... |
| bhmafibid1 15516 | The Brahmagupta-Fibonacci ... |
| bhmafibid2 15517 | The Brahmagupta-Fibonacci ... |
| limsupgord 15520 | Ordering property of the s... |
| limsupcl 15521 | Closure of the superior li... |
| limsupval 15522 | The superior limit of an i... |
| limsupgf 15523 | Closure of the superior li... |
| limsupgval 15524 | Value of the superior limi... |
| limsupgle 15525 | The defining property of t... |
| limsuple 15526 | The defining property of t... |
| limsuplt 15527 | The defining property of t... |
| limsupval2 15528 | The superior limit, relati... |
| limsupgre 15529 | If a sequence of real numb... |
| limsupbnd1 15530 | If a sequence is eventuall... |
| limsupbnd2 15531 | If a sequence is eventuall... |
| climrel 15540 | The limit relation is a re... |
| rlimrel 15541 | The limit relation is a re... |
| clim 15542 | Express the predicate: Th... |
| rlim 15543 | Express the predicate: Th... |
| rlim2 15544 | Rewrite ~ rlim for a mappi... |
| rlim2lt 15545 | Use strictly less-than in ... |
| rlim3 15546 | Restrict the range of the ... |
| climcl 15547 | Closure of the limit of a ... |
| rlimpm 15548 | Closure of a function with... |
| rlimf 15549 | Closure of a function with... |
| rlimss 15550 | Domain closure of a functi... |
| rlimcl 15551 | Closure of the limit of a ... |
| clim2 15552 | Express the predicate: Th... |
| clim2c 15553 | Express the predicate ` F ... |
| clim0 15554 | Express the predicate ` F ... |
| clim0c 15555 | Express the predicate ` F ... |
| rlim0 15556 | Express the predicate ` B ... |
| rlim0lt 15557 | Use strictly less-than in ... |
| climi 15558 | Convergence of a sequence ... |
| climi2 15559 | Convergence of a sequence ... |
| climi0 15560 | Convergence of a sequence ... |
| rlimi 15561 | Convergence at infinity of... |
| rlimi2 15562 | Convergence at infinity of... |
| ello1 15563 | Elementhood in the set of ... |
| ello12 15564 | Elementhood in the set of ... |
| ello12r 15565 | Sufficient condition for e... |
| lo1f 15566 | An eventually upper bounde... |
| lo1dm 15567 | An eventually upper bounde... |
| lo1bdd 15568 | The defining property of a... |
| ello1mpt 15569 | Elementhood in the set of ... |
| ello1mpt2 15570 | Elementhood in the set of ... |
| ello1d 15571 | Sufficient condition for e... |
| lo1bdd2 15572 | If an eventually bounded f... |
| lo1bddrp 15573 | Refine ~ o1bdd2 to give a ... |
| elo1 15574 | Elementhood in the set of ... |
| elo12 15575 | Elementhood in the set of ... |
| elo12r 15576 | Sufficient condition for e... |
| o1f 15577 | An eventually bounded func... |
| o1dm 15578 | An eventually bounded func... |
| o1bdd 15579 | The defining property of a... |
| lo1o1 15580 | A function is eventually b... |
| lo1o12 15581 | A function is eventually b... |
| elo1mpt 15582 | Elementhood in the set of ... |
| elo1mpt2 15583 | Elementhood in the set of ... |
| elo1d 15584 | Sufficient condition for e... |
| o1lo1 15585 | A real function is eventua... |
| o1lo12 15586 | A lower bounded real funct... |
| o1lo1d 15587 | A real eventually bounded ... |
| icco1 15588 | Derive eventual boundednes... |
| o1bdd2 15589 | If an eventually bounded f... |
| o1bddrp 15590 | Refine ~ o1bdd2 to give a ... |
| climconst 15591 | An (eventually) constant s... |
| rlimconst 15592 | A constant sequence conver... |
| rlimclim1 15593 | Forward direction of ~ rli... |
| rlimclim 15594 | A sequence on an upper int... |
| climrlim2 15595 | Produce a real limit from ... |
| climconst2 15596 | A constant sequence conver... |
| climz 15597 | The zero sequence converge... |
| rlimuni 15598 | A real function whose doma... |
| rlimdm 15599 | Two ways to express that a... |
| climuni 15600 | An infinite sequence of co... |
| fclim 15601 | The limit relation is func... |
| climdm 15602 | Two ways to express that a... |
| climeu 15603 | An infinite sequence of co... |
| climreu 15604 | An infinite sequence of co... |
| climmo 15605 | An infinite sequence of co... |
| rlimres 15606 | The restriction of a funct... |
| lo1res 15607 | The restriction of an even... |
| o1res 15608 | The restriction of an even... |
| rlimres2 15609 | The restriction of a funct... |
| lo1res2 15610 | The restriction of a funct... |
| o1res2 15611 | The restriction of a funct... |
| lo1resb 15612 | The restriction of a funct... |
| rlimresb 15613 | The restriction of a funct... |
| o1resb 15614 | The restriction of a funct... |
| climeq 15615 | Two functions that are eve... |
| lo1eq 15616 | Two functions that are eve... |
| rlimeq 15617 | Two functions that are eve... |
| o1eq 15618 | Two functions that are eve... |
| climmpt 15619 | Exhibit a function ` G ` w... |
| 2clim 15620 | If two sequences converge ... |
| climmpt2 15621 | Relate an integer limit on... |
| climshftlem 15622 | A shifted function converg... |
| climres 15623 | A function restricted to u... |
| climshft 15624 | A shifted function converg... |
| serclim0 15625 | The zero series converges ... |
| rlimcld2 15626 | If ` D ` is a closed set i... |
| rlimrege0 15627 | The limit of a sequence of... |
| rlimrecl 15628 | The limit of a real sequen... |
| rlimge0 15629 | The limit of a sequence of... |
| climshft2 15630 | A shifted function converg... |
| climrecl 15631 | The limit of a convergent ... |
| climge0 15632 | A nonnegative sequence con... |
| climabs0 15633 | Convergence to zero of the... |
| o1co 15634 | Sufficient condition for t... |
| o1compt 15635 | Sufficient condition for t... |
| rlimcn1 15636 | Image of a limit under a c... |
| rlimcn1b 15637 | Image of a limit under a c... |
| rlimcn3 15638 | Image of a limit under a c... |
| rlimcn2 15639 | Image of a limit under a c... |
| climcn1 15640 | Image of a limit under a c... |
| climcn2 15641 | Image of a limit under a c... |
| addcn2 15642 | Complex number addition is... |
| subcn2 15643 | Complex number subtraction... |
| mulcn2 15644 | Complex number multiplicat... |
| reccn2 15645 | The reciprocal function is... |
| cn1lem 15646 | A sufficient condition for... |
| abscn2 15647 | The absolute value functio... |
| cjcn2 15648 | The complex conjugate func... |
| recn2 15649 | The real part function is ... |
| imcn2 15650 | The imaginary part functio... |
| climcn1lem 15651 | The limit of a continuous ... |
| climabs 15652 | Limit of the absolute valu... |
| climcj 15653 | Limit of the complex conju... |
| climre 15654 | Limit of the real part of ... |
| climim 15655 | Limit of the imaginary par... |
| rlimmptrcl 15656 | Reverse closure for a real... |
| rlimabs 15657 | Limit of the absolute valu... |
| rlimcj 15658 | Limit of the complex conju... |
| rlimre 15659 | Limit of the real part of ... |
| rlimim 15660 | Limit of the imaginary par... |
| o1of2 15661 | Show that a binary operati... |
| o1add 15662 | The sum of two eventually ... |
| o1mul 15663 | The product of two eventua... |
| o1sub 15664 | The difference of two even... |
| rlimo1 15665 | Any function with a finite... |
| rlimdmo1 15666 | A convergent function is e... |
| o1rlimmul 15667 | The product of an eventual... |
| o1const 15668 | A constant function is eve... |
| lo1const 15669 | A constant function is eve... |
| lo1mptrcl 15670 | Reverse closure for an eve... |
| o1mptrcl 15671 | Reverse closure for an eve... |
| o1add2 15672 | The sum of two eventually ... |
| o1mul2 15673 | The product of two eventua... |
| o1sub2 15674 | The product of two eventua... |
| lo1add 15675 | The sum of two eventually ... |
| lo1mul 15676 | The product of an eventual... |
| lo1mul2 15677 | The product of an eventual... |
| o1dif 15678 | If the difference of two f... |
| lo1sub 15679 | The difference of an event... |
| climadd 15680 | Limit of the sum of two co... |
| climmul 15681 | Limit of the product of tw... |
| climsub 15682 | Limit of the difference of... |
| climaddc1 15683 | Limit of a constant ` C ` ... |
| climaddc2 15684 | Limit of a constant ` C ` ... |
| climmulc2 15685 | Limit of a sequence multip... |
| climsubc1 15686 | Limit of a constant ` C ` ... |
| climsubc2 15687 | Limit of a constant ` C ` ... |
| climle 15688 | Comparison of the limits o... |
| climsqz 15689 | Convergence of a sequence ... |
| climsqz2 15690 | Convergence of a sequence ... |
| rlimadd 15691 | Limit of the sum of two co... |
| rlimaddOLD 15692 | Obsolete version of ~ rlim... |
| rlimsub 15693 | Limit of the difference of... |
| rlimmul 15694 | Limit of the product of tw... |
| rlimmulOLD 15695 | Obsolete version of ~ rlim... |
| rlimdiv 15696 | Limit of the quotient of t... |
| rlimneg 15697 | Limit of the negative of a... |
| rlimle 15698 | Comparison of the limits o... |
| rlimsqzlem 15699 | Lemma for ~ rlimsqz and ~ ... |
| rlimsqz 15700 | Convergence of a sequence ... |
| rlimsqz2 15701 | Convergence of a sequence ... |
| lo1le 15702 | Transfer eventual upper bo... |
| o1le 15703 | Transfer eventual boundedn... |
| rlimno1 15704 | A function whose inverse c... |
| clim2ser 15705 | The limit of an infinite s... |
| clim2ser2 15706 | The limit of an infinite s... |
| iserex 15707 | An infinite series converg... |
| isermulc2 15708 | Multiplication of an infin... |
| climlec2 15709 | Comparison of a constant t... |
| iserle 15710 | Comparison of the limits o... |
| iserge0 15711 | The limit of an infinite s... |
| climub 15712 | The limit of a monotonic s... |
| climserle 15713 | The partial sums of a conv... |
| isershft 15714 | Index shift of the limit o... |
| isercolllem1 15715 | Lemma for ~ isercoll . (C... |
| isercolllem2 15716 | Lemma for ~ isercoll . (C... |
| isercolllem3 15717 | Lemma for ~ isercoll . (C... |
| isercoll 15718 | Rearrange an infinite seri... |
| isercoll2 15719 | Generalize ~ isercoll so t... |
| climsup 15720 | A bounded monotonic sequen... |
| climcau 15721 | A converging sequence of c... |
| climbdd 15722 | A converging sequence of c... |
| caucvgrlem 15723 | Lemma for ~ caurcvgr . (C... |
| caurcvgr 15724 | A Cauchy sequence of real ... |
| caucvgrlem2 15725 | Lemma for ~ caucvgr . (Co... |
| caucvgr 15726 | A Cauchy sequence of compl... |
| caurcvg 15727 | A Cauchy sequence of real ... |
| caurcvg2 15728 | A Cauchy sequence of real ... |
| caucvg 15729 | A Cauchy sequence of compl... |
| caucvgb 15730 | A function is convergent i... |
| serf0 15731 | If an infinite series conv... |
| iseraltlem1 15732 | Lemma for ~ iseralt . A d... |
| iseraltlem2 15733 | Lemma for ~ iseralt . The... |
| iseraltlem3 15734 | Lemma for ~ iseralt . Fro... |
| iseralt 15735 | The alternating series tes... |
| sumex 15738 | A sum is a set. (Contribu... |
| sumeq1 15739 | Equality theorem for a sum... |
| nfsum1 15740 | Bound-variable hypothesis ... |
| nfsum 15741 | Bound-variable hypothesis ... |
| sumeq2w 15742 | Equality theorem for sum, ... |
| sumeq2ii 15743 | Equality theorem for sum, ... |
| sumeq2 15744 | Equality theorem for sum. ... |
| cbvsum 15745 | Change bound variable in a... |
| cbvsumv 15746 | Change bound variable in a... |
| sumeq1i 15747 | Equality inference for sum... |
| sumeq2i 15748 | Equality inference for sum... |
| sumeq12i 15749 | Equality inference for sum... |
| sumeq1d 15750 | Equality deduction for sum... |
| sumeq2d 15751 | Equality deduction for sum... |
| sumeq2dv 15752 | Equality deduction for sum... |
| sumeq2sdv 15753 | Equality deduction for sum... |
| sumeq2sdvOLD 15754 | Obsolete version of ~ sume... |
| 2sumeq2dv 15755 | Equality deduction for dou... |
| sumeq12dv 15756 | Equality deduction for sum... |
| sumeq12rdv 15757 | Equality deduction for sum... |
| sum2id 15758 | The second class argument ... |
| sumfc 15759 | A lemma to facilitate conv... |
| fz1f1o 15760 | A lemma for working with f... |
| sumrblem 15761 | Lemma for ~ sumrb . (Cont... |
| fsumcvg 15762 | The sequence of partial su... |
| sumrb 15763 | Rebase the starting point ... |
| summolem3 15764 | Lemma for ~ summo . (Cont... |
| summolem2a 15765 | Lemma for ~ summo . (Cont... |
| summolem2 15766 | Lemma for ~ summo . (Cont... |
| summo 15767 | A sum has at most one limi... |
| zsum 15768 | Series sum with index set ... |
| isum 15769 | Series sum with an upper i... |
| fsum 15770 | The value of a sum over a ... |
| sum0 15771 | Any sum over the empty set... |
| sumz 15772 | Any sum of zero over a sum... |
| fsumf1o 15773 | Re-index a finite sum usin... |
| sumss 15774 | Change the index set to a ... |
| fsumss 15775 | Change the index set to a ... |
| sumss2 15776 | Change the index set of a ... |
| fsumcvg2 15777 | The sequence of partial su... |
| fsumsers 15778 | Special case of series sum... |
| fsumcvg3 15779 | A finite sum is convergent... |
| fsumser 15780 | A finite sum expressed in ... |
| fsumcl2lem 15781 | - Lemma for finite sum clo... |
| fsumcllem 15782 | - Lemma for finite sum clo... |
| fsumcl 15783 | Closure of a finite sum of... |
| fsumrecl 15784 | Closure of a finite sum of... |
| fsumzcl 15785 | Closure of a finite sum of... |
| fsumnn0cl 15786 | Closure of a finite sum of... |
| fsumrpcl 15787 | Closure of a finite sum of... |
| fsumclf 15788 | Closure of a finite sum of... |
| fsumzcl2 15789 | A finite sum with integer ... |
| fsumadd 15790 | The sum of two finite sums... |
| fsumsplit 15791 | Split a sum into two parts... |
| fsumsplitf 15792 | Split a sum into two parts... |
| sumsnf 15793 | A sum of a singleton is th... |
| fsumsplitsn 15794 | Separate out a term in a f... |
| fsumsplit1 15795 | Separate out a term in a f... |
| sumsn 15796 | A sum of a singleton is th... |
| fsum1 15797 | The finite sum of ` A ( k ... |
| sumpr 15798 | A sum over a pair is the s... |
| sumtp 15799 | A sum over a triple is the... |
| sumsns 15800 | A sum of a singleton is th... |
| fsumm1 15801 | Separate out the last term... |
| fzosump1 15802 | Separate out the last term... |
| fsum1p 15803 | Separate out the first ter... |
| fsummsnunz 15804 | A finite sum all of whose ... |
| fsumsplitsnun 15805 | Separate out a term in a f... |
| fsump1 15806 | The addition of the next t... |
| isumclim 15807 | An infinite sum equals the... |
| isumclim2 15808 | A converging series conver... |
| isumclim3 15809 | The sequence of partial fi... |
| sumnul 15810 | The sum of a non-convergen... |
| isumcl 15811 | The sum of a converging in... |
| isummulc2 15812 | An infinite sum multiplied... |
| isummulc1 15813 | An infinite sum multiplied... |
| isumdivc 15814 | An infinite sum divided by... |
| isumrecl 15815 | The sum of a converging in... |
| isumge0 15816 | An infinite sum of nonnega... |
| isumadd 15817 | Addition of infinite sums.... |
| sumsplit 15818 | Split a sum into two parts... |
| fsump1i 15819 | Optimized version of ~ fsu... |
| fsum2dlem 15820 | Lemma for ~ fsum2d - induc... |
| fsum2d 15821 | Write a double sum as a su... |
| fsumxp 15822 | Combine two sums into a si... |
| fsumcnv 15823 | Transform a region of summ... |
| fsumcom2 15824 | Interchange order of summa... |
| fsumcom 15825 | Interchange order of summa... |
| fsum0diaglem 15826 | Lemma for ~ fsum0diag . (... |
| fsum0diag 15827 | Two ways to express "the s... |
| mptfzshft 15828 | 1-1 onto function in maps-... |
| fsumrev 15829 | Reversal of a finite sum. ... |
| fsumshft 15830 | Index shift of a finite su... |
| fsumshftm 15831 | Negative index shift of a ... |
| fsumrev2 15832 | Reversal of a finite sum. ... |
| fsum0diag2 15833 | Two ways to express "the s... |
| fsummulc2 15834 | A finite sum multiplied by... |
| fsummulc1 15835 | A finite sum multiplied by... |
| fsumdivc 15836 | A finite sum divided by a ... |
| fsumneg 15837 | Negation of a finite sum. ... |
| fsumsub 15838 | Split a finite sum over a ... |
| fsum2mul 15839 | Separate the nested sum of... |
| fsumconst 15840 | The sum of constant terms ... |
| fsumdifsnconst 15841 | The sum of constant terms ... |
| modfsummodslem1 15842 | Lemma 1 for ~ modfsummods ... |
| modfsummods 15843 | Induction step for ~ modfs... |
| modfsummod 15844 | A finite sum modulo a posi... |
| fsumge0 15845 | If all of the terms of a f... |
| fsumless 15846 | A shorter sum of nonnegati... |
| fsumge1 15847 | A sum of nonnegative numbe... |
| fsum00 15848 | A sum of nonnegative numbe... |
| fsumle 15849 | If all of the terms of fin... |
| fsumlt 15850 | If every term in one finit... |
| fsumabs 15851 | Generalized triangle inequ... |
| telfsumo 15852 | Sum of a telescoping serie... |
| telfsumo2 15853 | Sum of a telescoping serie... |
| telfsum 15854 | Sum of a telescoping serie... |
| telfsum2 15855 | Sum of a telescoping serie... |
| fsumparts 15856 | Summation by parts. (Cont... |
| fsumrelem 15857 | Lemma for ~ fsumre , ~ fsu... |
| fsumre 15858 | The real part of a sum. (... |
| fsumim 15859 | The imaginary part of a su... |
| fsumcj 15860 | The complex conjugate of a... |
| fsumrlim 15861 | Limit of a finite sum of c... |
| fsumo1 15862 | The finite sum of eventual... |
| o1fsum 15863 | If ` A ( k ) ` is O(1), th... |
| seqabs 15864 | Generalized triangle inequ... |
| iserabs 15865 | Generalized triangle inequ... |
| cvgcmp 15866 | A comparison test for conv... |
| cvgcmpub 15867 | An upper bound for the lim... |
| cvgcmpce 15868 | A comparison test for conv... |
| abscvgcvg 15869 | An absolutely convergent s... |
| climfsum 15870 | Limit of a finite sum of c... |
| fsumiun 15871 | Sum over a disjoint indexe... |
| hashiun 15872 | The cardinality of a disjo... |
| hash2iun 15873 | The cardinality of a neste... |
| hash2iun1dif1 15874 | The cardinality of a neste... |
| hashrabrex 15875 | The number of elements in ... |
| hashuni 15876 | The cardinality of a disjo... |
| qshash 15877 | The cardinality of a set w... |
| ackbijnn 15878 | Translate the Ackermann bi... |
| binomlem 15879 | Lemma for ~ binom (binomia... |
| binom 15880 | The binomial theorem: ` ( ... |
| binom1p 15881 | Special case of the binomi... |
| binom11 15882 | Special case of the binomi... |
| binom1dif 15883 | A summation for the differ... |
| bcxmaslem1 15884 | Lemma for ~ bcxmas . (Con... |
| bcxmas 15885 | Parallel summation (Christ... |
| incexclem 15886 | Lemma for ~ incexc . (Con... |
| incexc 15887 | The inclusion/exclusion pr... |
| incexc2 15888 | The inclusion/exclusion pr... |
| isumshft 15889 | Index shift of an infinite... |
| isumsplit 15890 | Split off the first ` N ` ... |
| isum1p 15891 | The infinite sum of a conv... |
| isumnn0nn 15892 | Sum from 0 to infinity in ... |
| isumrpcl 15893 | The infinite sum of positi... |
| isumle 15894 | Comparison of two infinite... |
| isumless 15895 | A finite sum of nonnegativ... |
| isumsup2 15896 | An infinite sum of nonnega... |
| isumsup 15897 | An infinite sum of nonnega... |
| isumltss 15898 | A partial sum of a series ... |
| climcndslem1 15899 | Lemma for ~ climcnds : bou... |
| climcndslem2 15900 | Lemma for ~ climcnds : bou... |
| climcnds 15901 | The Cauchy condensation te... |
| divrcnv 15902 | The sequence of reciprocal... |
| divcnv 15903 | The sequence of reciprocal... |
| flo1 15904 | The floor function satisfi... |
| divcnvshft 15905 | Limit of a ratio function.... |
| supcvg 15906 | Extract a sequence ` f ` i... |
| infcvgaux1i 15907 | Auxiliary theorem for appl... |
| infcvgaux2i 15908 | Auxiliary theorem for appl... |
| harmonic 15909 | The harmonic series ` H ` ... |
| arisum 15910 | Arithmetic series sum of t... |
| arisum2 15911 | Arithmetic series sum of t... |
| trireciplem 15912 | Lemma for ~ trirecip . Sh... |
| trirecip 15913 | The sum of the reciprocals... |
| expcnv 15914 | A sequence of powers of a ... |
| explecnv 15915 | A sequence of terms conver... |
| geoserg 15916 | The value of the finite ge... |
| geoser 15917 | The value of the finite ge... |
| pwdif 15918 | The difference of two numb... |
| pwm1geoser 15919 | The n-th power of a number... |
| geolim 15920 | The partial sums in the in... |
| geolim2 15921 | The partial sums in the ge... |
| georeclim 15922 | The limit of a geometric s... |
| geo2sum 15923 | The value of the finite ge... |
| geo2sum2 15924 | The value of the finite ge... |
| geo2lim 15925 | The value of the infinite ... |
| geomulcvg 15926 | The geometric series conve... |
| geoisum 15927 | The infinite sum of ` 1 + ... |
| geoisumr 15928 | The infinite sum of recipr... |
| geoisum1 15929 | The infinite sum of ` A ^ ... |
| geoisum1c 15930 | The infinite sum of ` A x.... |
| 0.999... 15931 | The recurring decimal 0.99... |
| geoihalfsum 15932 | Prove that the infinite ge... |
| cvgrat 15933 | Ratio test for convergence... |
| mertenslem1 15934 | Lemma for ~ mertens . (Co... |
| mertenslem2 15935 | Lemma for ~ mertens . (Co... |
| mertens 15936 | Mertens' theorem. If ` A ... |
| prodf 15937 | An infinite product of com... |
| clim2prod 15938 | The limit of an infinite p... |
| clim2div 15939 | The limit of an infinite p... |
| prodfmul 15940 | The product of two infinit... |
| prodf1 15941 | The value of the partial p... |
| prodf1f 15942 | A one-valued infinite prod... |
| prodfclim1 15943 | The constant one product c... |
| prodfn0 15944 | No term of a nonzero infin... |
| prodfrec 15945 | The reciprocal of an infin... |
| prodfdiv 15946 | The quotient of two infini... |
| ntrivcvg 15947 | A non-trivially converging... |
| ntrivcvgn0 15948 | A product that converges t... |
| ntrivcvgfvn0 15949 | Any value of a product seq... |
| ntrivcvgtail 15950 | A tail of a non-trivially ... |
| ntrivcvgmullem 15951 | Lemma for ~ ntrivcvgmul . ... |
| ntrivcvgmul 15952 | The product of two non-tri... |
| prodex 15955 | A product is a set. (Cont... |
| prodeq1f 15956 | Equality theorem for a pro... |
| prodeq1 15957 | Equality theorem for a pro... |
| nfcprod1 15958 | Bound-variable hypothesis ... |
| nfcprod 15959 | Bound-variable hypothesis ... |
| prodeq2w 15960 | Equality theorem for produ... |
| prodeq2ii 15961 | Equality theorem for produ... |
| prodeq2 15962 | Equality theorem for produ... |
| cbvprod 15963 | Change bound variable in a... |
| cbvprodv 15964 | Change bound variable in a... |
| cbvprodi 15965 | Change bound variable in a... |
| prodeq1i 15966 | Equality inference for pro... |
| prodeq1iOLD 15967 | Obsolete version of ~ prod... |
| prodeq2i 15968 | Equality inference for pro... |
| prodeq12i 15969 | Equality inference for pro... |
| prodeq1d 15970 | Equality deduction for pro... |
| prodeq2d 15971 | Equality deduction for pro... |
| prodeq2dv 15972 | Equality deduction for pro... |
| prodeq2sdv 15973 | Equality deduction for pro... |
| prodeq2sdvOLD 15974 | Obsolete version of ~ prod... |
| 2cprodeq2dv 15975 | Equality deduction for dou... |
| prodeq12dv 15976 | Equality deduction for pro... |
| prodeq12rdv 15977 | Equality deduction for pro... |
| prod2id 15978 | The second class argument ... |
| prodrblem 15979 | Lemma for ~ prodrb . (Con... |
| fprodcvg 15980 | The sequence of partial pr... |
| prodrblem2 15981 | Lemma for ~ prodrb . (Con... |
| prodrb 15982 | Rebase the starting point ... |
| prodmolem3 15983 | Lemma for ~ prodmo . (Con... |
| prodmolem2a 15984 | Lemma for ~ prodmo . (Con... |
| prodmolem2 15985 | Lemma for ~ prodmo . (Con... |
| prodmo 15986 | A product has at most one ... |
| zprod 15987 | Series product with index ... |
| iprod 15988 | Series product with an upp... |
| zprodn0 15989 | Nonzero series product wit... |
| iprodn0 15990 | Nonzero series product wit... |
| fprod 15991 | The value of a product ove... |
| fprodntriv 15992 | A non-triviality lemma for... |
| prod0 15993 | A product over the empty s... |
| prod1 15994 | Any product of one over a ... |
| prodfc 15995 | A lemma to facilitate conv... |
| fprodf1o 15996 | Re-index a finite product ... |
| prodss 15997 | Change the index set to a ... |
| fprodss 15998 | Change the index set to a ... |
| fprodser 15999 | A finite product expressed... |
| fprodcl2lem 16000 | Finite product closure lem... |
| fprodcllem 16001 | Finite product closure lem... |
| fprodcl 16002 | Closure of a finite produc... |
| fprodrecl 16003 | Closure of a finite produc... |
| fprodzcl 16004 | Closure of a finite produc... |
| fprodnncl 16005 | Closure of a finite produc... |
| fprodrpcl 16006 | Closure of a finite produc... |
| fprodnn0cl 16007 | Closure of a finite produc... |
| fprodcllemf 16008 | Finite product closure lem... |
| fprodreclf 16009 | Closure of a finite produc... |
| fprodmul 16010 | The product of two finite ... |
| fproddiv 16011 | The quotient of two finite... |
| prodsn 16012 | A product of a singleton i... |
| fprod1 16013 | A finite product of only o... |
| prodsnf 16014 | A product of a singleton i... |
| climprod1 16015 | The limit of a product ove... |
| fprodsplit 16016 | Split a finite product int... |
| fprodm1 16017 | Separate out the last term... |
| fprod1p 16018 | Separate out the first ter... |
| fprodp1 16019 | Multiply in the last term ... |
| fprodm1s 16020 | Separate out the last term... |
| fprodp1s 16021 | Multiply in the last term ... |
| prodsns 16022 | A product of the singleton... |
| fprodfac 16023 | Factorial using product no... |
| fprodabs 16024 | The absolute value of a fi... |
| fprodeq0 16025 | Any finite product contain... |
| fprodshft 16026 | Shift the index of a finit... |
| fprodrev 16027 | Reversal of a finite produ... |
| fprodconst 16028 | The product of constant te... |
| fprodn0 16029 | A finite product of nonzer... |
| fprod2dlem 16030 | Lemma for ~ fprod2d - indu... |
| fprod2d 16031 | Write a double product as ... |
| fprodxp 16032 | Combine two products into ... |
| fprodcnv 16033 | Transform a product region... |
| fprodcom2 16034 | Interchange order of multi... |
| fprodcom 16035 | Interchange product order.... |
| fprod0diag 16036 | Two ways to express "the p... |
| fproddivf 16037 | The quotient of two finite... |
| fprodsplitf 16038 | Split a finite product int... |
| fprodsplitsn 16039 | Separate out a term in a f... |
| fprodsplit1f 16040 | Separate out a term in a f... |
| fprodn0f 16041 | A finite product of nonzer... |
| fprodclf 16042 | Closure of a finite produc... |
| fprodge0 16043 | If all the terms of a fini... |
| fprodeq0g 16044 | Any finite product contain... |
| fprodge1 16045 | If all of the terms of a f... |
| fprodle 16046 | If all the terms of two fi... |
| fprodmodd 16047 | If all factors of two fini... |
| iprodclim 16048 | An infinite product equals... |
| iprodclim2 16049 | A converging product conve... |
| iprodclim3 16050 | The sequence of partial fi... |
| iprodcl 16051 | The product of a non-trivi... |
| iprodrecl 16052 | The product of a non-trivi... |
| iprodmul 16053 | Multiplication of infinite... |
| risefacval 16058 | The value of the rising fa... |
| fallfacval 16059 | The value of the falling f... |
| risefacval2 16060 | One-based value of rising ... |
| fallfacval2 16061 | One-based value of falling... |
| fallfacval3 16062 | A product representation o... |
| risefaccllem 16063 | Lemma for rising factorial... |
| fallfaccllem 16064 | Lemma for falling factoria... |
| risefaccl 16065 | Closure law for rising fac... |
| fallfaccl 16066 | Closure law for falling fa... |
| rerisefaccl 16067 | Closure law for rising fac... |
| refallfaccl 16068 | Closure law for falling fa... |
| nnrisefaccl 16069 | Closure law for rising fac... |
| zrisefaccl 16070 | Closure law for rising fac... |
| zfallfaccl 16071 | Closure law for falling fa... |
| nn0risefaccl 16072 | Closure law for rising fac... |
| rprisefaccl 16073 | Closure law for rising fac... |
| risefallfac 16074 | A relationship between ris... |
| fallrisefac 16075 | A relationship between fal... |
| risefall0lem 16076 | Lemma for ~ risefac0 and ~... |
| risefac0 16077 | The value of the rising fa... |
| fallfac0 16078 | The value of the falling f... |
| risefacp1 16079 | The value of the rising fa... |
| fallfacp1 16080 | The value of the falling f... |
| risefacp1d 16081 | The value of the rising fa... |
| fallfacp1d 16082 | The value of the falling f... |
| risefac1 16083 | The value of rising factor... |
| fallfac1 16084 | The value of falling facto... |
| risefacfac 16085 | Relate rising factorial to... |
| fallfacfwd 16086 | The forward difference of ... |
| 0fallfac 16087 | The value of the zero fall... |
| 0risefac 16088 | The value of the zero risi... |
| binomfallfaclem1 16089 | Lemma for ~ binomfallfac .... |
| binomfallfaclem2 16090 | Lemma for ~ binomfallfac .... |
| binomfallfac 16091 | A version of the binomial ... |
| binomrisefac 16092 | A version of the binomial ... |
| fallfacval4 16093 | Represent the falling fact... |
| bcfallfac 16094 | Binomial coefficient in te... |
| fallfacfac 16095 | Relate falling factorial t... |
| bpolylem 16098 | Lemma for ~ bpolyval . (C... |
| bpolyval 16099 | The value of the Bernoulli... |
| bpoly0 16100 | The value of the Bernoulli... |
| bpoly1 16101 | The value of the Bernoulli... |
| bpolycl 16102 | Closure law for Bernoulli ... |
| bpolysum 16103 | A sum for Bernoulli polyno... |
| bpolydiflem 16104 | Lemma for ~ bpolydif . (C... |
| bpolydif 16105 | Calculate the difference b... |
| fsumkthpow 16106 | A closed-form expression f... |
| bpoly2 16107 | The Bernoulli polynomials ... |
| bpoly3 16108 | The Bernoulli polynomials ... |
| bpoly4 16109 | The Bernoulli polynomials ... |
| fsumcube 16110 | Express the sum of cubes i... |
| eftcl 16123 | Closure of a term in the s... |
| reeftcl 16124 | The terms of the series ex... |
| eftabs 16125 | The absolute value of a te... |
| eftval 16126 | The value of a term in the... |
| efcllem 16127 | Lemma for ~ efcl . The se... |
| ef0lem 16128 | The series defining the ex... |
| efval 16129 | Value of the exponential f... |
| esum 16130 | Value of Euler's constant ... |
| eff 16131 | Domain and codomain of the... |
| efcl 16132 | Closure law for the expone... |
| efcld 16133 | Closure law for the expone... |
| efval2 16134 | Value of the exponential f... |
| efcvg 16135 | The series that defines th... |
| efcvgfsum 16136 | Exponential function conve... |
| reefcl 16137 | The exponential function i... |
| reefcld 16138 | The exponential function i... |
| ere 16139 | Euler's constant ` _e ` = ... |
| ege2le3 16140 | Lemma for ~ egt2lt3 . (Co... |
| ef0 16141 | Value of the exponential f... |
| efcj 16142 | The exponential of a compl... |
| efaddlem 16143 | Lemma for ~ efadd (exponen... |
| efadd 16144 | Sum of exponents law for e... |
| fprodefsum 16145 | Move the exponential funct... |
| efcan 16146 | Cancellation law for expon... |
| efne0 16147 | The exponential of a compl... |
| efneg 16148 | The exponential of the opp... |
| eff2 16149 | The exponential function m... |
| efsub 16150 | Difference of exponents la... |
| efexp 16151 | The exponential of an inte... |
| efzval 16152 | Value of the exponential f... |
| efgt0 16153 | The exponential of a real ... |
| rpefcl 16154 | The exponential of a real ... |
| rpefcld 16155 | The exponential of a real ... |
| eftlcvg 16156 | The tail series of the exp... |
| eftlcl 16157 | Closure of the sum of an i... |
| reeftlcl 16158 | Closure of the sum of an i... |
| eftlub 16159 | An upper bound on the abso... |
| efsep 16160 | Separate out the next term... |
| effsumlt 16161 | The partial sums of the se... |
| eft0val 16162 | The value of the first ter... |
| ef4p 16163 | Separate out the first fou... |
| efgt1p2 16164 | The exponential of a posit... |
| efgt1p 16165 | The exponential of a posit... |
| efgt1 16166 | The exponential of a posit... |
| eflt 16167 | The exponential function o... |
| efle 16168 | The exponential function o... |
| reef11 16169 | The exponential function o... |
| reeff1 16170 | The exponential function m... |
| eflegeo 16171 | The exponential function o... |
| sinval 16172 | Value of the sine function... |
| cosval 16173 | Value of the cosine functi... |
| sinf 16174 | Domain and codomain of the... |
| cosf 16175 | Domain and codomain of the... |
| sincl 16176 | Closure of the sine functi... |
| coscl 16177 | Closure of the cosine func... |
| tanval 16178 | Value of the tangent funct... |
| tancl 16179 | The closure of the tangent... |
| sincld 16180 | Closure of the sine functi... |
| coscld 16181 | Closure of the cosine func... |
| tancld 16182 | Closure of the tangent fun... |
| tanval2 16183 | Express the tangent functi... |
| tanval3 16184 | Express the tangent functi... |
| resinval 16185 | The sine of a real number ... |
| recosval 16186 | The cosine of a real numbe... |
| efi4p 16187 | Separate out the first fou... |
| resin4p 16188 | Separate out the first fou... |
| recos4p 16189 | Separate out the first fou... |
| resincl 16190 | The sine of a real number ... |
| recoscl 16191 | The cosine of a real numbe... |
| retancl 16192 | The closure of the tangent... |
| resincld 16193 | Closure of the sine functi... |
| recoscld 16194 | Closure of the cosine func... |
| retancld 16195 | Closure of the tangent fun... |
| sinneg 16196 | The sine of a negative is ... |
| cosneg 16197 | The cosines of a number an... |
| tanneg 16198 | The tangent of a negative ... |
| sin0 16199 | Value of the sine function... |
| cos0 16200 | Value of the cosine functi... |
| tan0 16201 | The value of the tangent f... |
| efival 16202 | The exponential function i... |
| efmival 16203 | The exponential function i... |
| sinhval 16204 | Value of the hyperbolic si... |
| coshval 16205 | Value of the hyperbolic co... |
| resinhcl 16206 | The hyperbolic sine of a r... |
| rpcoshcl 16207 | The hyperbolic cosine of a... |
| recoshcl 16208 | The hyperbolic cosine of a... |
| retanhcl 16209 | The hyperbolic tangent of ... |
| tanhlt1 16210 | The hyperbolic tangent of ... |
| tanhbnd 16211 | The hyperbolic tangent of ... |
| efeul 16212 | Eulerian representation of... |
| efieq 16213 | The exponentials of two im... |
| sinadd 16214 | Addition formula for sine.... |
| cosadd 16215 | Addition formula for cosin... |
| tanaddlem 16216 | A useful intermediate step... |
| tanadd 16217 | Addition formula for tange... |
| sinsub 16218 | Sine of difference. (Cont... |
| cossub 16219 | Cosine of difference. (Co... |
| addsin 16220 | Sum of sines. (Contribute... |
| subsin 16221 | Difference of sines. (Con... |
| sinmul 16222 | Product of sines can be re... |
| cosmul 16223 | Product of cosines can be ... |
| addcos 16224 | Sum of cosines. (Contribu... |
| subcos 16225 | Difference of cosines. (C... |
| sincossq 16226 | Sine squared plus cosine s... |
| sin2t 16227 | Double-angle formula for s... |
| cos2t 16228 | Double-angle formula for c... |
| cos2tsin 16229 | Double-angle formula for c... |
| sinbnd 16230 | The sine of a real number ... |
| cosbnd 16231 | The cosine of a real numbe... |
| sinbnd2 16232 | The sine of a real number ... |
| cosbnd2 16233 | The cosine of a real numbe... |
| ef01bndlem 16234 | Lemma for ~ sin01bnd and ~... |
| sin01bnd 16235 | Bounds on the sine of a po... |
| cos01bnd 16236 | Bounds on the cosine of a ... |
| cos1bnd 16237 | Bounds on the cosine of 1.... |
| cos2bnd 16238 | Bounds on the cosine of 2.... |
| sinltx 16239 | The sine of a positive rea... |
| sin01gt0 16240 | The sine of a positive rea... |
| cos01gt0 16241 | The cosine of a positive r... |
| sin02gt0 16242 | The sine of a positive rea... |
| sincos1sgn 16243 | The signs of the sine and ... |
| sincos2sgn 16244 | The signs of the sine and ... |
| sin4lt0 16245 | The sine of 4 is negative.... |
| absefi 16246 | The absolute value of the ... |
| absef 16247 | The absolute value of the ... |
| absefib 16248 | A complex number is real i... |
| efieq1re 16249 | A number whose imaginary e... |
| demoivre 16250 | De Moivre's Formula. Proo... |
| demoivreALT 16251 | Alternate proof of ~ demoi... |
| eirrlem 16254 | Lemma for ~ eirr . (Contr... |
| eirr 16255 | ` _e ` is irrational. (Co... |
| egt2lt3 16256 | Euler's constant ` _e ` = ... |
| epos 16257 | Euler's constant ` _e ` is... |
| epr 16258 | Euler's constant ` _e ` is... |
| ene0 16259 | ` _e ` is not 0. (Contrib... |
| ene1 16260 | ` _e ` is not 1. (Contrib... |
| xpnnen 16261 | The Cartesian product of t... |
| znnen 16262 | The set of integers and th... |
| qnnen 16263 | The rational numbers are c... |
| rpnnen2lem1 16264 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem2 16265 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem3 16266 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem4 16267 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem5 16268 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem6 16269 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem7 16270 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem8 16271 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem9 16272 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem10 16273 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem11 16274 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem12 16275 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2 16276 | The other half of ~ rpnnen... |
| rpnnen 16277 | The cardinality of the con... |
| rexpen 16278 | The real numbers are equin... |
| cpnnen 16279 | The complex numbers are eq... |
| rucALT 16280 | Alternate proof of ~ ruc .... |
| ruclem1 16281 | Lemma for ~ ruc (the reals... |
| ruclem2 16282 | Lemma for ~ ruc . Orderin... |
| ruclem3 16283 | Lemma for ~ ruc . The con... |
| ruclem4 16284 | Lemma for ~ ruc . Initial... |
| ruclem6 16285 | Lemma for ~ ruc . Domain ... |
| ruclem7 16286 | Lemma for ~ ruc . Success... |
| ruclem8 16287 | Lemma for ~ ruc . The int... |
| ruclem9 16288 | Lemma for ~ ruc . The fir... |
| ruclem10 16289 | Lemma for ~ ruc . Every f... |
| ruclem11 16290 | Lemma for ~ ruc . Closure... |
| ruclem12 16291 | Lemma for ~ ruc . The sup... |
| ruclem13 16292 | Lemma for ~ ruc . There i... |
| ruc 16293 | The set of positive intege... |
| resdomq 16294 | The set of rationals is st... |
| aleph1re 16295 | There are at least aleph-o... |
| aleph1irr 16296 | There are at least aleph-o... |
| cnso 16297 | The complex numbers can be... |
| sqrt2irrlem 16298 | Lemma for ~ sqrt2irr . Th... |
| sqrt2irr 16299 | The square root of 2 is ir... |
| sqrt2re 16300 | The square root of 2 exist... |
| sqrt2irr0 16301 | The square root of 2 is an... |
| nthruc 16302 | The sequence ` NN ` , ` ZZ... |
| nthruz 16303 | The sequence ` NN ` , ` NN... |
| divides 16306 | Define the divides relatio... |
| dvdsval2 16307 | One nonzero integer divide... |
| dvdsval3 16308 | One nonzero integer divide... |
| dvdszrcl 16309 | Reverse closure for the di... |
| dvdsmod0 16310 | If a positive integer divi... |
| p1modz1 16311 | If a number greater than 1... |
| dvdsmodexp 16312 | If a positive integer divi... |
| nndivdvds 16313 | Strong form of ~ dvdsval2 ... |
| nndivides 16314 | Definition of the divides ... |
| moddvds 16315 | Two ways to say ` A == B `... |
| modm1div 16316 | An integer greater than on... |
| dvds0lem 16317 | A lemma to assist theorems... |
| dvds1lem 16318 | A lemma to assist theorems... |
| dvds2lem 16319 | A lemma to assist theorems... |
| iddvds 16320 | An integer divides itself.... |
| 1dvds 16321 | 1 divides any integer. Th... |
| dvds0 16322 | Any integer divides 0. Th... |
| negdvdsb 16323 | An integer divides another... |
| dvdsnegb 16324 | An integer divides another... |
| absdvdsb 16325 | An integer divides another... |
| dvdsabsb 16326 | An integer divides another... |
| 0dvds 16327 | Only 0 is divisible by 0. ... |
| dvdsmul1 16328 | An integer divides a multi... |
| dvdsmul2 16329 | An integer divides a multi... |
| iddvdsexp 16330 | An integer divides a posit... |
| muldvds1 16331 | If a product divides an in... |
| muldvds2 16332 | If a product divides an in... |
| dvdscmul 16333 | Multiplication by a consta... |
| dvdsmulc 16334 | Multiplication by a consta... |
| dvdscmulr 16335 | Cancellation law for the d... |
| dvdsmulcr 16336 | Cancellation law for the d... |
| summodnegmod 16337 | The sum of two integers mo... |
| modmulconst 16338 | Constant multiplication in... |
| dvds2ln 16339 | If an integer divides each... |
| dvds2add 16340 | If an integer divides each... |
| dvds2sub 16341 | If an integer divides each... |
| dvds2addd 16342 | Deduction form of ~ dvds2a... |
| dvds2subd 16343 | Deduction form of ~ dvds2s... |
| dvdstr 16344 | The divides relation is tr... |
| dvdstrd 16345 | The divides relation is tr... |
| dvdsmultr1 16346 | If an integer divides anot... |
| dvdsmultr1d 16347 | Deduction form of ~ dvdsmu... |
| dvdsmultr2 16348 | If an integer divides anot... |
| dvdsmultr2d 16349 | Deduction form of ~ dvdsmu... |
| ordvdsmul 16350 | If an integer divides eith... |
| dvdssub2 16351 | If an integer divides a di... |
| dvdsadd 16352 | An integer divides another... |
| dvdsaddr 16353 | An integer divides another... |
| dvdssub 16354 | An integer divides another... |
| dvdssubr 16355 | An integer divides another... |
| dvdsadd2b 16356 | Adding a multiple of the b... |
| dvdsaddre2b 16357 | Adding a multiple of the b... |
| fsumdvds 16358 | If every term in a sum is ... |
| dvdslelem 16359 | Lemma for ~ dvdsle . (Con... |
| dvdsle 16360 | The divisors of a positive... |
| dvdsleabs 16361 | The divisors of a nonzero ... |
| dvdsleabs2 16362 | Transfer divisibility to a... |
| dvdsabseq 16363 | If two integers divide eac... |
| dvdseq 16364 | If two nonnegative integer... |
| divconjdvds 16365 | If a nonzero integer ` M `... |
| dvdsdivcl 16366 | The complement of a diviso... |
| dvdsflip 16367 | An involution of the divis... |
| dvdsssfz1 16368 | The set of divisors of a n... |
| dvds1 16369 | The only nonnegative integ... |
| alzdvds 16370 | Only 0 is divisible by all... |
| dvdsext 16371 | Poset extensionality for d... |
| fzm1ndvds 16372 | No number between ` 1 ` an... |
| fzo0dvdseq 16373 | Zero is the only one of th... |
| fzocongeq 16374 | Two different elements of ... |
| addmodlteqALT 16375 | Two nonnegative integers l... |
| dvdsfac 16376 | A positive integer divides... |
| dvdsexp2im 16377 | If an integer divides anot... |
| dvdsexp 16378 | A power divides a power wi... |
| dvdsmod 16379 | Any number ` K ` whose mod... |
| mulmoddvds 16380 | If an integer is divisible... |
| 3dvds 16381 | A rule for divisibility by... |
| 3dvdsdec 16382 | A decimal number is divisi... |
| 3dvds2dec 16383 | A decimal number is divisi... |
| fprodfvdvdsd 16384 | A finite product of intege... |
| fproddvdsd 16385 | A finite product of intege... |
| evenelz 16386 | An even number is an integ... |
| zeo3 16387 | An integer is even or odd.... |
| zeo4 16388 | An integer is even or odd ... |
| zeneo 16389 | No even integer equals an ... |
| odd2np1lem 16390 | Lemma for ~ odd2np1 . (Co... |
| odd2np1 16391 | An integer is odd iff it i... |
| even2n 16392 | An integer is even iff it ... |
| oddm1even 16393 | An integer is odd iff its ... |
| oddp1even 16394 | An integer is odd iff its ... |
| oexpneg 16395 | The exponential of the neg... |
| mod2eq0even 16396 | An integer is 0 modulo 2 i... |
| mod2eq1n2dvds 16397 | An integer is 1 modulo 2 i... |
| oddnn02np1 16398 | A nonnegative integer is o... |
| oddge22np1 16399 | An integer greater than on... |
| evennn02n 16400 | A nonnegative integer is e... |
| evennn2n 16401 | A positive integer is even... |
| 2tp1odd 16402 | A number which is twice an... |
| mulsucdiv2z 16403 | An integer multiplied with... |
| sqoddm1div8z 16404 | A squared odd number minus... |
| 2teven 16405 | A number which is twice an... |
| zeo5 16406 | An integer is either even ... |
| evend2 16407 | An integer is even iff its... |
| oddp1d2 16408 | An integer is odd iff its ... |
| zob 16409 | Alternate characterization... |
| oddm1d2 16410 | An integer is odd iff its ... |
| ltoddhalfle 16411 | An integer is less than ha... |
| halfleoddlt 16412 | An integer is greater than... |
| opoe 16413 | The sum of two odds is eve... |
| omoe 16414 | The difference of two odds... |
| opeo 16415 | The sum of an odd and an e... |
| omeo 16416 | The difference of an odd a... |
| z0even 16417 | 2 divides 0. That means 0... |
| n2dvds1 16418 | 2 does not divide 1. That... |
| n2dvdsm1 16419 | 2 does not divide -1. Tha... |
| z2even 16420 | 2 divides 2. That means 2... |
| n2dvds3 16421 | 2 does not divide 3. That... |
| z4even 16422 | 2 divides 4. That means 4... |
| 4dvdseven 16423 | An integer which is divisi... |
| m1expe 16424 | Exponentiation of -1 by an... |
| m1expo 16425 | Exponentiation of -1 by an... |
| m1exp1 16426 | Exponentiation of negative... |
| nn0enne 16427 | A positive integer is an e... |
| nn0ehalf 16428 | The half of an even nonneg... |
| nnehalf 16429 | The half of an even positi... |
| nn0onn 16430 | An odd nonnegative integer... |
| nn0o1gt2 16431 | An odd nonnegative integer... |
| nno 16432 | An alternate characterizat... |
| nn0o 16433 | An alternate characterizat... |
| nn0ob 16434 | Alternate characterization... |
| nn0oddm1d2 16435 | A positive integer is odd ... |
| nnoddm1d2 16436 | A positive integer is odd ... |
| sumeven 16437 | If every term in a sum is ... |
| sumodd 16438 | If every term in a sum is ... |
| evensumodd 16439 | If every term in a sum wit... |
| oddsumodd 16440 | If every term in a sum wit... |
| pwp1fsum 16441 | The n-th power of a number... |
| oddpwp1fsum 16442 | An odd power of a number i... |
| divalglem0 16443 | Lemma for ~ divalg . (Con... |
| divalglem1 16444 | Lemma for ~ divalg . (Con... |
| divalglem2 16445 | Lemma for ~ divalg . (Con... |
| divalglem4 16446 | Lemma for ~ divalg . (Con... |
| divalglem5 16447 | Lemma for ~ divalg . (Con... |
| divalglem6 16448 | Lemma for ~ divalg . (Con... |
| divalglem7 16449 | Lemma for ~ divalg . (Con... |
| divalglem8 16450 | Lemma for ~ divalg . (Con... |
| divalglem9 16451 | Lemma for ~ divalg . (Con... |
| divalglem10 16452 | Lemma for ~ divalg . (Con... |
| divalg 16453 | The division algorithm (th... |
| divalgb 16454 | Express the division algor... |
| divalg2 16455 | The division algorithm (th... |
| divalgmod 16456 | The result of the ` mod ` ... |
| divalgmodcl 16457 | The result of the ` mod ` ... |
| modremain 16458 | The result of the modulo o... |
| ndvdssub 16459 | Corollary of the division ... |
| ndvdsadd 16460 | Corollary of the division ... |
| ndvdsp1 16461 | Special case of ~ ndvdsadd... |
| ndvdsi 16462 | A quick test for non-divis... |
| flodddiv4 16463 | The floor of an odd intege... |
| fldivndvdslt 16464 | The floor of an integer di... |
| flodddiv4lt 16465 | The floor of an odd number... |
| flodddiv4t2lthalf 16466 | The floor of an odd number... |
| bitsfval 16471 | Expand the definition of t... |
| bitsval 16472 | Expand the definition of t... |
| bitsval2 16473 | Expand the definition of t... |
| bitsss 16474 | The set of bits of an inte... |
| bitsf 16475 | The ` bits ` function is a... |
| bits0 16476 | Value of the zeroth bit. ... |
| bits0e 16477 | The zeroth bit of an even ... |
| bits0o 16478 | The zeroth bit of an odd n... |
| bitsp1 16479 | The ` M + 1 ` -th bit of `... |
| bitsp1e 16480 | The ` M + 1 ` -th bit of `... |
| bitsp1o 16481 | The ` M + 1 ` -th bit of `... |
| bitsfzolem 16482 | Lemma for ~ bitsfzo . (Co... |
| bitsfzo 16483 | The bits of a number are a... |
| bitsmod 16484 | Truncating the bit sequenc... |
| bitsfi 16485 | Every number is associated... |
| bitscmp 16486 | The bit complement of ` N ... |
| 0bits 16487 | The bits of zero. (Contri... |
| m1bits 16488 | The bits of negative one. ... |
| bitsinv1lem 16489 | Lemma for ~ bitsinv1 . (C... |
| bitsinv1 16490 | There is an explicit inver... |
| bitsinv2 16491 | There is an explicit inver... |
| bitsf1ocnv 16492 | The ` bits ` function rest... |
| bitsf1o 16493 | The ` bits ` function rest... |
| bitsf1 16494 | The ` bits ` function is a... |
| 2ebits 16495 | The bits of a power of two... |
| bitsinv 16496 | The inverse of the ` bits ... |
| bitsinvp1 16497 | Recursive definition of th... |
| sadadd2lem2 16498 | The core of the proof of ~... |
| sadfval 16500 | Define the addition of two... |
| sadcf 16501 | The carry sequence is a se... |
| sadc0 16502 | The initial element of the... |
| sadcp1 16503 | The carry sequence (which ... |
| sadval 16504 | The full adder sequence is... |
| sadcaddlem 16505 | Lemma for ~ sadcadd . (Co... |
| sadcadd 16506 | Non-recursive definition o... |
| sadadd2lem 16507 | Lemma for ~ sadadd2 . (Co... |
| sadadd2 16508 | Sum of initial segments of... |
| sadadd3 16509 | Sum of initial segments of... |
| sadcl 16510 | The sum of two sequences i... |
| sadcom 16511 | The adder sequence functio... |
| saddisjlem 16512 | Lemma for ~ sadadd . (Con... |
| saddisj 16513 | The sum of disjoint sequen... |
| sadaddlem 16514 | Lemma for ~ sadadd . (Con... |
| sadadd 16515 | For sequences that corresp... |
| sadid1 16516 | The adder sequence functio... |
| sadid2 16517 | The adder sequence functio... |
| sadasslem 16518 | Lemma for ~ sadass . (Con... |
| sadass 16519 | Sequence addition is assoc... |
| sadeq 16520 | Any element of a sequence ... |
| bitsres 16521 | Restrict the bits of a num... |
| bitsuz 16522 | The bits of a number are a... |
| bitsshft 16523 | Shifting a bit sequence to... |
| smufval 16525 | The multiplication of two ... |
| smupf 16526 | The sequence of partial su... |
| smup0 16527 | The initial element of the... |
| smupp1 16528 | The initial element of the... |
| smuval 16529 | Define the addition of two... |
| smuval2 16530 | The partial sum sequence s... |
| smupvallem 16531 | If ` A ` only has elements... |
| smucl 16532 | The product of two sequenc... |
| smu01lem 16533 | Lemma for ~ smu01 and ~ sm... |
| smu01 16534 | Multiplication of a sequen... |
| smu02 16535 | Multiplication of a sequen... |
| smupval 16536 | Rewrite the elements of th... |
| smup1 16537 | Rewrite ~ smupp1 using onl... |
| smueqlem 16538 | Any element of a sequence ... |
| smueq 16539 | Any element of a sequence ... |
| smumullem 16540 | Lemma for ~ smumul . (Con... |
| smumul 16541 | For sequences that corresp... |
| gcdval 16544 | The value of the ` gcd ` o... |
| gcd0val 16545 | The value, by convention, ... |
| gcdn0val 16546 | The value of the ` gcd ` o... |
| gcdcllem1 16547 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem2 16548 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem3 16549 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdn0cl 16550 | Closure of the ` gcd ` ope... |
| gcddvds 16551 | The gcd of two integers di... |
| dvdslegcd 16552 | An integer which divides b... |
| nndvdslegcd 16553 | A positive integer which d... |
| gcdcl 16554 | Closure of the ` gcd ` ope... |
| gcdnncl 16555 | Closure of the ` gcd ` ope... |
| gcdcld 16556 | Closure of the ` gcd ` ope... |
| gcd2n0cl 16557 | Closure of the ` gcd ` ope... |
| zeqzmulgcd 16558 | An integer is the product ... |
| divgcdz 16559 | An integer divided by the ... |
| gcdf 16560 | Domain and codomain of the... |
| gcdcom 16561 | The ` gcd ` operator is co... |
| gcdcomd 16562 | The ` gcd ` operator is co... |
| divgcdnn 16563 | A positive integer divided... |
| divgcdnnr 16564 | A positive integer divided... |
| gcdeq0 16565 | The gcd of two integers is... |
| gcdn0gt0 16566 | The gcd of two integers is... |
| gcd0id 16567 | The gcd of 0 and an intege... |
| gcdid0 16568 | The gcd of an integer and ... |
| nn0gcdid0 16569 | The gcd of a nonnegative i... |
| gcdneg 16570 | Negating one operand of th... |
| neggcd 16571 | Negating one operand of th... |
| gcdaddmlem 16572 | Lemma for ~ gcdaddm . (Co... |
| gcdaddm 16573 | Adding a multiple of one o... |
| gcdadd 16574 | The GCD of two numbers is ... |
| gcdid 16575 | The gcd of a number and it... |
| gcd1 16576 | The gcd of a number with 1... |
| gcdabs1 16577 | ` gcd ` of the absolute va... |
| gcdabs2 16578 | ` gcd ` of the absolute va... |
| gcdabs 16579 | The gcd of two integers is... |
| gcdabsOLD 16580 | Obsolete version of ~ gcda... |
| modgcd 16581 | The gcd remains unchanged ... |
| 1gcd 16582 | The GCD of one and an inte... |
| gcdmultipled 16583 | The greatest common diviso... |
| gcdmultiplez 16584 | The GCD of a multiple of a... |
| gcdmultiple 16585 | The GCD of a multiple of a... |
| dvdsgcdidd 16586 | The greatest common diviso... |
| 6gcd4e2 16587 | The greatest common diviso... |
| bezoutlem1 16588 | Lemma for ~ bezout . (Con... |
| bezoutlem2 16589 | Lemma for ~ bezout . (Con... |
| bezoutlem3 16590 | Lemma for ~ bezout . (Con... |
| bezoutlem4 16591 | Lemma for ~ bezout . (Con... |
| bezout 16592 | Bézout's identity: ... |
| dvdsgcd 16593 | An integer which divides e... |
| dvdsgcdb 16594 | Biconditional form of ~ dv... |
| dfgcd2 16595 | Alternate definition of th... |
| gcdass 16596 | Associative law for ` gcd ... |
| mulgcd 16597 | Distribute multiplication ... |
| absmulgcd 16598 | Distribute absolute value ... |
| mulgcdr 16599 | Reverse distribution law f... |
| gcddiv 16600 | Division law for GCD. (Con... |
| gcdzeq 16601 | A positive integer ` A ` i... |
| gcdeq 16602 | ` A ` is equal to its gcd ... |
| dvdssqim 16603 | Unidirectional form of ~ d... |
| dvdsexpim 16604 | If two numbers are divisib... |
| dvdsmulgcd 16605 | A divisibility equivalent ... |
| rpmulgcd 16606 | If ` K ` and ` M ` are rel... |
| rplpwr 16607 | If ` A ` and ` B ` are rel... |
| rprpwr 16608 | If ` A ` and ` B ` are rel... |
| rppwr 16609 | If ` A ` and ` B ` are rel... |
| nn0rppwr 16610 | If ` A ` and ` B ` are rel... |
| sqgcd 16611 | Square distributes over gc... |
| expgcd 16612 | Exponentiation distributes... |
| nn0expgcd 16613 | Exponentiation distributes... |
| zexpgcd 16614 | Exponentiation distributes... |
| dvdssqlem 16615 | Lemma for ~ dvdssq . (Con... |
| dvdssq 16616 | Two numbers are divisible ... |
| bezoutr 16617 | Partial converse to ~ bezo... |
| bezoutr1 16618 | Converse of ~ bezout for w... |
| nn0seqcvgd 16619 | A strictly-decreasing nonn... |
| seq1st 16620 | A sequence whose iteration... |
| algr0 16621 | The value of the algorithm... |
| algrf 16622 | An algorithm is a step fun... |
| algrp1 16623 | The value of the algorithm... |
| alginv 16624 | If ` I ` is an invariant o... |
| algcvg 16625 | One way to prove that an a... |
| algcvgblem 16626 | Lemma for ~ algcvgb . (Co... |
| algcvgb 16627 | Two ways of expressing tha... |
| algcvga 16628 | The countdown function ` C... |
| algfx 16629 | If ` F ` reaches a fixed p... |
| eucalgval2 16630 | The value of the step func... |
| eucalgval 16631 | Euclid's Algorithm ~ eucal... |
| eucalgf 16632 | Domain and codomain of the... |
| eucalginv 16633 | The invariant of the step ... |
| eucalglt 16634 | The second member of the s... |
| eucalgcvga 16635 | Once Euclid's Algorithm ha... |
| eucalg 16636 | Euclid's Algorithm compute... |
| lcmval 16641 | Value of the ` lcm ` opera... |
| lcmcom 16642 | The ` lcm ` operator is co... |
| lcm0val 16643 | The value, by convention, ... |
| lcmn0val 16644 | The value of the ` lcm ` o... |
| lcmcllem 16645 | Lemma for ~ lcmn0cl and ~ ... |
| lcmn0cl 16646 | Closure of the ` lcm ` ope... |
| dvdslcm 16647 | The lcm of two integers is... |
| lcmledvds 16648 | A positive integer which b... |
| lcmeq0 16649 | The lcm of two integers is... |
| lcmcl 16650 | Closure of the ` lcm ` ope... |
| gcddvdslcm 16651 | The greatest common diviso... |
| lcmneg 16652 | Negating one operand of th... |
| neglcm 16653 | Negating one operand of th... |
| lcmabs 16654 | The lcm of two integers is... |
| lcmgcdlem 16655 | Lemma for ~ lcmgcd and ~ l... |
| lcmgcd 16656 | The product of two numbers... |
| lcmdvds 16657 | The lcm of two integers di... |
| lcmid 16658 | The lcm of an integer and ... |
| lcm1 16659 | The lcm of an integer and ... |
| lcmgcdnn 16660 | The product of two positiv... |
| lcmgcdeq 16661 | Two integers' absolute val... |
| lcmdvdsb 16662 | Biconditional form of ~ lc... |
| lcmass 16663 | Associative law for ` lcm ... |
| 3lcm2e6woprm 16664 | The least common multiple ... |
| 6lcm4e12 16665 | The least common multiple ... |
| absproddvds 16666 | The absolute value of the ... |
| absprodnn 16667 | The absolute value of the ... |
| fissn0dvds 16668 | For each finite subset of ... |
| fissn0dvdsn0 16669 | For each finite subset of ... |
| lcmfval 16670 | Value of the ` _lcm ` func... |
| lcmf0val 16671 | The value, by convention, ... |
| lcmfn0val 16672 | The value of the ` _lcm ` ... |
| lcmfnnval 16673 | The value of the ` _lcm ` ... |
| lcmfcllem 16674 | Lemma for ~ lcmfn0cl and ~... |
| lcmfn0cl 16675 | Closure of the ` _lcm ` fu... |
| lcmfpr 16676 | The value of the ` _lcm ` ... |
| lcmfcl 16677 | Closure of the ` _lcm ` fu... |
| lcmfnncl 16678 | Closure of the ` _lcm ` fu... |
| lcmfeq0b 16679 | The least common multiple ... |
| dvdslcmf 16680 | The least common multiple ... |
| lcmfledvds 16681 | A positive integer which i... |
| lcmf 16682 | Characterization of the le... |
| lcmf0 16683 | The least common multiple ... |
| lcmfsn 16684 | The least common multiple ... |
| lcmftp 16685 | The least common multiple ... |
| lcmfunsnlem1 16686 | Lemma for ~ lcmfdvds and ~... |
| lcmfunsnlem2lem1 16687 | Lemma 1 for ~ lcmfunsnlem2... |
| lcmfunsnlem2lem2 16688 | Lemma 2 for ~ lcmfunsnlem2... |
| lcmfunsnlem2 16689 | Lemma for ~ lcmfunsn and ~... |
| lcmfunsnlem 16690 | Lemma for ~ lcmfdvds and ~... |
| lcmfdvds 16691 | The least common multiple ... |
| lcmfdvdsb 16692 | Biconditional form of ~ lc... |
| lcmfunsn 16693 | The ` _lcm ` function for ... |
| lcmfun 16694 | The ` _lcm ` function for ... |
| lcmfass 16695 | Associative law for the ` ... |
| lcmf2a3a4e12 16696 | The least common multiple ... |
| lcmflefac 16697 | The least common multiple ... |
| coprmgcdb 16698 | Two positive integers are ... |
| ncoprmgcdne1b 16699 | Two positive integers are ... |
| ncoprmgcdgt1b 16700 | Two positive integers are ... |
| coprmdvds1 16701 | If two positive integers a... |
| coprmdvds 16702 | Euclid's Lemma (see ProofW... |
| coprmdvds2 16703 | If an integer is divisible... |
| mulgcddvds 16704 | One half of ~ rpmulgcd2 , ... |
| rpmulgcd2 16705 | If ` M ` is relatively pri... |
| qredeq 16706 | Two equal reduced fraction... |
| qredeu 16707 | Every rational number has ... |
| rpmul 16708 | If ` K ` is relatively pri... |
| rpdvds 16709 | If ` K ` is relatively pri... |
| coprmprod 16710 | The product of the element... |
| coprmproddvdslem 16711 | Lemma for ~ coprmproddvds ... |
| coprmproddvds 16712 | If a positive integer is d... |
| congr 16713 | Definition of congruence b... |
| divgcdcoprm0 16714 | Integers divided by gcd ar... |
| divgcdcoprmex 16715 | Integers divided by gcd ar... |
| cncongr1 16716 | One direction of the bicon... |
| cncongr2 16717 | The other direction of the... |
| cncongr 16718 | Cancellability of Congruen... |
| cncongrcoprm 16719 | Corollary 1 of Cancellabil... |
| isprm 16722 | The predicate "is a prime ... |
| prmnn 16723 | A prime number is a positi... |
| prmz 16724 | A prime number is an integ... |
| prmssnn 16725 | The prime numbers are a su... |
| prmex 16726 | The set of prime numbers e... |
| 0nprm 16727 | 0 is not a prime number. ... |
| 1nprm 16728 | 1 is not a prime number. ... |
| 1idssfct 16729 | The positive divisors of a... |
| isprm2lem 16730 | Lemma for ~ isprm2 . (Con... |
| isprm2 16731 | The predicate "is a prime ... |
| isprm3 16732 | The predicate "is a prime ... |
| isprm4 16733 | The predicate "is a prime ... |
| prmind2 16734 | A variation on ~ prmind as... |
| prmind 16735 | Perform induction over the... |
| dvdsprime 16736 | If ` M ` divides a prime, ... |
| nprm 16737 | A product of two integers ... |
| nprmi 16738 | An inference for composite... |
| dvdsnprmd 16739 | If a number is divisible b... |
| prm2orodd 16740 | A prime number is either 2... |
| 2prm 16741 | 2 is a prime number. (Con... |
| 2mulprm 16742 | A multiple of two is prime... |
| 3prm 16743 | 3 is a prime number. (Con... |
| 4nprm 16744 | 4 is not a prime number. ... |
| prmuz2 16745 | A prime number is an integ... |
| prmgt1 16746 | A prime number is an integ... |
| prmm2nn0 16747 | Subtracting 2 from a prime... |
| oddprmgt2 16748 | An odd prime is greater th... |
| oddprmge3 16749 | An odd prime is greater th... |
| ge2nprmge4 16750 | A composite integer greate... |
| sqnprm 16751 | A square is never prime. ... |
| dvdsprm 16752 | An integer greater than or... |
| exprmfct 16753 | Every integer greater than... |
| prmdvdsfz 16754 | Each integer greater than ... |
| nprmdvds1 16755 | No prime number divides 1.... |
| isprm5 16756 | One need only check prime ... |
| isprm7 16757 | One need only check prime ... |
| maxprmfct 16758 | The set of prime factors o... |
| divgcdodd 16759 | Either ` A / ( A gcd B ) `... |
| coprm 16760 | A prime number either divi... |
| prmrp 16761 | Unequal prime numbers are ... |
| euclemma 16762 | Euclid's lemma. A prime n... |
| isprm6 16763 | A number is prime iff it s... |
| prmdvdsexp 16764 | A prime divides a positive... |
| prmdvdsexpb 16765 | A prime divides a positive... |
| prmdvdsexpr 16766 | If a prime divides a nonne... |
| prmdvdssq 16767 | Condition for a prime divi... |
| prmexpb 16768 | Two positive prime powers ... |
| prmfac1 16769 | The factorial of a number ... |
| dvdszzq 16770 | Divisibility for an intege... |
| rpexp 16771 | If two numbers ` A ` and `... |
| rpexp1i 16772 | Relative primality passes ... |
| rpexp12i 16773 | Relative primality passes ... |
| prmndvdsfaclt 16774 | A prime number does not di... |
| prmdvdsbc 16775 | Condition for a prime numb... |
| prmdvdsncoprmbd 16776 | Two positive integers are ... |
| ncoprmlnprm 16777 | If two positive integers a... |
| cncongrprm 16778 | Corollary 2 of Cancellabil... |
| isevengcd2 16779 | The predicate "is an even ... |
| isoddgcd1 16780 | The predicate "is an odd n... |
| 3lcm2e6 16781 | The least common multiple ... |
| qnumval 16786 | Value of the canonical num... |
| qdenval 16787 | Value of the canonical den... |
| qnumdencl 16788 | Lemma for ~ qnumcl and ~ q... |
| qnumcl 16789 | The canonical numerator of... |
| qdencl 16790 | The canonical denominator ... |
| fnum 16791 | Canonical numerator define... |
| fden 16792 | Canonical denominator defi... |
| qnumdenbi 16793 | Two numbers are the canoni... |
| qnumdencoprm 16794 | The canonical representati... |
| qeqnumdivden 16795 | Recover a rational number ... |
| qmuldeneqnum 16796 | Multiplying a rational by ... |
| divnumden 16797 | Calculate the reduced form... |
| divdenle 16798 | Reducing a quotient never ... |
| qnumgt0 16799 | A rational is positive iff... |
| qgt0numnn 16800 | A rational is positive iff... |
| nn0gcdsq 16801 | Squaring commutes with GCD... |
| zgcdsq 16802 | ~ nn0gcdsq extended to int... |
| numdensq 16803 | Squaring a rational square... |
| numsq 16804 | Square commutes with canon... |
| densq 16805 | Square commutes with canon... |
| qden1elz 16806 | A rational is an integer i... |
| zsqrtelqelz 16807 | If an integer has a ration... |
| nonsq 16808 | Any integer strictly betwe... |
| numdenexp 16809 | Elevating a rational numbe... |
| numexp 16810 | Elevating to a nonnegative... |
| denexp 16811 | Elevating to a nonnegative... |
| phival 16816 | Value of the Euler ` phi `... |
| phicl2 16817 | Bounds and closure for the... |
| phicl 16818 | Closure for the value of t... |
| phibndlem 16819 | Lemma for ~ phibnd . (Con... |
| phibnd 16820 | A slightly tighter bound o... |
| phicld 16821 | Closure for the value of t... |
| phi1 16822 | Value of the Euler ` phi `... |
| dfphi2 16823 | Alternate definition of th... |
| hashdvds 16824 | The number of numbers in a... |
| phiprmpw 16825 | Value of the Euler ` phi `... |
| phiprm 16826 | Value of the Euler ` phi `... |
| crth 16827 | The Chinese Remainder Theo... |
| phimullem 16828 | Lemma for ~ phimul . (Con... |
| phimul 16829 | The Euler ` phi ` function... |
| eulerthlem1 16830 | Lemma for ~ eulerth . (Co... |
| eulerthlem2 16831 | Lemma for ~ eulerth . (Co... |
| eulerth 16832 | Euler's theorem, a general... |
| fermltl 16833 | Fermat's little theorem. ... |
| prmdiv 16834 | Show an explicit expressio... |
| prmdiveq 16835 | The modular inverse of ` A... |
| prmdivdiv 16836 | The (modular) inverse of t... |
| hashgcdlem 16837 | A correspondence between e... |
| hashgcdeq 16838 | Number of initial positive... |
| phisum 16839 | The divisor sum identity o... |
| odzval 16840 | Value of the order functio... |
| odzcllem 16841 | - Lemma for ~ odzcl , show... |
| odzcl 16842 | The order of a group eleme... |
| odzid 16843 | Any element raised to the ... |
| odzdvds 16844 | The only powers of ` A ` t... |
| odzphi 16845 | The order of any group ele... |
| modprm1div 16846 | A prime number divides an ... |
| m1dvdsndvds 16847 | If an integer minus 1 is d... |
| modprminv 16848 | Show an explicit expressio... |
| modprminveq 16849 | The modular inverse of ` A... |
| vfermltl 16850 | Variant of Fermat's little... |
| vfermltlALT 16851 | Alternate proof of ~ vferm... |
| powm2modprm 16852 | If an integer minus 1 is d... |
| reumodprminv 16853 | For any prime number and f... |
| modprm0 16854 | For two positive integers ... |
| nnnn0modprm0 16855 | For a positive integer and... |
| modprmn0modprm0 16856 | For an integer not being 0... |
| coprimeprodsq 16857 | If three numbers are copri... |
| coprimeprodsq2 16858 | If three numbers are copri... |
| oddprm 16859 | A prime not equal to ` 2 `... |
| nnoddn2prm 16860 | A prime not equal to ` 2 `... |
| oddn2prm 16861 | A prime not equal to ` 2 `... |
| nnoddn2prmb 16862 | A number is a prime number... |
| prm23lt5 16863 | A prime less than 5 is eit... |
| prm23ge5 16864 | A prime is either 2 or 3 o... |
| pythagtriplem1 16865 | Lemma for ~ pythagtrip . ... |
| pythagtriplem2 16866 | Lemma for ~ pythagtrip . ... |
| pythagtriplem3 16867 | Lemma for ~ pythagtrip . ... |
| pythagtriplem4 16868 | Lemma for ~ pythagtrip . ... |
| pythagtriplem10 16869 | Lemma for ~ pythagtrip . ... |
| pythagtriplem6 16870 | Lemma for ~ pythagtrip . ... |
| pythagtriplem7 16871 | Lemma for ~ pythagtrip . ... |
| pythagtriplem8 16872 | Lemma for ~ pythagtrip . ... |
| pythagtriplem9 16873 | Lemma for ~ pythagtrip . ... |
| pythagtriplem11 16874 | Lemma for ~ pythagtrip . ... |
| pythagtriplem12 16875 | Lemma for ~ pythagtrip . ... |
| pythagtriplem13 16876 | Lemma for ~ pythagtrip . ... |
| pythagtriplem14 16877 | Lemma for ~ pythagtrip . ... |
| pythagtriplem15 16878 | Lemma for ~ pythagtrip . ... |
| pythagtriplem16 16879 | Lemma for ~ pythagtrip . ... |
| pythagtriplem17 16880 | Lemma for ~ pythagtrip . ... |
| pythagtriplem18 16881 | Lemma for ~ pythagtrip . ... |
| pythagtriplem19 16882 | Lemma for ~ pythagtrip . ... |
| pythagtrip 16883 | Parameterize the Pythagore... |
| iserodd 16884 | Collect the odd terms in a... |
| pclem 16887 | - Lemma for the prime powe... |
| pcprecl 16888 | Closure of the prime power... |
| pcprendvds 16889 | Non-divisibility property ... |
| pcprendvds2 16890 | Non-divisibility property ... |
| pcpre1 16891 | Value of the prime power p... |
| pcpremul 16892 | Multiplicative property of... |
| pcval 16893 | The value of the prime pow... |
| pceulem 16894 | Lemma for ~ pceu . (Contr... |
| pceu 16895 | Uniqueness for the prime p... |
| pczpre 16896 | Connect the prime count pr... |
| pczcl 16897 | Closure of the prime power... |
| pccl 16898 | Closure of the prime power... |
| pccld 16899 | Closure of the prime power... |
| pcmul 16900 | Multiplication property of... |
| pcdiv 16901 | Division property of the p... |
| pcqmul 16902 | Multiplication property of... |
| pc0 16903 | The value of the prime pow... |
| pc1 16904 | Value of the prime count f... |
| pcqcl 16905 | Closure of the general pri... |
| pcqdiv 16906 | Division property of the p... |
| pcrec 16907 | Prime power of a reciproca... |
| pcexp 16908 | Prime power of an exponent... |
| pcxnn0cl 16909 | Extended nonnegative integ... |
| pcxcl 16910 | Extended real closure of t... |
| pcge0 16911 | The prime count of an inte... |
| pczdvds 16912 | Defining property of the p... |
| pcdvds 16913 | Defining property of the p... |
| pczndvds 16914 | Defining property of the p... |
| pcndvds 16915 | Defining property of the p... |
| pczndvds2 16916 | The remainder after dividi... |
| pcndvds2 16917 | The remainder after dividi... |
| pcdvdsb 16918 | ` P ^ A ` divides ` N ` if... |
| pcelnn 16919 | There are a positive numbe... |
| pceq0 16920 | There are zero powers of a... |
| pcidlem 16921 | The prime count of a prime... |
| pcid 16922 | The prime count of a prime... |
| pcneg 16923 | The prime count of a negat... |
| pcabs 16924 | The prime count of an abso... |
| pcdvdstr 16925 | The prime count increases ... |
| pcgcd1 16926 | The prime count of a GCD i... |
| pcgcd 16927 | The prime count of a GCD i... |
| pc2dvds 16928 | A characterization of divi... |
| pc11 16929 | The prime count function, ... |
| pcz 16930 | The prime count function c... |
| pcprmpw2 16931 | Self-referential expressio... |
| pcprmpw 16932 | Self-referential expressio... |
| dvdsprmpweq 16933 | If a positive integer divi... |
| dvdsprmpweqnn 16934 | If an integer greater than... |
| dvdsprmpweqle 16935 | If a positive integer divi... |
| difsqpwdvds 16936 | If the difference of two s... |
| pcaddlem 16937 | Lemma for ~ pcadd . The o... |
| pcadd 16938 | An inequality for the prim... |
| pcadd2 16939 | The inequality of ~ pcadd ... |
| pcmptcl 16940 | Closure for the prime powe... |
| pcmpt 16941 | Construct a function with ... |
| pcmpt2 16942 | Dividing two prime count m... |
| pcmptdvds 16943 | The partial products of th... |
| pcprod 16944 | The product of the primes ... |
| sumhash 16945 | The sum of 1 over a set is... |
| fldivp1 16946 | The difference between the... |
| pcfaclem 16947 | Lemma for ~ pcfac . (Cont... |
| pcfac 16948 | Calculate the prime count ... |
| pcbc 16949 | Calculate the prime count ... |
| qexpz 16950 | If a power of a rational n... |
| expnprm 16951 | A second or higher power o... |
| oddprmdvds 16952 | Every positive integer whi... |
| prmpwdvds 16953 | A relation involving divis... |
| pockthlem 16954 | Lemma for ~ pockthg . (Co... |
| pockthg 16955 | The generalized Pocklingto... |
| pockthi 16956 | Pocklington's theorem, whi... |
| unbenlem 16957 | Lemma for ~ unben . (Cont... |
| unben 16958 | An unbounded set of positi... |
| infpnlem1 16959 | Lemma for ~ infpn . The s... |
| infpnlem2 16960 | Lemma for ~ infpn . For a... |
| infpn 16961 | There exist infinitely man... |
| infpn2 16962 | There exist infinitely man... |
| prmunb 16963 | The primes are unbounded. ... |
| prminf 16964 | There are an infinite numb... |
| prmreclem1 16965 | Lemma for ~ prmrec . Prop... |
| prmreclem2 16966 | Lemma for ~ prmrec . Ther... |
| prmreclem3 16967 | Lemma for ~ prmrec . The ... |
| prmreclem4 16968 | Lemma for ~ prmrec . Show... |
| prmreclem5 16969 | Lemma for ~ prmrec . Here... |
| prmreclem6 16970 | Lemma for ~ prmrec . If t... |
| prmrec 16971 | The sum of the reciprocals... |
| 1arithlem1 16972 | Lemma for ~ 1arith . (Con... |
| 1arithlem2 16973 | Lemma for ~ 1arith . (Con... |
| 1arithlem3 16974 | Lemma for ~ 1arith . (Con... |
| 1arithlem4 16975 | Lemma for ~ 1arith . (Con... |
| 1arith 16976 | Fundamental theorem of ari... |
| 1arith2 16977 | Fundamental theorem of ari... |
| elgz 16980 | Elementhood in the gaussia... |
| gzcn 16981 | A gaussian integer is a co... |
| zgz 16982 | An integer is a gaussian i... |
| igz 16983 | ` _i ` is a gaussian integ... |
| gznegcl 16984 | The gaussian integers are ... |
| gzcjcl 16985 | The gaussian integers are ... |
| gzaddcl 16986 | The gaussian integers are ... |
| gzmulcl 16987 | The gaussian integers are ... |
| gzreim 16988 | Construct a gaussian integ... |
| gzsubcl 16989 | The gaussian integers are ... |
| gzabssqcl 16990 | The squared norm of a gaus... |
| 4sqlem5 16991 | Lemma for ~ 4sq . (Contri... |
| 4sqlem6 16992 | Lemma for ~ 4sq . (Contri... |
| 4sqlem7 16993 | Lemma for ~ 4sq . (Contri... |
| 4sqlem8 16994 | Lemma for ~ 4sq . (Contri... |
| 4sqlem9 16995 | Lemma for ~ 4sq . (Contri... |
| 4sqlem10 16996 | Lemma for ~ 4sq . (Contri... |
| 4sqlem1 16997 | Lemma for ~ 4sq . The set... |
| 4sqlem2 16998 | Lemma for ~ 4sq . Change ... |
| 4sqlem3 16999 | Lemma for ~ 4sq . Suffici... |
| 4sqlem4a 17000 | Lemma for ~ 4sqlem4 . (Co... |
| 4sqlem4 17001 | Lemma for ~ 4sq . We can ... |
| mul4sqlem 17002 | Lemma for ~ mul4sq : algeb... |
| mul4sq 17003 | Euler's four-square identi... |
| 4sqlem11 17004 | Lemma for ~ 4sq . Use the... |
| 4sqlem12 17005 | Lemma for ~ 4sq . For any... |
| 4sqlem13 17006 | Lemma for ~ 4sq . (Contri... |
| 4sqlem14 17007 | Lemma for ~ 4sq . (Contri... |
| 4sqlem15 17008 | Lemma for ~ 4sq . (Contri... |
| 4sqlem16 17009 | Lemma for ~ 4sq . (Contri... |
| 4sqlem17 17010 | Lemma for ~ 4sq . (Contri... |
| 4sqlem18 17011 | Lemma for ~ 4sq . Inducti... |
| 4sqlem19 17012 | Lemma for ~ 4sq . The pro... |
| 4sq 17013 | Lagrange's four-square the... |
| vdwapfval 17020 | Define the arithmetic prog... |
| vdwapf 17021 | The arithmetic progression... |
| vdwapval 17022 | Value of the arithmetic pr... |
| vdwapun 17023 | Remove the first element o... |
| vdwapid1 17024 | The first element of an ar... |
| vdwap0 17025 | Value of a length-1 arithm... |
| vdwap1 17026 | Value of a length-1 arithm... |
| vdwmc 17027 | The predicate " The ` <. R... |
| vdwmc2 17028 | Expand out the definition ... |
| vdwpc 17029 | The predicate " The colori... |
| vdwlem1 17030 | Lemma for ~ vdw . (Contri... |
| vdwlem2 17031 | Lemma for ~ vdw . (Contri... |
| vdwlem3 17032 | Lemma for ~ vdw . (Contri... |
| vdwlem4 17033 | Lemma for ~ vdw . (Contri... |
| vdwlem5 17034 | Lemma for ~ vdw . (Contri... |
| vdwlem6 17035 | Lemma for ~ vdw . (Contri... |
| vdwlem7 17036 | Lemma for ~ vdw . (Contri... |
| vdwlem8 17037 | Lemma for ~ vdw . (Contri... |
| vdwlem9 17038 | Lemma for ~ vdw . (Contri... |
| vdwlem10 17039 | Lemma for ~ vdw . Set up ... |
| vdwlem11 17040 | Lemma for ~ vdw . (Contri... |
| vdwlem12 17041 | Lemma for ~ vdw . ` K = 2 ... |
| vdwlem13 17042 | Lemma for ~ vdw . Main in... |
| vdw 17043 | Van der Waerden's theorem.... |
| vdwnnlem1 17044 | Corollary of ~ vdw , and l... |
| vdwnnlem2 17045 | Lemma for ~ vdwnn . The s... |
| vdwnnlem3 17046 | Lemma for ~ vdwnn . (Cont... |
| vdwnn 17047 | Van der Waerden's theorem,... |
| ramtlecl 17049 | The set ` T ` of numbers w... |
| hashbcval 17051 | Value of the "binomial set... |
| hashbccl 17052 | The binomial set is a fini... |
| hashbcss 17053 | Subset relation for the bi... |
| hashbc0 17054 | The set of subsets of size... |
| hashbc2 17055 | The size of the binomial s... |
| 0hashbc 17056 | There are no subsets of th... |
| ramval 17057 | The value of the Ramsey nu... |
| ramcl2lem 17058 | Lemma for extended real cl... |
| ramtcl 17059 | The Ramsey number has the ... |
| ramtcl2 17060 | The Ramsey number is an in... |
| ramtub 17061 | The Ramsey number is a low... |
| ramub 17062 | The Ramsey number is a low... |
| ramub2 17063 | It is sufficient to check ... |
| rami 17064 | The defining property of a... |
| ramcl2 17065 | The Ramsey number is eithe... |
| ramxrcl 17066 | The Ramsey number is an ex... |
| ramubcl 17067 | If the Ramsey number is up... |
| ramlb 17068 | Establish a lower bound on... |
| 0ram 17069 | The Ramsey number when ` M... |
| 0ram2 17070 | The Ramsey number when ` M... |
| ram0 17071 | The Ramsey number when ` R... |
| 0ramcl 17072 | Lemma for ~ ramcl : Exist... |
| ramz2 17073 | The Ramsey number when ` F... |
| ramz 17074 | The Ramsey number when ` F... |
| ramub1lem1 17075 | Lemma for ~ ramub1 . (Con... |
| ramub1lem2 17076 | Lemma for ~ ramub1 . (Con... |
| ramub1 17077 | Inductive step for Ramsey'... |
| ramcl 17078 | Ramsey's theorem: the Rams... |
| ramsey 17079 | Ramsey's theorem with the ... |
| prmoval 17082 | Value of the primorial fun... |
| prmocl 17083 | Closure of the primorial f... |
| prmone0 17084 | The primorial function is ... |
| prmo0 17085 | The primorial of 0. (Cont... |
| prmo1 17086 | The primorial of 1. (Cont... |
| prmop1 17087 | The primorial of a success... |
| prmonn2 17088 | Value of the primorial fun... |
| prmo2 17089 | The primorial of 2. (Cont... |
| prmo3 17090 | The primorial of 3. (Cont... |
| prmdvdsprmo 17091 | The primorial of a number ... |
| prmdvdsprmop 17092 | The primorial of a number ... |
| fvprmselelfz 17093 | The value of the prime sel... |
| fvprmselgcd1 17094 | The greatest common diviso... |
| prmolefac 17095 | The primorial of a positiv... |
| prmodvdslcmf 17096 | The primorial of a nonnega... |
| prmolelcmf 17097 | The primorial of a positiv... |
| prmgaplem1 17098 | Lemma for ~ prmgap : The ... |
| prmgaplem2 17099 | Lemma for ~ prmgap : The ... |
| prmgaplcmlem1 17100 | Lemma for ~ prmgaplcm : T... |
| prmgaplcmlem2 17101 | Lemma for ~ prmgaplcm : T... |
| prmgaplem3 17102 | Lemma for ~ prmgap . (Con... |
| prmgaplem4 17103 | Lemma for ~ prmgap . (Con... |
| prmgaplem5 17104 | Lemma for ~ prmgap : for e... |
| prmgaplem6 17105 | Lemma for ~ prmgap : for e... |
| prmgaplem7 17106 | Lemma for ~ prmgap . (Con... |
| prmgaplem8 17107 | Lemma for ~ prmgap . (Con... |
| prmgap 17108 | The prime gap theorem: for... |
| prmgaplcm 17109 | Alternate proof of ~ prmga... |
| prmgapprmolem 17110 | Lemma for ~ prmgapprmo : ... |
| prmgapprmo 17111 | Alternate proof of ~ prmga... |
| dec2dvds 17112 | Divisibility by two is obv... |
| dec5dvds 17113 | Divisibility by five is ob... |
| dec5dvds2 17114 | Divisibility by five is ob... |
| dec5nprm 17115 | Divisibility by five is ob... |
| dec2nprm 17116 | Divisibility by two is obv... |
| modxai 17117 | Add exponents in a power m... |
| mod2xi 17118 | Double exponents in a powe... |
| modxp1i 17119 | Add one to an exponent in ... |
| mod2xnegi 17120 | Version of ~ mod2xi with a... |
| modsubi 17121 | Subtract from within a mod... |
| gcdi 17122 | Calculate a GCD via Euclid... |
| gcdmodi 17123 | Calculate a GCD via Euclid... |
| decexp2 17124 | Calculate a power of two. ... |
| numexp0 17125 | Calculate an integer power... |
| numexp1 17126 | Calculate an integer power... |
| numexpp1 17127 | Calculate an integer power... |
| numexp2x 17128 | Double an integer power. ... |
| decsplit0b 17129 | Split a decimal number int... |
| decsplit0 17130 | Split a decimal number int... |
| decsplit1 17131 | Split a decimal number int... |
| decsplit 17132 | Split a decimal number int... |
| karatsuba 17133 | The Karatsuba multiplicati... |
| 2exp4 17134 | Two to the fourth power is... |
| 2exp5 17135 | Two to the fifth power is ... |
| 2exp6 17136 | Two to the sixth power is ... |
| 2exp7 17137 | Two to the seventh power i... |
| 2exp8 17138 | Two to the eighth power is... |
| 2exp11 17139 | Two to the eleventh power ... |
| 2exp16 17140 | Two to the sixteenth power... |
| 3exp3 17141 | Three to the third power i... |
| 2expltfac 17142 | The factorial grows faster... |
| cshwsidrepsw 17143 | If cyclically shifting a w... |
| cshwsidrepswmod0 17144 | If cyclically shifting a w... |
| cshwshashlem1 17145 | If cyclically shifting a w... |
| cshwshashlem2 17146 | If cyclically shifting a w... |
| cshwshashlem3 17147 | If cyclically shifting a w... |
| cshwsdisj 17148 | The singletons resulting b... |
| cshwsiun 17149 | The set of (different!) wo... |
| cshwsex 17150 | The class of (different!) ... |
| cshws0 17151 | The size of the set of (di... |
| cshwrepswhash1 17152 | The size of the set of (di... |
| cshwshashnsame 17153 | If a word (not consisting ... |
| cshwshash 17154 | If a word has a length bei... |
| prmlem0 17155 | Lemma for ~ prmlem1 and ~ ... |
| prmlem1a 17156 | A quick proof skeleton to ... |
| prmlem1 17157 | A quick proof skeleton to ... |
| 5prm 17158 | 5 is a prime number. (Con... |
| 6nprm 17159 | 6 is not a prime number. ... |
| 7prm 17160 | 7 is a prime number. (Con... |
| 8nprm 17161 | 8 is not a prime number. ... |
| 9nprm 17162 | 9 is not a prime number. ... |
| 10nprm 17163 | 10 is not a prime number. ... |
| 11prm 17164 | 11 is a prime number. (Co... |
| 13prm 17165 | 13 is a prime number. (Co... |
| 17prm 17166 | 17 is a prime number. (Co... |
| 19prm 17167 | 19 is a prime number. (Co... |
| 23prm 17168 | 23 is a prime number. (Co... |
| prmlem2 17169 | Our last proving session g... |
| 37prm 17170 | 37 is a prime number. (Co... |
| 43prm 17171 | 43 is a prime number. (Co... |
| 83prm 17172 | 83 is a prime number. (Co... |
| 139prm 17173 | 139 is a prime number. (C... |
| 163prm 17174 | 163 is a prime number. (C... |
| 317prm 17175 | 317 is a prime number. (C... |
| 631prm 17176 | 631 is a prime number. (C... |
| prmo4 17177 | The primorial of 4. (Cont... |
| prmo5 17178 | The primorial of 5. (Cont... |
| prmo6 17179 | The primorial of 6. (Cont... |
| 1259lem1 17180 | Lemma for ~ 1259prm . Cal... |
| 1259lem2 17181 | Lemma for ~ 1259prm . Cal... |
| 1259lem3 17182 | Lemma for ~ 1259prm . Cal... |
| 1259lem4 17183 | Lemma for ~ 1259prm . Cal... |
| 1259lem5 17184 | Lemma for ~ 1259prm . Cal... |
| 1259prm 17185 | 1259 is a prime number. (... |
| 2503lem1 17186 | Lemma for ~ 2503prm . Cal... |
| 2503lem2 17187 | Lemma for ~ 2503prm . Cal... |
| 2503lem3 17188 | Lemma for ~ 2503prm . Cal... |
| 2503prm 17189 | 2503 is a prime number. (... |
| 4001lem1 17190 | Lemma for ~ 4001prm . Cal... |
| 4001lem2 17191 | Lemma for ~ 4001prm . Cal... |
| 4001lem3 17192 | Lemma for ~ 4001prm . Cal... |
| 4001lem4 17193 | Lemma for ~ 4001prm . Cal... |
| 4001prm 17194 | 4001 is a prime number. (... |
| brstruct 17197 | The structure relation is ... |
| isstruct2 17198 | The property of being a st... |
| structex 17199 | A structure is a set. (Co... |
| structn0fun 17200 | A structure without the em... |
| isstruct 17201 | The property of being a st... |
| structcnvcnv 17202 | Two ways to express the re... |
| structfung 17203 | The converse of the conver... |
| structfun 17204 | Convert between two kinds ... |
| structfn 17205 | Convert between two kinds ... |
| strleun 17206 | Combine two structures int... |
| strle1 17207 | Make a structure from a si... |
| strle2 17208 | Make a structure from a pa... |
| strle3 17209 | Make a structure from a tr... |
| sbcie2s 17210 | A special version of class... |
| sbcie3s 17211 | A special version of class... |
| reldmsets 17214 | The structure override ope... |
| setsvalg 17215 | Value of the structure rep... |
| setsval 17216 | Value of the structure rep... |
| fvsetsid 17217 | The value of the structure... |
| fsets 17218 | The structure replacement ... |
| setsdm 17219 | The domain of a structure ... |
| setsfun 17220 | A structure with replaceme... |
| setsfun0 17221 | A structure with replaceme... |
| setsn0fun 17222 | The value of the structure... |
| setsstruct2 17223 | An extensible structure wi... |
| setsexstruct2 17224 | An extensible structure wi... |
| setsstruct 17225 | An extensible structure wi... |
| wunsets 17226 | Closure of structure repla... |
| setsres 17227 | The structure replacement ... |
| setsabs 17228 | Replacing the same compone... |
| setscom 17229 | Different components can b... |
| sloteq 17232 | Equality theorem for the `... |
| slotfn 17233 | A slot is a function on se... |
| strfvnd 17234 | Deduction version of ~ str... |
| strfvn 17235 | Value of a structure compo... |
| strfvss 17236 | A structure component extr... |
| wunstr 17237 | Closure of a structure ind... |
| str0 17238 | All components of the empt... |
| strfvi 17239 | Structure slot extractors ... |
| fveqprc 17240 | Lemma for showing the equa... |
| oveqprc 17241 | Lemma for showing the equa... |
| wunndx 17244 | Closure of the index extra... |
| ndxarg 17245 | Get the numeric argument f... |
| ndxid 17246 | A structure component extr... |
| strndxid 17247 | The value of a structure c... |
| setsidvald 17248 | Value of the structure rep... |
| setsidvaldOLD 17249 | Obsolete version of ~ sets... |
| strfvd 17250 | Deduction version of ~ str... |
| strfv2d 17251 | Deduction version of ~ str... |
| strfv2 17252 | A variation on ~ strfv to ... |
| strfv 17253 | Extract a structure compon... |
| strfv3 17254 | Variant on ~ strfv for lar... |
| strssd 17255 | Deduction version of ~ str... |
| strss 17256 | Propagate component extrac... |
| setsid 17257 | Value of the structure rep... |
| setsnid 17258 | Value of the structure rep... |
| setsnidOLD 17259 | Obsolete proof of ~ setsni... |
| baseval 17262 | Value of the base set extr... |
| baseid 17263 | Utility theorem: index-ind... |
| basfn 17264 | The base set extractor is ... |
| base0 17265 | The base set of the empty ... |
| elbasfv 17266 | Utility theorem: reverse c... |
| elbasov 17267 | Utility theorem: reverse c... |
| strov2rcl 17268 | Partial reverse closure fo... |
| basendx 17269 | Index value of the base se... |
| basendxnn 17270 | The index value of the bas... |
| basendxnnOLD 17271 | Obsolete proof of ~ basend... |
| basndxelwund 17272 | The index of the base set ... |
| basprssdmsets 17273 | The pair of the base index... |
| opelstrbas 17274 | The base set of a structur... |
| 1strstr 17275 | A constructed one-slot str... |
| 1strstr1 17276 | A constructed one-slot str... |
| 1strbas 17277 | The base set of a construc... |
| 1strbasOLD 17278 | Obsolete proof of ~ 1strba... |
| 1strwunbndx 17279 | A constructed one-slot str... |
| 1strwun 17280 | A constructed one-slot str... |
| 1strwunOLD 17281 | Obsolete version of ~ 1str... |
| 2strstr 17282 | A constructed two-slot str... |
| 2strbas 17283 | The base set of a construc... |
| 2strop 17284 | The other slot of a constr... |
| 2strstr1 17285 | A constructed two-slot str... |
| 2strstr1OLD 17286 | Obsolete version of ~ 2str... |
| 2strbas1 17287 | The base set of a construc... |
| 2strop1 17288 | The other slot of a constr... |
| reldmress 17291 | The structure restriction ... |
| ressval 17292 | Value of structure restric... |
| ressid2 17293 | General behavior of trivia... |
| ressval2 17294 | Value of nontrivial struct... |
| ressbas 17295 | Base set of a structure re... |
| ressbasOLD 17296 | Obsolete proof of ~ ressba... |
| ressbasssg 17297 | The base set of a restrict... |
| ressbas2 17298 | Base set of a structure re... |
| ressbasss 17299 | The base set of a restrict... |
| ressbasssOLD 17300 | Obsolete proof of ~ ressba... |
| ressbasss2 17301 | The base set of a restrict... |
| resseqnbas 17302 | The components of an exten... |
| resslemOLD 17303 | Obsolete version of ~ ress... |
| ress0 17304 | All restrictions of the nu... |
| ressid 17305 | Behavior of trivial restri... |
| ressinbas 17306 | Restriction only cares abo... |
| ressval3d 17307 | Value of structure restric... |
| ressval3dOLD 17308 | Obsolete version of ~ ress... |
| ressress 17309 | Restriction composition la... |
| ressabs 17310 | Restriction absorption law... |
| wunress 17311 | Closure of structure restr... |
| wunressOLD 17312 | Obsolete proof of ~ wunres... |
| plusgndx 17339 | Index value of the ~ df-pl... |
| plusgid 17340 | Utility theorem: index-ind... |
| plusgndxnn 17341 | The index of the slot for ... |
| basendxltplusgndx 17342 | The index of the slot for ... |
| basendxnplusgndx 17343 | The slot for the base set ... |
| basendxnplusgndxOLD 17344 | Obsolete version of ~ base... |
| grpstr 17345 | A constructed group is a s... |
| grpstrndx 17346 | A constructed group is a s... |
| grpbase 17347 | The base set of a construc... |
| grpbaseOLD 17348 | Obsolete version of ~ grpb... |
| grpplusg 17349 | The operation of a constru... |
| grpplusgOLD 17350 | Obsolete version of ~ grpp... |
| ressplusg 17351 | ` +g ` is unaffected by re... |
| grpbasex 17352 | The base of an explicitly ... |
| grpplusgx 17353 | The operation of an explic... |
| mulrndx 17354 | Index value of the ~ df-mu... |
| mulridx 17355 | Utility theorem: index-ind... |
| basendxnmulrndx 17356 | The slot for the base set ... |
| basendxnmulrndxOLD 17357 | Obsolete proof of ~ basend... |
| plusgndxnmulrndx 17358 | The slot for the group (ad... |
| rngstr 17359 | A constructed ring is a st... |
| rngbase 17360 | The base set of a construc... |
| rngplusg 17361 | The additive operation of ... |
| rngmulr 17362 | The multiplicative operati... |
| starvndx 17363 | Index value of the ~ df-st... |
| starvid 17364 | Utility theorem: index-ind... |
| starvndxnbasendx 17365 | The slot for the involutio... |
| starvndxnplusgndx 17366 | The slot for the involutio... |
| starvndxnmulrndx 17367 | The slot for the involutio... |
| ressmulr 17368 | ` .r ` is unaffected by re... |
| ressstarv 17369 | ` *r ` is unaffected by re... |
| srngstr 17370 | A constructed star ring is... |
| srngbase 17371 | The base set of a construc... |
| srngplusg 17372 | The addition operation of ... |
| srngmulr 17373 | The multiplication operati... |
| srnginvl 17374 | The involution function of... |
| scandx 17375 | Index value of the ~ df-sc... |
| scaid 17376 | Utility theorem: index-ind... |
| scandxnbasendx 17377 | The slot for the scalar is... |
| scandxnplusgndx 17378 | The slot for the scalar fi... |
| scandxnmulrndx 17379 | The slot for the scalar fi... |
| vscandx 17380 | Index value of the ~ df-vs... |
| vscaid 17381 | Utility theorem: index-ind... |
| vscandxnbasendx 17382 | The slot for the scalar pr... |
| vscandxnplusgndx 17383 | The slot for the scalar pr... |
| vscandxnmulrndx 17384 | The slot for the scalar pr... |
| vscandxnscandx 17385 | The slot for the scalar pr... |
| lmodstr 17386 | A constructed left module ... |
| lmodbase 17387 | The base set of a construc... |
| lmodplusg 17388 | The additive operation of ... |
| lmodsca 17389 | The set of scalars of a co... |
| lmodvsca 17390 | The scalar product operati... |
| ipndx 17391 | Index value of the ~ df-ip... |
| ipid 17392 | Utility theorem: index-ind... |
| ipndxnbasendx 17393 | The slot for the inner pro... |
| ipndxnplusgndx 17394 | The slot for the inner pro... |
| ipndxnmulrndx 17395 | The slot for the inner pro... |
| slotsdifipndx 17396 | The slot for the scalar is... |
| ipsstr 17397 | Lemma to shorten proofs of... |
| ipsbase 17398 | The base set of a construc... |
| ipsaddg 17399 | The additive operation of ... |
| ipsmulr 17400 | The multiplicative operati... |
| ipssca 17401 | The set of scalars of a co... |
| ipsvsca 17402 | The scalar product operati... |
| ipsip 17403 | The multiplicative operati... |
| resssca 17404 | ` Scalar ` is unaffected b... |
| ressvsca 17405 | ` .s ` is unaffected by re... |
| ressip 17406 | The inner product is unaff... |
| phlstr 17407 | A constructed pre-Hilbert ... |
| phlbase 17408 | The base set of a construc... |
| phlplusg 17409 | The additive operation of ... |
| phlsca 17410 | The ring of scalars of a c... |
| phlvsca 17411 | The scalar product operati... |
| phlip 17412 | The inner product (Hermiti... |
| tsetndx 17413 | Index value of the ~ df-ts... |
| tsetid 17414 | Utility theorem: index-ind... |
| tsetndxnn 17415 | The index of the slot for ... |
| basendxlttsetndx 17416 | The index of the slot for ... |
| tsetndxnbasendx 17417 | The slot for the topology ... |
| tsetndxnplusgndx 17418 | The slot for the topology ... |
| tsetndxnmulrndx 17419 | The slot for the topology ... |
| tsetndxnstarvndx 17420 | The slot for the topology ... |
| slotstnscsi 17421 | The slots ` Scalar ` , ` .... |
| topgrpstr 17422 | A constructed topological ... |
| topgrpbas 17423 | The base set of a construc... |
| topgrpplusg 17424 | The additive operation of ... |
| topgrptset 17425 | The topology of a construc... |
| resstset 17426 | ` TopSet ` is unaffected b... |
| plendx 17427 | Index value of the ~ df-pl... |
| pleid 17428 | Utility theorem: self-refe... |
| plendxnn 17429 | The index value of the ord... |
| basendxltplendx 17430 | The index value of the ` B... |
| plendxnbasendx 17431 | The slot for the order is ... |
| plendxnplusgndx 17432 | The slot for the "less tha... |
| plendxnmulrndx 17433 | The slot for the "less tha... |
| plendxnscandx 17434 | The slot for the "less tha... |
| plendxnvscandx 17435 | The slot for the "less tha... |
| slotsdifplendx 17436 | The index of the slot for ... |
| otpsstr 17437 | Functionality of a topolog... |
| otpsbas 17438 | The base set of a topologi... |
| otpstset 17439 | The open sets of a topolog... |
| otpsle 17440 | The order of a topological... |
| ressle 17441 | ` le ` is unaffected by re... |
| ocndx 17442 | Index value of the ~ df-oc... |
| ocid 17443 | Utility theorem: index-ind... |
| basendxnocndx 17444 | The slot for the orthocomp... |
| plendxnocndx 17445 | The slot for the orthocomp... |
| dsndx 17446 | Index value of the ~ df-ds... |
| dsid 17447 | Utility theorem: index-ind... |
| dsndxnn 17448 | The index of the slot for ... |
| basendxltdsndx 17449 | The index of the slot for ... |
| dsndxnbasendx 17450 | The slot for the distance ... |
| dsndxnplusgndx 17451 | The slot for the distance ... |
| dsndxnmulrndx 17452 | The slot for the distance ... |
| slotsdnscsi 17453 | The slots ` Scalar ` , ` .... |
| dsndxntsetndx 17454 | The slot for the distance ... |
| slotsdifdsndx 17455 | The index of the slot for ... |
| unifndx 17456 | Index value of the ~ df-un... |
| unifid 17457 | Utility theorem: index-ind... |
| unifndxnn 17458 | The index of the slot for ... |
| basendxltunifndx 17459 | The index of the slot for ... |
| unifndxnbasendx 17460 | The slot for the uniform s... |
| unifndxntsetndx 17461 | The slot for the uniform s... |
| slotsdifunifndx 17462 | The index of the slot for ... |
| ressunif 17463 | ` UnifSet ` is unaffected ... |
| odrngstr 17464 | Functionality of an ordere... |
| odrngbas 17465 | The base set of an ordered... |
| odrngplusg 17466 | The addition operation of ... |
| odrngmulr 17467 | The multiplication operati... |
| odrngtset 17468 | The open sets of an ordere... |
| odrngle 17469 | The order of an ordered me... |
| odrngds 17470 | The metric of an ordered m... |
| ressds 17471 | ` dist ` is unaffected by ... |
| homndx 17472 | Index value of the ~ df-ho... |
| homid 17473 | Utility theorem: index-ind... |
| ccondx 17474 | Index value of the ~ df-cc... |
| ccoid 17475 | Utility theorem: index-ind... |
| slotsbhcdif 17476 | The slots ` Base ` , ` Hom... |
| slotsbhcdifOLD 17477 | Obsolete proof of ~ slotsb... |
| slotsdifplendx2 17478 | The index of the slot for ... |
| slotsdifocndx 17479 | The index of the slot for ... |
| resshom 17480 | ` Hom ` is unaffected by r... |
| ressco 17481 | ` comp ` is unaffected by ... |
| restfn 17486 | The subspace topology oper... |
| topnfn 17487 | The topology extractor fun... |
| restval 17488 | The subspace topology indu... |
| elrest 17489 | The predicate "is an open ... |
| elrestr 17490 | Sufficient condition for b... |
| 0rest 17491 | Value of the structure res... |
| restid2 17492 | The subspace topology over... |
| restsspw 17493 | The subspace topology is a... |
| firest 17494 | The finite intersections o... |
| restid 17495 | The subspace topology of t... |
| topnval 17496 | Value of the topology extr... |
| topnid 17497 | Value of the topology extr... |
| topnpropd 17498 | The topology extractor fun... |
| reldmprds 17510 | The structure product is a... |
| prdsbasex 17512 | Lemma for structure produc... |
| imasvalstr 17513 | An image structure value i... |
| prdsvalstr 17514 | Structure product value is... |
| prdsbaslem 17515 | Lemma for ~ prdsbas and si... |
| prdsvallem 17516 | Lemma for ~ prdsval . (Co... |
| prdsval 17517 | Value of the structure pro... |
| prdssca 17518 | Scalar ring of a structure... |
| prdsbas 17519 | Base set of a structure pr... |
| prdsplusg 17520 | Addition in a structure pr... |
| prdsmulr 17521 | Multiplication in a struct... |
| prdsvsca 17522 | Scalar multiplication in a... |
| prdsip 17523 | Inner product in a structu... |
| prdsle 17524 | Structure product weak ord... |
| prdsless 17525 | Closure of the order relat... |
| prdsds 17526 | Structure product distance... |
| prdsdsfn 17527 | Structure product distance... |
| prdstset 17528 | Structure product topology... |
| prdshom 17529 | Structure product hom-sets... |
| prdsco 17530 | Structure product composit... |
| prdsbas2 17531 | The base set of a structur... |
| prdsbasmpt 17532 | A constructed tuple is a p... |
| prdsbasfn 17533 | Points in the structure pr... |
| prdsbasprj 17534 | Each point in a structure ... |
| prdsplusgval 17535 | Value of a componentwise s... |
| prdsplusgfval 17536 | Value of a structure produ... |
| prdsmulrval 17537 | Value of a componentwise r... |
| prdsmulrfval 17538 | Value of a structure produ... |
| prdsleval 17539 | Value of the product order... |
| prdsdsval 17540 | Value of the metric in a s... |
| prdsvscaval 17541 | Scalar multiplication in a... |
| prdsvscafval 17542 | Scalar multiplication of a... |
| prdsbas3 17543 | The base set of an indexed... |
| prdsbasmpt2 17544 | A constructed tuple is a p... |
| prdsbascl 17545 | An element of the base has... |
| prdsdsval2 17546 | Value of the metric in a s... |
| prdsdsval3 17547 | Value of the metric in a s... |
| pwsval 17548 | Value of a structure power... |
| pwsbas 17549 | Base set of a structure po... |
| pwselbasb 17550 | Membership in the base set... |
| pwselbas 17551 | An element of a structure ... |
| pwsplusgval 17552 | Value of addition in a str... |
| pwsmulrval 17553 | Value of multiplication in... |
| pwsle 17554 | Ordering in a structure po... |
| pwsleval 17555 | Ordering in a structure po... |
| pwsvscafval 17556 | Scalar multiplication in a... |
| pwsvscaval 17557 | Scalar multiplication of a... |
| pwssca 17558 | The ring of scalars of a s... |
| pwsdiagel 17559 | Membership of diagonal ele... |
| pwssnf1o 17560 | Triviality of singleton po... |
| imasval 17573 | Value of an image structur... |
| imasbas 17574 | The base set of an image s... |
| imasds 17575 | The distance function of a... |
| imasdsfn 17576 | The distance function is a... |
| imasdsval 17577 | The distance function of a... |
| imasdsval2 17578 | The distance function of a... |
| imasplusg 17579 | The group operation in an ... |
| imasmulr 17580 | The ring multiplication in... |
| imassca 17581 | The scalar field of an ima... |
| imasvsca 17582 | The scalar multiplication ... |
| imasip 17583 | The inner product of an im... |
| imastset 17584 | The topology of an image s... |
| imasle 17585 | The ordering of an image s... |
| f1ocpbllem 17586 | Lemma for ~ f1ocpbl . (Co... |
| f1ocpbl 17587 | An injection is compatible... |
| f1ovscpbl 17588 | An injection is compatible... |
| f1olecpbl 17589 | An injection is compatible... |
| imasaddfnlem 17590 | The image structure operat... |
| imasaddvallem 17591 | The operation of an image ... |
| imasaddflem 17592 | The image set operations a... |
| imasaddfn 17593 | The image structure's grou... |
| imasaddval 17594 | The value of an image stru... |
| imasaddf 17595 | The image structure's grou... |
| imasmulfn 17596 | The image structure's ring... |
| imasmulval 17597 | The value of an image stru... |
| imasmulf 17598 | The image structure's ring... |
| imasvscafn 17599 | The image structure's scal... |
| imasvscaval 17600 | The value of an image stru... |
| imasvscaf 17601 | The image structure's scal... |
| imasless 17602 | The order relation defined... |
| imasleval 17603 | The value of the image str... |
| qusval 17604 | Value of a quotient struct... |
| quslem 17605 | The function in ~ qusval i... |
| qusin 17606 | Restrict the equivalence r... |
| qusbas 17607 | Base set of a quotient str... |
| quss 17608 | The scalar field of a quot... |
| divsfval 17609 | Value of the function in ~... |
| ercpbllem 17610 | Lemma for ~ ercpbl . (Con... |
| ercpbl 17611 | Translate the function com... |
| erlecpbl 17612 | Translate the relation com... |
| qusaddvallem 17613 | Value of an operation defi... |
| qusaddflem 17614 | The operation of a quotien... |
| qusaddval 17615 | The addition in a quotient... |
| qusaddf 17616 | The addition in a quotient... |
| qusmulval 17617 | The multiplication in a qu... |
| qusmulf 17618 | The multiplication in a qu... |
| fnpr2o 17619 | Function with a domain of ... |
| fnpr2ob 17620 | Biconditional version of ~... |
| fvpr0o 17621 | The value of a function wi... |
| fvpr1o 17622 | The value of a function wi... |
| fvprif 17623 | The value of the pair func... |
| xpsfrnel 17624 | Elementhood in the target ... |
| xpsfeq 17625 | A function on ` 2o ` is de... |
| xpsfrnel2 17626 | Elementhood in the target ... |
| xpscf 17627 | Equivalent condition for t... |
| xpsfval 17628 | The value of the function ... |
| xpsff1o 17629 | The function appearing in ... |
| xpsfrn 17630 | A short expression for the... |
| xpsff1o2 17631 | The function appearing in ... |
| xpsval 17632 | Value of the binary struct... |
| xpsrnbas 17633 | The indexed structure prod... |
| xpsbas 17634 | The base set of the binary... |
| xpsaddlem 17635 | Lemma for ~ xpsadd and ~ x... |
| xpsadd 17636 | Value of the addition oper... |
| xpsmul 17637 | Value of the multiplicatio... |
| xpssca 17638 | Value of the scalar field ... |
| xpsvsca 17639 | Value of the scalar multip... |
| xpsless 17640 | Closure of the ordering in... |
| xpsle 17641 | Value of the ordering in a... |
| ismre 17650 | Property of being a Moore ... |
| fnmre 17651 | The Moore collection gener... |
| mresspw 17652 | A Moore collection is a su... |
| mress 17653 | A Moore-closed subset is a... |
| mre1cl 17654 | In any Moore collection th... |
| mreintcl 17655 | A nonempty collection of c... |
| mreiincl 17656 | A nonempty indexed interse... |
| mrerintcl 17657 | The relative intersection ... |
| mreriincl 17658 | The relative intersection ... |
| mreincl 17659 | Two closed sets have a clo... |
| mreuni 17660 | Since the entire base set ... |
| mreunirn 17661 | Two ways to express the no... |
| ismred 17662 | Properties that determine ... |
| ismred2 17663 | Properties that determine ... |
| mremre 17664 | The Moore collections of s... |
| submre 17665 | The subcollection of a clo... |
| mrcflem 17666 | The domain and codomain of... |
| fnmrc 17667 | Moore-closure is a well-be... |
| mrcfval 17668 | Value of the function expr... |
| mrcf 17669 | The Moore closure is a fun... |
| mrcval 17670 | Evaluation of the Moore cl... |
| mrccl 17671 | The Moore closure of a set... |
| mrcsncl 17672 | The Moore closure of a sin... |
| mrcid 17673 | The closure of a closed se... |
| mrcssv 17674 | The closure of a set is a ... |
| mrcidb 17675 | A set is closed iff it is ... |
| mrcss 17676 | Closure preserves subset o... |
| mrcssid 17677 | The closure of a set is a ... |
| mrcidb2 17678 | A set is closed iff it con... |
| mrcidm 17679 | The closure operation is i... |
| mrcsscl 17680 | The closure is the minimal... |
| mrcuni 17681 | Idempotence of closure und... |
| mrcun 17682 | Idempotence of closure und... |
| mrcssvd 17683 | The Moore closure of a set... |
| mrcssd 17684 | Moore closure preserves su... |
| mrcssidd 17685 | A set is contained in its ... |
| mrcidmd 17686 | Moore closure is idempoten... |
| mressmrcd 17687 | In a Moore system, if a se... |
| submrc 17688 | In a closure system which ... |
| mrieqvlemd 17689 | In a Moore system, if ` Y ... |
| mrisval 17690 | Value of the set of indepe... |
| ismri 17691 | Criterion for a set to be ... |
| ismri2 17692 | Criterion for a subset of ... |
| ismri2d 17693 | Criterion for a subset of ... |
| ismri2dd 17694 | Definition of independence... |
| mriss 17695 | An independent set of a Mo... |
| mrissd 17696 | An independent set of a Mo... |
| ismri2dad 17697 | Consequence of a set in a ... |
| mrieqvd 17698 | In a Moore system, a set i... |
| mrieqv2d 17699 | In a Moore system, a set i... |
| mrissmrcd 17700 | In a Moore system, if an i... |
| mrissmrid 17701 | In a Moore system, subsets... |
| mreexd 17702 | In a Moore system, the clo... |
| mreexmrid 17703 | In a Moore system whose cl... |
| mreexexlemd 17704 | This lemma is used to gene... |
| mreexexlem2d 17705 | Used in ~ mreexexlem4d to ... |
| mreexexlem3d 17706 | Base case of the induction... |
| mreexexlem4d 17707 | Induction step of the indu... |
| mreexexd 17708 | Exchange-type theorem. In... |
| mreexdomd 17709 | In a Moore system whose cl... |
| mreexfidimd 17710 | In a Moore system whose cl... |
| isacs 17711 | A set is an algebraic clos... |
| acsmre 17712 | Algebraic closure systems ... |
| isacs2 17713 | In the definition of an al... |
| acsfiel 17714 | A set is closed in an alge... |
| acsfiel2 17715 | A set is closed in an alge... |
| acsmred 17716 | An algebraic closure syste... |
| isacs1i 17717 | A closure system determine... |
| mreacs 17718 | Algebraicity is a composab... |
| acsfn 17719 | Algebraicity of a conditio... |
| acsfn0 17720 | Algebraicity of a point cl... |
| acsfn1 17721 | Algebraicity of a one-argu... |
| acsfn1c 17722 | Algebraicity of a one-argu... |
| acsfn2 17723 | Algebraicity of a two-argu... |
| iscat 17732 | The predicate "is a catego... |
| iscatd 17733 | Properties that determine ... |
| catidex 17734 | Each object in a category ... |
| catideu 17735 | Each object in a category ... |
| cidfval 17736 | Each object in a category ... |
| cidval 17737 | Each object in a category ... |
| cidffn 17738 | The identity arrow constru... |
| cidfn 17739 | The identity arrow operato... |
| catidd 17740 | Deduce the identity arrow ... |
| iscatd2 17741 | Version of ~ iscatd with a... |
| catidcl 17742 | Each object in a category ... |
| catlid 17743 | Left identity property of ... |
| catrid 17744 | Right identity property of... |
| catcocl 17745 | Closure of a composition a... |
| catass 17746 | Associativity of compositi... |
| catcone0 17747 | Composition of non-empty h... |
| 0catg 17748 | Any structure with an empt... |
| 0cat 17749 | The empty set is a categor... |
| homffval 17750 | Value of the functionalize... |
| fnhomeqhomf 17751 | If the Hom-set operation i... |
| homfval 17752 | Value of the functionalize... |
| homffn 17753 | The functionalized Hom-set... |
| homfeq 17754 | Condition for two categori... |
| homfeqd 17755 | If two structures have the... |
| homfeqbas 17756 | Deduce equality of base se... |
| homfeqval 17757 | Value of the functionalize... |
| comfffval 17758 | Value of the functionalize... |
| comffval 17759 | Value of the functionalize... |
| comfval 17760 | Value of the functionalize... |
| comfffval2 17761 | Value of the functionalize... |
| comffval2 17762 | Value of the functionalize... |
| comfval2 17763 | Value of the functionalize... |
| comfffn 17764 | The functionalized composi... |
| comffn 17765 | The functionalized composi... |
| comfeq 17766 | Condition for two categori... |
| comfeqd 17767 | Condition for two categori... |
| comfeqval 17768 | Equality of two compositio... |
| catpropd 17769 | Two structures with the sa... |
| cidpropd 17770 | Two structures with the sa... |
| oppcval 17773 | Value of the opposite cate... |
| oppchomfval 17774 | Hom-sets of the opposite c... |
| oppchomfvalOLD 17775 | Obsolete proof of ~ oppcho... |
| oppchom 17776 | Hom-sets of the opposite c... |
| oppccofval 17777 | Composition in the opposit... |
| oppcco 17778 | Composition in the opposit... |
| oppcbas 17779 | Base set of an opposite ca... |
| oppcbasOLD 17780 | Obsolete version of ~ oppc... |
| oppccatid 17781 | Lemma for ~ oppccat . (Co... |
| oppchomf 17782 | Hom-sets of the opposite c... |
| oppcid 17783 | Identity function of an op... |
| oppccat 17784 | An opposite category is a ... |
| 2oppcbas 17785 | The double opposite catego... |
| 2oppchomf 17786 | The double opposite catego... |
| 2oppccomf 17787 | The double opposite catego... |
| oppchomfpropd 17788 | If two categories have the... |
| oppccomfpropd 17789 | If two categories have the... |
| oppccatf 17790 | ` oppCat ` restricted to `... |
| monfval 17795 | Definition of a monomorphi... |
| ismon 17796 | Definition of a monomorphi... |
| ismon2 17797 | Write out the monomorphism... |
| monhom 17798 | A monomorphism is a morphi... |
| moni 17799 | Property of a monomorphism... |
| monpropd 17800 | If two categories have the... |
| oppcmon 17801 | A monomorphism in the oppo... |
| oppcepi 17802 | An epimorphism in the oppo... |
| isepi 17803 | Definition of an epimorphi... |
| isepi2 17804 | Write out the epimorphism ... |
| epihom 17805 | An epimorphism is a morphi... |
| epii 17806 | Property of an epimorphism... |
| sectffval 17813 | Value of the section opera... |
| sectfval 17814 | Value of the section relat... |
| sectss 17815 | The section relation is a ... |
| issect 17816 | The property " ` F ` is a ... |
| issect2 17817 | Property of being a sectio... |
| sectcan 17818 | If ` G ` is a section of `... |
| sectco 17819 | Composition of two section... |
| isofval 17820 | Function value of the func... |
| invffval 17821 | Value of the inverse relat... |
| invfval 17822 | Value of the inverse relat... |
| isinv 17823 | Value of the inverse relat... |
| invss 17824 | The inverse relation is a ... |
| invsym 17825 | The inverse relation is sy... |
| invsym2 17826 | The inverse relation is sy... |
| invfun 17827 | The inverse relation is a ... |
| isoval 17828 | The isomorphisms are the d... |
| inviso1 17829 | If ` G ` is an inverse to ... |
| inviso2 17830 | If ` G ` is an inverse to ... |
| invf 17831 | The inverse relation is a ... |
| invf1o 17832 | The inverse relation is a ... |
| invinv 17833 | The inverse of the inverse... |
| invco 17834 | The composition of two iso... |
| dfiso2 17835 | Alternate definition of an... |
| dfiso3 17836 | Alternate definition of an... |
| inveq 17837 | If there are two inverses ... |
| isofn 17838 | The function value of the ... |
| isohom 17839 | An isomorphism is a homomo... |
| isoco 17840 | The composition of two iso... |
| oppcsect 17841 | A section in the opposite ... |
| oppcsect2 17842 | A section in the opposite ... |
| oppcinv 17843 | An inverse in the opposite... |
| oppciso 17844 | An isomorphism in the oppo... |
| sectmon 17845 | If ` F ` is a section of `... |
| monsect 17846 | If ` F ` is a monomorphism... |
| sectepi 17847 | If ` F ` is a section of `... |
| episect 17848 | If ` F ` is an epimorphism... |
| sectid 17849 | The identity is a section ... |
| invid 17850 | The inverse of the identit... |
| idiso 17851 | The identity is an isomorp... |
| idinv 17852 | The inverse of the identit... |
| invisoinvl 17853 | The inverse of an isomorph... |
| invisoinvr 17854 | The inverse of an isomorph... |
| invcoisoid 17855 | The inverse of an isomorph... |
| isocoinvid 17856 | The inverse of an isomorph... |
| rcaninv 17857 | Right cancellation of an i... |
| cicfval 17860 | The set of isomorphic obje... |
| brcic 17861 | The relation "is isomorphi... |
| cic 17862 | Objects ` X ` and ` Y ` in... |
| brcici 17863 | Prove that two objects are... |
| cicref 17864 | Isomorphism is reflexive. ... |
| ciclcl 17865 | Isomorphism implies the le... |
| cicrcl 17866 | Isomorphism implies the ri... |
| cicsym 17867 | Isomorphism is symmetric. ... |
| cictr 17868 | Isomorphism is transitive.... |
| cicer 17869 | Isomorphism is an equivale... |
| sscrel 17876 | The subcategory subset rel... |
| brssc 17877 | The subcategory subset rel... |
| sscpwex 17878 | An analogue of ~ pwex for ... |
| subcrcl 17879 | Reverse closure for the su... |
| sscfn1 17880 | The subcategory subset rel... |
| sscfn2 17881 | The subcategory subset rel... |
| ssclem 17882 | Lemma for ~ ssc1 and simil... |
| isssc 17883 | Value of the subcategory s... |
| ssc1 17884 | Infer subset relation on o... |
| ssc2 17885 | Infer subset relation on m... |
| sscres 17886 | Any function restricted to... |
| sscid 17887 | The subcategory subset rel... |
| ssctr 17888 | The subcategory subset rel... |
| ssceq 17889 | The subcategory subset rel... |
| rescval 17890 | Value of the category rest... |
| rescval2 17891 | Value of the category rest... |
| rescbas 17892 | Base set of the category r... |
| rescbasOLD 17893 | Obsolete version of ~ resc... |
| reschom 17894 | Hom-sets of the category r... |
| reschomf 17895 | Hom-sets of the category r... |
| rescco 17896 | Composition in the categor... |
| resccoOLD 17897 | Obsolete proof of ~ rescco... |
| rescabs 17898 | Restriction absorption law... |
| rescabsOLD 17899 | Obsolete proof of ~ seqp1d... |
| rescabs2 17900 | Restriction absorption law... |
| issubc 17901 | Elementhood in the set of ... |
| issubc2 17902 | Elementhood in the set of ... |
| 0ssc 17903 | For any category ` C ` , t... |
| 0subcat 17904 | For any category ` C ` , t... |
| catsubcat 17905 | For any category ` C ` , `... |
| subcssc 17906 | An element in the set of s... |
| subcfn 17907 | An element in the set of s... |
| subcss1 17908 | The objects of a subcatego... |
| subcss2 17909 | The morphisms of a subcate... |
| subcidcl 17910 | The identity of the origin... |
| subccocl 17911 | A subcategory is closed un... |
| subccatid 17912 | A subcategory is a categor... |
| subcid 17913 | The identity in a subcateg... |
| subccat 17914 | A subcategory is a categor... |
| issubc3 17915 | Alternate definition of a ... |
| fullsubc 17916 | The full subcategory gener... |
| fullresc 17917 | The category formed by str... |
| resscat 17918 | A category restricted to a... |
| subsubc 17919 | A subcategory of a subcate... |
| relfunc 17928 | The set of functors is a r... |
| funcrcl 17929 | Reverse closure for a func... |
| isfunc 17930 | Value of the set of functo... |
| isfuncd 17931 | Deduce that an operation i... |
| funcf1 17932 | The object part of a funct... |
| funcixp 17933 | The morphism part of a fun... |
| funcf2 17934 | The morphism part of a fun... |
| funcfn2 17935 | The morphism part of a fun... |
| funcid 17936 | A functor maps each identi... |
| funcco 17937 | A functor maps composition... |
| funcsect 17938 | The image of a section und... |
| funcinv 17939 | The image of an inverse un... |
| funciso 17940 | The image of an isomorphis... |
| funcoppc 17941 | A functor on categories yi... |
| idfuval 17942 | Value of the identity func... |
| idfu2nd 17943 | Value of the morphism part... |
| idfu2 17944 | Value of the morphism part... |
| idfu1st 17945 | Value of the object part o... |
| idfu1 17946 | Value of the object part o... |
| idfucl 17947 | The identity functor is a ... |
| cofuval 17948 | Value of the composition o... |
| cofu1st 17949 | Value of the object part o... |
| cofu1 17950 | Value of the object part o... |
| cofu2nd 17951 | Value of the morphism part... |
| cofu2 17952 | Value of the morphism part... |
| cofuval2 17953 | Value of the composition o... |
| cofucl 17954 | The composition of two fun... |
| cofuass 17955 | Functor composition is ass... |
| cofulid 17956 | The identity functor is a ... |
| cofurid 17957 | The identity functor is a ... |
| resfval 17958 | Value of the functor restr... |
| resfval2 17959 | Value of the functor restr... |
| resf1st 17960 | Value of the functor restr... |
| resf2nd 17961 | Value of the functor restr... |
| funcres 17962 | A functor restricted to a ... |
| funcres2b 17963 | Condition for a functor to... |
| funcres2 17964 | A functor into a restricte... |
| idfusubc0 17965 | The identity functor for a... |
| idfusubc 17966 | The identity functor for a... |
| wunfunc 17967 | A weak universe is closed ... |
| wunfuncOLD 17968 | Obsolete proof of ~ wunfun... |
| funcpropd 17969 | If two categories have the... |
| funcres2c 17970 | Condition for a functor to... |
| fullfunc 17975 | A full functor is a functo... |
| fthfunc 17976 | A faithful functor is a fu... |
| relfull 17977 | The set of full functors i... |
| relfth 17978 | The set of faithful functo... |
| isfull 17979 | Value of the set of full f... |
| isfull2 17980 | Equivalent condition for a... |
| fullfo 17981 | The morphism map of a full... |
| fulli 17982 | The morphism map of a full... |
| isfth 17983 | Value of the set of faithf... |
| isfth2 17984 | Equivalent condition for a... |
| isffth2 17985 | A fully faithful functor i... |
| fthf1 17986 | The morphism map of a fait... |
| fthi 17987 | The morphism map of a fait... |
| ffthf1o 17988 | The morphism map of a full... |
| fullpropd 17989 | If two categories have the... |
| fthpropd 17990 | If two categories have the... |
| fulloppc 17991 | The opposite functor of a ... |
| fthoppc 17992 | The opposite functor of a ... |
| ffthoppc 17993 | The opposite functor of a ... |
| fthsect 17994 | A faithful functor reflect... |
| fthinv 17995 | A faithful functor reflect... |
| fthmon 17996 | A faithful functor reflect... |
| fthepi 17997 | A faithful functor reflect... |
| ffthiso 17998 | A fully faithful functor r... |
| fthres2b 17999 | Condition for a faithful f... |
| fthres2c 18000 | Condition for a faithful f... |
| fthres2 18001 | A faithful functor into a ... |
| idffth 18002 | The identity functor is a ... |
| cofull 18003 | The composition of two ful... |
| cofth 18004 | The composition of two fai... |
| coffth 18005 | The composition of two ful... |
| rescfth 18006 | The inclusion functor from... |
| ressffth 18007 | The inclusion functor from... |
| fullres2c 18008 | Condition for a full funct... |
| ffthres2c 18009 | Condition for a fully fait... |
| inclfusubc 18010 | The "inclusion functor" fr... |
| fnfuc 18015 | The ` FuncCat ` operation ... |
| natfval 18016 | Value of the function givi... |
| isnat 18017 | Property of being a natura... |
| isnat2 18018 | Property of being a natura... |
| natffn 18019 | The natural transformation... |
| natrcl 18020 | Reverse closure for a natu... |
| nat1st2nd 18021 | Rewrite the natural transf... |
| natixp 18022 | A natural transformation i... |
| natcl 18023 | A component of a natural t... |
| natfn 18024 | A natural transformation i... |
| nati 18025 | Naturality property of a n... |
| wunnat 18026 | A weak universe is closed ... |
| wunnatOLD 18027 | Obsolete proof of ~ wunnat... |
| catstr 18028 | A category structure is a ... |
| fucval 18029 | Value of the functor categ... |
| fuccofval 18030 | Value of the functor categ... |
| fucbas 18031 | The objects of the functor... |
| fuchom 18032 | The morphisms in the funct... |
| fuchomOLD 18033 | Obsolete proof of ~ fuchom... |
| fucco 18034 | Value of the composition o... |
| fuccoval 18035 | Value of the functor categ... |
| fuccocl 18036 | The composition of two nat... |
| fucidcl 18037 | The identity natural trans... |
| fuclid 18038 | Left identity of natural t... |
| fucrid 18039 | Right identity of natural ... |
| fucass 18040 | Associativity of natural t... |
| fuccatid 18041 | The functor category is a ... |
| fuccat 18042 | The functor category is a ... |
| fucid 18043 | The identity morphism in t... |
| fucsect 18044 | Two natural transformation... |
| fucinv 18045 | Two natural transformation... |
| invfuc 18046 | If ` V ( x ) ` is an inver... |
| fuciso 18047 | A natural transformation i... |
| natpropd 18048 | If two categories have the... |
| fucpropd 18049 | If two categories have the... |
| initofn 18056 | ` InitO ` is a function on... |
| termofn 18057 | ` TermO ` is a function on... |
| zeroofn 18058 | ` ZeroO ` is a function on... |
| initorcl 18059 | Reverse closure for an ini... |
| termorcl 18060 | Reverse closure for a term... |
| zeroorcl 18061 | Reverse closure for a zero... |
| initoval 18062 | The value of the initial o... |
| termoval 18063 | The value of the terminal ... |
| zerooval 18064 | The value of the zero obje... |
| isinito 18065 | The predicate "is an initi... |
| istermo 18066 | The predicate "is a termin... |
| iszeroo 18067 | The predicate "is a zero o... |
| isinitoi 18068 | Implication of a class bei... |
| istermoi 18069 | Implication of a class bei... |
| initoid 18070 | For an initial object, the... |
| termoid 18071 | For a terminal object, the... |
| dfinito2 18072 | An initial object is a ter... |
| dftermo2 18073 | A terminal object is an in... |
| dfinito3 18074 | An alternate definition of... |
| dftermo3 18075 | An alternate definition of... |
| initoo 18076 | An initial object is an ob... |
| termoo 18077 | A terminal object is an ob... |
| iszeroi 18078 | Implication of a class bei... |
| 2initoinv 18079 | Morphisms between two init... |
| initoeu1 18080 | Initial objects are essent... |
| initoeu1w 18081 | Initial objects are essent... |
| initoeu2lem0 18082 | Lemma 0 for ~ initoeu2 . ... |
| initoeu2lem1 18083 | Lemma 1 for ~ initoeu2 . ... |
| initoeu2lem2 18084 | Lemma 2 for ~ initoeu2 . ... |
| initoeu2 18085 | Initial objects are essent... |
| 2termoinv 18086 | Morphisms between two term... |
| termoeu1 18087 | Terminal objects are essen... |
| termoeu1w 18088 | Terminal objects are essen... |
| homarcl 18097 | Reverse closure for an arr... |
| homafval 18098 | Value of the disjointified... |
| homaf 18099 | Functionality of the disjo... |
| homaval 18100 | Value of the disjointified... |
| elhoma 18101 | Value of the disjointified... |
| elhomai 18102 | Produce an arrow from a mo... |
| elhomai2 18103 | Produce an arrow from a mo... |
| homarcl2 18104 | Reverse closure for the do... |
| homarel 18105 | An arrow is an ordered pai... |
| homa1 18106 | The first component of an ... |
| homahom2 18107 | The second component of an... |
| homahom 18108 | The second component of an... |
| homadm 18109 | The domain of an arrow wit... |
| homacd 18110 | The codomain of an arrow w... |
| homadmcd 18111 | Decompose an arrow into do... |
| arwval 18112 | The set of arrows is the u... |
| arwrcl 18113 | The first component of an ... |
| arwhoma 18114 | An arrow is contained in t... |
| homarw 18115 | A hom-set is a subset of t... |
| arwdm 18116 | The domain of an arrow is ... |
| arwcd 18117 | The codomain of an arrow i... |
| dmaf 18118 | The domain function is a f... |
| cdaf 18119 | The codomain function is a... |
| arwhom 18120 | The second component of an... |
| arwdmcd 18121 | Decompose an arrow into do... |
| idafval 18126 | Value of the identity arro... |
| idaval 18127 | Value of the identity arro... |
| ida2 18128 | Morphism part of the ident... |
| idahom 18129 | Domain and codomain of the... |
| idadm 18130 | Domain of the identity arr... |
| idacd 18131 | Codomain of the identity a... |
| idaf 18132 | The identity arrow functio... |
| coafval 18133 | The value of the compositi... |
| eldmcoa 18134 | A pair ` <. G , F >. ` is ... |
| dmcoass 18135 | The domain of composition ... |
| homdmcoa 18136 | If ` F : X --> Y ` and ` G... |
| coaval 18137 | Value of composition for c... |
| coa2 18138 | The morphism part of arrow... |
| coahom 18139 | The composition of two com... |
| coapm 18140 | Composition of arrows is a... |
| arwlid 18141 | Left identity of a categor... |
| arwrid 18142 | Right identity of a catego... |
| arwass 18143 | Associativity of compositi... |
| setcval 18146 | Value of the category of s... |
| setcbas 18147 | Set of objects of the cate... |
| setchomfval 18148 | Set of arrows of the categ... |
| setchom 18149 | Set of arrows of the categ... |
| elsetchom 18150 | A morphism of sets is a fu... |
| setccofval 18151 | Composition in the categor... |
| setcco 18152 | Composition in the categor... |
| setccatid 18153 | Lemma for ~ setccat . (Co... |
| setccat 18154 | The category of sets is a ... |
| setcid 18155 | The identity arrow in the ... |
| setcmon 18156 | A monomorphism of sets is ... |
| setcepi 18157 | An epimorphism of sets is ... |
| setcsect 18158 | A section in the category ... |
| setcinv 18159 | An inverse in the category... |
| setciso 18160 | An isomorphism in the cate... |
| resssetc 18161 | The restriction of the cat... |
| funcsetcres2 18162 | A functor into a smaller c... |
| setc2obas 18163 | ` (/) ` and ` 1o ` are dis... |
| setc2ohom 18164 | ` ( SetCat `` 2o ) ` is a ... |
| cat1lem 18165 | The category of sets in a ... |
| cat1 18166 | The definition of category... |
| catcval 18169 | Value of the category of c... |
| catcbas 18170 | Set of objects of the cate... |
| catchomfval 18171 | Set of arrows of the categ... |
| catchom 18172 | Set of arrows of the categ... |
| catccofval 18173 | Composition in the categor... |
| catcco 18174 | Composition in the categor... |
| catccatid 18175 | Lemma for ~ catccat . (Co... |
| catcid 18176 | The identity arrow in the ... |
| catccat 18177 | The category of categories... |
| resscatc 18178 | The restriction of the cat... |
| catcisolem 18179 | Lemma for ~ catciso . (Co... |
| catciso 18180 | A functor is an isomorphis... |
| catcbascl 18181 | An element of the base set... |
| catcslotelcl 18182 | A slot entry of an element... |
| catcbaselcl 18183 | The base set of an element... |
| catchomcl 18184 | The Hom-set of an element ... |
| catcccocl 18185 | The composition operation ... |
| catcoppccl 18186 | The category of categories... |
| catcoppcclOLD 18187 | Obsolete proof of ~ catcop... |
| catcfuccl 18188 | The category of categories... |
| catcfucclOLD 18189 | Obsolete proof of ~ catcfu... |
| fncnvimaeqv 18190 | The inverse images of the ... |
| bascnvimaeqv 18191 | The inverse image of the u... |
| estrcval 18194 | Value of the category of e... |
| estrcbas 18195 | Set of objects of the cate... |
| estrchomfval 18196 | Set of morphisms ("arrows"... |
| estrchom 18197 | The morphisms between exte... |
| elestrchom 18198 | A morphism between extensi... |
| estrccofval 18199 | Composition in the categor... |
| estrcco 18200 | Composition in the categor... |
| estrcbasbas 18201 | An element of the base set... |
| estrccatid 18202 | Lemma for ~ estrccat . (C... |
| estrccat 18203 | The category of extensible... |
| estrcid 18204 | The identity arrow in the ... |
| estrchomfn 18205 | The Hom-set operation in t... |
| estrchomfeqhom 18206 | The functionalized Hom-set... |
| estrreslem1 18207 | Lemma 1 for ~ estrres . (... |
| estrreslem1OLD 18208 | Obsolete version of ~ estr... |
| estrreslem2 18209 | Lemma 2 for ~ estrres . (... |
| estrres 18210 | Any restriction of a categ... |
| funcestrcsetclem1 18211 | Lemma 1 for ~ funcestrcset... |
| funcestrcsetclem2 18212 | Lemma 2 for ~ funcestrcset... |
| funcestrcsetclem3 18213 | Lemma 3 for ~ funcestrcset... |
| funcestrcsetclem4 18214 | Lemma 4 for ~ funcestrcset... |
| funcestrcsetclem5 18215 | Lemma 5 for ~ funcestrcset... |
| funcestrcsetclem6 18216 | Lemma 6 for ~ funcestrcset... |
| funcestrcsetclem7 18217 | Lemma 7 for ~ funcestrcset... |
| funcestrcsetclem8 18218 | Lemma 8 for ~ funcestrcset... |
| funcestrcsetclem9 18219 | Lemma 9 for ~ funcestrcset... |
| funcestrcsetc 18220 | The "natural forgetful fun... |
| fthestrcsetc 18221 | The "natural forgetful fun... |
| fullestrcsetc 18222 | The "natural forgetful fun... |
| equivestrcsetc 18223 | The "natural forgetful fun... |
| setc1strwun 18224 | A constructed one-slot str... |
| funcsetcestrclem1 18225 | Lemma 1 for ~ funcsetcestr... |
| funcsetcestrclem2 18226 | Lemma 2 for ~ funcsetcestr... |
| funcsetcestrclem3 18227 | Lemma 3 for ~ funcsetcestr... |
| embedsetcestrclem 18228 | Lemma for ~ embedsetcestrc... |
| funcsetcestrclem4 18229 | Lemma 4 for ~ funcsetcestr... |
| funcsetcestrclem5 18230 | Lemma 5 for ~ funcsetcestr... |
| funcsetcestrclem6 18231 | Lemma 6 for ~ funcsetcestr... |
| funcsetcestrclem7 18232 | Lemma 7 for ~ funcsetcestr... |
| funcsetcestrclem8 18233 | Lemma 8 for ~ funcsetcestr... |
| funcsetcestrclem9 18234 | Lemma 9 for ~ funcsetcestr... |
| funcsetcestrc 18235 | The "embedding functor" fr... |
| fthsetcestrc 18236 | The "embedding functor" fr... |
| fullsetcestrc 18237 | The "embedding functor" fr... |
| embedsetcestrc 18238 | The "embedding functor" fr... |
| fnxpc 18247 | The binary product of cate... |
| xpcval 18248 | Value of the binary produc... |
| xpcbas 18249 | Set of objects of the bina... |
| xpchomfval 18250 | Set of morphisms of the bi... |
| xpchom 18251 | Set of morphisms of the bi... |
| relxpchom 18252 | A hom-set in the binary pr... |
| xpccofval 18253 | Value of composition in th... |
| xpcco 18254 | Value of composition in th... |
| xpcco1st 18255 | Value of composition in th... |
| xpcco2nd 18256 | Value of composition in th... |
| xpchom2 18257 | Value of the set of morphi... |
| xpcco2 18258 | Value of composition in th... |
| xpccatid 18259 | The product of two categor... |
| xpcid 18260 | The identity morphism in t... |
| xpccat 18261 | The product of two categor... |
| 1stfval 18262 | Value of the first project... |
| 1stf1 18263 | Value of the first project... |
| 1stf2 18264 | Value of the first project... |
| 2ndfval 18265 | Value of the first project... |
| 2ndf1 18266 | Value of the first project... |
| 2ndf2 18267 | Value of the first project... |
| 1stfcl 18268 | The first projection funct... |
| 2ndfcl 18269 | The second projection func... |
| prfval 18270 | Value of the pairing funct... |
| prf1 18271 | Value of the pairing funct... |
| prf2fval 18272 | Value of the pairing funct... |
| prf2 18273 | Value of the pairing funct... |
| prfcl 18274 | The pairing of functors ` ... |
| prf1st 18275 | Cancellation of pairing wi... |
| prf2nd 18276 | Cancellation of pairing wi... |
| 1st2ndprf 18277 | Break a functor into a pro... |
| catcxpccl 18278 | The category of categories... |
| catcxpcclOLD 18279 | Obsolete proof of ~ catcxp... |
| xpcpropd 18280 | If two categories have the... |
| evlfval 18289 | Value of the evaluation fu... |
| evlf2 18290 | Value of the evaluation fu... |
| evlf2val 18291 | Value of the evaluation na... |
| evlf1 18292 | Value of the evaluation fu... |
| evlfcllem 18293 | Lemma for ~ evlfcl . (Con... |
| evlfcl 18294 | The evaluation functor is ... |
| curfval 18295 | Value of the curry functor... |
| curf1fval 18296 | Value of the object part o... |
| curf1 18297 | Value of the object part o... |
| curf11 18298 | Value of the double evalua... |
| curf12 18299 | The partially evaluated cu... |
| curf1cl 18300 | The partially evaluated cu... |
| curf2 18301 | Value of the curry functor... |
| curf2val 18302 | Value of a component of th... |
| curf2cl 18303 | The curry functor at a mor... |
| curfcl 18304 | The curry functor of a fun... |
| curfpropd 18305 | If two categories have the... |
| uncfval 18306 | Value of the uncurry funct... |
| uncfcl 18307 | The uncurry operation take... |
| uncf1 18308 | Value of the uncurry funct... |
| uncf2 18309 | Value of the uncurry funct... |
| curfuncf 18310 | Cancellation of curry with... |
| uncfcurf 18311 | Cancellation of uncurry wi... |
| diagval 18312 | Define the diagonal functo... |
| diagcl 18313 | The diagonal functor is a ... |
| diag1cl 18314 | The constant functor of ` ... |
| diag11 18315 | Value of the constant func... |
| diag12 18316 | Value of the constant func... |
| diag2 18317 | Value of the diagonal func... |
| diag2cl 18318 | The diagonal functor at a ... |
| curf2ndf 18319 | As shown in ~ diagval , th... |
| hofval 18324 | Value of the Hom functor, ... |
| hof1fval 18325 | The object part of the Hom... |
| hof1 18326 | The object part of the Hom... |
| hof2fval 18327 | The morphism part of the H... |
| hof2val 18328 | The morphism part of the H... |
| hof2 18329 | The morphism part of the H... |
| hofcllem 18330 | Lemma for ~ hofcl . (Cont... |
| hofcl 18331 | Closure of the Hom functor... |
| oppchofcl 18332 | Closure of the opposite Ho... |
| yonval 18333 | Value of the Yoneda embedd... |
| yoncl 18334 | The Yoneda embedding is a ... |
| yon1cl 18335 | The Yoneda embedding at an... |
| yon11 18336 | Value of the Yoneda embedd... |
| yon12 18337 | Value of the Yoneda embedd... |
| yon2 18338 | Value of the Yoneda embedd... |
| hofpropd 18339 | If two categories have the... |
| yonpropd 18340 | If two categories have the... |
| oppcyon 18341 | Value of the opposite Yone... |
| oyoncl 18342 | The opposite Yoneda embedd... |
| oyon1cl 18343 | The opposite Yoneda embedd... |
| yonedalem1 18344 | Lemma for ~ yoneda . (Con... |
| yonedalem21 18345 | Lemma for ~ yoneda . (Con... |
| yonedalem3a 18346 | Lemma for ~ yoneda . (Con... |
| yonedalem4a 18347 | Lemma for ~ yoneda . (Con... |
| yonedalem4b 18348 | Lemma for ~ yoneda . (Con... |
| yonedalem4c 18349 | Lemma for ~ yoneda . (Con... |
| yonedalem22 18350 | Lemma for ~ yoneda . (Con... |
| yonedalem3b 18351 | Lemma for ~ yoneda . (Con... |
| yonedalem3 18352 | Lemma for ~ yoneda . (Con... |
| yonedainv 18353 | The Yoneda Lemma with expl... |
| yonffthlem 18354 | Lemma for ~ yonffth . (Co... |
| yoneda 18355 | The Yoneda Lemma. There i... |
| yonffth 18356 | The Yoneda Lemma. The Yon... |
| yoniso 18357 | If the codomain is recover... |
| oduval 18360 | Value of an order dual str... |
| oduleval 18361 | Value of the less-equal re... |
| oduleg 18362 | Truth of the less-equal re... |
| odubas 18363 | Base set of an order dual ... |
| odubasOLD 18364 | Obsolete proof of ~ odubas... |
| isprs 18369 | Property of being a preord... |
| prslem 18370 | Lemma for ~ prsref and ~ p... |
| prsref 18371 | "Less than or equal to" is... |
| prstr 18372 | "Less than or equal to" is... |
| isdrs 18373 | Property of being a direct... |
| drsdir 18374 | Direction of a directed se... |
| drsprs 18375 | A directed set is a proset... |
| drsbn0 18376 | The base of a directed set... |
| drsdirfi 18377 | Any _finite_ number of ele... |
| isdrs2 18378 | Directed sets may be defin... |
| ispos 18386 | The predicate "is a poset"... |
| ispos2 18387 | A poset is an antisymmetri... |
| posprs 18388 | A poset is a proset. (Con... |
| posi 18389 | Lemma for poset properties... |
| posref 18390 | A poset ordering is reflex... |
| posasymb 18391 | A poset ordering is asymme... |
| postr 18392 | A poset ordering is transi... |
| 0pos 18393 | Technical lemma to simplif... |
| 0posOLD 18394 | Obsolete proof of ~ 0pos a... |
| isposd 18395 | Properties that determine ... |
| isposi 18396 | Properties that determine ... |
| isposix 18397 | Properties that determine ... |
| isposixOLD 18398 | Obsolete proof of ~ isposi... |
| pospropd 18399 | Posethood is determined on... |
| odupos 18400 | Being a poset is a self-du... |
| oduposb 18401 | Being a poset is a self-du... |
| pltfval 18403 | Value of the less-than rel... |
| pltval 18404 | Less-than relation. ( ~ d... |
| pltle 18405 | "Less than" implies "less ... |
| pltne 18406 | The "less than" relation i... |
| pltirr 18407 | The "less than" relation i... |
| pleval2i 18408 | One direction of ~ pleval2... |
| pleval2 18409 | "Less than or equal to" in... |
| pltnle 18410 | "Less than" implies not co... |
| pltval3 18411 | Alternate expression for t... |
| pltnlt 18412 | The less-than relation imp... |
| pltn2lp 18413 | The less-than relation has... |
| plttr 18414 | The less-than relation is ... |
| pltletr 18415 | Transitive law for chained... |
| plelttr 18416 | Transitive law for chained... |
| pospo 18417 | Write a poset structure in... |
| lubfval 18422 | Value of the least upper b... |
| lubdm 18423 | Domain of the least upper ... |
| lubfun 18424 | The LUB is a function. (C... |
| lubeldm 18425 | Member of the domain of th... |
| lubelss 18426 | A member of the domain of ... |
| lubeu 18427 | Unique existence proper of... |
| lubval 18428 | Value of the least upper b... |
| lubcl 18429 | The least upper bound func... |
| lubprop 18430 | Properties of greatest low... |
| luble 18431 | The greatest lower bound i... |
| lublecllem 18432 | Lemma for ~ lublecl and ~ ... |
| lublecl 18433 | The set of all elements le... |
| lubid 18434 | The LUB of elements less t... |
| glbfval 18435 | Value of the greatest lowe... |
| glbdm 18436 | Domain of the greatest low... |
| glbfun 18437 | The GLB is a function. (C... |
| glbeldm 18438 | Member of the domain of th... |
| glbelss 18439 | A member of the domain of ... |
| glbeu 18440 | Unique existence proper of... |
| glbval 18441 | Value of the greatest lowe... |
| glbcl 18442 | The least upper bound func... |
| glbprop 18443 | Properties of greatest low... |
| glble 18444 | The greatest lower bound i... |
| joinfval 18445 | Value of join function for... |
| joinfval2 18446 | Value of join function for... |
| joindm 18447 | Domain of join function fo... |
| joindef 18448 | Two ways to say that a joi... |
| joinval 18449 | Join value. Since both si... |
| joincl 18450 | Closure of join of element... |
| joindmss 18451 | Subset property of domain ... |
| joinval2lem 18452 | Lemma for ~ joinval2 and ~... |
| joinval2 18453 | Value of join for a poset ... |
| joineu 18454 | Uniqueness of join of elem... |
| joinlem 18455 | Lemma for join properties.... |
| lejoin1 18456 | A join's first argument is... |
| lejoin2 18457 | A join's second argument i... |
| joinle 18458 | A join is less than or equ... |
| meetfval 18459 | Value of meet function for... |
| meetfval2 18460 | Value of meet function for... |
| meetdm 18461 | Domain of meet function fo... |
| meetdef 18462 | Two ways to say that a mee... |
| meetval 18463 | Meet value. Since both si... |
| meetcl 18464 | Closure of meet of element... |
| meetdmss 18465 | Subset property of domain ... |
| meetval2lem 18466 | Lemma for ~ meetval2 and ~... |
| meetval2 18467 | Value of meet for a poset ... |
| meeteu 18468 | Uniqueness of meet of elem... |
| meetlem 18469 | Lemma for meet properties.... |
| lemeet1 18470 | A meet's first argument is... |
| lemeet2 18471 | A meet's second argument i... |
| meetle 18472 | A meet is less than or equ... |
| joincomALT 18473 | The join of a poset is com... |
| joincom 18474 | The join of a poset is com... |
| meetcomALT 18475 | The meet of a poset is com... |
| meetcom 18476 | The meet of a poset is com... |
| join0 18477 | Lemma for ~ odumeet . (Co... |
| meet0 18478 | Lemma for ~ odujoin . (Co... |
| odulub 18479 | Least upper bounds in a du... |
| odujoin 18480 | Joins in a dual order are ... |
| oduglb 18481 | Greatest lower bounds in a... |
| odumeet 18482 | Meets in a dual order are ... |
| poslubmo 18483 | Least upper bounds in a po... |
| posglbmo 18484 | Greatest lower bounds in a... |
| poslubd 18485 | Properties which determine... |
| poslubdg 18486 | Properties which determine... |
| posglbdg 18487 | Properties which determine... |
| istos 18490 | The predicate "is a toset"... |
| tosso 18491 | Write the totally ordered ... |
| tospos 18492 | A Toset is a Poset. (Cont... |
| tleile 18493 | In a Toset, any two elemen... |
| tltnle 18494 | In a Toset, "less than" is... |
| p0val 18499 | Value of poset zero. (Con... |
| p1val 18500 | Value of poset zero. (Con... |
| p0le 18501 | Any element is less than o... |
| ple1 18502 | Any element is less than o... |
| islat 18505 | The predicate "is a lattic... |
| odulatb 18506 | Being a lattice is self-du... |
| odulat 18507 | Being a lattice is self-du... |
| latcl2 18508 | The join and meet of any t... |
| latlem 18509 | Lemma for lattice properti... |
| latpos 18510 | A lattice is a poset. (Co... |
| latjcl 18511 | Closure of join operation ... |
| latmcl 18512 | Closure of meet operation ... |
| latref 18513 | A lattice ordering is refl... |
| latasymb 18514 | A lattice ordering is asym... |
| latasym 18515 | A lattice ordering is asym... |
| lattr 18516 | A lattice ordering is tran... |
| latasymd 18517 | Deduce equality from latti... |
| lattrd 18518 | A lattice ordering is tran... |
| latjcom 18519 | The join of a lattice comm... |
| latlej1 18520 | A join's first argument is... |
| latlej2 18521 | A join's second argument i... |
| latjle12 18522 | A join is less than or equ... |
| latleeqj1 18523 | "Less than or equal to" in... |
| latleeqj2 18524 | "Less than or equal to" in... |
| latjlej1 18525 | Add join to both sides of ... |
| latjlej2 18526 | Add join to both sides of ... |
| latjlej12 18527 | Add join to both sides of ... |
| latnlej 18528 | An idiom to express that a... |
| latnlej1l 18529 | An idiom to express that a... |
| latnlej1r 18530 | An idiom to express that a... |
| latnlej2 18531 | An idiom to express that a... |
| latnlej2l 18532 | An idiom to express that a... |
| latnlej2r 18533 | An idiom to express that a... |
| latjidm 18534 | Lattice join is idempotent... |
| latmcom 18535 | The join of a lattice comm... |
| latmle1 18536 | A meet is less than or equ... |
| latmle2 18537 | A meet is less than or equ... |
| latlem12 18538 | An element is less than or... |
| latleeqm1 18539 | "Less than or equal to" in... |
| latleeqm2 18540 | "Less than or equal to" in... |
| latmlem1 18541 | Add meet to both sides of ... |
| latmlem2 18542 | Add meet to both sides of ... |
| latmlem12 18543 | Add join to both sides of ... |
| latnlemlt 18544 | Negation of "less than or ... |
| latnle 18545 | Equivalent expressions for... |
| latmidm 18546 | Lattice meet is idempotent... |
| latabs1 18547 | Lattice absorption law. F... |
| latabs2 18548 | Lattice absorption law. F... |
| latledi 18549 | An ortholattice is distrib... |
| latmlej11 18550 | Ordering of a meet and joi... |
| latmlej12 18551 | Ordering of a meet and joi... |
| latmlej21 18552 | Ordering of a meet and joi... |
| latmlej22 18553 | Ordering of a meet and joi... |
| lubsn 18554 | The least upper bound of a... |
| latjass 18555 | Lattice join is associativ... |
| latj12 18556 | Swap 1st and 2nd members o... |
| latj32 18557 | Swap 2nd and 3rd members o... |
| latj13 18558 | Swap 1st and 3rd members o... |
| latj31 18559 | Swap 2nd and 3rd members o... |
| latjrot 18560 | Rotate lattice join of 3 c... |
| latj4 18561 | Rearrangement of lattice j... |
| latj4rot 18562 | Rotate lattice join of 4 c... |
| latjjdi 18563 | Lattice join distributes o... |
| latjjdir 18564 | Lattice join distributes o... |
| mod1ile 18565 | The weak direction of the ... |
| mod2ile 18566 | The weak direction of the ... |
| latmass 18567 | Lattice meet is associativ... |
| latdisdlem 18568 | Lemma for ~ latdisd . (Co... |
| latdisd 18569 | In a lattice, joins distri... |
| isclat 18572 | The predicate "is a comple... |
| clatpos 18573 | A complete lattice is a po... |
| clatlem 18574 | Lemma for properties of a ... |
| clatlubcl 18575 | Any subset of the base set... |
| clatlubcl2 18576 | Any subset of the base set... |
| clatglbcl 18577 | Any subset of the base set... |
| clatglbcl2 18578 | Any subset of the base set... |
| oduclatb 18579 | Being a complete lattice i... |
| clatl 18580 | A complete lattice is a la... |
| isglbd 18581 | Properties that determine ... |
| lublem 18582 | Lemma for the least upper ... |
| lubub 18583 | The LUB of a complete latt... |
| lubl 18584 | The LUB of a complete latt... |
| lubss 18585 | Subset law for least upper... |
| lubel 18586 | An element of a set is les... |
| lubun 18587 | The LUB of a union. (Cont... |
| clatglb 18588 | Properties of greatest low... |
| clatglble 18589 | The greatest lower bound i... |
| clatleglb 18590 | Two ways of expressing "le... |
| clatglbss 18591 | Subset law for greatest lo... |
| isdlat 18594 | Property of being a distri... |
| dlatmjdi 18595 | In a distributive lattice,... |
| dlatl 18596 | A distributive lattice is ... |
| odudlatb 18597 | The dual of a distributive... |
| dlatjmdi 18598 | In a distributive lattice,... |
| ipostr 18601 | The structure of ~ df-ipo ... |
| ipoval 18602 | Value of the inclusion pos... |
| ipobas 18603 | Base set of the inclusion ... |
| ipolerval 18604 | Relation of the inclusion ... |
| ipotset 18605 | Topology of the inclusion ... |
| ipole 18606 | Weak order condition of th... |
| ipolt 18607 | Strict order condition of ... |
| ipopos 18608 | The inclusion poset on a f... |
| isipodrs 18609 | Condition for a family of ... |
| ipodrscl 18610 | Direction by inclusion as ... |
| ipodrsfi 18611 | Finite upper bound propert... |
| fpwipodrs 18612 | The finite subsets of any ... |
| ipodrsima 18613 | The monotone image of a di... |
| isacs3lem 18614 | An algebraic closure syste... |
| acsdrsel 18615 | An algebraic closure syste... |
| isacs4lem 18616 | In a closure system in whi... |
| isacs5lem 18617 | If closure commutes with d... |
| acsdrscl 18618 | In an algebraic closure sy... |
| acsficl 18619 | A closure in an algebraic ... |
| isacs5 18620 | A closure system is algebr... |
| isacs4 18621 | A closure system is algebr... |
| isacs3 18622 | A closure system is algebr... |
| acsficld 18623 | In an algebraic closure sy... |
| acsficl2d 18624 | In an algebraic closure sy... |
| acsfiindd 18625 | In an algebraic closure sy... |
| acsmapd 18626 | In an algebraic closure sy... |
| acsmap2d 18627 | In an algebraic closure sy... |
| acsinfd 18628 | In an algebraic closure sy... |
| acsdomd 18629 | In an algebraic closure sy... |
| acsinfdimd 18630 | In an algebraic closure sy... |
| acsexdimd 18631 | In an algebraic closure sy... |
| mrelatglb 18632 | Greatest lower bounds in a... |
| mrelatglb0 18633 | The empty intersection in ... |
| mrelatlub 18634 | Least upper bounds in a Mo... |
| mreclatBAD 18635 | A Moore space is a complet... |
| isps 18640 | The predicate "is a poset"... |
| psrel 18641 | A poset is a relation. (C... |
| psref2 18642 | A poset is antisymmetric a... |
| pstr2 18643 | A poset is transitive. (C... |
| pslem 18644 | Lemma for ~ psref and othe... |
| psdmrn 18645 | The domain and range of a ... |
| psref 18646 | A poset is reflexive. (Co... |
| psrn 18647 | The range of a poset equal... |
| psasym 18648 | A poset is antisymmetric. ... |
| pstr 18649 | A poset is transitive. (C... |
| cnvps 18650 | The converse of a poset is... |
| cnvpsb 18651 | The converse of a poset is... |
| psss 18652 | Any subset of a partially ... |
| psssdm2 18653 | Field of a subposet. (Con... |
| psssdm 18654 | Field of a subposet. (Con... |
| istsr 18655 | The predicate is a toset. ... |
| istsr2 18656 | The predicate is a toset. ... |
| tsrlin 18657 | A toset is a linear order.... |
| tsrlemax 18658 | Two ways of saying a numbe... |
| tsrps 18659 | A toset is a poset. (Cont... |
| cnvtsr 18660 | The converse of a toset is... |
| tsrss 18661 | Any subset of a totally or... |
| ledm 18662 | The domain of ` <_ ` is ` ... |
| lern 18663 | The range of ` <_ ` is ` R... |
| lefld 18664 | The field of the 'less or ... |
| letsr 18665 | The "less than or equal to... |
| isdir 18670 | A condition for a relation... |
| reldir 18671 | A direction is a relation.... |
| dirdm 18672 | A direction's domain is eq... |
| dirref 18673 | A direction is reflexive. ... |
| dirtr 18674 | A direction is transitive.... |
| dirge 18675 | For any two elements of a ... |
| tsrdir 18676 | A totally ordered set is a... |
| ismgm 18681 | The predicate "is a magma"... |
| ismgmn0 18682 | The predicate "is a magma"... |
| mgmcl 18683 | Closure of the operation o... |
| isnmgm 18684 | A condition for a structur... |
| mgmsscl 18685 | If the base set of a magma... |
| plusffval 18686 | The group addition operati... |
| plusfval 18687 | The group addition operati... |
| plusfeq 18688 | If the addition operation ... |
| plusffn 18689 | The group addition operati... |
| mgmplusf 18690 | The group addition functio... |
| mgmpropd 18691 | If two structures have the... |
| ismgmd 18692 | Deduce a magma from its pr... |
| issstrmgm 18693 | Characterize a substructur... |
| intopsn 18694 | The internal operation for... |
| mgmb1mgm1 18695 | The only magma with a base... |
| mgm0 18696 | Any set with an empty base... |
| mgm0b 18697 | The structure with an empt... |
| mgm1 18698 | The structure with one ele... |
| opifismgm 18699 | A structure with a group a... |
| mgmidmo 18700 | A two-sided identity eleme... |
| grpidval 18701 | The value of the identity ... |
| grpidpropd 18702 | If two structures have the... |
| fn0g 18703 | The group zero extractor i... |
| 0g0 18704 | The identity element funct... |
| ismgmid 18705 | The identity element of a ... |
| mgmidcl 18706 | The identity element of a ... |
| mgmlrid 18707 | The identity element of a ... |
| ismgmid2 18708 | Show that a given element ... |
| lidrideqd 18709 | If there is a left and rig... |
| lidrididd 18710 | If there is a left and rig... |
| grpidd 18711 | Deduce the identity elemen... |
| mgmidsssn0 18712 | Property of the set of ide... |
| grpinvalem 18713 | Lemma for ~ grpinva . (Co... |
| grpinva 18714 | Deduce right inverse from ... |
| grprida 18715 | Deduce right identity from... |
| gsumvalx 18716 | Expand out the substitutio... |
| gsumval 18717 | Expand out the substitutio... |
| gsumpropd 18718 | The group sum depends only... |
| gsumpropd2lem 18719 | Lemma for ~ gsumpropd2 . ... |
| gsumpropd2 18720 | A stronger version of ~ gs... |
| gsummgmpropd 18721 | A stronger version of ~ gs... |
| gsumress 18722 | The group sum in a substru... |
| gsumval1 18723 | Value of the group sum ope... |
| gsum0 18724 | Value of the empty group s... |
| gsumval2a 18725 | Value of the group sum ope... |
| gsumval2 18726 | Value of the group sum ope... |
| gsumsplit1r 18727 | Splitting off the rightmos... |
| gsumprval 18728 | Value of the group sum ope... |
| gsumpr12val 18729 | Value of the group sum ope... |
| mgmhmrcl 18734 | Reverse closure of a magma... |
| submgmrcl 18735 | Reverse closure for submag... |
| ismgmhm 18736 | Property of a magma homomo... |
| mgmhmf 18737 | A magma homomorphism is a ... |
| mgmhmpropd 18738 | Magma homomorphism depends... |
| mgmhmlin 18739 | A magma homomorphism prese... |
| mgmhmf1o 18740 | A magma homomorphism is bi... |
| idmgmhm 18741 | The identity homomorphism ... |
| issubmgm 18742 | Expand definition of a sub... |
| issubmgm2 18743 | Submagmas are subsets that... |
| rabsubmgmd 18744 | Deduction for proving that... |
| submgmss 18745 | Submagmas are subsets of t... |
| submgmid 18746 | Every magma is trivially a... |
| submgmcl 18747 | Submagmas are closed under... |
| submgmmgm 18748 | Submagmas are themselves m... |
| submgmbas 18749 | The base set of a submagma... |
| subsubmgm 18750 | A submagma of a submagma i... |
| resmgmhm 18751 | Restriction of a magma hom... |
| resmgmhm2 18752 | One direction of ~ resmgmh... |
| resmgmhm2b 18753 | Restriction of the codomai... |
| mgmhmco 18754 | The composition of magma h... |
| mgmhmima 18755 | The homomorphic image of a... |
| mgmhmeql 18756 | The equalizer of two magma... |
| submgmacs 18757 | Submagmas are an algebraic... |
| issgrp 18760 | The predicate "is a semigr... |
| issgrpv 18761 | The predicate "is a semigr... |
| issgrpn0 18762 | The predicate "is a semigr... |
| isnsgrp 18763 | A condition for a structur... |
| sgrpmgm 18764 | A semigroup is a magma. (... |
| sgrpass 18765 | A semigroup operation is a... |
| sgrpcl 18766 | Closure of the operation o... |
| sgrp0 18767 | Any set with an empty base... |
| sgrp0b 18768 | The structure with an empt... |
| sgrp1 18769 | The structure with one ele... |
| issgrpd 18770 | Deduce a semigroup from it... |
| sgrppropd 18771 | If two structures are sets... |
| prdsplusgsgrpcl 18772 | Structure product pointwis... |
| prdssgrpd 18773 | The product of a family of... |
| ismnddef 18776 | The predicate "is a monoid... |
| ismnd 18777 | The predicate "is a monoid... |
| isnmnd 18778 | A condition for a structur... |
| sgrpidmnd 18779 | A semigroup with an identi... |
| mndsgrp 18780 | A monoid is a semigroup. ... |
| mndmgm 18781 | A monoid is a magma. (Con... |
| mndcl 18782 | Closure of the operation o... |
| mndass 18783 | A monoid operation is asso... |
| mndid 18784 | A monoid has a two-sided i... |
| mndideu 18785 | The two-sided identity ele... |
| mnd32g 18786 | Commutative/associative la... |
| mnd12g 18787 | Commutative/associative la... |
| mnd4g 18788 | Commutative/associative la... |
| mndidcl 18789 | The identity element of a ... |
| mndbn0 18790 | The base set of a monoid i... |
| hashfinmndnn 18791 | A finite monoid has positi... |
| mndplusf 18792 | The group addition operati... |
| mndlrid 18793 | A monoid's identity elemen... |
| mndlid 18794 | The identity element of a ... |
| mndrid 18795 | The identity element of a ... |
| ismndd 18796 | Deduce a monoid from its p... |
| mndpfo 18797 | The addition operation of ... |
| mndfo 18798 | The addition operation of ... |
| mndpropd 18799 | If two structures have the... |
| mndprop 18800 | If two structures have the... |
| issubmnd 18801 | Characterize a submonoid b... |
| ress0g 18802 | ` 0g ` is unaffected by re... |
| submnd0 18803 | The zero of a submonoid is... |
| mndinvmod 18804 | Uniqueness of an inverse e... |
| prdsplusgcl 18805 | Structure product pointwis... |
| prdsidlem 18806 | Characterization of identi... |
| prdsmndd 18807 | The product of a family of... |
| prds0g 18808 | Zero in a product of monoi... |
| pwsmnd 18809 | The structure power of a m... |
| pws0g 18810 | Zero in a structure power ... |
| imasmnd2 18811 | The image structure of a m... |
| imasmnd 18812 | The image structure of a m... |
| imasmndf1 18813 | The image of a monoid unde... |
| xpsmnd 18814 | The binary product of mono... |
| xpsmnd0 18815 | The identity element of a ... |
| mnd1 18816 | The (smallest) structure r... |
| mnd1id 18817 | The singleton element of a... |
| ismhm 18822 | Property of a monoid homom... |
| ismhmd 18823 | Deduction version of ~ ism... |
| mhmrcl1 18824 | Reverse closure of a monoi... |
| mhmrcl2 18825 | Reverse closure of a monoi... |
| mhmf 18826 | A monoid homomorphism is a... |
| ismhm0 18827 | Property of a monoid homom... |
| mhmismgmhm 18828 | Each monoid homomorphism i... |
| mhmpropd 18829 | Monoid homomorphism depend... |
| mhmlin 18830 | A monoid homomorphism comm... |
| mhm0 18831 | A monoid homomorphism pres... |
| idmhm 18832 | The identity homomorphism ... |
| mhmf1o 18833 | A monoid homomorphism is b... |
| mndvcl 18834 | Tuple-wise additive closur... |
| mndvass 18835 | Tuple-wise associativity i... |
| mndvlid 18836 | Tuple-wise left identity i... |
| mndvrid 18837 | Tuple-wise right identity ... |
| mhmvlin 18838 | Tuple extension of monoid ... |
| submrcl 18839 | Reverse closure for submon... |
| issubm 18840 | Expand definition of a sub... |
| issubm2 18841 | Submonoids are subsets tha... |
| issubmndb 18842 | The submonoid predicate. ... |
| issubmd 18843 | Deduction for proving a su... |
| mndissubm 18844 | If the base set of a monoi... |
| resmndismnd 18845 | If the base set of a monoi... |
| submss 18846 | Submonoids are subsets of ... |
| submid 18847 | Every monoid is trivially ... |
| subm0cl 18848 | Submonoids contain zero. ... |
| submcl 18849 | Submonoids are closed unde... |
| submmnd 18850 | Submonoids are themselves ... |
| submbas 18851 | The base set of a submonoi... |
| subm0 18852 | Submonoids have the same i... |
| subsubm 18853 | A submonoid of a submonoid... |
| 0subm 18854 | The zero submonoid of an a... |
| insubm 18855 | The intersection of two su... |
| 0mhm 18856 | The constant zero linear f... |
| resmhm 18857 | Restriction of a monoid ho... |
| resmhm2 18858 | One direction of ~ resmhm2... |
| resmhm2b 18859 | Restriction of the codomai... |
| mhmco 18860 | The composition of monoid ... |
| mhmimalem 18861 | Lemma for ~ mhmima and sim... |
| mhmima 18862 | The homomorphic image of a... |
| mhmeql 18863 | The equalizer of two monoi... |
| submacs 18864 | Submonoids are an algebrai... |
| mndind 18865 | Induction in a monoid. In... |
| prdspjmhm 18866 | A projection from a produc... |
| pwspjmhm 18867 | A projection from a struct... |
| pwsdiagmhm 18868 | Diagonal monoid homomorphi... |
| pwsco1mhm 18869 | Right composition with a f... |
| pwsco2mhm 18870 | Left composition with a mo... |
| gsumvallem2 18871 | Lemma for properties of th... |
| gsumsubm 18872 | Evaluate a group sum in a ... |
| gsumz 18873 | Value of a group sum over ... |
| gsumwsubmcl 18874 | Closure of the composite i... |
| gsumws1 18875 | A singleton composite reco... |
| gsumwcl 18876 | Closure of the composite o... |
| gsumsgrpccat 18877 | Homomorphic property of no... |
| gsumccat 18878 | Homomorphic property of co... |
| gsumws2 18879 | Valuation of a pair in a m... |
| gsumccatsn 18880 | Homomorphic property of co... |
| gsumspl 18881 | The primary purpose of the... |
| gsumwmhm 18882 | Behavior of homomorphisms ... |
| gsumwspan 18883 | The submonoid generated by... |
| frmdval 18888 | Value of the free monoid c... |
| frmdbas 18889 | The base set of a free mon... |
| frmdelbas 18890 | An element of the base set... |
| frmdplusg 18891 | The monoid operation of a ... |
| frmdadd 18892 | Value of the monoid operat... |
| vrmdfval 18893 | The canonical injection fr... |
| vrmdval 18894 | The value of the generatin... |
| vrmdf 18895 | The mapping from the index... |
| frmdmnd 18896 | A free monoid is a monoid.... |
| frmd0 18897 | The identity of the free m... |
| frmdsssubm 18898 | The set of words taking va... |
| frmdgsum 18899 | Any word in a free monoid ... |
| frmdss2 18900 | A subset of generators is ... |
| frmdup1 18901 | Any assignment of the gene... |
| frmdup2 18902 | The evaluation map has the... |
| frmdup3lem 18903 | Lemma for ~ frmdup3 . (Co... |
| frmdup3 18904 | Universal property of the ... |
| efmnd 18907 | The monoid of endofunction... |
| efmndbas 18908 | The base set of the monoid... |
| efmndbasabf 18909 | The base set of the monoid... |
| elefmndbas 18910 | Two ways of saying a funct... |
| elefmndbas2 18911 | Two ways of saying a funct... |
| efmndbasf 18912 | Elements in the monoid of ... |
| efmndhash 18913 | The monoid of endofunction... |
| efmndbasfi 18914 | The monoid of endofunction... |
| efmndfv 18915 | The function value of an e... |
| efmndtset 18916 | The topology of the monoid... |
| efmndplusg 18917 | The group operation of a m... |
| efmndov 18918 | The value of the group ope... |
| efmndcl 18919 | The group operation of the... |
| efmndtopn 18920 | The topology of the monoid... |
| symggrplem 18921 | Lemma for ~ symggrp and ~ ... |
| efmndmgm 18922 | The monoid of endofunction... |
| efmndsgrp 18923 | The monoid of endofunction... |
| ielefmnd 18924 | The identity function rest... |
| efmndid 18925 | The identity function rest... |
| efmndmnd 18926 | The monoid of endofunction... |
| efmnd0nmnd 18927 | Even the monoid of endofun... |
| efmndbas0 18928 | The base set of the monoid... |
| efmnd1hash 18929 | The monoid of endofunction... |
| efmnd1bas 18930 | The monoid of endofunction... |
| efmnd2hash 18931 | The monoid of endofunction... |
| submefmnd 18932 | If the base set of a monoi... |
| sursubmefmnd 18933 | The set of surjective endo... |
| injsubmefmnd 18934 | The set of injective endof... |
| idressubmefmnd 18935 | The singleton containing o... |
| idresefmnd 18936 | The structure with the sin... |
| smndex1ibas 18937 | The modulo function ` I ` ... |
| smndex1iidm 18938 | The modulo function ` I ` ... |
| smndex1gbas 18939 | The constant functions ` (... |
| smndex1gid 18940 | The composition of a const... |
| smndex1igid 18941 | The composition of the mod... |
| smndex1basss 18942 | The modulo function ` I ` ... |
| smndex1bas 18943 | The base set of the monoid... |
| smndex1mgm 18944 | The monoid of endofunction... |
| smndex1sgrp 18945 | The monoid of endofunction... |
| smndex1mndlem 18946 | Lemma for ~ smndex1mnd and... |
| smndex1mnd 18947 | The monoid of endofunction... |
| smndex1id 18948 | The modulo function ` I ` ... |
| smndex1n0mnd 18949 | The identity of the monoid... |
| nsmndex1 18950 | The base set ` B ` of the ... |
| smndex2dbas 18951 | The doubling function ` D ... |
| smndex2dnrinv 18952 | The doubling function ` D ... |
| smndex2hbas 18953 | The halving functions ` H ... |
| smndex2dlinvh 18954 | The halving functions ` H ... |
| mgm2nsgrplem1 18955 | Lemma 1 for ~ mgm2nsgrp : ... |
| mgm2nsgrplem2 18956 | Lemma 2 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem3 18957 | Lemma 3 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem4 18958 | Lemma 4 for ~ mgm2nsgrp : ... |
| mgm2nsgrp 18959 | A small magma (with two el... |
| sgrp2nmndlem1 18960 | Lemma 1 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem2 18961 | Lemma 2 for ~ sgrp2nmnd . ... |
| sgrp2nmndlem3 18962 | Lemma 3 for ~ sgrp2nmnd . ... |
| sgrp2rid2 18963 | A small semigroup (with tw... |
| sgrp2rid2ex 18964 | A small semigroup (with tw... |
| sgrp2nmndlem4 18965 | Lemma 4 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem5 18966 | Lemma 5 for ~ sgrp2nmnd : ... |
| sgrp2nmnd 18967 | A small semigroup (with tw... |
| mgmnsgrpex 18968 | There is a magma which is ... |
| sgrpnmndex 18969 | There is a semigroup which... |
| sgrpssmgm 18970 | The class of all semigroup... |
| mndsssgrp 18971 | The class of all monoids i... |
| pwmndgplus 18972 | The operation of the monoi... |
| pwmndid 18973 | The identity of the monoid... |
| pwmnd 18974 | The power set of a class `... |
| isgrp 18981 | The predicate "is a group"... |
| grpmnd 18982 | A group is a monoid. (Con... |
| grpcl 18983 | Closure of the operation o... |
| grpass 18984 | A group operation is assoc... |
| grpinvex 18985 | Every member of a group ha... |
| grpideu 18986 | The two-sided identity ele... |
| grpassd 18987 | A group operation is assoc... |
| grpmndd 18988 | A group is a monoid. (Con... |
| grpcld 18989 | Closure of the operation o... |
| grpplusf 18990 | The group addition operati... |
| grpplusfo 18991 | The group addition operati... |
| resgrpplusfrn 18992 | The underlying set of a gr... |
| grppropd 18993 | If two structures have the... |
| grpprop 18994 | If two structures have the... |
| grppropstr 18995 | Generalize a specific 2-el... |
| grpss 18996 | Show that a structure exte... |
| isgrpd2e 18997 | Deduce a group from its pr... |
| isgrpd2 18998 | Deduce a group from its pr... |
| isgrpde 18999 | Deduce a group from its pr... |
| isgrpd 19000 | Deduce a group from its pr... |
| isgrpi 19001 | Properties that determine ... |
| grpsgrp 19002 | A group is a semigroup. (... |
| grpmgmd 19003 | A group is a magma, deduct... |
| dfgrp2 19004 | Alternate definition of a ... |
| dfgrp2e 19005 | Alternate definition of a ... |
| isgrpix 19006 | Properties that determine ... |
| grpidcl 19007 | The identity element of a ... |
| grpbn0 19008 | The base set of a group is... |
| grplid 19009 | The identity element of a ... |
| grprid 19010 | The identity element of a ... |
| grplidd 19011 | The identity element of a ... |
| grpridd 19012 | The identity element of a ... |
| grpn0 19013 | A group is not empty. (Co... |
| hashfingrpnn 19014 | A finite group has positiv... |
| grprcan 19015 | Right cancellation law for... |
| grpinveu 19016 | The left inverse element o... |
| grpid 19017 | Two ways of saying that an... |
| isgrpid2 19018 | Properties showing that an... |
| grpidd2 19019 | Deduce the identity elemen... |
| grpinvfval 19020 | The inverse function of a ... |
| grpinvfvalALT 19021 | Shorter proof of ~ grpinvf... |
| grpinvval 19022 | The inverse of a group ele... |
| grpinvfn 19023 | Functionality of the group... |
| grpinvfvi 19024 | The group inverse function... |
| grpsubfval 19025 | Group subtraction (divisio... |
| grpsubfvalALT 19026 | Shorter proof of ~ grpsubf... |
| grpsubval 19027 | Group subtraction (divisio... |
| grpinvf 19028 | The group inversion operat... |
| grpinvcl 19029 | A group element's inverse ... |
| grpinvcld 19030 | A group element's inverse ... |
| grplinv 19031 | The left inverse of a grou... |
| grprinv 19032 | The right inverse of a gro... |
| grpinvid1 19033 | The inverse of a group ele... |
| grpinvid2 19034 | The inverse of a group ele... |
| isgrpinv 19035 | Properties showing that a ... |
| grplinvd 19036 | The left inverse of a grou... |
| grprinvd 19037 | The right inverse of a gro... |
| grplrinv 19038 | In a group, every member h... |
| grpidinv2 19039 | A group's properties using... |
| grpidinv 19040 | A group has a left and rig... |
| grpinvid 19041 | The inverse of the identit... |
| grplcan 19042 | Left cancellation law for ... |
| grpasscan1 19043 | An associative cancellatio... |
| grpasscan2 19044 | An associative cancellatio... |
| grpidrcan 19045 | If right adding an element... |
| grpidlcan 19046 | If left adding an element ... |
| grpinvinv 19047 | Double inverse law for gro... |
| grpinvcnv 19048 | The group inverse is its o... |
| grpinv11 19049 | The group inverse is one-t... |
| grpinv11OLD 19050 | Obsolete version of ~ grpi... |
| grpinvf1o 19051 | The group inverse is a one... |
| grpinvnz 19052 | The inverse of a nonzero g... |
| grpinvnzcl 19053 | The inverse of a nonzero g... |
| grpsubinv 19054 | Subtraction of an inverse.... |
| grplmulf1o 19055 | Left multiplication by a g... |
| grpraddf1o 19056 | Right addition by a group ... |
| grpinvpropd 19057 | If two structures have the... |
| grpidssd 19058 | If the base set of a group... |
| grpinvssd 19059 | If the base set of a group... |
| grpinvadd 19060 | The inverse of the group o... |
| grpsubf 19061 | Functionality of group sub... |
| grpsubcl 19062 | Closure of group subtracti... |
| grpsubrcan 19063 | Right cancellation law for... |
| grpinvsub 19064 | Inverse of a group subtrac... |
| grpinvval2 19065 | A ~ df-neg -like equation ... |
| grpsubid 19066 | Subtraction of a group ele... |
| grpsubid1 19067 | Subtraction of the identit... |
| grpsubeq0 19068 | If the difference between ... |
| grpsubadd0sub 19069 | Subtraction expressed as a... |
| grpsubadd 19070 | Relationship between group... |
| grpsubsub 19071 | Double group subtraction. ... |
| grpaddsubass 19072 | Associative-type law for g... |
| grppncan 19073 | Cancellation law for subtr... |
| grpnpcan 19074 | Cancellation law for subtr... |
| grpsubsub4 19075 | Double group subtraction (... |
| grppnpcan2 19076 | Cancellation law for mixed... |
| grpnpncan 19077 | Cancellation law for group... |
| grpnpncan0 19078 | Cancellation law for group... |
| grpnnncan2 19079 | Cancellation law for group... |
| dfgrp3lem 19080 | Lemma for ~ dfgrp3 . (Con... |
| dfgrp3 19081 | Alternate definition of a ... |
| dfgrp3e 19082 | Alternate definition of a ... |
| grplactfval 19083 | The left group action of e... |
| grplactval 19084 | The value of the left grou... |
| grplactcnv 19085 | The left group action of e... |
| grplactf1o 19086 | The left group action of e... |
| grpsubpropd 19087 | Weak property deduction fo... |
| grpsubpropd2 19088 | Strong property deduction ... |
| grp1 19089 | The (smallest) structure r... |
| grp1inv 19090 | The inverse function of th... |
| prdsinvlem 19091 | Characterization of invers... |
| prdsgrpd 19092 | The product of a family of... |
| prdsinvgd 19093 | Negation in a product of g... |
| pwsgrp 19094 | A structure power of a gro... |
| pwsinvg 19095 | Negation in a group power.... |
| pwssub 19096 | Subtraction in a group pow... |
| imasgrp2 19097 | The image structure of a g... |
| imasgrp 19098 | The image structure of a g... |
| imasgrpf1 19099 | The image of a group under... |
| qusgrp2 19100 | Prove that a quotient stru... |
| xpsgrp 19101 | The binary product of grou... |
| xpsinv 19102 | Value of the negation oper... |
| xpsgrpsub 19103 | Value of the subtraction o... |
| mhmlem 19104 | Lemma for ~ mhmmnd and ~ g... |
| mhmid 19105 | A surjective monoid morphi... |
| mhmmnd 19106 | The image of a monoid ` G ... |
| mhmfmhm 19107 | The function fulfilling th... |
| ghmgrp 19108 | The image of a group ` G `... |
| mulgfval 19111 | Group multiple (exponentia... |
| mulgfvalALT 19112 | Shorter proof of ~ mulgfva... |
| mulgval 19113 | Value of the group multipl... |
| mulgfn 19114 | Functionality of the group... |
| mulgfvi 19115 | The group multiple operati... |
| mulg0 19116 | Group multiple (exponentia... |
| mulgnn 19117 | Group multiple (exponentia... |
| ressmulgnn 19118 | Values for the group multi... |
| ressmulgnn0 19119 | Values for the group multi... |
| ressmulgnnd 19120 | Values for the group multi... |
| mulgnngsum 19121 | Group multiple (exponentia... |
| mulgnn0gsum 19122 | Group multiple (exponentia... |
| mulg1 19123 | Group multiple (exponentia... |
| mulgnnp1 19124 | Group multiple (exponentia... |
| mulg2 19125 | Group multiple (exponentia... |
| mulgnegnn 19126 | Group multiple (exponentia... |
| mulgnn0p1 19127 | Group multiple (exponentia... |
| mulgnnsubcl 19128 | Closure of the group multi... |
| mulgnn0subcl 19129 | Closure of the group multi... |
| mulgsubcl 19130 | Closure of the group multi... |
| mulgnncl 19131 | Closure of the group multi... |
| mulgnn0cl 19132 | Closure of the group multi... |
| mulgcl 19133 | Closure of the group multi... |
| mulgneg 19134 | Group multiple (exponentia... |
| mulgnegneg 19135 | The inverse of a negative ... |
| mulgm1 19136 | Group multiple (exponentia... |
| mulgnn0cld 19137 | Closure of the group multi... |
| mulgcld 19138 | Deduction associated with ... |
| mulgaddcomlem 19139 | Lemma for ~ mulgaddcom . ... |
| mulgaddcom 19140 | The group multiple operato... |
| mulginvcom 19141 | The group multiple operato... |
| mulginvinv 19142 | The group multiple operato... |
| mulgnn0z 19143 | A group multiple of the id... |
| mulgz 19144 | A group multiple of the id... |
| mulgnndir 19145 | Sum of group multiples, fo... |
| mulgnn0dir 19146 | Sum of group multiples, ge... |
| mulgdirlem 19147 | Lemma for ~ mulgdir . (Co... |
| mulgdir 19148 | Sum of group multiples, ge... |
| mulgp1 19149 | Group multiple (exponentia... |
| mulgneg2 19150 | Group multiple (exponentia... |
| mulgnnass 19151 | Product of group multiples... |
| mulgnn0ass 19152 | Product of group multiples... |
| mulgass 19153 | Product of group multiples... |
| mulgassr 19154 | Reversed product of group ... |
| mulgmodid 19155 | Casting out multiples of t... |
| mulgsubdir 19156 | Distribution of group mult... |
| mhmmulg 19157 | A homomorphism of monoids ... |
| mulgpropd 19158 | Two structures with the sa... |
| submmulgcl 19159 | Closure of the group multi... |
| submmulg 19160 | A group multiple is the sa... |
| pwsmulg 19161 | Value of a group multiple ... |
| issubg 19168 | The subgroup predicate. (... |
| subgss 19169 | A subgroup is a subset. (... |
| subgid 19170 | A group is a subgroup of i... |
| subggrp 19171 | A subgroup is a group. (C... |
| subgbas 19172 | The base of the restricted... |
| subgrcl 19173 | Reverse closure for the su... |
| subg0 19174 | A subgroup of a group must... |
| subginv 19175 | The inverse of an element ... |
| subg0cl 19176 | The group identity is an e... |
| subginvcl 19177 | The inverse of an element ... |
| subgcl 19178 | A subgroup is closed under... |
| subgsubcl 19179 | A subgroup is closed under... |
| subgsub 19180 | The subtraction of element... |
| subgmulgcl 19181 | Closure of the group multi... |
| subgmulg 19182 | A group multiple is the sa... |
| issubg2 19183 | Characterize the subgroups... |
| issubgrpd2 19184 | Prove a subgroup by closur... |
| issubgrpd 19185 | Prove a subgroup by closur... |
| issubg3 19186 | A subgroup is a symmetric ... |
| issubg4 19187 | A subgroup is a nonempty s... |
| grpissubg 19188 | If the base set of a group... |
| resgrpisgrp 19189 | If the base set of a group... |
| subgsubm 19190 | A subgroup is a submonoid.... |
| subsubg 19191 | A subgroup of a subgroup i... |
| subgint 19192 | The intersection of a none... |
| 0subg 19193 | The zero subgroup of an ar... |
| 0subgOLD 19194 | Obsolete version of ~ 0sub... |
| trivsubgd 19195 | The only subgroup of a tri... |
| trivsubgsnd 19196 | The only subgroup of a tri... |
| isnsg 19197 | Property of being a normal... |
| isnsg2 19198 | Weaken the condition of ~ ... |
| nsgbi 19199 | Defining property of a nor... |
| nsgsubg 19200 | A normal subgroup is a sub... |
| nsgconj 19201 | The conjugation of an elem... |
| isnsg3 19202 | A subgroup is normal iff t... |
| subgacs 19203 | Subgroups are an algebraic... |
| nsgacs 19204 | Normal subgroups form an a... |
| elnmz 19205 | Elementhood in the normali... |
| nmzbi 19206 | Defining property of the n... |
| nmzsubg 19207 | The normalizer N_G(S) of a... |
| ssnmz 19208 | A subgroup is a subset of ... |
| isnsg4 19209 | A subgroup is normal iff i... |
| nmznsg 19210 | Any subgroup is a normal s... |
| 0nsg 19211 | The zero subgroup is norma... |
| nsgid 19212 | The whole group is a norma... |
| 0idnsgd 19213 | The whole group and the ze... |
| trivnsgd 19214 | The only normal subgroup o... |
| triv1nsgd 19215 | A trivial group has exactl... |
| 1nsgtrivd 19216 | A group with exactly one n... |
| releqg 19217 | The left coset equivalence... |
| eqgfval 19218 | Value of the subgroup left... |
| eqgval 19219 | Value of the subgroup left... |
| eqger 19220 | The subgroup coset equival... |
| eqglact 19221 | A left coset can be expres... |
| eqgid 19222 | The left coset containing ... |
| eqgen 19223 | Each coset is equipotent t... |
| eqgcpbl 19224 | The subgroup coset equival... |
| eqg0el 19225 | Equivalence class of a quo... |
| quselbas 19226 | Membership in the base set... |
| quseccl0 19227 | Closure of the quotient ma... |
| qusgrp 19228 | If ` Y ` is a normal subgr... |
| quseccl 19229 | Closure of the quotient ma... |
| qusadd 19230 | Value of the group operati... |
| qus0 19231 | Value of the group identit... |
| qusinv 19232 | Value of the group inverse... |
| qussub 19233 | Value of the group subtrac... |
| ecqusaddd 19234 | Addition of equivalence cl... |
| ecqusaddcl 19235 | Closure of the addition in... |
| lagsubg2 19236 | Lagrange's theorem for fin... |
| lagsubg 19237 | Lagrange's theorem for Gro... |
| eqg0subg 19238 | The coset equivalence rela... |
| eqg0subgecsn 19239 | The equivalence classes mo... |
| qus0subgbas 19240 | The base set of a quotient... |
| qus0subgadd 19241 | The addition in a quotient... |
| cycsubmel 19242 | Characterization of an ele... |
| cycsubmcl 19243 | The set of nonnegative int... |
| cycsubm 19244 | The set of nonnegative int... |
| cyccom 19245 | Condition for an operation... |
| cycsubmcom 19246 | The operation of a monoid ... |
| cycsubggend 19247 | The cyclic subgroup genera... |
| cycsubgcl 19248 | The set of integer powers ... |
| cycsubgss 19249 | The cyclic subgroup genera... |
| cycsubg 19250 | The cyclic group generated... |
| cycsubgcld 19251 | The cyclic subgroup genera... |
| cycsubg2 19252 | The subgroup generated by ... |
| cycsubg2cl 19253 | Any multiple of an element... |
| reldmghm 19256 | Lemma for group homomorphi... |
| isghm 19257 | Property of being a homomo... |
| isghmOLD 19258 | Obsolete version of ~ isgh... |
| isghm3 19259 | Property of a group homomo... |
| ghmgrp1 19260 | A group homomorphism is on... |
| ghmgrp2 19261 | A group homomorphism is on... |
| ghmf 19262 | A group homomorphism is a ... |
| ghmlin 19263 | A homomorphism of groups i... |
| ghmid 19264 | A homomorphism of groups p... |
| ghminv 19265 | A homomorphism of groups p... |
| ghmsub 19266 | Linearity of subtraction t... |
| isghmd 19267 | Deduction for a group homo... |
| ghmmhm 19268 | A group homomorphism is a ... |
| ghmmhmb 19269 | Group homomorphisms and mo... |
| ghmmulg 19270 | A group homomorphism prese... |
| ghmrn 19271 | The range of a homomorphis... |
| 0ghm 19272 | The constant zero linear f... |
| idghm 19273 | The identity homomorphism ... |
| resghm 19274 | Restriction of a homomorph... |
| resghm2 19275 | One direction of ~ resghm2... |
| resghm2b 19276 | Restriction of the codomai... |
| ghmghmrn 19277 | A group homomorphism from ... |
| ghmco 19278 | The composition of group h... |
| ghmima 19279 | The image of a subgroup un... |
| ghmpreima 19280 | The inverse image of a sub... |
| ghmeql 19281 | The equalizer of two group... |
| ghmnsgima 19282 | The image of a normal subg... |
| ghmnsgpreima 19283 | The inverse image of a nor... |
| ghmker 19284 | The kernel of a homomorphi... |
| ghmeqker 19285 | Two source points map to t... |
| pwsdiagghm 19286 | Diagonal homomorphism into... |
| f1ghm0to0 19287 | If a group homomorphism ` ... |
| ghmf1 19288 | Two ways of saying a group... |
| kerf1ghm 19289 | A group homomorphism ` F `... |
| ghmf1o 19290 | A bijective group homomorp... |
| conjghm 19291 | Conjugation is an automorp... |
| conjsubg 19292 | A conjugated subgroup is a... |
| conjsubgen 19293 | A conjugated subgroup is e... |
| conjnmz 19294 | A subgroup is unchanged un... |
| conjnmzb 19295 | Alternative condition for ... |
| conjnsg 19296 | A normal subgroup is uncha... |
| qusghm 19297 | If ` Y ` is a normal subgr... |
| ghmpropd 19298 | Group homomorphism depends... |
| gimfn 19303 | The group isomorphism func... |
| isgim 19304 | An isomorphism of groups i... |
| gimf1o 19305 | An isomorphism of groups i... |
| gimghm 19306 | An isomorphism of groups i... |
| isgim2 19307 | A group isomorphism is a h... |
| subggim 19308 | Behavior of subgroups unde... |
| gimcnv 19309 | The converse of a group is... |
| gimco 19310 | The composition of group i... |
| gim0to0 19311 | A group isomorphism maps t... |
| brgic 19312 | The relation "is isomorphi... |
| brgici 19313 | Prove isomorphic by an exp... |
| gicref 19314 | Isomorphism is reflexive. ... |
| giclcl 19315 | Isomorphism implies the le... |
| gicrcl 19316 | Isomorphism implies the ri... |
| gicsym 19317 | Isomorphism is symmetric. ... |
| gictr 19318 | Isomorphism is transitive.... |
| gicer 19319 | Isomorphism is an equivale... |
| gicen 19320 | Isomorphic groups have equ... |
| gicsubgen 19321 | A less trivial example of ... |
| ghmqusnsglem1 19322 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsglem2 19323 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsg 19324 | The mapping ` H ` induced ... |
| ghmquskerlem1 19325 | Lemma for ~ ghmqusker . (... |
| ghmquskerco 19326 | In the case of theorem ~ g... |
| ghmquskerlem2 19327 | Lemma for ~ ghmqusker . (... |
| ghmquskerlem3 19328 | The mapping ` H ` induced ... |
| ghmqusker 19329 | A surjective group homomor... |
| gicqusker 19330 | The image ` H ` of a group... |
| isga 19333 | The predicate "is a (left)... |
| gagrp 19334 | The left argument of a gro... |
| gaset 19335 | The right argument of a gr... |
| gagrpid 19336 | The identity of the group ... |
| gaf 19337 | The mapping of the group a... |
| gafo 19338 | A group action is onto its... |
| gaass 19339 | An "associative" property ... |
| ga0 19340 | The action of a group on t... |
| gaid 19341 | The trivial action of a gr... |
| subgga 19342 | A subgroup acts on its par... |
| gass 19343 | A subset of a group action... |
| gasubg 19344 | The restriction of a group... |
| gaid2 19345 | A group operation is a lef... |
| galcan 19346 | The action of a particular... |
| gacan 19347 | Group inverses cancel in a... |
| gapm 19348 | The action of a particular... |
| gaorb 19349 | The orbit equivalence rela... |
| gaorber 19350 | The orbit equivalence rela... |
| gastacl 19351 | The stabilizer subgroup in... |
| gastacos 19352 | Write the coset relation f... |
| orbstafun 19353 | Existence and uniqueness f... |
| orbstaval 19354 | Value of the function at a... |
| orbsta 19355 | The Orbit-Stabilizer theor... |
| orbsta2 19356 | Relation between the size ... |
| cntrval 19361 | Substitute definition of t... |
| cntzfval 19362 | First level substitution f... |
| cntzval 19363 | Definition substitution fo... |
| elcntz 19364 | Elementhood in the central... |
| cntzel 19365 | Membership in a centralize... |
| cntzsnval 19366 | Special substitution for t... |
| elcntzsn 19367 | Value of the centralizer o... |
| sscntz 19368 | A centralizer expression f... |
| cntzrcl 19369 | Reverse closure for elemen... |
| cntzssv 19370 | The centralizer is uncondi... |
| cntzi 19371 | Membership in a centralize... |
| elcntr 19372 | Elementhood in the center ... |
| cntrss 19373 | The center is a subset of ... |
| cntri 19374 | Defining property of the c... |
| resscntz 19375 | Centralizer in a substruct... |
| cntzsgrpcl 19376 | Centralizers are closed un... |
| cntz2ss 19377 | Centralizers reverse the s... |
| cntzrec 19378 | Reciprocity relationship f... |
| cntziinsn 19379 | Express any centralizer as... |
| cntzsubm 19380 | Centralizers in a monoid a... |
| cntzsubg 19381 | Centralizers in a group ar... |
| cntzidss 19382 | If the elements of ` S ` c... |
| cntzmhm 19383 | Centralizers in a monoid a... |
| cntzmhm2 19384 | Centralizers in a monoid a... |
| cntrsubgnsg 19385 | A central subgroup is norm... |
| cntrnsg 19386 | The center of a group is a... |
| oppgval 19389 | Value of the opposite grou... |
| oppgplusfval 19390 | Value of the addition oper... |
| oppgplus 19391 | Value of the addition oper... |
| setsplusg 19392 | The other components of an... |
| oppglemOLD 19393 | Obsolete version of ~ sets... |
| oppgbas 19394 | Base set of an opposite gr... |
| oppgbasOLD 19395 | Obsolete version of ~ oppg... |
| oppgtset 19396 | Topology of an opposite gr... |
| oppgtsetOLD 19397 | Obsolete version of ~ oppg... |
| oppgtopn 19398 | Topology of an opposite gr... |
| oppgmnd 19399 | The opposite of a monoid i... |
| oppgmndb 19400 | Bidirectional form of ~ op... |
| oppgid 19401 | Zero in a monoid is a symm... |
| oppggrp 19402 | The opposite of a group is... |
| oppggrpb 19403 | Bidirectional form of ~ op... |
| oppginv 19404 | Inverses in a group are a ... |
| invoppggim 19405 | The inverse is an antiauto... |
| oppggic 19406 | Every group is (naturally)... |
| oppgsubm 19407 | Being a submonoid is a sym... |
| oppgsubg 19408 | Being a subgroup is a symm... |
| oppgcntz 19409 | A centralizer in a group i... |
| oppgcntr 19410 | The center of a group is t... |
| gsumwrev 19411 | A sum in an opposite monoi... |
| symgval 19414 | The value of the symmetric... |
| symgbas 19415 | The base set of the symmet... |
| elsymgbas2 19416 | Two ways of saying a funct... |
| elsymgbas 19417 | Two ways of saying a funct... |
| symgbasf1o 19418 | Elements in the symmetric ... |
| symgbasf 19419 | A permutation (element of ... |
| symgbasmap 19420 | A permutation (element of ... |
| symghash 19421 | The symmetric group on ` n... |
| symgbasfi 19422 | The symmetric group on a f... |
| symgfv 19423 | The function value of a pe... |
| symgfvne 19424 | The function values of a p... |
| symgressbas 19425 | The symmetric group on ` A... |
| symgplusg 19426 | The group operation of a s... |
| symgov 19427 | The value of the group ope... |
| symgcl 19428 | The group operation of the... |
| idresperm 19429 | The identity function rest... |
| symgmov1 19430 | For a permutation of a set... |
| symgmov2 19431 | For a permutation of a set... |
| symgbas0 19432 | The base set of the symmet... |
| symg1hash 19433 | The symmetric group on a s... |
| symg1bas 19434 | The symmetric group on a s... |
| symg2hash 19435 | The symmetric group on a (... |
| symg2bas 19436 | The symmetric group on a p... |
| 0symgefmndeq 19437 | The symmetric group on the... |
| snsymgefmndeq 19438 | The symmetric group on a s... |
| symgpssefmnd 19439 | For a set ` A ` with more ... |
| symgvalstruct 19440 | The value of the symmetric... |
| symgvalstructOLD 19441 | Obsolete proof of ~ symgva... |
| symgsubmefmnd 19442 | The symmetric group on a s... |
| symgtset 19443 | The topology of the symmet... |
| symggrp 19444 | The symmetric group on a s... |
| symgid 19445 | The group identity element... |
| symginv 19446 | The group inverse in the s... |
| symgsubmefmndALT 19447 | The symmetric group on a s... |
| galactghm 19448 | The currying of a group ac... |
| lactghmga 19449 | The converse of ~ galactgh... |
| symgtopn 19450 | The topology of the symmet... |
| symgga 19451 | The symmetric group induce... |
| pgrpsubgsymgbi 19452 | Every permutation group is... |
| pgrpsubgsymg 19453 | Every permutation group is... |
| idressubgsymg 19454 | The singleton containing o... |
| idrespermg 19455 | The structure with the sin... |
| cayleylem1 19456 | Lemma for ~ cayley . (Con... |
| cayleylem2 19457 | Lemma for ~ cayley . (Con... |
| cayley 19458 | Cayley's Theorem (construc... |
| cayleyth 19459 | Cayley's Theorem (existenc... |
| symgfix2 19460 | If a permutation does not ... |
| symgextf 19461 | The extension of a permuta... |
| symgextfv 19462 | The function value of the ... |
| symgextfve 19463 | The function value of the ... |
| symgextf1lem 19464 | Lemma for ~ symgextf1 . (... |
| symgextf1 19465 | The extension of a permuta... |
| symgextfo 19466 | The extension of a permuta... |
| symgextf1o 19467 | The extension of a permuta... |
| symgextsymg 19468 | The extension of a permuta... |
| symgextres 19469 | The restriction of the ext... |
| gsumccatsymgsn 19470 | Homomorphic property of co... |
| gsmsymgrfixlem1 19471 | Lemma 1 for ~ gsmsymgrfix ... |
| gsmsymgrfix 19472 | The composition of permuta... |
| fvcosymgeq 19473 | The values of two composit... |
| gsmsymgreqlem1 19474 | Lemma 1 for ~ gsmsymgreq .... |
| gsmsymgreqlem2 19475 | Lemma 2 for ~ gsmsymgreq .... |
| gsmsymgreq 19476 | Two combination of permuta... |
| symgfixelq 19477 | A permutation of a set fix... |
| symgfixels 19478 | The restriction of a permu... |
| symgfixelsi 19479 | The restriction of a permu... |
| symgfixf 19480 | The mapping of a permutati... |
| symgfixf1 19481 | The mapping of a permutati... |
| symgfixfolem1 19482 | Lemma 1 for ~ symgfixfo . ... |
| symgfixfo 19483 | The mapping of a permutati... |
| symgfixf1o 19484 | The mapping of a permutati... |
| f1omvdmvd 19487 | A permutation of any class... |
| f1omvdcnv 19488 | A permutation and its inve... |
| mvdco 19489 | Composing two permutations... |
| f1omvdconj 19490 | Conjugation of a permutati... |
| f1otrspeq 19491 | A transposition is charact... |
| f1omvdco2 19492 | If exactly one of two perm... |
| f1omvdco3 19493 | If a point is moved by exa... |
| pmtrfval 19494 | The function generating tr... |
| pmtrval 19495 | A generated transposition,... |
| pmtrfv 19496 | General value of mapping a... |
| pmtrprfv 19497 | In a transposition of two ... |
| pmtrprfv3 19498 | In a transposition of two ... |
| pmtrf 19499 | Functionality of a transpo... |
| pmtrmvd 19500 | A transposition moves prec... |
| pmtrrn 19501 | Transposing two points giv... |
| pmtrfrn 19502 | A transposition (as a kind... |
| pmtrffv 19503 | Mapping of a point under a... |
| pmtrrn2 19504 | For any transposition ther... |
| pmtrfinv 19505 | A transposition function i... |
| pmtrfmvdn0 19506 | A transposition moves at l... |
| pmtrff1o 19507 | A transposition function i... |
| pmtrfcnv 19508 | A transposition function i... |
| pmtrfb 19509 | An intrinsic characterizat... |
| pmtrfconj 19510 | Any conjugate of a transpo... |
| symgsssg 19511 | The symmetric group has su... |
| symgfisg 19512 | The symmetric group has a ... |
| symgtrf 19513 | Transpositions are element... |
| symggen 19514 | The span of the transposit... |
| symggen2 19515 | A finite permutation group... |
| symgtrinv 19516 | To invert a permutation re... |
| pmtr3ncomlem1 19517 | Lemma 1 for ~ pmtr3ncom . ... |
| pmtr3ncomlem2 19518 | Lemma 2 for ~ pmtr3ncom . ... |
| pmtr3ncom 19519 | Transpositions over sets w... |
| pmtrdifellem1 19520 | Lemma 1 for ~ pmtrdifel . ... |
| pmtrdifellem2 19521 | Lemma 2 for ~ pmtrdifel . ... |
| pmtrdifellem3 19522 | Lemma 3 for ~ pmtrdifel . ... |
| pmtrdifellem4 19523 | Lemma 4 for ~ pmtrdifel . ... |
| pmtrdifel 19524 | A transposition of element... |
| pmtrdifwrdellem1 19525 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem2 19526 | Lemma 2 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem3 19527 | Lemma 3 for ~ pmtrdifwrdel... |
| pmtrdifwrdel2lem1 19528 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdel 19529 | A sequence of transpositio... |
| pmtrdifwrdel2 19530 | A sequence of transpositio... |
| pmtrprfval 19531 | The transpositions on a pa... |
| pmtrprfvalrn 19532 | The range of the transposi... |
| psgnunilem1 19537 | Lemma for ~ psgnuni . Giv... |
| psgnunilem5 19538 | Lemma for ~ psgnuni . It ... |
| psgnunilem2 19539 | Lemma for ~ psgnuni . Ind... |
| psgnunilem3 19540 | Lemma for ~ psgnuni . Any... |
| psgnunilem4 19541 | Lemma for ~ psgnuni . An ... |
| m1expaddsub 19542 | Addition and subtraction o... |
| psgnuni 19543 | If the same permutation ca... |
| psgnfval 19544 | Function definition of the... |
| psgnfn 19545 | Functionality and domain o... |
| psgndmsubg 19546 | The finitary permutations ... |
| psgneldm 19547 | Property of being a finita... |
| psgneldm2 19548 | The finitary permutations ... |
| psgneldm2i 19549 | A sequence of transpositio... |
| psgneu 19550 | A finitary permutation has... |
| psgnval 19551 | Value of the permutation s... |
| psgnvali 19552 | A finitary permutation has... |
| psgnvalii 19553 | Any representation of a pe... |
| psgnpmtr 19554 | All transpositions are odd... |
| psgn0fv0 19555 | The permutation sign funct... |
| sygbasnfpfi 19556 | The class of non-fixed poi... |
| psgnfvalfi 19557 | Function definition of the... |
| psgnvalfi 19558 | Value of the permutation s... |
| psgnran 19559 | The range of the permutati... |
| gsmtrcl 19560 | The group sum of transposi... |
| psgnfitr 19561 | A permutation of a finite ... |
| psgnfieu 19562 | A permutation of a finite ... |
| pmtrsn 19563 | The value of the transposi... |
| psgnsn 19564 | The permutation sign funct... |
| psgnprfval 19565 | The permutation sign funct... |
| psgnprfval1 19566 | The permutation sign of th... |
| psgnprfval2 19567 | The permutation sign of th... |
| odfval 19576 | Value of the order functio... |
| odfvalALT 19577 | Shorter proof of ~ odfval ... |
| odval 19578 | Second substitution for th... |
| odlem1 19579 | The group element order is... |
| odcl 19580 | The order of a group eleme... |
| odf 19581 | Functionality of the group... |
| odid 19582 | Any element to the power o... |
| odlem2 19583 | Any positive annihilator o... |
| odmodnn0 19584 | Reduce the argument of a g... |
| mndodconglem 19585 | Lemma for ~ mndodcong . (... |
| mndodcong 19586 | If two multipliers are con... |
| mndodcongi 19587 | If two multipliers are con... |
| oddvdsnn0 19588 | The only multiples of ` A ... |
| odnncl 19589 | If a nonzero multiple of a... |
| odmod 19590 | Reduce the argument of a g... |
| oddvds 19591 | The only multiples of ` A ... |
| oddvdsi 19592 | Any group element is annih... |
| odcong 19593 | If two multipliers are con... |
| odeq 19594 | The ~ oddvds property uniq... |
| odval2 19595 | A non-conditional definiti... |
| odcld 19596 | The order of a group eleme... |
| odm1inv 19597 | The (order-1)th multiple o... |
| odmulgid 19598 | A relationship between the... |
| odmulg2 19599 | The order of a multiple di... |
| odmulg 19600 | Relationship between the o... |
| odmulgeq 19601 | A multiple of a point of f... |
| odbezout 19602 | If ` N ` is coprime to the... |
| od1 19603 | The order of the group ide... |
| odeq1 19604 | The group identity is the ... |
| odinv 19605 | The order of the inverse o... |
| odf1 19606 | The multiples of an elemen... |
| odinf 19607 | The multiples of an elemen... |
| dfod2 19608 | An alternative definition ... |
| odcl2 19609 | The order of an element of... |
| oddvds2 19610 | The order of an element of... |
| finodsubmsubg 19611 | A submonoid whose elements... |
| 0subgALT 19612 | A shorter proof of ~ 0subg... |
| submod 19613 | The order of an element is... |
| subgod 19614 | The order of an element is... |
| odsubdvds 19615 | The order of an element of... |
| odf1o1 19616 | An element with zero order... |
| odf1o2 19617 | An element with nonzero or... |
| odhash 19618 | An element of zero order g... |
| odhash2 19619 | If an element has nonzero ... |
| odhash3 19620 | An element which generates... |
| odngen 19621 | A cyclic subgroup of size ... |
| gexval 19622 | Value of the exponent of a... |
| gexlem1 19623 | The group element order is... |
| gexcl 19624 | The exponent of a group is... |
| gexid 19625 | Any element to the power o... |
| gexlem2 19626 | Any positive annihilator o... |
| gexdvdsi 19627 | Any group element is annih... |
| gexdvds 19628 | The only ` N ` that annihi... |
| gexdvds2 19629 | An integer divides the gro... |
| gexod 19630 | Any group element is annih... |
| gexcl3 19631 | If the order of every grou... |
| gexnnod 19632 | Every group element has fi... |
| gexcl2 19633 | The exponent of a finite g... |
| gexdvds3 19634 | The exponent of a finite g... |
| gex1 19635 | A group or monoid has expo... |
| ispgp 19636 | A group is a ` P ` -group ... |
| pgpprm 19637 | Reverse closure for the fi... |
| pgpgrp 19638 | Reverse closure for the se... |
| pgpfi1 19639 | A finite group with order ... |
| pgp0 19640 | The identity subgroup is a... |
| subgpgp 19641 | A subgroup of a p-group is... |
| sylow1lem1 19642 | Lemma for ~ sylow1 . The ... |
| sylow1lem2 19643 | Lemma for ~ sylow1 . The ... |
| sylow1lem3 19644 | Lemma for ~ sylow1 . One ... |
| sylow1lem4 19645 | Lemma for ~ sylow1 . The ... |
| sylow1lem5 19646 | Lemma for ~ sylow1 . Usin... |
| sylow1 19647 | Sylow's first theorem. If... |
| odcau 19648 | Cauchy's theorem for the o... |
| pgpfi 19649 | The converse to ~ pgpfi1 .... |
| pgpfi2 19650 | Alternate version of ~ pgp... |
| pgphash 19651 | The order of a p-group. (... |
| isslw 19652 | The property of being a Sy... |
| slwprm 19653 | Reverse closure for the fi... |
| slwsubg 19654 | A Sylow ` P ` -subgroup is... |
| slwispgp 19655 | Defining property of a Syl... |
| slwpss 19656 | A proper superset of a Syl... |
| slwpgp 19657 | A Sylow ` P ` -subgroup is... |
| pgpssslw 19658 | Every ` P ` -subgroup is c... |
| slwn0 19659 | Every finite group contain... |
| subgslw 19660 | A Sylow subgroup that is c... |
| sylow2alem1 19661 | Lemma for ~ sylow2a . An ... |
| sylow2alem2 19662 | Lemma for ~ sylow2a . All... |
| sylow2a 19663 | A named lemma of Sylow's s... |
| sylow2blem1 19664 | Lemma for ~ sylow2b . Eva... |
| sylow2blem2 19665 | Lemma for ~ sylow2b . Lef... |
| sylow2blem3 19666 | Sylow's second theorem. P... |
| sylow2b 19667 | Sylow's second theorem. A... |
| slwhash 19668 | A sylow subgroup has cardi... |
| fislw 19669 | The sylow subgroups of a f... |
| sylow2 19670 | Sylow's second theorem. S... |
| sylow3lem1 19671 | Lemma for ~ sylow3 , first... |
| sylow3lem2 19672 | Lemma for ~ sylow3 , first... |
| sylow3lem3 19673 | Lemma for ~ sylow3 , first... |
| sylow3lem4 19674 | Lemma for ~ sylow3 , first... |
| sylow3lem5 19675 | Lemma for ~ sylow3 , secon... |
| sylow3lem6 19676 | Lemma for ~ sylow3 , secon... |
| sylow3 19677 | Sylow's third theorem. Th... |
| lsmfval 19682 | The subgroup sum function ... |
| lsmvalx 19683 | Subspace sum value (for a ... |
| lsmelvalx 19684 | Subspace sum membership (f... |
| lsmelvalix 19685 | Subspace sum membership (f... |
| oppglsm 19686 | The subspace sum operation... |
| lsmssv 19687 | Subgroup sum is a subset o... |
| lsmless1x 19688 | Subset implies subgroup su... |
| lsmless2x 19689 | Subset implies subgroup su... |
| lsmub1x 19690 | Subgroup sum is an upper b... |
| lsmub2x 19691 | Subgroup sum is an upper b... |
| lsmval 19692 | Subgroup sum value (for a ... |
| lsmelval 19693 | Subgroup sum membership (f... |
| lsmelvali 19694 | Subgroup sum membership (f... |
| lsmelvalm 19695 | Subgroup sum membership an... |
| lsmelvalmi 19696 | Membership of vector subtr... |
| lsmsubm 19697 | The sum of two commuting s... |
| lsmsubg 19698 | The sum of two commuting s... |
| lsmcom2 19699 | Subgroup sum commutes. (C... |
| smndlsmidm 19700 | The direct product is idem... |
| lsmub1 19701 | Subgroup sum is an upper b... |
| lsmub2 19702 | Subgroup sum is an upper b... |
| lsmunss 19703 | Union of subgroups is a su... |
| lsmless1 19704 | Subset implies subgroup su... |
| lsmless2 19705 | Subset implies subgroup su... |
| lsmless12 19706 | Subset implies subgroup su... |
| lsmidm 19707 | Subgroup sum is idempotent... |
| lsmlub 19708 | The least upper bound prop... |
| lsmss1 19709 | Subgroup sum with a subset... |
| lsmss1b 19710 | Subgroup sum with a subset... |
| lsmss2 19711 | Subgroup sum with a subset... |
| lsmss2b 19712 | Subgroup sum with a subset... |
| lsmass 19713 | Subgroup sum is associativ... |
| mndlsmidm 19714 | Subgroup sum is idempotent... |
| lsm01 19715 | Subgroup sum with the zero... |
| lsm02 19716 | Subgroup sum with the zero... |
| subglsm 19717 | The subgroup sum evaluated... |
| lssnle 19718 | Equivalent expressions for... |
| lsmmod 19719 | The modular law holds for ... |
| lsmmod2 19720 | Modular law dual for subgr... |
| lsmpropd 19721 | If two structures have the... |
| cntzrecd 19722 | Commute the "subgroups com... |
| lsmcntz 19723 | The "subgroups commute" pr... |
| lsmcntzr 19724 | The "subgroups commute" pr... |
| lsmdisj 19725 | Disjointness from a subgro... |
| lsmdisj2 19726 | Association of the disjoin... |
| lsmdisj3 19727 | Association of the disjoin... |
| lsmdisjr 19728 | Disjointness from a subgro... |
| lsmdisj2r 19729 | Association of the disjoin... |
| lsmdisj3r 19730 | Association of the disjoin... |
| lsmdisj2a 19731 | Association of the disjoin... |
| lsmdisj2b 19732 | Association of the disjoin... |
| lsmdisj3a 19733 | Association of the disjoin... |
| lsmdisj3b 19734 | Association of the disjoin... |
| subgdisj1 19735 | Vectors belonging to disjo... |
| subgdisj2 19736 | Vectors belonging to disjo... |
| subgdisjb 19737 | Vectors belonging to disjo... |
| pj1fval 19738 | The left projection functi... |
| pj1val 19739 | The left projection functi... |
| pj1eu 19740 | Uniqueness of a left proje... |
| pj1f 19741 | The left projection functi... |
| pj2f 19742 | The right projection funct... |
| pj1id 19743 | Any element of a direct su... |
| pj1eq 19744 | Any element of a direct su... |
| pj1lid 19745 | The left projection functi... |
| pj1rid 19746 | The left projection functi... |
| pj1ghm 19747 | The left projection functi... |
| pj1ghm2 19748 | The left projection functi... |
| lsmhash 19749 | The order of the direct pr... |
| efgmval 19756 | Value of the formal invers... |
| efgmf 19757 | The formal inverse operati... |
| efgmnvl 19758 | The inversion function on ... |
| efgrcl 19759 | Lemma for ~ efgval . (Con... |
| efglem 19760 | Lemma for ~ efgval . (Con... |
| efgval 19761 | Value of the free group co... |
| efger 19762 | Value of the free group co... |
| efgi 19763 | Value of the free group co... |
| efgi0 19764 | Value of the free group co... |
| efgi1 19765 | Value of the free group co... |
| efgtf 19766 | Value of the free group co... |
| efgtval 19767 | Value of the extension fun... |
| efgval2 19768 | Value of the free group co... |
| efgi2 19769 | Value of the free group co... |
| efgtlen 19770 | Value of the free group co... |
| efginvrel2 19771 | The inverse of the reverse... |
| efginvrel1 19772 | The inverse of the reverse... |
| efgsf 19773 | Value of the auxiliary fun... |
| efgsdm 19774 | Elementhood in the domain ... |
| efgsval 19775 | Value of the auxiliary fun... |
| efgsdmi 19776 | Property of the last link ... |
| efgsval2 19777 | Value of the auxiliary fun... |
| efgsrel 19778 | The start and end of any e... |
| efgs1 19779 | A singleton of an irreduci... |
| efgs1b 19780 | Every extension sequence e... |
| efgsp1 19781 | If ` F ` is an extension s... |
| efgsres 19782 | An initial segment of an e... |
| efgsfo 19783 | For any word, there is a s... |
| efgredlema 19784 | The reduced word that form... |
| efgredlemf 19785 | Lemma for ~ efgredleme . ... |
| efgredlemg 19786 | Lemma for ~ efgred . (Con... |
| efgredleme 19787 | Lemma for ~ efgred . (Con... |
| efgredlemd 19788 | The reduced word that form... |
| efgredlemc 19789 | The reduced word that form... |
| efgredlemb 19790 | The reduced word that form... |
| efgredlem 19791 | The reduced word that form... |
| efgred 19792 | The reduced word that form... |
| efgrelexlema 19793 | If two words ` A , B ` are... |
| efgrelexlemb 19794 | If two words ` A , B ` are... |
| efgrelex 19795 | If two words ` A , B ` are... |
| efgredeu 19796 | There is a unique reduced ... |
| efgred2 19797 | Two extension sequences ha... |
| efgcpbllema 19798 | Lemma for ~ efgrelex . De... |
| efgcpbllemb 19799 | Lemma for ~ efgrelex . Sh... |
| efgcpbl 19800 | Two extension sequences ha... |
| efgcpbl2 19801 | Two extension sequences ha... |
| frgpval 19802 | Value of the free group co... |
| frgpcpbl 19803 | Compatibility of the group... |
| frgp0 19804 | The free group is a group.... |
| frgpeccl 19805 | Closure of the quotient ma... |
| frgpgrp 19806 | The free group is a group.... |
| frgpadd 19807 | Addition in the free group... |
| frgpinv 19808 | The inverse of an element ... |
| frgpmhm 19809 | The "natural map" from wor... |
| vrgpfval 19810 | The canonical injection fr... |
| vrgpval 19811 | The value of the generatin... |
| vrgpf 19812 | The mapping from the index... |
| vrgpinv 19813 | The inverse of a generatin... |
| frgpuptf 19814 | Any assignment of the gene... |
| frgpuptinv 19815 | Any assignment of the gene... |
| frgpuplem 19816 | Any assignment of the gene... |
| frgpupf 19817 | Any assignment of the gene... |
| frgpupval 19818 | Any assignment of the gene... |
| frgpup1 19819 | Any assignment of the gene... |
| frgpup2 19820 | The evaluation map has the... |
| frgpup3lem 19821 | The evaluation map has the... |
| frgpup3 19822 | Universal property of the ... |
| 0frgp 19823 | The free group on zero gen... |
| isabl 19828 | The predicate "is an Abeli... |
| ablgrp 19829 | An Abelian group is a grou... |
| ablgrpd 19830 | An Abelian group is a grou... |
| ablcmn 19831 | An Abelian group is a comm... |
| ablcmnd 19832 | An Abelian group is a comm... |
| iscmn 19833 | The predicate "is a commut... |
| isabl2 19834 | The predicate "is an Abeli... |
| cmnpropd 19835 | If two structures have the... |
| ablpropd 19836 | If two structures have the... |
| ablprop 19837 | If two structures have the... |
| iscmnd 19838 | Properties that determine ... |
| isabld 19839 | Properties that determine ... |
| isabli 19840 | Properties that determine ... |
| cmnmnd 19841 | A commutative monoid is a ... |
| cmncom 19842 | A commutative monoid is co... |
| ablcom 19843 | An Abelian group operation... |
| cmn32 19844 | Commutative/associative la... |
| cmn4 19845 | Commutative/associative la... |
| cmn12 19846 | Commutative/associative la... |
| abl32 19847 | Commutative/associative la... |
| cmnmndd 19848 | A commutative monoid is a ... |
| cmnbascntr 19849 | The base set of a commutat... |
| rinvmod 19850 | Uniqueness of a right inve... |
| ablinvadd 19851 | The inverse of an Abelian ... |
| ablsub2inv 19852 | Abelian group subtraction ... |
| ablsubadd 19853 | Relationship between Abeli... |
| ablsub4 19854 | Commutative/associative su... |
| abladdsub4 19855 | Abelian group addition/sub... |
| abladdsub 19856 | Associative-type law for g... |
| ablsubadd23 19857 | Commutative/associative la... |
| ablsubaddsub 19858 | Double subtraction and add... |
| ablpncan2 19859 | Cancellation law for subtr... |
| ablpncan3 19860 | A cancellation law for Abe... |
| ablsubsub 19861 | Law for double subtraction... |
| ablsubsub4 19862 | Law for double subtraction... |
| ablpnpcan 19863 | Cancellation law for mixed... |
| ablnncan 19864 | Cancellation law for group... |
| ablsub32 19865 | Swap the second and third ... |
| ablnnncan 19866 | Cancellation law for group... |
| ablnnncan1 19867 | Cancellation law for group... |
| ablsubsub23 19868 | Swap subtrahend and result... |
| mulgnn0di 19869 | Group multiple of a sum, f... |
| mulgdi 19870 | Group multiple of a sum. ... |
| mulgmhm 19871 | The map from ` x ` to ` n ... |
| mulgghm 19872 | The map from ` x ` to ` n ... |
| mulgsubdi 19873 | Group multiple of a differ... |
| ghmfghm 19874 | The function fulfilling th... |
| ghmcmn 19875 | The image of a commutative... |
| ghmabl 19876 | The image of an abelian gr... |
| invghm 19877 | The inversion map is a gro... |
| eqgabl 19878 | Value of the subgroup cose... |
| qusecsub 19879 | Two subgroup cosets are eq... |
| subgabl 19880 | A subgroup of an abelian g... |
| subcmn 19881 | A submonoid of a commutati... |
| submcmn 19882 | A submonoid of a commutati... |
| submcmn2 19883 | A submonoid is commutative... |
| cntzcmn 19884 | The centralizer of any sub... |
| cntzcmnss 19885 | Any subset in a commutativ... |
| cntrcmnd 19886 | The center of a monoid is ... |
| cntrabl 19887 | The center of a group is a... |
| cntzspan 19888 | If the generators commute,... |
| cntzcmnf 19889 | Discharge the centralizer ... |
| ghmplusg 19890 | The pointwise sum of two l... |
| ablnsg 19891 | Every subgroup of an abeli... |
| odadd1 19892 | The order of a product in ... |
| odadd2 19893 | The order of a product in ... |
| odadd 19894 | The order of a product is ... |
| gex2abl 19895 | A group with exponent 2 (o... |
| gexexlem 19896 | Lemma for ~ gexex . (Cont... |
| gexex 19897 | In an abelian group with f... |
| torsubg 19898 | The set of all elements of... |
| oddvdssubg 19899 | The set of all elements wh... |
| lsmcomx 19900 | Subgroup sum commutes (ext... |
| ablcntzd 19901 | All subgroups in an abelia... |
| lsmcom 19902 | Subgroup sum commutes. (C... |
| lsmsubg2 19903 | The sum of two subgroups i... |
| lsm4 19904 | Commutative/associative la... |
| prdscmnd 19905 | The product of a family of... |
| prdsabld 19906 | The product of a family of... |
| pwscmn 19907 | The structure power on a c... |
| pwsabl 19908 | The structure power on an ... |
| qusabl 19909 | If ` Y ` is a subgroup of ... |
| abl1 19910 | The (smallest) structure r... |
| abln0 19911 | Abelian groups (and theref... |
| cnaddablx 19912 | The complex numbers are an... |
| cnaddabl 19913 | The complex numbers are an... |
| cnaddid 19914 | The group identity element... |
| cnaddinv 19915 | Value of the group inverse... |
| zaddablx 19916 | The integers are an Abelia... |
| frgpnabllem1 19917 | Lemma for ~ frgpnabl . (C... |
| frgpnabllem2 19918 | Lemma for ~ frgpnabl . (C... |
| frgpnabl 19919 | The free group on two or m... |
| imasabl 19920 | The image structure of an ... |
| iscyg 19923 | Definition of a cyclic gro... |
| iscyggen 19924 | The property of being a cy... |
| iscyggen2 19925 | The property of being a cy... |
| iscyg2 19926 | A cyclic group is a group ... |
| cyggeninv 19927 | The inverse of a cyclic ge... |
| cyggenod 19928 | An element is the generato... |
| cyggenod2 19929 | In an infinite cyclic grou... |
| iscyg3 19930 | Definition of a cyclic gro... |
| iscygd 19931 | Definition of a cyclic gro... |
| iscygodd 19932 | Show that a group with an ... |
| cycsubmcmn 19933 | The set of nonnegative int... |
| cyggrp 19934 | A cyclic group is a group.... |
| cygabl 19935 | A cyclic group is abelian.... |
| cygctb 19936 | A cyclic group is countabl... |
| 0cyg 19937 | The trivial group is cycli... |
| prmcyg 19938 | A group with prime order i... |
| lt6abl 19939 | A group with fewer than ` ... |
| ghmcyg 19940 | The image of a cyclic grou... |
| cyggex2 19941 | The exponent of a cyclic g... |
| cyggex 19942 | The exponent of a finite c... |
| cyggexb 19943 | A finite abelian group is ... |
| giccyg 19944 | Cyclicity is a group prope... |
| cycsubgcyg 19945 | The cyclic subgroup genera... |
| cycsubgcyg2 19946 | The cyclic subgroup genera... |
| gsumval3a 19947 | Value of the group sum ope... |
| gsumval3eu 19948 | The group sum as defined i... |
| gsumval3lem1 19949 | Lemma 1 for ~ gsumval3 . ... |
| gsumval3lem2 19950 | Lemma 2 for ~ gsumval3 . ... |
| gsumval3 19951 | Value of the group sum ope... |
| gsumcllem 19952 | Lemma for ~ gsumcl and rel... |
| gsumzres 19953 | Extend a finite group sum ... |
| gsumzcl2 19954 | Closure of a finite group ... |
| gsumzcl 19955 | Closure of a finite group ... |
| gsumzf1o 19956 | Re-index a finite group su... |
| gsumres 19957 | Extend a finite group sum ... |
| gsumcl2 19958 | Closure of a finite group ... |
| gsumcl 19959 | Closure of a finite group ... |
| gsumf1o 19960 | Re-index a finite group su... |
| gsumreidx 19961 | Re-index a finite group su... |
| gsumzsubmcl 19962 | Closure of a group sum in ... |
| gsumsubmcl 19963 | Closure of a group sum in ... |
| gsumsubgcl 19964 | Closure of a group sum in ... |
| gsumzaddlem 19965 | The sum of two group sums.... |
| gsumzadd 19966 | The sum of two group sums.... |
| gsumadd 19967 | The sum of two group sums.... |
| gsummptfsadd 19968 | The sum of two group sums ... |
| gsummptfidmadd 19969 | The sum of two group sums ... |
| gsummptfidmadd2 19970 | The sum of two group sums ... |
| gsumzsplit 19971 | Split a group sum into two... |
| gsumsplit 19972 | Split a group sum into two... |
| gsumsplit2 19973 | Split a group sum into two... |
| gsummptfidmsplit 19974 | Split a group sum expresse... |
| gsummptfidmsplitres 19975 | Split a group sum expresse... |
| gsummptfzsplit 19976 | Split a group sum expresse... |
| gsummptfzsplitl 19977 | Split a group sum expresse... |
| gsumconst 19978 | Sum of a constant series. ... |
| gsumconstf 19979 | Sum of a constant series. ... |
| gsummptshft 19980 | Index shift of a finite gr... |
| gsumzmhm 19981 | Apply a group homomorphism... |
| gsummhm 19982 | Apply a group homomorphism... |
| gsummhm2 19983 | Apply a group homomorphism... |
| gsummptmhm 19984 | Apply a group homomorphism... |
| gsummulglem 19985 | Lemma for ~ gsummulg and ~... |
| gsummulg 19986 | Nonnegative multiple of a ... |
| gsummulgz 19987 | Integer multiple of a grou... |
| gsumzoppg 19988 | The opposite of a group su... |
| gsumzinv 19989 | Inverse of a group sum. (... |
| gsuminv 19990 | Inverse of a group sum. (... |
| gsummptfidminv 19991 | Inverse of a group sum exp... |
| gsumsub 19992 | The difference of two grou... |
| gsummptfssub 19993 | The difference of two grou... |
| gsummptfidmsub 19994 | The difference of two grou... |
| gsumsnfd 19995 | Group sum of a singleton, ... |
| gsumsnd 19996 | Group sum of a singleton, ... |
| gsumsnf 19997 | Group sum of a singleton, ... |
| gsumsn 19998 | Group sum of a singleton. ... |
| gsumpr 19999 | Group sum of a pair. (Con... |
| gsumzunsnd 20000 | Append an element to a fin... |
| gsumunsnfd 20001 | Append an element to a fin... |
| gsumunsnd 20002 | Append an element to a fin... |
| gsumunsnf 20003 | Append an element to a fin... |
| gsumunsn 20004 | Append an element to a fin... |
| gsumdifsnd 20005 | Extract a summand from a f... |
| gsumpt 20006 | Sum of a family that is no... |
| gsummptf1o 20007 | Re-index a finite group su... |
| gsummptun 20008 | Group sum of a disjoint un... |
| gsummpt1n0 20009 | If only one summand in a f... |
| gsummptif1n0 20010 | If only one summand in a f... |
| gsummptcl 20011 | Closure of a finite group ... |
| gsummptfif1o 20012 | Re-index a finite group su... |
| gsummptfzcl 20013 | Closure of a finite group ... |
| gsum2dlem1 20014 | Lemma 1 for ~ gsum2d . (C... |
| gsum2dlem2 20015 | Lemma for ~ gsum2d . (Con... |
| gsum2d 20016 | Write a sum over a two-dim... |
| gsum2d2lem 20017 | Lemma for ~ gsum2d2 : show... |
| gsum2d2 20018 | Write a group sum over a t... |
| gsumcom2 20019 | Two-dimensional commutatio... |
| gsumxp 20020 | Write a group sum over a c... |
| gsumcom 20021 | Commute the arguments of a... |
| gsumcom3 20022 | A commutative law for fini... |
| gsumcom3fi 20023 | A commutative law for fini... |
| gsumxp2 20024 | Write a group sum over a c... |
| prdsgsum 20025 | Finite commutative sums in... |
| pwsgsum 20026 | Finite commutative sums in... |
| fsfnn0gsumfsffz 20027 | Replacing a finitely suppo... |
| nn0gsumfz 20028 | Replacing a finitely suppo... |
| nn0gsumfz0 20029 | Replacing a finitely suppo... |
| gsummptnn0fz 20030 | A final group sum over a f... |
| gsummptnn0fzfv 20031 | A final group sum over a f... |
| telgsumfzslem 20032 | Lemma for ~ telgsumfzs (in... |
| telgsumfzs 20033 | Telescoping group sum rang... |
| telgsumfz 20034 | Telescoping group sum rang... |
| telgsumfz0s 20035 | Telescoping finite group s... |
| telgsumfz0 20036 | Telescoping finite group s... |
| telgsums 20037 | Telescoping finitely suppo... |
| telgsum 20038 | Telescoping finitely suppo... |
| reldmdprd 20043 | The domain of the internal... |
| dmdprd 20044 | The domain of definition o... |
| dmdprdd 20045 | Show that a given family i... |
| dprddomprc 20046 | A family of subgroups inde... |
| dprddomcld 20047 | If a family of subgroups i... |
| dprdval0prc 20048 | The internal direct produc... |
| dprdval 20049 | The value of the internal ... |
| eldprd 20050 | A class ` A ` is an intern... |
| dprdgrp 20051 | Reverse closure for the in... |
| dprdf 20052 | The function ` S ` is a fa... |
| dprdf2 20053 | The function ` S ` is a fa... |
| dprdcntz 20054 | The function ` S ` is a fa... |
| dprddisj 20055 | The function ` S ` is a fa... |
| dprdw 20056 | The property of being a fi... |
| dprdwd 20057 | A mapping being a finitely... |
| dprdff 20058 | A finitely supported funct... |
| dprdfcl 20059 | A finitely supported funct... |
| dprdffsupp 20060 | A finitely supported funct... |
| dprdfcntz 20061 | A function on the elements... |
| dprdssv 20062 | The internal direct produc... |
| dprdfid 20063 | A function mapping all but... |
| eldprdi 20064 | The domain of definition o... |
| dprdfinv 20065 | Take the inverse of a grou... |
| dprdfadd 20066 | Take the sum of group sums... |
| dprdfsub 20067 | Take the difference of gro... |
| dprdfeq0 20068 | The zero function is the o... |
| dprdf11 20069 | Two group sums over a dire... |
| dprdsubg 20070 | The internal direct produc... |
| dprdub 20071 | Each factor is a subset of... |
| dprdlub 20072 | The direct product is smal... |
| dprdspan 20073 | The direct product is the ... |
| dprdres 20074 | Restriction of a direct pr... |
| dprdss 20075 | Create a direct product by... |
| dprdz 20076 | A family consisting entire... |
| dprd0 20077 | The empty family is an int... |
| dprdf1o 20078 | Rearrange the index set of... |
| dprdf1 20079 | Rearrange the index set of... |
| subgdmdprd 20080 | A direct product in a subg... |
| subgdprd 20081 | A direct product in a subg... |
| dprdsn 20082 | A singleton family is an i... |
| dmdprdsplitlem 20083 | Lemma for ~ dmdprdsplit . ... |
| dprdcntz2 20084 | The function ` S ` is a fa... |
| dprddisj2 20085 | The function ` S ` is a fa... |
| dprd2dlem2 20086 | The direct product of a co... |
| dprd2dlem1 20087 | The direct product of a co... |
| dprd2da 20088 | The direct product of a co... |
| dprd2db 20089 | The direct product of a co... |
| dprd2d2 20090 | The direct product of a co... |
| dmdprdsplit2lem 20091 | Lemma for ~ dmdprdsplit . ... |
| dmdprdsplit2 20092 | The direct product splits ... |
| dmdprdsplit 20093 | The direct product splits ... |
| dprdsplit 20094 | The direct product is the ... |
| dmdprdpr 20095 | A singleton family is an i... |
| dprdpr 20096 | A singleton family is an i... |
| dpjlem 20097 | Lemma for theorems about d... |
| dpjcntz 20098 | The two subgroups that app... |
| dpjdisj 20099 | The two subgroups that app... |
| dpjlsm 20100 | The two subgroups that app... |
| dpjfval 20101 | Value of the direct produc... |
| dpjval 20102 | Value of the direct produc... |
| dpjf 20103 | The ` X ` -th index projec... |
| dpjidcl 20104 | The key property of projec... |
| dpjeq 20105 | Decompose a group sum into... |
| dpjid 20106 | The key property of projec... |
| dpjlid 20107 | The ` X ` -th index projec... |
| dpjrid 20108 | The ` Y ` -th index projec... |
| dpjghm 20109 | The direct product is the ... |
| dpjghm2 20110 | The direct product is the ... |
| ablfacrplem 20111 | Lemma for ~ ablfacrp2 . (... |
| ablfacrp 20112 | A finite abelian group who... |
| ablfacrp2 20113 | The factors ` K , L ` of ~... |
| ablfac1lem 20114 | Lemma for ~ ablfac1b . Sa... |
| ablfac1a 20115 | The factors of ~ ablfac1b ... |
| ablfac1b 20116 | Any abelian group is the d... |
| ablfac1c 20117 | The factors of ~ ablfac1b ... |
| ablfac1eulem 20118 | Lemma for ~ ablfac1eu . (... |
| ablfac1eu 20119 | The factorization of ~ abl... |
| pgpfac1lem1 20120 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem2 20121 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3a 20122 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3 20123 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem4 20124 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem5 20125 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1 20126 | Factorization of a finite ... |
| pgpfaclem1 20127 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem2 20128 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem3 20129 | Lemma for ~ pgpfac . (Con... |
| pgpfac 20130 | Full factorization of a fi... |
| ablfaclem1 20131 | Lemma for ~ ablfac . (Con... |
| ablfaclem2 20132 | Lemma for ~ ablfac . (Con... |
| ablfaclem3 20133 | Lemma for ~ ablfac . (Con... |
| ablfac 20134 | The Fundamental Theorem of... |
| ablfac2 20135 | Choose generators for each... |
| issimpg 20138 | The predicate "is a simple... |
| issimpgd 20139 | Deduce a simple group from... |
| simpggrp 20140 | A simple group is a group.... |
| simpggrpd 20141 | A simple group is a group.... |
| simpg2nsg 20142 | A simple group has two nor... |
| trivnsimpgd 20143 | Trivial groups are not sim... |
| simpgntrivd 20144 | Simple groups are nontrivi... |
| simpgnideld 20145 | A simple group contains a ... |
| simpgnsgd 20146 | The only normal subgroups ... |
| simpgnsgeqd 20147 | A normal subgroup of a sim... |
| 2nsgsimpgd 20148 | If any normal subgroup of ... |
| simpgnsgbid 20149 | A nontrivial group is simp... |
| ablsimpnosubgd 20150 | A subgroup of an abelian s... |
| ablsimpg1gend 20151 | An abelian simple group is... |
| ablsimpgcygd 20152 | An abelian simple group is... |
| ablsimpgfindlem1 20153 | Lemma for ~ ablsimpgfind .... |
| ablsimpgfindlem2 20154 | Lemma for ~ ablsimpgfind .... |
| cycsubggenodd 20155 | Relationship between the o... |
| ablsimpgfind 20156 | An abelian simple group is... |
| fincygsubgd 20157 | The subgroup referenced in... |
| fincygsubgodd 20158 | Calculate the order of a s... |
| fincygsubgodexd 20159 | A finite cyclic group has ... |
| prmgrpsimpgd 20160 | A group of prime order is ... |
| ablsimpgprmd 20161 | An abelian simple group ha... |
| ablsimpgd 20162 | An abelian group is simple... |
| fnmgp 20165 | The multiplicative group o... |
| mgpval 20166 | Value of the multiplicatio... |
| mgpplusg 20167 | Value of the group operati... |
| mgplemOLD 20168 | Obsolete version of ~ sets... |
| mgpbas 20169 | Base set of the multiplica... |
| mgpbasOLD 20170 | Obsolete version of ~ mgpb... |
| mgpsca 20171 | The multiplication monoid ... |
| mgpscaOLD 20172 | Obsolete version of ~ mgps... |
| mgptset 20173 | Topology component of the ... |
| mgptsetOLD 20174 | Obsolete version of ~ mgpt... |
| mgptopn 20175 | Topology of the multiplica... |
| mgpds 20176 | Distance function of the m... |
| mgpdsOLD 20177 | Obsolete version of ~ mgpd... |
| mgpress 20178 | Subgroup commutes with the... |
| mgpressOLD 20179 | Obsolete version of ~ mgpr... |
| prdsmgp 20180 | The multiplicative monoid ... |
| isrng 20183 | The predicate "is a non-un... |
| rngabl 20184 | A non-unital ring is an (a... |
| rngmgp 20185 | A non-unital ring is a sem... |
| rngmgpf 20186 | Restricted functionality o... |
| rnggrp 20187 | A non-unital ring is a (ad... |
| rngass 20188 | Associative law for the mu... |
| rngdi 20189 | Distributive law for the m... |
| rngdir 20190 | Distributive law for the m... |
| rngacl 20191 | Closure of the addition op... |
| rng0cl 20192 | The zero element of a non-... |
| rngcl 20193 | Closure of the multiplicat... |
| rnglz 20194 | The zero of a non-unital r... |
| rngrz 20195 | The zero of a non-unital r... |
| rngmneg1 20196 | Negation of a product in a... |
| rngmneg2 20197 | Negation of a product in a... |
| rngm2neg 20198 | Double negation of a produ... |
| rngansg 20199 | Every additive subgroup of... |
| rngsubdi 20200 | Ring multiplication distri... |
| rngsubdir 20201 | Ring multiplication distri... |
| isrngd 20202 | Properties that determine ... |
| rngpropd 20203 | If two structures have the... |
| prdsmulrngcl 20204 | Closure of the multiplicat... |
| prdsrngd 20205 | A product of non-unital ri... |
| imasrng 20206 | The image structure of a n... |
| imasrngf1 20207 | The image of a non-unital ... |
| xpsrngd 20208 | A product of two non-unita... |
| qusrng 20209 | The quotient structure of ... |
| ringidval 20212 | The value of the unity ele... |
| dfur2 20213 | The multiplicative identit... |
| ringurd 20214 | Deduce the unity element o... |
| issrg 20217 | The predicate "is a semiri... |
| srgcmn 20218 | A semiring is a commutativ... |
| srgmnd 20219 | A semiring is a monoid. (... |
| srgmgp 20220 | A semiring is a monoid und... |
| srgdilem 20221 | Lemma for ~ srgdi and ~ sr... |
| srgcl 20222 | Closure of the multiplicat... |
| srgass 20223 | Associative law for the mu... |
| srgideu 20224 | The unity element of a sem... |
| srgfcl 20225 | Functionality of the multi... |
| srgdi 20226 | Distributive law for the m... |
| srgdir 20227 | Distributive law for the m... |
| srgidcl 20228 | The unity element of a sem... |
| srg0cl 20229 | The zero element of a semi... |
| srgidmlem 20230 | Lemma for ~ srglidm and ~ ... |
| srglidm 20231 | The unity element of a sem... |
| srgridm 20232 | The unity element of a sem... |
| issrgid 20233 | Properties showing that an... |
| srgacl 20234 | Closure of the addition op... |
| srgcom 20235 | Commutativity of the addit... |
| srgrz 20236 | The zero of a semiring is ... |
| srglz 20237 | The zero of a semiring is ... |
| srgisid 20238 | In a semiring, the only le... |
| o2timesd 20239 | An element of a ring-like ... |
| rglcom4d 20240 | Restricted commutativity o... |
| srgo2times 20241 | A semiring element plus it... |
| srgcom4lem 20242 | Lemma for ~ srgcom4 . Thi... |
| srgcom4 20243 | Restricted commutativity o... |
| srg1zr 20244 | The only semiring with a b... |
| srgen1zr 20245 | The only semiring with one... |
| srgmulgass 20246 | An associative property be... |
| srgpcomp 20247 | If two elements of a semir... |
| srgpcompp 20248 | If two elements of a semir... |
| srgpcomppsc 20249 | If two elements of a semir... |
| srglmhm 20250 | Left-multiplication in a s... |
| srgrmhm 20251 | Right-multiplication in a ... |
| srgsummulcr 20252 | A finite semiring sum mult... |
| sgsummulcl 20253 | A finite semiring sum mult... |
| srg1expzeq1 20254 | The exponentiation (by a n... |
| srgbinomlem1 20255 | Lemma 1 for ~ srgbinomlem ... |
| srgbinomlem2 20256 | Lemma 2 for ~ srgbinomlem ... |
| srgbinomlem3 20257 | Lemma 3 for ~ srgbinomlem ... |
| srgbinomlem4 20258 | Lemma 4 for ~ srgbinomlem ... |
| srgbinomlem 20259 | Lemma for ~ srgbinom . In... |
| srgbinom 20260 | The binomial theorem for c... |
| csrgbinom 20261 | The binomial theorem for c... |
| isring 20266 | The predicate "is a (unita... |
| ringgrp 20267 | A ring is a group. (Contr... |
| ringmgp 20268 | A ring is a monoid under m... |
| iscrng 20269 | A commutative ring is a ri... |
| crngmgp 20270 | A commutative ring's multi... |
| ringgrpd 20271 | A ring is a group. (Contr... |
| ringmnd 20272 | A ring is a monoid under a... |
| ringmgm 20273 | A ring is a magma. (Contr... |
| crngring 20274 | A commutative ring is a ri... |
| crngringd 20275 | A commutative ring is a ri... |
| crnggrpd 20276 | A commutative ring is a gr... |
| mgpf 20277 | Restricted functionality o... |
| ringdilem 20278 | Properties of a unital rin... |
| ringcl 20279 | Closure of the multiplicat... |
| crngcom 20280 | A commutative ring's multi... |
| iscrng2 20281 | A commutative ring is a ri... |
| ringass 20282 | Associative law for multip... |
| ringideu 20283 | The unity element of a rin... |
| crngcomd 20284 | Multiplication is commutat... |
| crngbascntr 20285 | The base set of a commutat... |
| ringassd 20286 | Associative law for multip... |
| crng12d 20287 | Commutative/associative la... |
| ringcld 20288 | Closure of the multiplicat... |
| ringdi 20289 | Distributive law for the m... |
| ringdir 20290 | Distributive law for the m... |
| ringidcl 20291 | The unity element of a rin... |
| ring0cl 20292 | The zero element of a ring... |
| ringidmlem 20293 | Lemma for ~ ringlidm and ~... |
| ringlidm 20294 | The unity element of a rin... |
| ringridm 20295 | The unity element of a rin... |
| isringid 20296 | Properties showing that an... |
| ringlidmd 20297 | The unity element of a rin... |
| ringridmd 20298 | The unity element of a rin... |
| ringid 20299 | The multiplication operati... |
| ringo2times 20300 | A ring element plus itself... |
| ringadd2 20301 | A ring element plus itself... |
| ringidss 20302 | A subset of the multiplica... |
| ringacl 20303 | Closure of the addition op... |
| ringcomlem 20304 | Lemma for ~ ringcom . Thi... |
| ringcom 20305 | Commutativity of the addit... |
| ringabl 20306 | A ring is an Abelian group... |
| ringcmn 20307 | A ring is a commutative mo... |
| ringabld 20308 | A ring is an Abelian group... |
| ringcmnd 20309 | A ring is a commutative mo... |
| ringrng 20310 | A unital ring is a non-uni... |
| ringssrng 20311 | The unital rings are non-u... |
| isringrng 20312 | The predicate "is a unital... |
| ringpropd 20313 | If two structures have the... |
| crngpropd 20314 | If two structures have the... |
| ringprop 20315 | If two structures have the... |
| isringd 20316 | Properties that determine ... |
| iscrngd 20317 | Properties that determine ... |
| ringlz 20318 | The zero of a unital ring ... |
| ringrz 20319 | The zero of a unital ring ... |
| ringlzd 20320 | The zero of a unital ring ... |
| ringrzd 20321 | The zero of a unital ring ... |
| ringsrg 20322 | Any ring is also a semirin... |
| ring1eq0 20323 | If one and zero are equal,... |
| ring1ne0 20324 | If a ring has at least two... |
| ringinvnz1ne0 20325 | In a unital ring, a left i... |
| ringinvnzdiv 20326 | In a unital ring, a left i... |
| ringnegl 20327 | Negation in a ring is the ... |
| ringnegr 20328 | Negation in a ring is the ... |
| ringmneg1 20329 | Negation of a product in a... |
| ringmneg2 20330 | Negation of a product in a... |
| ringm2neg 20331 | Double negation of a produ... |
| ringsubdi 20332 | Ring multiplication distri... |
| ringsubdir 20333 | Ring multiplication distri... |
| mulgass2 20334 | An associative property be... |
| ring1 20335 | The (smallest) structure r... |
| ringn0 20336 | Rings exist. (Contributed... |
| ringlghm 20337 | Left-multiplication in a r... |
| ringrghm 20338 | Right-multiplication in a ... |
| gsummulc1OLD 20339 | Obsolete version of ~ gsum... |
| gsummulc2OLD 20340 | Obsolete version of ~ gsum... |
| gsummulc1 20341 | A finite ring sum multipli... |
| gsummulc2 20342 | A finite ring sum multipli... |
| gsummgp0 20343 | If one factor in a finite ... |
| gsumdixp 20344 | Distribute a binary produc... |
| prdsmulrcl 20345 | A structure product of rin... |
| prdsringd 20346 | A product of rings is a ri... |
| prdscrngd 20347 | A product of commutative r... |
| prds1 20348 | Value of the ring unity in... |
| pwsring 20349 | A structure power of a rin... |
| pws1 20350 | Value of the ring unity in... |
| pwscrng 20351 | A structure power of a com... |
| pwsmgp 20352 | The multiplicative group o... |
| pwspjmhmmgpd 20353 | The projection given by ~ ... |
| pwsexpg 20354 | Value of a group exponenti... |
| imasring 20355 | The image structure of a r... |
| imasringf1 20356 | The image of a ring under ... |
| xpsringd 20357 | A product of two rings is ... |
| xpsring1d 20358 | The multiplicative identit... |
| qusring2 20359 | The quotient structure of ... |
| crngbinom 20360 | The binomial theorem for c... |
| opprval 20363 | Value of the opposite ring... |
| opprmulfval 20364 | Value of the multiplicatio... |
| opprmul 20365 | Value of the multiplicatio... |
| crngoppr 20366 | In a commutative ring, the... |
| opprlem 20367 | Lemma for ~ opprbas and ~ ... |
| opprlemOLD 20368 | Obsolete version of ~ oppr... |
| opprbas 20369 | Base set of an opposite ri... |
| opprbasOLD 20370 | Obsolete proof of ~ opprba... |
| oppradd 20371 | Addition operation of an o... |
| oppraddOLD 20372 | Obsolete proof of ~ opprba... |
| opprrng 20373 | An opposite non-unital rin... |
| opprrngb 20374 | A class is a non-unital ri... |
| opprring 20375 | An opposite ring is a ring... |
| opprringb 20376 | Bidirectional form of ~ op... |
| oppr0 20377 | Additive identity of an op... |
| oppr1 20378 | Multiplicative identity of... |
| opprneg 20379 | The negative function in a... |
| opprsubg 20380 | Being a subgroup is a symm... |
| mulgass3 20381 | An associative property be... |
| reldvdsr 20388 | The divides relation is a ... |
| dvdsrval 20389 | Value of the divides relat... |
| dvdsr 20390 | Value of the divides relat... |
| dvdsr2 20391 | Value of the divides relat... |
| dvdsrmul 20392 | A left-multiple of ` X ` i... |
| dvdsrcl 20393 | Closure of a dividing elem... |
| dvdsrcl2 20394 | Closure of a dividing elem... |
| dvdsrid 20395 | An element in a (unital) r... |
| dvdsrtr 20396 | Divisibility is transitive... |
| dvdsrmul1 20397 | The divisibility relation ... |
| dvdsrneg 20398 | An element divides its neg... |
| dvdsr01 20399 | In a ring, zero is divisib... |
| dvdsr02 20400 | Only zero is divisible by ... |
| isunit 20401 | Property of being a unit o... |
| 1unit 20402 | The multiplicative identit... |
| unitcl 20403 | A unit is an element of th... |
| unitss 20404 | The set of units is contai... |
| opprunit 20405 | Being a unit is a symmetri... |
| crngunit 20406 | Property of being a unit i... |
| dvdsunit 20407 | A divisor of a unit is a u... |
| unitmulcl 20408 | The product of units is a ... |
| unitmulclb 20409 | Reversal of ~ unitmulcl in... |
| unitgrpbas 20410 | The base set of the group ... |
| unitgrp 20411 | The group of units is a gr... |
| unitabl 20412 | The group of units of a co... |
| unitgrpid 20413 | The identity of the group ... |
| unitsubm 20414 | The group of units is a su... |
| invrfval 20417 | Multiplicative inverse fun... |
| unitinvcl 20418 | The inverse of a unit exis... |
| unitinvinv 20419 | The inverse of the inverse... |
| ringinvcl 20420 | The inverse of a unit is a... |
| unitlinv 20421 | A unit times its inverse i... |
| unitrinv 20422 | A unit times its inverse i... |
| 1rinv 20423 | The inverse of the ring un... |
| 0unit 20424 | The additive identity is a... |
| unitnegcl 20425 | The negative of a unit is ... |
| ringunitnzdiv 20426 | In a unitary ring, a unit ... |
| ring1nzdiv 20427 | In a unitary ring, the rin... |
| dvrfval 20430 | Division operation in a ri... |
| dvrval 20431 | Division operation in a ri... |
| dvrcl 20432 | Closure of division operat... |
| unitdvcl 20433 | The units are closed under... |
| dvrid 20434 | A ring element divided by ... |
| dvr1 20435 | A ring element divided by ... |
| dvrass 20436 | An associative law for div... |
| dvrcan1 20437 | A cancellation law for div... |
| dvrcan3 20438 | A cancellation law for div... |
| dvreq1 20439 | Equality in terms of ratio... |
| dvrdir 20440 | Distributive law for the d... |
| rdivmuldivd 20441 | Multiplication of two rati... |
| ringinvdv 20442 | Write the inverse function... |
| rngidpropd 20443 | The ring unity depends onl... |
| dvdsrpropd 20444 | The divisibility relation ... |
| unitpropd 20445 | The set of units depends o... |
| invrpropd 20446 | The ring inverse function ... |
| isirred 20447 | An irreducible element of ... |
| isnirred 20448 | The property of being a no... |
| isirred2 20449 | Expand out the class diffe... |
| opprirred 20450 | Irreducibility is symmetri... |
| irredn0 20451 | The additive identity is n... |
| irredcl 20452 | An irreducible element is ... |
| irrednu 20453 | An irreducible element is ... |
| irredn1 20454 | The multiplicative identit... |
| irredrmul 20455 | The product of an irreduci... |
| irredlmul 20456 | The product of a unit and ... |
| irredmul 20457 | If product of two elements... |
| irredneg 20458 | The negative of an irreduc... |
| irrednegb 20459 | An element is irreducible ... |
| rnghmrcl 20466 | Reverse closure of a non-u... |
| rnghmfn 20467 | The mapping of two non-uni... |
| rnghmval 20468 | The set of the non-unital ... |
| isrnghm 20469 | A function is a non-unital... |
| isrnghmmul 20470 | A function is a non-unital... |
| rnghmmgmhm 20471 | A non-unital ring homomorp... |
| rnghmval2 20472 | The non-unital ring homomo... |
| isrngim 20473 | An isomorphism of non-unit... |
| rngimrcl 20474 | Reverse closure for an iso... |
| rnghmghm 20475 | A non-unital ring homomorp... |
| rnghmf 20476 | A ring homomorphism is a f... |
| rnghmmul 20477 | A homomorphism of non-unit... |
| isrnghm2d 20478 | Demonstration of non-unita... |
| isrnghmd 20479 | Demonstration of non-unita... |
| rnghmf1o 20480 | A non-unital ring homomorp... |
| isrngim2 20481 | An isomorphism of non-unit... |
| rngimf1o 20482 | An isomorphism of non-unit... |
| rngimrnghm 20483 | An isomorphism of non-unit... |
| rngimcnv 20484 | The converse of an isomorp... |
| rnghmco 20485 | The composition of non-uni... |
| idrnghm 20486 | The identity homomorphism ... |
| c0mgm 20487 | The constant mapping to ze... |
| c0mhm 20488 | The constant mapping to ze... |
| c0ghm 20489 | The constant mapping to ze... |
| c0snmgmhm 20490 | The constant mapping to ze... |
| c0snmhm 20491 | The constant mapping to ze... |
| c0snghm 20492 | The constant mapping to ze... |
| rngisomfv1 20493 | If there is a non-unital r... |
| rngisom1 20494 | If there is a non-unital r... |
| rngisomring 20495 | If there is a non-unital r... |
| rngisomring1 20496 | If there is a non-unital r... |
| dfrhm2 20502 | The property of a ring hom... |
| rhmrcl1 20504 | Reverse closure of a ring ... |
| rhmrcl2 20505 | Reverse closure of a ring ... |
| isrhm 20506 | A function is a ring homom... |
| rhmmhm 20507 | A ring homomorphism is a h... |
| rhmisrnghm 20508 | Each unital ring homomorph... |
| isrim0OLD 20509 | Obsolete version of ~ isri... |
| rimrcl 20510 | Reverse closure for an iso... |
| isrim0 20511 | A ring isomorphism is a ho... |
| rhmghm 20512 | A ring homomorphism is an ... |
| rhmf 20513 | A ring homomorphism is a f... |
| rhmmul 20514 | A homomorphism of rings pr... |
| isrhm2d 20515 | Demonstration of ring homo... |
| isrhmd 20516 | Demonstration of ring homo... |
| rhm1 20517 | Ring homomorphisms are req... |
| idrhm 20518 | The identity homomorphism ... |
| rhmf1o 20519 | A ring homomorphism is bij... |
| isrim 20520 | An isomorphism of rings is... |
| isrimOLD 20521 | Obsolete version of ~ isri... |
| rimf1o 20522 | An isomorphism of rings is... |
| rimrhmOLD 20523 | Obsolete version of ~ rimr... |
| rimrhm 20524 | A ring isomorphism is a ho... |
| rimgim 20525 | An isomorphism of rings is... |
| rimisrngim 20526 | Each unital ring isomorphi... |
| rhmfn 20527 | The mapping of two rings t... |
| rhmval 20528 | The ring homomorphisms bet... |
| rhmco 20529 | The composition of ring ho... |
| pwsco1rhm 20530 | Right composition with a f... |
| pwsco2rhm 20531 | Left composition with a ri... |
| brric 20532 | The relation "is isomorphi... |
| brrici 20533 | Prove isomorphic by an exp... |
| brric2 20534 | The relation "is isomorphi... |
| ricgic 20535 | If two rings are (ring) is... |
| rhmdvdsr 20536 | A ring homomorphism preser... |
| rhmopp 20537 | A ring homomorphism is als... |
| elrhmunit 20538 | Ring homomorphisms preserv... |
| rhmunitinv 20539 | Ring homomorphisms preserv... |
| isnzr 20542 | Property of a nonzero ring... |
| nzrnz 20543 | One and zero are different... |
| nzrring 20544 | A nonzero ring is a ring. ... |
| nzrringOLD 20545 | Obsolete version of ~ nzrr... |
| isnzr2 20546 | Equivalent characterizatio... |
| isnzr2hash 20547 | Equivalent characterizatio... |
| nzrpropd 20548 | If two structures have the... |
| opprnzrb 20549 | The opposite of a nonzero ... |
| opprnzr 20550 | The opposite of a nonzero ... |
| ringelnzr 20551 | A ring is nonzero if it ha... |
| nzrunit 20552 | A unit is nonzero in any n... |
| 0ringnnzr 20553 | A ring is a zero ring iff ... |
| 0ring 20554 | If a ring has only one ele... |
| 0ringdif 20555 | A zero ring is a ring whic... |
| 0ringbas 20556 | The base set of a zero rin... |
| 0ring01eq 20557 | In a ring with only one el... |
| 01eq0ring 20558 | If the zero and the identi... |
| 01eq0ringOLD 20559 | Obsolete version of ~ 01eq... |
| 0ring01eqbi 20560 | In a unital ring the zero ... |
| 0ring1eq0 20561 | In a zero ring, a ring whi... |
| c0rhm 20562 | The constant mapping to ze... |
| c0rnghm 20563 | The constant mapping to ze... |
| zrrnghm 20564 | The constant mapping to ze... |
| nrhmzr 20565 | There is no ring homomorph... |
| islring 20568 | The predicate "is a local ... |
| lringnzr 20569 | A local ring is a nonzero ... |
| lringring 20570 | A local ring is a ring. (... |
| lringnz 20571 | A local ring is a nonzero ... |
| lringuplu 20572 | If the sum of two elements... |
| issubrng 20575 | The subring of non-unital ... |
| subrngss 20576 | A subring is a subset. (C... |
| subrngid 20577 | Every non-unital ring is a... |
| subrngrng 20578 | A subring is a non-unital ... |
| subrngrcl 20579 | Reverse closure for a subr... |
| subrngsubg 20580 | A subring is a subgroup. ... |
| subrngringnsg 20581 | A subring is a normal subg... |
| subrngbas 20582 | Base set of a subring stru... |
| subrng0 20583 | A subring always has the s... |
| subrngacl 20584 | A subring is closed under ... |
| subrngmcl 20585 | A subring is closed under ... |
| issubrng2 20586 | Characterize the subrings ... |
| opprsubrng 20587 | Being a subring is a symme... |
| subrngint 20588 | The intersection of a none... |
| subrngin 20589 | The intersection of two su... |
| subrngmre 20590 | The subrings of a non-unit... |
| subsubrng 20591 | A subring of a subring is ... |
| subsubrng2 20592 | The set of subrings of a s... |
| rhmimasubrnglem 20593 | Lemma for ~ rhmimasubrng :... |
| rhmimasubrng 20594 | The homomorphic image of a... |
| cntzsubrng 20595 | Centralizers in a non-unit... |
| subrngpropd 20596 | If two structures have the... |
| issubrg 20601 | The subring predicate. (C... |
| subrgss 20602 | A subring is a subset. (C... |
| subrgid 20603 | Every ring is a subring of... |
| subrgring 20604 | A subring is a ring. (Con... |
| subrgcrng 20605 | A subring of a commutative... |
| subrgrcl 20606 | Reverse closure for a subr... |
| subrgsubg 20607 | A subring is a subgroup. ... |
| subrgsubrng 20608 | A subring of a unital ring... |
| subrg0 20609 | A subring always has the s... |
| subrg1cl 20610 | A subring contains the mul... |
| subrgbas 20611 | Base set of a subring stru... |
| subrg1 20612 | A subring always has the s... |
| subrgacl 20613 | A subring is closed under ... |
| subrgmcl 20614 | A subring is closed under ... |
| subrgsubm 20615 | A subring is a submonoid o... |
| subrgdvds 20616 | If an element divides anot... |
| subrguss 20617 | A unit of a subring is a u... |
| subrginv 20618 | A subring always has the s... |
| subrgdv 20619 | A subring always has the s... |
| subrgunit 20620 | An element of a ring is a ... |
| subrgugrp 20621 | The units of a subring for... |
| issubrg2 20622 | Characterize the subrings ... |
| opprsubrg 20623 | Being a subring is a symme... |
| subrgnzr 20624 | A subring of a nonzero rin... |
| subrgint 20625 | The intersection of a none... |
| subrgin 20626 | The intersection of two su... |
| subrgmre 20627 | The subrings of a ring are... |
| subsubrg 20628 | A subring of a subring is ... |
| subsubrg2 20629 | The set of subrings of a s... |
| issubrg3 20630 | A subring is an additive s... |
| resrhm 20631 | Restriction of a ring homo... |
| resrhm2b 20632 | Restriction of the codomai... |
| rhmeql 20633 | The equalizer of two ring ... |
| rhmima 20634 | The homomorphic image of a... |
| rnrhmsubrg 20635 | The range of a ring homomo... |
| cntzsubr 20636 | Centralizers in a ring are... |
| pwsdiagrhm 20637 | Diagonal homomorphism into... |
| subrgpropd 20638 | If two structures have the... |
| rhmpropd 20639 | Ring homomorphism depends ... |
| rngcval 20642 | Value of the category of n... |
| rnghmresfn 20643 | The class of non-unital ri... |
| rnghmresel 20644 | An element of the non-unit... |
| rngcbas 20645 | Set of objects of the cate... |
| rngchomfval 20646 | Set of arrows of the categ... |
| rngchom 20647 | Set of arrows of the categ... |
| elrngchom 20648 | A morphism of non-unital r... |
| rngchomfeqhom 20649 | The functionalized Hom-set... |
| rngccofval 20650 | Composition in the categor... |
| rngcco 20651 | Composition in the categor... |
| dfrngc2 20652 | Alternate definition of th... |
| rnghmsscmap2 20653 | The non-unital ring homomo... |
| rnghmsscmap 20654 | The non-unital ring homomo... |
| rnghmsubcsetclem1 20655 | Lemma 1 for ~ rnghmsubcset... |
| rnghmsubcsetclem2 20656 | Lemma 2 for ~ rnghmsubcset... |
| rnghmsubcsetc 20657 | The non-unital ring homomo... |
| rngccat 20658 | The category of non-unital... |
| rngcid 20659 | The identity arrow in the ... |
| rngcsect 20660 | A section in the category ... |
| rngcinv 20661 | An inverse in the category... |
| rngciso 20662 | An isomorphism in the cate... |
| rngcifuestrc 20663 | The "inclusion functor" fr... |
| funcrngcsetc 20664 | The "natural forgetful fun... |
| funcrngcsetcALT 20665 | Alternate proof of ~ funcr... |
| zrinitorngc 20666 | The zero ring is an initia... |
| zrtermorngc 20667 | The zero ring is a termina... |
| zrzeroorngc 20668 | The zero ring is a zero ob... |
| ringcval 20671 | Value of the category of u... |
| rhmresfn 20672 | The class of unital ring h... |
| rhmresel 20673 | An element of the unital r... |
| ringcbas 20674 | Set of objects of the cate... |
| ringchomfval 20675 | Set of arrows of the categ... |
| ringchom 20676 | Set of arrows of the categ... |
| elringchom 20677 | A morphism of unital rings... |
| ringchomfeqhom 20678 | The functionalized Hom-set... |
| ringccofval 20679 | Composition in the categor... |
| ringcco 20680 | Composition in the categor... |
| dfringc2 20681 | Alternate definition of th... |
| rhmsscmap2 20682 | The unital ring homomorphi... |
| rhmsscmap 20683 | The unital ring homomorphi... |
| rhmsubcsetclem1 20684 | Lemma 1 for ~ rhmsubcsetc ... |
| rhmsubcsetclem2 20685 | Lemma 2 for ~ rhmsubcsetc ... |
| rhmsubcsetc 20686 | The unital ring homomorphi... |
| ringccat 20687 | The category of unital rin... |
| ringcid 20688 | The identity arrow in the ... |
| rhmsscrnghm 20689 | The unital ring homomorphi... |
| rhmsubcrngclem1 20690 | Lemma 1 for ~ rhmsubcrngc ... |
| rhmsubcrngclem2 20691 | Lemma 2 for ~ rhmsubcrngc ... |
| rhmsubcrngc 20692 | The unital ring homomorphi... |
| rngcresringcat 20693 | The restriction of the cat... |
| ringcsect 20694 | A section in the category ... |
| ringcinv 20695 | An inverse in the category... |
| ringciso 20696 | An isomorphism in the cate... |
| ringcbasbas 20697 | An element of the base set... |
| funcringcsetc 20698 | The "natural forgetful fun... |
| zrtermoringc 20699 | The zero ring is a termina... |
| zrninitoringc 20700 | The zero ring is not an in... |
| srhmsubclem1 20701 | Lemma 1 for ~ srhmsubc . ... |
| srhmsubclem2 20702 | Lemma 2 for ~ srhmsubc . ... |
| srhmsubclem3 20703 | Lemma 3 for ~ srhmsubc . ... |
| srhmsubc 20704 | According to ~ df-subc , t... |
| sringcat 20705 | The restriction of the cat... |
| crhmsubc 20706 | According to ~ df-subc , t... |
| cringcat 20707 | The restriction of the cat... |
| rngcrescrhm 20708 | The category of non-unital... |
| rhmsubclem1 20709 | Lemma 1 for ~ rhmsubc . (... |
| rhmsubclem2 20710 | Lemma 2 for ~ rhmsubc . (... |
| rhmsubclem3 20711 | Lemma 3 for ~ rhmsubc . (... |
| rhmsubclem4 20712 | Lemma 4 for ~ rhmsubc . (... |
| rhmsubc 20713 | According to ~ df-subc , t... |
| rhmsubccat 20714 | The restriction of the cat... |
| rrgval 20721 | Value of the set or left-r... |
| isrrg 20722 | Membership in the set of l... |
| rrgeq0i 20723 | Property of a left-regular... |
| rrgeq0 20724 | Left-multiplication by a l... |
| rrgsupp 20725 | Left multiplication by a l... |
| rrgss 20726 | Left-regular elements are ... |
| unitrrg 20727 | Units are regular elements... |
| rrgnz 20728 | In a nonzero ring, the zer... |
| isdomn 20729 | Expand definition of a dom... |
| domnnzr 20730 | A domain is a nonzero ring... |
| domnring 20731 | A domain is a ring. (Cont... |
| domneq0 20732 | In a domain, a product is ... |
| domnmuln0 20733 | In a domain, a product of ... |
| isdomn5 20734 | The equivalence between th... |
| isdomn2 20735 | A ring is a domain iff all... |
| isdomn2OLD 20736 | Obsolete version of ~ isdo... |
| domnrrg 20737 | In a domain, a nonzero ele... |
| isdomn6 20738 | A ring is a domain iff the... |
| isdomn3 20739 | Nonzero elements form a mu... |
| isdomn4 20740 | A ring is a domain iff it ... |
| opprdomnb 20741 | A class is a domain if and... |
| opprdomn 20742 | The opposite of a domain i... |
| isdomn4r 20743 | A ring is a domain iff it ... |
| domnlcanb 20744 | Left-cancellation law for ... |
| domnlcan 20745 | Left-cancellation law for ... |
| domnrcanb 20746 | Right-cancellation law for... |
| domnrcan 20747 | Right-cancellation law for... |
| domneq0r 20748 | Right multiplication by a ... |
| isidom 20749 | An integral domain is a co... |
| idomdomd 20750 | An integral domain is a do... |
| idomcringd 20751 | An integral domain is a co... |
| idomringd 20752 | An integral domain is a ri... |
| isdrng 20757 | The predicate "is a divisi... |
| drngunit 20758 | Elementhood in the set of ... |
| drngui 20759 | The set of units of a divi... |
| drngring 20760 | A division ring is a ring.... |
| drngringd 20761 | A division ring is a ring.... |
| drnggrpd 20762 | A division ring is a group... |
| drnggrp 20763 | A division ring is a group... |
| isfld 20764 | A field is a commutative d... |
| flddrngd 20765 | A field is a division ring... |
| fldcrngd 20766 | A field is a commutative r... |
| isdrng2 20767 | A division ring can equiva... |
| drngprop 20768 | If two structures have the... |
| drngmgp 20769 | A division ring contains a... |
| drngid 20770 | A division ring's unity is... |
| drngunz 20771 | A division ring's unity is... |
| drngnzr 20772 | A division ring is a nonze... |
| drngdomn 20773 | A division ring is a domai... |
| drngmcl 20774 | The product of two nonzero... |
| drngmclOLD 20775 | Obsolete version of ~ drng... |
| drngid2 20776 | Properties showing that an... |
| drnginvrcl 20777 | Closure of the multiplicat... |
| drnginvrn0 20778 | The multiplicative inverse... |
| drnginvrcld 20779 | Closure of the multiplicat... |
| drnginvrl 20780 | Property of the multiplica... |
| drnginvrr 20781 | Property of the multiplica... |
| drnginvrld 20782 | Property of the multiplica... |
| drnginvrrd 20783 | Property of the multiplica... |
| drngmul0or 20784 | A product is zero iff one ... |
| drngmul0orOLD 20785 | Obsolete version of ~ drng... |
| drngmulne0 20786 | A product is nonzero iff b... |
| drngmuleq0 20787 | An element is zero iff its... |
| opprdrng 20788 | The opposite of a division... |
| isdrngd 20789 | Properties that characteri... |
| isdrngrd 20790 | Properties that characteri... |
| isdrngdOLD 20791 | Obsolete version of ~ isdr... |
| isdrngrdOLD 20792 | Obsolete version of ~ isdr... |
| drngpropd 20793 | If two structures have the... |
| fldpropd 20794 | If two structures have the... |
| fldidom 20795 | A field is an integral dom... |
| fldidomOLD 20796 | Obsolete version of ~ fldi... |
| fidomndrnglem 20797 | Lemma for ~ fidomndrng . ... |
| fidomndrng 20798 | A finite domain is a divis... |
| fiidomfld 20799 | A finite integral domain i... |
| rng1nnzr 20800 | The (smallest) structure r... |
| ring1zr 20801 | The only (unital) ring wit... |
| rngen1zr 20802 | The only (unital) ring wit... |
| ringen1zr 20803 | The only unital ring with ... |
| rng1nfld 20804 | The zero ring is not a fie... |
| issubdrg 20805 | Characterize the subfields... |
| drhmsubc 20806 | According to ~ df-subc , t... |
| drngcat 20807 | The restriction of the cat... |
| fldcat 20808 | The restriction of the cat... |
| fldc 20809 | The restriction of the cat... |
| fldhmsubc 20810 | According to ~ df-subc , t... |
| issdrg 20813 | Property of a division sub... |
| sdrgrcl 20814 | Reverse closure for a sub-... |
| sdrgdrng 20815 | A sub-division-ring is a d... |
| sdrgsubrg 20816 | A sub-division-ring is a s... |
| sdrgid 20817 | Every division ring is a d... |
| sdrgss 20818 | A division subring is a su... |
| sdrgbas 20819 | Base set of a sub-division... |
| issdrg2 20820 | Property of a division sub... |
| sdrgunit 20821 | A unit of a sub-division-r... |
| imadrhmcl 20822 | The image of a (nontrivial... |
| fldsdrgfld 20823 | A sub-division-ring of a f... |
| acsfn1p 20824 | Construction of a closure ... |
| subrgacs 20825 | Closure property of subrin... |
| sdrgacs 20826 | Closure property of divisi... |
| cntzsdrg 20827 | Centralizers in division r... |
| subdrgint 20828 | The intersection of a none... |
| sdrgint 20829 | The intersection of a none... |
| primefld 20830 | The smallest sub division ... |
| primefld0cl 20831 | The prime field contains t... |
| primefld1cl 20832 | The prime field contains t... |
| abvfval 20835 | Value of the set of absolu... |
| isabv 20836 | Elementhood in the set of ... |
| isabvd 20837 | Properties that determine ... |
| abvrcl 20838 | Reverse closure for the ab... |
| abvfge0 20839 | An absolute value is a fun... |
| abvf 20840 | An absolute value is a fun... |
| abvcl 20841 | An absolute value is a fun... |
| abvge0 20842 | The absolute value of a nu... |
| abveq0 20843 | The value of an absolute v... |
| abvne0 20844 | The absolute value of a no... |
| abvgt0 20845 | The absolute value of a no... |
| abvmul 20846 | An absolute value distribu... |
| abvtri 20847 | An absolute value satisfie... |
| abv0 20848 | The absolute value of zero... |
| abv1z 20849 | The absolute value of one ... |
| abv1 20850 | The absolute value of one ... |
| abvneg 20851 | The absolute value of a ne... |
| abvsubtri 20852 | An absolute value satisfie... |
| abvrec 20853 | The absolute value distrib... |
| abvdiv 20854 | The absolute value distrib... |
| abvdom 20855 | Any ring with an absolute ... |
| abvres 20856 | The restriction of an abso... |
| abvtrivd 20857 | The trivial absolute value... |
| abvtrivg 20858 | The trivial absolute value... |
| abvtriv 20859 | The trivial absolute value... |
| abvpropd 20860 | If two structures have the... |
| abvn0b 20861 | Another characterization o... |
| staffval 20866 | The functionalization of t... |
| stafval 20867 | The functionalization of t... |
| staffn 20868 | The functionalization is e... |
| issrng 20869 | The predicate "is a star r... |
| srngrhm 20870 | The involution function in... |
| srngring 20871 | A star ring is a ring. (C... |
| srngcnv 20872 | The involution function in... |
| srngf1o 20873 | The involution function in... |
| srngcl 20874 | The involution function in... |
| srngnvl 20875 | The involution function in... |
| srngadd 20876 | The involution function in... |
| srngmul 20877 | The involution function in... |
| srng1 20878 | The conjugate of the ring ... |
| srng0 20879 | The conjugate of the ring ... |
| issrngd 20880 | Properties that determine ... |
| idsrngd 20881 | A commutative ring is a st... |
| islmod 20886 | The predicate "is a left m... |
| lmodlema 20887 | Lemma for properties of a ... |
| islmodd 20888 | Properties that determine ... |
| lmodgrp 20889 | A left module is a group. ... |
| lmodring 20890 | The scalar component of a ... |
| lmodfgrp 20891 | The scalar component of a ... |
| lmodgrpd 20892 | A left module is a group. ... |
| lmodbn0 20893 | The base set of a left mod... |
| lmodacl 20894 | Closure of ring addition f... |
| lmodmcl 20895 | Closure of ring multiplica... |
| lmodsn0 20896 | The set of scalars in a le... |
| lmodvacl 20897 | Closure of vector addition... |
| lmodass 20898 | Left module vector sum is ... |
| lmodlcan 20899 | Left cancellation law for ... |
| lmodvscl 20900 | Closure of scalar product ... |
| lmodvscld 20901 | Closure of scalar product ... |
| scaffval 20902 | The scalar multiplication ... |
| scafval 20903 | The scalar multiplication ... |
| scafeq 20904 | If the scalar multiplicati... |
| scaffn 20905 | The scalar multiplication ... |
| lmodscaf 20906 | The scalar multiplication ... |
| lmodvsdi 20907 | Distributive law for scala... |
| lmodvsdir 20908 | Distributive law for scala... |
| lmodvsass 20909 | Associative law for scalar... |
| lmod0cl 20910 | The ring zero in a left mo... |
| lmod1cl 20911 | The ring unity in a left m... |
| lmodvs1 20912 | Scalar product with the ri... |
| lmod0vcl 20913 | The zero vector is a vecto... |
| lmod0vlid 20914 | Left identity law for the ... |
| lmod0vrid 20915 | Right identity law for the... |
| lmod0vid 20916 | Identity equivalent to the... |
| lmod0vs 20917 | Zero times a vector is the... |
| lmodvs0 20918 | Anything times the zero ve... |
| lmodvsmmulgdi 20919 | Distributive law for a gro... |
| lmodfopnelem1 20920 | Lemma 1 for ~ lmodfopne . ... |
| lmodfopnelem2 20921 | Lemma 2 for ~ lmodfopne . ... |
| lmodfopne 20922 | The (functionalized) opera... |
| lcomf 20923 | A linear-combination sum i... |
| lcomfsupp 20924 | A linear-combination sum i... |
| lmodvnegcl 20925 | Closure of vector negative... |
| lmodvnegid 20926 | Addition of a vector with ... |
| lmodvneg1 20927 | Minus 1 times a vector is ... |
| lmodvsneg 20928 | Multiplication of a vector... |
| lmodvsubcl 20929 | Closure of vector subtract... |
| lmodcom 20930 | Left module vector sum is ... |
| lmodabl 20931 | A left module is an abelia... |
| lmodcmn 20932 | A left module is a commuta... |
| lmodnegadd 20933 | Distribute negation throug... |
| lmod4 20934 | Commutative/associative la... |
| lmodvsubadd 20935 | Relationship between vecto... |
| lmodvaddsub4 20936 | Vector addition/subtractio... |
| lmodvpncan 20937 | Addition/subtraction cance... |
| lmodvnpcan 20938 | Cancellation law for vecto... |
| lmodvsubval2 20939 | Value of vector subtractio... |
| lmodsubvs 20940 | Subtraction of a scalar pr... |
| lmodsubdi 20941 | Scalar multiplication dist... |
| lmodsubdir 20942 | Scalar multiplication dist... |
| lmodsubeq0 20943 | If the difference between ... |
| lmodsubid 20944 | Subtraction of a vector fr... |
| lmodvsghm 20945 | Scalar multiplication of t... |
| lmodprop2d 20946 | If two structures have the... |
| lmodpropd 20947 | If two structures have the... |
| gsumvsmul 20948 | Pull a scalar multiplicati... |
| mptscmfsupp0 20949 | A mapping to a scalar prod... |
| mptscmfsuppd 20950 | A function mapping to a sc... |
| rmodislmodlem 20951 | Lemma for ~ rmodislmod . ... |
| rmodislmod 20952 | The right module ` R ` ind... |
| rmodislmodOLD 20953 | Obsolete version of ~ rmod... |
| lssset 20956 | The set of all (not necess... |
| islss 20957 | The predicate "is a subspa... |
| islssd 20958 | Properties that determine ... |
| lssss 20959 | A subspace is a set of vec... |
| lssel 20960 | A subspace member is a vec... |
| lss1 20961 | The set of vectors in a le... |
| lssuni 20962 | The union of all subspaces... |
| lssn0 20963 | A subspace is not empty. ... |
| 00lss 20964 | The empty structure has no... |
| lsscl 20965 | Closure property of a subs... |
| lssvacl 20966 | Closure of vector addition... |
| lssvsubcl 20967 | Closure of vector subtract... |
| lssvancl1 20968 | Non-closure: if one vector... |
| lssvancl2 20969 | Non-closure: if one vector... |
| lss0cl 20970 | The zero vector belongs to... |
| lsssn0 20971 | The singleton of the zero ... |
| lss0ss 20972 | The zero subspace is inclu... |
| lssle0 20973 | No subspace is smaller tha... |
| lssne0 20974 | A nonzero subspace has a n... |
| lssvneln0 20975 | A vector ` X ` which doesn... |
| lssneln0 20976 | A vector ` X ` which doesn... |
| lssssr 20977 | Conclude subspace ordering... |
| lssvscl 20978 | Closure of scalar product ... |
| lssvnegcl 20979 | Closure of negative vector... |
| lsssubg 20980 | All subspaces are subgroup... |
| lsssssubg 20981 | All subspaces are subgroup... |
| islss3 20982 | A linear subspace of a mod... |
| lsslmod 20983 | A submodule is a module. ... |
| lsslss 20984 | The subspaces of a subspac... |
| islss4 20985 | A linear subspace is a sub... |
| lss1d 20986 | One-dimensional subspace (... |
| lssintcl 20987 | The intersection of a none... |
| lssincl 20988 | The intersection of two su... |
| lssmre 20989 | The subspaces of a module ... |
| lssacs 20990 | Submodules are an algebrai... |
| prdsvscacl 20991 | Pointwise scalar multiplic... |
| prdslmodd 20992 | The product of a family of... |
| pwslmod 20993 | A structure power of a lef... |
| lspfval 20996 | The span function for a le... |
| lspf 20997 | The span function on a lef... |
| lspval 20998 | The span of a set of vecto... |
| lspcl 20999 | The span of a set of vecto... |
| lspsncl 21000 | The span of a singleton is... |
| lspprcl 21001 | The span of a pair is a su... |
| lsptpcl 21002 | The span of an unordered t... |
| lspsnsubg 21003 | The span of a singleton is... |
| 00lsp 21004 | ~ fvco4i lemma for linear ... |
| lspid 21005 | The span of a subspace is ... |
| lspssv 21006 | A span is a set of vectors... |
| lspss 21007 | Span preserves subset orde... |
| lspssid 21008 | A set of vectors is a subs... |
| lspidm 21009 | The span of a set of vecto... |
| lspun 21010 | The span of union is the s... |
| lspssp 21011 | If a set of vectors is a s... |
| mrclsp 21012 | Moore closure generalizes ... |
| lspsnss 21013 | The span of the singleton ... |
| ellspsn3 21014 | A member of the span of th... |
| lspprss 21015 | The span of a pair of vect... |
| lspsnid 21016 | A vector belongs to the sp... |
| ellspsn6 21017 | Relationship between a vec... |
| ellspsn5b 21018 | Relationship between a vec... |
| ellspsn5 21019 | Relationship between a vec... |
| lspprid1 21020 | A member of a pair of vect... |
| lspprid2 21021 | A member of a pair of vect... |
| lspprvacl 21022 | The sum of two vectors bel... |
| lssats2 21023 | A way to express atomistic... |
| ellspsni 21024 | A scalar product with a ve... |
| lspsn 21025 | Span of the singleton of a... |
| ellspsn 21026 | Member of span of the sing... |
| lspsnvsi 21027 | Span of a scalar product o... |
| lspsnss2 21028 | Comparable spans of single... |
| lspsnneg 21029 | Negation does not change t... |
| lspsnsub 21030 | Swapping subtraction order... |
| lspsn0 21031 | Span of the singleton of t... |
| lsp0 21032 | Span of the empty set. (C... |
| lspuni0 21033 | Union of the span of the e... |
| lspun0 21034 | The span of a union with t... |
| lspsneq0 21035 | Span of the singleton is t... |
| lspsneq0b 21036 | Equal singleton spans impl... |
| lmodindp1 21037 | Two independent (non-colin... |
| lsslsp 21038 | Spans in submodules corres... |
| lsslspOLD 21039 | Obsolete version of ~ lssl... |
| lss0v 21040 | The zero vector in a submo... |
| lsspropd 21041 | If two structures have the... |
| lsppropd 21042 | If two structures have the... |
| reldmlmhm 21049 | Lemma for module homomorph... |
| lmimfn 21050 | Lemma for module isomorphi... |
| islmhm 21051 | Property of being a homomo... |
| islmhm3 21052 | Property of a module homom... |
| lmhmlem 21053 | Non-quantified consequence... |
| lmhmsca 21054 | A homomorphism of left mod... |
| lmghm 21055 | A homomorphism of left mod... |
| lmhmlmod2 21056 | A homomorphism of left mod... |
| lmhmlmod1 21057 | A homomorphism of left mod... |
| lmhmf 21058 | A homomorphism of left mod... |
| lmhmlin 21059 | A homomorphism of left mod... |
| lmodvsinv 21060 | Multiplication of a vector... |
| lmodvsinv2 21061 | Multiplying a negated vect... |
| islmhm2 21062 | A one-equation proof of li... |
| islmhmd 21063 | Deduction for a module hom... |
| 0lmhm 21064 | The constant zero linear f... |
| idlmhm 21065 | The identity function on a... |
| invlmhm 21066 | The negative function on a... |
| lmhmco 21067 | The composition of two mod... |
| lmhmplusg 21068 | The pointwise sum of two l... |
| lmhmvsca 21069 | The pointwise scalar produ... |
| lmhmf1o 21070 | A bijective module homomor... |
| lmhmima 21071 | The image of a subspace un... |
| lmhmpreima 21072 | The inverse image of a sub... |
| lmhmlsp 21073 | Homomorphisms preserve spa... |
| lmhmrnlss 21074 | The range of a homomorphis... |
| lmhmkerlss 21075 | The kernel of a homomorphi... |
| reslmhm 21076 | Restriction of a homomorph... |
| reslmhm2 21077 | Expansion of the codomain ... |
| reslmhm2b 21078 | Expansion of the codomain ... |
| lmhmeql 21079 | The equalizer of two modul... |
| lspextmo 21080 | A linear function is compl... |
| pwsdiaglmhm 21081 | Diagonal homomorphism into... |
| pwssplit0 21082 | Splitting for structure po... |
| pwssplit1 21083 | Splitting for structure po... |
| pwssplit2 21084 | Splitting for structure po... |
| pwssplit3 21085 | Splitting for structure po... |
| islmim 21086 | An isomorphism of left mod... |
| lmimf1o 21087 | An isomorphism of left mod... |
| lmimlmhm 21088 | An isomorphism of modules ... |
| lmimgim 21089 | An isomorphism of modules ... |
| islmim2 21090 | An isomorphism of left mod... |
| lmimcnv 21091 | The converse of a bijectiv... |
| brlmic 21092 | The relation "is isomorphi... |
| brlmici 21093 | Prove isomorphic by an exp... |
| lmiclcl 21094 | Isomorphism implies the le... |
| lmicrcl 21095 | Isomorphism implies the ri... |
| lmicsym 21096 | Module isomorphism is symm... |
| lmhmpropd 21097 | Module homomorphism depend... |
| islbs 21100 | The predicate " ` B ` is a... |
| lbsss 21101 | A basis is a set of vector... |
| lbsel 21102 | An element of a basis is a... |
| lbssp 21103 | The span of a basis is the... |
| lbsind 21104 | A basis is linearly indepe... |
| lbsind2 21105 | A basis is linearly indepe... |
| lbspss 21106 | No proper subset of a basi... |
| lsmcl 21107 | The sum of two subspaces i... |
| lsmspsn 21108 | Member of subspace sum of ... |
| lsmelval2 21109 | Subspace sum membership in... |
| lsmsp 21110 | Subspace sum in terms of s... |
| lsmsp2 21111 | Subspace sum of spans of s... |
| lsmssspx 21112 | Subspace sum (in its exten... |
| lsmpr 21113 | The span of a pair of vect... |
| lsppreli 21114 | A vector expressed as a su... |
| lsmelpr 21115 | Two ways to say that a vec... |
| lsppr0 21116 | The span of a vector paire... |
| lsppr 21117 | Span of a pair of vectors.... |
| lspprel 21118 | Member of the span of a pa... |
| lspprabs 21119 | Absorption of vector sum i... |
| lspvadd 21120 | The span of a vector sum i... |
| lspsntri 21121 | Triangle-type inequality f... |
| lspsntrim 21122 | Triangle-type inequality f... |
| lbspropd 21123 | If two structures have the... |
| pj1lmhm 21124 | The left projection functi... |
| pj1lmhm2 21125 | The left projection functi... |
| islvec 21128 | The predicate "is a left v... |
| lvecdrng 21129 | The set of scalars of a le... |
| lveclmod 21130 | A left vector space is a l... |
| lveclmodd 21131 | A vector space is a left m... |
| lvecgrpd 21132 | A vector space is a group.... |
| lsslvec 21133 | A vector subspace is a vec... |
| lmhmlvec 21134 | The property for modules t... |
| lvecvs0or 21135 | If a scalar product is zer... |
| lvecvsn0 21136 | A scalar product is nonzer... |
| lssvs0or 21137 | If a scalar product belong... |
| lvecvscan 21138 | Cancellation law for scala... |
| lvecvscan2 21139 | Cancellation law for scala... |
| lvecinv 21140 | Invert coefficient of scal... |
| lspsnvs 21141 | A nonzero scalar product d... |
| lspsneleq 21142 | Membership relation that i... |
| lspsncmp 21143 | Comparable spans of nonzer... |
| lspsnne1 21144 | Two ways to express that v... |
| lspsnne2 21145 | Two ways to express that v... |
| lspsnnecom 21146 | Swap two vectors with diff... |
| lspabs2 21147 | Absorption law for span of... |
| lspabs3 21148 | Absorption law for span of... |
| lspsneq 21149 | Equal spans of singletons ... |
| lspsneu 21150 | Nonzero vectors with equal... |
| ellspsn4 21151 | A member of the span of th... |
| lspdisj 21152 | The span of a vector not i... |
| lspdisjb 21153 | A nonzero vector is not in... |
| lspdisj2 21154 | Unequal spans are disjoint... |
| lspfixed 21155 | Show membership in the spa... |
| lspexch 21156 | Exchange property for span... |
| lspexchn1 21157 | Exchange property for span... |
| lspexchn2 21158 | Exchange property for span... |
| lspindpi 21159 | Partial independence prope... |
| lspindp1 21160 | Alternate way to say 3 vec... |
| lspindp2l 21161 | Alternate way to say 3 vec... |
| lspindp2 21162 | Alternate way to say 3 vec... |
| lspindp3 21163 | Independence of 2 vectors ... |
| lspindp4 21164 | (Partial) independence of ... |
| lvecindp 21165 | Compute the ` X ` coeffici... |
| lvecindp2 21166 | Sums of independent vector... |
| lspsnsubn0 21167 | Unequal singleton spans im... |
| lsmcv 21168 | Subspace sum has the cover... |
| lspsolvlem 21169 | Lemma for ~ lspsolv . (Co... |
| lspsolv 21170 | If ` X ` is in the span of... |
| lssacsex 21171 | In a vector space, subspac... |
| lspsnat 21172 | There is no subspace stric... |
| lspsncv0 21173 | The span of a singleton co... |
| lsppratlem1 21174 | Lemma for ~ lspprat . Let... |
| lsppratlem2 21175 | Lemma for ~ lspprat . Sho... |
| lsppratlem3 21176 | Lemma for ~ lspprat . In ... |
| lsppratlem4 21177 | Lemma for ~ lspprat . In ... |
| lsppratlem5 21178 | Lemma for ~ lspprat . Com... |
| lsppratlem6 21179 | Lemma for ~ lspprat . Neg... |
| lspprat 21180 | A proper subspace of the s... |
| islbs2 21181 | An equivalent formulation ... |
| islbs3 21182 | An equivalent formulation ... |
| lbsacsbs 21183 | Being a basis in a vector ... |
| lvecdim 21184 | The dimension theorem for ... |
| lbsextlem1 21185 | Lemma for ~ lbsext . The ... |
| lbsextlem2 21186 | Lemma for ~ lbsext . Sinc... |
| lbsextlem3 21187 | Lemma for ~ lbsext . A ch... |
| lbsextlem4 21188 | Lemma for ~ lbsext . ~ lbs... |
| lbsextg 21189 | For any linearly independe... |
| lbsext 21190 | For any linearly independe... |
| lbsexg 21191 | Every vector space has a b... |
| lbsex 21192 | Every vector space has a b... |
| lvecprop2d 21193 | If two structures have the... |
| lvecpropd 21194 | If two structures have the... |
| sraval 21199 | Lemma for ~ srabase throug... |
| sralem 21200 | Lemma for ~ srabase and si... |
| sralemOLD 21201 | Obsolete version of ~ sral... |
| srabase 21202 | Base set of a subring alge... |
| srabaseOLD 21203 | Obsolete proof of ~ srabas... |
| sraaddg 21204 | Additive operation of a su... |
| sraaddgOLD 21205 | Obsolete proof of ~ sraadd... |
| sramulr 21206 | Multiplicative operation o... |
| sramulrOLD 21207 | Obsolete proof of ~ sramul... |
| srasca 21208 | The set of scalars of a su... |
| srascaOLD 21209 | Obsolete proof of ~ srasca... |
| sravsca 21210 | The scalar product operati... |
| sravscaOLD 21211 | Obsolete proof of ~ sravsc... |
| sraip 21212 | The inner product operatio... |
| sratset 21213 | Topology component of a su... |
| sratsetOLD 21214 | Obsolete proof of ~ sratse... |
| sratopn 21215 | Topology component of a su... |
| srads 21216 | Distance function of a sub... |
| sradsOLD 21217 | Obsolete proof of ~ srads ... |
| sraring 21218 | Condition for a subring al... |
| sralmod 21219 | The subring algebra is a l... |
| sralmod0 21220 | The subring module inherit... |
| issubrgd 21221 | Prove a subring by closure... |
| rlmfn 21222 | ` ringLMod ` is a function... |
| rlmval 21223 | Value of the ring module. ... |
| rlmval2 21224 | Value of the ring module e... |
| rlmbas 21225 | Base set of the ring modul... |
| rlmplusg 21226 | Vector addition in the rin... |
| rlm0 21227 | Zero vector in the ring mo... |
| rlmsub 21228 | Subtraction in the ring mo... |
| rlmmulr 21229 | Ring multiplication in the... |
| rlmsca 21230 | Scalars in the ring module... |
| rlmsca2 21231 | Scalars in the ring module... |
| rlmvsca 21232 | Scalar multiplication in t... |
| rlmtopn 21233 | Topology component of the ... |
| rlmds 21234 | Metric component of the ri... |
| rlmlmod 21235 | The ring module is a modul... |
| rlmlvec 21236 | The ring module over a div... |
| rlmlsm 21237 | Subgroup sum of the ring m... |
| rlmvneg 21238 | Vector negation in the rin... |
| rlmscaf 21239 | Functionalized scalar mult... |
| ixpsnbasval 21240 | The value of an infinite C... |
| lidlval 21245 | Value of the set of ring i... |
| rspval 21246 | Value of the ring span fun... |
| lidlss 21247 | An ideal is a subset of th... |
| lidlssbas 21248 | The base set of the restri... |
| lidlbas 21249 | A (left) ideal of a ring i... |
| islidl 21250 | Predicate of being a (left... |
| rnglidlmcl 21251 | A (left) ideal containing ... |
| rngridlmcl 21252 | A right ideal (which is a ... |
| dflidl2rng 21253 | Alternate (the usual textb... |
| isridlrng 21254 | A right ideal is a left id... |
| lidl0cl 21255 | An ideal contains 0. (Con... |
| lidlacl 21256 | An ideal is closed under a... |
| lidlnegcl 21257 | An ideal contains negative... |
| lidlsubg 21258 | An ideal is a subgroup of ... |
| lidlsubcl 21259 | An ideal is closed under s... |
| lidlmcl 21260 | An ideal is closed under l... |
| lidl1el 21261 | An ideal contains 1 iff it... |
| dflidl2 21262 | Alternate (the usual textb... |
| lidl0ALT 21263 | Alternate proof for ~ lidl... |
| rnglidl0 21264 | Every non-unital ring cont... |
| lidl0 21265 | Every ring contains a zero... |
| lidl1ALT 21266 | Alternate proof for ~ lidl... |
| rnglidl1 21267 | The base set of every non-... |
| lidl1 21268 | Every ring contains a unit... |
| lidlacs 21269 | The ideal system is an alg... |
| rspcl 21270 | The span of a set of ring ... |
| rspssid 21271 | The span of a set of ring ... |
| rsp1 21272 | The span of the identity e... |
| rsp0 21273 | The span of the zero eleme... |
| rspssp 21274 | The ideal span of a set of... |
| elrspsn 21275 | Membership in a principal ... |
| mrcrsp 21276 | Moore closure generalizes ... |
| lidlnz 21277 | A nonzero ideal contains a... |
| drngnidl 21278 | A division ring has only t... |
| lidlrsppropd 21279 | The left ideals and ring s... |
| rnglidlmmgm 21280 | The multiplicative group o... |
| rnglidlmsgrp 21281 | The multiplicative group o... |
| rnglidlrng 21282 | A (left) ideal of a non-un... |
| lidlnsg 21283 | An ideal is a normal subgr... |
| 2idlval 21286 | Definition of a two-sided ... |
| isridl 21287 | A right ideal is a left id... |
| 2idlelb 21288 | Membership in a two-sided ... |
| 2idllidld 21289 | A two-sided ideal is a lef... |
| 2idlridld 21290 | A two-sided ideal is a rig... |
| df2idl2rng 21291 | Alternate (the usual textb... |
| df2idl2 21292 | Alternate (the usual textb... |
| ridl0 21293 | Every ring contains a zero... |
| ridl1 21294 | Every ring contains a unit... |
| 2idl0 21295 | Every ring contains a zero... |
| 2idl1 21296 | Every ring contains a unit... |
| 2idlss 21297 | A two-sided ideal is a sub... |
| 2idlbas 21298 | The base set of a two-side... |
| 2idlelbas 21299 | The base set of a two-side... |
| rng2idlsubrng 21300 | A two-sided ideal of a non... |
| rng2idlnsg 21301 | A two-sided ideal of a non... |
| rng2idl0 21302 | The zero (additive identit... |
| rng2idlsubgsubrng 21303 | A two-sided ideal of a non... |
| rng2idlsubgnsg 21304 | A two-sided ideal of a non... |
| rng2idlsubg0 21305 | The zero (additive identit... |
| 2idlcpblrng 21306 | The coset equivalence rela... |
| 2idlcpbl 21307 | The coset equivalence rela... |
| qus2idrng 21308 | The quotient of a non-unit... |
| qus1 21309 | The multiplicative identit... |
| qusring 21310 | If ` S ` is a two-sided id... |
| qusrhm 21311 | If ` S ` is a two-sided id... |
| rhmpreimaidl 21312 | The preimage of an ideal b... |
| kerlidl 21313 | The kernel of a ring homom... |
| qusmul2idl 21314 | Value of the ring operatio... |
| crngridl 21315 | In a commutative ring, the... |
| crng2idl 21316 | In a commutative ring, a t... |
| qusmulrng 21317 | Value of the multiplicatio... |
| quscrng 21318 | The quotient of a commutat... |
| qusmulcrng 21319 | Value of the ring operatio... |
| rhmqusnsg 21320 | The mapping ` J ` induced ... |
| rngqiprng1elbas 21321 | The ring unity of a two-si... |
| rngqiprngghmlem1 21322 | Lemma 1 for ~ rngqiprngghm... |
| rngqiprngghmlem2 21323 | Lemma 2 for ~ rngqiprngghm... |
| rngqiprngghmlem3 21324 | Lemma 3 for ~ rngqiprngghm... |
| rngqiprngimfolem 21325 | Lemma for ~ rngqiprngimfo ... |
| rngqiprnglinlem1 21326 | Lemma 1 for ~ rngqiprnglin... |
| rngqiprnglinlem2 21327 | Lemma 2 for ~ rngqiprnglin... |
| rngqiprnglinlem3 21328 | Lemma 3 for ~ rngqiprnglin... |
| rngqiprngimf1lem 21329 | Lemma for ~ rngqiprngimf1 ... |
| rngqipbas 21330 | The base set of the produc... |
| rngqiprng 21331 | The product of the quotien... |
| rngqiprngimf 21332 | ` F ` is a function from (... |
| rngqiprngimfv 21333 | The value of the function ... |
| rngqiprngghm 21334 | ` F ` is a homomorphism of... |
| rngqiprngimf1 21335 | ` F ` is a one-to-one func... |
| rngqiprngimfo 21336 | ` F ` is a function from (... |
| rngqiprnglin 21337 | ` F ` is linear with respe... |
| rngqiprngho 21338 | ` F ` is a homomorphism of... |
| rngqiprngim 21339 | ` F ` is an isomorphism of... |
| rng2idl1cntr 21340 | The unity of a two-sided i... |
| rngringbdlem1 21341 | In a unital ring, the quot... |
| rngringbdlem2 21342 | A non-unital ring is unita... |
| rngringbd 21343 | A non-unital ring is unita... |
| ring2idlqus 21344 | For every unital ring ther... |
| ring2idlqusb 21345 | A non-unital ring is unita... |
| rngqiprngfulem1 21346 | Lemma 1 for ~ rngqiprngfu ... |
| rngqiprngfulem2 21347 | Lemma 2 for ~ rngqiprngfu ... |
| rngqiprngfulem3 21348 | Lemma 3 for ~ rngqiprngfu ... |
| rngqiprngfulem4 21349 | Lemma 4 for ~ rngqiprngfu ... |
| rngqiprngfulem5 21350 | Lemma 5 for ~ rngqiprngfu ... |
| rngqipring1 21351 | The ring unity of the prod... |
| rngqiprngfu 21352 | The function value of ` F ... |
| rngqiprngu 21353 | If a non-unital ring has a... |
| ring2idlqus1 21354 | If a non-unital ring has a... |
| lpival 21359 | Value of the set of princi... |
| islpidl 21360 | Property of being a princi... |
| lpi0 21361 | The zero ideal is always p... |
| lpi1 21362 | The unit ideal is always p... |
| islpir 21363 | Principal ideal rings are ... |
| lpiss 21364 | Principal ideals are a sub... |
| islpir2 21365 | Principal ideal rings are ... |
| lpirring 21366 | Principal ideal rings are ... |
| drnglpir 21367 | Division rings are princip... |
| rspsn 21368 | Membership in principal id... |
| lidldvgen 21369 | An element generates an id... |
| lpigen 21370 | An ideal is principal iff ... |
| cnfldstr 21391 | The field of complex numbe... |
| cnfldex 21392 | The field of complex numbe... |
| cnfldbas 21393 | The base set of the field ... |
| mpocnfldadd 21394 | The addition operation of ... |
| cnfldadd 21395 | The addition operation of ... |
| mpocnfldmul 21396 | The multiplication operati... |
| cnfldmul 21397 | The multiplication operati... |
| cnfldcj 21398 | The conjugation operation ... |
| cnfldtset 21399 | The topology component of ... |
| cnfldle 21400 | The ordering of the field ... |
| cnfldds 21401 | The metric of the field of... |
| cnfldunif 21402 | The uniform structure comp... |
| cnfldfun 21403 | The field of complex numbe... |
| cnfldfunALT 21404 | The field of complex numbe... |
| dfcnfldOLD 21405 | Obsolete version of ~ df-c... |
| cnfldstrOLD 21406 | Obsolete version of ~ cnfl... |
| cnfldexOLD 21407 | Obsolete version of ~ cnfl... |
| cnfldbasOLD 21408 | Obsolete version of ~ cnfl... |
| cnfldaddOLD 21409 | Obsolete version of ~ cnfl... |
| cnfldmulOLD 21410 | Obsolete version of ~ cnfl... |
| cnfldcjOLD 21411 | Obsolete version of ~ cnfl... |
| cnfldtsetOLD 21412 | Obsolete version of ~ cnfl... |
| cnfldleOLD 21413 | Obsolete version of ~ cnfl... |
| cnflddsOLD 21414 | Obsolete version of ~ cnfl... |
| cnfldunifOLD 21415 | Obsolete version of ~ cnfl... |
| cnfldfunOLD 21416 | Obsolete version of ~ cnfl... |
| cnfldfunALTOLD 21417 | Obsolete version of ~ cnfl... |
| cnfldfunALTOLDOLD 21418 | Obsolete proof of ~ cnfldf... |
| xrsstr 21419 | The extended real structur... |
| xrsex 21420 | The extended real structur... |
| xrsbas 21421 | The base set of the extend... |
| xrsadd 21422 | The addition operation of ... |
| xrsmul 21423 | The multiplication operati... |
| xrstset 21424 | The topology component of ... |
| xrsle 21425 | The ordering of the extend... |
| cncrng 21426 | The complex numbers form a... |
| cncrngOLD 21427 | Obsolete version of ~ cncr... |
| cnring 21428 | The complex numbers form a... |
| xrsmcmn 21429 | The "multiplicative group"... |
| cnfld0 21430 | Zero is the zero element o... |
| cnfld1 21431 | One is the unity element o... |
| cnfld1OLD 21432 | Obsolete version of ~ cnfl... |
| cnfldneg 21433 | The additive inverse in th... |
| cnfldplusf 21434 | The functionalized additio... |
| cnfldsub 21435 | The subtraction operator i... |
| cndrng 21436 | The complex numbers form a... |
| cndrngOLD 21437 | Obsolete version of ~ cndr... |
| cnflddiv 21438 | The division operation in ... |
| cnflddivOLD 21439 | Obsolete version of ~ cnfl... |
| cnfldinv 21440 | The multiplicative inverse... |
| cnfldmulg 21441 | The group multiple functio... |
| cnfldexp 21442 | The exponentiation operato... |
| cnsrng 21443 | The complex numbers form a... |
| xrsmgm 21444 | The "additive group" of th... |
| xrsnsgrp 21445 | The "additive group" of th... |
| xrsmgmdifsgrp 21446 | The "additive group" of th... |
| xrs1mnd 21447 | The extended real numbers,... |
| xrs10 21448 | The zero of the extended r... |
| xrs1cmn 21449 | The extended real numbers ... |
| xrge0subm 21450 | The nonnegative extended r... |
| xrge0cmn 21451 | The nonnegative extended r... |
| xrsds 21452 | The metric of the extended... |
| xrsdsval 21453 | The metric of the extended... |
| xrsdsreval 21454 | The metric of the extended... |
| xrsdsreclblem 21455 | Lemma for ~ xrsdsreclb . ... |
| xrsdsreclb 21456 | The metric of the extended... |
| cnsubmlem 21457 | Lemma for ~ nn0subm and fr... |
| cnsubglem 21458 | Lemma for ~ resubdrg and f... |
| cnsubrglem 21459 | Lemma for ~ resubdrg and f... |
| cnsubrglemOLD 21460 | Obsolete version of ~ cnsu... |
| cnsubdrglem 21461 | Lemma for ~ resubdrg and f... |
| qsubdrg 21462 | The rational numbers form ... |
| zsubrg 21463 | The integers form a subrin... |
| gzsubrg 21464 | The gaussian integers form... |
| nn0subm 21465 | The nonnegative integers f... |
| rege0subm 21466 | The nonnegative reals form... |
| absabv 21467 | The regular absolute value... |
| zsssubrg 21468 | The integers are a subset ... |
| qsssubdrg 21469 | The rational numbers are a... |
| cnsubrg 21470 | There are no subrings of t... |
| cnmgpabl 21471 | The unit group of the comp... |
| cnmgpid 21472 | The group identity element... |
| cnmsubglem 21473 | Lemma for ~ rpmsubg and fr... |
| rpmsubg 21474 | The positive reals form a ... |
| gzrngunitlem 21475 | Lemma for ~ gzrngunit . (... |
| gzrngunit 21476 | The units on ` ZZ [ _i ] `... |
| gsumfsum 21477 | Relate a group sum on ` CC... |
| regsumfsum 21478 | Relate a group sum on ` ( ... |
| expmhm 21479 | Exponentiation is a monoid... |
| nn0srg 21480 | The nonnegative integers f... |
| rge0srg 21481 | The nonnegative real numbe... |
| zringcrng 21484 | The ring of integers is a ... |
| zringring 21485 | The ring of integers is a ... |
| zringrng 21486 | The ring of integers is a ... |
| zringabl 21487 | The ring of integers is an... |
| zringgrp 21488 | The ring of integers is an... |
| zringbas 21489 | The integers are the base ... |
| zringplusg 21490 | The addition operation of ... |
| zringsub 21491 | The subtraction of element... |
| zringmulg 21492 | The multiplication (group ... |
| zringmulr 21493 | The multiplication operati... |
| zring0 21494 | The zero element of the ri... |
| zring1 21495 | The unity element of the r... |
| zringnzr 21496 | The ring of integers is a ... |
| dvdsrzring 21497 | Ring divisibility in the r... |
| zringlpirlem1 21498 | Lemma for ~ zringlpir . A... |
| zringlpirlem2 21499 | Lemma for ~ zringlpir . A... |
| zringlpirlem3 21500 | Lemma for ~ zringlpir . A... |
| zringinvg 21501 | The additive inverse of an... |
| zringunit 21502 | The units of ` ZZ ` are th... |
| zringlpir 21503 | The integers are a princip... |
| zringndrg 21504 | The integers are not a div... |
| zringcyg 21505 | The integers are a cyclic ... |
| zringsubgval 21506 | Subtraction in the ring of... |
| zringmpg 21507 | The multiplicative group o... |
| prmirredlem 21508 | A positive integer is irre... |
| dfprm2 21509 | The positive irreducible e... |
| prmirred 21510 | The irreducible elements o... |
| expghm 21511 | Exponentiation is a group ... |
| mulgghm2 21512 | The powers of a group elem... |
| mulgrhm 21513 | The powers of the element ... |
| mulgrhm2 21514 | The powers of the element ... |
| irinitoringc 21515 | The ring of integers is an... |
| nzerooringczr 21516 | There is no zero object in... |
| pzriprnglem1 21517 | Lemma 1 for ~ pzriprng : `... |
| pzriprnglem2 21518 | Lemma 2 for ~ pzriprng : ... |
| pzriprnglem3 21519 | Lemma 3 for ~ pzriprng : ... |
| pzriprnglem4 21520 | Lemma 4 for ~ pzriprng : `... |
| pzriprnglem5 21521 | Lemma 5 for ~ pzriprng : `... |
| pzriprnglem6 21522 | Lemma 6 for ~ pzriprng : `... |
| pzriprnglem7 21523 | Lemma 7 for ~ pzriprng : `... |
| pzriprnglem8 21524 | Lemma 8 for ~ pzriprng : `... |
| pzriprnglem9 21525 | Lemma 9 for ~ pzriprng : ... |
| pzriprnglem10 21526 | Lemma 10 for ~ pzriprng : ... |
| pzriprnglem11 21527 | Lemma 11 for ~ pzriprng : ... |
| pzriprnglem12 21528 | Lemma 12 for ~ pzriprng : ... |
| pzriprnglem13 21529 | Lemma 13 for ~ pzriprng : ... |
| pzriprnglem14 21530 | Lemma 14 for ~ pzriprng : ... |
| pzriprngALT 21531 | The non-unital ring ` ( ZZ... |
| pzriprng1ALT 21532 | The ring unity of the ring... |
| pzriprng 21533 | The non-unital ring ` ( ZZ... |
| pzriprng1 21534 | The ring unity of the ring... |
| zrhval 21543 | Define the unique homomorp... |
| zrhval2 21544 | Alternate value of the ` Z... |
| zrhmulg 21545 | Value of the ` ZRHom ` hom... |
| zrhrhmb 21546 | The ` ZRHom ` homomorphism... |
| zrhrhm 21547 | The ` ZRHom ` homomorphism... |
| zrh1 21548 | Interpretation of 1 in a r... |
| zrh0 21549 | Interpretation of 0 in a r... |
| zrhpropd 21550 | The ` ZZ ` ring homomorphi... |
| zlmval 21551 | Augment an abelian group w... |
| zlmlem 21552 | Lemma for ~ zlmbas and ~ z... |
| zlmlemOLD 21553 | Obsolete version of ~ zlml... |
| zlmbas 21554 | Base set of a ` ZZ ` -modu... |
| zlmbasOLD 21555 | Obsolete version of ~ zlmb... |
| zlmplusg 21556 | Group operation of a ` ZZ ... |
| zlmplusgOLD 21557 | Obsolete version of ~ zlmb... |
| zlmmulr 21558 | Ring operation of a ` ZZ `... |
| zlmmulrOLD 21559 | Obsolete version of ~ zlmb... |
| zlmsca 21560 | Scalar ring of a ` ZZ ` -m... |
| zlmvsca 21561 | Scalar multiplication oper... |
| zlmlmod 21562 | The ` ZZ ` -module operati... |
| chrval 21563 | Definition substitution of... |
| chrcl 21564 | Closure of the characteris... |
| chrid 21565 | The canonical ` ZZ ` ring ... |
| chrdvds 21566 | The ` ZZ ` ring homomorphi... |
| chrcong 21567 | If two integers are congru... |
| dvdschrmulg 21568 | In a ring, any multiple of... |
| fermltlchr 21569 | A generalization of Fermat... |
| chrnzr 21570 | Nonzero rings are precisel... |
| chrrhm 21571 | The characteristic restric... |
| domnchr 21572 | The characteristic of a do... |
| znlidl 21573 | The set ` n ZZ ` is an ide... |
| zncrng2 21574 | Making a commutative ring ... |
| znval 21575 | The value of the ` Z/nZ ` ... |
| znle 21576 | The value of the ` Z/nZ ` ... |
| znval2 21577 | Self-referential expressio... |
| znbaslem 21578 | Lemma for ~ znbas . (Cont... |
| znbaslemOLD 21579 | Obsolete version of ~ znba... |
| znbas2 21580 | The base set of ` Z/nZ ` i... |
| znbas2OLD 21581 | Obsolete version of ~ znba... |
| znadd 21582 | The additive structure of ... |
| znaddOLD 21583 | Obsolete version of ~ znad... |
| znmul 21584 | The multiplicative structu... |
| znmulOLD 21585 | Obsolete version of ~ znad... |
| znzrh 21586 | The ` ZZ ` ring homomorphi... |
| znbas 21587 | The base set of ` Z/nZ ` s... |
| zncrng 21588 | ` Z/nZ ` is a commutative ... |
| znzrh2 21589 | The ` ZZ ` ring homomorphi... |
| znzrhval 21590 | The ` ZZ ` ring homomorphi... |
| znzrhfo 21591 | The ` ZZ ` ring homomorphi... |
| zncyg 21592 | The group ` ZZ / n ZZ ` is... |
| zndvds 21593 | Express equality of equiva... |
| zndvds0 21594 | Special case of ~ zndvds w... |
| znf1o 21595 | The function ` F ` enumera... |
| zzngim 21596 | The ` ZZ ` ring homomorphi... |
| znle2 21597 | The ordering of the ` Z/nZ... |
| znleval 21598 | The ordering of the ` Z/nZ... |
| znleval2 21599 | The ordering of the ` Z/nZ... |
| zntoslem 21600 | Lemma for ~ zntos . (Cont... |
| zntos 21601 | The ` Z/nZ ` structure is ... |
| znhash 21602 | The ` Z/nZ ` structure has... |
| znfi 21603 | The ` Z/nZ ` structure is ... |
| znfld 21604 | The ` Z/nZ ` structure is ... |
| znidomb 21605 | The ` Z/nZ ` structure is ... |
| znchr 21606 | Cyclic rings are defined b... |
| znunit 21607 | The units of ` Z/nZ ` are ... |
| znunithash 21608 | The size of the unit group... |
| znrrg 21609 | The regular elements of ` ... |
| cygznlem1 21610 | Lemma for ~ cygzn . (Cont... |
| cygznlem2a 21611 | Lemma for ~ cygzn . (Cont... |
| cygznlem2 21612 | Lemma for ~ cygzn . (Cont... |
| cygznlem3 21613 | A cyclic group with ` n ` ... |
| cygzn 21614 | A cyclic group with ` n ` ... |
| cygth 21615 | The "fundamental theorem o... |
| cyggic 21616 | Cyclic groups are isomorph... |
| frgpcyg 21617 | A free group is cyclic iff... |
| freshmansdream 21618 | For a prime number ` P ` ,... |
| frobrhm 21619 | In a commutative ring with... |
| cnmsgnsubg 21620 | The signs form a multiplic... |
| cnmsgnbas 21621 | The base set of the sign s... |
| cnmsgngrp 21622 | The group of signs under m... |
| psgnghm 21623 | The sign is a homomorphism... |
| psgnghm2 21624 | The sign is a homomorphism... |
| psgninv 21625 | The sign of a permutation ... |
| psgnco 21626 | Multiplicativity of the pe... |
| zrhpsgnmhm 21627 | Embedding of permutation s... |
| zrhpsgninv 21628 | The embedded sign of a per... |
| evpmss 21629 | Even permutations are perm... |
| psgnevpmb 21630 | A class is an even permuta... |
| psgnodpm 21631 | A permutation which is odd... |
| psgnevpm 21632 | A permutation which is eve... |
| psgnodpmr 21633 | If a permutation has sign ... |
| zrhpsgnevpm 21634 | The sign of an even permut... |
| zrhpsgnodpm 21635 | The sign of an odd permuta... |
| cofipsgn 21636 | Composition of any class `... |
| zrhpsgnelbas 21637 | Embedding of permutation s... |
| zrhcopsgnelbas 21638 | Embedding of permutation s... |
| evpmodpmf1o 21639 | The function for performin... |
| pmtrodpm 21640 | A transposition is an odd ... |
| psgnfix1 21641 | A permutation of a finite ... |
| psgnfix2 21642 | A permutation of a finite ... |
| psgndiflemB 21643 | Lemma 1 for ~ psgndif . (... |
| psgndiflemA 21644 | Lemma 2 for ~ psgndif . (... |
| psgndif 21645 | Embedding of permutation s... |
| copsgndif 21646 | Embedding of permutation s... |
| rebase 21649 | The base of the field of r... |
| remulg 21650 | The multiplication (group ... |
| resubdrg 21651 | The real numbers form a di... |
| resubgval 21652 | Subtraction in the field o... |
| replusg 21653 | The addition operation of ... |
| remulr 21654 | The multiplication operati... |
| re0g 21655 | The zero element of the fi... |
| re1r 21656 | The unity element of the f... |
| rele2 21657 | The ordering relation of t... |
| relt 21658 | The ordering relation of t... |
| reds 21659 | The distance of the field ... |
| redvr 21660 | The division operation of ... |
| retos 21661 | The real numbers are a tot... |
| refld 21662 | The real numbers form a fi... |
| refldcj 21663 | The conjugation operation ... |
| resrng 21664 | The real numbers form a st... |
| regsumsupp 21665 | The group sum over the rea... |
| rzgrp 21666 | The quotient group ` RR / ... |
| isphl 21671 | The predicate "is a genera... |
| phllvec 21672 | A pre-Hilbert space is a l... |
| phllmod 21673 | A pre-Hilbert space is a l... |
| phlsrng 21674 | The scalar ring of a pre-H... |
| phllmhm 21675 | The inner product of a pre... |
| ipcl 21676 | Closure of the inner produ... |
| ipcj 21677 | Conjugate of an inner prod... |
| iporthcom 21678 | Orthogonality (meaning inn... |
| ip0l 21679 | Inner product with a zero ... |
| ip0r 21680 | Inner product with a zero ... |
| ipeq0 21681 | The inner product of a vec... |
| ipdir 21682 | Distributive law for inner... |
| ipdi 21683 | Distributive law for inner... |
| ip2di 21684 | Distributive law for inner... |
| ipsubdir 21685 | Distributive law for inner... |
| ipsubdi 21686 | Distributive law for inner... |
| ip2subdi 21687 | Distributive law for inner... |
| ipass 21688 | Associative law for inner ... |
| ipassr 21689 | "Associative" law for seco... |
| ipassr2 21690 | "Associative" law for inne... |
| ipffval 21691 | The inner product operatio... |
| ipfval 21692 | The inner product operatio... |
| ipfeq 21693 | If the inner product opera... |
| ipffn 21694 | The inner product operatio... |
| phlipf 21695 | The inner product operatio... |
| ip2eq 21696 | Two vectors are equal iff ... |
| isphld 21697 | Properties that determine ... |
| phlpropd 21698 | If two structures have the... |
| ssipeq 21699 | The inner product on a sub... |
| phssipval 21700 | The inner product on a sub... |
| phssip 21701 | The inner product (as a fu... |
| phlssphl 21702 | A subspace of an inner pro... |
| ocvfval 21709 | The orthocomplement operat... |
| ocvval 21710 | Value of the orthocompleme... |
| elocv 21711 | Elementhood in the orthoco... |
| ocvi 21712 | Property of a member of th... |
| ocvss 21713 | The orthocomplement of a s... |
| ocvocv 21714 | A set is contained in its ... |
| ocvlss 21715 | The orthocomplement of a s... |
| ocv2ss 21716 | Orthocomplements reverse s... |
| ocvin 21717 | An orthocomplement has tri... |
| ocvsscon 21718 | Two ways to say that ` S `... |
| ocvlsp 21719 | The orthocomplement of a l... |
| ocv0 21720 | The orthocomplement of the... |
| ocvz 21721 | The orthocomplement of the... |
| ocv1 21722 | The orthocomplement of the... |
| unocv 21723 | The orthocomplement of a u... |
| iunocv 21724 | The orthocomplement of an ... |
| cssval 21725 | The set of closed subspace... |
| iscss 21726 | The predicate "is a closed... |
| cssi 21727 | Property of a closed subsp... |
| cssss 21728 | A closed subspace is a sub... |
| iscss2 21729 | It is sufficient to prove ... |
| ocvcss 21730 | The orthocomplement of any... |
| cssincl 21731 | The zero subspace is a clo... |
| css0 21732 | The zero subspace is a clo... |
| css1 21733 | The whole space is a close... |
| csslss 21734 | A closed subspace of a pre... |
| lsmcss 21735 | A subset of a pre-Hilbert ... |
| cssmre 21736 | The closed subspaces of a ... |
| mrccss 21737 | The Moore closure correspo... |
| thlval 21738 | Value of the Hilbert latti... |
| thlbas 21739 | Base set of the Hilbert la... |
| thlbasOLD 21740 | Obsolete proof of ~ thlbas... |
| thlle 21741 | Ordering on the Hilbert la... |
| thlleOLD 21742 | Obsolete proof of ~ thlle ... |
| thlleval 21743 | Ordering on the Hilbert la... |
| thloc 21744 | Orthocomplement on the Hil... |
| pjfval 21751 | The value of the projectio... |
| pjdm 21752 | A subspace is in the domai... |
| pjpm 21753 | The projection map is a pa... |
| pjfval2 21754 | Value of the projection ma... |
| pjval 21755 | Value of the projection ma... |
| pjdm2 21756 | A subspace is in the domai... |
| pjff 21757 | A projection is a linear o... |
| pjf 21758 | A projection is a function... |
| pjf2 21759 | A projection is a function... |
| pjfo 21760 | A projection is a surjecti... |
| pjcss 21761 | A projection subspace is a... |
| ocvpj 21762 | The orthocomplement of a p... |
| ishil 21763 | The predicate "is a Hilber... |
| ishil2 21764 | The predicate "is a Hilber... |
| isobs 21765 | The predicate "is an ortho... |
| obsip 21766 | The inner product of two e... |
| obsipid 21767 | A basis element has length... |
| obsrcl 21768 | Reverse closure for an ort... |
| obsss 21769 | An orthonormal basis is a ... |
| obsne0 21770 | A basis element is nonzero... |
| obsocv 21771 | An orthonormal basis has t... |
| obs2ocv 21772 | The double orthocomplement... |
| obselocv 21773 | A basis element is in the ... |
| obs2ss 21774 | A basis has no proper subs... |
| obslbs 21775 | An orthogonal basis is a l... |
| reldmdsmm 21778 | The direct sum is a well-b... |
| dsmmval 21779 | Value of the module direct... |
| dsmmbase 21780 | Base set of the module dir... |
| dsmmval2 21781 | Self-referential definitio... |
| dsmmbas2 21782 | Base set of the direct sum... |
| dsmmfi 21783 | For finite products, the d... |
| dsmmelbas 21784 | Membership in the finitely... |
| dsmm0cl 21785 | The all-zero vector is con... |
| dsmmacl 21786 | The finite hull is closed ... |
| prdsinvgd2 21787 | Negation of a single coord... |
| dsmmsubg 21788 | The finite hull of a produ... |
| dsmmlss 21789 | The finite hull of a produ... |
| dsmmlmod 21790 | The direct sum of a family... |
| frlmval 21793 | Value of the "free module"... |
| frlmlmod 21794 | The free module is a modul... |
| frlmpws 21795 | The free module as a restr... |
| frlmlss 21796 | The base set of the free m... |
| frlmpwsfi 21797 | The finite free module is ... |
| frlmsca 21798 | The ring of scalars of a f... |
| frlm0 21799 | Zero in a free module (rin... |
| frlmbas 21800 | Base set of the free modul... |
| frlmelbas 21801 | Membership in the base set... |
| frlmrcl 21802 | If a free module is inhabi... |
| frlmbasfsupp 21803 | Elements of the free modul... |
| frlmbasmap 21804 | Elements of the free modul... |
| frlmbasf 21805 | Elements of the free modul... |
| frlmlvec 21806 | The free module over a div... |
| frlmfibas 21807 | The base set of the finite... |
| elfrlmbasn0 21808 | If the dimension of a free... |
| frlmplusgval 21809 | Addition in a free module.... |
| frlmsubgval 21810 | Subtraction in a free modu... |
| frlmvscafval 21811 | Scalar multiplication in a... |
| frlmvplusgvalc 21812 | Coordinates of a sum with ... |
| frlmvscaval 21813 | Coordinates of a scalar mu... |
| frlmplusgvalb 21814 | Addition in a free module ... |
| frlmvscavalb 21815 | Scalar multiplication in a... |
| frlmvplusgscavalb 21816 | Addition combined with sca... |
| frlmgsum 21817 | Finite commutative sums in... |
| frlmsplit2 21818 | Restriction is homomorphic... |
| frlmsslss 21819 | A subset of a free module ... |
| frlmsslss2 21820 | A subset of a free module ... |
| frlmbas3 21821 | An element of the base set... |
| mpofrlmd 21822 | Elements of the free modul... |
| frlmip 21823 | The inner product of a fre... |
| frlmipval 21824 | The inner product of a fre... |
| frlmphllem 21825 | Lemma for ~ frlmphl . (Co... |
| frlmphl 21826 | Conditions for a free modu... |
| uvcfval 21829 | Value of the unit-vector g... |
| uvcval 21830 | Value of a single unit vec... |
| uvcvval 21831 | Value of a unit vector coo... |
| uvcvvcl 21832 | A coordinate of a unit vec... |
| uvcvvcl2 21833 | A unit vector coordinate i... |
| uvcvv1 21834 | The unit vector is one at ... |
| uvcvv0 21835 | The unit vector is zero at... |
| uvcff 21836 | Domain and codomain of the... |
| uvcf1 21837 | In a nonzero ring, each un... |
| uvcresum 21838 | Any element of a free modu... |
| frlmssuvc1 21839 | A scalar multiple of a uni... |
| frlmssuvc2 21840 | A nonzero scalar multiple ... |
| frlmsslsp 21841 | A subset of a free module ... |
| frlmlbs 21842 | The unit vectors comprise ... |
| frlmup1 21843 | Any assignment of unit vec... |
| frlmup2 21844 | The evaluation map has the... |
| frlmup3 21845 | The range of such an evalu... |
| frlmup4 21846 | Universal property of the ... |
| ellspd 21847 | The elements of the span o... |
| elfilspd 21848 | Simplified version of ~ el... |
| rellindf 21853 | The independent-family pre... |
| islinds 21854 | Property of an independent... |
| linds1 21855 | An independent set of vect... |
| linds2 21856 | An independent set of vect... |
| islindf 21857 | Property of an independent... |
| islinds2 21858 | Expanded property of an in... |
| islindf2 21859 | Property of an independent... |
| lindff 21860 | Functional property of a l... |
| lindfind 21861 | A linearly independent fam... |
| lindsind 21862 | A linearly independent set... |
| lindfind2 21863 | In a linearly independent ... |
| lindsind2 21864 | In a linearly independent ... |
| lindff1 21865 | A linearly independent fam... |
| lindfrn 21866 | The range of an independen... |
| f1lindf 21867 | Rearranging and deleting e... |
| lindfres 21868 | Any restriction of an inde... |
| lindsss 21869 | Any subset of an independe... |
| f1linds 21870 | A family constructed from ... |
| islindf3 21871 | In a nonzero ring, indepen... |
| lindfmm 21872 | Linear independence of a f... |
| lindsmm 21873 | Linear independence of a s... |
| lindsmm2 21874 | The monomorphic image of a... |
| lsslindf 21875 | Linear independence is unc... |
| lsslinds 21876 | Linear independence is unc... |
| islbs4 21877 | A basis is an independent ... |
| lbslinds 21878 | A basis is independent. (... |
| islinds3 21879 | A subset is linearly indep... |
| islinds4 21880 | A set is independent in a ... |
| lmimlbs 21881 | The isomorphic image of a ... |
| lmiclbs 21882 | Having a basis is an isomo... |
| islindf4 21883 | A family is independent if... |
| islindf5 21884 | A family is independent if... |
| indlcim 21885 | An independent, spanning f... |
| lbslcic 21886 | A module with a basis is i... |
| lmisfree 21887 | A module has a basis iff i... |
| lvecisfrlm 21888 | Every vector space is isom... |
| lmimco 21889 | The composition of two iso... |
| lmictra 21890 | Module isomorphism is tran... |
| uvcf1o 21891 | In a nonzero ring, the map... |
| uvcendim 21892 | In a nonzero ring, the num... |
| frlmisfrlm 21893 | A free module is isomorphi... |
| frlmiscvec 21894 | Every free module is isomo... |
| isassa 21901 | The properties of an assoc... |
| assalem 21902 | The properties of an assoc... |
| assaass 21903 | Left-associative property ... |
| assaassr 21904 | Right-associative property... |
| assalmod 21905 | An associative algebra is ... |
| assaring 21906 | An associative algebra is ... |
| assasca 21907 | The scalars of an associat... |
| assa2ass 21908 | Left- and right-associativ... |
| assa2ass2 21909 | Left- and right-associativ... |
| isassad 21910 | Sufficient condition for b... |
| issubassa3 21911 | A subring that is also a s... |
| issubassa 21912 | The subalgebras of an asso... |
| sraassab 21913 | A subring algebra is an as... |
| sraassa 21914 | The subring algebra over a... |
| sraassaOLD 21915 | Obsolete version of ~ sraa... |
| rlmassa 21916 | The ring module over a com... |
| assapropd 21917 | If two structures have the... |
| aspval 21918 | Value of the algebraic clo... |
| asplss 21919 | The algebraic span of a se... |
| aspid 21920 | The algebraic span of a su... |
| aspsubrg 21921 | The algebraic span of a se... |
| aspss 21922 | Span preserves subset orde... |
| aspssid 21923 | A set of vectors is a subs... |
| asclfval 21924 | Function value of the alge... |
| asclval 21925 | Value of a mapped algebra ... |
| asclfn 21926 | Unconditional functionalit... |
| asclf 21927 | The algebra scalars functi... |
| asclghm 21928 | The algebra scalars functi... |
| ascl0 21929 | The scalar 0 embedded into... |
| ascl1 21930 | The scalar 1 embedded into... |
| asclmul1 21931 | Left multiplication by a l... |
| asclmul2 21932 | Right multiplication by a ... |
| ascldimul 21933 | The algebra scalars functi... |
| asclinvg 21934 | The group inverse (negatio... |
| asclrhm 21935 | The algebra scalars functi... |
| rnascl 21936 | The set of lifted scalars ... |
| issubassa2 21937 | A subring of a unital alge... |
| rnasclsubrg 21938 | The scalar multiples of th... |
| rnasclmulcl 21939 | (Vector) multiplication is... |
| rnasclassa 21940 | The scalar multiples of th... |
| ressascl 21941 | The lifting of scalars is ... |
| asclpropd 21942 | If two structures have the... |
| aspval2 21943 | The algebraic closure is t... |
| assamulgscmlem1 21944 | Lemma 1 for ~ assamulgscm ... |
| assamulgscmlem2 21945 | Lemma for ~ assamulgscm (i... |
| assamulgscm 21946 | Exponentiation of a scalar... |
| asclmulg 21947 | Apply group multiplication... |
| zlmassa 21948 | The ` ZZ ` -module operati... |
| reldmpsr 21959 | The multivariate power ser... |
| psrval 21960 | Value of the multivariate ... |
| psrvalstr 21961 | The multivariate power ser... |
| psrbag 21962 | Elementhood in the set of ... |
| psrbagf 21963 | A finite bag is a function... |
| psrbagfsupp 21964 | Finite bags have finite su... |
| snifpsrbag 21965 | A bag containing one eleme... |
| fczpsrbag 21966 | The constant function equa... |
| psrbaglesupp 21967 | The support of a dominated... |
| psrbaglecl 21968 | The set of finite bags is ... |
| psrbagaddcl 21969 | The sum of two finite bags... |
| psrbagcon 21970 | The analogue of the statem... |
| psrbaglefi 21971 | There are finitely many ba... |
| psrbagconcl 21972 | The complement of a bag is... |
| psrbagleadd1 21973 | The analogue of " ` X <_ F... |
| psrbagconf1o 21974 | Bag complementation is a b... |
| gsumbagdiaglem 21975 | Lemma for ~ gsumbagdiag . ... |
| gsumbagdiag 21976 | Two-dimensional commutatio... |
| psrass1lem 21977 | A group sum commutation us... |
| psrbas 21978 | The base set of the multiv... |
| psrelbas 21979 | An element of the set of p... |
| psrelbasfun 21980 | An element of the set of p... |
| psrplusg 21981 | The addition operation of ... |
| psradd 21982 | The addition operation of ... |
| psraddcl 21983 | Closure of the power serie... |
| psraddclOLD 21984 | Obsolete version of ~ psra... |
| rhmpsrlem1 21985 | Lemma for ~ rhmpsr et al. ... |
| rhmpsrlem2 21986 | Lemma for ~ rhmpsr et al. ... |
| psrmulr 21987 | The multiplication operati... |
| psrmulfval 21988 | The multiplication operati... |
| psrmulval 21989 | The multiplication operati... |
| psrmulcllem 21990 | Closure of the power serie... |
| psrmulcl 21991 | Closure of the power serie... |
| psrsca 21992 | The scalar field of the mu... |
| psrvscafval 21993 | The scalar multiplication ... |
| psrvsca 21994 | The scalar multiplication ... |
| psrvscaval 21995 | The scalar multiplication ... |
| psrvscacl 21996 | Closure of the power serie... |
| psr0cl 21997 | The zero element of the ri... |
| psr0lid 21998 | The zero element of the ri... |
| psrnegcl 21999 | The negative function in t... |
| psrlinv 22000 | The negative function in t... |
| psrgrp 22001 | The ring of power series i... |
| psrgrpOLD 22002 | Obsolete proof of ~ psrgrp... |
| psr0 22003 | The zero element of the ri... |
| psrneg 22004 | The negative function of t... |
| psrlmod 22005 | The ring of power series i... |
| psr1cl 22006 | The identity element of th... |
| psrlidm 22007 | The identity element of th... |
| psrridm 22008 | The identity element of th... |
| psrass1 22009 | Associative identity for t... |
| psrdi 22010 | Distributive law for the r... |
| psrdir 22011 | Distributive law for the r... |
| psrass23l 22012 | Associative identity for t... |
| psrcom 22013 | Commutative law for the ri... |
| psrass23 22014 | Associative identities for... |
| psrring 22015 | The ring of power series i... |
| psr1 22016 | The identity element of th... |
| psrcrng 22017 | The ring of power series i... |
| psrassa 22018 | The ring of power series i... |
| resspsrbas 22019 | A restricted power series ... |
| resspsradd 22020 | A restricted power series ... |
| resspsrmul 22021 | A restricted power series ... |
| resspsrvsca 22022 | A restricted power series ... |
| subrgpsr 22023 | A subring of the base ring... |
| psrascl 22024 | Value of the scalar inject... |
| psrasclcl 22025 | A scalar is lifted into a ... |
| mvrfval 22026 | Value of the generating el... |
| mvrval 22027 | Value of the generating el... |
| mvrval2 22028 | Value of the generating el... |
| mvrid 22029 | The ` X i ` -th coefficien... |
| mvrf 22030 | The power series variable ... |
| mvrf1 22031 | The power series variable ... |
| mvrcl2 22032 | A power series variable is... |
| reldmmpl 22033 | The multivariate polynomia... |
| mplval 22034 | Value of the set of multiv... |
| mplbas 22035 | Base set of the set of mul... |
| mplelbas 22036 | Property of being a polyno... |
| mvrcl 22037 | A power series variable is... |
| mvrf2 22038 | The power series/polynomia... |
| mplrcl 22039 | Reverse closure for the po... |
| mplelsfi 22040 | A polynomial treated as a ... |
| mplval2 22041 | Self-referential expressio... |
| mplbasss 22042 | The set of polynomials is ... |
| mplelf 22043 | A polynomial is defined as... |
| mplsubglem 22044 | If ` A ` is an ideal of se... |
| mpllsslem 22045 | If ` A ` is an ideal of su... |
| mplsubglem2 22046 | Lemma for ~ mplsubg and ~ ... |
| mplsubg 22047 | The set of polynomials is ... |
| mpllss 22048 | The set of polynomials is ... |
| mplsubrglem 22049 | Lemma for ~ mplsubrg . (C... |
| mplsubrg 22050 | The set of polynomials is ... |
| mpl0 22051 | The zero polynomial. (Con... |
| mplplusg 22052 | Value of addition in a pol... |
| mplmulr 22053 | Value of multiplication in... |
| mpladd 22054 | The addition operation on ... |
| mplneg 22055 | The negative function on m... |
| mplmul 22056 | The multiplication operati... |
| mpl1 22057 | The identity element of th... |
| mplsca 22058 | The scalar field of a mult... |
| mplvsca2 22059 | The scalar multiplication ... |
| mplvsca 22060 | The scalar multiplication ... |
| mplvscaval 22061 | The scalar multiplication ... |
| mplgrp 22062 | The polynomial ring is a g... |
| mpllmod 22063 | The polynomial ring is a l... |
| mplring 22064 | The polynomial ring is a r... |
| mpllvec 22065 | The polynomial ring is a v... |
| mplcrng 22066 | The polynomial ring is a c... |
| mplassa 22067 | The polynomial ring is an ... |
| mplringd 22068 | The polynomial ring is a r... |
| mpllmodd 22069 | The polynomial ring is a l... |
| ressmplbas2 22070 | The base set of a restrict... |
| ressmplbas 22071 | A restricted polynomial al... |
| ressmpladd 22072 | A restricted polynomial al... |
| ressmplmul 22073 | A restricted polynomial al... |
| ressmplvsca 22074 | A restricted power series ... |
| subrgmpl 22075 | A subring of the base ring... |
| subrgmvr 22076 | The variables in a subring... |
| subrgmvrf 22077 | The variables in a polynom... |
| mplmon 22078 | A monomial is a polynomial... |
| mplmonmul 22079 | The product of two monomia... |
| mplcoe1 22080 | Decompose a polynomial int... |
| mplcoe3 22081 | Decompose a monomial in on... |
| mplcoe5lem 22082 | Lemma for ~ mplcoe4 . (Co... |
| mplcoe5 22083 | Decompose a monomial into ... |
| mplcoe2 22084 | Decompose a monomial into ... |
| mplbas2 22085 | An alternative expression ... |
| ltbval 22086 | Value of the well-order on... |
| ltbwe 22087 | The finite bag order is a ... |
| reldmopsr 22088 | Lemma for ordered power se... |
| opsrval 22089 | The value of the "ordered ... |
| opsrle 22090 | An alternative expression ... |
| opsrval2 22091 | Self-referential expressio... |
| opsrbaslem 22092 | Get a component of the ord... |
| opsrbaslemOLD 22093 | Obsolete version of ~ opsr... |
| opsrbas 22094 | The base set of the ordere... |
| opsrbasOLD 22095 | Obsolete version of ~ opsr... |
| opsrplusg 22096 | The addition operation of ... |
| opsrplusgOLD 22097 | Obsolete version of ~ opsr... |
| opsrmulr 22098 | The multiplication operati... |
| opsrmulrOLD 22099 | Obsolete version of ~ opsr... |
| opsrvsca 22100 | The scalar product operati... |
| opsrvscaOLD 22101 | Obsolete version of ~ opsr... |
| opsrsca 22102 | The scalar ring of the ord... |
| opsrscaOLD 22103 | Obsolete version of ~ opsr... |
| opsrtoslem1 22104 | Lemma for ~ opsrtos . (Co... |
| opsrtoslem2 22105 | Lemma for ~ opsrtos . (Co... |
| opsrtos 22106 | The ordered power series s... |
| opsrso 22107 | The ordered power series s... |
| opsrcrng 22108 | The ring of ordered power ... |
| opsrassa 22109 | The ring of ordered power ... |
| mplmon2 22110 | Express a scaled monomial.... |
| psrbag0 22111 | The empty bag is a bag. (... |
| psrbagsn 22112 | A singleton bag is a bag. ... |
| mplascl 22113 | Value of the scalar inject... |
| mplasclf 22114 | The scalar injection is a ... |
| subrgascl 22115 | The scalar injection funct... |
| subrgasclcl 22116 | The scalars in a polynomia... |
| mplmon2cl 22117 | A scaled monomial is a pol... |
| mplmon2mul 22118 | Product of scaled monomial... |
| mplind 22119 | Prove a property of polyno... |
| mplcoe4 22120 | Decompose a polynomial int... |
| evlslem4 22125 | The support of a tensor pr... |
| psrbagev1 22126 | A bag of multipliers provi... |
| psrbagev2 22127 | Closure of a sum using a b... |
| evlslem2 22128 | A linear function on the p... |
| evlslem3 22129 | Lemma for ~ evlseu . Poly... |
| evlslem6 22130 | Lemma for ~ evlseu . Fini... |
| evlslem1 22131 | Lemma for ~ evlseu , give ... |
| evlseu 22132 | For a given interpretation... |
| reldmevls 22133 | Well-behaved binary operat... |
| mpfrcl 22134 | Reverse closure for the se... |
| evlsval 22135 | Value of the polynomial ev... |
| evlsval2 22136 | Characterizing properties ... |
| evlsrhm 22137 | Polynomial evaluation is a... |
| evlssca 22138 | Polynomial evaluation maps... |
| evlsvar 22139 | Polynomial evaluation maps... |
| evlsgsumadd 22140 | Polynomial evaluation maps... |
| evlsgsummul 22141 | Polynomial evaluation maps... |
| evlspw 22142 | Polynomial evaluation for ... |
| evlsvarpw 22143 | Polynomial evaluation for ... |
| evlval 22144 | Value of the simple/same r... |
| evlrhm 22145 | The simple evaluation map ... |
| evlsscasrng 22146 | The evaluation of a scalar... |
| evlsca 22147 | Simple polynomial evaluati... |
| evlsvarsrng 22148 | The evaluation of the vari... |
| evlvar 22149 | Simple polynomial evaluati... |
| mpfconst 22150 | Constants are multivariate... |
| mpfproj 22151 | Projections are multivaria... |
| mpfsubrg 22152 | Polynomial functions are a... |
| mpff 22153 | Polynomial functions are f... |
| mpfaddcl 22154 | The sum of multivariate po... |
| mpfmulcl 22155 | The product of multivariat... |
| mpfind 22156 | Prove a property of polyno... |
| selvffval 22162 | Value of the "variable sel... |
| selvfval 22163 | Value of the "variable sel... |
| selvval 22164 | Value of the "variable sel... |
| reldmmhp 22166 | The domain of the homogene... |
| mhpfval 22167 | Value of the "homogeneous ... |
| mhpval 22168 | Value of the "homogeneous ... |
| ismhp 22169 | Property of being a homoge... |
| ismhp2 22170 | Deduce a homogeneous polyn... |
| ismhp3 22171 | A polynomial is homogeneou... |
| mhprcl 22172 | Reverse closure for homoge... |
| mhpmpl 22173 | A homogeneous polynomial i... |
| mhpdeg 22174 | All nonzero terms of a hom... |
| mhp0cl 22175 | The zero polynomial is hom... |
| mhpsclcl 22176 | A scalar (or constant) pol... |
| mhpvarcl 22177 | A power series variable is... |
| mhpmulcl 22178 | A product of homogeneous p... |
| mhppwdeg 22179 | Degree of a homogeneous po... |
| mhpaddcl 22180 | Homogeneous polynomials ar... |
| mhpinvcl 22181 | Homogeneous polynomials ar... |
| mhpsubg 22182 | Homogeneous polynomials fo... |
| mhpvscacl 22183 | Homogeneous polynomials ar... |
| mhplss 22184 | Homogeneous polynomials fo... |
| psdffval 22186 | Value of the power series ... |
| psdfval 22187 | Give a map between power s... |
| psdval 22188 | Evaluate the partial deriv... |
| psdcoef 22189 | Coefficient of a term of t... |
| psdcl 22190 | The derivative of a power ... |
| psdmplcl 22191 | The derivative of a polyno... |
| psdadd 22192 | The derivative of a sum is... |
| psdvsca 22193 | The derivative of a scaled... |
| psdmullem 22194 | Lemma for ~ psdmul . Tran... |
| psdmul 22195 | Product rule for power ser... |
| psd1 22196 | The derivative of one is z... |
| psdascl 22197 | The derivative of a consta... |
| psr1baslem 22209 | The set of finite bags on ... |
| psr1val 22210 | Value of the ring of univa... |
| psr1crng 22211 | The ring of univariate pow... |
| psr1assa 22212 | The ring of univariate pow... |
| psr1tos 22213 | The ordered power series s... |
| psr1bas2 22214 | The base set of the ring o... |
| psr1bas 22215 | The base set of the ring o... |
| vr1val 22216 | The value of the generator... |
| vr1cl2 22217 | The variable ` X ` is a me... |
| ply1val 22218 | The value of the set of un... |
| ply1bas 22219 | The value of the base set ... |
| ply1basOLD 22220 | Obsolete version of ~ ply1... |
| ply1lss 22221 | Univariate polynomials for... |
| ply1subrg 22222 | Univariate polynomials for... |
| ply1crng 22223 | The ring of univariate pol... |
| ply1assa 22224 | The ring of univariate pol... |
| psr1bascl 22225 | A univariate power series ... |
| psr1basf 22226 | Univariate power series ba... |
| ply1basf 22227 | Univariate polynomial base... |
| ply1bascl 22228 | A univariate polynomial is... |
| ply1bascl2 22229 | A univariate polynomial is... |
| coe1fval 22230 | Value of the univariate po... |
| coe1fv 22231 | Value of an evaluated coef... |
| fvcoe1 22232 | Value of a multivariate co... |
| coe1fval3 22233 | Univariate power series co... |
| coe1f2 22234 | Functionality of univariat... |
| coe1fval2 22235 | Univariate polynomial coef... |
| coe1f 22236 | Functionality of univariat... |
| coe1fvalcl 22237 | A coefficient of a univari... |
| coe1sfi 22238 | Finite support of univaria... |
| coe1fsupp 22239 | The coefficient vector of ... |
| mptcoe1fsupp 22240 | A mapping involving coeffi... |
| coe1ae0 22241 | The coefficient vector of ... |
| vr1cl 22242 | The generator of a univari... |
| opsr0 22243 | Zero in the ordered power ... |
| opsr1 22244 | One in the ordered power s... |
| psr1plusg 22245 | Value of addition in a uni... |
| psr1vsca 22246 | Value of scalar multiplica... |
| psr1mulr 22247 | Value of multiplication in... |
| ply1plusg 22248 | Value of addition in a uni... |
| ply1vsca 22249 | Value of scalar multiplica... |
| ply1mulr 22250 | Value of multiplication in... |
| ply1ass23l 22251 | Associative identity with ... |
| ressply1bas2 22252 | The base set of a restrict... |
| ressply1bas 22253 | A restricted polynomial al... |
| ressply1add 22254 | A restricted polynomial al... |
| ressply1mul 22255 | A restricted polynomial al... |
| ressply1vsca 22256 | A restricted power series ... |
| subrgply1 22257 | A subring of the base ring... |
| gsumply1subr 22258 | Evaluate a group sum in a ... |
| psrbaspropd 22259 | Property deduction for pow... |
| psrplusgpropd 22260 | Property deduction for pow... |
| mplbaspropd 22261 | Property deduction for pol... |
| psropprmul 22262 | Reversing multiplication i... |
| ply1opprmul 22263 | Reversing multiplication i... |
| 00ply1bas 22264 | Lemma for ~ ply1basfvi and... |
| ply1basfvi 22265 | Protection compatibility o... |
| ply1plusgfvi 22266 | Protection compatibility o... |
| ply1baspropd 22267 | Property deduction for uni... |
| ply1plusgpropd 22268 | Property deduction for uni... |
| opsrring 22269 | Ordered power series form ... |
| opsrlmod 22270 | Ordered power series form ... |
| psr1ring 22271 | Univariate power series fo... |
| ply1ring 22272 | Univariate polynomials for... |
| psr1lmod 22273 | Univariate power series fo... |
| psr1sca 22274 | Scalars of a univariate po... |
| psr1sca2 22275 | Scalars of a univariate po... |
| ply1lmod 22276 | Univariate polynomials for... |
| ply1sca 22277 | Scalars of a univariate po... |
| ply1sca2 22278 | Scalars of a univariate po... |
| ply1ascl0 22279 | The zero scalar as a polyn... |
| ply1ascl1 22280 | The multiplicative identit... |
| ply1mpl0 22281 | The univariate polynomial ... |
| ply10s0 22282 | Zero times a univariate po... |
| ply1mpl1 22283 | The univariate polynomial ... |
| ply1ascl 22284 | The univariate polynomial ... |
| subrg1ascl 22285 | The scalar injection funct... |
| subrg1asclcl 22286 | The scalars in a polynomia... |
| subrgvr1 22287 | The variables in a subring... |
| subrgvr1cl 22288 | The variables in a polynom... |
| coe1z 22289 | The coefficient vector of ... |
| coe1add 22290 | The coefficient vector of ... |
| coe1addfv 22291 | A particular coefficient o... |
| coe1subfv 22292 | A particular coefficient o... |
| coe1mul2lem1 22293 | An equivalence for ~ coe1m... |
| coe1mul2lem2 22294 | An equivalence for ~ coe1m... |
| coe1mul2 22295 | The coefficient vector of ... |
| coe1mul 22296 | The coefficient vector of ... |
| ply1moncl 22297 | Closure of the expression ... |
| ply1tmcl 22298 | Closure of the expression ... |
| coe1tm 22299 | Coefficient vector of a po... |
| coe1tmfv1 22300 | Nonzero coefficient of a p... |
| coe1tmfv2 22301 | Zero coefficient of a poly... |
| coe1tmmul2 22302 | Coefficient vector of a po... |
| coe1tmmul 22303 | Coefficient vector of a po... |
| coe1tmmul2fv 22304 | Function value of a right-... |
| coe1pwmul 22305 | Coefficient vector of a po... |
| coe1pwmulfv 22306 | Function value of a right-... |
| ply1scltm 22307 | A scalar is a term with ze... |
| coe1sclmul 22308 | Coefficient vector of a po... |
| coe1sclmulfv 22309 | A single coefficient of a ... |
| coe1sclmul2 22310 | Coefficient vector of a po... |
| ply1sclf 22311 | A scalar polynomial is a p... |
| ply1sclcl 22312 | The value of the algebra s... |
| coe1scl 22313 | Coefficient vector of a sc... |
| ply1sclid 22314 | Recover the base scalar fr... |
| ply1sclf1 22315 | The polynomial scalar func... |
| ply1scl0 22316 | The zero scalar is zero. ... |
| ply1scl0OLD 22317 | Obsolete version of ~ ply1... |
| ply1scln0 22318 | Nonzero scalars create non... |
| ply1scl1 22319 | The one scalar is the unit... |
| ply1scl1OLD 22320 | Obsolete version of ~ ply1... |
| ply1idvr1 22321 | The identity of a polynomi... |
| ply1idvr1OLD 22322 | Obsolete version of ~ ply1... |
| cply1mul 22323 | The product of two constan... |
| ply1coefsupp 22324 | The decomposition of a uni... |
| ply1coe 22325 | Decompose a univariate pol... |
| eqcoe1ply1eq 22326 | Two polynomials over the s... |
| ply1coe1eq 22327 | Two polynomials over the s... |
| cply1coe0 22328 | All but the first coeffici... |
| cply1coe0bi 22329 | A polynomial is constant (... |
| coe1fzgsumdlem 22330 | Lemma for ~ coe1fzgsumd (i... |
| coe1fzgsumd 22331 | Value of an evaluated coef... |
| ply1scleq 22332 | Equality of a constant pol... |
| ply1chr 22333 | The characteristic of a po... |
| gsumsmonply1 22334 | A finite group sum of scal... |
| gsummoncoe1 22335 | A coefficient of the polyn... |
| gsumply1eq 22336 | Two univariate polynomials... |
| lply1binom 22337 | The binomial theorem for l... |
| lply1binomsc 22338 | The binomial theorem for l... |
| ply1fermltlchr 22339 | Fermat's little theorem fo... |
| reldmevls1 22344 | Well-behaved binary operat... |
| ply1frcl 22345 | Reverse closure for the se... |
| evls1fval 22346 | Value of the univariate po... |
| evls1val 22347 | Value of the univariate po... |
| evls1rhmlem 22348 | Lemma for ~ evl1rhm and ~ ... |
| evls1rhm 22349 | Polynomial evaluation is a... |
| evls1sca 22350 | Univariate polynomial eval... |
| evls1gsumadd 22351 | Univariate polynomial eval... |
| evls1gsummul 22352 | Univariate polynomial eval... |
| evls1pw 22353 | Univariate polynomial eval... |
| evls1varpw 22354 | Univariate polynomial eval... |
| evl1fval 22355 | Value of the simple/same r... |
| evl1val 22356 | Value of the simple/same r... |
| evl1fval1lem 22357 | Lemma for ~ evl1fval1 . (... |
| evl1fval1 22358 | Value of the simple/same r... |
| evl1rhm 22359 | Polynomial evaluation is a... |
| fveval1fvcl 22360 | The function value of the ... |
| evl1sca 22361 | Polynomial evaluation maps... |
| evl1scad 22362 | Polynomial evaluation buil... |
| evl1var 22363 | Polynomial evaluation maps... |
| evl1vard 22364 | Polynomial evaluation buil... |
| evls1var 22365 | Univariate polynomial eval... |
| evls1scasrng 22366 | The evaluation of a scalar... |
| evls1varsrng 22367 | The evaluation of the vari... |
| evl1addd 22368 | Polynomial evaluation buil... |
| evl1subd 22369 | Polynomial evaluation buil... |
| evl1muld 22370 | Polynomial evaluation buil... |
| evl1vsd 22371 | Polynomial evaluation buil... |
| evl1expd 22372 | Polynomial evaluation buil... |
| pf1const 22373 | Constants are polynomial f... |
| pf1id 22374 | The identity is a polynomi... |
| pf1subrg 22375 | Polynomial functions are a... |
| pf1rcl 22376 | Reverse closure for the se... |
| pf1f 22377 | Polynomial functions are f... |
| mpfpf1 22378 | Convert a multivariate pol... |
| pf1mpf 22379 | Convert a univariate polyn... |
| pf1addcl 22380 | The sum of multivariate po... |
| pf1mulcl 22381 | The product of multivariat... |
| pf1ind 22382 | Prove a property of polyno... |
| evl1gsumdlem 22383 | Lemma for ~ evl1gsumd (ind... |
| evl1gsumd 22384 | Polynomial evaluation buil... |
| evl1gsumadd 22385 | Univariate polynomial eval... |
| evl1gsumaddval 22386 | Value of a univariate poly... |
| evl1gsummul 22387 | Univariate polynomial eval... |
| evl1varpw 22388 | Univariate polynomial eval... |
| evl1varpwval 22389 | Value of a univariate poly... |
| evl1scvarpw 22390 | Univariate polynomial eval... |
| evl1scvarpwval 22391 | Value of a univariate poly... |
| evl1gsummon 22392 | Value of a univariate poly... |
| evls1scafv 22393 | Value of the univariate po... |
| evls1expd 22394 | Univariate polynomial eval... |
| evls1varpwval 22395 | Univariate polynomial eval... |
| evls1fpws 22396 | Evaluation of a univariate... |
| ressply1evl 22397 | Evaluation of a univariate... |
| evls1addd 22398 | Univariate polynomial eval... |
| evls1muld 22399 | Univariate polynomial eval... |
| evls1vsca 22400 | Univariate polynomial eval... |
| asclply1subcl 22401 | Closure of the algebra sca... |
| evls1fvcl 22402 | Variant of ~ fveval1fvcl f... |
| evls1maprhm 22403 | The function ` F ` mapping... |
| evls1maplmhm 22404 | The function ` F ` mapping... |
| evls1maprnss 22405 | The function ` F ` mapping... |
| evl1maprhm 22406 | The function ` F ` mapping... |
| mhmcompl 22407 | The composition of a monoi... |
| mhmcoaddmpl 22408 | Show that the ring homomor... |
| rhmcomulmpl 22409 | Show that the ring homomor... |
| rhmmpl 22410 | Provide a ring homomorphis... |
| ply1vscl 22411 | Closure of scalar multipli... |
| mhmcoply1 22412 | The composition of a monoi... |
| rhmply1 22413 | Provide a ring homomorphis... |
| rhmply1vr1 22414 | A ring homomorphism betwee... |
| rhmply1vsca 22415 | Apply a ring homomorphism ... |
| rhmply1mon 22416 | Apply a ring homomorphism ... |
| mamufval 22419 | Functional value of the ma... |
| mamuval 22420 | Multiplication of two matr... |
| mamufv 22421 | A cell in the multiplicati... |
| mamudm 22422 | The domain of the matrix m... |
| mamufacex 22423 | Every solution of the equa... |
| mamures 22424 | Rows in a matrix product a... |
| grpvlinv 22425 | Tuple-wise left inverse in... |
| grpvrinv 22426 | Tuple-wise right inverse i... |
| ringvcl 22427 | Tuple-wise multiplication ... |
| mamucl 22428 | Operation closure of matri... |
| mamuass 22429 | Matrix multiplication is a... |
| mamudi 22430 | Matrix multiplication dist... |
| mamudir 22431 | Matrix multiplication dist... |
| mamuvs1 22432 | Matrix multiplication dist... |
| mamuvs2 22433 | Matrix multiplication dist... |
| matbas0pc 22436 | There is no matrix with a ... |
| matbas0 22437 | There is no matrix for a n... |
| matval 22438 | Value of the matrix algebr... |
| matrcl 22439 | Reverse closure for the ma... |
| matbas 22440 | The matrix ring has the sa... |
| matplusg 22441 | The matrix ring has the sa... |
| matsca 22442 | The matrix ring has the sa... |
| matscaOLD 22443 | Obsolete proof of ~ matsca... |
| matvsca 22444 | The matrix ring has the sa... |
| matvscaOLD 22445 | Obsolete proof of ~ matvsc... |
| mat0 22446 | The matrix ring has the sa... |
| matinvg 22447 | The matrix ring has the sa... |
| mat0op 22448 | Value of a zero matrix as ... |
| matsca2 22449 | The scalars of the matrix ... |
| matbas2 22450 | The base set of the matrix... |
| matbas2i 22451 | A matrix is a function. (... |
| matbas2d 22452 | The base set of the matrix... |
| eqmat 22453 | Two square matrices of the... |
| matecl 22454 | Each entry (according to W... |
| matecld 22455 | Each entry (according to W... |
| matplusg2 22456 | Addition in the matrix rin... |
| matvsca2 22457 | Scalar multiplication in t... |
| matlmod 22458 | The matrix ring is a linea... |
| matgrp 22459 | The matrix ring is a group... |
| matvscl 22460 | Closure of the scalar mult... |
| matsubg 22461 | The matrix ring has the sa... |
| matplusgcell 22462 | Addition in the matrix rin... |
| matsubgcell 22463 | Subtraction in the matrix ... |
| matinvgcell 22464 | Additive inversion in the ... |
| matvscacell 22465 | Scalar multiplication in t... |
| matgsum 22466 | Finite commutative sums in... |
| matmulr 22467 | Multiplication in the matr... |
| mamumat1cl 22468 | The identity matrix (as op... |
| mat1comp 22469 | The components of the iden... |
| mamulid 22470 | The identity matrix (as op... |
| mamurid 22471 | The identity matrix (as op... |
| matring 22472 | Existence of the matrix ri... |
| matassa 22473 | Existence of the matrix al... |
| matmulcell 22474 | Multiplication in the matr... |
| mpomatmul 22475 | Multiplication of two N x ... |
| mat1 22476 | Value of an identity matri... |
| mat1ov 22477 | Entries of an identity mat... |
| mat1bas 22478 | The identity matrix is a m... |
| matsc 22479 | The identity matrix multip... |
| ofco2 22480 | Distribution law for the f... |
| oftpos 22481 | The transposition of the v... |
| mattposcl 22482 | The transpose of a square ... |
| mattpostpos 22483 | The transpose of the trans... |
| mattposvs 22484 | The transposition of a mat... |
| mattpos1 22485 | The transposition of the i... |
| tposmap 22486 | The transposition of an I ... |
| mamutpos 22487 | Behavior of transposes in ... |
| mattposm 22488 | Multiplying two transposed... |
| matgsumcl 22489 | Closure of a group sum ove... |
| madetsumid 22490 | The identity summand in th... |
| matepmcl 22491 | Each entry of a matrix wit... |
| matepm2cl 22492 | Each entry of a matrix wit... |
| madetsmelbas 22493 | A summand of the determina... |
| madetsmelbas2 22494 | A summand of the determina... |
| mat0dimbas0 22495 | The empty set is the one a... |
| mat0dim0 22496 | The zero of the algebra of... |
| mat0dimid 22497 | The identity of the algebr... |
| mat0dimscm 22498 | The scalar multiplication ... |
| mat0dimcrng 22499 | The algebra of matrices wi... |
| mat1dimelbas 22500 | A matrix with dimension 1 ... |
| mat1dimbas 22501 | A matrix with dimension 1 ... |
| mat1dim0 22502 | The zero of the algebra of... |
| mat1dimid 22503 | The identity of the algebr... |
| mat1dimscm 22504 | The scalar multiplication ... |
| mat1dimmul 22505 | The ring multiplication in... |
| mat1dimcrng 22506 | The algebra of matrices wi... |
| mat1f1o 22507 | There is a 1-1 function fr... |
| mat1rhmval 22508 | The value of the ring homo... |
| mat1rhmelval 22509 | The value of the ring homo... |
| mat1rhmcl 22510 | The value of the ring homo... |
| mat1f 22511 | There is a function from a... |
| mat1ghm 22512 | There is a group homomorph... |
| mat1mhm 22513 | There is a monoid homomorp... |
| mat1rhm 22514 | There is a ring homomorphi... |
| mat1rngiso 22515 | There is a ring isomorphis... |
| mat1ric 22516 | A ring is isomorphic to th... |
| dmatval 22521 | The set of ` N ` x ` N ` d... |
| dmatel 22522 | A ` N ` x ` N ` diagonal m... |
| dmatmat 22523 | An ` N ` x ` N ` diagonal ... |
| dmatid 22524 | The identity matrix is a d... |
| dmatelnd 22525 | An extradiagonal entry of ... |
| dmatmul 22526 | The product of two diagona... |
| dmatsubcl 22527 | The difference of two diag... |
| dmatsgrp 22528 | The set of diagonal matric... |
| dmatmulcl 22529 | The product of two diagona... |
| dmatsrng 22530 | The set of diagonal matric... |
| dmatcrng 22531 | The subring of diagonal ma... |
| dmatscmcl 22532 | The multiplication of a di... |
| scmatval 22533 | The set of ` N ` x ` N ` s... |
| scmatel 22534 | An ` N ` x ` N ` scalar ma... |
| scmatscmid 22535 | A scalar matrix can be exp... |
| scmatscmide 22536 | An entry of a scalar matri... |
| scmatscmiddistr 22537 | Distributive law for scala... |
| scmatmat 22538 | An ` N ` x ` N ` scalar ma... |
| scmate 22539 | An entry of an ` N ` x ` N... |
| scmatmats 22540 | The set of an ` N ` x ` N ... |
| scmateALT 22541 | Alternate proof of ~ scmat... |
| scmatscm 22542 | The multiplication of a ma... |
| scmatid 22543 | The identity matrix is a s... |
| scmatdmat 22544 | A scalar matrix is a diago... |
| scmataddcl 22545 | The sum of two scalar matr... |
| scmatsubcl 22546 | The difference of two scal... |
| scmatmulcl 22547 | The product of two scalar ... |
| scmatsgrp 22548 | The set of scalar matrices... |
| scmatsrng 22549 | The set of scalar matrices... |
| scmatcrng 22550 | The subring of scalar matr... |
| scmatsgrp1 22551 | The set of scalar matrices... |
| scmatsrng1 22552 | The set of scalar matrices... |
| smatvscl 22553 | Closure of the scalar mult... |
| scmatlss 22554 | The set of scalar matrices... |
| scmatstrbas 22555 | The set of scalar matrices... |
| scmatrhmval 22556 | The value of the ring homo... |
| scmatrhmcl 22557 | The value of the ring homo... |
| scmatf 22558 | There is a function from a... |
| scmatfo 22559 | There is a function from a... |
| scmatf1 22560 | There is a 1-1 function fr... |
| scmatf1o 22561 | There is a bijection betwe... |
| scmatghm 22562 | There is a group homomorph... |
| scmatmhm 22563 | There is a monoid homomorp... |
| scmatrhm 22564 | There is a ring homomorphi... |
| scmatrngiso 22565 | There is a ring isomorphis... |
| scmatric 22566 | A ring is isomorphic to ev... |
| mat0scmat 22567 | The empty matrix over a ri... |
| mat1scmat 22568 | A 1-dimensional matrix ove... |
| mvmulfval 22571 | Functional value of the ma... |
| mvmulval 22572 | Multiplication of a vector... |
| mvmulfv 22573 | A cell/element in the vect... |
| mavmulval 22574 | Multiplication of a vector... |
| mavmulfv 22575 | A cell/element in the vect... |
| mavmulcl 22576 | Multiplication of an NxN m... |
| 1mavmul 22577 | Multiplication of the iden... |
| mavmulass 22578 | Associativity of the multi... |
| mavmuldm 22579 | The domain of the matrix v... |
| mavmulsolcl 22580 | Every solution of the equa... |
| mavmul0 22581 | Multiplication of a 0-dime... |
| mavmul0g 22582 | The result of the 0-dimens... |
| mvmumamul1 22583 | The multiplication of an M... |
| mavmumamul1 22584 | The multiplication of an N... |
| marrepfval 22589 | First substitution for the... |
| marrepval0 22590 | Second substitution for th... |
| marrepval 22591 | Third substitution for the... |
| marrepeval 22592 | An entry of a matrix with ... |
| marrepcl 22593 | Closure of the row replace... |
| marepvfval 22594 | First substitution for the... |
| marepvval0 22595 | Second substitution for th... |
| marepvval 22596 | Third substitution for the... |
| marepveval 22597 | An entry of a matrix with ... |
| marepvcl 22598 | Closure of the column repl... |
| ma1repvcl 22599 | Closure of the column repl... |
| ma1repveval 22600 | An entry of an identity ma... |
| mulmarep1el 22601 | Element by element multipl... |
| mulmarep1gsum1 22602 | The sum of element by elem... |
| mulmarep1gsum2 22603 | The sum of element by elem... |
| 1marepvmarrepid 22604 | Replacing the ith row by 0... |
| submabas 22607 | Any subset of the index se... |
| submafval 22608 | First substitution for a s... |
| submaval0 22609 | Second substitution for a ... |
| submaval 22610 | Third substitution for a s... |
| submaeval 22611 | An entry of a submatrix of... |
| 1marepvsma1 22612 | The submatrix of the ident... |
| mdetfval 22615 | First substitution for the... |
| mdetleib 22616 | Full substitution of our d... |
| mdetleib2 22617 | Leibniz' formula can also ... |
| nfimdetndef 22618 | The determinant is not def... |
| mdetfval1 22619 | First substitution of an a... |
| mdetleib1 22620 | Full substitution of an al... |
| mdet0pr 22621 | The determinant function f... |
| mdet0f1o 22622 | The determinant function f... |
| mdet0fv0 22623 | The determinant of the emp... |
| mdetf 22624 | Functionality of the deter... |
| mdetcl 22625 | The determinant evaluates ... |
| m1detdiag 22626 | The determinant of a 1-dim... |
| mdetdiaglem 22627 | Lemma for ~ mdetdiag . Pr... |
| mdetdiag 22628 | The determinant of a diago... |
| mdetdiagid 22629 | The determinant of a diago... |
| mdet1 22630 | The determinant of the ide... |
| mdetrlin 22631 | The determinant function i... |
| mdetrsca 22632 | The determinant function i... |
| mdetrsca2 22633 | The determinant function i... |
| mdetr0 22634 | The determinant of a matri... |
| mdet0 22635 | The determinant of the zer... |
| mdetrlin2 22636 | The determinant function i... |
| mdetralt 22637 | The determinant function i... |
| mdetralt2 22638 | The determinant function i... |
| mdetero 22639 | The determinant function i... |
| mdettpos 22640 | Determinant is invariant u... |
| mdetunilem1 22641 | Lemma for ~ mdetuni . (Co... |
| mdetunilem2 22642 | Lemma for ~ mdetuni . (Co... |
| mdetunilem3 22643 | Lemma for ~ mdetuni . (Co... |
| mdetunilem4 22644 | Lemma for ~ mdetuni . (Co... |
| mdetunilem5 22645 | Lemma for ~ mdetuni . (Co... |
| mdetunilem6 22646 | Lemma for ~ mdetuni . (Co... |
| mdetunilem7 22647 | Lemma for ~ mdetuni . (Co... |
| mdetunilem8 22648 | Lemma for ~ mdetuni . (Co... |
| mdetunilem9 22649 | Lemma for ~ mdetuni . (Co... |
| mdetuni0 22650 | Lemma for ~ mdetuni . (Co... |
| mdetuni 22651 | According to the definitio... |
| mdetmul 22652 | Multiplicativity of the de... |
| m2detleiblem1 22653 | Lemma 1 for ~ m2detleib . ... |
| m2detleiblem5 22654 | Lemma 5 for ~ m2detleib . ... |
| m2detleiblem6 22655 | Lemma 6 for ~ m2detleib . ... |
| m2detleiblem7 22656 | Lemma 7 for ~ m2detleib . ... |
| m2detleiblem2 22657 | Lemma 2 for ~ m2detleib . ... |
| m2detleiblem3 22658 | Lemma 3 for ~ m2detleib . ... |
| m2detleiblem4 22659 | Lemma 4 for ~ m2detleib . ... |
| m2detleib 22660 | Leibniz' Formula for 2x2-m... |
| mndifsplit 22665 | Lemma for ~ maducoeval2 . ... |
| madufval 22666 | First substitution for the... |
| maduval 22667 | Second substitution for th... |
| maducoeval 22668 | An entry of the adjunct (c... |
| maducoeval2 22669 | An entry of the adjunct (c... |
| maduf 22670 | Creating the adjunct of ma... |
| madutpos 22671 | The adjuct of a transposed... |
| madugsum 22672 | The determinant of a matri... |
| madurid 22673 | Multiplying a matrix with ... |
| madulid 22674 | Multiplying the adjunct of... |
| minmar1fval 22675 | First substitution for the... |
| minmar1val0 22676 | Second substitution for th... |
| minmar1val 22677 | Third substitution for the... |
| minmar1eval 22678 | An entry of a matrix for a... |
| minmar1marrep 22679 | The minor matrix is a spec... |
| minmar1cl 22680 | Closure of the row replace... |
| maducoevalmin1 22681 | The coefficients of an adj... |
| symgmatr01lem 22682 | Lemma for ~ symgmatr01 . ... |
| symgmatr01 22683 | Applying a permutation tha... |
| gsummatr01lem1 22684 | Lemma A for ~ gsummatr01 .... |
| gsummatr01lem2 22685 | Lemma B for ~ gsummatr01 .... |
| gsummatr01lem3 22686 | Lemma 1 for ~ gsummatr01 .... |
| gsummatr01lem4 22687 | Lemma 2 for ~ gsummatr01 .... |
| gsummatr01 22688 | Lemma 1 for ~ smadiadetlem... |
| marep01ma 22689 | Replacing a row of a squar... |
| smadiadetlem0 22690 | Lemma 0 for ~ smadiadet : ... |
| smadiadetlem1 22691 | Lemma 1 for ~ smadiadet : ... |
| smadiadetlem1a 22692 | Lemma 1a for ~ smadiadet :... |
| smadiadetlem2 22693 | Lemma 2 for ~ smadiadet : ... |
| smadiadetlem3lem0 22694 | Lemma 0 for ~ smadiadetlem... |
| smadiadetlem3lem1 22695 | Lemma 1 for ~ smadiadetlem... |
| smadiadetlem3lem2 22696 | Lemma 2 for ~ smadiadetlem... |
| smadiadetlem3 22697 | Lemma 3 for ~ smadiadet . ... |
| smadiadetlem4 22698 | Lemma 4 for ~ smadiadet . ... |
| smadiadet 22699 | The determinant of a subma... |
| smadiadetglem1 22700 | Lemma 1 for ~ smadiadetg .... |
| smadiadetglem2 22701 | Lemma 2 for ~ smadiadetg .... |
| smadiadetg 22702 | The determinant of a squar... |
| smadiadetg0 22703 | Lemma for ~ smadiadetr : v... |
| smadiadetr 22704 | The determinant of a squar... |
| invrvald 22705 | If a matrix multiplied wit... |
| matinv 22706 | The inverse of a matrix is... |
| matunit 22707 | A matrix is a unit in the ... |
| slesolvec 22708 | Every solution of a system... |
| slesolinv 22709 | The solution of a system o... |
| slesolinvbi 22710 | The solution of a system o... |
| slesolex 22711 | Every system of linear equ... |
| cramerimplem1 22712 | Lemma 1 for ~ cramerimp : ... |
| cramerimplem2 22713 | Lemma 2 for ~ cramerimp : ... |
| cramerimplem3 22714 | Lemma 3 for ~ cramerimp : ... |
| cramerimp 22715 | One direction of Cramer's ... |
| cramerlem1 22716 | Lemma 1 for ~ cramer . (C... |
| cramerlem2 22717 | Lemma 2 for ~ cramer . (C... |
| cramerlem3 22718 | Lemma 3 for ~ cramer . (C... |
| cramer0 22719 | Special case of Cramer's r... |
| cramer 22720 | Cramer's rule. According ... |
| pmatring 22721 | The set of polynomial matr... |
| pmatlmod 22722 | The set of polynomial matr... |
| pmatassa 22723 | The set of polynomial matr... |
| pmat0op 22724 | The zero polynomial matrix... |
| pmat1op 22725 | The identity polynomial ma... |
| pmat1ovd 22726 | Entries of the identity po... |
| pmat0opsc 22727 | The zero polynomial matrix... |
| pmat1opsc 22728 | The identity polynomial ma... |
| pmat1ovscd 22729 | Entries of the identity po... |
| pmatcoe1fsupp 22730 | For a polynomial matrix th... |
| 1pmatscmul 22731 | The scalar product of the ... |
| cpmat 22738 | Value of the constructor o... |
| cpmatpmat 22739 | A constant polynomial matr... |
| cpmatel 22740 | Property of a constant pol... |
| cpmatelimp 22741 | Implication of a set being... |
| cpmatel2 22742 | Another property of a cons... |
| cpmatelimp2 22743 | Another implication of a s... |
| 1elcpmat 22744 | The identity of the ring o... |
| cpmatacl 22745 | The set of all constant po... |
| cpmatinvcl 22746 | The set of all constant po... |
| cpmatmcllem 22747 | Lemma for ~ cpmatmcl . (C... |
| cpmatmcl 22748 | The set of all constant po... |
| cpmatsubgpmat 22749 | The set of all constant po... |
| cpmatsrgpmat 22750 | The set of all constant po... |
| 0elcpmat 22751 | The zero of the ring of al... |
| mat2pmatfval 22752 | Value of the matrix transf... |
| mat2pmatval 22753 | The result of a matrix tra... |
| mat2pmatvalel 22754 | A (matrix) element of the ... |
| mat2pmatbas 22755 | The result of a matrix tra... |
| mat2pmatbas0 22756 | The result of a matrix tra... |
| mat2pmatf 22757 | The matrix transformation ... |
| mat2pmatf1 22758 | The matrix transformation ... |
| mat2pmatghm 22759 | The transformation of matr... |
| mat2pmatmul 22760 | The transformation of matr... |
| mat2pmat1 22761 | The transformation of the ... |
| mat2pmatmhm 22762 | The transformation of matr... |
| mat2pmatrhm 22763 | The transformation of matr... |
| mat2pmatlin 22764 | The transformation of matr... |
| 0mat2pmat 22765 | The transformed zero matri... |
| idmatidpmat 22766 | The transformed identity m... |
| d0mat2pmat 22767 | The transformed empty set ... |
| d1mat2pmat 22768 | The transformation of a ma... |
| mat2pmatscmxcl 22769 | A transformed matrix multi... |
| m2cpm 22770 | The result of a matrix tra... |
| m2cpmf 22771 | The matrix transformation ... |
| m2cpmf1 22772 | The matrix transformation ... |
| m2cpmghm 22773 | The transformation of matr... |
| m2cpmmhm 22774 | The transformation of matr... |
| m2cpmrhm 22775 | The transformation of matr... |
| m2pmfzmap 22776 | The transformed values of ... |
| m2pmfzgsumcl 22777 | Closure of the sum of scal... |
| cpm2mfval 22778 | Value of the inverse matri... |
| cpm2mval 22779 | The result of an inverse m... |
| cpm2mvalel 22780 | A (matrix) element of the ... |
| cpm2mf 22781 | The inverse matrix transfo... |
| m2cpminvid 22782 | The inverse transformation... |
| m2cpminvid2lem 22783 | Lemma for ~ m2cpminvid2 . ... |
| m2cpminvid2 22784 | The transformation applied... |
| m2cpmfo 22785 | The matrix transformation ... |
| m2cpmf1o 22786 | The matrix transformation ... |
| m2cpmrngiso 22787 | The transformation of matr... |
| matcpmric 22788 | The ring of matrices over ... |
| m2cpminv 22789 | The inverse matrix transfo... |
| m2cpminv0 22790 | The inverse matrix transfo... |
| decpmatval0 22793 | The matrix consisting of t... |
| decpmatval 22794 | The matrix consisting of t... |
| decpmate 22795 | An entry of the matrix con... |
| decpmatcl 22796 | Closure of the decompositi... |
| decpmataa0 22797 | The matrix consisting of t... |
| decpmatfsupp 22798 | The mapping to the matrice... |
| decpmatid 22799 | The matrix consisting of t... |
| decpmatmullem 22800 | Lemma for ~ decpmatmul . ... |
| decpmatmul 22801 | The matrix consisting of t... |
| decpmatmulsumfsupp 22802 | Lemma 0 for ~ pm2mpmhm . ... |
| pmatcollpw1lem1 22803 | Lemma 1 for ~ pmatcollpw1 ... |
| pmatcollpw1lem2 22804 | Lemma 2 for ~ pmatcollpw1 ... |
| pmatcollpw1 22805 | Write a polynomial matrix ... |
| pmatcollpw2lem 22806 | Lemma for ~ pmatcollpw2 . ... |
| pmatcollpw2 22807 | Write a polynomial matrix ... |
| monmatcollpw 22808 | The matrix consisting of t... |
| pmatcollpwlem 22809 | Lemma for ~ pmatcollpw . ... |
| pmatcollpw 22810 | Write a polynomial matrix ... |
| pmatcollpwfi 22811 | Write a polynomial matrix ... |
| pmatcollpw3lem 22812 | Lemma for ~ pmatcollpw3 an... |
| pmatcollpw3 22813 | Write a polynomial matrix ... |
| pmatcollpw3fi 22814 | Write a polynomial matrix ... |
| pmatcollpw3fi1lem1 22815 | Lemma 1 for ~ pmatcollpw3f... |
| pmatcollpw3fi1lem2 22816 | Lemma 2 for ~ pmatcollpw3f... |
| pmatcollpw3fi1 22817 | Write a polynomial matrix ... |
| pmatcollpwscmatlem1 22818 | Lemma 1 for ~ pmatcollpwsc... |
| pmatcollpwscmatlem2 22819 | Lemma 2 for ~ pmatcollpwsc... |
| pmatcollpwscmat 22820 | Write a scalar matrix over... |
| pm2mpf1lem 22823 | Lemma for ~ pm2mpf1 . (Co... |
| pm2mpval 22824 | Value of the transformatio... |
| pm2mpfval 22825 | A polynomial matrix transf... |
| pm2mpcl 22826 | The transformation of poly... |
| pm2mpf 22827 | The transformation of poly... |
| pm2mpf1 22828 | The transformation of poly... |
| pm2mpcoe1 22829 | A coefficient of the polyn... |
| idpm2idmp 22830 | The transformation of the ... |
| mptcoe1matfsupp 22831 | The mapping extracting the... |
| mply1topmatcllem 22832 | Lemma for ~ mply1topmatcl ... |
| mply1topmatval 22833 | A polynomial over matrices... |
| mply1topmatcl 22834 | A polynomial over matrices... |
| mp2pm2mplem1 22835 | Lemma 1 for ~ mp2pm2mp . ... |
| mp2pm2mplem2 22836 | Lemma 2 for ~ mp2pm2mp . ... |
| mp2pm2mplem3 22837 | Lemma 3 for ~ mp2pm2mp . ... |
| mp2pm2mplem4 22838 | Lemma 4 for ~ mp2pm2mp . ... |
| mp2pm2mplem5 22839 | Lemma 5 for ~ mp2pm2mp . ... |
| mp2pm2mp 22840 | A polynomial over matrices... |
| pm2mpghmlem2 22841 | Lemma 2 for ~ pm2mpghm . ... |
| pm2mpghmlem1 22842 | Lemma 1 for pm2mpghm . (C... |
| pm2mpfo 22843 | The transformation of poly... |
| pm2mpf1o 22844 | The transformation of poly... |
| pm2mpghm 22845 | The transformation of poly... |
| pm2mpgrpiso 22846 | The transformation of poly... |
| pm2mpmhmlem1 22847 | Lemma 1 for ~ pm2mpmhm . ... |
| pm2mpmhmlem2 22848 | Lemma 2 for ~ pm2mpmhm . ... |
| pm2mpmhm 22849 | The transformation of poly... |
| pm2mprhm 22850 | The transformation of poly... |
| pm2mprngiso 22851 | The transformation of poly... |
| pmmpric 22852 | The ring of polynomial mat... |
| monmat2matmon 22853 | The transformation of a po... |
| pm2mp 22854 | The transformation of a su... |
| chmatcl 22857 | Closure of the characteris... |
| chmatval 22858 | The entries of the charact... |
| chpmatfval 22859 | Value of the characteristi... |
| chpmatval 22860 | The characteristic polynom... |
| chpmatply1 22861 | The characteristic polynom... |
| chpmatval2 22862 | The characteristic polynom... |
| chpmat0d 22863 | The characteristic polynom... |
| chpmat1dlem 22864 | Lemma for ~ chpmat1d . (C... |
| chpmat1d 22865 | The characteristic polynom... |
| chpdmatlem0 22866 | Lemma 0 for ~ chpdmat . (... |
| chpdmatlem1 22867 | Lemma 1 for ~ chpdmat . (... |
| chpdmatlem2 22868 | Lemma 2 for ~ chpdmat . (... |
| chpdmatlem3 22869 | Lemma 3 for ~ chpdmat . (... |
| chpdmat 22870 | The characteristic polynom... |
| chpscmat 22871 | The characteristic polynom... |
| chpscmat0 22872 | The characteristic polynom... |
| chpscmatgsumbin 22873 | The characteristic polynom... |
| chpscmatgsummon 22874 | The characteristic polynom... |
| chp0mat 22875 | The characteristic polynom... |
| chpidmat 22876 | The characteristic polynom... |
| chmaidscmat 22877 | The characteristic polynom... |
| fvmptnn04if 22878 | The function values of a m... |
| fvmptnn04ifa 22879 | The function value of a ma... |
| fvmptnn04ifb 22880 | The function value of a ma... |
| fvmptnn04ifc 22881 | The function value of a ma... |
| fvmptnn04ifd 22882 | The function value of a ma... |
| chfacfisf 22883 | The "characteristic factor... |
| chfacfisfcpmat 22884 | The "characteristic factor... |
| chfacffsupp 22885 | The "characteristic factor... |
| chfacfscmulcl 22886 | Closure of a scaled value ... |
| chfacfscmul0 22887 | A scaled value of the "cha... |
| chfacfscmulfsupp 22888 | A mapping of scaled values... |
| chfacfscmulgsum 22889 | Breaking up a sum of value... |
| chfacfpmmulcl 22890 | Closure of the value of th... |
| chfacfpmmul0 22891 | The value of the "characte... |
| chfacfpmmulfsupp 22892 | A mapping of values of the... |
| chfacfpmmulgsum 22893 | Breaking up a sum of value... |
| chfacfpmmulgsum2 22894 | Breaking up a sum of value... |
| cayhamlem1 22895 | Lemma 1 for ~ cayleyhamilt... |
| cpmadurid 22896 | The right-hand fundamental... |
| cpmidgsum 22897 | Representation of the iden... |
| cpmidgsumm2pm 22898 | Representation of the iden... |
| cpmidpmatlem1 22899 | Lemma 1 for ~ cpmidpmat . ... |
| cpmidpmatlem2 22900 | Lemma 2 for ~ cpmidpmat . ... |
| cpmidpmatlem3 22901 | Lemma 3 for ~ cpmidpmat . ... |
| cpmidpmat 22902 | Representation of the iden... |
| cpmadugsumlemB 22903 | Lemma B for ~ cpmadugsum .... |
| cpmadugsumlemC 22904 | Lemma C for ~ cpmadugsum .... |
| cpmadugsumlemF 22905 | Lemma F for ~ cpmadugsum .... |
| cpmadugsumfi 22906 | The product of the charact... |
| cpmadugsum 22907 | The product of the charact... |
| cpmidgsum2 22908 | Representation of the iden... |
| cpmidg2sum 22909 | Equality of two sums repre... |
| cpmadumatpolylem1 22910 | Lemma 1 for ~ cpmadumatpol... |
| cpmadumatpolylem2 22911 | Lemma 2 for ~ cpmadumatpol... |
| cpmadumatpoly 22912 | The product of the charact... |
| cayhamlem2 22913 | Lemma for ~ cayhamlem3 . ... |
| chcoeffeqlem 22914 | Lemma for ~ chcoeffeq . (... |
| chcoeffeq 22915 | The coefficients of the ch... |
| cayhamlem3 22916 | Lemma for ~ cayhamlem4 . ... |
| cayhamlem4 22917 | Lemma for ~ cayleyhamilton... |
| cayleyhamilton0 22918 | The Cayley-Hamilton theore... |
| cayleyhamilton 22919 | The Cayley-Hamilton theore... |
| cayleyhamiltonALT 22920 | Alternate proof of ~ cayle... |
| cayleyhamilton1 22921 | The Cayley-Hamilton theore... |
| istopg 22924 | Express the predicate " ` ... |
| istop2g 22925 | Express the predicate " ` ... |
| uniopn 22926 | The union of a subset of a... |
| iunopn 22927 | The indexed union of a sub... |
| inopn 22928 | The intersection of two op... |
| fitop 22929 | A topology is closed under... |
| fiinopn 22930 | The intersection of a none... |
| iinopn 22931 | The intersection of a none... |
| unopn 22932 | The union of two open sets... |
| 0opn 22933 | The empty set is an open s... |
| 0ntop 22934 | The empty set is not a top... |
| topopn 22935 | The underlying set of a to... |
| eltopss 22936 | A member of a topology is ... |
| riinopn 22937 | A finite indexed relative ... |
| rintopn 22938 | A finite relative intersec... |
| istopon 22941 | Property of being a topolo... |
| topontop 22942 | A topology on a given base... |
| toponuni 22943 | The base set of a topology... |
| topontopi 22944 | A topology on a given base... |
| toponunii 22945 | The base set of a topology... |
| toptopon 22946 | Alternative definition of ... |
| toptopon2 22947 | A topology is the same thi... |
| topontopon 22948 | A topology on a set is a t... |
| funtopon 22949 | The class ` TopOn ` is a f... |
| toponrestid 22950 | Given a topology on a set,... |
| toponsspwpw 22951 | The set of topologies on a... |
| dmtopon 22952 | The domain of ` TopOn ` is... |
| fntopon 22953 | The class ` TopOn ` is a f... |
| toprntopon 22954 | A topology is the same thi... |
| toponmax 22955 | The base set of a topology... |
| toponss 22956 | A member of a topology is ... |
| toponcom 22957 | If ` K ` is a topology on ... |
| toponcomb 22958 | Biconditional form of ~ to... |
| topgele 22959 | The topologies over the sa... |
| topsn 22960 | The only topology on a sin... |
| istps 22963 | Express the predicate "is ... |
| istps2 22964 | Express the predicate "is ... |
| tpsuni 22965 | The base set of a topologi... |
| tpstop 22966 | The topology extractor on ... |
| tpspropd 22967 | A topological space depend... |
| tpsprop2d 22968 | A topological space depend... |
| topontopn 22969 | Express the predicate "is ... |
| tsettps 22970 | If the topology component ... |
| istpsi 22971 | Properties that determine ... |
| eltpsg 22972 | Properties that determine ... |
| eltpsgOLD 22973 | Obsolete version of ~ eltp... |
| eltpsi 22974 | Properties that determine ... |
| isbasisg 22977 | Express the predicate "the... |
| isbasis2g 22978 | Express the predicate "the... |
| isbasis3g 22979 | Express the predicate "the... |
| basis1 22980 | Property of a basis. (Con... |
| basis2 22981 | Property of a basis. (Con... |
| fiinbas 22982 | If a set is closed under f... |
| basdif0 22983 | A basis is not affected by... |
| baspartn 22984 | A disjoint system of sets ... |
| tgval 22985 | The topology generated by ... |
| tgval2 22986 | Definition of a topology g... |
| eltg 22987 | Membership in a topology g... |
| eltg2 22988 | Membership in a topology g... |
| eltg2b 22989 | Membership in a topology g... |
| eltg4i 22990 | An open set in a topology ... |
| eltg3i 22991 | The union of a set of basi... |
| eltg3 22992 | Membership in a topology g... |
| tgval3 22993 | Alternate expression for t... |
| tg1 22994 | Property of a member of a ... |
| tg2 22995 | Property of a member of a ... |
| bastg 22996 | A member of a basis is a s... |
| unitg 22997 | The topology generated by ... |
| tgss 22998 | Subset relation for genera... |
| tgcl 22999 | Show that a basis generate... |
| tgclb 23000 | The property ~ tgcl can be... |
| tgtopon 23001 | A basis generates a topolo... |
| topbas 23002 | A topology is its own basi... |
| tgtop 23003 | A topology is its own basi... |
| eltop 23004 | Membership in a topology, ... |
| eltop2 23005 | Membership in a topology. ... |
| eltop3 23006 | Membership in a topology. ... |
| fibas 23007 | A collection of finite int... |
| tgdom 23008 | A space has no more open s... |
| tgiun 23009 | The indexed union of a set... |
| tgidm 23010 | The topology generator fun... |
| bastop 23011 | Two ways to express that a... |
| tgtop11 23012 | The topology generation fu... |
| 0top 23013 | The singleton of the empty... |
| en1top 23014 | ` { (/) } ` is the only to... |
| en2top 23015 | If a topology has two elem... |
| tgss3 23016 | A criterion for determinin... |
| tgss2 23017 | A criterion for determinin... |
| basgen 23018 | Given a topology ` J ` , s... |
| basgen2 23019 | Given a topology ` J ` , s... |
| 2basgen 23020 | Conditions that determine ... |
| tgfiss 23021 | If a subbase is included i... |
| tgdif0 23022 | A generated topology is no... |
| bastop1 23023 | A subset of a topology is ... |
| bastop2 23024 | A version of ~ bastop1 tha... |
| distop 23025 | The discrete topology on a... |
| topnex 23026 | The class of all topologie... |
| distopon 23027 | The discrete topology on a... |
| sn0topon 23028 | The singleton of the empty... |
| sn0top 23029 | The singleton of the empty... |
| indislem 23030 | A lemma to eliminate some ... |
| indistopon 23031 | The indiscrete topology on... |
| indistop 23032 | The indiscrete topology on... |
| indisuni 23033 | The base set of the indisc... |
| fctop 23034 | The finite complement topo... |
| fctop2 23035 | The finite complement topo... |
| cctop 23036 | The countable complement t... |
| ppttop 23037 | The particular point topol... |
| pptbas 23038 | The particular point topol... |
| epttop 23039 | The excluded point topolog... |
| indistpsx 23040 | The indiscrete topology on... |
| indistps 23041 | The indiscrete topology on... |
| indistps2 23042 | The indiscrete topology on... |
| indistpsALT 23043 | The indiscrete topology on... |
| indistpsALTOLD 23044 | Obsolete version of ~ indi... |
| indistps2ALT 23045 | The indiscrete topology on... |
| distps 23046 | The discrete topology on a... |
| fncld 23053 | The closed-set generator i... |
| cldval 23054 | The set of closed sets of ... |
| ntrfval 23055 | The interior function on t... |
| clsfval 23056 | The closure function on th... |
| cldrcl 23057 | Reverse closure of the clo... |
| iscld 23058 | The predicate "the class `... |
| iscld2 23059 | A subset of the underlying... |
| cldss 23060 | A closed set is a subset o... |
| cldss2 23061 | The set of closed sets is ... |
| cldopn 23062 | The complement of a closed... |
| isopn2 23063 | A subset of the underlying... |
| opncld 23064 | The complement of an open ... |
| difopn 23065 | The difference of a closed... |
| topcld 23066 | The underlying set of a to... |
| ntrval 23067 | The interior of a subset o... |
| clsval 23068 | The closure of a subset of... |
| 0cld 23069 | The empty set is closed. ... |
| iincld 23070 | The indexed intersection o... |
| intcld 23071 | The intersection of a set ... |
| uncld 23072 | The union of two closed se... |
| cldcls 23073 | A closed subset equals its... |
| incld 23074 | The intersection of two cl... |
| riincld 23075 | An indexed relative inters... |
| iuncld 23076 | A finite indexed union of ... |
| unicld 23077 | A finite union of closed s... |
| clscld 23078 | The closure of a subset of... |
| clsf 23079 | The closure function is a ... |
| ntropn 23080 | The interior of a subset o... |
| clsval2 23081 | Express closure in terms o... |
| ntrval2 23082 | Interior expressed in term... |
| ntrdif 23083 | An interior of a complemen... |
| clsdif 23084 | A closure of a complement ... |
| clsss 23085 | Subset relationship for cl... |
| ntrss 23086 | Subset relationship for in... |
| sscls 23087 | A subset of a topology's u... |
| ntrss2 23088 | A subset includes its inte... |
| ssntr 23089 | An open subset of a set is... |
| clsss3 23090 | The closure of a subset of... |
| ntrss3 23091 | The interior of a subset o... |
| ntrin 23092 | A pairwise intersection of... |
| cmclsopn 23093 | The complement of a closur... |
| cmntrcld 23094 | The complement of an inter... |
| iscld3 23095 | A subset is closed iff it ... |
| iscld4 23096 | A subset is closed iff it ... |
| isopn3 23097 | A subset is open iff it eq... |
| clsidm 23098 | The closure operation is i... |
| ntridm 23099 | The interior operation is ... |
| clstop 23100 | The closure of a topology'... |
| ntrtop 23101 | The interior of a topology... |
| 0ntr 23102 | A subset with an empty int... |
| clsss2 23103 | If a subset is included in... |
| elcls 23104 | Membership in a closure. ... |
| elcls2 23105 | Membership in a closure. ... |
| clsndisj 23106 | Any open set containing a ... |
| ntrcls0 23107 | A subset whose closure has... |
| ntreq0 23108 | Two ways to say that a sub... |
| cldmre 23109 | The closed sets of a topol... |
| mrccls 23110 | Moore closure generalizes ... |
| cls0 23111 | The closure of the empty s... |
| ntr0 23112 | The interior of the empty ... |
| isopn3i 23113 | An open subset equals its ... |
| elcls3 23114 | Membership in a closure in... |
| opncldf1 23115 | A bijection useful for con... |
| opncldf2 23116 | The values of the open-clo... |
| opncldf3 23117 | The values of the converse... |
| isclo 23118 | A set ` A ` is clopen iff ... |
| isclo2 23119 | A set ` A ` is clopen iff ... |
| discld 23120 | The open sets of a discret... |
| sn0cld 23121 | The closed sets of the top... |
| indiscld 23122 | The closed sets of an indi... |
| mretopd 23123 | A Moore collection which i... |
| toponmre 23124 | The topologies over a give... |
| cldmreon 23125 | The closed sets of a topol... |
| iscldtop 23126 | A family is the closed set... |
| mreclatdemoBAD 23127 | The closed subspaces of a ... |
| neifval 23130 | Value of the neighborhood ... |
| neif 23131 | The neighborhood function ... |
| neiss2 23132 | A set with a neighborhood ... |
| neival 23133 | Value of the set of neighb... |
| isnei 23134 | The predicate "the class `... |
| neiint 23135 | An intuitive definition of... |
| isneip 23136 | The predicate "the class `... |
| neii1 23137 | A neighborhood is included... |
| neisspw 23138 | The neighborhoods of any s... |
| neii2 23139 | Property of a neighborhood... |
| neiss 23140 | Any neighborhood of a set ... |
| ssnei 23141 | A set is included in any o... |
| elnei 23142 | A point belongs to any of ... |
| 0nnei 23143 | The empty set is not a nei... |
| neips 23144 | A neighborhood of a set is... |
| opnneissb 23145 | An open set is a neighborh... |
| opnssneib 23146 | Any superset of an open se... |
| ssnei2 23147 | Any subset ` M ` of ` X ` ... |
| neindisj 23148 | Any neighborhood of an ele... |
| opnneiss 23149 | An open set is a neighborh... |
| opnneip 23150 | An open set is a neighborh... |
| opnnei 23151 | A set is open iff it is a ... |
| tpnei 23152 | The underlying set of a to... |
| neiuni 23153 | The union of the neighborh... |
| neindisj2 23154 | A point ` P ` belongs to t... |
| topssnei 23155 | A finer topology has more ... |
| innei 23156 | The intersection of two ne... |
| opnneiid 23157 | Only an open set is a neig... |
| neissex 23158 | For any neighborhood ` N `... |
| 0nei 23159 | The empty set is a neighbo... |
| neipeltop 23160 | Lemma for ~ neiptopreu . ... |
| neiptopuni 23161 | Lemma for ~ neiptopreu . ... |
| neiptoptop 23162 | Lemma for ~ neiptopreu . ... |
| neiptopnei 23163 | Lemma for ~ neiptopreu . ... |
| neiptopreu 23164 | If, to each element ` P ` ... |
| lpfval 23169 | The limit point function o... |
| lpval 23170 | The set of limit points of... |
| islp 23171 | The predicate "the class `... |
| lpsscls 23172 | The limit points of a subs... |
| lpss 23173 | The limit points of a subs... |
| lpdifsn 23174 | ` P ` is a limit point of ... |
| lpss3 23175 | Subset relationship for li... |
| islp2 23176 | The predicate " ` P ` is a... |
| islp3 23177 | The predicate " ` P ` is a... |
| maxlp 23178 | A point is a limit point o... |
| clslp 23179 | The closure of a subset of... |
| islpi 23180 | A point belonging to a set... |
| cldlp 23181 | A subset of a topological ... |
| isperf 23182 | Definition of a perfect sp... |
| isperf2 23183 | Definition of a perfect sp... |
| isperf3 23184 | A perfect space is a topol... |
| perflp 23185 | The limit points of a perf... |
| perfi 23186 | Property of a perfect spac... |
| perftop 23187 | A perfect space is a topol... |
| restrcl 23188 | Reverse closure for the su... |
| restbas 23189 | A subspace topology basis ... |
| tgrest 23190 | A subspace can be generate... |
| resttop 23191 | A subspace topology is a t... |
| resttopon 23192 | A subspace topology is a t... |
| restuni 23193 | The underlying set of a su... |
| stoig 23194 | The topological space buil... |
| restco 23195 | Composition of subspaces. ... |
| restabs 23196 | Equivalence of being a sub... |
| restin 23197 | When the subspace region i... |
| restuni2 23198 | The underlying set of a su... |
| resttopon2 23199 | The underlying set of a su... |
| rest0 23200 | The subspace topology indu... |
| restsn 23201 | The only subspace topology... |
| restsn2 23202 | The subspace topology indu... |
| restcld 23203 | A closed set of a subspace... |
| restcldi 23204 | A closed set is closed in ... |
| restcldr 23205 | A set which is closed in t... |
| restopnb 23206 | If ` B ` is an open subset... |
| ssrest 23207 | If ` K ` is a finer topolo... |
| restopn2 23208 | If ` A ` is open, then ` B... |
| restdis 23209 | A subspace of a discrete t... |
| restfpw 23210 | The restriction of the set... |
| neitr 23211 | The neighborhood of a trac... |
| restcls 23212 | A closure in a subspace to... |
| restntr 23213 | An interior in a subspace ... |
| restlp 23214 | The limit points of a subs... |
| restperf 23215 | Perfection of a subspace. ... |
| perfopn 23216 | An open subset of a perfec... |
| resstopn 23217 | The topology of a restrict... |
| resstps 23218 | A restricted topological s... |
| ordtbaslem 23219 | Lemma for ~ ordtbas . In ... |
| ordtval 23220 | Value of the order topolog... |
| ordtuni 23221 | Value of the order topolog... |
| ordtbas2 23222 | Lemma for ~ ordtbas . (Co... |
| ordtbas 23223 | In a total order, the fini... |
| ordttopon 23224 | Value of the order topolog... |
| ordtopn1 23225 | An upward ray ` ( P , +oo ... |
| ordtopn2 23226 | A downward ray ` ( -oo , P... |
| ordtopn3 23227 | An open interval ` ( A , B... |
| ordtcld1 23228 | A downward ray ` ( -oo , P... |
| ordtcld2 23229 | An upward ray ` [ P , +oo ... |
| ordtcld3 23230 | A closed interval ` [ A , ... |
| ordttop 23231 | The order topology is a to... |
| ordtcnv 23232 | The order dual generates t... |
| ordtrest 23233 | The subspace topology of a... |
| ordtrest2lem 23234 | Lemma for ~ ordtrest2 . (... |
| ordtrest2 23235 | An interval-closed set ` A... |
| letopon 23236 | The topology of the extend... |
| letop 23237 | The topology of the extend... |
| letopuni 23238 | The topology of the extend... |
| xrstopn 23239 | The topology component of ... |
| xrstps 23240 | The extended real number s... |
| leordtvallem1 23241 | Lemma for ~ leordtval . (... |
| leordtvallem2 23242 | Lemma for ~ leordtval . (... |
| leordtval2 23243 | The topology of the extend... |
| leordtval 23244 | The topology of the extend... |
| iccordt 23245 | A closed interval is close... |
| iocpnfordt 23246 | An unbounded above open in... |
| icomnfordt 23247 | An unbounded above open in... |
| iooordt 23248 | An open interval is open i... |
| reordt 23249 | The real numbers are an op... |
| lecldbas 23250 | The set of closed interval... |
| pnfnei 23251 | A neighborhood of ` +oo ` ... |
| mnfnei 23252 | A neighborhood of ` -oo ` ... |
| ordtrestixx 23253 | The restriction of the les... |
| ordtresticc 23254 | The restriction of the les... |
| lmrel 23261 | The topological space conv... |
| lmrcl 23262 | Reverse closure for the co... |
| lmfval 23263 | The relation "sequence ` f... |
| cnfval 23264 | The set of all continuous ... |
| cnpfval 23265 | The function mapping the p... |
| iscn 23266 | The predicate "the class `... |
| cnpval 23267 | The set of all functions f... |
| iscnp 23268 | The predicate "the class `... |
| iscn2 23269 | The predicate "the class `... |
| iscnp2 23270 | The predicate "the class `... |
| cntop1 23271 | Reverse closure for a cont... |
| cntop2 23272 | Reverse closure for a cont... |
| cnptop1 23273 | Reverse closure for a func... |
| cnptop2 23274 | Reverse closure for a func... |
| iscnp3 23275 | The predicate "the class `... |
| cnprcl 23276 | Reverse closure for a func... |
| cnf 23277 | A continuous function is a... |
| cnpf 23278 | A continuous function at p... |
| cnpcl 23279 | The value of a continuous ... |
| cnf2 23280 | A continuous function is a... |
| cnpf2 23281 | A continuous function at p... |
| cnprcl2 23282 | Reverse closure for a func... |
| tgcn 23283 | The continuity predicate w... |
| tgcnp 23284 | The "continuous at a point... |
| subbascn 23285 | The continuity predicate w... |
| ssidcn 23286 | The identity function is a... |
| cnpimaex 23287 | Property of a function con... |
| idcn 23288 | A restricted identity func... |
| lmbr 23289 | Express the binary relatio... |
| lmbr2 23290 | Express the binary relatio... |
| lmbrf 23291 | Express the binary relatio... |
| lmconst 23292 | A constant sequence conver... |
| lmcvg 23293 | Convergence property of a ... |
| iscnp4 23294 | The predicate "the class `... |
| cnpnei 23295 | A condition for continuity... |
| cnima 23296 | An open subset of the codo... |
| cnco 23297 | The composition of two con... |
| cnpco 23298 | The composition of a funct... |
| cnclima 23299 | A closed subset of the cod... |
| iscncl 23300 | A characterization of a co... |
| cncls2i 23301 | Property of the preimage o... |
| cnntri 23302 | Property of the preimage o... |
| cnclsi 23303 | Property of the image of a... |
| cncls2 23304 | Continuity in terms of clo... |
| cncls 23305 | Continuity in terms of clo... |
| cnntr 23306 | Continuity in terms of int... |
| cnss1 23307 | If the topology ` K ` is f... |
| cnss2 23308 | If the topology ` K ` is f... |
| cncnpi 23309 | A continuous function is c... |
| cnsscnp 23310 | The set of continuous func... |
| cncnp 23311 | A continuous function is c... |
| cncnp2 23312 | A continuous function is c... |
| cnnei 23313 | Continuity in terms of nei... |
| cnconst2 23314 | A constant function is con... |
| cnconst 23315 | A constant function is con... |
| cnrest 23316 | Continuity of a restrictio... |
| cnrest2 23317 | Equivalence of continuity ... |
| cnrest2r 23318 | Equivalence of continuity ... |
| cnpresti 23319 | One direction of ~ cnprest... |
| cnprest 23320 | Equivalence of continuity ... |
| cnprest2 23321 | Equivalence of point-conti... |
| cndis 23322 | Every function is continuo... |
| cnindis 23323 | Every function is continuo... |
| cnpdis 23324 | If ` A ` is an isolated po... |
| paste 23325 | Pasting lemma. If ` A ` a... |
| lmfpm 23326 | If ` F ` converges, then `... |
| lmfss 23327 | Inclusion of a function ha... |
| lmcl 23328 | Closure of a limit. (Cont... |
| lmss 23329 | Limit on a subspace. (Con... |
| sslm 23330 | A finer topology has fewer... |
| lmres 23331 | A function converges iff i... |
| lmff 23332 | If ` F ` converges, there ... |
| lmcls 23333 | Any convergent sequence of... |
| lmcld 23334 | Any convergent sequence of... |
| lmcnp 23335 | The image of a convergent ... |
| lmcn 23336 | The image of a convergent ... |
| ist0 23351 | The predicate "is a T_0 sp... |
| ist1 23352 | The predicate "is a T_1 sp... |
| ishaus 23353 | The predicate "is a Hausdo... |
| iscnrm 23354 | The property of being comp... |
| t0sep 23355 | Any two topologically indi... |
| t0dist 23356 | Any two distinct points in... |
| t1sncld 23357 | In a T_1 space, singletons... |
| t1ficld 23358 | In a T_1 space, finite set... |
| hausnei 23359 | Neighborhood property of a... |
| t0top 23360 | A T_0 space is a topologic... |
| t1top 23361 | A T_1 space is a topologic... |
| haustop 23362 | A Hausdorff space is a top... |
| isreg 23363 | The predicate "is a regula... |
| regtop 23364 | A regular space is a topol... |
| regsep 23365 | In a regular space, every ... |
| isnrm 23366 | The predicate "is a normal... |
| nrmtop 23367 | A normal space is a topolo... |
| cnrmtop 23368 | A completely normal space ... |
| iscnrm2 23369 | The property of being comp... |
| ispnrm 23370 | The property of being perf... |
| pnrmnrm 23371 | A perfectly normal space i... |
| pnrmtop 23372 | A perfectly normal space i... |
| pnrmcld 23373 | A closed set in a perfectl... |
| pnrmopn 23374 | An open set in a perfectly... |
| ist0-2 23375 | The predicate "is a T_0 sp... |
| ist0-3 23376 | The predicate "is a T_0 sp... |
| cnt0 23377 | The preimage of a T_0 topo... |
| ist1-2 23378 | An alternate characterizat... |
| t1t0 23379 | A T_1 space is a T_0 space... |
| ist1-3 23380 | A space is T_1 iff every p... |
| cnt1 23381 | The preimage of a T_1 topo... |
| ishaus2 23382 | Express the predicate " ` ... |
| haust1 23383 | A Hausdorff space is a T_1... |
| hausnei2 23384 | The Hausdorff condition st... |
| cnhaus 23385 | The preimage of a Hausdorf... |
| nrmsep3 23386 | In a normal space, given a... |
| nrmsep2 23387 | In a normal space, any two... |
| nrmsep 23388 | In a normal space, disjoin... |
| isnrm2 23389 | An alternate characterizat... |
| isnrm3 23390 | A topological space is nor... |
| cnrmi 23391 | A subspace of a completely... |
| cnrmnrm 23392 | A completely normal space ... |
| restcnrm 23393 | A subspace of a completely... |
| resthauslem 23394 | Lemma for ~ resthaus and s... |
| lpcls 23395 | The limit points of the cl... |
| perfcls 23396 | A subset of a perfect spac... |
| restt0 23397 | A subspace of a T_0 topolo... |
| restt1 23398 | A subspace of a T_1 topolo... |
| resthaus 23399 | A subspace of a Hausdorff ... |
| t1sep2 23400 | Any two points in a T_1 sp... |
| t1sep 23401 | Any two distinct points in... |
| sncld 23402 | A singleton is closed in a... |
| sshauslem 23403 | Lemma for ~ sshaus and sim... |
| sst0 23404 | A topology finer than a T_... |
| sst1 23405 | A topology finer than a T_... |
| sshaus 23406 | A topology finer than a Ha... |
| regsep2 23407 | In a regular space, a clos... |
| isreg2 23408 | A topological space is reg... |
| dnsconst 23409 | If a continuous mapping to... |
| ordtt1 23410 | The order topology is T_1 ... |
| lmmo 23411 | A sequence in a Hausdorff ... |
| lmfun 23412 | The convergence relation i... |
| dishaus 23413 | A discrete topology is Hau... |
| ordthauslem 23414 | Lemma for ~ ordthaus . (C... |
| ordthaus 23415 | The order topology of a to... |
| xrhaus 23416 | The topology of the extend... |
| iscmp 23419 | The predicate "is a compac... |
| cmpcov 23420 | An open cover of a compact... |
| cmpcov2 23421 | Rewrite ~ cmpcov for the c... |
| cmpcovf 23422 | Combine ~ cmpcov with ~ ac... |
| cncmp 23423 | Compactness is respected b... |
| fincmp 23424 | A finite topology is compa... |
| 0cmp 23425 | The singleton of the empty... |
| cmptop 23426 | A compact topology is a to... |
| rncmp 23427 | The image of a compact set... |
| imacmp 23428 | The image of a compact set... |
| discmp 23429 | A discrete topology is com... |
| cmpsublem 23430 | Lemma for ~ cmpsub . (Con... |
| cmpsub 23431 | Two equivalent ways of des... |
| tgcmp 23432 | A topology generated by a ... |
| cmpcld 23433 | A closed subset of a compa... |
| uncmp 23434 | The union of two compact s... |
| fiuncmp 23435 | A finite union of compact ... |
| sscmp 23436 | A subset of a compact topo... |
| hauscmplem 23437 | Lemma for ~ hauscmp . (Co... |
| hauscmp 23438 | A compact subspace of a T2... |
| cmpfi 23439 | If a topology is compact a... |
| cmpfii 23440 | In a compact topology, a s... |
| bwth 23441 | The glorious Bolzano-Weier... |
| isconn 23444 | The predicate ` J ` is a c... |
| isconn2 23445 | The predicate ` J ` is a c... |
| connclo 23446 | The only nonempty clopen s... |
| conndisj 23447 | If a topology is connected... |
| conntop 23448 | A connected topology is a ... |
| indisconn 23449 | The indiscrete topology (o... |
| dfconn2 23450 | An alternate definition of... |
| connsuba 23451 | Connectedness for a subspa... |
| connsub 23452 | Two equivalent ways of say... |
| cnconn 23453 | Connectedness is respected... |
| nconnsubb 23454 | Disconnectedness for a sub... |
| connsubclo 23455 | If a clopen set meets a co... |
| connima 23456 | The image of a connected s... |
| conncn 23457 | A continuous function from... |
| iunconnlem 23458 | Lemma for ~ iunconn . (Co... |
| iunconn 23459 | The indexed union of conne... |
| unconn 23460 | The union of two connected... |
| clsconn 23461 | The closure of a connected... |
| conncompid 23462 | The connected component co... |
| conncompconn 23463 | The connected component co... |
| conncompss 23464 | The connected component co... |
| conncompcld 23465 | The connected component co... |
| conncompclo 23466 | The connected component co... |
| t1connperf 23467 | A connected T_1 space is p... |
| is1stc 23472 | The predicate "is a first-... |
| is1stc2 23473 | An equivalent way of sayin... |
| 1stctop 23474 | A first-countable topology... |
| 1stcclb 23475 | A property of points in a ... |
| 1stcfb 23476 | For any point ` A ` in a f... |
| is2ndc 23477 | The property of being seco... |
| 2ndctop 23478 | A second-countable topolog... |
| 2ndci 23479 | A countable basis generate... |
| 2ndcsb 23480 | Having a countable subbase... |
| 2ndcredom 23481 | A second-countable space h... |
| 2ndc1stc 23482 | A second-countable space i... |
| 1stcrestlem 23483 | Lemma for ~ 1stcrest . (C... |
| 1stcrest 23484 | A subspace of a first-coun... |
| 2ndcrest 23485 | A subspace of a second-cou... |
| 2ndcctbss 23486 | If a topology is second-co... |
| 2ndcdisj 23487 | Any disjoint family of ope... |
| 2ndcdisj2 23488 | Any disjoint collection of... |
| 2ndcomap 23489 | A surjective continuous op... |
| 2ndcsep 23490 | A second-countable topolog... |
| dis2ndc 23491 | A discrete space is second... |
| 1stcelcls 23492 | A point belongs to the clo... |
| 1stccnp 23493 | A mapping is continuous at... |
| 1stccn 23494 | A mapping ` X --> Y ` , wh... |
| islly 23499 | The property of being a lo... |
| isnlly 23500 | The property of being an n... |
| llyeq 23501 | Equality theorem for the `... |
| nllyeq 23502 | Equality theorem for the `... |
| llytop 23503 | A locally ` A ` space is a... |
| nllytop 23504 | A locally ` A ` space is a... |
| llyi 23505 | The property of a locally ... |
| nllyi 23506 | The property of an n-local... |
| nlly2i 23507 | Eliminate the neighborhood... |
| llynlly 23508 | A locally ` A ` space is n... |
| llyssnlly 23509 | A locally ` A ` space is n... |
| llyss 23510 | The "locally" predicate re... |
| nllyss 23511 | The "n-locally" predicate ... |
| subislly 23512 | The property of a subspace... |
| restnlly 23513 | If the property ` A ` pass... |
| restlly 23514 | If the property ` A ` pass... |
| islly2 23515 | An alternative expression ... |
| llyrest 23516 | An open subspace of a loca... |
| nllyrest 23517 | An open subspace of an n-l... |
| loclly 23518 | If ` A ` is a local proper... |
| llyidm 23519 | Idempotence of the "locall... |
| nllyidm 23520 | Idempotence of the "n-loca... |
| toplly 23521 | A topology is locally a to... |
| topnlly 23522 | A topology is n-locally a ... |
| hauslly 23523 | A Hausdorff space is local... |
| hausnlly 23524 | A Hausdorff space is n-loc... |
| hausllycmp 23525 | A compact Hausdorff space ... |
| cldllycmp 23526 | A closed subspace of a loc... |
| lly1stc 23527 | First-countability is a lo... |
| dislly 23528 | The discrete space ` ~P X ... |
| disllycmp 23529 | A discrete space is locall... |
| dis1stc 23530 | A discrete space is first-... |
| hausmapdom 23531 | If ` X ` is a first-counta... |
| hauspwdom 23532 | Simplify the cardinal ` A ... |
| refrel 23539 | Refinement is a relation. ... |
| isref 23540 | The property of being a re... |
| refbas 23541 | A refinement covers the sa... |
| refssex 23542 | Every set in a refinement ... |
| ssref 23543 | A subcover is a refinement... |
| refref 23544 | Reflexivity of refinement.... |
| reftr 23545 | Refinement is transitive. ... |
| refun0 23546 | Adding the empty set prese... |
| isptfin 23547 | The statement "is a point-... |
| islocfin 23548 | The statement "is a locall... |
| finptfin 23549 | A finite cover is a point-... |
| ptfinfin 23550 | A point covered by a point... |
| finlocfin 23551 | A finite cover of a topolo... |
| locfintop 23552 | A locally finite cover cov... |
| locfinbas 23553 | A locally finite cover mus... |
| locfinnei 23554 | A point covered by a local... |
| lfinpfin 23555 | A locally finite cover is ... |
| lfinun 23556 | Adding a finite set preser... |
| locfincmp 23557 | For a compact space, the l... |
| unisngl 23558 | Taking the union of the se... |
| dissnref 23559 | The set of singletons is a... |
| dissnlocfin 23560 | The set of singletons is l... |
| locfindis 23561 | The locally finite covers ... |
| locfincf 23562 | A locally finite cover in ... |
| comppfsc 23563 | A space where every open c... |
| kgenval 23566 | Value of the compact gener... |
| elkgen 23567 | Value of the compact gener... |
| kgeni 23568 | Property of the open sets ... |
| kgentopon 23569 | The compact generator gene... |
| kgenuni 23570 | The base set of the compac... |
| kgenftop 23571 | The compact generator gene... |
| kgenf 23572 | The compact generator is a... |
| kgentop 23573 | A compactly generated spac... |
| kgenss 23574 | The compact generator gene... |
| kgenhaus 23575 | The compact generator gene... |
| kgencmp 23576 | The compact generator topo... |
| kgencmp2 23577 | The compact generator topo... |
| kgenidm 23578 | The compact generator is i... |
| iskgen2 23579 | A space is compactly gener... |
| iskgen3 23580 | Derive the usual definitio... |
| llycmpkgen2 23581 | A locally compact space is... |
| cmpkgen 23582 | A compact space is compact... |
| llycmpkgen 23583 | A locally compact space is... |
| 1stckgenlem 23584 | The one-point compactifica... |
| 1stckgen 23585 | A first-countable space is... |
| kgen2ss 23586 | The compact generator pres... |
| kgencn 23587 | A function from a compactl... |
| kgencn2 23588 | A function ` F : J --> K `... |
| kgencn3 23589 | The set of continuous func... |
| kgen2cn 23590 | A continuous function is a... |
| txval 23595 | Value of the binary topolo... |
| txuni2 23596 | The underlying set of the ... |
| txbasex 23597 | The basis for the product ... |
| txbas 23598 | The set of Cartesian produ... |
| eltx 23599 | A set in a product is open... |
| txtop 23600 | The product of two topolog... |
| ptval 23601 | The value of the product t... |
| ptpjpre1 23602 | The preimage of a projecti... |
| elpt 23603 | Elementhood in the bases o... |
| elptr 23604 | A basic open set in the pr... |
| elptr2 23605 | A basic open set in the pr... |
| ptbasid 23606 | The base set of the produc... |
| ptuni2 23607 | The base set for the produ... |
| ptbasin 23608 | The basis for a product to... |
| ptbasin2 23609 | The basis for a product to... |
| ptbas 23610 | The basis for a product to... |
| ptpjpre2 23611 | The basis for a product to... |
| ptbasfi 23612 | The basis for the product ... |
| pttop 23613 | The product topology is a ... |
| ptopn 23614 | A basic open set in the pr... |
| ptopn2 23615 | A sub-basic open set in th... |
| xkotf 23616 | Functionality of function ... |
| xkobval 23617 | Alternative expression for... |
| xkoval 23618 | Value of the compact-open ... |
| xkotop 23619 | The compact-open topology ... |
| xkoopn 23620 | A basic open set of the co... |
| txtopi 23621 | The product of two topolog... |
| txtopon 23622 | The underlying set of the ... |
| txuni 23623 | The underlying set of the ... |
| txunii 23624 | The underlying set of the ... |
| ptuni 23625 | The base set for the produ... |
| ptunimpt 23626 | Base set of a product topo... |
| pttopon 23627 | The base set for the produ... |
| pttoponconst 23628 | The base set for a product... |
| ptuniconst 23629 | The base set for a product... |
| xkouni 23630 | The base set of the compac... |
| xkotopon 23631 | The base set of the compac... |
| ptval2 23632 | The value of the product t... |
| txopn 23633 | The product of two open se... |
| txcld 23634 | The product of two closed ... |
| txcls 23635 | Closure of a rectangle in ... |
| txss12 23636 | Subset property of the top... |
| txbasval 23637 | It is sufficient to consid... |
| neitx 23638 | The Cartesian product of t... |
| txcnpi 23639 | Continuity of a two-argume... |
| tx1cn 23640 | Continuity of the first pr... |
| tx2cn 23641 | Continuity of the second p... |
| ptpjcn 23642 | Continuity of a projection... |
| ptpjopn 23643 | The projection map is an o... |
| ptcld 23644 | A closed box in the produc... |
| ptcldmpt 23645 | A closed box in the produc... |
| ptclsg 23646 | The closure of a box in th... |
| ptcls 23647 | The closure of a box in th... |
| dfac14lem 23648 | Lemma for ~ dfac14 . By e... |
| dfac14 23649 | Theorem ~ ptcls is an equi... |
| xkoccn 23650 | The "constant function" fu... |
| txcnp 23651 | If two functions are conti... |
| ptcnplem 23652 | Lemma for ~ ptcnp . (Cont... |
| ptcnp 23653 | If every projection of a f... |
| upxp 23654 | Universal property of the ... |
| txcnmpt 23655 | A map into the product of ... |
| uptx 23656 | Universal property of the ... |
| txcn 23657 | A map into the product of ... |
| ptcn 23658 | If every projection of a f... |
| prdstopn 23659 | Topology of a structure pr... |
| prdstps 23660 | A structure product of top... |
| pwstps 23661 | A structure power of a top... |
| txrest 23662 | The subspace of a topologi... |
| txdis 23663 | The topological product of... |
| txindislem 23664 | Lemma for ~ txindis . (Co... |
| txindis 23665 | The topological product of... |
| txdis1cn 23666 | A function is jointly cont... |
| txlly 23667 | If the property ` A ` is p... |
| txnlly 23668 | If the property ` A ` is p... |
| pthaus 23669 | The product of a collectio... |
| ptrescn 23670 | Restriction is a continuou... |
| txtube 23671 | The "tube lemma". If ` X ... |
| txcmplem1 23672 | Lemma for ~ txcmp . (Cont... |
| txcmplem2 23673 | Lemma for ~ txcmp . (Cont... |
| txcmp 23674 | The topological product of... |
| txcmpb 23675 | The topological product of... |
| hausdiag 23676 | A topology is Hausdorff if... |
| hauseqlcld 23677 | In a Hausdorff topology, t... |
| txhaus 23678 | The topological product of... |
| txlm 23679 | Two sequences converge iff... |
| lmcn2 23680 | The image of a convergent ... |
| tx1stc 23681 | The topological product of... |
| tx2ndc 23682 | The topological product of... |
| txkgen 23683 | The topological product of... |
| xkohaus 23684 | If the codomain space is H... |
| xkoptsub 23685 | The compact-open topology ... |
| xkopt 23686 | The compact-open topology ... |
| xkopjcn 23687 | Continuity of a projection... |
| xkoco1cn 23688 | If ` F ` is a continuous f... |
| xkoco2cn 23689 | If ` F ` is a continuous f... |
| xkococnlem 23690 | Continuity of the composit... |
| xkococn 23691 | Continuity of the composit... |
| cnmptid 23692 | The identity function is c... |
| cnmptc 23693 | A constant function is con... |
| cnmpt11 23694 | The composition of continu... |
| cnmpt11f 23695 | The composition of continu... |
| cnmpt1t 23696 | The composition of continu... |
| cnmpt12f 23697 | The composition of continu... |
| cnmpt12 23698 | The composition of continu... |
| cnmpt1st 23699 | The projection onto the fi... |
| cnmpt2nd 23700 | The projection onto the se... |
| cnmpt2c 23701 | A constant function is con... |
| cnmpt21 23702 | The composition of continu... |
| cnmpt21f 23703 | The composition of continu... |
| cnmpt2t 23704 | The composition of continu... |
| cnmpt22 23705 | The composition of continu... |
| cnmpt22f 23706 | The composition of continu... |
| cnmpt1res 23707 | The restriction of a conti... |
| cnmpt2res 23708 | The restriction of a conti... |
| cnmptcom 23709 | The argument converse of a... |
| cnmptkc 23710 | The curried first projecti... |
| cnmptkp 23711 | The evaluation of the inne... |
| cnmptk1 23712 | The composition of a curri... |
| cnmpt1k 23713 | The composition of a one-a... |
| cnmptkk 23714 | The composition of two cur... |
| xkofvcn 23715 | Joint continuity of the fu... |
| cnmptk1p 23716 | The evaluation of a currie... |
| cnmptk2 23717 | The uncurrying of a currie... |
| xkoinjcn 23718 | Continuity of "injection",... |
| cnmpt2k 23719 | The currying of a two-argu... |
| txconn 23720 | The topological product of... |
| imasnopn 23721 | If a relation graph is ope... |
| imasncld 23722 | If a relation graph is clo... |
| imasncls 23723 | If a relation graph is clo... |
| qtopval 23726 | Value of the quotient topo... |
| qtopval2 23727 | Value of the quotient topo... |
| elqtop 23728 | Value of the quotient topo... |
| qtopres 23729 | The quotient topology is u... |
| qtoptop2 23730 | The quotient topology is a... |
| qtoptop 23731 | The quotient topology is a... |
| elqtop2 23732 | Value of the quotient topo... |
| qtopuni 23733 | The base set of the quotie... |
| elqtop3 23734 | Value of the quotient topo... |
| qtoptopon 23735 | The base set of the quotie... |
| qtopid 23736 | A quotient map is a contin... |
| idqtop 23737 | The quotient topology indu... |
| qtopcmplem 23738 | Lemma for ~ qtopcmp and ~ ... |
| qtopcmp 23739 | A quotient of a compact sp... |
| qtopconn 23740 | A quotient of a connected ... |
| qtopkgen 23741 | A quotient of a compactly ... |
| basqtop 23742 | An injection maps bases to... |
| tgqtop 23743 | An injection maps generate... |
| qtopcld 23744 | The property of being a cl... |
| qtopcn 23745 | Universal property of a qu... |
| qtopss 23746 | A surjective continuous fu... |
| qtopeu 23747 | Universal property of the ... |
| qtoprest 23748 | If ` A ` is a saturated op... |
| qtopomap 23749 | If ` F ` is a surjective c... |
| qtopcmap 23750 | If ` F ` is a surjective c... |
| imastopn 23751 | The topology of an image s... |
| imastps 23752 | The image of a topological... |
| qustps 23753 | A quotient structure is a ... |
| kqfval 23754 | Value of the function appe... |
| kqfeq 23755 | Two points in the Kolmogor... |
| kqffn 23756 | The topological indistingu... |
| kqval 23757 | Value of the quotient topo... |
| kqtopon 23758 | The Kolmogorov quotient is... |
| kqid 23759 | The topological indistingu... |
| ist0-4 23760 | The topological indistingu... |
| kqfvima 23761 | When the image set is open... |
| kqsat 23762 | Any open set is saturated ... |
| kqdisj 23763 | A version of ~ imain for t... |
| kqcldsat 23764 | Any closed set is saturate... |
| kqopn 23765 | The topological indistingu... |
| kqcld 23766 | The topological indistingu... |
| kqt0lem 23767 | Lemma for ~ kqt0 . (Contr... |
| isr0 23768 | The property " ` J ` is an... |
| r0cld 23769 | The analogue of the T_1 ax... |
| regr1lem 23770 | Lemma for ~ regr1 . (Cont... |
| regr1lem2 23771 | A Kolmogorov quotient of a... |
| kqreglem1 23772 | A Kolmogorov quotient of a... |
| kqreglem2 23773 | If the Kolmogorov quotient... |
| kqnrmlem1 23774 | A Kolmogorov quotient of a... |
| kqnrmlem2 23775 | If the Kolmogorov quotient... |
| kqtop 23776 | The Kolmogorov quotient is... |
| kqt0 23777 | The Kolmogorov quotient is... |
| kqf 23778 | The Kolmogorov quotient is... |
| r0sep 23779 | The separation property of... |
| nrmr0reg 23780 | A normal R_0 space is also... |
| regr1 23781 | A regular space is R_1, wh... |
| kqreg 23782 | The Kolmogorov quotient of... |
| kqnrm 23783 | The Kolmogorov quotient of... |
| hmeofn 23788 | The set of homeomorphisms ... |
| hmeofval 23789 | The set of all the homeomo... |
| ishmeo 23790 | The predicate F is a homeo... |
| hmeocn 23791 | A homeomorphism is continu... |
| hmeocnvcn 23792 | The converse of a homeomor... |
| hmeocnv 23793 | The converse of a homeomor... |
| hmeof1o2 23794 | A homeomorphism is a 1-1-o... |
| hmeof1o 23795 | A homeomorphism is a 1-1-o... |
| hmeoima 23796 | The image of an open set b... |
| hmeoopn 23797 | Homeomorphisms preserve op... |
| hmeocld 23798 | Homeomorphisms preserve cl... |
| hmeocls 23799 | Homeomorphisms preserve cl... |
| hmeontr 23800 | Homeomorphisms preserve in... |
| hmeoimaf1o 23801 | The function mapping open ... |
| hmeores 23802 | The restriction of a homeo... |
| hmeoco 23803 | The composite of two homeo... |
| idhmeo 23804 | The identity function is a... |
| hmeocnvb 23805 | The converse of a homeomor... |
| hmeoqtop 23806 | A homeomorphism is a quoti... |
| hmph 23807 | Express the predicate ` J ... |
| hmphi 23808 | If there is a homeomorphis... |
| hmphtop 23809 | Reverse closure for the ho... |
| hmphtop1 23810 | The relation "being homeom... |
| hmphtop2 23811 | The relation "being homeom... |
| hmphref 23812 | "Is homeomorphic to" is re... |
| hmphsym 23813 | "Is homeomorphic to" is sy... |
| hmphtr 23814 | "Is homeomorphic to" is tr... |
| hmpher 23815 | "Is homeomorphic to" is an... |
| hmphen 23816 | Homeomorphisms preserve th... |
| hmphsymb 23817 | "Is homeomorphic to" is sy... |
| haushmphlem 23818 | Lemma for ~ haushmph and s... |
| cmphmph 23819 | Compactness is a topologic... |
| connhmph 23820 | Connectedness is a topolog... |
| t0hmph 23821 | T_0 is a topological prope... |
| t1hmph 23822 | T_1 is a topological prope... |
| haushmph 23823 | Hausdorff-ness is a topolo... |
| reghmph 23824 | Regularity is a topologica... |
| nrmhmph 23825 | Normality is a topological... |
| hmph0 23826 | A topology homeomorphic to... |
| hmphdis 23827 | Homeomorphisms preserve to... |
| hmphindis 23828 | Homeomorphisms preserve to... |
| indishmph 23829 | Equinumerous sets equipped... |
| hmphen2 23830 | Homeomorphisms preserve th... |
| cmphaushmeo 23831 | A continuous bijection fro... |
| ordthmeolem 23832 | Lemma for ~ ordthmeo . (C... |
| ordthmeo 23833 | An order isomorphism is a ... |
| txhmeo 23834 | Lift a pair of homeomorphi... |
| txswaphmeolem 23835 | Show inverse for the "swap... |
| txswaphmeo 23836 | There is a homeomorphism f... |
| pt1hmeo 23837 | The canonical homeomorphis... |
| ptuncnv 23838 | Exhibit the converse funct... |
| ptunhmeo 23839 | Define a homeomorphism fro... |
| xpstopnlem1 23840 | The function ` F ` used in... |
| xpstps 23841 | A binary product of topolo... |
| xpstopnlem2 23842 | Lemma for ~ xpstopn . (Co... |
| xpstopn 23843 | The topology on a binary p... |
| ptcmpfi 23844 | A topological product of f... |
| xkocnv 23845 | The inverse of the "curryi... |
| xkohmeo 23846 | The Exponential Law for to... |
| qtopf1 23847 | If a quotient map is injec... |
| qtophmeo 23848 | If two functions on a base... |
| t0kq 23849 | A topological space is T_0... |
| kqhmph 23850 | A topological space is T_0... |
| ist1-5lem 23851 | Lemma for ~ ist1-5 and sim... |
| t1r0 23852 | A T_1 space is R_0. That ... |
| ist1-5 23853 | A topological space is T_1... |
| ishaus3 23854 | A topological space is Hau... |
| nrmreg 23855 | A normal T_1 space is regu... |
| reghaus 23856 | A regular T_0 space is Hau... |
| nrmhaus 23857 | A T_1 normal space is Haus... |
| elmptrab 23858 | Membership in a one-parame... |
| elmptrab2 23859 | Membership in a one-parame... |
| isfbas 23860 | The predicate " ` F ` is a... |
| fbasne0 23861 | There are no empty filter ... |
| 0nelfb 23862 | No filter base contains th... |
| fbsspw 23863 | A filter base on a set is ... |
| fbelss 23864 | An element of the filter b... |
| fbdmn0 23865 | The domain of a filter bas... |
| isfbas2 23866 | The predicate " ` F ` is a... |
| fbasssin 23867 | A filter base contains sub... |
| fbssfi 23868 | A filter base contains sub... |
| fbssint 23869 | A filter base contains sub... |
| fbncp 23870 | A filter base does not con... |
| fbun 23871 | A necessary and sufficient... |
| fbfinnfr 23872 | No filter base containing ... |
| opnfbas 23873 | The collection of open sup... |
| trfbas2 23874 | Conditions for the trace o... |
| trfbas 23875 | Conditions for the trace o... |
| isfil 23878 | The predicate "is a filter... |
| filfbas 23879 | A filter is a filter base.... |
| 0nelfil 23880 | The empty set doesn't belo... |
| fileln0 23881 | An element of a filter is ... |
| filsspw 23882 | A filter is a subset of th... |
| filelss 23883 | An element of a filter is ... |
| filss 23884 | A filter is closed under t... |
| filin 23885 | A filter is closed under t... |
| filtop 23886 | The underlying set belongs... |
| isfil2 23887 | Derive the standard axioms... |
| isfildlem 23888 | Lemma for ~ isfild . (Con... |
| isfild 23889 | Sufficient condition for a... |
| filfi 23890 | A filter is closed under t... |
| filinn0 23891 | The intersection of two el... |
| filintn0 23892 | A filter has the finite in... |
| filn0 23893 | The empty set is not a fil... |
| infil 23894 | The intersection of two fi... |
| snfil 23895 | A singleton is a filter. ... |
| fbasweak 23896 | A filter base on any set i... |
| snfbas 23897 | Condition for a singleton ... |
| fsubbas 23898 | A condition for a set to g... |
| fbasfip 23899 | A filter base has the fini... |
| fbunfip 23900 | A helpful lemma for showin... |
| fgval 23901 | The filter generating clas... |
| elfg 23902 | A condition for elements o... |
| ssfg 23903 | A filter base is a subset ... |
| fgss 23904 | A bigger base generates a ... |
| fgss2 23905 | A condition for a filter t... |
| fgfil 23906 | A filter generates itself.... |
| elfilss 23907 | An element belongs to a fi... |
| filfinnfr 23908 | No filter containing a fin... |
| fgcl 23909 | A generated filter is a fi... |
| fgabs 23910 | Absorption law for filter ... |
| neifil 23911 | The neighborhoods of a non... |
| filunibas 23912 | Recover the base set from ... |
| filunirn 23913 | Two ways to express a filt... |
| filconn 23914 | A filter gives rise to a c... |
| fbasrn 23915 | Given a filter on a domain... |
| filuni 23916 | The union of a nonempty se... |
| trfil1 23917 | Conditions for the trace o... |
| trfil2 23918 | Conditions for the trace o... |
| trfil3 23919 | Conditions for the trace o... |
| trfilss 23920 | If ` A ` is a member of th... |
| fgtr 23921 | If ` A ` is a member of th... |
| trfg 23922 | The trace operation and th... |
| trnei 23923 | The trace, over a set ` A ... |
| cfinfil 23924 | Relative complements of th... |
| csdfil 23925 | The set of all elements wh... |
| supfil 23926 | The supersets of a nonempt... |
| zfbas 23927 | The set of upper sets of i... |
| uzrest 23928 | The restriction of the set... |
| uzfbas 23929 | The set of upper sets of i... |
| isufil 23934 | The property of being an u... |
| ufilfil 23935 | An ultrafilter is a filter... |
| ufilss 23936 | For any subset of the base... |
| ufilb 23937 | The complement is in an ul... |
| ufilmax 23938 | Any filter finer than an u... |
| isufil2 23939 | The maximal property of an... |
| ufprim 23940 | An ultrafilter is a prime ... |
| trufil 23941 | Conditions for the trace o... |
| filssufilg 23942 | A filter is contained in s... |
| filssufil 23943 | A filter is contained in s... |
| isufl 23944 | Define the (strong) ultraf... |
| ufli 23945 | Property of a set that sat... |
| numufl 23946 | Consequence of ~ filssufil... |
| fiufl 23947 | A finite set satisfies the... |
| acufl 23948 | The axiom of choice implie... |
| ssufl 23949 | If ` Y ` is a subset of ` ... |
| ufileu 23950 | If the ultrafilter contain... |
| filufint 23951 | A filter is equal to the i... |
| uffix 23952 | Lemma for ~ fixufil and ~ ... |
| fixufil 23953 | The condition describing a... |
| uffixfr 23954 | An ultrafilter is either f... |
| uffix2 23955 | A classification of fixed ... |
| uffixsn 23956 | The singleton of the gener... |
| ufildom1 23957 | An ultrafilter is generate... |
| uffinfix 23958 | An ultrafilter containing ... |
| cfinufil 23959 | An ultrafilter is free iff... |
| ufinffr 23960 | An infinite subset is cont... |
| ufilen 23961 | Any infinite set has an ul... |
| ufildr 23962 | An ultrafilter gives rise ... |
| fin1aufil 23963 | There are no definable fre... |
| fmval 23974 | Introduce a function that ... |
| fmfil 23975 | A mapping filter is a filt... |
| fmf 23976 | Pushing-forward via a func... |
| fmss 23977 | A finer filter produces a ... |
| elfm 23978 | An element of a mapping fi... |
| elfm2 23979 | An element of a mapping fi... |
| fmfg 23980 | The image filter of a filt... |
| elfm3 23981 | An alternate formulation o... |
| imaelfm 23982 | An image of a filter eleme... |
| rnelfmlem 23983 | Lemma for ~ rnelfm . (Con... |
| rnelfm 23984 | A condition for a filter t... |
| fmfnfmlem1 23985 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem2 23986 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem3 23987 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem4 23988 | Lemma for ~ fmfnfm . (Con... |
| fmfnfm 23989 | A filter finer than an ima... |
| fmufil 23990 | An image filter of an ultr... |
| fmid 23991 | The filter map applied to ... |
| fmco 23992 | Composition of image filte... |
| ufldom 23993 | The ultrafilter lemma prop... |
| flimval 23994 | The set of limit points of... |
| elflim2 23995 | The predicate "is a limit ... |
| flimtop 23996 | Reverse closure for the li... |
| flimneiss 23997 | A filter contains the neig... |
| flimnei 23998 | A filter contains all of t... |
| flimelbas 23999 | A limit point of a filter ... |
| flimfil 24000 | Reverse closure for the li... |
| flimtopon 24001 | Reverse closure for the li... |
| elflim 24002 | The predicate "is a limit ... |
| flimss2 24003 | A limit point of a filter ... |
| flimss1 24004 | A limit point of a filter ... |
| neiflim 24005 | A point is a limit point o... |
| flimopn 24006 | The condition for being a ... |
| fbflim 24007 | A condition for a filter t... |
| fbflim2 24008 | A condition for a filter b... |
| flimclsi 24009 | The convergent points of a... |
| hausflimlem 24010 | If ` A ` and ` B ` are bot... |
| hausflimi 24011 | One direction of ~ hausfli... |
| hausflim 24012 | A condition for a topology... |
| flimcf 24013 | Fineness is properly chara... |
| flimrest 24014 | The set of limit points in... |
| flimclslem 24015 | Lemma for ~ flimcls . (Co... |
| flimcls 24016 | Closure in terms of filter... |
| flimsncls 24017 | If ` A ` is a limit point ... |
| hauspwpwf1 24018 | Lemma for ~ hauspwpwdom . ... |
| hauspwpwdom 24019 | If ` X ` is a Hausdorff sp... |
| flffval 24020 | Given a topology and a fil... |
| flfval 24021 | Given a function from a fi... |
| flfnei 24022 | The property of being a li... |
| flfneii 24023 | A neighborhood of a limit ... |
| isflf 24024 | The property of being a li... |
| flfelbas 24025 | A limit point of a functio... |
| flffbas 24026 | Limit points of a function... |
| flftg 24027 | Limit points of a function... |
| hausflf 24028 | If a function has its valu... |
| hausflf2 24029 | If a convergent function h... |
| cnpflfi 24030 | Forward direction of ~ cnp... |
| cnpflf2 24031 | ` F ` is continuous at poi... |
| cnpflf 24032 | Continuity of a function a... |
| cnflf 24033 | A function is continuous i... |
| cnflf2 24034 | A function is continuous i... |
| flfcnp 24035 | A continuous function pres... |
| lmflf 24036 | The topological limit rela... |
| txflf 24037 | Two sequences converge in ... |
| flfcnp2 24038 | The image of a convergent ... |
| fclsval 24039 | The set of all cluster poi... |
| isfcls 24040 | A cluster point of a filte... |
| fclsfil 24041 | Reverse closure for the cl... |
| fclstop 24042 | Reverse closure for the cl... |
| fclstopon 24043 | Reverse closure for the cl... |
| isfcls2 24044 | A cluster point of a filte... |
| fclsopn 24045 | Write the cluster point co... |
| fclsopni 24046 | An open neighborhood of a ... |
| fclselbas 24047 | A cluster point is in the ... |
| fclsneii 24048 | A neighborhood of a cluste... |
| fclssscls 24049 | The set of cluster points ... |
| fclsnei 24050 | Cluster points in terms of... |
| supnfcls 24051 | The filter of supersets of... |
| fclsbas 24052 | Cluster points in terms of... |
| fclsss1 24053 | A finer topology has fewer... |
| fclsss2 24054 | A finer filter has fewer c... |
| fclsrest 24055 | The set of cluster points ... |
| fclscf 24056 | Characterization of finene... |
| flimfcls 24057 | A limit point is a cluster... |
| fclsfnflim 24058 | A filter clusters at a poi... |
| flimfnfcls 24059 | A filter converges to a po... |
| fclscmpi 24060 | Forward direction of ~ fcl... |
| fclscmp 24061 | A space is compact iff eve... |
| uffclsflim 24062 | The cluster points of an u... |
| ufilcmp 24063 | A space is compact iff eve... |
| fcfval 24064 | The set of cluster points ... |
| isfcf 24065 | The property of being a cl... |
| fcfnei 24066 | The property of being a cl... |
| fcfelbas 24067 | A cluster point of a funct... |
| fcfneii 24068 | A neighborhood of a cluste... |
| flfssfcf 24069 | A limit point of a functio... |
| uffcfflf 24070 | If the domain filter is an... |
| cnpfcfi 24071 | Lemma for ~ cnpfcf . If a... |
| cnpfcf 24072 | A function ` F ` is contin... |
| cnfcf 24073 | Continuity of a function i... |
| flfcntr 24074 | A continuous function's va... |
| alexsublem 24075 | Lemma for ~ alexsub . (Co... |
| alexsub 24076 | The Alexander Subbase Theo... |
| alexsubb 24077 | Biconditional form of the ... |
| alexsubALTlem1 24078 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem2 24079 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem3 24080 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem4 24081 | Lemma for ~ alexsubALT . ... |
| alexsubALT 24082 | The Alexander Subbase Theo... |
| ptcmplem1 24083 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem2 24084 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem3 24085 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem4 24086 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem5 24087 | Lemma for ~ ptcmp . (Cont... |
| ptcmpg 24088 | Tychonoff's theorem: The ... |
| ptcmp 24089 | Tychonoff's theorem: The ... |
| cnextval 24092 | The function applying cont... |
| cnextfval 24093 | The continuous extension o... |
| cnextrel 24094 | In the general case, a con... |
| cnextfun 24095 | If the target space is Hau... |
| cnextfvval 24096 | The value of the continuou... |
| cnextf 24097 | Extension by continuity. ... |
| cnextcn 24098 | Extension by continuity. ... |
| cnextfres1 24099 | ` F ` and its extension by... |
| cnextfres 24100 | ` F ` and its extension by... |
| istmd 24105 | The predicate "is a topolo... |
| tmdmnd 24106 | A topological monoid is a ... |
| tmdtps 24107 | A topological monoid is a ... |
| istgp 24108 | The predicate "is a topolo... |
| tgpgrp 24109 | A topological group is a g... |
| tgptmd 24110 | A topological group is a t... |
| tgptps 24111 | A topological group is a t... |
| tmdtopon 24112 | The topology of a topologi... |
| tgptopon 24113 | The topology of a topologi... |
| tmdcn 24114 | In a topological monoid, t... |
| tgpcn 24115 | In a topological group, th... |
| tgpinv 24116 | In a topological group, th... |
| grpinvhmeo 24117 | The inverse function in a ... |
| cnmpt1plusg 24118 | Continuity of the group su... |
| cnmpt2plusg 24119 | Continuity of the group su... |
| tmdcn2 24120 | Write out the definition o... |
| tgpsubcn 24121 | In a topological group, th... |
| istgp2 24122 | A group with a topology is... |
| tmdmulg 24123 | In a topological monoid, t... |
| tgpmulg 24124 | In a topological group, th... |
| tgpmulg2 24125 | In a topological monoid, t... |
| tmdgsum 24126 | In a topological monoid, t... |
| tmdgsum2 24127 | For any neighborhood ` U `... |
| oppgtmd 24128 | The opposite of a topologi... |
| oppgtgp 24129 | The opposite of a topologi... |
| distgp 24130 | Any group equipped with th... |
| indistgp 24131 | Any group equipped with th... |
| efmndtmd 24132 | The monoid of endofunction... |
| tmdlactcn 24133 | The left group action of e... |
| tgplacthmeo 24134 | The left group action of e... |
| submtmd 24135 | A submonoid of a topologic... |
| subgtgp 24136 | A subgroup of a topologica... |
| symgtgp 24137 | The symmetric group is a t... |
| subgntr 24138 | A subgroup of a topologica... |
| opnsubg 24139 | An open subgroup of a topo... |
| clssubg 24140 | The closure of a subgroup ... |
| clsnsg 24141 | The closure of a normal su... |
| cldsubg 24142 | A subgroup of finite index... |
| tgpconncompeqg 24143 | The connected component co... |
| tgpconncomp 24144 | The identity component, th... |
| tgpconncompss 24145 | The identity component is ... |
| ghmcnp 24146 | A group homomorphism on to... |
| snclseqg 24147 | The coset of the closure o... |
| tgphaus 24148 | A topological group is Hau... |
| tgpt1 24149 | Hausdorff and T1 are equiv... |
| tgpt0 24150 | Hausdorff and T0 are equiv... |
| qustgpopn 24151 | A quotient map in a topolo... |
| qustgplem 24152 | Lemma for ~ qustgp . (Con... |
| qustgp 24153 | The quotient of a topologi... |
| qustgphaus 24154 | The quotient of a topologi... |
| prdstmdd 24155 | The product of a family of... |
| prdstgpd 24156 | The product of a family of... |
| tsmsfbas 24159 | The collection of all sets... |
| tsmslem1 24160 | The finite partial sums of... |
| tsmsval2 24161 | Definition of the topologi... |
| tsmsval 24162 | Definition of the topologi... |
| tsmspropd 24163 | The group sum depends only... |
| eltsms 24164 | The property of being a su... |
| tsmsi 24165 | The property of being a su... |
| tsmscl 24166 | A sum in a topological gro... |
| haustsms 24167 | In a Hausdorff topological... |
| haustsms2 24168 | In a Hausdorff topological... |
| tsmscls 24169 | One half of ~ tgptsmscls ,... |
| tsmsgsum 24170 | The convergent points of a... |
| tsmsid 24171 | If a sum is finite, the us... |
| haustsmsid 24172 | In a Hausdorff topological... |
| tsms0 24173 | The sum of zero is zero. ... |
| tsmssubm 24174 | Evaluate an infinite group... |
| tsmsres 24175 | Extend an infinite group s... |
| tsmsf1o 24176 | Re-index an infinite group... |
| tsmsmhm 24177 | Apply a continuous group h... |
| tsmsadd 24178 | The sum of two infinite gr... |
| tsmsinv 24179 | Inverse of an infinite gro... |
| tsmssub 24180 | The difference of two infi... |
| tgptsmscls 24181 | A sum in a topological gro... |
| tgptsmscld 24182 | The set of limit points to... |
| tsmssplit 24183 | Split a topological group ... |
| tsmsxplem1 24184 | Lemma for ~ tsmsxp . (Con... |
| tsmsxplem2 24185 | Lemma for ~ tsmsxp . (Con... |
| tsmsxp 24186 | Write a sum over a two-dim... |
| istrg 24195 | Express the predicate " ` ... |
| trgtmd 24196 | The multiplicative monoid ... |
| istdrg 24197 | Express the predicate " ` ... |
| tdrgunit 24198 | The unit group of a topolo... |
| trgtgp 24199 | A topological ring is a to... |
| trgtmd2 24200 | A topological ring is a to... |
| trgtps 24201 | A topological ring is a to... |
| trgring 24202 | A topological ring is a ri... |
| trggrp 24203 | A topological ring is a gr... |
| tdrgtrg 24204 | A topological division rin... |
| tdrgdrng 24205 | A topological division rin... |
| tdrgring 24206 | A topological division rin... |
| tdrgtmd 24207 | A topological division rin... |
| tdrgtps 24208 | A topological division rin... |
| istdrg2 24209 | A topological-ring divisio... |
| mulrcn 24210 | The functionalization of t... |
| invrcn2 24211 | The multiplicative inverse... |
| invrcn 24212 | The multiplicative inverse... |
| cnmpt1mulr 24213 | Continuity of ring multipl... |
| cnmpt2mulr 24214 | Continuity of ring multipl... |
| dvrcn 24215 | The division function is c... |
| istlm 24216 | The predicate " ` W ` is a... |
| vscacn 24217 | The scalar multiplication ... |
| tlmtmd 24218 | A topological module is a ... |
| tlmtps 24219 | A topological module is a ... |
| tlmlmod 24220 | A topological module is a ... |
| tlmtrg 24221 | The scalar ring of a topol... |
| tlmscatps 24222 | The scalar ring of a topol... |
| istvc 24223 | A topological vector space... |
| tvctdrg 24224 | The scalar field of a topo... |
| cnmpt1vsca 24225 | Continuity of scalar multi... |
| cnmpt2vsca 24226 | Continuity of scalar multi... |
| tlmtgp 24227 | A topological vector space... |
| tvctlm 24228 | A topological vector space... |
| tvclmod 24229 | A topological vector space... |
| tvclvec 24230 | A topological vector space... |
| ustfn 24233 | The defined uniform struct... |
| ustval 24234 | The class of all uniform s... |
| isust 24235 | The predicate " ` U ` is a... |
| ustssxp 24236 | Entourages are subsets of ... |
| ustssel 24237 | A uniform structure is upw... |
| ustbasel 24238 | The full set is always an ... |
| ustincl 24239 | A uniform structure is clo... |
| ustdiag 24240 | The diagonal set is includ... |
| ustinvel 24241 | If ` V ` is an entourage, ... |
| ustexhalf 24242 | For each entourage ` V ` t... |
| ustrel 24243 | The elements of uniform st... |
| ustfilxp 24244 | A uniform structure on a n... |
| ustne0 24245 | A uniform structure cannot... |
| ustssco 24246 | In an uniform structure, a... |
| ustexsym 24247 | In an uniform structure, f... |
| ustex2sym 24248 | In an uniform structure, f... |
| ustex3sym 24249 | In an uniform structure, f... |
| ustref 24250 | Any element of the base se... |
| ust0 24251 | The unique uniform structu... |
| ustn0 24252 | The empty set is not an un... |
| ustund 24253 | If two intersecting sets `... |
| ustelimasn 24254 | Any point ` A ` is near en... |
| ustneism 24255 | For a point ` A ` in ` X `... |
| elrnustOLD 24256 | Obsolete version of ~ elfv... |
| ustbas2 24257 | Second direction for ~ ust... |
| ustuni 24258 | The set union of a uniform... |
| ustbas 24259 | Recover the base of an uni... |
| ustimasn 24260 | Lemma for ~ ustuqtop . (C... |
| trust 24261 | The trace of a uniform str... |
| utopval 24264 | The topology induced by a ... |
| elutop 24265 | Open sets in the topology ... |
| utoptop 24266 | The topology induced by a ... |
| utopbas 24267 | The base of the topology i... |
| utoptopon 24268 | Topology induced by a unif... |
| restutop 24269 | Restriction of a topology ... |
| restutopopn 24270 | The restriction of the top... |
| ustuqtoplem 24271 | Lemma for ~ ustuqtop . (C... |
| ustuqtop0 24272 | Lemma for ~ ustuqtop . (C... |
| ustuqtop1 24273 | Lemma for ~ ustuqtop , sim... |
| ustuqtop2 24274 | Lemma for ~ ustuqtop . (C... |
| ustuqtop3 24275 | Lemma for ~ ustuqtop , sim... |
| ustuqtop4 24276 | Lemma for ~ ustuqtop . (C... |
| ustuqtop5 24277 | Lemma for ~ ustuqtop . (C... |
| ustuqtop 24278 | For a given uniform struct... |
| utopsnneiplem 24279 | The neighborhoods of a poi... |
| utopsnneip 24280 | The neighborhoods of a poi... |
| utopsnnei 24281 | Images of singletons by en... |
| utop2nei 24282 | For any symmetrical entour... |
| utop3cls 24283 | Relation between a topolog... |
| utopreg 24284 | All Hausdorff uniform spac... |
| ussval 24291 | The uniform structure on u... |
| ussid 24292 | In case the base of the ` ... |
| isusp 24293 | The predicate ` W ` is a u... |
| ressuss 24294 | Value of the uniform struc... |
| ressust 24295 | The uniform structure of a... |
| ressusp 24296 | The restriction of a unifo... |
| tusval 24297 | The value of the uniform s... |
| tuslem 24298 | Lemma for ~ tusbas , ~ tus... |
| tuslemOLD 24299 | Obsolete proof of ~ tuslem... |
| tusbas 24300 | The base set of a construc... |
| tusunif 24301 | The uniform structure of a... |
| tususs 24302 | The uniform structure of a... |
| tustopn 24303 | The topology induced by a ... |
| tususp 24304 | A constructed uniform spac... |
| tustps 24305 | A constructed uniform spac... |
| uspreg 24306 | If a uniform space is Haus... |
| ucnval 24309 | The set of all uniformly c... |
| isucn 24310 | The predicate " ` F ` is a... |
| isucn2 24311 | The predicate " ` F ` is a... |
| ucnimalem 24312 | Reformulate the ` G ` func... |
| ucnima 24313 | An equivalent statement of... |
| ucnprima 24314 | The preimage by a uniforml... |
| iducn 24315 | The identity is uniformly ... |
| cstucnd 24316 | A constant function is uni... |
| ucncn 24317 | Uniform continuity implies... |
| iscfilu 24320 | The predicate " ` F ` is a... |
| cfilufbas 24321 | A Cauchy filter base is a ... |
| cfiluexsm 24322 | For a Cauchy filter base a... |
| fmucndlem 24323 | Lemma for ~ fmucnd . (Con... |
| fmucnd 24324 | The image of a Cauchy filt... |
| cfilufg 24325 | The filter generated by a ... |
| trcfilu 24326 | Condition for the trace of... |
| cfiluweak 24327 | A Cauchy filter base is al... |
| neipcfilu 24328 | In an uniform space, a nei... |
| iscusp 24331 | The predicate " ` W ` is a... |
| cuspusp 24332 | A complete uniform space i... |
| cuspcvg 24333 | In a complete uniform spac... |
| iscusp2 24334 | The predicate " ` W ` is a... |
| cnextucn 24335 | Extension by continuity. ... |
| ucnextcn 24336 | Extension by continuity. ... |
| ispsmet 24337 | Express the predicate " ` ... |
| psmetdmdm 24338 | Recover the base set from ... |
| psmetf 24339 | The distance function of a... |
| psmetcl 24340 | Closure of the distance fu... |
| psmet0 24341 | The distance function of a... |
| psmettri2 24342 | Triangle inequality for th... |
| psmetsym 24343 | The distance function of a... |
| psmettri 24344 | Triangle inequality for th... |
| psmetge0 24345 | The distance function of a... |
| psmetxrge0 24346 | The distance function of a... |
| psmetres2 24347 | Restriction of a pseudomet... |
| psmetlecl 24348 | Real closure of an extende... |
| distspace 24349 | A set ` X ` together with ... |
| ismet 24356 | Express the predicate " ` ... |
| isxmet 24357 | Express the predicate " ` ... |
| ismeti 24358 | Properties that determine ... |
| isxmetd 24359 | Properties that determine ... |
| isxmet2d 24360 | It is safe to only require... |
| metflem 24361 | Lemma for ~ metf and other... |
| xmetf 24362 | Mapping of the distance fu... |
| metf 24363 | Mapping of the distance fu... |
| xmetcl 24364 | Closure of the distance fu... |
| metcl 24365 | Closure of the distance fu... |
| ismet2 24366 | An extended metric is a me... |
| metxmet 24367 | A metric is an extended me... |
| xmetdmdm 24368 | Recover the base set from ... |
| metdmdm 24369 | Recover the base set from ... |
| xmetunirn 24370 | Two ways to express an ext... |
| xmeteq0 24371 | The value of an extended m... |
| meteq0 24372 | The value of a metric is z... |
| xmettri2 24373 | Triangle inequality for th... |
| mettri2 24374 | Triangle inequality for th... |
| xmet0 24375 | The distance function of a... |
| met0 24376 | The distance function of a... |
| xmetge0 24377 | The distance function of a... |
| metge0 24378 | The distance function of a... |
| xmetlecl 24379 | Real closure of an extende... |
| xmetsym 24380 | The distance function of a... |
| xmetpsmet 24381 | An extended metric is a ps... |
| xmettpos 24382 | The distance function of a... |
| metsym 24383 | The distance function of a... |
| xmettri 24384 | Triangle inequality for th... |
| mettri 24385 | Triangle inequality for th... |
| xmettri3 24386 | Triangle inequality for th... |
| mettri3 24387 | Triangle inequality for th... |
| xmetrtri 24388 | One half of the reverse tr... |
| xmetrtri2 24389 | The reverse triangle inequ... |
| metrtri 24390 | Reverse triangle inequalit... |
| xmetgt0 24391 | The distance function of a... |
| metgt0 24392 | The distance function of a... |
| metn0 24393 | A metric space is nonempty... |
| xmetres2 24394 | Restriction of an extended... |
| metreslem 24395 | Lemma for ~ metres . (Con... |
| metres2 24396 | Lemma for ~ metres . (Con... |
| xmetres 24397 | A restriction of an extend... |
| metres 24398 | A restriction of a metric ... |
| 0met 24399 | The empty metric. (Contri... |
| prdsdsf 24400 | The product metric is a fu... |
| prdsxmetlem 24401 | The product metric is an e... |
| prdsxmet 24402 | The product metric is an e... |
| prdsmet 24403 | The product metric is a me... |
| ressprdsds 24404 | Restriction of a product m... |
| resspwsds 24405 | Restriction of a power met... |
| imasdsf1olem 24406 | Lemma for ~ imasdsf1o . (... |
| imasdsf1o 24407 | The distance function is t... |
| imasf1oxmet 24408 | The image of an extended m... |
| imasf1omet 24409 | The image of a metric is a... |
| xpsdsfn 24410 | Closure of the metric in a... |
| xpsdsfn2 24411 | Closure of the metric in a... |
| xpsxmetlem 24412 | Lemma for ~ xpsxmet . (Co... |
| xpsxmet 24413 | A product metric of extend... |
| xpsdsval 24414 | Value of the metric in a b... |
| xpsmet 24415 | The direct product of two ... |
| blfvalps 24416 | The value of the ball func... |
| blfval 24417 | The value of the ball func... |
| blvalps 24418 | The ball around a point ` ... |
| blval 24419 | The ball around a point ` ... |
| elblps 24420 | Membership in a ball. (Co... |
| elbl 24421 | Membership in a ball. (Co... |
| elbl2ps 24422 | Membership in a ball. (Co... |
| elbl2 24423 | Membership in a ball. (Co... |
| elbl3ps 24424 | Membership in a ball, with... |
| elbl3 24425 | Membership in a ball, with... |
| blcomps 24426 | Commute the arguments to t... |
| blcom 24427 | Commute the arguments to t... |
| xblpnfps 24428 | The infinity ball in an ex... |
| xblpnf 24429 | The infinity ball in an ex... |
| blpnf 24430 | The infinity ball in a sta... |
| bldisj 24431 | Two balls are disjoint if ... |
| blgt0 24432 | A nonempty ball implies th... |
| bl2in 24433 | Two balls are disjoint if ... |
| xblss2ps 24434 | One ball is contained in a... |
| xblss2 24435 | One ball is contained in a... |
| blss2ps 24436 | One ball is contained in a... |
| blss2 24437 | One ball is contained in a... |
| blhalf 24438 | A ball of radius ` R / 2 `... |
| blfps 24439 | Mapping of a ball. (Contr... |
| blf 24440 | Mapping of a ball. (Contr... |
| blrnps 24441 | Membership in the range of... |
| blrn 24442 | Membership in the range of... |
| xblcntrps 24443 | A ball contains its center... |
| xblcntr 24444 | A ball contains its center... |
| blcntrps 24445 | A ball contains its center... |
| blcntr 24446 | A ball contains its center... |
| xbln0 24447 | A ball is nonempty iff the... |
| bln0 24448 | A ball is not empty. (Con... |
| blelrnps 24449 | A ball belongs to the set ... |
| blelrn 24450 | A ball belongs to the set ... |
| blssm 24451 | A ball is a subset of the ... |
| unirnblps 24452 | The union of the set of ba... |
| unirnbl 24453 | The union of the set of ba... |
| blin 24454 | The intersection of two ba... |
| ssblps 24455 | The size of a ball increas... |
| ssbl 24456 | The size of a ball increas... |
| blssps 24457 | Any point ` P ` in a ball ... |
| blss 24458 | Any point ` P ` in a ball ... |
| blssexps 24459 | Two ways to express the ex... |
| blssex 24460 | Two ways to express the ex... |
| ssblex 24461 | A nested ball exists whose... |
| blin2 24462 | Given any two balls and a ... |
| blbas 24463 | The balls of a metric spac... |
| blres 24464 | A ball in a restricted met... |
| xmeterval 24465 | Value of the "finitely sep... |
| xmeter 24466 | The "finitely separated" r... |
| xmetec 24467 | The equivalence classes un... |
| blssec 24468 | A ball centered at ` P ` i... |
| blpnfctr 24469 | The infinity ball in an ex... |
| xmetresbl 24470 | An extended metric restric... |
| mopnval 24471 | An open set is a subset of... |
| mopntopon 24472 | The set of open sets of a ... |
| mopntop 24473 | The set of open sets of a ... |
| mopnuni 24474 | The union of all open sets... |
| elmopn 24475 | The defining property of a... |
| mopnfss 24476 | The family of open sets of... |
| mopnm 24477 | The base set of a metric s... |
| elmopn2 24478 | A defining property of an ... |
| mopnss 24479 | An open set of a metric sp... |
| isxms 24480 | Express the predicate " ` ... |
| isxms2 24481 | Express the predicate " ` ... |
| isms 24482 | Express the predicate " ` ... |
| isms2 24483 | Express the predicate " ` ... |
| xmstopn 24484 | The topology component of ... |
| mstopn 24485 | The topology component of ... |
| xmstps 24486 | An extended metric space i... |
| msxms 24487 | A metric space is an exten... |
| mstps 24488 | A metric space is a topolo... |
| xmsxmet 24489 | The distance function, sui... |
| msmet 24490 | The distance function, sui... |
| msf 24491 | The distance function of a... |
| xmsxmet2 24492 | The distance function, sui... |
| msmet2 24493 | The distance function, sui... |
| mscl 24494 | Closure of the distance fu... |
| xmscl 24495 | Closure of the distance fu... |
| xmsge0 24496 | The distance function in a... |
| xmseq0 24497 | The distance between two p... |
| xmssym 24498 | The distance function in a... |
| xmstri2 24499 | Triangle inequality for th... |
| mstri2 24500 | Triangle inequality for th... |
| xmstri 24501 | Triangle inequality for th... |
| mstri 24502 | Triangle inequality for th... |
| xmstri3 24503 | Triangle inequality for th... |
| mstri3 24504 | Triangle inequality for th... |
| msrtri 24505 | Reverse triangle inequalit... |
| xmspropd 24506 | Property deduction for an ... |
| mspropd 24507 | Property deduction for a m... |
| setsmsbas 24508 | The base set of a construc... |
| setsmsbasOLD 24509 | Obsolete proof of ~ setsms... |
| setsmsds 24510 | The distance function of a... |
| setsmsdsOLD 24511 | Obsolete proof of ~ setsms... |
| setsmstset 24512 | The topology of a construc... |
| setsmstopn 24513 | The topology of a construc... |
| setsxms 24514 | The constructed metric spa... |
| setsms 24515 | The constructed metric spa... |
| tmsval 24516 | For any metric there is an... |
| tmslem 24517 | Lemma for ~ tmsbas , ~ tms... |
| tmslemOLD 24518 | Obsolete version of ~ tmsl... |
| tmsbas 24519 | The base set of a construc... |
| tmsds 24520 | The metric of a constructe... |
| tmstopn 24521 | The topology of a construc... |
| tmsxms 24522 | The constructed metric spa... |
| tmsms 24523 | The constructed metric spa... |
| imasf1obl 24524 | The image of a metric spac... |
| imasf1oxms 24525 | The image of a metric spac... |
| imasf1oms 24526 | The image of a metric spac... |
| prdsbl 24527 | A ball in the product metr... |
| mopni 24528 | An open set of a metric sp... |
| mopni2 24529 | An open set of a metric sp... |
| mopni3 24530 | An open set of a metric sp... |
| blssopn 24531 | The balls of a metric spac... |
| unimopn 24532 | The union of a collection ... |
| mopnin 24533 | The intersection of two op... |
| mopn0 24534 | The empty set is an open s... |
| rnblopn 24535 | A ball of a metric space i... |
| blopn 24536 | A ball of a metric space i... |
| neibl 24537 | The neighborhoods around a... |
| blnei 24538 | A ball around a point is a... |
| lpbl 24539 | Every ball around a limit ... |
| blsscls2 24540 | A smaller closed ball is c... |
| blcld 24541 | A "closed ball" in a metri... |
| blcls 24542 | The closure of an open bal... |
| blsscls 24543 | If two concentric balls ha... |
| metss 24544 | Two ways of saying that me... |
| metequiv 24545 | Two ways of saying that tw... |
| metequiv2 24546 | If there is a sequence of ... |
| metss2lem 24547 | Lemma for ~ metss2 . (Con... |
| metss2 24548 | If the metric ` D ` is "st... |
| comet 24549 | The composition of an exte... |
| stdbdmetval 24550 | Value of the standard boun... |
| stdbdxmet 24551 | The standard bounded metri... |
| stdbdmet 24552 | The standard bounded metri... |
| stdbdbl 24553 | The standard bounded metri... |
| stdbdmopn 24554 | The standard bounded metri... |
| mopnex 24555 | The topology generated by ... |
| methaus 24556 | The topology generated by ... |
| met1stc 24557 | The topology generated by ... |
| met2ndci 24558 | A separable metric space (... |
| met2ndc 24559 | A metric space is second-c... |
| metrest 24560 | Two alternate formulations... |
| ressxms 24561 | The restriction of a metri... |
| ressms 24562 | The restriction of a metri... |
| prdsmslem1 24563 | Lemma for ~ prdsms . The ... |
| prdsxmslem1 24564 | Lemma for ~ prdsms . The ... |
| prdsxmslem2 24565 | Lemma for ~ prdsxms . The... |
| prdsxms 24566 | The indexed product struct... |
| prdsms 24567 | The indexed product struct... |
| pwsxms 24568 | A power of an extended met... |
| pwsms 24569 | A power of a metric space ... |
| xpsxms 24570 | A binary product of metric... |
| xpsms 24571 | A binary product of metric... |
| tmsxps 24572 | Express the product of two... |
| tmsxpsmopn 24573 | Express the product of two... |
| tmsxpsval 24574 | Value of the product of tw... |
| tmsxpsval2 24575 | Value of the product of tw... |
| metcnp3 24576 | Two ways to express that `... |
| metcnp 24577 | Two ways to say a mapping ... |
| metcnp2 24578 | Two ways to say a mapping ... |
| metcn 24579 | Two ways to say a mapping ... |
| metcnpi 24580 | Epsilon-delta property of ... |
| metcnpi2 24581 | Epsilon-delta property of ... |
| metcnpi3 24582 | Epsilon-delta property of ... |
| txmetcnp 24583 | Continuity of a binary ope... |
| txmetcn 24584 | Continuity of a binary ope... |
| metuval 24585 | Value of the uniform struc... |
| metustel 24586 | Define a filter base ` F `... |
| metustss 24587 | Range of the elements of t... |
| metustrel 24588 | Elements of the filter bas... |
| metustto 24589 | Any two elements of the fi... |
| metustid 24590 | The identity diagonal is i... |
| metustsym 24591 | Elements of the filter bas... |
| metustexhalf 24592 | For any element ` A ` of t... |
| metustfbas 24593 | The filter base generated ... |
| metust 24594 | The uniform structure gene... |
| cfilucfil 24595 | Given a metric ` D ` and a... |
| metuust 24596 | The uniform structure gene... |
| cfilucfil2 24597 | Given a metric ` D ` and a... |
| blval2 24598 | The ball around a point ` ... |
| elbl4 24599 | Membership in a ball, alte... |
| metuel 24600 | Elementhood in the uniform... |
| metuel2 24601 | Elementhood in the uniform... |
| metustbl 24602 | The "section" image of an ... |
| psmetutop 24603 | The topology induced by a ... |
| xmetutop 24604 | The topology induced by a ... |
| xmsusp 24605 | If the uniform set of a me... |
| restmetu 24606 | The uniform structure gene... |
| metucn 24607 | Uniform continuity in metr... |
| dscmet 24608 | The discrete metric on any... |
| dscopn 24609 | The discrete metric genera... |
| nrmmetd 24610 | Show that a group norm gen... |
| abvmet 24611 | An absolute value ` F ` ge... |
| nmfval 24624 | The value of the norm func... |
| nmval 24625 | The value of the norm as t... |
| nmfval0 24626 | The value of the norm func... |
| nmfval2 24627 | The value of the norm func... |
| nmval2 24628 | The value of the norm on a... |
| nmf2 24629 | The norm on a metric group... |
| nmpropd 24630 | Weak property deduction fo... |
| nmpropd2 24631 | Strong property deduction ... |
| isngp 24632 | The property of being a no... |
| isngp2 24633 | The property of being a no... |
| isngp3 24634 | The property of being a no... |
| ngpgrp 24635 | A normed group is a group.... |
| ngpms 24636 | A normed group is a metric... |
| ngpxms 24637 | A normed group is an exten... |
| ngptps 24638 | A normed group is a topolo... |
| ngpmet 24639 | The (induced) metric of a ... |
| ngpds 24640 | Value of the distance func... |
| ngpdsr 24641 | Value of the distance func... |
| ngpds2 24642 | Write the distance between... |
| ngpds2r 24643 | Write the distance between... |
| ngpds3 24644 | Write the distance between... |
| ngpds3r 24645 | Write the distance between... |
| ngprcan 24646 | Cancel right addition insi... |
| ngplcan 24647 | Cancel left addition insid... |
| isngp4 24648 | Express the property of be... |
| ngpinvds 24649 | Two elements are the same ... |
| ngpsubcan 24650 | Cancel right subtraction i... |
| nmf 24651 | The norm on a normed group... |
| nmcl 24652 | The norm of a normed group... |
| nmge0 24653 | The norm of a normed group... |
| nmeq0 24654 | The identity is the only e... |
| nmne0 24655 | The norm of a nonzero elem... |
| nmrpcl 24656 | The norm of a nonzero elem... |
| nminv 24657 | The norm of a negated elem... |
| nmmtri 24658 | The triangle inequality fo... |
| nmsub 24659 | The norm of the difference... |
| nmrtri 24660 | Reverse triangle inequalit... |
| nm2dif 24661 | Inequality for the differe... |
| nmtri 24662 | The triangle inequality fo... |
| nmtri2 24663 | Triangle inequality for th... |
| ngpi 24664 | The properties of a normed... |
| nm0 24665 | Norm of the identity eleme... |
| nmgt0 24666 | The norm of a nonzero elem... |
| sgrim 24667 | The induced metric on a su... |
| sgrimval 24668 | The induced metric on a su... |
| subgnm 24669 | The norm in a subgroup. (... |
| subgnm2 24670 | A substructure assigns the... |
| subgngp 24671 | A normed group restricted ... |
| ngptgp 24672 | A normed abelian group is ... |
| ngppropd 24673 | Property deduction for a n... |
| reldmtng 24674 | The function ` toNrmGrp ` ... |
| tngval 24675 | Value of the function whic... |
| tnglem 24676 | Lemma for ~ tngbas and sim... |
| tnglemOLD 24677 | Obsolete version of ~ tngl... |
| tngbas 24678 | The base set of a structur... |
| tngbasOLD 24679 | Obsolete proof of ~ tngbas... |
| tngplusg 24680 | The group addition of a st... |
| tngplusgOLD 24681 | Obsolete proof of ~ tngplu... |
| tng0 24682 | The group identity of a st... |
| tngmulr 24683 | The ring multiplication of... |
| tngmulrOLD 24684 | Obsolete proof of ~ tngmul... |
| tngsca 24685 | The scalar ring of a struc... |
| tngscaOLD 24686 | Obsolete proof of ~ tngsca... |
| tngvsca 24687 | The scalar multiplication ... |
| tngvscaOLD 24688 | Obsolete proof of ~ tngvsc... |
| tngip 24689 | The inner product operatio... |
| tngipOLD 24690 | Obsolete proof of ~ tngip ... |
| tngds 24691 | The metric function of a s... |
| tngdsOLD 24692 | Obsolete proof of ~ tngds ... |
| tngtset 24693 | The topology generated by ... |
| tngtopn 24694 | The topology generated by ... |
| tngnm 24695 | The topology generated by ... |
| tngngp2 24696 | A norm turns a group into ... |
| tngngpd 24697 | Derive the axioms for a no... |
| tngngp 24698 | Derive the axioms for a no... |
| tnggrpr 24699 | If a structure equipped wi... |
| tngngp3 24700 | Alternate definition of a ... |
| nrmtngdist 24701 | The augmentation of a norm... |
| nrmtngnrm 24702 | The augmentation of a norm... |
| tngngpim 24703 | The induced metric of a no... |
| isnrg 24704 | A normed ring is a ring wi... |
| nrgabv 24705 | The norm of a normed ring ... |
| nrgngp 24706 | A normed ring is a normed ... |
| nrgring 24707 | A normed ring is a ring. ... |
| nmmul 24708 | The norm of a product in a... |
| nrgdsdi 24709 | Distribute a distance calc... |
| nrgdsdir 24710 | Distribute a distance calc... |
| nm1 24711 | The norm of one in a nonze... |
| unitnmn0 24712 | The norm of a unit is nonz... |
| nminvr 24713 | The norm of an inverse in ... |
| nmdvr 24714 | The norm of a division in ... |
| nrgdomn 24715 | A nonzero normed ring is a... |
| nrgtgp 24716 | A normed ring is a topolog... |
| subrgnrg 24717 | A normed ring restricted t... |
| tngnrg 24718 | Given any absolute value o... |
| isnlm 24719 | A normed (left) module is ... |
| nmvs 24720 | Defining property of a nor... |
| nlmngp 24721 | A normed module is a norme... |
| nlmlmod 24722 | A normed module is a left ... |
| nlmnrg 24723 | The scalar component of a ... |
| nlmngp2 24724 | The scalar component of a ... |
| nlmdsdi 24725 | Distribute a distance calc... |
| nlmdsdir 24726 | Distribute a distance calc... |
| nlmmul0or 24727 | If a scalar product is zer... |
| sranlm 24728 | The subring algebra over a... |
| nlmvscnlem2 24729 | Lemma for ~ nlmvscn . Com... |
| nlmvscnlem1 24730 | Lemma for ~ nlmvscn . (Co... |
| nlmvscn 24731 | The scalar multiplication ... |
| rlmnlm 24732 | The ring module over a nor... |
| rlmnm 24733 | The norm function in the r... |
| nrgtrg 24734 | A normed ring is a topolog... |
| nrginvrcnlem 24735 | Lemma for ~ nrginvrcn . C... |
| nrginvrcn 24736 | The ring inverse function ... |
| nrgtdrg 24737 | A normed division ring is ... |
| nlmtlm 24738 | A normed module is a topol... |
| isnvc 24739 | A normed vector space is j... |
| nvcnlm 24740 | A normed vector space is a... |
| nvclvec 24741 | A normed vector space is a... |
| nvclmod 24742 | A normed vector space is a... |
| isnvc2 24743 | A normed vector space is j... |
| nvctvc 24744 | A normed vector space is a... |
| lssnlm 24745 | A subspace of a normed mod... |
| lssnvc 24746 | A subspace of a normed vec... |
| rlmnvc 24747 | The ring module over a nor... |
| ngpocelbl 24748 | Membership of an off-cente... |
| nmoffn 24755 | The function producing ope... |
| reldmnghm 24756 | Lemma for normed group hom... |
| reldmnmhm 24757 | Lemma for module homomorph... |
| nmofval 24758 | Value of the operator norm... |
| nmoval 24759 | Value of the operator norm... |
| nmogelb 24760 | Property of the operator n... |
| nmolb 24761 | Any upper bound on the val... |
| nmolb2d 24762 | Any upper bound on the val... |
| nmof 24763 | The operator norm is a fun... |
| nmocl 24764 | The operator norm of an op... |
| nmoge0 24765 | The operator norm of an op... |
| nghmfval 24766 | A normed group homomorphis... |
| isnghm 24767 | A normed group homomorphis... |
| isnghm2 24768 | A normed group homomorphis... |
| isnghm3 24769 | A normed group homomorphis... |
| bddnghm 24770 | A bounded group homomorphi... |
| nghmcl 24771 | A normed group homomorphis... |
| nmoi 24772 | The operator norm achieves... |
| nmoix 24773 | The operator norm is a bou... |
| nmoi2 24774 | The operator norm is a bou... |
| nmoleub 24775 | The operator norm, defined... |
| nghmrcl1 24776 | Reverse closure for a norm... |
| nghmrcl2 24777 | Reverse closure for a norm... |
| nghmghm 24778 | A normed group homomorphis... |
| nmo0 24779 | The operator norm of the z... |
| nmoeq0 24780 | The operator norm is zero ... |
| nmoco 24781 | An upper bound on the oper... |
| nghmco 24782 | The composition of normed ... |
| nmotri 24783 | Triangle inequality for th... |
| nghmplusg 24784 | The sum of two bounded lin... |
| 0nghm 24785 | The zero operator is a nor... |
| nmoid 24786 | The operator norm of the i... |
| idnghm 24787 | The identity operator is a... |
| nmods 24788 | Upper bound for the distan... |
| nghmcn 24789 | A normed group homomorphis... |
| isnmhm 24790 | A normed module homomorphi... |
| nmhmrcl1 24791 | Reverse closure for a norm... |
| nmhmrcl2 24792 | Reverse closure for a norm... |
| nmhmlmhm 24793 | A normed module homomorphi... |
| nmhmnghm 24794 | A normed module homomorphi... |
| nmhmghm 24795 | A normed module homomorphi... |
| isnmhm2 24796 | A normed module homomorphi... |
| nmhmcl 24797 | A normed module homomorphi... |
| idnmhm 24798 | The identity operator is a... |
| 0nmhm 24799 | The zero operator is a bou... |
| nmhmco 24800 | The composition of bounded... |
| nmhmplusg 24801 | The sum of two bounded lin... |
| qtopbaslem 24802 | The set of open intervals ... |
| qtopbas 24803 | The set of open intervals ... |
| retopbas 24804 | A basis for the standard t... |
| retop 24805 | The standard topology on t... |
| uniretop 24806 | The underlying set of the ... |
| retopon 24807 | The standard topology on t... |
| retps 24808 | The standard topological s... |
| iooretop 24809 | Open intervals are open se... |
| icccld 24810 | Closed intervals are close... |
| icopnfcld 24811 | Right-unbounded closed int... |
| iocmnfcld 24812 | Left-unbounded closed inte... |
| qdensere 24813 | ` QQ ` is dense in the sta... |
| cnmetdval 24814 | Value of the distance func... |
| cnmet 24815 | The absolute value metric ... |
| cnxmet 24816 | The absolute value metric ... |
| cnbl0 24817 | Two ways to write the open... |
| cnblcld 24818 | Two ways to write the clos... |
| cnfldms 24819 | The complex number field i... |
| cnfldxms 24820 | The complex number field i... |
| cnfldtps 24821 | The complex number field i... |
| cnfldnm 24822 | The norm of the field of c... |
| cnngp 24823 | The complex numbers form a... |
| cnnrg 24824 | The complex numbers form a... |
| cnfldtopn 24825 | The topology of the comple... |
| cnfldtopon 24826 | The topology of the comple... |
| cnfldtop 24827 | The topology of the comple... |
| cnfldhaus 24828 | The topology of the comple... |
| unicntop 24829 | The underlying set of the ... |
| cnopn 24830 | The set of complex numbers... |
| zringnrg 24831 | The ring of integers is a ... |
| remetdval 24832 | Value of the distance func... |
| remet 24833 | The absolute value metric ... |
| rexmet 24834 | The absolute value metric ... |
| bl2ioo 24835 | A ball in terms of an open... |
| ioo2bl 24836 | An open interval of reals ... |
| ioo2blex 24837 | An open interval of reals ... |
| blssioo 24838 | The balls of the standard ... |
| tgioo 24839 | The topology generated by ... |
| qdensere2 24840 | ` QQ ` is dense in ` RR ` ... |
| blcvx 24841 | An open ball in the comple... |
| rehaus 24842 | The standard topology on t... |
| tgqioo 24843 | The topology generated by ... |
| re2ndc 24844 | The standard topology on t... |
| resubmet 24845 | The subspace topology indu... |
| tgioo2 24846 | The standard topology on t... |
| rerest 24847 | The subspace topology indu... |
| tgioo3 24848 | The standard topology on t... |
| xrtgioo 24849 | The topology on the extend... |
| xrrest 24850 | The subspace topology indu... |
| xrrest2 24851 | The subspace topology indu... |
| xrsxmet 24852 | The metric on the extended... |
| xrsdsre 24853 | The metric on the extended... |
| xrsblre 24854 | Any ball of the metric of ... |
| xrsmopn 24855 | The metric on the extended... |
| zcld 24856 | The integers are a closed ... |
| recld2 24857 | The real numbers are a clo... |
| zcld2 24858 | The integers are a closed ... |
| zdis 24859 | The integers are a discret... |
| sszcld 24860 | Every subset of the intege... |
| reperflem 24861 | A subset of the real numbe... |
| reperf 24862 | The real numbers are a per... |
| cnperf 24863 | The complex numbers are a ... |
| iccntr 24864 | The interior of a closed i... |
| icccmplem1 24865 | Lemma for ~ icccmp . (Con... |
| icccmplem2 24866 | Lemma for ~ icccmp . (Con... |
| icccmplem3 24867 | Lemma for ~ icccmp . (Con... |
| icccmp 24868 | A closed interval in ` RR ... |
| reconnlem1 24869 | Lemma for ~ reconn . Conn... |
| reconnlem2 24870 | Lemma for ~ reconn . (Con... |
| reconn 24871 | A subset of the reals is c... |
| retopconn 24872 | Corollary of ~ reconn . T... |
| iccconn 24873 | A closed interval is conne... |
| opnreen 24874 | Every nonempty open set is... |
| rectbntr0 24875 | A countable subset of the ... |
| xrge0gsumle 24876 | A finite sum in the nonneg... |
| xrge0tsms 24877 | Any finite or infinite sum... |
| xrge0tsms2 24878 | Any finite or infinite sum... |
| metdcnlem 24879 | The metric function of a m... |
| xmetdcn2 24880 | The metric function of an ... |
| xmetdcn 24881 | The metric function of an ... |
| metdcn2 24882 | The metric function of a m... |
| metdcn 24883 | The metric function of a m... |
| msdcn 24884 | The metric function of a m... |
| cnmpt1ds 24885 | Continuity of the metric f... |
| cnmpt2ds 24886 | Continuity of the metric f... |
| nmcn 24887 | The norm of a normed group... |
| ngnmcncn 24888 | The norm of a normed group... |
| abscn 24889 | The absolute value functio... |
| metdsval 24890 | Value of the "distance to ... |
| metdsf 24891 | The distance from a point ... |
| metdsge 24892 | The distance from the poin... |
| metds0 24893 | If a point is in a set, it... |
| metdstri 24894 | A generalization of the tr... |
| metdsle 24895 | The distance from a point ... |
| metdsre 24896 | The distance from a point ... |
| metdseq0 24897 | The distance from a point ... |
| metdscnlem 24898 | Lemma for ~ metdscn . (Co... |
| metdscn 24899 | The function ` F ` which g... |
| metdscn2 24900 | The function ` F ` which g... |
| metnrmlem1a 24901 | Lemma for ~ metnrm . (Con... |
| metnrmlem1 24902 | Lemma for ~ metnrm . (Con... |
| metnrmlem2 24903 | Lemma for ~ metnrm . (Con... |
| metnrmlem3 24904 | Lemma for ~ metnrm . (Con... |
| metnrm 24905 | A metric space is normal. ... |
| metreg 24906 | A metric space is regular.... |
| addcnlem 24907 | Lemma for ~ addcn , ~ subc... |
| addcn 24908 | Complex number addition is... |
| subcn 24909 | Complex number subtraction... |
| mulcn 24910 | Complex number multiplicat... |
| divcnOLD 24911 | Obsolete version of ~ divc... |
| mpomulcn 24912 | Complex number multiplicat... |
| divcn 24913 | Complex number division is... |
| cnfldtgp 24914 | The complex numbers form a... |
| fsumcn 24915 | A finite sum of functions ... |
| fsum2cn 24916 | Version of ~ fsumcn for tw... |
| expcn 24917 | The power function on comp... |
| divccn 24918 | Division by a nonzero cons... |
| expcnOLD 24919 | Obsolete version of ~ expc... |
| divccnOLD 24920 | Obsolete version of ~ divc... |
| sqcn 24921 | The square function on com... |
| iitopon 24926 | The unit interval is a top... |
| iitop 24927 | The unit interval is a top... |
| iiuni 24928 | The base set of the unit i... |
| dfii2 24929 | Alternate definition of th... |
| dfii3 24930 | Alternate definition of th... |
| dfii4 24931 | Alternate definition of th... |
| dfii5 24932 | The unit interval expresse... |
| iicmp 24933 | The unit interval is compa... |
| iiconn 24934 | The unit interval is conne... |
| cncfval 24935 | The value of the continuou... |
| elcncf 24936 | Membership in the set of c... |
| elcncf2 24937 | Version of ~ elcncf with a... |
| cncfrss 24938 | Reverse closure of the con... |
| cncfrss2 24939 | Reverse closure of the con... |
| cncff 24940 | A continuous complex funct... |
| cncfi 24941 | Defining property of a con... |
| elcncf1di 24942 | Membership in the set of c... |
| elcncf1ii 24943 | Membership in the set of c... |
| rescncf 24944 | A continuous complex funct... |
| cncfcdm 24945 | Change the codomain of a c... |
| cncfss 24946 | The set of continuous func... |
| climcncf 24947 | Image of a limit under a c... |
| abscncf 24948 | Absolute value is continuo... |
| recncf 24949 | Real part is continuous. ... |
| imcncf 24950 | Imaginary part is continuo... |
| cjcncf 24951 | Complex conjugate is conti... |
| mulc1cncf 24952 | Multiplication by a consta... |
| divccncf 24953 | Division by a constant is ... |
| cncfco 24954 | The composition of two con... |
| cncfcompt2 24955 | Composition of continuous ... |
| cncfmet 24956 | Relate complex function co... |
| cncfcn 24957 | Relate complex function co... |
| cncfcn1 24958 | Relate complex function co... |
| cncfmptc 24959 | A constant function is a c... |
| cncfmptid 24960 | The identity function is a... |
| cncfmpt1f 24961 | Composition of continuous ... |
| cncfmpt2f 24962 | Composition of continuous ... |
| cncfmpt2ss 24963 | Composition of continuous ... |
| addccncf 24964 | Adding a constant is a con... |
| idcncf 24965 | The identity function is a... |
| sub1cncf 24966 | Subtracting a constant is ... |
| sub2cncf 24967 | Subtraction from a constan... |
| cdivcncf 24968 | Division with a constant n... |
| negcncf 24969 | The negative function is c... |
| negcncfOLD 24970 | Obsolete version of ~ negc... |
| negfcncf 24971 | The negative of a continuo... |
| abscncfALT 24972 | Absolute value is continuo... |
| cncfcnvcn 24973 | Rewrite ~ cmphaushmeo for ... |
| expcncf 24974 | The power function on comp... |
| cnmptre 24975 | Lemma for ~ iirevcn and re... |
| cnmpopc 24976 | Piecewise definition of a ... |
| iirev 24977 | Reverse the unit interval.... |
| iirevcn 24978 | The reversion function is ... |
| iihalf1 24979 | Map the first half of ` II... |
| iihalf1cn 24980 | The first half function is... |
| iihalf1cnOLD 24981 | Obsolete version of ~ iiha... |
| iihalf2 24982 | Map the second half of ` I... |
| iihalf2cn 24983 | The second half function i... |
| iihalf2cnOLD 24984 | Obsolete version of ~ iiha... |
| elii1 24985 | Divide the unit interval i... |
| elii2 24986 | Divide the unit interval i... |
| iimulcl 24987 | The unit interval is close... |
| iimulcn 24988 | Multiplication is a contin... |
| iimulcnOLD 24989 | Obsolete version of ~ iimu... |
| icoopnst 24990 | A half-open interval start... |
| iocopnst 24991 | A half-open interval endin... |
| icchmeo 24992 | The natural bijection from... |
| icchmeoOLD 24993 | Obsolete version of ~ icch... |
| icopnfcnv 24994 | Define a bijection from ` ... |
| icopnfhmeo 24995 | The defined bijection from... |
| iccpnfcnv 24996 | Define a bijection from ` ... |
| iccpnfhmeo 24997 | The defined bijection from... |
| xrhmeo 24998 | The bijection from ` [ -u ... |
| xrhmph 24999 | The extended reals are hom... |
| xrcmp 25000 | The topology of the extend... |
| xrconn 25001 | The topology of the extend... |
| icccvx 25002 | A linear combination of tw... |
| oprpiece1res1 25003 | Restriction to the first p... |
| oprpiece1res2 25004 | Restriction to the second ... |
| cnrehmeo 25005 | The canonical bijection fr... |
| cnrehmeoOLD 25006 | Obsolete version of ~ cnre... |
| cnheiborlem 25007 | Lemma for ~ cnheibor . (C... |
| cnheibor 25008 | Heine-Borel theorem for co... |
| cnllycmp 25009 | The topology on the comple... |
| rellycmp 25010 | The topology on the reals ... |
| bndth 25011 | The Boundedness Theorem. ... |
| evth 25012 | The Extreme Value Theorem.... |
| evth2 25013 | The Extreme Value Theorem,... |
| lebnumlem1 25014 | Lemma for ~ lebnum . The ... |
| lebnumlem2 25015 | Lemma for ~ lebnum . As a... |
| lebnumlem3 25016 | Lemma for ~ lebnum . By t... |
| lebnum 25017 | The Lebesgue number lemma,... |
| xlebnum 25018 | Generalize ~ lebnum to ext... |
| lebnumii 25019 | Specialize the Lebesgue nu... |
| ishtpy 25025 | Membership in the class of... |
| htpycn 25026 | A homotopy is a continuous... |
| htpyi 25027 | A homotopy evaluated at it... |
| ishtpyd 25028 | Deduction for membership i... |
| htpycom 25029 | Given a homotopy from ` F ... |
| htpyid 25030 | A homotopy from a function... |
| htpyco1 25031 | Compose a homotopy with a ... |
| htpyco2 25032 | Compose a homotopy with a ... |
| htpycc 25033 | Concatenate two homotopies... |
| isphtpy 25034 | Membership in the class of... |
| phtpyhtpy 25035 | A path homotopy is a homot... |
| phtpycn 25036 | A path homotopy is a conti... |
| phtpyi 25037 | Membership in the class of... |
| phtpy01 25038 | Two path-homotopic paths h... |
| isphtpyd 25039 | Deduction for membership i... |
| isphtpy2d 25040 | Deduction for membership i... |
| phtpycom 25041 | Given a homotopy from ` F ... |
| phtpyid 25042 | A homotopy from a path to ... |
| phtpyco2 25043 | Compose a path homotopy wi... |
| phtpycc 25044 | Concatenate two path homot... |
| phtpcrel 25046 | The path homotopy relation... |
| isphtpc 25047 | The relation "is path homo... |
| phtpcer 25048 | Path homotopy is an equiva... |
| phtpc01 25049 | Path homotopic paths have ... |
| reparphti 25050 | Lemma for ~ reparpht . (C... |
| reparphtiOLD 25051 | Obsolete version of ~ repa... |
| reparpht 25052 | Reparametrization lemma. ... |
| phtpcco2 25053 | Compose a path homotopy wi... |
| pcofval 25064 | The value of the path conc... |
| pcoval 25065 | The concatenation of two p... |
| pcovalg 25066 | Evaluate the concatenation... |
| pcoval1 25067 | Evaluate the concatenation... |
| pco0 25068 | The starting point of a pa... |
| pco1 25069 | The ending point of a path... |
| pcoval2 25070 | Evaluate the concatenation... |
| pcocn 25071 | The concatenation of two p... |
| copco 25072 | The composition of a conca... |
| pcohtpylem 25073 | Lemma for ~ pcohtpy . (Co... |
| pcohtpy 25074 | Homotopy invariance of pat... |
| pcoptcl 25075 | A constant function is a p... |
| pcopt 25076 | Concatenation with a point... |
| pcopt2 25077 | Concatenation with a point... |
| pcoass 25078 | Order of concatenation doe... |
| pcorevcl 25079 | Closure for a reversed pat... |
| pcorevlem 25080 | Lemma for ~ pcorev . Prov... |
| pcorev 25081 | Concatenation with the rev... |
| pcorev2 25082 | Concatenation with the rev... |
| pcophtb 25083 | The path homotopy equivale... |
| om1val 25084 | The definition of the loop... |
| om1bas 25085 | The base set of the loop s... |
| om1elbas 25086 | Elementhood in the base se... |
| om1addcl 25087 | Closure of the group opera... |
| om1plusg 25088 | The group operation (which... |
| om1tset 25089 | The topology of the loop s... |
| om1opn 25090 | The topology of the loop s... |
| pi1val 25091 | The definition of the fund... |
| pi1bas 25092 | The base set of the fundam... |
| pi1blem 25093 | Lemma for ~ pi1buni . (Co... |
| pi1buni 25094 | Another way to write the l... |
| pi1bas2 25095 | The base set of the fundam... |
| pi1eluni 25096 | Elementhood in the base se... |
| pi1bas3 25097 | The base set of the fundam... |
| pi1cpbl 25098 | The group operation, loop ... |
| elpi1 25099 | The elements of the fundam... |
| elpi1i 25100 | The elements of the fundam... |
| pi1addf 25101 | The group operation of ` p... |
| pi1addval 25102 | The concatenation of two p... |
| pi1grplem 25103 | Lemma for ~ pi1grp . (Con... |
| pi1grp 25104 | The fundamental group is a... |
| pi1id 25105 | The identity element of th... |
| pi1inv 25106 | An inverse in the fundamen... |
| pi1xfrf 25107 | Functionality of the loop ... |
| pi1xfrval 25108 | The value of the loop tran... |
| pi1xfr 25109 | Given a path ` F ` and its... |
| pi1xfrcnvlem 25110 | Given a path ` F ` between... |
| pi1xfrcnv 25111 | Given a path ` F ` between... |
| pi1xfrgim 25112 | The mapping ` G ` between ... |
| pi1cof 25113 | Functionality of the loop ... |
| pi1coval 25114 | The value of the loop tran... |
| pi1coghm 25115 | The mapping ` G ` between ... |
| isclm 25118 | A subcomplex module is a l... |
| clmsca 25119 | The ring of scalars ` F ` ... |
| clmsubrg 25120 | The base set of the ring o... |
| clmlmod 25121 | A subcomplex module is a l... |
| clmgrp 25122 | A subcomplex module is an ... |
| clmabl 25123 | A subcomplex module is an ... |
| clmring 25124 | The scalar ring of a subco... |
| clmfgrp 25125 | The scalar ring of a subco... |
| clm0 25126 | The zero of the scalar rin... |
| clm1 25127 | The identity of the scalar... |
| clmadd 25128 | The addition of the scalar... |
| clmmul 25129 | The multiplication of the ... |
| clmcj 25130 | The conjugation of the sca... |
| isclmi 25131 | Reverse direction of ~ isc... |
| clmzss 25132 | The scalar ring of a subco... |
| clmsscn 25133 | The scalar ring of a subco... |
| clmsub 25134 | Subtraction in the scalar ... |
| clmneg 25135 | Negation in the scalar rin... |
| clmneg1 25136 | Minus one is in the scalar... |
| clmabs 25137 | Norm in the scalar ring of... |
| clmacl 25138 | Closure of ring addition f... |
| clmmcl 25139 | Closure of ring multiplica... |
| clmsubcl 25140 | Closure of ring subtractio... |
| lmhmclm 25141 | The domain of a linear ope... |
| clmvscl 25142 | Closure of scalar product ... |
| clmvsass 25143 | Associative law for scalar... |
| clmvscom 25144 | Commutative law for the sc... |
| clmvsdir 25145 | Distributive law for scala... |
| clmvsdi 25146 | Distributive law for scala... |
| clmvs1 25147 | Scalar product with ring u... |
| clmvs2 25148 | A vector plus itself is tw... |
| clm0vs 25149 | Zero times a vector is the... |
| clmopfne 25150 | The (functionalized) opera... |
| isclmp 25151 | The predicate "is a subcom... |
| isclmi0 25152 | Properties that determine ... |
| clmvneg1 25153 | Minus 1 times a vector is ... |
| clmvsneg 25154 | Multiplication of a vector... |
| clmmulg 25155 | The group multiple functio... |
| clmsubdir 25156 | Scalar multiplication dist... |
| clmpm1dir 25157 | Subtractive distributive l... |
| clmnegneg 25158 | Double negative of a vecto... |
| clmnegsubdi2 25159 | Distribution of negative o... |
| clmsub4 25160 | Rearrangement of 4 terms i... |
| clmvsrinv 25161 | A vector minus itself. (C... |
| clmvslinv 25162 | Minus a vector plus itself... |
| clmvsubval 25163 | Value of vector subtractio... |
| clmvsubval2 25164 | Value of vector subtractio... |
| clmvz 25165 | Two ways to express the ne... |
| zlmclm 25166 | The ` ZZ ` -module operati... |
| clmzlmvsca 25167 | The scalar product of a su... |
| nmoleub2lem 25168 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem3 25169 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem2 25170 | Lemma for ~ nmoleub2a and ... |
| nmoleub2a 25171 | The operator norm is the s... |
| nmoleub2b 25172 | The operator norm is the s... |
| nmoleub3 25173 | The operator norm is the s... |
| nmhmcn 25174 | A linear operator over a n... |
| cmodscexp 25175 | The powers of ` _i ` belon... |
| cmodscmulexp 25176 | The scalar product of a ve... |
| cvslvec 25179 | A subcomplex vector space ... |
| cvsclm 25180 | A subcomplex vector space ... |
| iscvs 25181 | A subcomplex vector space ... |
| iscvsp 25182 | The predicate "is a subcom... |
| iscvsi 25183 | Properties that determine ... |
| cvsi 25184 | The properties of a subcom... |
| cvsunit 25185 | Unit group of the scalar r... |
| cvsdiv 25186 | Division of the scalar rin... |
| cvsdivcl 25187 | The scalar field of a subc... |
| cvsmuleqdivd 25188 | An equality involving rati... |
| cvsdiveqd 25189 | An equality involving rati... |
| cnlmodlem1 25190 | Lemma 1 for ~ cnlmod . (C... |
| cnlmodlem2 25191 | Lemma 2 for ~ cnlmod . (C... |
| cnlmodlem3 25192 | Lemma 3 for ~ cnlmod . (C... |
| cnlmod4 25193 | Lemma 4 for ~ cnlmod . (C... |
| cnlmod 25194 | The set of complex numbers... |
| cnstrcvs 25195 | The set of complex numbers... |
| cnrbas 25196 | The set of complex numbers... |
| cnrlmod 25197 | The complex left module of... |
| cnrlvec 25198 | The complex left module of... |
| cncvs 25199 | The complex left module of... |
| recvs 25200 | The field of the real numb... |
| recvsOLD 25201 | Obsolete version of ~ recv... |
| qcvs 25202 | The field of rational numb... |
| zclmncvs 25203 | The ring of integers as le... |
| isncvsngp 25204 | A normed subcomplex vector... |
| isncvsngpd 25205 | Properties that determine ... |
| ncvsi 25206 | The properties of a normed... |
| ncvsprp 25207 | Proportionality property o... |
| ncvsge0 25208 | The norm of a scalar produ... |
| ncvsm1 25209 | The norm of the opposite o... |
| ncvsdif 25210 | The norm of the difference... |
| ncvspi 25211 | The norm of a vector plus ... |
| ncvs1 25212 | From any nonzero vector of... |
| cnrnvc 25213 | The module of complex numb... |
| cnncvs 25214 | The module of complex numb... |
| cnnm 25215 | The norm of the normed sub... |
| ncvspds 25216 | Value of the distance func... |
| cnindmet 25217 | The metric induced on the ... |
| cnncvsaddassdemo 25218 | Derive the associative law... |
| cnncvsmulassdemo 25219 | Derive the associative law... |
| cnncvsabsnegdemo 25220 | Derive the absolute value ... |
| iscph 25225 | A subcomplex pre-Hilbert s... |
| cphphl 25226 | A subcomplex pre-Hilbert s... |
| cphnlm 25227 | A subcomplex pre-Hilbert s... |
| cphngp 25228 | A subcomplex pre-Hilbert s... |
| cphlmod 25229 | A subcomplex pre-Hilbert s... |
| cphlvec 25230 | A subcomplex pre-Hilbert s... |
| cphnvc 25231 | A subcomplex pre-Hilbert s... |
| cphsubrglem 25232 | Lemma for ~ cphsubrg . (C... |
| cphreccllem 25233 | Lemma for ~ cphreccl . (C... |
| cphsca 25234 | A subcomplex pre-Hilbert s... |
| cphsubrg 25235 | The scalar field of a subc... |
| cphreccl 25236 | The scalar field of a subc... |
| cphdivcl 25237 | The scalar field of a subc... |
| cphcjcl 25238 | The scalar field of a subc... |
| cphsqrtcl 25239 | The scalar field of a subc... |
| cphabscl 25240 | The scalar field of a subc... |
| cphsqrtcl2 25241 | The scalar field of a subc... |
| cphsqrtcl3 25242 | If the scalar field of a s... |
| cphqss 25243 | The scalar field of a subc... |
| cphclm 25244 | A subcomplex pre-Hilbert s... |
| cphnmvs 25245 | Norm of a scalar product. ... |
| cphipcl 25246 | An inner product is a memb... |
| cphnmfval 25247 | The value of the norm in a... |
| cphnm 25248 | The square of the norm is ... |
| nmsq 25249 | The square of the norm is ... |
| cphnmf 25250 | The norm of a vector is a ... |
| cphnmcl 25251 | The norm of a vector is a ... |
| reipcl 25252 | An inner product of an ele... |
| ipge0 25253 | The inner product in a sub... |
| cphipcj 25254 | Conjugate of an inner prod... |
| cphipipcj 25255 | An inner product times its... |
| cphorthcom 25256 | Orthogonality (meaning inn... |
| cphip0l 25257 | Inner product with a zero ... |
| cphip0r 25258 | Inner product with a zero ... |
| cphipeq0 25259 | The inner product of a vec... |
| cphdir 25260 | Distributive law for inner... |
| cphdi 25261 | Distributive law for inner... |
| cph2di 25262 | Distributive law for inner... |
| cphsubdir 25263 | Distributive law for inner... |
| cphsubdi 25264 | Distributive law for inner... |
| cph2subdi 25265 | Distributive law for inner... |
| cphass 25266 | Associative law for inner ... |
| cphassr 25267 | "Associative" law for seco... |
| cph2ass 25268 | Move scalar multiplication... |
| cphassi 25269 | Associative law for the fi... |
| cphassir 25270 | "Associative" law for the ... |
| cphpyth 25271 | The pythagorean theorem fo... |
| tcphex 25272 | Lemma for ~ tcphbas and si... |
| tcphval 25273 | Define a function to augme... |
| tcphbas 25274 | The base set of a subcompl... |
| tchplusg 25275 | The addition operation of ... |
| tcphsub 25276 | The subtraction operation ... |
| tcphmulr 25277 | The ring operation of a su... |
| tcphsca 25278 | The scalar field of a subc... |
| tcphvsca 25279 | The scalar multiplication ... |
| tcphip 25280 | The inner product of a sub... |
| tcphtopn 25281 | The topology of a subcompl... |
| tcphphl 25282 | Augmentation of a subcompl... |
| tchnmfval 25283 | The norm of a subcomplex p... |
| tcphnmval 25284 | The norm of a subcomplex p... |
| cphtcphnm 25285 | The norm of a norm-augment... |
| tcphds 25286 | The distance of a pre-Hilb... |
| phclm 25287 | A pre-Hilbert space whose ... |
| tcphcphlem3 25288 | Lemma for ~ tcphcph : real... |
| ipcau2 25289 | The Cauchy-Schwarz inequal... |
| tcphcphlem1 25290 | Lemma for ~ tcphcph : the ... |
| tcphcphlem2 25291 | Lemma for ~ tcphcph : homo... |
| tcphcph 25292 | The standard definition of... |
| ipcau 25293 | The Cauchy-Schwarz inequal... |
| nmparlem 25294 | Lemma for ~ nmpar . (Cont... |
| nmpar 25295 | A subcomplex pre-Hilbert s... |
| cphipval2 25296 | Value of the inner product... |
| 4cphipval2 25297 | Four times the inner produ... |
| cphipval 25298 | Value of the inner product... |
| ipcnlem2 25299 | The inner product operatio... |
| ipcnlem1 25300 | The inner product operatio... |
| ipcn 25301 | The inner product operatio... |
| cnmpt1ip 25302 | Continuity of inner produc... |
| cnmpt2ip 25303 | Continuity of inner produc... |
| csscld 25304 | A "closed subspace" in a s... |
| clsocv 25305 | The orthogonal complement ... |
| cphsscph 25306 | A subspace of a subcomplex... |
| lmmbr 25313 | Express the binary relatio... |
| lmmbr2 25314 | Express the binary relatio... |
| lmmbr3 25315 | Express the binary relatio... |
| lmmcvg 25316 | Convergence property of a ... |
| lmmbrf 25317 | Express the binary relatio... |
| lmnn 25318 | A condition that implies c... |
| cfilfval 25319 | The set of Cauchy filters ... |
| iscfil 25320 | The property of being a Ca... |
| iscfil2 25321 | The property of being a Ca... |
| cfilfil 25322 | A Cauchy filter is a filte... |
| cfili 25323 | Property of a Cauchy filte... |
| cfil3i 25324 | A Cauchy filter contains b... |
| cfilss 25325 | A filter finer than a Cauc... |
| fgcfil 25326 | The Cauchy filter conditio... |
| fmcfil 25327 | The Cauchy filter conditio... |
| iscfil3 25328 | A filter is Cauchy iff it ... |
| cfilfcls 25329 | Similar to ultrafilters ( ... |
| caufval 25330 | The set of Cauchy sequence... |
| iscau 25331 | Express the property " ` F... |
| iscau2 25332 | Express the property " ` F... |
| iscau3 25333 | Express the Cauchy sequenc... |
| iscau4 25334 | Express the property " ` F... |
| iscauf 25335 | Express the property " ` F... |
| caun0 25336 | A metric with a Cauchy seq... |
| caufpm 25337 | Inclusion of a Cauchy sequ... |
| caucfil 25338 | A Cauchy sequence predicat... |
| iscmet 25339 | The property " ` D ` is a ... |
| cmetcvg 25340 | The convergence of a Cauch... |
| cmetmet 25341 | A complete metric space is... |
| cmetmeti 25342 | A complete metric space is... |
| cmetcaulem 25343 | Lemma for ~ cmetcau . (Co... |
| cmetcau 25344 | The convergence of a Cauch... |
| iscmet3lem3 25345 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem1 25346 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem2 25347 | Lemma for ~ iscmet3 . (Co... |
| iscmet3 25348 | The property " ` D ` is a ... |
| iscmet2 25349 | A metric ` D ` is complete... |
| cfilresi 25350 | A Cauchy filter on a metri... |
| cfilres 25351 | Cauchy filter on a metric ... |
| caussi 25352 | Cauchy sequence on a metri... |
| causs 25353 | Cauchy sequence on a metri... |
| equivcfil 25354 | If the metric ` D ` is "st... |
| equivcau 25355 | If the metric ` D ` is "st... |
| lmle 25356 | If the distance from each ... |
| nglmle 25357 | If the norm of each member... |
| lmclim 25358 | Relate a limit on the metr... |
| lmclimf 25359 | Relate a limit on the metr... |
| metelcls 25360 | A point belongs to the clo... |
| metcld 25361 | A subset of a metric space... |
| metcld2 25362 | A subset of a metric space... |
| caubl 25363 | Sufficient condition to en... |
| caublcls 25364 | The convergent point of a ... |
| metcnp4 25365 | Two ways to say a mapping ... |
| metcn4 25366 | Two ways to say a mapping ... |
| iscmet3i 25367 | Properties that determine ... |
| lmcau 25368 | Every convergent sequence ... |
| flimcfil 25369 | Every convergent filter in... |
| metsscmetcld 25370 | A complete subspace of a m... |
| cmetss 25371 | A subspace of a complete m... |
| equivcmet 25372 | If two metrics are strongl... |
| relcmpcmet 25373 | If ` D ` is a metric space... |
| cmpcmet 25374 | A compact metric space is ... |
| cfilucfil3 25375 | Given a metric ` D ` and a... |
| cfilucfil4 25376 | Given a metric ` D ` and a... |
| cncmet 25377 | The set of complex numbers... |
| recmet 25378 | The real numbers are a com... |
| bcthlem1 25379 | Lemma for ~ bcth . Substi... |
| bcthlem2 25380 | Lemma for ~ bcth . The ba... |
| bcthlem3 25381 | Lemma for ~ bcth . The li... |
| bcthlem4 25382 | Lemma for ~ bcth . Given ... |
| bcthlem5 25383 | Lemma for ~ bcth . The pr... |
| bcth 25384 | Baire's Category Theorem. ... |
| bcth2 25385 | Baire's Category Theorem, ... |
| bcth3 25386 | Baire's Category Theorem, ... |
| isbn 25393 | A Banach space is a normed... |
| bnsca 25394 | The scalar field of a Bana... |
| bnnvc 25395 | A Banach space is a normed... |
| bnnlm 25396 | A Banach space is a normed... |
| bnngp 25397 | A Banach space is a normed... |
| bnlmod 25398 | A Banach space is a left m... |
| bncms 25399 | A Banach space is a comple... |
| iscms 25400 | A complete metric space is... |
| cmscmet 25401 | The induced metric on a co... |
| bncmet 25402 | The induced metric on Bana... |
| cmsms 25403 | A complete metric space is... |
| cmspropd 25404 | Property deduction for a c... |
| cmssmscld 25405 | The restriction of a metri... |
| cmsss 25406 | The restriction of a compl... |
| lssbn 25407 | A subspace of a Banach spa... |
| cmetcusp1 25408 | If the uniform set of a co... |
| cmetcusp 25409 | The uniform space generate... |
| cncms 25410 | The field of complex numbe... |
| cnflduss 25411 | The uniform structure of t... |
| cnfldcusp 25412 | The field of complex numbe... |
| resscdrg 25413 | The real numbers are a sub... |
| cncdrg 25414 | The only complete subfield... |
| srabn 25415 | The subring algebra over a... |
| rlmbn 25416 | The ring module over a com... |
| ishl 25417 | The predicate "is a subcom... |
| hlbn 25418 | Every subcomplex Hilbert s... |
| hlcph 25419 | Every subcomplex Hilbert s... |
| hlphl 25420 | Every subcomplex Hilbert s... |
| hlcms 25421 | Every subcomplex Hilbert s... |
| hlprlem 25422 | Lemma for ~ hlpr . (Contr... |
| hlress 25423 | The scalar field of a subc... |
| hlpr 25424 | The scalar field of a subc... |
| ishl2 25425 | A Hilbert space is a compl... |
| cphssphl 25426 | A Banach subspace of a sub... |
| cmslssbn 25427 | A complete linear subspace... |
| cmscsscms 25428 | A closed subspace of a com... |
| bncssbn 25429 | A closed subspace of a Ban... |
| cssbn 25430 | A complete subspace of a n... |
| csschl 25431 | A complete subspace of a c... |
| cmslsschl 25432 | A complete linear subspace... |
| chlcsschl 25433 | A closed subspace of a sub... |
| retopn 25434 | The topology of the real n... |
| recms 25435 | The real numbers form a co... |
| reust 25436 | The Uniform structure of t... |
| recusp 25437 | The real numbers form a co... |
| rrxval 25442 | Value of the generalized E... |
| rrxbase 25443 | The base of the generalize... |
| rrxprds 25444 | Expand the definition of t... |
| rrxip 25445 | The inner product of the g... |
| rrxnm 25446 | The norm of the generalize... |
| rrxcph 25447 | Generalized Euclidean real... |
| rrxds 25448 | The distance over generali... |
| rrxvsca 25449 | The scalar product over ge... |
| rrxplusgvscavalb 25450 | The result of the addition... |
| rrxsca 25451 | The field of real numbers ... |
| rrx0 25452 | The zero ("origin") in a g... |
| rrx0el 25453 | The zero ("origin") in a g... |
| csbren 25454 | Cauchy-Schwarz-Bunjakovsky... |
| trirn 25455 | Triangle inequality in R^n... |
| rrxf 25456 | Euclidean vectors as funct... |
| rrxfsupp 25457 | Euclidean vectors are of f... |
| rrxsuppss 25458 | Support of Euclidean vecto... |
| rrxmvallem 25459 | Support of the function us... |
| rrxmval 25460 | The value of the Euclidean... |
| rrxmfval 25461 | The value of the Euclidean... |
| rrxmetlem 25462 | Lemma for ~ rrxmet . (Con... |
| rrxmet 25463 | Euclidean space is a metri... |
| rrxdstprj1 25464 | The distance between two p... |
| rrxbasefi 25465 | The base of the generalize... |
| rrxdsfi 25466 | The distance over generali... |
| rrxmetfi 25467 | Euclidean space is a metri... |
| rrxdsfival 25468 | The value of the Euclidean... |
| ehlval 25469 | Value of the Euclidean spa... |
| ehlbase 25470 | The base of the Euclidean ... |
| ehl0base 25471 | The base of the Euclidean ... |
| ehl0 25472 | The Euclidean space of dim... |
| ehleudis 25473 | The Euclidean distance fun... |
| ehleudisval 25474 | The value of the Euclidean... |
| ehl1eudis 25475 | The Euclidean distance fun... |
| ehl1eudisval 25476 | The value of the Euclidean... |
| ehl2eudis 25477 | The Euclidean distance fun... |
| ehl2eudisval 25478 | The value of the Euclidean... |
| minveclem1 25479 | Lemma for ~ minvec . The ... |
| minveclem4c 25480 | Lemma for ~ minvec . The ... |
| minveclem2 25481 | Lemma for ~ minvec . Any ... |
| minveclem3a 25482 | Lemma for ~ minvec . ` D `... |
| minveclem3b 25483 | Lemma for ~ minvec . The ... |
| minveclem3 25484 | Lemma for ~ minvec . The ... |
| minveclem4a 25485 | Lemma for ~ minvec . ` F `... |
| minveclem4b 25486 | Lemma for ~ minvec . The ... |
| minveclem4 25487 | Lemma for ~ minvec . The ... |
| minveclem5 25488 | Lemma for ~ minvec . Disc... |
| minveclem6 25489 | Lemma for ~ minvec . Any ... |
| minveclem7 25490 | Lemma for ~ minvec . Sinc... |
| minvec 25491 | Minimizing vector theorem,... |
| pjthlem1 25492 | Lemma for ~ pjth . (Contr... |
| pjthlem2 25493 | Lemma for ~ pjth . (Contr... |
| pjth 25494 | Projection Theorem: Any H... |
| pjth2 25495 | Projection Theorem with ab... |
| cldcss 25496 | Corollary of the Projectio... |
| cldcss2 25497 | Corollary of the Projectio... |
| hlhil 25498 | Corollary of the Projectio... |
| addcncf 25499 | The addition of two contin... |
| subcncf 25500 | The subtraction of two con... |
| mulcncf 25501 | The multiplication of two ... |
| mulcncfOLD 25502 | Obsolete version of ~ mulc... |
| divcncf 25503 | The quotient of two contin... |
| pmltpclem1 25504 | Lemma for ~ pmltpc . (Con... |
| pmltpclem2 25505 | Lemma for ~ pmltpc . (Con... |
| pmltpc 25506 | Any function on the reals ... |
| ivthlem1 25507 | Lemma for ~ ivth . The se... |
| ivthlem2 25508 | Lemma for ~ ivth . Show t... |
| ivthlem3 25509 | Lemma for ~ ivth , the int... |
| ivth 25510 | The intermediate value the... |
| ivth2 25511 | The intermediate value the... |
| ivthle 25512 | The intermediate value the... |
| ivthle2 25513 | The intermediate value the... |
| ivthicc 25514 | The interval between any t... |
| evthicc 25515 | Specialization of the Extr... |
| evthicc2 25516 | Combine ~ ivthicc with ~ e... |
| cniccbdd 25517 | A continuous function on a... |
| ovolfcl 25522 | Closure for the interval e... |
| ovolfioo 25523 | Unpack the interval coveri... |
| ovolficc 25524 | Unpack the interval coveri... |
| ovolficcss 25525 | Any (closed) interval cove... |
| ovolfsval 25526 | The value of the interval ... |
| ovolfsf 25527 | Closure for the interval l... |
| ovolsf 25528 | Closure for the partial su... |
| ovolval 25529 | The value of the outer mea... |
| elovolmlem 25530 | Lemma for ~ elovolm and re... |
| elovolm 25531 | Elementhood in the set ` M... |
| elovolmr 25532 | Sufficient condition for e... |
| ovolmge0 25533 | The set ` M ` is composed ... |
| ovolcl 25534 | The volume of a set is an ... |
| ovollb 25535 | The outer volume is a lowe... |
| ovolgelb 25536 | The outer volume is the gr... |
| ovolge0 25537 | The volume of a set is alw... |
| ovolf 25538 | The domain and codomain of... |
| ovollecl 25539 | If an outer volume is boun... |
| ovolsslem 25540 | Lemma for ~ ovolss . (Con... |
| ovolss 25541 | The volume of a set is mon... |
| ovolsscl 25542 | If a set is contained in a... |
| ovolssnul 25543 | A subset of a nullset is n... |
| ovollb2lem 25544 | Lemma for ~ ovollb2 . (Co... |
| ovollb2 25545 | It is often more convenien... |
| ovolctb 25546 | The volume of a denumerabl... |
| ovolq 25547 | The rational numbers have ... |
| ovolctb2 25548 | The volume of a countable ... |
| ovol0 25549 | The empty set has 0 outer ... |
| ovolfi 25550 | A finite set has 0 outer L... |
| ovolsn 25551 | A singleton has 0 outer Le... |
| ovolunlem1a 25552 | Lemma for ~ ovolun . (Con... |
| ovolunlem1 25553 | Lemma for ~ ovolun . (Con... |
| ovolunlem2 25554 | Lemma for ~ ovolun . (Con... |
| ovolun 25555 | The Lebesgue outer measure... |
| ovolunnul 25556 | Adding a nullset does not ... |
| ovolfiniun 25557 | The Lebesgue outer measure... |
| ovoliunlem1 25558 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem2 25559 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem3 25560 | Lemma for ~ ovoliun . (Co... |
| ovoliun 25561 | The Lebesgue outer measure... |
| ovoliun2 25562 | The Lebesgue outer measure... |
| ovoliunnul 25563 | A countable union of nulls... |
| shft2rab 25564 | If ` B ` is a shift of ` A... |
| ovolshftlem1 25565 | Lemma for ~ ovolshft . (C... |
| ovolshftlem2 25566 | Lemma for ~ ovolshft . (C... |
| ovolshft 25567 | The Lebesgue outer measure... |
| sca2rab 25568 | If ` B ` is a scale of ` A... |
| ovolscalem1 25569 | Lemma for ~ ovolsca . (Co... |
| ovolscalem2 25570 | Lemma for ~ ovolshft . (C... |
| ovolsca 25571 | The Lebesgue outer measure... |
| ovolicc1 25572 | The measure of a closed in... |
| ovolicc2lem1 25573 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem2 25574 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem3 25575 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem4 25576 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem5 25577 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2 25578 | The measure of a closed in... |
| ovolicc 25579 | The measure of a closed in... |
| ovolicopnf 25580 | The measure of a right-unb... |
| ovolre 25581 | The measure of the real nu... |
| ismbl 25582 | The predicate " ` A ` is L... |
| ismbl2 25583 | From ~ ovolun , it suffice... |
| volres 25584 | A self-referencing abbrevi... |
| volf 25585 | The domain and codomain of... |
| mblvol 25586 | The volume of a measurable... |
| mblss 25587 | A measurable set is a subs... |
| mblsplit 25588 | The defining property of m... |
| volss 25589 | The Lebesgue measure is mo... |
| cmmbl 25590 | The complement of a measur... |
| nulmbl 25591 | A nullset is measurable. ... |
| nulmbl2 25592 | A set of outer measure zer... |
| unmbl 25593 | A union of measurable sets... |
| shftmbl 25594 | A shift of a measurable se... |
| 0mbl 25595 | The empty set is measurabl... |
| rembl 25596 | The set of all real number... |
| unidmvol 25597 | The union of the Lebesgue ... |
| inmbl 25598 | An intersection of measura... |
| difmbl 25599 | A difference of measurable... |
| finiunmbl 25600 | A finite union of measurab... |
| volun 25601 | The Lebesgue measure funct... |
| volinun 25602 | Addition of non-disjoint s... |
| volfiniun 25603 | The volume of a disjoint f... |
| iundisj 25604 | Rewrite a countable union ... |
| iundisj2 25605 | A disjoint union is disjoi... |
| voliunlem1 25606 | Lemma for ~ voliun . (Con... |
| voliunlem2 25607 | Lemma for ~ voliun . (Con... |
| voliunlem3 25608 | Lemma for ~ voliun . (Con... |
| iunmbl 25609 | The measurable sets are cl... |
| voliun 25610 | The Lebesgue measure funct... |
| volsuplem 25611 | Lemma for ~ volsup . (Con... |
| volsup 25612 | The volume of the limit of... |
| iunmbl2 25613 | The measurable sets are cl... |
| ioombl1lem1 25614 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem2 25615 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem3 25616 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem4 25617 | Lemma for ~ ioombl1 . (Co... |
| ioombl1 25618 | An open right-unbounded in... |
| icombl1 25619 | A closed unbounded-above i... |
| icombl 25620 | A closed-below, open-above... |
| ioombl 25621 | An open real interval is m... |
| iccmbl 25622 | A closed real interval is ... |
| iccvolcl 25623 | A closed real interval has... |
| ovolioo 25624 | The measure of an open int... |
| volioo 25625 | The measure of an open int... |
| ioovolcl 25626 | An open real interval has ... |
| ovolfs2 25627 | Alternative expression for... |
| ioorcl2 25628 | An open interval with fini... |
| ioorf 25629 | Define a function from ope... |
| ioorval 25630 | Define a function from ope... |
| ioorinv2 25631 | The function ` F ` is an "... |
| ioorinv 25632 | The function ` F ` is an "... |
| ioorcl 25633 | The function ` F ` does no... |
| uniiccdif 25634 | A union of closed interval... |
| uniioovol 25635 | A disjoint union of open i... |
| uniiccvol 25636 | An almost-disjoint union o... |
| uniioombllem1 25637 | Lemma for ~ uniioombl . (... |
| uniioombllem2a 25638 | Lemma for ~ uniioombl . (... |
| uniioombllem2 25639 | Lemma for ~ uniioombl . (... |
| uniioombllem3a 25640 | Lemma for ~ uniioombl . (... |
| uniioombllem3 25641 | Lemma for ~ uniioombl . (... |
| uniioombllem4 25642 | Lemma for ~ uniioombl . (... |
| uniioombllem5 25643 | Lemma for ~ uniioombl . (... |
| uniioombllem6 25644 | Lemma for ~ uniioombl . (... |
| uniioombl 25645 | A disjoint union of open i... |
| uniiccmbl 25646 | An almost-disjoint union o... |
| dyadf 25647 | The function ` F ` returns... |
| dyadval 25648 | Value of the dyadic ration... |
| dyadovol 25649 | Volume of a dyadic rationa... |
| dyadss 25650 | Two closed dyadic rational... |
| dyaddisjlem 25651 | Lemma for ~ dyaddisj . (C... |
| dyaddisj 25652 | Two closed dyadic rational... |
| dyadmaxlem 25653 | Lemma for ~ dyadmax . (Co... |
| dyadmax 25654 | Any nonempty set of dyadic... |
| dyadmbllem 25655 | Lemma for ~ dyadmbl . (Co... |
| dyadmbl 25656 | Any union of dyadic ration... |
| opnmbllem 25657 | Lemma for ~ opnmbl . (Con... |
| opnmbl 25658 | All open sets are measurab... |
| opnmblALT 25659 | All open sets are measurab... |
| subopnmbl 25660 | Sets which are open in a m... |
| volsup2 25661 | The volume of ` A ` is the... |
| volcn 25662 | The function formed by res... |
| volivth 25663 | The Intermediate Value The... |
| vitalilem1 25664 | Lemma for ~ vitali . (Con... |
| vitalilem2 25665 | Lemma for ~ vitali . (Con... |
| vitalilem3 25666 | Lemma for ~ vitali . (Con... |
| vitalilem4 25667 | Lemma for ~ vitali . (Con... |
| vitalilem5 25668 | Lemma for ~ vitali . (Con... |
| vitali 25669 | If the reals can be well-o... |
| ismbf1 25680 | The predicate " ` F ` is a... |
| mbff 25681 | A measurable function is a... |
| mbfdm 25682 | The domain of a measurable... |
| mbfconstlem 25683 | Lemma for ~ mbfconst and r... |
| ismbf 25684 | The predicate " ` F ` is a... |
| ismbfcn 25685 | A complex function is meas... |
| mbfima 25686 | Definitional property of a... |
| mbfimaicc 25687 | The preimage of any closed... |
| mbfimasn 25688 | The preimage of a point un... |
| mbfconst 25689 | A constant function is mea... |
| mbf0 25690 | The empty function is meas... |
| mbfid 25691 | The identity function is m... |
| mbfmptcl 25692 | Lemma for the ` MblFn ` pr... |
| mbfdm2 25693 | The domain of a measurable... |
| ismbfcn2 25694 | A complex function is meas... |
| ismbfd 25695 | Deduction to prove measura... |
| ismbf2d 25696 | Deduction to prove measura... |
| mbfeqalem1 25697 | Lemma for ~ mbfeqalem2 . ... |
| mbfeqalem2 25698 | Lemma for ~ mbfeqa . (Con... |
| mbfeqa 25699 | If two functions are equal... |
| mbfres 25700 | The restriction of a measu... |
| mbfres2 25701 | Measurability of a piecewi... |
| mbfss 25702 | Change the domain of a mea... |
| mbfmulc2lem 25703 | Multiplication by a consta... |
| mbfmulc2re 25704 | Multiplication by a consta... |
| mbfmax 25705 | The maximum of two functio... |
| mbfneg 25706 | The negative of a measurab... |
| mbfpos 25707 | The positive part of a mea... |
| mbfposr 25708 | Converse to ~ mbfpos . (C... |
| mbfposb 25709 | A function is measurable i... |
| ismbf3d 25710 | Simplified form of ~ ismbf... |
| mbfimaopnlem 25711 | Lemma for ~ mbfimaopn . (... |
| mbfimaopn 25712 | The preimage of any open s... |
| mbfimaopn2 25713 | The preimage of any set op... |
| cncombf 25714 | The composition of a conti... |
| cnmbf 25715 | A continuous function is m... |
| mbfaddlem 25716 | The sum of two measurable ... |
| mbfadd 25717 | The sum of two measurable ... |
| mbfsub 25718 | The difference of two meas... |
| mbfmulc2 25719 | A complex constant times a... |
| mbfsup 25720 | The supremum of a sequence... |
| mbfinf 25721 | The infimum of a sequence ... |
| mbflimsup 25722 | The limit supremum of a se... |
| mbflimlem 25723 | The pointwise limit of a s... |
| mbflim 25724 | The pointwise limit of a s... |
| 0pval 25727 | The zero function evaluate... |
| 0plef 25728 | Two ways to say that the f... |
| 0pledm 25729 | Adjust the domain of the l... |
| isi1f 25730 | The predicate " ` F ` is a... |
| i1fmbf 25731 | Simple functions are measu... |
| i1ff 25732 | A simple function is a fun... |
| i1frn 25733 | A simple function has fini... |
| i1fima 25734 | Any preimage of a simple f... |
| i1fima2 25735 | Any preimage of a simple f... |
| i1fima2sn 25736 | Preimage of a singleton. ... |
| i1fd 25737 | A simplified set of assump... |
| i1f0rn 25738 | Any simple function takes ... |
| itg1val 25739 | The value of the integral ... |
| itg1val2 25740 | The value of the integral ... |
| itg1cl 25741 | Closure of the integral on... |
| itg1ge0 25742 | Closure of the integral on... |
| i1f0 25743 | The zero function is simpl... |
| itg10 25744 | The zero function has zero... |
| i1f1lem 25745 | Lemma for ~ i1f1 and ~ itg... |
| i1f1 25746 | Base case simple functions... |
| itg11 25747 | The integral of an indicat... |
| itg1addlem1 25748 | Decompose a preimage, whic... |
| i1faddlem 25749 | Decompose the preimage of ... |
| i1fmullem 25750 | Decompose the preimage of ... |
| i1fadd 25751 | The sum of two simple func... |
| i1fmul 25752 | The pointwise product of t... |
| itg1addlem2 25753 | Lemma for ~ itg1add . The... |
| itg1addlem3 25754 | Lemma for ~ itg1add . (Co... |
| itg1addlem4 25755 | Lemma for ~ itg1add . (Co... |
| itg1addlem4OLD 25756 | Obsolete version of ~ itg1... |
| itg1addlem5 25757 | Lemma for ~ itg1add . (Co... |
| itg1add 25758 | The integral of a sum of s... |
| i1fmulclem 25759 | Decompose the preimage of ... |
| i1fmulc 25760 | A nonnegative constant tim... |
| itg1mulc 25761 | The integral of a constant... |
| i1fres 25762 | The "restriction" of a sim... |
| i1fpos 25763 | The positive part of a sim... |
| i1fposd 25764 | Deduction form of ~ i1fpos... |
| i1fsub 25765 | The difference of two simp... |
| itg1sub 25766 | The integral of a differen... |
| itg10a 25767 | The integral of a simple f... |
| itg1ge0a 25768 | The integral of an almost ... |
| itg1lea 25769 | Approximate version of ~ i... |
| itg1le 25770 | If one simple function dom... |
| itg1climres 25771 | Restricting the simple fun... |
| mbfi1fseqlem1 25772 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem2 25773 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem3 25774 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem4 25775 | Lemma for ~ mbfi1fseq . T... |
| mbfi1fseqlem5 25776 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseqlem6 25777 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseq 25778 | A characterization of meas... |
| mbfi1flimlem 25779 | Lemma for ~ mbfi1flim . (... |
| mbfi1flim 25780 | Any real measurable functi... |
| mbfmullem2 25781 | Lemma for ~ mbfmul . (Con... |
| mbfmullem 25782 | Lemma for ~ mbfmul . (Con... |
| mbfmul 25783 | The product of two measura... |
| itg2lcl 25784 | The set of lower sums is a... |
| itg2val 25785 | Value of the integral on n... |
| itg2l 25786 | Elementhood in the set ` L... |
| itg2lr 25787 | Sufficient condition for e... |
| xrge0f 25788 | A real function is a nonne... |
| itg2cl 25789 | The integral of a nonnegat... |
| itg2ub 25790 | The integral of a nonnegat... |
| itg2leub 25791 | Any upper bound on the int... |
| itg2ge0 25792 | The integral of a nonnegat... |
| itg2itg1 25793 | The integral of a nonnegat... |
| itg20 25794 | The integral of the zero f... |
| itg2lecl 25795 | If an ` S.2 ` integral is ... |
| itg2le 25796 | If one function dominates ... |
| itg2const 25797 | Integral of a constant fun... |
| itg2const2 25798 | When the base set of a con... |
| itg2seq 25799 | Definitional property of t... |
| itg2uba 25800 | Approximate version of ~ i... |
| itg2lea 25801 | Approximate version of ~ i... |
| itg2eqa 25802 | Approximate equality of in... |
| itg2mulclem 25803 | Lemma for ~ itg2mulc . (C... |
| itg2mulc 25804 | The integral of a nonnegat... |
| itg2splitlem 25805 | Lemma for ~ itg2split . (... |
| itg2split 25806 | The ` S.2 ` integral split... |
| itg2monolem1 25807 | Lemma for ~ itg2mono . We... |
| itg2monolem2 25808 | Lemma for ~ itg2mono . (C... |
| itg2monolem3 25809 | Lemma for ~ itg2mono . (C... |
| itg2mono 25810 | The Monotone Convergence T... |
| itg2i1fseqle 25811 | Subject to the conditions ... |
| itg2i1fseq 25812 | Subject to the conditions ... |
| itg2i1fseq2 25813 | In an extension to the res... |
| itg2i1fseq3 25814 | Special case of ~ itg2i1fs... |
| itg2addlem 25815 | Lemma for ~ itg2add . (Co... |
| itg2add 25816 | The ` S.2 ` integral is li... |
| itg2gt0 25817 | If the function ` F ` is s... |
| itg2cnlem1 25818 | Lemma for ~ itgcn . (Cont... |
| itg2cnlem2 25819 | Lemma for ~ itgcn . (Cont... |
| itg2cn 25820 | A sort of absolute continu... |
| ibllem 25821 | Conditioned equality theor... |
| isibl 25822 | The predicate " ` F ` is i... |
| isibl2 25823 | The predicate " ` F ` is i... |
| iblmbf 25824 | An integrable function is ... |
| iblitg 25825 | If a function is integrabl... |
| dfitg 25826 | Evaluate the class substit... |
| itgex 25827 | An integral is a set. (Co... |
| itgeq1f 25828 | Equality theorem for an in... |
| itgeq1fOLD 25829 | Obsolete version of ~ itge... |
| itgeq1 25830 | Equality theorem for an in... |
| nfitg1 25831 | Bound-variable hypothesis ... |
| nfitg 25832 | Bound-variable hypothesis ... |
| cbvitg 25833 | Change bound variable in a... |
| cbvitgv 25834 | Change bound variable in a... |
| itgeq2 25835 | Equality theorem for an in... |
| itgresr 25836 | The domain of an integral ... |
| itg0 25837 | The integral of anything o... |
| itgz 25838 | The integral of zero on an... |
| itgeq2dv 25839 | Equality theorem for an in... |
| itgmpt 25840 | Change bound variable in a... |
| itgcl 25841 | The integral of an integra... |
| itgvallem 25842 | Substitution lemma. (Cont... |
| itgvallem3 25843 | Lemma for ~ itgposval and ... |
| ibl0 25844 | The zero function is integ... |
| iblcnlem1 25845 | Lemma for ~ iblcnlem . (C... |
| iblcnlem 25846 | Expand out the universal q... |
| itgcnlem 25847 | Expand out the sum in ~ df... |
| iblrelem 25848 | Integrability of a real fu... |
| iblposlem 25849 | Lemma for ~ iblpos . (Con... |
| iblpos 25850 | Integrability of a nonnega... |
| iblre 25851 | Integrability of a real fu... |
| itgrevallem1 25852 | Lemma for ~ itgposval and ... |
| itgposval 25853 | The integral of a nonnegat... |
| itgreval 25854 | Decompose the integral of ... |
| itgrecl 25855 | Real closure of an integra... |
| iblcn 25856 | Integrability of a complex... |
| itgcnval 25857 | Decompose the integral of ... |
| itgre 25858 | Real part of an integral. ... |
| itgim 25859 | Imaginary part of an integ... |
| iblneg 25860 | The negative of an integra... |
| itgneg 25861 | Negation of an integral. ... |
| iblss 25862 | A subset of an integrable ... |
| iblss2 25863 | Change the domain of an in... |
| itgitg2 25864 | Transfer an integral using... |
| i1fibl 25865 | A simple function is integ... |
| itgitg1 25866 | Transfer an integral using... |
| itgle 25867 | Monotonicity of an integra... |
| itgge0 25868 | The integral of a positive... |
| itgss 25869 | Expand the set of an integ... |
| itgss2 25870 | Expand the set of an integ... |
| itgeqa 25871 | Approximate equality of in... |
| itgss3 25872 | Expand the set of an integ... |
| itgioo 25873 | Equality of integrals on o... |
| itgless 25874 | Expand the integral of a n... |
| iblconst 25875 | A constant function is int... |
| itgconst 25876 | Integral of a constant fun... |
| ibladdlem 25877 | Lemma for ~ ibladd . (Con... |
| ibladd 25878 | Add two integrals over the... |
| iblsub 25879 | Subtract two integrals ove... |
| itgaddlem1 25880 | Lemma for ~ itgadd . (Con... |
| itgaddlem2 25881 | Lemma for ~ itgadd . (Con... |
| itgadd 25882 | Add two integrals over the... |
| itgsub 25883 | Subtract two integrals ove... |
| itgfsum 25884 | Take a finite sum of integ... |
| iblabslem 25885 | Lemma for ~ iblabs . (Con... |
| iblabs 25886 | The absolute value of an i... |
| iblabsr 25887 | A measurable function is i... |
| iblmulc2 25888 | Multiply an integral by a ... |
| itgmulc2lem1 25889 | Lemma for ~ itgmulc2 : pos... |
| itgmulc2lem2 25890 | Lemma for ~ itgmulc2 : rea... |
| itgmulc2 25891 | Multiply an integral by a ... |
| itgabs 25892 | The triangle inequality fo... |
| itgsplit 25893 | The ` S. ` integral splits... |
| itgspliticc 25894 | The ` S. ` integral splits... |
| itgsplitioo 25895 | The ` S. ` integral splits... |
| bddmulibl 25896 | A bounded function times a... |
| bddibl 25897 | A bounded function is inte... |
| cniccibl 25898 | A continuous function on a... |
| bddiblnc 25899 | Choice-free proof of ~ bdd... |
| cnicciblnc 25900 | Choice-free proof of ~ cni... |
| itggt0 25901 | The integral of a strictly... |
| itgcn 25902 | Transfer ~ itg2cn to the f... |
| ditgeq1 25905 | Equality theorem for the d... |
| ditgeq2 25906 | Equality theorem for the d... |
| ditgeq3 25907 | Equality theorem for the d... |
| ditgeq3dv 25908 | Equality theorem for the d... |
| ditgex 25909 | A directed integral is a s... |
| ditg0 25910 | Value of the directed inte... |
| cbvditg 25911 | Change bound variable in a... |
| cbvditgv 25912 | Change bound variable in a... |
| ditgpos 25913 | Value of the directed inte... |
| ditgneg 25914 | Value of the directed inte... |
| ditgcl 25915 | Closure of a directed inte... |
| ditgswap 25916 | Reverse a directed integra... |
| ditgsplitlem 25917 | Lemma for ~ ditgsplit . (... |
| ditgsplit 25918 | This theorem is the raison... |
| reldv 25927 | The derivative function is... |
| limcvallem 25928 | Lemma for ~ ellimc . (Con... |
| limcfval 25929 | Value and set bounds on th... |
| ellimc 25930 | Value of the limit predica... |
| limcrcl 25931 | Reverse closure for the li... |
| limccl 25932 | Closure of the limit opera... |
| limcdif 25933 | It suffices to consider fu... |
| ellimc2 25934 | Write the definition of a ... |
| limcnlp 25935 | If ` B ` is not a limit po... |
| ellimc3 25936 | Write the epsilon-delta de... |
| limcflflem 25937 | Lemma for ~ limcflf . (Co... |
| limcflf 25938 | The limit operator can be ... |
| limcmo 25939 | If ` B ` is a limit point ... |
| limcmpt 25940 | Express the limit operator... |
| limcmpt2 25941 | Express the limit operator... |
| limcresi 25942 | Any limit of ` F ` is also... |
| limcres 25943 | If ` B ` is an interior po... |
| cnplimc 25944 | A function is continuous a... |
| cnlimc 25945 | ` F ` is a continuous func... |
| cnlimci 25946 | If ` F ` is a continuous f... |
| cnmptlimc 25947 | If ` F ` is a continuous f... |
| limccnp 25948 | If the limit of ` F ` at `... |
| limccnp2 25949 | The image of a convergent ... |
| limcco 25950 | Composition of two limits.... |
| limciun 25951 | A point is a limit of ` F ... |
| limcun 25952 | A point is a limit of ` F ... |
| dvlem 25953 | Closure for a difference q... |
| dvfval 25954 | Value and set bounds on th... |
| eldv 25955 | The differentiable predica... |
| dvcl 25956 | The derivative function ta... |
| dvbssntr 25957 | The set of differentiable ... |
| dvbss 25958 | The set of differentiable ... |
| dvbsss 25959 | The set of differentiable ... |
| perfdvf 25960 | The derivative is a functi... |
| recnprss 25961 | Both ` RR ` and ` CC ` are... |
| recnperf 25962 | Both ` RR ` and ` CC ` are... |
| dvfg 25963 | Explicitly write out the f... |
| dvf 25964 | The derivative is a functi... |
| dvfcn 25965 | The derivative is a functi... |
| dvreslem 25966 | Lemma for ~ dvres . (Cont... |
| dvres2lem 25967 | Lemma for ~ dvres2 . (Con... |
| dvres 25968 | Restriction of a derivativ... |
| dvres2 25969 | Restriction of the base se... |
| dvres3 25970 | Restriction of a complex d... |
| dvres3a 25971 | Restriction of a complex d... |
| dvidlem 25972 | Lemma for ~ dvid and ~ dvc... |
| dvmptresicc 25973 | Derivative of a function r... |
| dvconst 25974 | Derivative of a constant f... |
| dvid 25975 | Derivative of the identity... |
| dvcnp 25976 | The difference quotient is... |
| dvcnp2 25977 | A function is continuous a... |
| dvcnp2OLD 25978 | Obsolete version of ~ dvcn... |
| dvcn 25979 | A differentiable function ... |
| dvnfval 25980 | Value of the iterated deri... |
| dvnff 25981 | The iterated derivative is... |
| dvn0 25982 | Zero times iterated deriva... |
| dvnp1 25983 | Successor iterated derivat... |
| dvn1 25984 | One times iterated derivat... |
| dvnf 25985 | The N-times derivative is ... |
| dvnbss 25986 | The set of N-times differe... |
| dvnadd 25987 | The ` N ` -th derivative o... |
| dvn2bss 25988 | An N-times differentiable ... |
| dvnres 25989 | Multiple derivative versio... |
| cpnfval 25990 | Condition for n-times cont... |
| fncpn 25991 | The ` C^n ` object is a fu... |
| elcpn 25992 | Condition for n-times cont... |
| cpnord 25993 | ` C^n ` conditions are ord... |
| cpncn 25994 | A ` C^n ` function is cont... |
| cpnres 25995 | The restriction of a ` C^n... |
| dvaddbr 25996 | The sum rule for derivativ... |
| dvmulbr 25997 | The product rule for deriv... |
| dvmulbrOLD 25998 | Obsolete version of ~ dvmu... |
| dvadd 25999 | The sum rule for derivativ... |
| dvmul 26000 | The product rule for deriv... |
| dvaddf 26001 | The sum rule for everywher... |
| dvmulf 26002 | The product rule for every... |
| dvcmul 26003 | The product rule when one ... |
| dvcmulf 26004 | The product rule when one ... |
| dvcobr 26005 | The chain rule for derivat... |
| dvcobrOLD 26006 | Obsolete version of ~ dvco... |
| dvco 26007 | The chain rule for derivat... |
| dvcof 26008 | The chain rule for everywh... |
| dvcjbr 26009 | The derivative of the conj... |
| dvcj 26010 | The derivative of the conj... |
| dvfre 26011 | The derivative of a real f... |
| dvnfre 26012 | The ` N ` -th derivative o... |
| dvexp 26013 | Derivative of a power func... |
| dvexp2 26014 | Derivative of an exponenti... |
| dvrec 26015 | Derivative of the reciproc... |
| dvmptres3 26016 | Function-builder for deriv... |
| dvmptid 26017 | Function-builder for deriv... |
| dvmptc 26018 | Function-builder for deriv... |
| dvmptcl 26019 | Closure lemma for ~ dvmptc... |
| dvmptadd 26020 | Function-builder for deriv... |
| dvmptmul 26021 | Function-builder for deriv... |
| dvmptres2 26022 | Function-builder for deriv... |
| dvmptres 26023 | Function-builder for deriv... |
| dvmptcmul 26024 | Function-builder for deriv... |
| dvmptdivc 26025 | Function-builder for deriv... |
| dvmptneg 26026 | Function-builder for deriv... |
| dvmptsub 26027 | Function-builder for deriv... |
| dvmptcj 26028 | Function-builder for deriv... |
| dvmptre 26029 | Function-builder for deriv... |
| dvmptim 26030 | Function-builder for deriv... |
| dvmptntr 26031 | Function-builder for deriv... |
| dvmptco 26032 | Function-builder for deriv... |
| dvrecg 26033 | Derivative of the reciproc... |
| dvmptdiv 26034 | Function-builder for deriv... |
| dvmptfsum 26035 | Function-builder for deriv... |
| dvcnvlem 26036 | Lemma for ~ dvcnvre . (Co... |
| dvcnv 26037 | A weak version of ~ dvcnvr... |
| dvexp3 26038 | Derivative of an exponenti... |
| dveflem 26039 | Derivative of the exponent... |
| dvef 26040 | Derivative of the exponent... |
| dvsincos 26041 | Derivative of the sine and... |
| dvsin 26042 | Derivative of the sine fun... |
| dvcos 26043 | Derivative of the cosine f... |
| dvferm1lem 26044 | Lemma for ~ dvferm . (Con... |
| dvferm1 26045 | One-sided version of ~ dvf... |
| dvferm2lem 26046 | Lemma for ~ dvferm . (Con... |
| dvferm2 26047 | One-sided version of ~ dvf... |
| dvferm 26048 | Fermat's theorem on statio... |
| rollelem 26049 | Lemma for ~ rolle . (Cont... |
| rolle 26050 | Rolle's theorem. If ` F `... |
| cmvth 26051 | Cauchy's Mean Value Theore... |
| cmvthOLD 26052 | Obsolete version of ~ cmvt... |
| mvth 26053 | The Mean Value Theorem. I... |
| dvlip 26054 | A function with derivative... |
| dvlipcn 26055 | A complex function with de... |
| dvlip2 26056 | Combine the results of ~ d... |
| c1liplem1 26057 | Lemma for ~ c1lip1 . (Con... |
| c1lip1 26058 | C^1 functions are Lipschit... |
| c1lip2 26059 | C^1 functions are Lipschit... |
| c1lip3 26060 | C^1 functions are Lipschit... |
| dveq0 26061 | If a continuous function h... |
| dv11cn 26062 | Two functions defined on a... |
| dvgt0lem1 26063 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0lem2 26064 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0 26065 | A function on a closed int... |
| dvlt0 26066 | A function on a closed int... |
| dvge0 26067 | A function on a closed int... |
| dvle 26068 | If ` A ( x ) , C ( x ) ` a... |
| dvivthlem1 26069 | Lemma for ~ dvivth . (Con... |
| dvivthlem2 26070 | Lemma for ~ dvivth . (Con... |
| dvivth 26071 | Darboux' theorem, or the i... |
| dvne0 26072 | A function on a closed int... |
| dvne0f1 26073 | A function on a closed int... |
| lhop1lem 26074 | Lemma for ~ lhop1 . (Cont... |
| lhop1 26075 | L'Hôpital's Rule for... |
| lhop2 26076 | L'Hôpital's Rule for... |
| lhop 26077 | L'Hôpital's Rule. I... |
| dvcnvrelem1 26078 | Lemma for ~ dvcnvre . (Co... |
| dvcnvrelem2 26079 | Lemma for ~ dvcnvre . (Co... |
| dvcnvre 26080 | The derivative rule for in... |
| dvcvx 26081 | A real function with stric... |
| dvfsumle 26082 | Compare a finite sum to an... |
| dvfsumleOLD 26083 | Obsolete version of ~ dvfs... |
| dvfsumge 26084 | Compare a finite sum to an... |
| dvfsumabs 26085 | Compare a finite sum to an... |
| dvmptrecl 26086 | Real closure of a derivati... |
| dvfsumrlimf 26087 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem1 26088 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2 26089 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2OLD 26090 | Obsolete version of ~ dvfs... |
| dvfsumlem3 26091 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem4 26092 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlimge0 26093 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlim 26094 | Compare a finite sum to an... |
| dvfsumrlim2 26095 | Compare a finite sum to an... |
| dvfsumrlim3 26096 | Conjoin the statements of ... |
| dvfsum2 26097 | The reverse of ~ dvfsumrli... |
| ftc1lem1 26098 | Lemma for ~ ftc1a and ~ ft... |
| ftc1lem2 26099 | Lemma for ~ ftc1 . (Contr... |
| ftc1a 26100 | The Fundamental Theorem of... |
| ftc1lem3 26101 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem4 26102 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem5 26103 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem6 26104 | Lemma for ~ ftc1 . (Contr... |
| ftc1 26105 | The Fundamental Theorem of... |
| ftc1cn 26106 | Strengthen the assumptions... |
| ftc2 26107 | The Fundamental Theorem of... |
| ftc2ditglem 26108 | Lemma for ~ ftc2ditg . (C... |
| ftc2ditg 26109 | Directed integral analogue... |
| itgparts 26110 | Integration by parts. If ... |
| itgsubstlem 26111 | Lemma for ~ itgsubst . (C... |
| itgsubst 26112 | Integration by ` u ` -subs... |
| itgpowd 26113 | The integral of a monomial... |
| reldmmdeg 26118 | Multivariate degree is a b... |
| tdeglem1 26119 | Functionality of the total... |
| tdeglem3 26120 | Additivity of the total de... |
| tdeglem4 26121 | There is only one multi-in... |
| tdeglem2 26122 | Simplification of total de... |
| mdegfval 26123 | Value of the multivariate ... |
| mdegval 26124 | Value of the multivariate ... |
| mdegleb 26125 | Property of being of limit... |
| mdeglt 26126 | If there is an upper limit... |
| mdegldg 26127 | A nonzero polynomial has s... |
| mdegxrcl 26128 | Closure of polynomial degr... |
| mdegxrf 26129 | Functionality of polynomia... |
| mdegcl 26130 | Sharp closure for multivar... |
| mdeg0 26131 | Degree of the zero polynom... |
| mdegnn0cl 26132 | Degree of a nonzero polyno... |
| degltlem1 26133 | Theorem on arithmetic of e... |
| degltp1le 26134 | Theorem on arithmetic of e... |
| mdegaddle 26135 | The degree of a sum is at ... |
| mdegvscale 26136 | The degree of a scalar mul... |
| mdegvsca 26137 | The degree of a scalar mul... |
| mdegle0 26138 | A polynomial has nonpositi... |
| mdegmullem 26139 | Lemma for ~ mdegmulle2 . ... |
| mdegmulle2 26140 | The multivariate degree of... |
| deg1fval 26141 | Relate univariate polynomi... |
| deg1xrf 26142 | Functionality of univariat... |
| deg1xrcl 26143 | Closure of univariate poly... |
| deg1cl 26144 | Sharp closure of univariat... |
| mdegpropd 26145 | Property deduction for pol... |
| deg1fvi 26146 | Univariate polynomial degr... |
| deg1propd 26147 | Property deduction for pol... |
| deg1z 26148 | Degree of the zero univari... |
| deg1nn0cl 26149 | Degree of a nonzero univar... |
| deg1n0ima 26150 | Degree image of a set of p... |
| deg1nn0clb 26151 | A polynomial is nonzero if... |
| deg1lt0 26152 | A polynomial is zero iff i... |
| deg1ldg 26153 | A nonzero univariate polyn... |
| deg1ldgn 26154 | An index at which a polyno... |
| deg1ldgdomn 26155 | A nonzero univariate polyn... |
| deg1leb 26156 | Property of being of limit... |
| deg1val 26157 | Value of the univariate de... |
| deg1lt 26158 | If the degree of a univari... |
| deg1ge 26159 | Conversely, a nonzero coef... |
| coe1mul3 26160 | The coefficient vector of ... |
| coe1mul4 26161 | Value of the "leading" coe... |
| deg1addle 26162 | The degree of a sum is at ... |
| deg1addle2 26163 | If both factors have degre... |
| deg1add 26164 | Exact degree of a sum of t... |
| deg1vscale 26165 | The degree of a scalar tim... |
| deg1vsca 26166 | The degree of a scalar tim... |
| deg1invg 26167 | The degree of the negated ... |
| deg1suble 26168 | The degree of a difference... |
| deg1sub 26169 | Exact degree of a differen... |
| deg1mulle2 26170 | Produce a bound on the pro... |
| deg1sublt 26171 | Subtraction of two polynom... |
| deg1le0 26172 | A polynomial has nonpositi... |
| deg1sclle 26173 | A scalar polynomial has no... |
| deg1scl 26174 | A nonzero scalar polynomia... |
| deg1mul2 26175 | Degree of multiplication o... |
| deg1mul 26176 | Degree of multiplication o... |
| deg1mul3 26177 | Degree of multiplication o... |
| deg1mul3le 26178 | Degree of multiplication o... |
| deg1tmle 26179 | Limiting degree of a polyn... |
| deg1tm 26180 | Exact degree of a polynomi... |
| deg1pwle 26181 | Limiting degree of a varia... |
| deg1pw 26182 | Exact degree of a variable... |
| ply1nz 26183 | Univariate polynomials ove... |
| ply1nzb 26184 | Univariate polynomials are... |
| ply1domn 26185 | Corollary of ~ deg1mul2 : ... |
| ply1idom 26186 | The ring of univariate pol... |
| ply1divmo 26197 | Uniqueness of a quotient i... |
| ply1divex 26198 | Lemma for ~ ply1divalg : e... |
| ply1divalg 26199 | The division algorithm for... |
| ply1divalg2 26200 | Reverse the order of multi... |
| uc1pval 26201 | Value of the set of unitic... |
| isuc1p 26202 | Being a unitic polynomial.... |
| mon1pval 26203 | Value of the set of monic ... |
| ismon1p 26204 | Being a monic polynomial. ... |
| uc1pcl 26205 | Unitic polynomials are pol... |
| mon1pcl 26206 | Monic polynomials are poly... |
| uc1pn0 26207 | Unitic polynomials are not... |
| mon1pn0 26208 | Monic polynomials are not ... |
| uc1pdeg 26209 | Unitic polynomials have no... |
| uc1pldg 26210 | Unitic polynomials have un... |
| mon1pldg 26211 | Unitic polynomials have on... |
| mon1puc1p 26212 | Monic polynomials are unit... |
| uc1pmon1p 26213 | Make a unitic polynomial m... |
| deg1submon1p 26214 | The difference of two moni... |
| mon1pid 26215 | Monicity and degree of the... |
| q1pval 26216 | Value of the univariate po... |
| q1peqb 26217 | Characterizing property of... |
| q1pcl 26218 | Closure of the quotient by... |
| r1pval 26219 | Value of the polynomial re... |
| r1pcl 26220 | Closure of remainder follo... |
| r1pdeglt 26221 | The remainder has a degree... |
| r1pid 26222 | Express the original polyn... |
| r1pid2 26223 | Identity law for polynomia... |
| dvdsq1p 26224 | Divisibility in a polynomi... |
| dvdsr1p 26225 | Divisibility in a polynomi... |
| ply1remlem 26226 | A term of the form ` x - N... |
| ply1rem 26227 | The polynomial remainder t... |
| facth1 26228 | The factor theorem and its... |
| fta1glem1 26229 | Lemma for ~ fta1g . (Cont... |
| fta1glem2 26230 | Lemma for ~ fta1g . (Cont... |
| fta1g 26231 | The one-sided fundamental ... |
| fta1blem 26232 | Lemma for ~ fta1b . (Cont... |
| fta1b 26233 | The assumption that ` R ` ... |
| idomrootle 26234 | No element of an integral ... |
| drnguc1p 26235 | Over a division ring, all ... |
| ig1peu 26236 | There is a unique monic po... |
| ig1pval 26237 | Substitutions for the poly... |
| ig1pval2 26238 | Generator of the zero idea... |
| ig1pval3 26239 | Characterizing properties ... |
| ig1pcl 26240 | The monic generator of an ... |
| ig1pdvds 26241 | The monic generator of an ... |
| ig1prsp 26242 | Any ideal of polynomials o... |
| ply1lpir 26243 | The ring of polynomials ov... |
| ply1pid 26244 | The polynomials over a fie... |
| plyco0 26253 | Two ways to say that a fun... |
| plyval 26254 | Value of the polynomial se... |
| plybss 26255 | Reverse closure of the par... |
| elply 26256 | Definition of a polynomial... |
| elply2 26257 | The coefficient function c... |
| plyun0 26258 | The set of polynomials is ... |
| plyf 26259 | A polynomial is a function... |
| plyss 26260 | The polynomial set functio... |
| plyssc 26261 | Every polynomial ring is c... |
| elplyr 26262 | Sufficient condition for e... |
| elplyd 26263 | Sufficient condition for e... |
| ply1termlem 26264 | Lemma for ~ ply1term . (C... |
| ply1term 26265 | A one-term polynomial. (C... |
| plypow 26266 | A power is a polynomial. ... |
| plyconst 26267 | A constant function is a p... |
| ne0p 26268 | A test to show that a poly... |
| ply0 26269 | The zero function is a pol... |
| plyid 26270 | The identity function is a... |
| plyeq0lem 26271 | Lemma for ~ plyeq0 . If `... |
| plyeq0 26272 | If a polynomial is zero at... |
| plypf1 26273 | Write the set of complex p... |
| plyaddlem1 26274 | Derive the coefficient fun... |
| plymullem1 26275 | Derive the coefficient fun... |
| plyaddlem 26276 | Lemma for ~ plyadd . (Con... |
| plymullem 26277 | Lemma for ~ plymul . (Con... |
| plyadd 26278 | The sum of two polynomials... |
| plymul 26279 | The product of two polynom... |
| plysub 26280 | The difference of two poly... |
| plyaddcl 26281 | The sum of two polynomials... |
| plymulcl 26282 | The product of two polynom... |
| plysubcl 26283 | The difference of two poly... |
| coeval 26284 | Value of the coefficient f... |
| coeeulem 26285 | Lemma for ~ coeeu . (Cont... |
| coeeu 26286 | Uniqueness of the coeffici... |
| coelem 26287 | Lemma for properties of th... |
| coeeq 26288 | If ` A ` satisfies the pro... |
| dgrval 26289 | Value of the degree functi... |
| dgrlem 26290 | Lemma for ~ dgrcl and simi... |
| coef 26291 | The domain and codomain of... |
| coef2 26292 | The domain and codomain of... |
| coef3 26293 | The domain and codomain of... |
| dgrcl 26294 | The degree of any polynomi... |
| dgrub 26295 | If the ` M ` -th coefficie... |
| dgrub2 26296 | All the coefficients above... |
| dgrlb 26297 | If all the coefficients ab... |
| coeidlem 26298 | Lemma for ~ coeid . (Cont... |
| coeid 26299 | Reconstruct a polynomial a... |
| coeid2 26300 | Reconstruct a polynomial a... |
| coeid3 26301 | Reconstruct a polynomial a... |
| plyco 26302 | The composition of two pol... |
| coeeq2 26303 | Compute the coefficient fu... |
| dgrle 26304 | Given an explicit expressi... |
| dgreq 26305 | If the highest term in a p... |
| 0dgr 26306 | A constant function has de... |
| 0dgrb 26307 | A function has degree zero... |
| dgrnznn 26308 | A nonzero polynomial with ... |
| coefv0 26309 | The result of evaluating a... |
| coeaddlem 26310 | Lemma for ~ coeadd and ~ d... |
| coemullem 26311 | Lemma for ~ coemul and ~ d... |
| coeadd 26312 | The coefficient function o... |
| coemul 26313 | A coefficient of a product... |
| coe11 26314 | The coefficient function i... |
| coemulhi 26315 | The leading coefficient of... |
| coemulc 26316 | The coefficient function i... |
| coe0 26317 | The coefficients of the ze... |
| coesub 26318 | The coefficient function o... |
| coe1termlem 26319 | The coefficient function o... |
| coe1term 26320 | The coefficient function o... |
| dgr1term 26321 | The degree of a monomial. ... |
| plycn 26322 | A polynomial is a continuo... |
| plycnOLD 26323 | Obsolete version of ~ plyc... |
| dgr0 26324 | The degree of the zero pol... |
| coeidp 26325 | The coefficients of the id... |
| dgrid 26326 | The degree of the identity... |
| dgreq0 26327 | The leading coefficient of... |
| dgrlt 26328 | Two ways to say that the d... |
| dgradd 26329 | The degree of a sum of pol... |
| dgradd2 26330 | The degree of a sum of pol... |
| dgrmul2 26331 | The degree of a product of... |
| dgrmul 26332 | The degree of a product of... |
| dgrmulc 26333 | Scalar multiplication by a... |
| dgrsub 26334 | The degree of a difference... |
| dgrcolem1 26335 | The degree of a compositio... |
| dgrcolem2 26336 | Lemma for ~ dgrco . (Cont... |
| dgrco 26337 | The degree of a compositio... |
| plycjlem 26338 | Lemma for ~ plycj and ~ co... |
| plycj 26339 | The double conjugation of ... |
| coecj 26340 | Double conjugation of a po... |
| plyrecj 26341 | A polynomial with real coe... |
| plymul0or 26342 | Polynomial multiplication ... |
| ofmulrt 26343 | The set of roots of a prod... |
| plyreres 26344 | Real-coefficient polynomia... |
| dvply1 26345 | Derivative of a polynomial... |
| dvply2g 26346 | The derivative of a polyno... |
| dvply2gOLD 26347 | Obsolete version of ~ dvpl... |
| dvply2 26348 | The derivative of a polyno... |
| dvnply2 26349 | Polynomials have polynomia... |
| dvnply 26350 | Polynomials have polynomia... |
| plycpn 26351 | Polynomials are smooth. (... |
| quotval 26354 | Value of the quotient func... |
| plydivlem1 26355 | Lemma for ~ plydivalg . (... |
| plydivlem2 26356 | Lemma for ~ plydivalg . (... |
| plydivlem3 26357 | Lemma for ~ plydivex . Ba... |
| plydivlem4 26358 | Lemma for ~ plydivex . In... |
| plydivex 26359 | Lemma for ~ plydivalg . (... |
| plydiveu 26360 | Lemma for ~ plydivalg . (... |
| plydivalg 26361 | The division algorithm on ... |
| quotlem 26362 | Lemma for properties of th... |
| quotcl 26363 | The quotient of two polyno... |
| quotcl2 26364 | Closure of the quotient fu... |
| quotdgr 26365 | Remainder property of the ... |
| plyremlem 26366 | Closure of a linear factor... |
| plyrem 26367 | The polynomial remainder t... |
| facth 26368 | The factor theorem. If a ... |
| fta1lem 26369 | Lemma for ~ fta1 . (Contr... |
| fta1 26370 | The easy direction of the ... |
| quotcan 26371 | Exact division with a mult... |
| vieta1lem1 26372 | Lemma for ~ vieta1 . (Con... |
| vieta1lem2 26373 | Lemma for ~ vieta1 : induc... |
| vieta1 26374 | The first-order Vieta's fo... |
| plyexmo 26375 | An infinite set of values ... |
| elaa 26378 | Elementhood in the set of ... |
| aacn 26379 | An algebraic number is a c... |
| aasscn 26380 | The algebraic numbers are ... |
| elqaalem1 26381 | Lemma for ~ elqaa . The f... |
| elqaalem2 26382 | Lemma for ~ elqaa . (Cont... |
| elqaalem3 26383 | Lemma for ~ elqaa . (Cont... |
| elqaa 26384 | The set of numbers generat... |
| qaa 26385 | Every rational number is a... |
| qssaa 26386 | The rational numbers are c... |
| iaa 26387 | The imaginary unit is alge... |
| aareccl 26388 | The reciprocal of an algeb... |
| aacjcl 26389 | The conjugate of an algebr... |
| aannenlem1 26390 | Lemma for ~ aannen . (Con... |
| aannenlem2 26391 | Lemma for ~ aannen . (Con... |
| aannenlem3 26392 | The algebraic numbers are ... |
| aannen 26393 | The algebraic numbers are ... |
| aalioulem1 26394 | Lemma for ~ aaliou . An i... |
| aalioulem2 26395 | Lemma for ~ aaliou . (Con... |
| aalioulem3 26396 | Lemma for ~ aaliou . (Con... |
| aalioulem4 26397 | Lemma for ~ aaliou . (Con... |
| aalioulem5 26398 | Lemma for ~ aaliou . (Con... |
| aalioulem6 26399 | Lemma for ~ aaliou . (Con... |
| aaliou 26400 | Liouville's theorem on dio... |
| geolim3 26401 | Geometric series convergen... |
| aaliou2 26402 | Liouville's approximation ... |
| aaliou2b 26403 | Liouville's approximation ... |
| aaliou3lem1 26404 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem2 26405 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem3 26406 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem8 26407 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem4 26408 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem5 26409 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem6 26410 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem7 26411 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem9 26412 | Example of a "Liouville nu... |
| aaliou3 26413 | Example of a "Liouville nu... |
| taylfvallem1 26418 | Lemma for ~ taylfval . (C... |
| taylfvallem 26419 | Lemma for ~ taylfval . (C... |
| taylfval 26420 | Define the Taylor polynomi... |
| eltayl 26421 | Value of the Taylor series... |
| taylf 26422 | The Taylor series defines ... |
| tayl0 26423 | The Taylor series is alway... |
| taylplem1 26424 | Lemma for ~ taylpfval and ... |
| taylplem2 26425 | Lemma for ~ taylpfval and ... |
| taylpfval 26426 | Define the Taylor polynomi... |
| taylpf 26427 | The Taylor polynomial is a... |
| taylpval 26428 | Value of the Taylor polyno... |
| taylply2 26429 | The Taylor polynomial is a... |
| taylply2OLD 26430 | Obsolete version of ~ tayl... |
| taylply 26431 | The Taylor polynomial is a... |
| dvtaylp 26432 | The derivative of the Tayl... |
| dvntaylp 26433 | The ` M ` -th derivative o... |
| dvntaylp0 26434 | The first ` N ` derivative... |
| taylthlem1 26435 | Lemma for ~ taylth . This... |
| taylthlem2 26436 | Lemma for ~ taylth . (Con... |
| taylthlem2OLD 26437 | Obsolete version of ~ tayl... |
| taylth 26438 | Taylor's theorem. The Tay... |
| ulmrel 26441 | The uniform limit relation... |
| ulmscl 26442 | Closure of the base set in... |
| ulmval 26443 | Express the predicate: Th... |
| ulmcl 26444 | Closure of a uniform limit... |
| ulmf 26445 | Closure of a uniform limit... |
| ulmpm 26446 | Closure of a uniform limit... |
| ulmf2 26447 | Closure of a uniform limit... |
| ulm2 26448 | Simplify ~ ulmval when ` F... |
| ulmi 26449 | The uniform limit property... |
| ulmclm 26450 | A uniform limit of functio... |
| ulmres 26451 | A sequence of functions co... |
| ulmshftlem 26452 | Lemma for ~ ulmshft . (Co... |
| ulmshft 26453 | A sequence of functions co... |
| ulm0 26454 | Every function converges u... |
| ulmuni 26455 | A sequence of functions un... |
| ulmdm 26456 | Two ways to express that a... |
| ulmcaulem 26457 | Lemma for ~ ulmcau and ~ u... |
| ulmcau 26458 | A sequence of functions co... |
| ulmcau2 26459 | A sequence of functions co... |
| ulmss 26460 | A uniform limit of functio... |
| ulmbdd 26461 | A uniform limit of bounded... |
| ulmcn 26462 | A uniform limit of continu... |
| ulmdvlem1 26463 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem2 26464 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem3 26465 | Lemma for ~ ulmdv . (Cont... |
| ulmdv 26466 | If ` F ` is a sequence of ... |
| mtest 26467 | The Weierstrass M-test. I... |
| mtestbdd 26468 | Given the hypotheses of th... |
| mbfulm 26469 | A uniform limit of measura... |
| iblulm 26470 | A uniform limit of integra... |
| itgulm 26471 | A uniform limit of integra... |
| itgulm2 26472 | A uniform limit of integra... |
| pserval 26473 | Value of the function ` G ... |
| pserval2 26474 | Value of the function ` G ... |
| psergf 26475 | The sequence of terms in t... |
| radcnvlem1 26476 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem2 26477 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem3 26478 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnv0 26479 | Zero is always a convergen... |
| radcnvcl 26480 | The radius of convergence ... |
| radcnvlt1 26481 | If ` X ` is within the ope... |
| radcnvlt2 26482 | If ` X ` is within the ope... |
| radcnvle 26483 | If ` X ` is a convergent p... |
| dvradcnv 26484 | The radius of convergence ... |
| pserulm 26485 | If ` S ` is a region conta... |
| psercn2 26486 | Since by ~ pserulm the ser... |
| psercn2OLD 26487 | Obsolete version of ~ pser... |
| psercnlem2 26488 | Lemma for ~ psercn . (Con... |
| psercnlem1 26489 | Lemma for ~ psercn . (Con... |
| psercn 26490 | An infinite series converg... |
| pserdvlem1 26491 | Lemma for ~ pserdv . (Con... |
| pserdvlem2 26492 | Lemma for ~ pserdv . (Con... |
| pserdv 26493 | The derivative of a power ... |
| pserdv2 26494 | The derivative of a power ... |
| abelthlem1 26495 | Lemma for ~ abelth . (Con... |
| abelthlem2 26496 | Lemma for ~ abelth . The ... |
| abelthlem3 26497 | Lemma for ~ abelth . (Con... |
| abelthlem4 26498 | Lemma for ~ abelth . (Con... |
| abelthlem5 26499 | Lemma for ~ abelth . (Con... |
| abelthlem6 26500 | Lemma for ~ abelth . (Con... |
| abelthlem7a 26501 | Lemma for ~ abelth . (Con... |
| abelthlem7 26502 | Lemma for ~ abelth . (Con... |
| abelthlem8 26503 | Lemma for ~ abelth . (Con... |
| abelthlem9 26504 | Lemma for ~ abelth . By a... |
| abelth 26505 | Abel's theorem. If the po... |
| abelth2 26506 | Abel's theorem, restricted... |
| efcn 26507 | The exponential function i... |
| sincn 26508 | Sine is continuous. (Cont... |
| coscn 26509 | Cosine is continuous. (Co... |
| reeff1olem 26510 | Lemma for ~ reeff1o . (Co... |
| reeff1o 26511 | The real exponential funct... |
| reefiso 26512 | The exponential function o... |
| efcvx 26513 | The exponential function o... |
| reefgim 26514 | The exponential function i... |
| pilem1 26515 | Lemma for ~ pire , ~ pigt2... |
| pilem2 26516 | Lemma for ~ pire , ~ pigt2... |
| pilem3 26517 | Lemma for ~ pire , ~ pigt2... |
| pigt2lt4 26518 | ` _pi ` is between 2 and 4... |
| sinpi 26519 | The sine of ` _pi ` is 0. ... |
| pire 26520 | ` _pi ` is a real number. ... |
| picn 26521 | ` _pi ` is a complex numbe... |
| pipos 26522 | ` _pi ` is positive. (Con... |
| pirp 26523 | ` _pi ` is a positive real... |
| negpicn 26524 | ` -u _pi ` is a real numbe... |
| sinhalfpilem 26525 | Lemma for ~ sinhalfpi and ... |
| halfpire 26526 | ` _pi / 2 ` is real. (Con... |
| neghalfpire 26527 | ` -u _pi / 2 ` is real. (... |
| neghalfpirx 26528 | ` -u _pi / 2 ` is an exten... |
| pidiv2halves 26529 | Adding ` _pi / 2 ` to itse... |
| sinhalfpi 26530 | The sine of ` _pi / 2 ` is... |
| coshalfpi 26531 | The cosine of ` _pi / 2 ` ... |
| cosneghalfpi 26532 | The cosine of ` -u _pi / 2... |
| efhalfpi 26533 | The exponential of ` _i _p... |
| cospi 26534 | The cosine of ` _pi ` is `... |
| efipi 26535 | The exponential of ` _i x.... |
| eulerid 26536 | Euler's identity. (Contri... |
| sin2pi 26537 | The sine of ` 2 _pi ` is 0... |
| cos2pi 26538 | The cosine of ` 2 _pi ` is... |
| ef2pi 26539 | The exponential of ` 2 _pi... |
| ef2kpi 26540 | If ` K ` is an integer, th... |
| efper 26541 | The exponential function i... |
| sinperlem 26542 | Lemma for ~ sinper and ~ c... |
| sinper 26543 | The sine function is perio... |
| cosper 26544 | The cosine function is per... |
| sin2kpi 26545 | If ` K ` is an integer, th... |
| cos2kpi 26546 | If ` K ` is an integer, th... |
| sin2pim 26547 | Sine of a number subtracte... |
| cos2pim 26548 | Cosine of a number subtrac... |
| sinmpi 26549 | Sine of a number less ` _p... |
| cosmpi 26550 | Cosine of a number less ` ... |
| sinppi 26551 | Sine of a number plus ` _p... |
| cosppi 26552 | Cosine of a number plus ` ... |
| efimpi 26553 | The exponential function a... |
| sinhalfpip 26554 | The sine of ` _pi / 2 ` pl... |
| sinhalfpim 26555 | The sine of ` _pi / 2 ` mi... |
| coshalfpip 26556 | The cosine of ` _pi / 2 ` ... |
| coshalfpim 26557 | The cosine of ` _pi / 2 ` ... |
| ptolemy 26558 | Ptolemy's Theorem. This t... |
| sincosq1lem 26559 | Lemma for ~ sincosq1sgn . ... |
| sincosq1sgn 26560 | The signs of the sine and ... |
| sincosq2sgn 26561 | The signs of the sine and ... |
| sincosq3sgn 26562 | The signs of the sine and ... |
| sincosq4sgn 26563 | The signs of the sine and ... |
| coseq00topi 26564 | Location of the zeroes of ... |
| coseq0negpitopi 26565 | Location of the zeroes of ... |
| tanrpcl 26566 | Positive real closure of t... |
| tangtx 26567 | The tangent function is gr... |
| tanabsge 26568 | The tangent function is gr... |
| sinq12gt0 26569 | The sine of a number stric... |
| sinq12ge0 26570 | The sine of a number betwe... |
| sinq34lt0t 26571 | The sine of a number stric... |
| cosq14gt0 26572 | The cosine of a number str... |
| cosq14ge0 26573 | The cosine of a number bet... |
| sincosq1eq 26574 | Complementarity of the sin... |
| sincos4thpi 26575 | The sine and cosine of ` _... |
| tan4thpi 26576 | The tangent of ` _pi / 4 `... |
| tan4thpiOLD 26577 | Obsolete version of ~ tan4... |
| sincos6thpi 26578 | The sine and cosine of ` _... |
| sincos3rdpi 26579 | The sine and cosine of ` _... |
| pigt3 26580 | ` _pi ` is greater than 3.... |
| pige3 26581 | ` _pi ` is greater than or... |
| pige3ALT 26582 | Alternate proof of ~ pige3... |
| abssinper 26583 | The absolute value of sine... |
| sinkpi 26584 | The sine of an integer mul... |
| coskpi 26585 | The absolute value of the ... |
| sineq0 26586 | A complex number whose sin... |
| coseq1 26587 | A complex number whose cos... |
| cos02pilt1 26588 | Cosine is less than one be... |
| cosq34lt1 26589 | Cosine is less than one in... |
| efeq1 26590 | A complex number whose exp... |
| cosne0 26591 | The cosine function has no... |
| cosordlem 26592 | Lemma for ~ cosord . (Con... |
| cosord 26593 | Cosine is decreasing over ... |
| cos0pilt1 26594 | Cosine is between minus on... |
| cos11 26595 | Cosine is one-to-one over ... |
| sinord 26596 | Sine is increasing over th... |
| recosf1o 26597 | The cosine function is a b... |
| resinf1o 26598 | The sine function is a bij... |
| tanord1 26599 | The tangent function is st... |
| tanord 26600 | The tangent function is st... |
| tanregt0 26601 | The real part of the tange... |
| negpitopissre 26602 | The interval ` ( -u _pi (,... |
| efgh 26603 | The exponential function o... |
| efif1olem1 26604 | Lemma for ~ efif1o . (Con... |
| efif1olem2 26605 | Lemma for ~ efif1o . (Con... |
| efif1olem3 26606 | Lemma for ~ efif1o . (Con... |
| efif1olem4 26607 | The exponential function o... |
| efif1o 26608 | The exponential function o... |
| efifo 26609 | The exponential function o... |
| eff1olem 26610 | The exponential function m... |
| eff1o 26611 | The exponential function m... |
| efabl 26612 | The image of a subgroup of... |
| efsubm 26613 | The image of a subgroup of... |
| circgrp 26614 | The circle group ` T ` is ... |
| circsubm 26615 | The circle group ` T ` is ... |
| logrn 26620 | The range of the natural l... |
| ellogrn 26621 | Write out the property ` A... |
| dflog2 26622 | The natural logarithm func... |
| relogrn 26623 | The range of the natural l... |
| logrncn 26624 | The range of the natural l... |
| eff1o2 26625 | The exponential function r... |
| logf1o 26626 | The natural logarithm func... |
| dfrelog 26627 | The natural logarithm func... |
| relogf1o 26628 | The natural logarithm func... |
| logrncl 26629 | Closure of the natural log... |
| logcl 26630 | Closure of the natural log... |
| logimcl 26631 | Closure of the imaginary p... |
| logcld 26632 | The logarithm of a nonzero... |
| logimcld 26633 | The imaginary part of the ... |
| logimclad 26634 | The imaginary part of the ... |
| abslogimle 26635 | The imaginary part of the ... |
| logrnaddcl 26636 | The range of the natural l... |
| relogcl 26637 | Closure of the natural log... |
| eflog 26638 | Relationship between the n... |
| logeq0im1 26639 | If the logarithm of a numb... |
| logccne0 26640 | The logarithm isn't 0 if i... |
| logne0 26641 | Logarithm of a non-1 posit... |
| reeflog 26642 | Relationship between the n... |
| logef 26643 | Relationship between the n... |
| relogef 26644 | Relationship between the n... |
| logeftb 26645 | Relationship between the n... |
| relogeftb 26646 | Relationship between the n... |
| log1 26647 | The natural logarithm of `... |
| loge 26648 | The natural logarithm of `... |
| logi 26649 | The natural logarithm of `... |
| logneg 26650 | The natural logarithm of a... |
| logm1 26651 | The natural logarithm of n... |
| lognegb 26652 | If a number has imaginary ... |
| relogoprlem 26653 | Lemma for ~ relogmul and ~... |
| relogmul 26654 | The natural logarithm of t... |
| relogdiv 26655 | The natural logarithm of t... |
| explog 26656 | Exponentiation of a nonzer... |
| reexplog 26657 | Exponentiation of a positi... |
| relogexp 26658 | The natural logarithm of p... |
| relog 26659 | Real part of a logarithm. ... |
| relogiso 26660 | The natural logarithm func... |
| reloggim 26661 | The natural logarithm is a... |
| logltb 26662 | The natural logarithm func... |
| logfac 26663 | The logarithm of a factori... |
| eflogeq 26664 | Solve an equation involvin... |
| logleb 26665 | Natural logarithm preserve... |
| rplogcl 26666 | Closure of the logarithm f... |
| logge0 26667 | The logarithm of a number ... |
| logcj 26668 | The natural logarithm dist... |
| efiarg 26669 | The exponential of the "ar... |
| cosargd 26670 | The cosine of the argument... |
| cosarg0d 26671 | The cosine of the argument... |
| argregt0 26672 | Closure of the argument of... |
| argrege0 26673 | Closure of the argument of... |
| argimgt0 26674 | Closure of the argument of... |
| argimlt0 26675 | Closure of the argument of... |
| logimul 26676 | Multiplying a number by ` ... |
| logneg2 26677 | The logarithm of the negat... |
| logmul2 26678 | Generalization of ~ relogm... |
| logdiv2 26679 | Generalization of ~ relogd... |
| abslogle 26680 | Bound on the magnitude of ... |
| tanarg 26681 | The basic relation between... |
| logdivlti 26682 | The ` log x / x ` function... |
| logdivlt 26683 | The ` log x / x ` function... |
| logdivle 26684 | The ` log x / x ` function... |
| relogcld 26685 | Closure of the natural log... |
| reeflogd 26686 | Relationship between the n... |
| relogmuld 26687 | The natural logarithm of t... |
| relogdivd 26688 | The natural logarithm of t... |
| logled 26689 | Natural logarithm preserve... |
| relogefd 26690 | Relationship between the n... |
| rplogcld 26691 | Closure of the logarithm f... |
| logge0d 26692 | The logarithm of a number ... |
| logge0b 26693 | The logarithm of a number ... |
| loggt0b 26694 | The logarithm of a number ... |
| logle1b 26695 | The logarithm of a number ... |
| loglt1b 26696 | The logarithm of a number ... |
| divlogrlim 26697 | The inverse logarithm func... |
| logno1 26698 | The logarithm function is ... |
| dvrelog 26699 | The derivative of the real... |
| relogcn 26700 | The real logarithm functio... |
| ellogdm 26701 | Elementhood in the "contin... |
| logdmn0 26702 | A number in the continuous... |
| logdmnrp 26703 | A number in the continuous... |
| logdmss 26704 | The continuity domain of `... |
| logcnlem2 26705 | Lemma for ~ logcn . (Cont... |
| logcnlem3 26706 | Lemma for ~ logcn . (Cont... |
| logcnlem4 26707 | Lemma for ~ logcn . (Cont... |
| logcnlem5 26708 | Lemma for ~ logcn . (Cont... |
| logcn 26709 | The logarithm function is ... |
| dvloglem 26710 | Lemma for ~ dvlog . (Cont... |
| logdmopn 26711 | The "continuous domain" of... |
| logf1o2 26712 | The logarithm maps its con... |
| dvlog 26713 | The derivative of the comp... |
| dvlog2lem 26714 | Lemma for ~ dvlog2 . (Con... |
| dvlog2 26715 | The derivative of the comp... |
| advlog 26716 | The antiderivative of the ... |
| advlogexp 26717 | The antiderivative of a po... |
| efopnlem1 26718 | Lemma for ~ efopn . (Cont... |
| efopnlem2 26719 | Lemma for ~ efopn . (Cont... |
| efopn 26720 | The exponential map is an ... |
| logtayllem 26721 | Lemma for ~ logtayl . (Co... |
| logtayl 26722 | The Taylor series for ` -u... |
| logtaylsum 26723 | The Taylor series for ` -u... |
| logtayl2 26724 | Power series expression fo... |
| logccv 26725 | The natural logarithm func... |
| cxpval 26726 | Value of the complex power... |
| cxpef 26727 | Value of the complex power... |
| 0cxp 26728 | Value of the complex power... |
| cxpexpz 26729 | Relate the complex power f... |
| cxpexp 26730 | Relate the complex power f... |
| logcxp 26731 | Logarithm of a complex pow... |
| cxp0 26732 | Value of the complex power... |
| cxp1 26733 | Value of the complex power... |
| 1cxp 26734 | Value of the complex power... |
| ecxp 26735 | Write the exponential func... |
| cxpcl 26736 | Closure of the complex pow... |
| recxpcl 26737 | Real closure of the comple... |
| rpcxpcl 26738 | Positive real closure of t... |
| cxpne0 26739 | Complex exponentiation is ... |
| cxpeq0 26740 | Complex exponentiation is ... |
| cxpadd 26741 | Sum of exponents law for c... |
| cxpp1 26742 | Value of a nonzero complex... |
| cxpneg 26743 | Value of a complex number ... |
| cxpsub 26744 | Exponent subtraction law f... |
| cxpge0 26745 | Nonnegative exponentiation... |
| mulcxplem 26746 | Lemma for ~ mulcxp . (Con... |
| mulcxp 26747 | Complex exponentiation of ... |
| cxprec 26748 | Complex exponentiation of ... |
| divcxp 26749 | Complex exponentiation of ... |
| cxpmul 26750 | Product of exponents law f... |
| cxpmul2 26751 | Product of exponents law f... |
| cxproot 26752 | The complex power function... |
| cxpmul2z 26753 | Generalize ~ cxpmul2 to ne... |
| abscxp 26754 | Absolute value of a power,... |
| abscxp2 26755 | Absolute value of a power,... |
| cxplt 26756 | Ordering property for comp... |
| cxple 26757 | Ordering property for comp... |
| cxplea 26758 | Ordering property for comp... |
| cxple2 26759 | Ordering property for comp... |
| cxplt2 26760 | Ordering property for comp... |
| cxple2a 26761 | Ordering property for comp... |
| cxplt3 26762 | Ordering property for comp... |
| cxple3 26763 | Ordering property for comp... |
| cxpsqrtlem 26764 | Lemma for ~ cxpsqrt . (Co... |
| cxpsqrt 26765 | The complex exponential fu... |
| logsqrt 26766 | Logarithm of a square root... |
| cxp0d 26767 | Value of the complex power... |
| cxp1d 26768 | Value of the complex power... |
| 1cxpd 26769 | Value of the complex power... |
| cxpcld 26770 | Closure of the complex pow... |
| cxpmul2d 26771 | Product of exponents law f... |
| 0cxpd 26772 | Value of the complex power... |
| cxpexpzd 26773 | Relate the complex power f... |
| cxpefd 26774 | Value of the complex power... |
| cxpne0d 26775 | Complex exponentiation is ... |
| cxpp1d 26776 | Value of a nonzero complex... |
| cxpnegd 26777 | Value of a complex number ... |
| cxpmul2zd 26778 | Generalize ~ cxpmul2 to ne... |
| cxpaddd 26779 | Sum of exponents law for c... |
| cxpsubd 26780 | Exponent subtraction law f... |
| cxpltd 26781 | Ordering property for comp... |
| cxpled 26782 | Ordering property for comp... |
| cxplead 26783 | Ordering property for comp... |
| divcxpd 26784 | Complex exponentiation of ... |
| recxpcld 26785 | Positive real closure of t... |
| cxpge0d 26786 | Nonnegative exponentiation... |
| cxple2ad 26787 | Ordering property for comp... |
| cxplt2d 26788 | Ordering property for comp... |
| cxple2d 26789 | Ordering property for comp... |
| mulcxpd 26790 | Complex exponentiation of ... |
| recxpf1lem 26791 | Complex exponentiation on ... |
| cxpsqrtth 26792 | Square root theorem over t... |
| 2irrexpq 26793 | There exist irrational num... |
| cxprecd 26794 | Complex exponentiation of ... |
| rpcxpcld 26795 | Positive real closure of t... |
| logcxpd 26796 | Logarithm of a complex pow... |
| cxplt3d 26797 | Ordering property for comp... |
| cxple3d 26798 | Ordering property for comp... |
| cxpmuld 26799 | Product of exponents law f... |
| cxpgt0d 26800 | A positive real raised to ... |
| cxpcom 26801 | Commutative law for real e... |
| dvcxp1 26802 | The derivative of a comple... |
| dvcxp2 26803 | The derivative of a comple... |
| dvsqrt 26804 | The derivative of the real... |
| dvcncxp1 26805 | Derivative of complex powe... |
| dvcnsqrt 26806 | Derivative of square root ... |
| cxpcn 26807 | Domain of continuity of th... |
| cxpcnOLD 26808 | Obsolete version of ~ cxpc... |
| cxpcn2 26809 | Continuity of the complex ... |
| cxpcn3lem 26810 | Lemma for ~ cxpcn3 . (Con... |
| cxpcn3 26811 | Extend continuity of the c... |
| resqrtcn 26812 | Continuity of the real squ... |
| sqrtcn 26813 | Continuity of the square r... |
| cxpaddlelem 26814 | Lemma for ~ cxpaddle . (C... |
| cxpaddle 26815 | Ordering property for comp... |
| abscxpbnd 26816 | Bound on the absolute valu... |
| root1id 26817 | Property of an ` N ` -th r... |
| root1eq1 26818 | The only powers of an ` N ... |
| root1cj 26819 | Within the ` N ` -th roots... |
| cxpeq 26820 | Solve an equation involvin... |
| zrtelqelz 26821 | If the ` N ` -th root of a... |
| zrtdvds 26822 | A positive integer root di... |
| rtprmirr 26823 | The root of a prime number... |
| loglesqrt 26824 | An upper bound on the loga... |
| logreclem 26825 | Symmetry of the natural lo... |
| logrec 26826 | Logarithm of a reciprocal ... |
| logbval 26829 | Define the value of the ` ... |
| logbcl 26830 | General logarithm closure.... |
| logbid1 26831 | General logarithm is 1 whe... |
| logb1 26832 | The logarithm of ` 1 ` to ... |
| elogb 26833 | The general logarithm of a... |
| logbchbase 26834 | Change of base for logarit... |
| relogbval 26835 | Value of the general logar... |
| relogbcl 26836 | Closure of the general log... |
| relogbzcl 26837 | Closure of the general log... |
| relogbreexp 26838 | Power law for the general ... |
| relogbzexp 26839 | Power law for the general ... |
| relogbmul 26840 | The logarithm of the produ... |
| relogbmulexp 26841 | The logarithm of the produ... |
| relogbdiv 26842 | The logarithm of the quoti... |
| relogbexp 26843 | Identity law for general l... |
| nnlogbexp 26844 | Identity law for general l... |
| logbrec 26845 | Logarithm of a reciprocal ... |
| logbleb 26846 | The general logarithm func... |
| logblt 26847 | The general logarithm func... |
| relogbcxp 26848 | Identity law for the gener... |
| cxplogb 26849 | Identity law for the gener... |
| relogbcxpb 26850 | The logarithm is the inver... |
| logbmpt 26851 | The general logarithm to a... |
| logbf 26852 | The general logarithm to a... |
| logbfval 26853 | The general logarithm of a... |
| relogbf 26854 | The general logarithm to a... |
| logblog 26855 | The general logarithm to t... |
| logbgt0b 26856 | The logarithm of a positiv... |
| logbgcd1irr 26857 | The logarithm of an intege... |
| 2logb9irr 26858 | Example for ~ logbgcd1irr ... |
| logbprmirr 26859 | The logarithm of a prime t... |
| 2logb3irr 26860 | Example for ~ logbprmirr .... |
| 2logb9irrALT 26861 | Alternate proof of ~ 2logb... |
| sqrt2cxp2logb9e3 26862 | The square root of two to ... |
| 2irrexpqALT 26863 | Alternate proof of ~ 2irre... |
| angval 26864 | Define the angle function,... |
| angcan 26865 | Cancel a constant multipli... |
| angneg 26866 | Cancel a negative sign in ... |
| angvald 26867 | The (signed) angle between... |
| angcld 26868 | The (signed) angle between... |
| angrteqvd 26869 | Two vectors are at a right... |
| cosangneg2d 26870 | The cosine of the angle be... |
| angrtmuld 26871 | Perpendicularity of two ve... |
| ang180lem1 26872 | Lemma for ~ ang180 . Show... |
| ang180lem2 26873 | Lemma for ~ ang180 . Show... |
| ang180lem3 26874 | Lemma for ~ ang180 . Sinc... |
| ang180lem4 26875 | Lemma for ~ ang180 . Redu... |
| ang180lem5 26876 | Lemma for ~ ang180 : Redu... |
| ang180 26877 | The sum of angles ` m A B ... |
| lawcoslem1 26878 | Lemma for ~ lawcos . Here... |
| lawcos 26879 | Law of cosines (also known... |
| pythag 26880 | Pythagorean theorem. Give... |
| isosctrlem1 26881 | Lemma for ~ isosctr . (Co... |
| isosctrlem2 26882 | Lemma for ~ isosctr . Cor... |
| isosctrlem3 26883 | Lemma for ~ isosctr . Cor... |
| isosctr 26884 | Isosceles triangle theorem... |
| ssscongptld 26885 | If two triangles have equa... |
| affineequiv 26886 | Equivalence between two wa... |
| affineequiv2 26887 | Equivalence between two wa... |
| affineequiv3 26888 | Equivalence between two wa... |
| affineequiv4 26889 | Equivalence between two wa... |
| affineequivne 26890 | Equivalence between two wa... |
| angpieqvdlem 26891 | Equivalence used in the pr... |
| angpieqvdlem2 26892 | Equivalence used in ~ angp... |
| angpined 26893 | If the angle at ABC is ` _... |
| angpieqvd 26894 | The angle ABC is ` _pi ` i... |
| chordthmlem 26895 | If ` M ` is the midpoint o... |
| chordthmlem2 26896 | If M is the midpoint of AB... |
| chordthmlem3 26897 | If M is the midpoint of AB... |
| chordthmlem4 26898 | If P is on the segment AB ... |
| chordthmlem5 26899 | If P is on the segment AB ... |
| chordthm 26900 | The intersecting chords th... |
| heron 26901 | Heron's formula gives the ... |
| quad2 26902 | The quadratic equation, wi... |
| quad 26903 | The quadratic equation. (... |
| 1cubrlem 26904 | The cube roots of unity. ... |
| 1cubr 26905 | The cube roots of unity. ... |
| dcubic1lem 26906 | Lemma for ~ dcubic1 and ~ ... |
| dcubic2 26907 | Reverse direction of ~ dcu... |
| dcubic1 26908 | Forward direction of ~ dcu... |
| dcubic 26909 | Solutions to the depressed... |
| mcubic 26910 | Solutions to a monic cubic... |
| cubic2 26911 | The solution to the genera... |
| cubic 26912 | The cubic equation, which ... |
| binom4 26913 | Work out a quartic binomia... |
| dquartlem1 26914 | Lemma for ~ dquart . (Con... |
| dquartlem2 26915 | Lemma for ~ dquart . (Con... |
| dquart 26916 | Solve a depressed quartic ... |
| quart1cl 26917 | Closure lemmas for ~ quart... |
| quart1lem 26918 | Lemma for ~ quart1 . (Con... |
| quart1 26919 | Depress a quartic equation... |
| quartlem1 26920 | Lemma for ~ quart . (Cont... |
| quartlem2 26921 | Closure lemmas for ~ quart... |
| quartlem3 26922 | Closure lemmas for ~ quart... |
| quartlem4 26923 | Closure lemmas for ~ quart... |
| quart 26924 | The quartic equation, writ... |
| asinlem 26931 | The argument to the logari... |
| asinlem2 26932 | The argument to the logari... |
| asinlem3a 26933 | Lemma for ~ asinlem3 . (C... |
| asinlem3 26934 | The argument to the logari... |
| asinf 26935 | Domain and codomain of the... |
| asincl 26936 | Closure for the arcsin fun... |
| acosf 26937 | Domain and codoamin of the... |
| acoscl 26938 | Closure for the arccos fun... |
| atandm 26939 | Since the property is a li... |
| atandm2 26940 | This form of ~ atandm is a... |
| atandm3 26941 | A compact form of ~ atandm... |
| atandm4 26942 | A compact form of ~ atandm... |
| atanf 26943 | Domain and codoamin of the... |
| atancl 26944 | Closure for the arctan fun... |
| asinval 26945 | Value of the arcsin functi... |
| acosval 26946 | Value of the arccos functi... |
| atanval 26947 | Value of the arctan functi... |
| atanre 26948 | A real number is in the do... |
| asinneg 26949 | The arcsine function is od... |
| acosneg 26950 | The negative symmetry rela... |
| efiasin 26951 | The exponential of the arc... |
| sinasin 26952 | The arcsine function is an... |
| cosacos 26953 | The arccosine function is ... |
| asinsinlem 26954 | Lemma for ~ asinsin . (Co... |
| asinsin 26955 | The arcsine function compo... |
| acoscos 26956 | The arccosine function is ... |
| asin1 26957 | The arcsine of ` 1 ` is ` ... |
| acos1 26958 | The arccosine of ` 1 ` is ... |
| reasinsin 26959 | The arcsine function compo... |
| asinsinb 26960 | Relationship between sine ... |
| acoscosb 26961 | Relationship between cosin... |
| asinbnd 26962 | The arcsine function has r... |
| acosbnd 26963 | The arccosine function has... |
| asinrebnd 26964 | Bounds on the arcsine func... |
| asinrecl 26965 | The arcsine function is re... |
| acosrecl 26966 | The arccosine function is ... |
| cosasin 26967 | The cosine of the arcsine ... |
| sinacos 26968 | The sine of the arccosine ... |
| atandmneg 26969 | The domain of the arctange... |
| atanneg 26970 | The arctangent function is... |
| atan0 26971 | The arctangent of zero is ... |
| atandmcj 26972 | The arctangent function di... |
| atancj 26973 | The arctangent function di... |
| atanrecl 26974 | The arctangent function is... |
| efiatan 26975 | Value of the exponential o... |
| atanlogaddlem 26976 | Lemma for ~ atanlogadd . ... |
| atanlogadd 26977 | The rule ` sqrt ( z w ) = ... |
| atanlogsublem 26978 | Lemma for ~ atanlogsub . ... |
| atanlogsub 26979 | A variation on ~ atanlogad... |
| efiatan2 26980 | Value of the exponential o... |
| 2efiatan 26981 | Value of the exponential o... |
| tanatan 26982 | The arctangent function is... |
| atandmtan 26983 | The tangent function has r... |
| cosatan 26984 | The cosine of an arctangen... |
| cosatanne0 26985 | The arctangent function ha... |
| atantan 26986 | The arctangent function is... |
| atantanb 26987 | Relationship between tange... |
| atanbndlem 26988 | Lemma for ~ atanbnd . (Co... |
| atanbnd 26989 | The arctangent function is... |
| atanord 26990 | The arctangent function is... |
| atan1 26991 | The arctangent of ` 1 ` is... |
| bndatandm 26992 | A point in the open unit d... |
| atans 26993 | The "domain of continuity"... |
| atans2 26994 | It suffices to show that `... |
| atansopn 26995 | The domain of continuity o... |
| atansssdm 26996 | The domain of continuity o... |
| ressatans 26997 | The real number line is a ... |
| dvatan 26998 | The derivative of the arct... |
| atancn 26999 | The arctangent is a contin... |
| atantayl 27000 | The Taylor series for ` ar... |
| atantayl2 27001 | The Taylor series for ` ar... |
| atantayl3 27002 | The Taylor series for ` ar... |
| leibpilem1 27003 | Lemma for ~ leibpi . (Con... |
| leibpilem2 27004 | The Leibniz formula for ` ... |
| leibpi 27005 | The Leibniz formula for ` ... |
| leibpisum 27006 | The Leibniz formula for ` ... |
| log2cnv 27007 | Using the Taylor series fo... |
| log2tlbnd 27008 | Bound the error term in th... |
| log2ublem1 27009 | Lemma for ~ log2ub . The ... |
| log2ublem2 27010 | Lemma for ~ log2ub . (Con... |
| log2ublem3 27011 | Lemma for ~ log2ub . In d... |
| log2ub 27012 | ` log 2 ` is less than ` 2... |
| log2le1 27013 | ` log 2 ` is less than ` 1... |
| birthdaylem1 27014 | Lemma for ~ birthday . (C... |
| birthdaylem2 27015 | For general ` N ` and ` K ... |
| birthdaylem3 27016 | For general ` N ` and ` K ... |
| birthday 27017 | The Birthday Problem. The... |
| dmarea 27020 | The domain of the area fun... |
| areambl 27021 | The fibers of a measurable... |
| areass 27022 | A measurable region is a s... |
| dfarea 27023 | Rewrite ~ df-area self-ref... |
| areaf 27024 | Area measurement is a func... |
| areacl 27025 | The area of a measurable r... |
| areage0 27026 | The area of a measurable r... |
| areaval 27027 | The area of a measurable r... |
| rlimcnp 27028 | Relate a limit of a real-v... |
| rlimcnp2 27029 | Relate a limit of a real-v... |
| rlimcnp3 27030 | Relate a limit of a real-v... |
| xrlimcnp 27031 | Relate a limit of a real-v... |
| efrlim 27032 | The limit of the sequence ... |
| efrlimOLD 27033 | Obsolete version of ~ efrl... |
| dfef2 27034 | The limit of the sequence ... |
| cxplim 27035 | A power to a negative expo... |
| sqrtlim 27036 | The inverse square root fu... |
| rlimcxp 27037 | Any power to a positive ex... |
| o1cxp 27038 | An eventually bounded func... |
| cxp2limlem 27039 | A linear factor grows slow... |
| cxp2lim 27040 | Any power grows slower tha... |
| cxploglim 27041 | The logarithm grows slower... |
| cxploglim2 27042 | Every power of the logarit... |
| divsqrtsumlem 27043 | Lemma for ~ divsqrsum and ... |
| divsqrsumf 27044 | The function ` F ` used in... |
| divsqrsum 27045 | The sum ` sum_ n <_ x ( 1 ... |
| divsqrtsum2 27046 | A bound on the distance of... |
| divsqrtsumo1 27047 | The sum ` sum_ n <_ x ( 1 ... |
| cvxcl 27048 | Closure of a 0-1 linear co... |
| scvxcvx 27049 | A strictly convex function... |
| jensenlem1 27050 | Lemma for ~ jensen . (Con... |
| jensenlem2 27051 | Lemma for ~ jensen . (Con... |
| jensen 27052 | Jensen's inequality, a fin... |
| amgmlem 27053 | Lemma for ~ amgm . (Contr... |
| amgm 27054 | Inequality of arithmetic a... |
| logdifbnd 27057 | Bound on the difference of... |
| logdiflbnd 27058 | Lower bound on the differe... |
| emcllem1 27059 | Lemma for ~ emcl . The se... |
| emcllem2 27060 | Lemma for ~ emcl . ` F ` i... |
| emcllem3 27061 | Lemma for ~ emcl . The fu... |
| emcllem4 27062 | Lemma for ~ emcl . The di... |
| emcllem5 27063 | Lemma for ~ emcl . The pa... |
| emcllem6 27064 | Lemma for ~ emcl . By the... |
| emcllem7 27065 | Lemma for ~ emcl and ~ har... |
| emcl 27066 | Closure and bounds for the... |
| harmonicbnd 27067 | A bound on the harmonic se... |
| harmonicbnd2 27068 | A bound on the harmonic se... |
| emre 27069 | The Euler-Mascheroni const... |
| emgt0 27070 | The Euler-Mascheroni const... |
| harmonicbnd3 27071 | A bound on the harmonic se... |
| harmoniclbnd 27072 | A bound on the harmonic se... |
| harmonicubnd 27073 | A bound on the harmonic se... |
| harmonicbnd4 27074 | The asymptotic behavior of... |
| fsumharmonic 27075 | Bound a finite sum based o... |
| zetacvg 27078 | The zeta series is converg... |
| eldmgm 27085 | Elementhood in the set of ... |
| dmgmaddn0 27086 | If ` A ` is not a nonposit... |
| dmlogdmgm 27087 | If ` A ` is in the continu... |
| rpdmgm 27088 | A positive real number is ... |
| dmgmn0 27089 | If ` A ` is not a nonposit... |
| dmgmaddnn0 27090 | If ` A ` is not a nonposit... |
| dmgmdivn0 27091 | Lemma for ~ lgamf . (Cont... |
| lgamgulmlem1 27092 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem2 27093 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem3 27094 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem4 27095 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem5 27096 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem6 27097 | The series ` G ` is unifor... |
| lgamgulm 27098 | The series ` G ` is unifor... |
| lgamgulm2 27099 | Rewrite the limit of the s... |
| lgambdd 27100 | The log-Gamma function is ... |
| lgamucov 27101 | The ` U ` regions used in ... |
| lgamucov2 27102 | The ` U ` regions used in ... |
| lgamcvglem 27103 | Lemma for ~ lgamf and ~ lg... |
| lgamcl 27104 | The log-Gamma function is ... |
| lgamf 27105 | The log-Gamma function is ... |
| gamf 27106 | The Gamma function is a co... |
| gamcl 27107 | The exponential of the log... |
| eflgam 27108 | The exponential of the log... |
| gamne0 27109 | The Gamma function is neve... |
| igamval 27110 | Value of the inverse Gamma... |
| igamz 27111 | Value of the inverse Gamma... |
| igamgam 27112 | Value of the inverse Gamma... |
| igamlgam 27113 | Value of the inverse Gamma... |
| igamf 27114 | Closure of the inverse Gam... |
| igamcl 27115 | Closure of the inverse Gam... |
| gamigam 27116 | The Gamma function is the ... |
| lgamcvg 27117 | The series ` G ` converges... |
| lgamcvg2 27118 | The series ` G ` converges... |
| gamcvg 27119 | The pointwise exponential ... |
| lgamp1 27120 | The functional equation of... |
| gamp1 27121 | The functional equation of... |
| gamcvg2lem 27122 | Lemma for ~ gamcvg2 . (Co... |
| gamcvg2 27123 | An infinite product expres... |
| regamcl 27124 | The Gamma function is real... |
| relgamcl 27125 | The log-Gamma function is ... |
| rpgamcl 27126 | The log-Gamma function is ... |
| lgam1 27127 | The log-Gamma function at ... |
| gam1 27128 | The log-Gamma function at ... |
| facgam 27129 | The Gamma function general... |
| gamfac 27130 | The Gamma function general... |
| wilthlem1 27131 | The only elements that are... |
| wilthlem2 27132 | Lemma for ~ wilth : induct... |
| wilthlem3 27133 | Lemma for ~ wilth . Here ... |
| wilth 27134 | Wilson's theorem. A numbe... |
| wilthimp 27135 | The forward implication of... |
| ftalem1 27136 | Lemma for ~ fta : "growth... |
| ftalem2 27137 | Lemma for ~ fta . There e... |
| ftalem3 27138 | Lemma for ~ fta . There e... |
| ftalem4 27139 | Lemma for ~ fta : Closure... |
| ftalem5 27140 | Lemma for ~ fta : Main pr... |
| ftalem6 27141 | Lemma for ~ fta : Dischar... |
| ftalem7 27142 | Lemma for ~ fta . Shift t... |
| fta 27143 | The Fundamental Theorem of... |
| basellem1 27144 | Lemma for ~ basel . Closu... |
| basellem2 27145 | Lemma for ~ basel . Show ... |
| basellem3 27146 | Lemma for ~ basel . Using... |
| basellem4 27147 | Lemma for ~ basel . By ~ ... |
| basellem5 27148 | Lemma for ~ basel . Using... |
| basellem6 27149 | Lemma for ~ basel . The f... |
| basellem7 27150 | Lemma for ~ basel . The f... |
| basellem8 27151 | Lemma for ~ basel . The f... |
| basellem9 27152 | Lemma for ~ basel . Since... |
| basel 27153 | The sum of the inverse squ... |
| efnnfsumcl 27166 | Finite sum closure in the ... |
| ppisval 27167 | The set of primes less tha... |
| ppisval2 27168 | The set of primes less tha... |
| ppifi 27169 | The set of primes less tha... |
| prmdvdsfi 27170 | The set of prime divisors ... |
| chtf 27171 | Domain and codoamin of the... |
| chtcl 27172 | Real closure of the Chebys... |
| chtval 27173 | Value of the Chebyshev fun... |
| efchtcl 27174 | The Chebyshev function is ... |
| chtge0 27175 | The Chebyshev function is ... |
| vmaval 27176 | Value of the von Mangoldt ... |
| isppw 27177 | Two ways to say that ` A `... |
| isppw2 27178 | Two ways to say that ` A `... |
| vmappw 27179 | Value of the von Mangoldt ... |
| vmaprm 27180 | Value of the von Mangoldt ... |
| vmacl 27181 | Closure for the von Mangol... |
| vmaf 27182 | Functionality of the von M... |
| efvmacl 27183 | The von Mangoldt is closed... |
| vmage0 27184 | The von Mangoldt function ... |
| chpval 27185 | Value of the second Chebys... |
| chpf 27186 | Functionality of the secon... |
| chpcl 27187 | Closure for the second Che... |
| efchpcl 27188 | The second Chebyshev funct... |
| chpge0 27189 | The second Chebyshev funct... |
| ppival 27190 | Value of the prime-countin... |
| ppival2 27191 | Value of the prime-countin... |
| ppival2g 27192 | Value of the prime-countin... |
| ppif 27193 | Domain and codomain of the... |
| ppicl 27194 | Real closure of the prime-... |
| muval 27195 | The value of the Möbi... |
| muval1 27196 | The value of the Möbi... |
| muval2 27197 | The value of the Möbi... |
| isnsqf 27198 | Two ways to say that a num... |
| issqf 27199 | Two ways to say that a num... |
| sqfpc 27200 | The prime count of a squar... |
| dvdssqf 27201 | A divisor of a squarefree ... |
| sqf11 27202 | A squarefree number is com... |
| muf 27203 | The Möbius function i... |
| mucl 27204 | Closure of the Möbius... |
| sgmval 27205 | The value of the divisor f... |
| sgmval2 27206 | The value of the divisor f... |
| 0sgm 27207 | The value of the sum-of-di... |
| sgmf 27208 | The divisor function is a ... |
| sgmcl 27209 | Closure of the divisor fun... |
| sgmnncl 27210 | Closure of the divisor fun... |
| mule1 27211 | The Möbius function t... |
| chtfl 27212 | The Chebyshev function doe... |
| chpfl 27213 | The second Chebyshev funct... |
| ppiprm 27214 | The prime-counting functio... |
| ppinprm 27215 | The prime-counting functio... |
| chtprm 27216 | The Chebyshev function at ... |
| chtnprm 27217 | The Chebyshev function at ... |
| chpp1 27218 | The second Chebyshev funct... |
| chtwordi 27219 | The Chebyshev function is ... |
| chpwordi 27220 | The second Chebyshev funct... |
| chtdif 27221 | The difference of the Cheb... |
| efchtdvds 27222 | The exponentiated Chebyshe... |
| ppifl 27223 | The prime-counting functio... |
| ppip1le 27224 | The prime-counting functio... |
| ppiwordi 27225 | The prime-counting functio... |
| ppidif 27226 | The difference of the prim... |
| ppi1 27227 | The prime-counting functio... |
| cht1 27228 | The Chebyshev function at ... |
| vma1 27229 | The von Mangoldt function ... |
| chp1 27230 | The second Chebyshev funct... |
| ppi1i 27231 | Inference form of ~ ppiprm... |
| ppi2i 27232 | Inference form of ~ ppinpr... |
| ppi2 27233 | The prime-counting functio... |
| ppi3 27234 | The prime-counting functio... |
| cht2 27235 | The Chebyshev function at ... |
| cht3 27236 | The Chebyshev function at ... |
| ppinncl 27237 | Closure of the prime-count... |
| chtrpcl 27238 | Closure of the Chebyshev f... |
| ppieq0 27239 | The prime-counting functio... |
| ppiltx 27240 | The prime-counting functio... |
| prmorcht 27241 | Relate the primorial (prod... |
| mumullem1 27242 | Lemma for ~ mumul . A mul... |
| mumullem2 27243 | Lemma for ~ mumul . The p... |
| mumul 27244 | The Möbius function i... |
| sqff1o 27245 | There is a bijection from ... |
| fsumdvdsdiaglem 27246 | A "diagonal commutation" o... |
| fsumdvdsdiag 27247 | A "diagonal commutation" o... |
| fsumdvdscom 27248 | A double commutation of di... |
| dvdsppwf1o 27249 | A bijection from the divis... |
| dvdsflf1o 27250 | A bijection from the numbe... |
| dvdsflsumcom 27251 | A sum commutation from ` s... |
| fsumfldivdiaglem 27252 | Lemma for ~ fsumfldivdiag ... |
| fsumfldivdiag 27253 | The right-hand side of ~ d... |
| musum 27254 | The sum of the Möbius... |
| musumsum 27255 | Evaluate a collapsing sum ... |
| muinv 27256 | The Möbius inversion ... |
| mpodvdsmulf1o 27257 | If ` M ` and ` N ` are two... |
| fsumdvdsmul 27258 | Product of two divisor sum... |
| dvdsmulf1o 27259 | If ` M ` and ` N ` are two... |
| fsumdvdsmulOLD 27260 | Obsolete version of ~ fsum... |
| sgmppw 27261 | The value of the divisor f... |
| 0sgmppw 27262 | A prime power ` P ^ K ` ha... |
| 1sgmprm 27263 | The sum of divisors for a ... |
| 1sgm2ppw 27264 | The sum of the divisors of... |
| sgmmul 27265 | The divisor function for f... |
| ppiublem1 27266 | Lemma for ~ ppiub . (Cont... |
| ppiublem2 27267 | A prime greater than ` 3 `... |
| ppiub 27268 | An upper bound on the prim... |
| vmalelog 27269 | The von Mangoldt function ... |
| chtlepsi 27270 | The first Chebyshev functi... |
| chprpcl 27271 | Closure of the second Cheb... |
| chpeq0 27272 | The second Chebyshev funct... |
| chteq0 27273 | The first Chebyshev functi... |
| chtleppi 27274 | Upper bound on the ` theta... |
| chtublem 27275 | Lemma for ~ chtub . (Cont... |
| chtub 27276 | An upper bound on the Cheb... |
| fsumvma 27277 | Rewrite a sum over the von... |
| fsumvma2 27278 | Apply ~ fsumvma for the co... |
| pclogsum 27279 | The logarithmic analogue o... |
| vmasum 27280 | The sum of the von Mangold... |
| logfac2 27281 | Another expression for the... |
| chpval2 27282 | Express the second Chebysh... |
| chpchtsum 27283 | The second Chebyshev funct... |
| chpub 27284 | An upper bound on the seco... |
| logfacubnd 27285 | A simple upper bound on th... |
| logfaclbnd 27286 | A lower bound on the logar... |
| logfacbnd3 27287 | Show the stronger statemen... |
| logfacrlim 27288 | Combine the estimates ~ lo... |
| logexprlim 27289 | The sum ` sum_ n <_ x , lo... |
| logfacrlim2 27290 | Write out ~ logfacrlim as ... |
| mersenne 27291 | A Mersenne prime is a prim... |
| perfect1 27292 | Euclid's contribution to t... |
| perfectlem1 27293 | Lemma for ~ perfect . (Co... |
| perfectlem2 27294 | Lemma for ~ perfect . (Co... |
| perfect 27295 | The Euclid-Euler theorem, ... |
| dchrval 27298 | Value of the group of Diri... |
| dchrbas 27299 | Base set of the group of D... |
| dchrelbas 27300 | A Dirichlet character is a... |
| dchrelbas2 27301 | A Dirichlet character is a... |
| dchrelbas3 27302 | A Dirichlet character is a... |
| dchrelbasd 27303 | A Dirichlet character is a... |
| dchrrcl 27304 | Reverse closure for a Diri... |
| dchrmhm 27305 | A Dirichlet character is a... |
| dchrf 27306 | A Dirichlet character is a... |
| dchrelbas4 27307 | A Dirichlet character is a... |
| dchrzrh1 27308 | Value of a Dirichlet chara... |
| dchrzrhcl 27309 | A Dirichlet character take... |
| dchrzrhmul 27310 | A Dirichlet character is c... |
| dchrplusg 27311 | Group operation on the gro... |
| dchrmul 27312 | Group operation on the gro... |
| dchrmulcl 27313 | Closure of the group opera... |
| dchrn0 27314 | A Dirichlet character is n... |
| dchr1cl 27315 | Closure of the principal D... |
| dchrmullid 27316 | Left identity for the prin... |
| dchrinvcl 27317 | Closure of the group inver... |
| dchrabl 27318 | The set of Dirichlet chara... |
| dchrfi 27319 | The group of Dirichlet cha... |
| dchrghm 27320 | A Dirichlet character rest... |
| dchr1 27321 | Value of the principal Dir... |
| dchreq 27322 | A Dirichlet character is d... |
| dchrresb 27323 | A Dirichlet character is d... |
| dchrabs 27324 | A Dirichlet character take... |
| dchrinv 27325 | The inverse of a Dirichlet... |
| dchrabs2 27326 | A Dirichlet character take... |
| dchr1re 27327 | The principal Dirichlet ch... |
| dchrptlem1 27328 | Lemma for ~ dchrpt . (Con... |
| dchrptlem2 27329 | Lemma for ~ dchrpt . (Con... |
| dchrptlem3 27330 | Lemma for ~ dchrpt . (Con... |
| dchrpt 27331 | For any element other than... |
| dchrsum2 27332 | An orthogonality relation ... |
| dchrsum 27333 | An orthogonality relation ... |
| sumdchr2 27334 | Lemma for ~ sumdchr . (Co... |
| dchrhash 27335 | There are exactly ` phi ( ... |
| sumdchr 27336 | An orthogonality relation ... |
| dchr2sum 27337 | An orthogonality relation ... |
| sum2dchr 27338 | An orthogonality relation ... |
| bcctr 27339 | Value of the central binom... |
| pcbcctr 27340 | Prime count of a central b... |
| bcmono 27341 | The binomial coefficient i... |
| bcmax 27342 | The binomial coefficient t... |
| bcp1ctr 27343 | Ratio of two central binom... |
| bclbnd 27344 | A bound on the binomial co... |
| efexple 27345 | Convert a bound on a power... |
| bpos1lem 27346 | Lemma for ~ bpos1 . (Cont... |
| bpos1 27347 | Bertrand's postulate, chec... |
| bposlem1 27348 | An upper bound on the prim... |
| bposlem2 27349 | There are no odd primes in... |
| bposlem3 27350 | Lemma for ~ bpos . Since ... |
| bposlem4 27351 | Lemma for ~ bpos . (Contr... |
| bposlem5 27352 | Lemma for ~ bpos . Bound ... |
| bposlem6 27353 | Lemma for ~ bpos . By usi... |
| bposlem7 27354 | Lemma for ~ bpos . The fu... |
| bposlem8 27355 | Lemma for ~ bpos . Evalua... |
| bposlem9 27356 | Lemma for ~ bpos . Derive... |
| bpos 27357 | Bertrand's postulate: ther... |
| zabsle1 27360 | ` { -u 1 , 0 , 1 } ` is th... |
| lgslem1 27361 | When ` a ` is coprime to t... |
| lgslem2 27362 | The set ` Z ` of all integ... |
| lgslem3 27363 | The set ` Z ` of all integ... |
| lgslem4 27364 | Lemma for ~ lgsfcl2 . (Co... |
| lgsval 27365 | Value of the Legendre symb... |
| lgsfval 27366 | Value of the function ` F ... |
| lgsfcl2 27367 | The function ` F ` is clos... |
| lgscllem 27368 | The Legendre symbol is an ... |
| lgsfcl 27369 | Closure of the function ` ... |
| lgsfle1 27370 | The function ` F ` has mag... |
| lgsval2lem 27371 | Lemma for ~ lgsval2 . (Co... |
| lgsval4lem 27372 | Lemma for ~ lgsval4 . (Co... |
| lgscl2 27373 | The Legendre symbol is an ... |
| lgs0 27374 | The Legendre symbol when t... |
| lgscl 27375 | The Legendre symbol is an ... |
| lgsle1 27376 | The Legendre symbol has ab... |
| lgsval2 27377 | The Legendre symbol at a p... |
| lgs2 27378 | The Legendre symbol at ` 2... |
| lgsval3 27379 | The Legendre symbol at an ... |
| lgsvalmod 27380 | The Legendre symbol is equ... |
| lgsval4 27381 | Restate ~ lgsval for nonze... |
| lgsfcl3 27382 | Closure of the function ` ... |
| lgsval4a 27383 | Same as ~ lgsval4 for posi... |
| lgscl1 27384 | The value of the Legendre ... |
| lgsneg 27385 | The Legendre symbol is eit... |
| lgsneg1 27386 | The Legendre symbol for no... |
| lgsmod 27387 | The Legendre (Jacobi) symb... |
| lgsdilem 27388 | Lemma for ~ lgsdi and ~ lg... |
| lgsdir2lem1 27389 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem2 27390 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem3 27391 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem4 27392 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem5 27393 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2 27394 | The Legendre symbol is com... |
| lgsdirprm 27395 | The Legendre symbol is com... |
| lgsdir 27396 | The Legendre symbol is com... |
| lgsdilem2 27397 | Lemma for ~ lgsdi . (Cont... |
| lgsdi 27398 | The Legendre symbol is com... |
| lgsne0 27399 | The Legendre symbol is non... |
| lgsabs1 27400 | The Legendre symbol is non... |
| lgssq 27401 | The Legendre symbol at a s... |
| lgssq2 27402 | The Legendre symbol at a s... |
| lgsprme0 27403 | The Legendre symbol at any... |
| 1lgs 27404 | The Legendre symbol at ` 1... |
| lgs1 27405 | The Legendre symbol at ` 1... |
| lgsmodeq 27406 | The Legendre (Jacobi) symb... |
| lgsmulsqcoprm 27407 | The Legendre (Jacobi) symb... |
| lgsdirnn0 27408 | Variation on ~ lgsdir vali... |
| lgsdinn0 27409 | Variation on ~ lgsdi valid... |
| lgsqrlem1 27410 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem2 27411 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem3 27412 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem4 27413 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem5 27414 | Lemma for ~ lgsqr . (Cont... |
| lgsqr 27415 | The Legendre symbol for od... |
| lgsqrmod 27416 | If the Legendre symbol of ... |
| lgsqrmodndvds 27417 | If the Legendre symbol of ... |
| lgsdchrval 27418 | The Legendre symbol functi... |
| lgsdchr 27419 | The Legendre symbol functi... |
| gausslemma2dlem0a 27420 | Auxiliary lemma 1 for ~ ga... |
| gausslemma2dlem0b 27421 | Auxiliary lemma 2 for ~ ga... |
| gausslemma2dlem0c 27422 | Auxiliary lemma 3 for ~ ga... |
| gausslemma2dlem0d 27423 | Auxiliary lemma 4 for ~ ga... |
| gausslemma2dlem0e 27424 | Auxiliary lemma 5 for ~ ga... |
| gausslemma2dlem0f 27425 | Auxiliary lemma 6 for ~ ga... |
| gausslemma2dlem0g 27426 | Auxiliary lemma 7 for ~ ga... |
| gausslemma2dlem0h 27427 | Auxiliary lemma 8 for ~ ga... |
| gausslemma2dlem0i 27428 | Auxiliary lemma 9 for ~ ga... |
| gausslemma2dlem1a 27429 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem1 27430 | Lemma 1 for ~ gausslemma2d... |
| gausslemma2dlem2 27431 | Lemma 2 for ~ gausslemma2d... |
| gausslemma2dlem3 27432 | Lemma 3 for ~ gausslemma2d... |
| gausslemma2dlem4 27433 | Lemma 4 for ~ gausslemma2d... |
| gausslemma2dlem5a 27434 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem5 27435 | Lemma 5 for ~ gausslemma2d... |
| gausslemma2dlem6 27436 | Lemma 6 for ~ gausslemma2d... |
| gausslemma2dlem7 27437 | Lemma 7 for ~ gausslemma2d... |
| gausslemma2d 27438 | Gauss' Lemma (see also the... |
| lgseisenlem1 27439 | Lemma for ~ lgseisen . If... |
| lgseisenlem2 27440 | Lemma for ~ lgseisen . Th... |
| lgseisenlem3 27441 | Lemma for ~ lgseisen . (C... |
| lgseisenlem4 27442 | Lemma for ~ lgseisen . (C... |
| lgseisen 27443 | Eisenstein's lemma, an exp... |
| lgsquadlem1 27444 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem2 27445 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem3 27446 | Lemma for ~ lgsquad . (Co... |
| lgsquad 27447 | The Law of Quadratic Recip... |
| lgsquad2lem1 27448 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2lem2 27449 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2 27450 | Extend ~ lgsquad to coprim... |
| lgsquad3 27451 | Extend ~ lgsquad2 to integ... |
| m1lgs 27452 | The first supplement to th... |
| 2lgslem1a1 27453 | Lemma 1 for ~ 2lgslem1a . ... |
| 2lgslem1a2 27454 | Lemma 2 for ~ 2lgslem1a . ... |
| 2lgslem1a 27455 | Lemma 1 for ~ 2lgslem1 . ... |
| 2lgslem1b 27456 | Lemma 2 for ~ 2lgslem1 . ... |
| 2lgslem1c 27457 | Lemma 3 for ~ 2lgslem1 . ... |
| 2lgslem1 27458 | Lemma 1 for ~ 2lgs . (Con... |
| 2lgslem2 27459 | Lemma 2 for ~ 2lgs . (Con... |
| 2lgslem3a 27460 | Lemma for ~ 2lgslem3a1 . ... |
| 2lgslem3b 27461 | Lemma for ~ 2lgslem3b1 . ... |
| 2lgslem3c 27462 | Lemma for ~ 2lgslem3c1 . ... |
| 2lgslem3d 27463 | Lemma for ~ 2lgslem3d1 . ... |
| 2lgslem3a1 27464 | Lemma 1 for ~ 2lgslem3 . ... |
| 2lgslem3b1 27465 | Lemma 2 for ~ 2lgslem3 . ... |
| 2lgslem3c1 27466 | Lemma 3 for ~ 2lgslem3 . ... |
| 2lgslem3d1 27467 | Lemma 4 for ~ 2lgslem3 . ... |
| 2lgslem3 27468 | Lemma 3 for ~ 2lgs . (Con... |
| 2lgs2 27469 | The Legendre symbol for ` ... |
| 2lgslem4 27470 | Lemma 4 for ~ 2lgs : speci... |
| 2lgs 27471 | The second supplement to t... |
| 2lgsoddprmlem1 27472 | Lemma 1 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem2 27473 | Lemma 2 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem3a 27474 | Lemma 1 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3b 27475 | Lemma 2 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3c 27476 | Lemma 3 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3d 27477 | Lemma 4 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3 27478 | Lemma 3 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem4 27479 | Lemma 4 for ~ 2lgsoddprm .... |
| 2lgsoddprm 27480 | The second supplement to t... |
| 2sqlem1 27481 | Lemma for ~ 2sq . (Contri... |
| 2sqlem2 27482 | Lemma for ~ 2sq . (Contri... |
| mul2sq 27483 | Fibonacci's identity (actu... |
| 2sqlem3 27484 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem4 27485 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem5 27486 | Lemma for ~ 2sq . If a nu... |
| 2sqlem6 27487 | Lemma for ~ 2sq . If a nu... |
| 2sqlem7 27488 | Lemma for ~ 2sq . (Contri... |
| 2sqlem8a 27489 | Lemma for ~ 2sqlem8 . (Co... |
| 2sqlem8 27490 | Lemma for ~ 2sq . (Contri... |
| 2sqlem9 27491 | Lemma for ~ 2sq . (Contri... |
| 2sqlem10 27492 | Lemma for ~ 2sq . Every f... |
| 2sqlem11 27493 | Lemma for ~ 2sq . (Contri... |
| 2sq 27494 | All primes of the form ` 4... |
| 2sqblem 27495 | Lemma for ~ 2sqb . (Contr... |
| 2sqb 27496 | The converse to ~ 2sq . (... |
| 2sq2 27497 | ` 2 ` is the sum of square... |
| 2sqn0 27498 | If the sum of two squares ... |
| 2sqcoprm 27499 | If the sum of two squares ... |
| 2sqmod 27500 | Given two decompositions o... |
| 2sqmo 27501 | There exists at most one d... |
| 2sqnn0 27502 | All primes of the form ` 4... |
| 2sqnn 27503 | All primes of the form ` 4... |
| addsq2reu 27504 | For each complex number ` ... |
| addsqn2reu 27505 | For each complex number ` ... |
| addsqrexnreu 27506 | For each complex number, t... |
| addsqnreup 27507 | There is no unique decompo... |
| addsq2nreurex 27508 | For each complex number ` ... |
| addsqn2reurex2 27509 | For each complex number ` ... |
| 2sqreulem1 27510 | Lemma 1 for ~ 2sqreu . (C... |
| 2sqreultlem 27511 | Lemma for ~ 2sqreult . (C... |
| 2sqreultblem 27512 | Lemma for ~ 2sqreultb . (... |
| 2sqreunnlem1 27513 | Lemma 1 for ~ 2sqreunn . ... |
| 2sqreunnltlem 27514 | Lemma for ~ 2sqreunnlt . ... |
| 2sqreunnltblem 27515 | Lemma for ~ 2sqreunnltb . ... |
| 2sqreulem2 27516 | Lemma 2 for ~ 2sqreu etc. ... |
| 2sqreulem3 27517 | Lemma 3 for ~ 2sqreu etc. ... |
| 2sqreulem4 27518 | Lemma 4 for ~ 2sqreu et. ... |
| 2sqreunnlem2 27519 | Lemma 2 for ~ 2sqreunn . ... |
| 2sqreu 27520 | There exists a unique deco... |
| 2sqreunn 27521 | There exists a unique deco... |
| 2sqreult 27522 | There exists a unique deco... |
| 2sqreultb 27523 | There exists a unique deco... |
| 2sqreunnlt 27524 | There exists a unique deco... |
| 2sqreunnltb 27525 | There exists a unique deco... |
| 2sqreuop 27526 | There exists a unique deco... |
| 2sqreuopnn 27527 | There exists a unique deco... |
| 2sqreuoplt 27528 | There exists a unique deco... |
| 2sqreuopltb 27529 | There exists a unique deco... |
| 2sqreuopnnlt 27530 | There exists a unique deco... |
| 2sqreuopnnltb 27531 | There exists a unique deco... |
| 2sqreuopb 27532 | There exists a unique deco... |
| chebbnd1lem1 27533 | Lemma for ~ chebbnd1 : sho... |
| chebbnd1lem2 27534 | Lemma for ~ chebbnd1 : Sh... |
| chebbnd1lem3 27535 | Lemma for ~ chebbnd1 : get... |
| chebbnd1 27536 | The Chebyshev bound: The ... |
| chtppilimlem1 27537 | Lemma for ~ chtppilim . (... |
| chtppilimlem2 27538 | Lemma for ~ chtppilim . (... |
| chtppilim 27539 | The ` theta ` function is ... |
| chto1ub 27540 | The ` theta ` function is ... |
| chebbnd2 27541 | The Chebyshev bound, part ... |
| chto1lb 27542 | The ` theta ` function is ... |
| chpchtlim 27543 | The ` psi ` and ` theta ` ... |
| chpo1ub 27544 | The ` psi ` function is up... |
| chpo1ubb 27545 | The ` psi ` function is up... |
| vmadivsum 27546 | The sum of the von Mangold... |
| vmadivsumb 27547 | Give a total bound on the ... |
| rplogsumlem1 27548 | Lemma for ~ rplogsum . (C... |
| rplogsumlem2 27549 | Lemma for ~ rplogsum . Eq... |
| dchrisum0lem1a 27550 | Lemma for ~ dchrisum0lem1 ... |
| rpvmasumlem 27551 | Lemma for ~ rpvmasum . Ca... |
| dchrisumlema 27552 | Lemma for ~ dchrisum . Le... |
| dchrisumlem1 27553 | Lemma for ~ dchrisum . Le... |
| dchrisumlem2 27554 | Lemma for ~ dchrisum . Le... |
| dchrisumlem3 27555 | Lemma for ~ dchrisum . Le... |
| dchrisum 27556 | If ` n e. [ M , +oo ) |-> ... |
| dchrmusumlema 27557 | Lemma for ~ dchrmusum and ... |
| dchrmusum2 27558 | The sum of the Möbius... |
| dchrvmasumlem1 27559 | An alternative expression ... |
| dchrvmasum2lem 27560 | Give an expression for ` l... |
| dchrvmasum2if 27561 | Combine the results of ~ d... |
| dchrvmasumlem2 27562 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlem3 27563 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlema 27564 | Lemma for ~ dchrvmasum and... |
| dchrvmasumiflem1 27565 | Lemma for ~ dchrvmasumif .... |
| dchrvmasumiflem2 27566 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumif 27567 | An asymptotic approximatio... |
| dchrvmaeq0 27568 | The set ` W ` is the colle... |
| dchrisum0fval 27569 | Value of the function ` F ... |
| dchrisum0fmul 27570 | The function ` F ` , the d... |
| dchrisum0ff 27571 | The function ` F ` is a re... |
| dchrisum0flblem1 27572 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flblem2 27573 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flb 27574 | The divisor sum of a real ... |
| dchrisum0fno1 27575 | The sum ` sum_ k <_ x , F ... |
| rpvmasum2 27576 | A partial result along the... |
| dchrisum0re 27577 | Suppose ` X ` is a non-pri... |
| dchrisum0lema 27578 | Lemma for ~ dchrisum0 . A... |
| dchrisum0lem1b 27579 | Lemma for ~ dchrisum0lem1 ... |
| dchrisum0lem1 27580 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2a 27581 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2 27582 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem3 27583 | Lemma for ~ dchrisum0 . (... |
| dchrisum0 27584 | The sum ` sum_ n e. NN , X... |
| dchrisumn0 27585 | The sum ` sum_ n e. NN , X... |
| dchrmusumlem 27586 | The sum of the Möbius... |
| dchrvmasumlem 27587 | The sum of the Möbius... |
| dchrmusum 27588 | The sum of the Möbius... |
| dchrvmasum 27589 | The sum of the von Mangold... |
| rpvmasum 27590 | The sum of the von Mangold... |
| rplogsum 27591 | The sum of ` log p / p ` o... |
| dirith2 27592 | Dirichlet's theorem: there... |
| dirith 27593 | Dirichlet's theorem: there... |
| mudivsum 27594 | Asymptotic formula for ` s... |
| mulogsumlem 27595 | Lemma for ~ mulogsum . (C... |
| mulogsum 27596 | Asymptotic formula for ... |
| logdivsum 27597 | Asymptotic analysis of ... |
| mulog2sumlem1 27598 | Asymptotic formula for ... |
| mulog2sumlem2 27599 | Lemma for ~ mulog2sum . (... |
| mulog2sumlem3 27600 | Lemma for ~ mulog2sum . (... |
| mulog2sum 27601 | Asymptotic formula for ... |
| vmalogdivsum2 27602 | The sum ` sum_ n <_ x , La... |
| vmalogdivsum 27603 | The sum ` sum_ n <_ x , La... |
| 2vmadivsumlem 27604 | Lemma for ~ 2vmadivsum . ... |
| 2vmadivsum 27605 | The sum ` sum_ m n <_ x , ... |
| logsqvma 27606 | A formula for ` log ^ 2 ( ... |
| logsqvma2 27607 | The Möbius inverse of... |
| log2sumbnd 27608 | Bound on the difference be... |
| selberglem1 27609 | Lemma for ~ selberg . Est... |
| selberglem2 27610 | Lemma for ~ selberg . (Co... |
| selberglem3 27611 | Lemma for ~ selberg . Est... |
| selberg 27612 | Selberg's symmetry formula... |
| selbergb 27613 | Convert eventual boundedne... |
| selberg2lem 27614 | Lemma for ~ selberg2 . Eq... |
| selberg2 27615 | Selberg's symmetry formula... |
| selberg2b 27616 | Convert eventual boundedne... |
| chpdifbndlem1 27617 | Lemma for ~ chpdifbnd . (... |
| chpdifbndlem2 27618 | Lemma for ~ chpdifbnd . (... |
| chpdifbnd 27619 | A bound on the difference ... |
| logdivbnd 27620 | A bound on a sum of logs, ... |
| selberg3lem1 27621 | Introduce a log weighting ... |
| selberg3lem2 27622 | Lemma for ~ selberg3 . Eq... |
| selberg3 27623 | Introduce a log weighting ... |
| selberg4lem1 27624 | Lemma for ~ selberg4 . Eq... |
| selberg4 27625 | The Selberg symmetry formu... |
| pntrval 27626 | Define the residual of the... |
| pntrf 27627 | Functionality of the resid... |
| pntrmax 27628 | There is a bound on the re... |
| pntrsumo1 27629 | A bound on a sum over ` R ... |
| pntrsumbnd 27630 | A bound on a sum over ` R ... |
| pntrsumbnd2 27631 | A bound on a sum over ` R ... |
| selbergr 27632 | Selberg's symmetry formula... |
| selberg3r 27633 | Selberg's symmetry formula... |
| selberg4r 27634 | Selberg's symmetry formula... |
| selberg34r 27635 | The sum of ~ selberg3r and... |
| pntsval 27636 | Define the "Selberg functi... |
| pntsf 27637 | Functionality of the Selbe... |
| selbergs 27638 | Selberg's symmetry formula... |
| selbergsb 27639 | Selberg's symmetry formula... |
| pntsval2 27640 | The Selberg function can b... |
| pntrlog2bndlem1 27641 | The sum of ~ selberg3r and... |
| pntrlog2bndlem2 27642 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem3 27643 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem4 27644 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem5 27645 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem6a 27646 | Lemma for ~ pntrlog2bndlem... |
| pntrlog2bndlem6 27647 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bnd 27648 | A bound on ` R ( x ) log ^... |
| pntpbnd1a 27649 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd1 27650 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd2 27651 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd 27652 | Lemma for ~ pnt . Establi... |
| pntibndlem1 27653 | Lemma for ~ pntibnd . (Co... |
| pntibndlem2a 27654 | Lemma for ~ pntibndlem2 . ... |
| pntibndlem2 27655 | Lemma for ~ pntibnd . The... |
| pntibndlem3 27656 | Lemma for ~ pntibnd . Pac... |
| pntibnd 27657 | Lemma for ~ pnt . Establi... |
| pntlemd 27658 | Lemma for ~ pnt . Closure... |
| pntlemc 27659 | Lemma for ~ pnt . Closure... |
| pntlema 27660 | Lemma for ~ pnt . Closure... |
| pntlemb 27661 | Lemma for ~ pnt . Unpack ... |
| pntlemg 27662 | Lemma for ~ pnt . Closure... |
| pntlemh 27663 | Lemma for ~ pnt . Bounds ... |
| pntlemn 27664 | Lemma for ~ pnt . The "na... |
| pntlemq 27665 | Lemma for ~ pntlemj . (Co... |
| pntlemr 27666 | Lemma for ~ pntlemj . (Co... |
| pntlemj 27667 | Lemma for ~ pnt . The ind... |
| pntlemi 27668 | Lemma for ~ pnt . Elimina... |
| pntlemf 27669 | Lemma for ~ pnt . Add up ... |
| pntlemk 27670 | Lemma for ~ pnt . Evaluat... |
| pntlemo 27671 | Lemma for ~ pnt . Combine... |
| pntleme 27672 | Lemma for ~ pnt . Package... |
| pntlem3 27673 | Lemma for ~ pnt . Equatio... |
| pntlemp 27674 | Lemma for ~ pnt . Wrappin... |
| pntleml 27675 | Lemma for ~ pnt . Equatio... |
| pnt3 27676 | The Prime Number Theorem, ... |
| pnt2 27677 | The Prime Number Theorem, ... |
| pnt 27678 | The Prime Number Theorem: ... |
| abvcxp 27679 | Raising an absolute value ... |
| padicfval 27680 | Value of the p-adic absolu... |
| padicval 27681 | Value of the p-adic absolu... |
| ostth2lem1 27682 | Lemma for ~ ostth2 , altho... |
| qrngbas 27683 | The base set of the field ... |
| qdrng 27684 | The rationals form a divis... |
| qrng0 27685 | The zero element of the fi... |
| qrng1 27686 | The unity element of the f... |
| qrngneg 27687 | The additive inverse in th... |
| qrngdiv 27688 | The division operation in ... |
| qabvle 27689 | By using induction on ` N ... |
| qabvexp 27690 | Induct the product rule ~ ... |
| ostthlem1 27691 | Lemma for ~ ostth . If tw... |
| ostthlem2 27692 | Lemma for ~ ostth . Refin... |
| qabsabv 27693 | The regular absolute value... |
| padicabv 27694 | The p-adic absolute value ... |
| padicabvf 27695 | The p-adic absolute value ... |
| padicabvcxp 27696 | All positive powers of the... |
| ostth1 27697 | - Lemma for ~ ostth : triv... |
| ostth2lem2 27698 | Lemma for ~ ostth2 . (Con... |
| ostth2lem3 27699 | Lemma for ~ ostth2 . (Con... |
| ostth2lem4 27700 | Lemma for ~ ostth2 . (Con... |
| ostth2 27701 | - Lemma for ~ ostth : regu... |
| ostth3 27702 | - Lemma for ~ ostth : p-ad... |
| ostth 27703 | Ostrowski's theorem, which... |
| elno 27710 | Membership in the surreals... |
| elnoOLD 27711 | Obsolete version of ~ elno... |
| sltval 27712 | The value of the surreal l... |
| bdayval 27713 | The value of the birthday ... |
| nofun 27714 | A surreal is a function. ... |
| nodmon 27715 | The domain of a surreal is... |
| norn 27716 | The range of a surreal is ... |
| nofnbday 27717 | A surreal is a function ov... |
| nodmord 27718 | The domain of a surreal ha... |
| elno2 27719 | An alternative condition f... |
| elno3 27720 | Another condition for memb... |
| sltval2 27721 | Alternate expression for s... |
| nofv 27722 | The function value of a su... |
| nosgnn0 27723 | ` (/) ` is not a surreal s... |
| nosgnn0i 27724 | If ` X ` is a surreal sign... |
| noreson 27725 | The restriction of a surre... |
| sltintdifex 27726 |
If ` A |
| sltres 27727 | If the restrictions of two... |
| noxp1o 27728 | The Cartesian product of a... |
| noseponlem 27729 | Lemma for ~ nosepon . Con... |
| nosepon 27730 | Given two unequal surreals... |
| noextend 27731 | Extending a surreal by one... |
| noextendseq 27732 | Extend a surreal by a sequ... |
| noextenddif 27733 | Calculate the place where ... |
| noextendlt 27734 | Extending a surreal with a... |
| noextendgt 27735 | Extending a surreal with a... |
| nolesgn2o 27736 | Given ` A ` less-than or e... |
| nolesgn2ores 27737 | Given ` A ` less-than or e... |
| nogesgn1o 27738 | Given ` A ` greater than o... |
| nogesgn1ores 27739 | Given ` A ` greater than o... |
| sltsolem1 27740 | Lemma for ~ sltso . The "... |
| sltso 27741 | Less-than totally orders t... |
| bdayfo 27742 | The birthday function maps... |
| fvnobday 27743 | The value of a surreal at ... |
| nosepnelem 27744 | Lemma for ~ nosepne . (Co... |
| nosepne 27745 | The value of two non-equal... |
| nosep1o 27746 | If the value of a surreal ... |
| nosep2o 27747 | If the value of a surreal ... |
| nosepdmlem 27748 | Lemma for ~ nosepdm . (Co... |
| nosepdm 27749 | The first place two surrea... |
| nosepeq 27750 | The values of two surreals... |
| nosepssdm 27751 | Given two non-equal surrea... |
| nodenselem4 27752 | Lemma for ~ nodense . Sho... |
| nodenselem5 27753 | Lemma for ~ nodense . If ... |
| nodenselem6 27754 | The restriction of a surre... |
| nodenselem7 27755 | Lemma for ~ nodense . ` A ... |
| nodenselem8 27756 | Lemma for ~ nodense . Giv... |
| nodense 27757 | Given two distinct surreal... |
| bdayimaon 27758 | Lemma for full-eta propert... |
| nolt02olem 27759 | Lemma for ~ nolt02o . If ... |
| nolt02o 27760 | Given ` A ` less-than ` B ... |
| nogt01o 27761 | Given ` A ` greater than `... |
| noresle 27762 | Restriction law for surrea... |
| nomaxmo 27763 | A class of surreals has at... |
| nominmo 27764 | A class of surreals has at... |
| nosupprefixmo 27765 | In any class of surreals, ... |
| noinfprefixmo 27766 | In any class of surreals, ... |
| nosupcbv 27767 | Lemma to change bound vari... |
| nosupno 27768 | The next several theorems ... |
| nosupdm 27769 | The domain of the surreal ... |
| nosupbday 27770 | Birthday bounding law for ... |
| nosupfv 27771 | The value of surreal supre... |
| nosupres 27772 | A restriction law for surr... |
| nosupbnd1lem1 27773 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1lem2 27774 | Lemma for ~ nosupbnd1 . W... |
| nosupbnd1lem3 27775 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem4 27776 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem5 27777 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem6 27778 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1 27779 | Bounding law from below fo... |
| nosupbnd2lem1 27780 | Bounding law from above wh... |
| nosupbnd2 27781 | Bounding law from above fo... |
| noinfcbv 27782 | Change bound variables for... |
| noinfno 27783 | The next several theorems ... |
| noinfdm 27784 | Next, we calculate the dom... |
| noinfbday 27785 | Birthday bounding law for ... |
| noinffv 27786 | The value of surreal infim... |
| noinfres 27787 | The restriction of surreal... |
| noinfbnd1lem1 27788 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1lem2 27789 | Lemma for ~ noinfbnd1 . W... |
| noinfbnd1lem3 27790 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem4 27791 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem5 27792 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem6 27793 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1 27794 | Bounding law from above fo... |
| noinfbnd2lem1 27795 | Bounding law from below wh... |
| noinfbnd2 27796 | Bounding law from below fo... |
| nosupinfsep 27797 | Given two sets of surreals... |
| noetasuplem1 27798 | Lemma for ~ noeta . Estab... |
| noetasuplem2 27799 | Lemma for ~ noeta . The r... |
| noetasuplem3 27800 | Lemma for ~ noeta . ` Z ` ... |
| noetasuplem4 27801 | Lemma for ~ noeta . When ... |
| noetainflem1 27802 | Lemma for ~ noeta . Estab... |
| noetainflem2 27803 | Lemma for ~ noeta . The r... |
| noetainflem3 27804 | Lemma for ~ noeta . ` W ` ... |
| noetainflem4 27805 | Lemma for ~ noeta . If ` ... |
| noetalem1 27806 | Lemma for ~ noeta . Eithe... |
| noetalem2 27807 | Lemma for ~ noeta . The f... |
| noeta 27808 | The full-eta axiom for the... |
| sltirr 27811 | Surreal less-than is irref... |
| slttr 27812 | Surreal less-than is trans... |
| sltasym 27813 | Surreal less-than is asymm... |
| sltlin 27814 | Surreal less-than obeys tr... |
| slttrieq2 27815 | Trichotomy law for surreal... |
| slttrine 27816 | Trichotomy law for surreal... |
| slenlt 27817 | Surreal less-than or equal... |
| sltnle 27818 | Surreal less-than in terms... |
| sleloe 27819 | Surreal less-than or equal... |
| sletri3 27820 | Trichotomy law for surreal... |
| sltletr 27821 | Surreal transitive law. (... |
| slelttr 27822 | Surreal transitive law. (... |
| sletr 27823 | Surreal transitive law. (... |
| slttrd 27824 | Surreal less-than is trans... |
| sltletrd 27825 | Surreal less-than is trans... |
| slelttrd 27826 | Surreal less-than is trans... |
| sletrd 27827 | Surreal less-than or equal... |
| slerflex 27828 | Surreal less-than or equal... |
| sletric 27829 | Surreal trichotomy law. (... |
| maxs1 27830 | A surreal is less than or ... |
| maxs2 27831 | A surreal is less than or ... |
| mins1 27832 | The minimum of two surreal... |
| mins2 27833 | The minimum of two surreal... |
| sltled 27834 | Surreal less-than implies ... |
| sltne 27835 | Surreal less-than implies ... |
| sltlend 27836 | Surreal less-than in terms... |
| bdayfun 27837 | The birthday function is a... |
| bdayfn 27838 | The birthday function is a... |
| bdaydm 27839 | The birthday function's do... |
| bdayrn 27840 | The birthday function's ra... |
| bdayelon 27841 | The value of the birthday ... |
| nocvxminlem 27842 | Lemma for ~ nocvxmin . Gi... |
| nocvxmin 27843 | Given a nonempty convex cl... |
| noprc 27844 | The surreal numbers are a ... |
| noeta2 27849 | A version of ~ noeta with ... |
| brsslt 27850 | Binary relation form of th... |
| ssltex1 27851 | The first argument of surr... |
| ssltex2 27852 | The second argument of sur... |
| ssltss1 27853 | The first argument of surr... |
| ssltss2 27854 | The second argument of sur... |
| ssltsep 27855 | The separation property of... |
| ssltd 27856 | Deduce surreal set less-th... |
| ssltsn 27857 | Surreal set less-than of t... |
| ssltsepc 27858 | Two elements of separated ... |
| ssltsepcd 27859 | Two elements of separated ... |
| sssslt1 27860 | Relation between surreal s... |
| sssslt2 27861 | Relation between surreal s... |
| nulsslt 27862 | The empty set is less-than... |
| nulssgt 27863 | The empty set is greater t... |
| conway 27864 | Conway's Simplicity Theore... |
| scutval 27865 | The value of the surreal c... |
| scutcut 27866 | Cut properties of the surr... |
| scutcl 27867 | Closure law for surreal cu... |
| scutcld 27868 | Closure law for surreal cu... |
| scutbday 27869 | The birthday of the surrea... |
| eqscut 27870 | Condition for equality to ... |
| eqscut2 27871 | Condition for equality to ... |
| sslttr 27872 | Transitive law for surreal... |
| ssltun1 27873 | Union law for surreal set ... |
| ssltun2 27874 | Union law for surreal set ... |
| scutun12 27875 | Union law for surreal cuts... |
| dmscut 27876 | The domain of the surreal ... |
| scutf 27877 | Functionality statement fo... |
| etasslt 27878 | A restatement of ~ noeta u... |
| etasslt2 27879 | A version of ~ etasslt wit... |
| scutbdaybnd 27880 | An upper bound on the birt... |
| scutbdaybnd2 27881 | An upper bound on the birt... |
| scutbdaybnd2lim 27882 | An upper bound on the birt... |
| scutbdaylt 27883 | If a surreal lies in a gap... |
| slerec 27884 | A comparison law for surre... |
| sltrec 27885 | A comparison law for surre... |
| ssltdisj 27886 | If ` A ` preceeds ` B ` , ... |
| 0sno 27891 | Surreal zero is a surreal.... |
| 1sno 27892 | Surreal one is a surreal. ... |
| bday0s 27893 | Calculate the birthday of ... |
| 0slt1s 27894 | Surreal zero is less than ... |
| bday0b 27895 | The only surreal with birt... |
| bday1s 27896 | The birthday of surreal on... |
| cuteq0 27897 | Condition for a surreal cu... |
| cuteq1 27898 | Condition for a surreal cu... |
| sgt0ne0 27899 | A positive surreal is not ... |
| sgt0ne0d 27900 | A positive surreal is not ... |
| madeval 27911 | The value of the made by f... |
| madeval2 27912 | Alternative characterizati... |
| oldval 27913 | The value of the old optio... |
| newval 27914 | The value of the new optio... |
| madef 27915 | The made function is a fun... |
| oldf 27916 | The older function is a fu... |
| newf 27917 | The new function is a func... |
| old0 27918 | No surreal is older than `... |
| madessno 27919 | Made sets are surreals. (... |
| oldssno 27920 | Old sets are surreals. (C... |
| newssno 27921 | New sets are surreals. (C... |
| leftval 27922 | The value of the left opti... |
| rightval 27923 | The value of the right opt... |
| leftf 27924 | The functionality of the l... |
| rightf 27925 | The functionality of the r... |
| elmade 27926 | Membership in the made fun... |
| elmade2 27927 | Membership in the made fun... |
| elold 27928 | Membership in an old set. ... |
| ssltleft 27929 | A surreal is greater than ... |
| ssltright 27930 | A surreal is less than its... |
| lltropt 27931 | The left options of a surr... |
| made0 27932 | The only surreal made on d... |
| new0 27933 | The only surreal new on da... |
| old1 27934 | The only surreal older tha... |
| madess 27935 | If ` A ` is less than or e... |
| oldssmade 27936 | The older-than set is a su... |
| leftssold 27937 | The left options are a sub... |
| rightssold 27938 | The right options are a su... |
| leftssno 27939 | The left set of a surreal ... |
| rightssno 27940 | The right set of a surreal... |
| madecut 27941 | Given a section that is a ... |
| madeun 27942 | The made set is the union ... |
| madeoldsuc 27943 | The made set is the old se... |
| oldsuc 27944 | The value of the old set a... |
| oldlim 27945 | The value of the old set a... |
| madebdayim 27946 | If a surreal is a member o... |
| oldbdayim 27947 | If ` X ` is in the old set... |
| oldirr 27948 | No surreal is a member of ... |
| leftirr 27949 | No surreal is a member of ... |
| rightirr 27950 | No surreal is a member of ... |
| left0s 27951 | The left set of ` 0s ` is ... |
| right0s 27952 | The right set of ` 0s ` is... |
| left1s 27953 | The left set of ` 1s ` is ... |
| right1s 27954 | The right set of ` 1s ` is... |
| lrold 27955 | The union of the left and ... |
| madebdaylemold 27956 | Lemma for ~ madebday . If... |
| madebdaylemlrcut 27957 | Lemma for ~ madebday . If... |
| madebday 27958 | A surreal is part of the s... |
| oldbday 27959 | A surreal is part of the s... |
| newbday 27960 | A surreal is an element of... |
| lrcut 27961 | A surreal is equal to the ... |
| scutfo 27962 | The surreal cut function i... |
| sltn0 27963 | If ` X ` is less than ` Y ... |
| lruneq 27964 | If two surreals share a bi... |
| sltlpss 27965 | If two surreals share a bi... |
| slelss 27966 | If two surreals ` A ` and ... |
| 0elold 27967 | Zero is in the old set of ... |
| 0elleft 27968 | Zero is in the left set of... |
| 0elright 27969 | Zero is in the right set o... |
| madefi 27970 | The made set of an ordinal... |
| oldfi 27971 | The old set of an ordinal ... |
| cofsslt 27972 | If every element of ` A ` ... |
| coinitsslt 27973 | If ` B ` is coinitial with... |
| cofcut1 27974 | If ` C ` is cofinal with `... |
| cofcut1d 27975 | If ` C ` is cofinal with `... |
| cofcut2 27976 | If ` A ` and ` C ` are mut... |
| cofcut2d 27977 | If ` A ` and ` C ` are mut... |
| cofcutr 27978 | If ` X ` is the cut of ` A... |
| cofcutr1d 27979 | If ` X ` is the cut of ` A... |
| cofcutr2d 27980 | If ` X ` is the cut of ` A... |
| cofcutrtime 27981 | If ` X ` is the cut of ` A... |
| cofcutrtime1d 27982 | If ` X ` is a timely cut o... |
| cofcutrtime2d 27983 | If ` X ` is a timely cut o... |
| cofss 27984 | Cofinality for a subset. ... |
| coiniss 27985 | Coinitiality for a subset.... |
| cutlt 27986 | Eliminating all elements b... |
| cutpos 27987 | Reduce the elements of a c... |
| cutmax 27988 | If ` A ` has a maximum, th... |
| cutmin 27989 | If ` B ` has a minimum, th... |
| lrrecval 27992 | The next step in the devel... |
| lrrecval2 27993 | Next, we establish an alte... |
| lrrecpo 27994 | Now, we establish that ` R... |
| lrrecse 27995 | Next, we show that ` R ` i... |
| lrrecfr 27996 | Now we show that ` R ` is ... |
| lrrecpred 27997 | Finally, we calculate the ... |
| noinds 27998 | Induction principle for a ... |
| norecfn 27999 | Surreal recursion over one... |
| norecov 28000 | Calculate the value of the... |
| noxpordpo 28003 | To get through most of the... |
| noxpordfr 28004 | Next we establish the foun... |
| noxpordse 28005 | Next we establish the set-... |
| noxpordpred 28006 | Next we calculate the pred... |
| no2indslem 28007 | Double induction on surrea... |
| no2inds 28008 | Double induction on surrea... |
| norec2fn 28009 | The double-recursion opera... |
| norec2ov 28010 | The value of the double-re... |
| no3inds 28011 | Triple induction over surr... |
| addsfn 28014 | Surreal addition is a func... |
| addsval 28015 | The value of surreal addit... |
| addsval2 28016 | The value of surreal addit... |
| addsrid 28017 | Surreal addition to zero i... |
| addsridd 28018 | Surreal addition to zero i... |
| addscom 28019 | Surreal addition commutes.... |
| addscomd 28020 | Surreal addition commutes.... |
| addslid 28021 | Surreal addition to zero i... |
| addsproplem1 28022 | Lemma for surreal addition... |
| addsproplem2 28023 | Lemma for surreal addition... |
| addsproplem3 28024 | Lemma for surreal addition... |
| addsproplem4 28025 | Lemma for surreal addition... |
| addsproplem5 28026 | Lemma for surreal addition... |
| addsproplem6 28027 | Lemma for surreal addition... |
| addsproplem7 28028 | Lemma for surreal addition... |
| addsprop 28029 | Inductively show that surr... |
| addscutlem 28030 | Lemma for ~ addscut . Sho... |
| addscut 28031 | Demonstrate the cut proper... |
| addscut2 28032 | Show that the cut involved... |
| addscld 28033 | Surreal numbers are closed... |
| addscl 28034 | Surreal numbers are closed... |
| addsf 28035 | Function statement for sur... |
| addsfo 28036 | Surreal addition is onto. ... |
| peano2no 28037 | A theorem for surreals tha... |
| sltadd1im 28038 | Surreal less-than is prese... |
| sltadd2im 28039 | Surreal less-than is prese... |
| sleadd1im 28040 | Surreal less-than or equal... |
| sleadd2im 28041 | Surreal less-than or equal... |
| sleadd1 28042 | Addition to both sides of ... |
| sleadd2 28043 | Addition to both sides of ... |
| sltadd2 28044 | Addition to both sides of ... |
| sltadd1 28045 | Addition to both sides of ... |
| addscan2 28046 | Cancellation law for surre... |
| addscan1 28047 | Cancellation law for surre... |
| sleadd1d 28048 | Addition to both sides of ... |
| sleadd2d 28049 | Addition to both sides of ... |
| sltadd2d 28050 | Addition to both sides of ... |
| sltadd1d 28051 | Addition to both sides of ... |
| addscan2d 28052 | Cancellation law for surre... |
| addscan1d 28053 | Cancellation law for surre... |
| addsuniflem 28054 | Lemma for ~ addsunif . St... |
| addsunif 28055 | Uniformity theorem for sur... |
| addsasslem1 28056 | Lemma for addition associa... |
| addsasslem2 28057 | Lemma for addition associa... |
| addsass 28058 | Surreal addition is associ... |
| addsassd 28059 | Surreal addition is associ... |
| adds32d 28060 | Commutative/associative la... |
| adds12d 28061 | Commutative/associative la... |
| adds4d 28062 | Rearrangement of four term... |
| adds42d 28063 | Rearrangement of four term... |
| sltaddpos1d 28064 | Addition of a positive num... |
| sltaddpos2d 28065 | Addition of a positive num... |
| slt2addd 28066 | Adding both sides of two s... |
| addsgt0d 28067 | The sum of two positive su... |
| sltp1d 28068 | A surreal is less than its... |
| addsbdaylem 28069 | Lemma for ~ addsbday . (C... |
| addsbday 28070 | The birthday of the sum of... |
| negsfn 28075 | Surreal negation is a func... |
| subsfn 28076 | Surreal subtraction is a f... |
| negsval 28077 | The value of the surreal n... |
| negs0s 28078 | Negative surreal zero is s... |
| negs1s 28079 | An expression for negative... |
| negsproplem1 28080 | Lemma for surreal negation... |
| negsproplem2 28081 | Lemma for surreal negation... |
| negsproplem3 28082 | Lemma for surreal negation... |
| negsproplem4 28083 | Lemma for surreal negation... |
| negsproplem5 28084 | Lemma for surreal negation... |
| negsproplem6 28085 | Lemma for surreal negation... |
| negsproplem7 28086 | Lemma for surreal negation... |
| negsprop 28087 | Show closure and ordering ... |
| negscl 28088 | The surreals are closed un... |
| negscld 28089 | The surreals are closed un... |
| sltnegim 28090 | The forward direction of t... |
| negscut 28091 | The cut properties of surr... |
| negscut2 28092 | The cut that defines surre... |
| negsid 28093 | Surreal addition of a numb... |
| negsidd 28094 | Surreal addition of a numb... |
| negsex 28095 | Every surreal has a negati... |
| negnegs 28096 | A surreal is equal to the ... |
| sltneg 28097 | Negative of both sides of ... |
| sleneg 28098 | Negative of both sides of ... |
| sltnegd 28099 | Negative of both sides of ... |
| slenegd 28100 | Negative of both sides of ... |
| negs11 28101 | Surreal negation is one-to... |
| negsdi 28102 | Distribution of surreal ne... |
| slt0neg2d 28103 | Comparison of a surreal an... |
| negsf 28104 | Function statement for sur... |
| negsfo 28105 | Function statement for sur... |
| negsf1o 28106 | Surreal negation is a bije... |
| negsunif 28107 | Uniformity property for su... |
| negsbdaylem 28108 | Lemma for ~ negsbday . Bo... |
| negsbday 28109 | Negation of a surreal numb... |
| subsval 28110 | The value of surreal subtr... |
| subsvald 28111 | The value of surreal subtr... |
| subscl 28112 | Closure law for surreal su... |
| subscld 28113 | Closure law for surreal su... |
| subsf 28114 | Function statement for sur... |
| subsfo 28115 | Surreal subtraction is an ... |
| negsval2 28116 | Surreal negation in terms ... |
| negsval2d 28117 | Surreal negation in terms ... |
| subsid1 28118 | Identity law for subtracti... |
| subsid 28119 | Subtraction of a surreal f... |
| subadds 28120 | Relationship between addit... |
| subaddsd 28121 | Relationship between addit... |
| pncans 28122 | Cancellation law for surre... |
| pncan3s 28123 | Subtraction and addition o... |
| pncan2s 28124 | Cancellation law for surre... |
| npcans 28125 | Cancellation law for surre... |
| sltsub1 28126 | Subtraction from both side... |
| sltsub2 28127 | Subtraction from both side... |
| sltsub1d 28128 | Subtraction from both side... |
| sltsub2d 28129 | Subtraction from both side... |
| negsubsdi2d 28130 | Distribution of negative o... |
| addsubsassd 28131 | Associative-type law for s... |
| addsubsd 28132 | Law for surreal addition a... |
| sltsubsubbd 28133 | Equivalence for the surrea... |
| sltsubsub2bd 28134 | Equivalence for the surrea... |
| sltsubsub3bd 28135 | Equivalence for the surrea... |
| slesubsubbd 28136 | Equivalence for the surrea... |
| slesubsub2bd 28137 | Equivalence for the surrea... |
| slesubsub3bd 28138 | Equivalence for the surrea... |
| sltsubaddd 28139 | Surreal less-than relation... |
| sltsubadd2d 28140 | Surreal less-than relation... |
| sltaddsubd 28141 | Surreal less-than relation... |
| sltaddsub2d 28142 | Surreal less-than relation... |
| slesubaddd 28143 | Surreal less-than or equal... |
| subsubs4d 28144 | Law for double surreal sub... |
| subsubs2d 28145 | Law for double surreal sub... |
| nncansd 28146 | Cancellation law for surre... |
| posdifsd 28147 | Comparison of two surreals... |
| sltsubposd 28148 | Subtraction of a positive ... |
| subsge0d 28149 | Non-negative subtraction. ... |
| addsubs4d 28150 | Rearrangement of four term... |
| sltm1d 28151 | A surreal is greater than ... |
| mulsfn 28154 | Surreal multiplication is ... |
| mulsval 28155 | The value of surreal multi... |
| mulsval2lem 28156 | Lemma for ~ mulsval2 . Ch... |
| mulsval2 28157 | The value of surreal multi... |
| muls01 28158 | Surreal multiplication by ... |
| mulsrid 28159 | Surreal one is a right ide... |
| mulsridd 28160 | Surreal one is a right ide... |
| mulsproplemcbv 28161 | Lemma for surreal multipli... |
| mulsproplem1 28162 | Lemma for surreal multipli... |
| mulsproplem2 28163 | Lemma for surreal multipli... |
| mulsproplem3 28164 | Lemma for surreal multipli... |
| mulsproplem4 28165 | Lemma for surreal multipli... |
| mulsproplem5 28166 | Lemma for surreal multipli... |
| mulsproplem6 28167 | Lemma for surreal multipli... |
| mulsproplem7 28168 | Lemma for surreal multipli... |
| mulsproplem8 28169 | Lemma for surreal multipli... |
| mulsproplem9 28170 | Lemma for surreal multipli... |
| mulsproplem10 28171 | Lemma for surreal multipli... |
| mulsproplem11 28172 | Lemma for surreal multipli... |
| mulsproplem12 28173 | Lemma for surreal multipli... |
| mulsproplem13 28174 | Lemma for surreal multipli... |
| mulsproplem14 28175 | Lemma for surreal multipli... |
| mulsprop 28176 | Surreals are closed under ... |
| mulscutlem 28177 | Lemma for ~ mulscut . Sta... |
| mulscut 28178 | Show the cut properties of... |
| mulscut2 28179 | Show that the cut involved... |
| mulscl 28180 | The surreals are closed un... |
| mulscld 28181 | The surreals are closed un... |
| sltmul 28182 | An ordering relationship f... |
| sltmuld 28183 | An ordering relationship f... |
| slemuld 28184 | An ordering relationship f... |
| mulscom 28185 | Surreal multiplication com... |
| mulscomd 28186 | Surreal multiplication com... |
| muls02 28187 | Surreal multiplication by ... |
| mulslid 28188 | Surreal one is a left iden... |
| mulslidd 28189 | Surreal one is a left iden... |
| mulsgt0 28190 | The product of two positiv... |
| mulsgt0d 28191 | The product of two positiv... |
| mulsge0d 28192 | The product of two non-neg... |
| ssltmul1 28193 | One surreal set less-than ... |
| ssltmul2 28194 | One surreal set less-than ... |
| mulsuniflem 28195 | Lemma for ~ mulsunif . St... |
| mulsunif 28196 | Surreal multiplication has... |
| addsdilem1 28197 | Lemma for surreal distribu... |
| addsdilem2 28198 | Lemma for surreal distribu... |
| addsdilem3 28199 | Lemma for ~ addsdi . Show... |
| addsdilem4 28200 | Lemma for ~ addsdi . Show... |
| addsdi 28201 | Distributive law for surre... |
| addsdid 28202 | Distributive law for surre... |
| addsdird 28203 | Distributive law for surre... |
| subsdid 28204 | Distribution of surreal mu... |
| subsdird 28205 | Distribution of surreal mu... |
| mulnegs1d 28206 | Product with negative is n... |
| mulnegs2d 28207 | Product with negative is n... |
| mul2negsd 28208 | Surreal product of two neg... |
| mulsasslem1 28209 | Lemma for ~ mulsass . Exp... |
| mulsasslem2 28210 | Lemma for ~ mulsass . Exp... |
| mulsasslem3 28211 | Lemma for ~ mulsass . Dem... |
| mulsass 28212 | Associative law for surrea... |
| mulsassd 28213 | Associative law for surrea... |
| muls4d 28214 | Rearrangement of four surr... |
| mulsunif2lem 28215 | Lemma for ~ mulsunif2 . S... |
| mulsunif2 28216 | Alternate expression for s... |
| sltmul2 28217 | Multiplication of both sid... |
| sltmul2d 28218 | Multiplication of both sid... |
| sltmul1d 28219 | Multiplication of both sid... |
| slemul2d 28220 | Multiplication of both sid... |
| slemul1d 28221 | Multiplication of both sid... |
| sltmulneg1d 28222 | Multiplication of both sid... |
| sltmulneg2d 28223 | Multiplication of both sid... |
| mulscan2dlem 28224 | Lemma for ~ mulscan2d . C... |
| mulscan2d 28225 | Cancellation of surreal mu... |
| mulscan1d 28226 | Cancellation of surreal mu... |
| muls12d 28227 | Commutative/associative la... |
| slemul1ad 28228 | Multiplication of both sid... |
| sltmul12ad 28229 | Comparison of the product ... |
| divsmo 28230 | Uniqueness of surreal inve... |
| muls0ord 28231 | If a surreal product is ze... |
| mulsne0bd 28232 | The product of two non-zer... |
| divsval 28235 | The value of surreal divis... |
| norecdiv 28236 | If a surreal has a recipro... |
| noreceuw 28237 | If a surreal has a recipro... |
| divsmulw 28238 | Relationship between surre... |
| divsmulwd 28239 | Relationship between surre... |
| divsclw 28240 | Weak division closure law.... |
| divsclwd 28241 | Weak division closure law.... |
| divscan2wd 28242 | A weak cancellation law fo... |
| divscan1wd 28243 | A weak cancellation law fo... |
| sltdivmulwd 28244 | Surreal less-than relation... |
| sltdivmul2wd 28245 | Surreal less-than relation... |
| sltmuldivwd 28246 | Surreal less-than relation... |
| sltmuldiv2wd 28247 | Surreal less-than relation... |
| divsasswd 28248 | An associative law for sur... |
| divs1 28249 | A surreal divided by one i... |
| precsexlemcbv 28250 | Lemma for surreal reciproc... |
| precsexlem1 28251 | Lemma for surreal reciproc... |
| precsexlem2 28252 | Lemma for surreal reciproc... |
| precsexlem3 28253 | Lemma for surreal reciproc... |
| precsexlem4 28254 | Lemma for surreal reciproc... |
| precsexlem5 28255 | Lemma for surreal reciproc... |
| precsexlem6 28256 | Lemma for surreal reciproc... |
| precsexlem7 28257 | Lemma for surreal reciproc... |
| precsexlem8 28258 | Lemma for surreal reciproc... |
| precsexlem9 28259 | Lemma for surreal reciproc... |
| precsexlem10 28260 | Lemma for surreal reciproc... |
| precsexlem11 28261 | Lemma for surreal reciproc... |
| precsex 28262 | Every positive surreal has... |
| recsex 28263 | A non-zero surreal has a r... |
| recsexd 28264 | A non-zero surreal has a r... |
| divsmul 28265 | Relationship between surre... |
| divsmuld 28266 | Relationship between surre... |
| divscl 28267 | Surreal division closure l... |
| divscld 28268 | Surreal division closure l... |
| divscan2d 28269 | A cancellation law for sur... |
| divscan1d 28270 | A cancellation law for sur... |
| sltdivmuld 28271 | Surreal less-than relation... |
| sltdivmul2d 28272 | Surreal less-than relation... |
| sltmuldivd 28273 | Surreal less-than relation... |
| sltmuldiv2d 28274 | Surreal less-than relation... |
| divsassd 28275 | An associative law for sur... |
| divmuldivsd 28276 | Multiplication of two surr... |
| divdivs1d 28277 | Surreal division into a fr... |
| divsrecd 28278 | Relationship between surre... |
| divsdird 28279 | Distribution of surreal di... |
| divscan3d 28280 | A cancellation law for sur... |
| abssval 28283 | The value of surreal absol... |
| absscl 28284 | Closure law for surreal ab... |
| abssid 28285 | The absolute value of a no... |
| abs0s 28286 | The absolute value of surr... |
| abssnid 28287 | For a negative surreal, it... |
| absmuls 28288 | Surreal absolute value dis... |
| abssge0 28289 | The absolute value of a su... |
| abssor 28290 | The absolute value of a su... |
| abssneg 28291 | Surreal absolute value of ... |
| sleabs 28292 | A surreal is less than or ... |
| absslt 28293 | Surreal absolute value and... |
| elons 28296 | Membership in the class of... |
| onssno 28297 | The surreal ordinals are a... |
| onsno 28298 | A surreal ordinal is a sur... |
| 0ons 28299 | Surreal zero is a surreal ... |
| 1ons 28300 | Surreal one is a surreal o... |
| elons2 28301 | A surreal is ordinal iff i... |
| elons2d 28302 | The cut of any set of surr... |
| sltonold 28303 | The class of ordinals less... |
| sltonex 28304 | The class of ordinals less... |
| onscutleft 28305 | A surreal ordinal is equal... |
| onaddscl 28306 | The surreal ordinals are c... |
| onmulscl 28307 | The surreal ordinals are c... |
| peano2ons 28308 | The successor of a surreal... |
| seqsex 28311 | Existence of the surreal s... |
| seqseq123d 28312 | Equality deduction for the... |
| nfseqs 28313 | Hypothesis builder for the... |
| seqsval 28314 | The value of the surreal s... |
| noseqex 28315 | The next several theorems ... |
| noseq0 28316 | The surreal ` A ` is a mem... |
| noseqp1 28317 | One plus an element of ` Z... |
| noseqind 28318 | Peano's inductive postulat... |
| noseqinds 28319 | Induction schema for surre... |
| noseqssno 28320 | A surreal sequence is a su... |
| noseqno 28321 | An element of a surreal se... |
| om2noseq0 28322 | The mapping ` G ` is a one... |
| om2noseqsuc 28323 | The value of ` G ` at a su... |
| om2noseqfo 28324 | Function statement for ` G... |
| om2noseqlt 28325 | Surreal less-than relation... |
| om2noseqlt2 28326 | The mapping ` G ` preserve... |
| om2noseqf1o 28327 | ` G ` is a bijection. (Co... |
| om2noseqiso 28328 | ` G ` is an isomorphism fr... |
| om2noseqoi 28329 | An alternative definition ... |
| om2noseqrdg 28330 | A helper lemma for the val... |
| noseqrdglem 28331 | A helper lemma for the val... |
| noseqrdgfn 28332 | The recursive definition g... |
| noseqrdg0 28333 | Initial value of a recursi... |
| noseqrdgsuc 28334 | Successor value of a recur... |
| seqsfn 28335 | The surreal sequence build... |
| seqs1 28336 | The value of the surreal s... |
| seqsp1 28337 | The value of the surreal s... |
| n0sex 28342 | The set of all non-negativ... |
| nnsex 28343 | The set of all positive su... |
| peano5n0s 28344 | Peano's inductive postulat... |
| n0ssno 28345 | The non-negative surreal i... |
| nnssn0s 28346 | The positive surreal integ... |
| nnssno 28347 | The positive surreal integ... |
| n0sno 28348 | A non-negative surreal int... |
| nnsno 28349 | A positive surreal integer... |
| n0snod 28350 | A non-negative surreal int... |
| nnsnod 28351 | A positive surreal integer... |
| nnn0s 28352 | A positive surreal integer... |
| nnn0sd 28353 | A positive surreal integer... |
| 0n0s 28354 | Peano postulate: ` 0s ` is... |
| peano2n0s 28355 | Peano postulate: the succe... |
| dfn0s2 28356 | Alternate definition of th... |
| n0sind 28357 | Principle of Mathematical ... |
| n0scut 28358 | A cut form for surreal nat... |
| n0ons 28359 | A surreal natural is a sur... |
| nnne0s 28360 | A surreal positive integer... |
| n0sge0 28361 | A non-negative integer is ... |
| nnsgt0 28362 | A positive integer is grea... |
| elnns 28363 | Membership in the positive... |
| elnns2 28364 | A positive surreal integer... |
| n0s0suc 28365 | A non-negative surreal int... |
| nnsge1 28366 | A positive surreal integer... |
| n0addscl 28367 | The non-negative surreal i... |
| n0mulscl 28368 | The non-negative surreal i... |
| nnaddscl 28369 | The positive surreal integ... |
| nnmulscl 28370 | The positive surreal integ... |
| 1n0s 28371 | Surreal one is a non-negat... |
| 1nns 28372 | Surreal one is a positive ... |
| peano2nns 28373 | Peano postulate for positi... |
| n0sbday 28374 | A non-negative surreal int... |
| n0ssold 28375 | The non-negative surreal i... |
| nnsrecgt0d 28376 | The reciprocal of a positi... |
| seqn0sfn 28377 | The surreal sequence build... |
| eln0s 28378 | A non-negative surreal int... |
| n0s0m1 28379 | Every non-negative surreal... |
| n0subs 28380 | Subtraction of non-negativ... |
| n0p1nns 28381 | One plus a non-negative su... |
| dfnns2 28382 | Alternate definition of th... |
| nnsind 28383 | Principle of Mathematical ... |
| zsex 28386 | The surreal integers form ... |
| zssno 28387 | The surreal integers are a... |
| zno 28388 | A surreal integer is a sur... |
| znod 28389 | A surreal integer is a sur... |
| elzs 28390 | Membership in the set of s... |
| nnzsubs 28391 | The difference of two surr... |
| nnzs 28392 | A positive surreal integer... |
| nnzsd 28393 | A positive surreal integer... |
| 0zs 28394 | Zero is a surreal integer.... |
| n0zs 28395 | A non-negative surreal int... |
| n0zsd 28396 | A non-negative surreal int... |
| 1zs 28397 | One is a surreal integer. ... |
| znegscl 28398 | The surreal integers are c... |
| znegscld 28399 | The surreal integers are c... |
| zaddscl 28400 | The surreal integers are c... |
| zaddscld 28401 | The surreal integers are c... |
| zsubscld 28402 | The surreal integers are c... |
| zmulscld 28403 | The surreal integers are c... |
| elzn0s 28404 | A surreal integer is a sur... |
| elzs2 28405 | A surreal integer is eithe... |
| eln0zs 28406 | Non-negative surreal integ... |
| elnnzs 28407 | Positive surreal integer p... |
| elznns 28408 | Surreal integer property e... |
| zn0subs 28409 | The non-negative differenc... |
| peano5uzs 28410 | Peano's inductive postulat... |
| uzsind 28411 | Induction on the upper sur... |
| zsbday 28412 | A surreal integer has a fi... |
| zscut 28413 | A cut expression for surre... |
| 1p1e2s 28420 | One plus one is two. Surr... |
| no2times 28421 | Version of ~ 2times for su... |
| 2nns 28422 | Surreal two is a surreal n... |
| 2sno 28423 | Surreal two is a surreal n... |
| 2ne0s 28424 | Surreal two is non-zero. ... |
| n0seo 28425 | A non-negative surreal int... |
| zseo 28426 | A surreal integer is eithe... |
| nohalf 28427 | An explicit expression for... |
| expsval 28428 | The value of surreal expon... |
| expsnnval 28429 | Value of surreal exponenti... |
| exps0 28430 | Surreal exponentiation to ... |
| exps1 28431 | Surreal exponentiation to ... |
| expsp1 28432 | Value of a surreal number ... |
| expscl 28433 | Closure law for surreal ex... |
| expsne0 28434 | A non-negative surreal int... |
| expsgt0 28435 | A non-negative surreal int... |
| halfcut 28436 | Relate the cut of twice of... |
| cutpw2 28437 | A cut expression for inver... |
| pw2bday 28438 | The inverses of powers of ... |
| addhalfcut 28439 | The cut of a surreal non-n... |
| pw2cut 28440 | Extend ~ halfcut to arbitr... |
| elzs12 28441 | Membership in the dyadic f... |
| zs12ex 28442 | The class of dyadic fracti... |
| zzs12 28443 | A surreal integer is a dya... |
| zs12bday 28444 | A dyadic fraction has a fi... |
| elreno 28447 | Membership in the set of s... |
| recut 28448 | The cut involved in defini... |
| 0reno 28449 | Surreal zero is a surreal ... |
| renegscl 28450 | The surreal reals are clos... |
| readdscl 28451 | The surreal reals are clos... |
| remulscllem1 28452 | Lemma for ~ remulscl . Sp... |
| remulscllem2 28453 | Lemma for ~ remulscl . Bo... |
| remulscl 28454 | The surreal reals are clos... |
| itvndx 28465 | Index value of the Interva... |
| lngndx 28466 | Index value of the "line" ... |
| itvid 28467 | Utility theorem: index-ind... |
| lngid 28468 | Utility theorem: index-ind... |
| slotsinbpsd 28469 | The slots ` Base ` , ` +g ... |
| slotslnbpsd 28470 | The slots ` Base ` , ` +g ... |
| lngndxnitvndx 28471 | The slot for the line is n... |
| trkgstr 28472 | Functionality of a Tarski ... |
| trkgbas 28473 | The base set of a Tarski g... |
| trkgdist 28474 | The measure of a distance ... |
| trkgitv 28475 | The congruence relation in... |
| istrkgc 28482 | Property of being a Tarski... |
| istrkgb 28483 | Property of being a Tarski... |
| istrkgcb 28484 | Property of being a Tarski... |
| istrkge 28485 | Property of fulfilling Euc... |
| istrkgl 28486 | Building lines from the se... |
| istrkgld 28487 | Property of fulfilling the... |
| istrkg2ld 28488 | Property of fulfilling the... |
| istrkg3ld 28489 | Property of fulfilling the... |
| axtgcgrrflx 28490 | Axiom of reflexivity of co... |
| axtgcgrid 28491 | Axiom of identity of congr... |
| axtgsegcon 28492 | Axiom of segment construct... |
| axtg5seg 28493 | Five segments axiom, Axiom... |
| axtgbtwnid 28494 | Identity of Betweenness. ... |
| axtgpasch 28495 | Axiom of (Inner) Pasch, Ax... |
| axtgcont1 28496 | Axiom of Continuity. Axio... |
| axtgcont 28497 | Axiom of Continuity. Axio... |
| axtglowdim2 28498 | Lower dimension axiom for ... |
| axtgupdim2 28499 | Upper dimension axiom for ... |
| axtgeucl 28500 | Euclid's Axiom. Axiom A10... |
| tgjustf 28501 | Given any function ` F ` ,... |
| tgjustr 28502 | Given any equivalence rela... |
| tgjustc1 28503 | A justification for using ... |
| tgjustc2 28504 | A justification for using ... |
| tgcgrcomimp 28505 | Congruence commutes on the... |
| tgcgrcomr 28506 | Congruence commutes on the... |
| tgcgrcoml 28507 | Congruence commutes on the... |
| tgcgrcomlr 28508 | Congruence commutes on bot... |
| tgcgreqb 28509 | Congruence and equality. ... |
| tgcgreq 28510 | Congruence and equality. ... |
| tgcgrneq 28511 | Congruence and equality. ... |
| tgcgrtriv 28512 | Degenerate segments are co... |
| tgcgrextend 28513 | Link congruence over a pai... |
| tgsegconeq 28514 | Two points that satisfy th... |
| tgbtwntriv2 28515 | Betweenness always holds f... |
| tgbtwncom 28516 | Betweenness commutes. The... |
| tgbtwncomb 28517 | Betweenness commutes, bico... |
| tgbtwnne 28518 | Betweenness and inequality... |
| tgbtwntriv1 28519 | Betweenness always holds f... |
| tgbtwnswapid 28520 | If you can swap the first ... |
| tgbtwnintr 28521 | Inner transitivity law for... |
| tgbtwnexch3 28522 | Exchange the first endpoin... |
| tgbtwnouttr2 28523 | Outer transitivity law for... |
| tgbtwnexch2 28524 | Exchange the outer point o... |
| tgbtwnouttr 28525 | Outer transitivity law for... |
| tgbtwnexch 28526 | Outer transitivity law for... |
| tgtrisegint 28527 | A line segment between two... |
| tglowdim1 28528 | Lower dimension axiom for ... |
| tglowdim1i 28529 | Lower dimension axiom for ... |
| tgldimor 28530 | Excluded-middle like state... |
| tgldim0eq 28531 | In dimension zero, any two... |
| tgldim0itv 28532 | In dimension zero, any two... |
| tgldim0cgr 28533 | In dimension zero, any two... |
| tgbtwndiff 28534 | There is always a ` c ` di... |
| tgdim01 28535 | In geometries of dimension... |
| tgifscgr 28536 | Inner five segment congrue... |
| tgcgrsub 28537 | Removing identical parts f... |
| iscgrg 28540 | The congruence property fo... |
| iscgrgd 28541 | The property for two seque... |
| iscgrglt 28542 | The property for two seque... |
| trgcgrg 28543 | The property for two trian... |
| trgcgr 28544 | Triangle congruence. (Con... |
| ercgrg 28545 | The shape congruence relat... |
| tgcgrxfr 28546 | A line segment can be divi... |
| cgr3id 28547 | Reflexivity law for three-... |
| cgr3simp1 28548 | Deduce segment congruence ... |
| cgr3simp2 28549 | Deduce segment congruence ... |
| cgr3simp3 28550 | Deduce segment congruence ... |
| cgr3swap12 28551 | Permutation law for three-... |
| cgr3swap23 28552 | Permutation law for three-... |
| cgr3swap13 28553 | Permutation law for three-... |
| cgr3rotr 28554 | Permutation law for three-... |
| cgr3rotl 28555 | Permutation law for three-... |
| trgcgrcom 28556 | Commutative law for three-... |
| cgr3tr 28557 | Transitivity law for three... |
| tgbtwnxfr 28558 | A condition for extending ... |
| tgcgr4 28559 | Two quadrilaterals to be c... |
| isismt 28562 | Property of being an isome... |
| ismot 28563 | Property of being an isome... |
| motcgr 28564 | Property of a motion: dist... |
| idmot 28565 | The identity is a motion. ... |
| motf1o 28566 | Motions are bijections. (... |
| motcl 28567 | Closure of motions. (Cont... |
| motco 28568 | The composition of two mot... |
| cnvmot 28569 | The converse of a motion i... |
| motplusg 28570 | The operation for motions ... |
| motgrp 28571 | The motions of a geometry ... |
| motcgrg 28572 | Property of a motion: dist... |
| motcgr3 28573 | Property of a motion: dist... |
| tglng 28574 | Lines of a Tarski Geometry... |
| tglnfn 28575 | Lines as functions. (Cont... |
| tglnunirn 28576 | Lines are sets of points. ... |
| tglnpt 28577 | Lines are sets of points. ... |
| tglngne 28578 | It takes two different poi... |
| tglngval 28579 | The line going through poi... |
| tglnssp 28580 | Lines are subset of the ge... |
| tgellng 28581 | Property of lying on the l... |
| tgcolg 28582 | We choose the notation ` (... |
| btwncolg1 28583 | Betweenness implies coline... |
| btwncolg2 28584 | Betweenness implies coline... |
| btwncolg3 28585 | Betweenness implies coline... |
| colcom 28586 | Swapping the points defini... |
| colrot1 28587 | Rotating the points defini... |
| colrot2 28588 | Rotating the points defini... |
| ncolcom 28589 | Swapping non-colinear poin... |
| ncolrot1 28590 | Rotating non-colinear poin... |
| ncolrot2 28591 | Rotating non-colinear poin... |
| tgdim01ln 28592 | In geometries of dimension... |
| ncoltgdim2 28593 | If there are three non-col... |
| lnxfr 28594 | Transfer law for colineari... |
| lnext 28595 | Extend a line with a missi... |
| tgfscgr 28596 | Congruence law for the gen... |
| lncgr 28597 | Congruence rule for lines.... |
| lnid 28598 | Identity law for points on... |
| tgidinside 28599 | Law for finding a point in... |
| tgbtwnconn1lem1 28600 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem2 28601 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem3 28602 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1 28603 | Connectivity law for betwe... |
| tgbtwnconn2 28604 | Another connectivity law f... |
| tgbtwnconn3 28605 | Inner connectivity law for... |
| tgbtwnconnln3 28606 | Derive colinearity from be... |
| tgbtwnconn22 28607 | Double connectivity law fo... |
| tgbtwnconnln1 28608 | Derive colinearity from be... |
| tgbtwnconnln2 28609 | Derive colinearity from be... |
| legval 28612 | Value of the less-than rel... |
| legov 28613 | Value of the less-than rel... |
| legov2 28614 | An equivalent definition o... |
| legid 28615 | Reflexivity of the less-th... |
| btwnleg 28616 | Betweenness implies less-t... |
| legtrd 28617 | Transitivity of the less-t... |
| legtri3 28618 | Equality from the less-tha... |
| legtrid 28619 | Trichotomy law for the les... |
| leg0 28620 | Degenerated (zero-length) ... |
| legeq 28621 | Deduce equality from "less... |
| legbtwn 28622 | Deduce betweenness from "l... |
| tgcgrsub2 28623 | Removing identical parts f... |
| ltgseg 28624 | The set ` E ` denotes the ... |
| ltgov 28625 | Strict "shorter than" geom... |
| legov3 28626 | An equivalent definition o... |
| legso 28627 | The "shorter than" relatio... |
| ishlg 28630 | Rays : Definition 6.1 of ... |
| hlcomb 28631 | The half-line relation com... |
| hlcomd 28632 | The half-line relation com... |
| hlne1 28633 | The half-line relation imp... |
| hlne2 28634 | The half-line relation imp... |
| hlln 28635 | The half-line relation imp... |
| hleqnid 28636 | The endpoint does not belo... |
| hlid 28637 | The half-line relation is ... |
| hltr 28638 | The half-line relation is ... |
| hlbtwn 28639 | Betweenness is a sufficien... |
| btwnhl1 28640 | Deduce half-line from betw... |
| btwnhl2 28641 | Deduce half-line from betw... |
| btwnhl 28642 | Swap betweenness for a hal... |
| lnhl 28643 | Either a point ` C ` on th... |
| hlcgrex 28644 | Construct a point on a hal... |
| hlcgreulem 28645 | Lemma for ~ hlcgreu . (Co... |
| hlcgreu 28646 | The point constructed in ~... |
| btwnlng1 28647 | Betweenness implies coline... |
| btwnlng2 28648 | Betweenness implies coline... |
| btwnlng3 28649 | Betweenness implies coline... |
| lncom 28650 | Swapping the points defini... |
| lnrot1 28651 | Rotating the points defini... |
| lnrot2 28652 | Rotating the points defini... |
| ncolne1 28653 | Non-colinear points are di... |
| ncolne2 28654 | Non-colinear points are di... |
| tgisline 28655 | The property of being a pr... |
| tglnne 28656 | It takes two different poi... |
| tglndim0 28657 | There are no lines in dime... |
| tgelrnln 28658 | The property of being a pr... |
| tglineeltr 28659 | Transitivity law for lines... |
| tglineelsb2 28660 | If ` S ` lies on PQ , then... |
| tglinerflx1 28661 | Reflexivity law for line m... |
| tglinerflx2 28662 | Reflexivity law for line m... |
| tglinecom 28663 | Commutativity law for line... |
| tglinethru 28664 | If ` A ` is a line contain... |
| tghilberti1 28665 | There is a line through an... |
| tghilberti2 28666 | There is at most one line ... |
| tglinethrueu 28667 | There is a unique line goi... |
| tglnne0 28668 | A line ` A ` has at least ... |
| tglnpt2 28669 | Find a second point on a l... |
| tglineintmo 28670 | Two distinct lines interse... |
| tglineineq 28671 | Two distinct lines interse... |
| tglineneq 28672 | Given three non-colinear p... |
| tglineinteq 28673 | Two distinct lines interse... |
| ncolncol 28674 | Deduce non-colinearity fro... |
| coltr 28675 | A transitivity law for col... |
| coltr3 28676 | A transitivity law for col... |
| colline 28677 | Three points are colinear ... |
| tglowdim2l 28678 | Reformulation of the lower... |
| tglowdim2ln 28679 | There is always one point ... |
| mirreu3 28682 | Existential uniqueness of ... |
| mirval 28683 | Value of the point inversi... |
| mirfv 28684 | Value of the point inversi... |
| mircgr 28685 | Property of the image by t... |
| mirbtwn 28686 | Property of the image by t... |
| ismir 28687 | Property of the image by t... |
| mirf 28688 | Point inversion as functio... |
| mircl 28689 | Closure of the point inver... |
| mirmir 28690 | The point inversion functi... |
| mircom 28691 | Variation on ~ mirmir . (... |
| mirreu 28692 | Any point has a unique ant... |
| mireq 28693 | Equality deduction for poi... |
| mirinv 28694 | The only invariant point o... |
| mirne 28695 | Mirror of non-center point... |
| mircinv 28696 | The center point is invari... |
| mirf1o 28697 | The point inversion functi... |
| miriso 28698 | The point inversion functi... |
| mirbtwni 28699 | Point inversion preserves ... |
| mirbtwnb 28700 | Point inversion preserves ... |
| mircgrs 28701 | Point inversion preserves ... |
| mirmir2 28702 | Point inversion of a point... |
| mirmot 28703 | Point investion is a motio... |
| mirln 28704 | If two points are on the s... |
| mirln2 28705 | If a point and its mirror ... |
| mirconn 28706 | Point inversion of connect... |
| mirhl 28707 | If two points ` X ` and ` ... |
| mirbtwnhl 28708 | If the center of the point... |
| mirhl2 28709 | Deduce half-line relation ... |
| mircgrextend 28710 | Link congruence over a pai... |
| mirtrcgr 28711 | Point inversion of one poi... |
| mirauto 28712 | Point inversion preserves ... |
| miduniq 28713 | Uniqueness of the middle p... |
| miduniq1 28714 | Uniqueness of the middle p... |
| miduniq2 28715 | If two point inversions co... |
| colmid 28716 | Colinearity and equidistan... |
| symquadlem 28717 | Lemma of the symetrial qua... |
| krippenlem 28718 | Lemma for ~ krippen . We ... |
| krippen 28719 | Krippenlemma (German for c... |
| midexlem 28720 | Lemma for the existence of... |
| israg 28725 | Property for 3 points A, B... |
| ragcom 28726 | Commutative rule for right... |
| ragcol 28727 | The right angle property i... |
| ragmir 28728 | Right angle property is pr... |
| mirrag 28729 | Right angle is conserved b... |
| ragtrivb 28730 | Trivial right angle. Theo... |
| ragflat2 28731 | Deduce equality from two r... |
| ragflat 28732 | Deduce equality from two r... |
| ragtriva 28733 | Trivial right angle. Theo... |
| ragflat3 28734 | Right angle and colinearit... |
| ragcgr 28735 | Right angle and colinearit... |
| motrag 28736 | Right angles are preserved... |
| ragncol 28737 | Right angle implies non-co... |
| perpln1 28738 | Derive a line from perpend... |
| perpln2 28739 | Derive a line from perpend... |
| isperp 28740 | Property for 2 lines A, B ... |
| perpcom 28741 | The "perpendicular" relati... |
| perpneq 28742 | Two perpendicular lines ar... |
| isperp2 28743 | Property for 2 lines A, B,... |
| isperp2d 28744 | One direction of ~ isperp2... |
| ragperp 28745 | Deduce that two lines are ... |
| footexALT 28746 | Alternative version of ~ f... |
| footexlem1 28747 | Lemma for ~ footex . (Con... |
| footexlem2 28748 | Lemma for ~ footex . (Con... |
| footex 28749 | From a point ` C ` outside... |
| foot 28750 | From a point ` C ` outside... |
| footne 28751 | Uniqueness of the foot poi... |
| footeq 28752 | Uniqueness of the foot poi... |
| hlperpnel 28753 | A point on a half-line whi... |
| perprag 28754 | Deduce a right angle from ... |
| perpdragALT 28755 | Deduce a right angle from ... |
| perpdrag 28756 | Deduce a right angle from ... |
| colperp 28757 | Deduce a perpendicularity ... |
| colperpexlem1 28758 | Lemma for ~ colperp . Fir... |
| colperpexlem2 28759 | Lemma for ~ colperpex . S... |
| colperpexlem3 28760 | Lemma for ~ colperpex . C... |
| colperpex 28761 | In dimension 2 and above, ... |
| mideulem2 28762 | Lemma for ~ opphllem , whi... |
| opphllem 28763 | Lemma 8.24 of [Schwabhause... |
| mideulem 28764 | Lemma for ~ mideu . We ca... |
| midex 28765 | Existence of the midpoint,... |
| mideu 28766 | Existence and uniqueness o... |
| islnopp 28767 | The property for two point... |
| islnoppd 28768 | Deduce that ` A ` and ` B ... |
| oppne1 28769 | Points lying on opposite s... |
| oppne2 28770 | Points lying on opposite s... |
| oppne3 28771 | Points lying on opposite s... |
| oppcom 28772 | Commutativity rule for "op... |
| opptgdim2 28773 | If two points opposite to ... |
| oppnid 28774 | The "opposite to a line" r... |
| opphllem1 28775 | Lemma for ~ opphl . (Cont... |
| opphllem2 28776 | Lemma for ~ opphl . Lemma... |
| opphllem3 28777 | Lemma for ~ opphl : We as... |
| opphllem4 28778 | Lemma for ~ opphl . (Cont... |
| opphllem5 28779 | Second part of Lemma 9.4 o... |
| opphllem6 28780 | First part of Lemma 9.4 of... |
| oppperpex 28781 | Restating ~ colperpex usin... |
| opphl 28782 | If two points ` A ` and ` ... |
| outpasch 28783 | Axiom of Pasch, outer form... |
| hlpasch 28784 | An application of the axio... |
| ishpg 28787 | Value of the half-plane re... |
| hpgbr 28788 | Half-planes : property for... |
| hpgne1 28789 | Points on the open half pl... |
| hpgne2 28790 | Points on the open half pl... |
| lnopp2hpgb 28791 | Theorem 9.8 of [Schwabhaus... |
| lnoppnhpg 28792 | If two points lie on the o... |
| hpgerlem 28793 | Lemma for the proof that t... |
| hpgid 28794 | The half-plane relation is... |
| hpgcom 28795 | The half-plane relation co... |
| hpgtr 28796 | The half-plane relation is... |
| colopp 28797 | Opposite sides of a line f... |
| colhp 28798 | Half-plane relation for co... |
| hphl 28799 | If two points are on the s... |
| midf 28804 | Midpoint as a function. (... |
| midcl 28805 | Closure of the midpoint. ... |
| ismidb 28806 | Property of the midpoint. ... |
| midbtwn 28807 | Betweenness of midpoint. ... |
| midcgr 28808 | Congruence of midpoint. (... |
| midid 28809 | Midpoint of a null segment... |
| midcom 28810 | Commutativity rule for the... |
| mirmid 28811 | Point inversion preserves ... |
| lmieu 28812 | Uniqueness of the line mir... |
| lmif 28813 | Line mirror as a function.... |
| lmicl 28814 | Closure of the line mirror... |
| islmib 28815 | Property of the line mirro... |
| lmicom 28816 | The line mirroring functio... |
| lmilmi 28817 | Line mirroring is an invol... |
| lmireu 28818 | Any point has a unique ant... |
| lmieq 28819 | Equality deduction for lin... |
| lmiinv 28820 | The invariants of the line... |
| lmicinv 28821 | The mirroring line is an i... |
| lmimid 28822 | If we have a right angle, ... |
| lmif1o 28823 | The line mirroring functio... |
| lmiisolem 28824 | Lemma for ~ lmiiso . (Con... |
| lmiiso 28825 | The line mirroring functio... |
| lmimot 28826 | Line mirroring is a motion... |
| hypcgrlem1 28827 | Lemma for ~ hypcgr , case ... |
| hypcgrlem2 28828 | Lemma for ~ hypcgr , case ... |
| hypcgr 28829 | If the catheti of two righ... |
| lmiopp 28830 | Line mirroring produces po... |
| lnperpex 28831 | Existence of a perpendicul... |
| trgcopy 28832 | Triangle construction: a c... |
| trgcopyeulem 28833 | Lemma for ~ trgcopyeu . (... |
| trgcopyeu 28834 | Triangle construction: a c... |
| iscgra 28837 | Property for two angles AB... |
| iscgra1 28838 | A special version of ~ isc... |
| iscgrad 28839 | Sufficient conditions for ... |
| cgrane1 28840 | Angles imply inequality. ... |
| cgrane2 28841 | Angles imply inequality. ... |
| cgrane3 28842 | Angles imply inequality. ... |
| cgrane4 28843 | Angles imply inequality. ... |
| cgrahl1 28844 | Angle congruence is indepe... |
| cgrahl2 28845 | Angle congruence is indepe... |
| cgracgr 28846 | First direction of proposi... |
| cgraid 28847 | Angle congruence is reflex... |
| cgraswap 28848 | Swap rays in a congruence ... |
| cgrcgra 28849 | Triangle congruence implie... |
| cgracom 28850 | Angle congruence commutes.... |
| cgratr 28851 | Angle congruence is transi... |
| flatcgra 28852 | Flat angles are congruent.... |
| cgraswaplr 28853 | Swap both side of angle co... |
| cgrabtwn 28854 | Angle congruence preserves... |
| cgrahl 28855 | Angle congruence preserves... |
| cgracol 28856 | Angle congruence preserves... |
| cgrancol 28857 | Angle congruence preserves... |
| dfcgra2 28858 | This is the full statement... |
| sacgr 28859 | Supplementary angles of co... |
| oacgr 28860 | Vertical angle theorem. V... |
| acopy 28861 | Angle construction. Theor... |
| acopyeu 28862 | Angle construction. Theor... |
| isinag 28866 | Property for point ` X ` t... |
| isinagd 28867 | Sufficient conditions for ... |
| inagflat 28868 | Any point lies in a flat a... |
| inagswap 28869 | Swap the order of the half... |
| inagne1 28870 | Deduce inequality from the... |
| inagne2 28871 | Deduce inequality from the... |
| inagne3 28872 | Deduce inequality from the... |
| inaghl 28873 | The "point lie in angle" r... |
| isleag 28875 | Geometrical "less than" pr... |
| isleagd 28876 | Sufficient condition for "... |
| leagne1 28877 | Deduce inequality from the... |
| leagne2 28878 | Deduce inequality from the... |
| leagne3 28879 | Deduce inequality from the... |
| leagne4 28880 | Deduce inequality from the... |
| cgrg3col4 28881 | Lemma 11.28 of [Schwabhaus... |
| tgsas1 28882 | First congruence theorem: ... |
| tgsas 28883 | First congruence theorem: ... |
| tgsas2 28884 | First congruence theorem: ... |
| tgsas3 28885 | First congruence theorem: ... |
| tgasa1 28886 | Second congruence theorem:... |
| tgasa 28887 | Second congruence theorem:... |
| tgsss1 28888 | Third congruence theorem: ... |
| tgsss2 28889 | Third congruence theorem: ... |
| tgsss3 28890 | Third congruence theorem: ... |
| dfcgrg2 28891 | Congruence for two triangl... |
| isoas 28892 | Congruence theorem for iso... |
| iseqlg 28895 | Property of a triangle bei... |
| iseqlgd 28896 | Condition for a triangle t... |
| f1otrgds 28897 | Convenient lemma for ~ f1o... |
| f1otrgitv 28898 | Convenient lemma for ~ f1o... |
| f1otrg 28899 | A bijection between bases ... |
| f1otrge 28900 | A bijection between bases ... |
| ttgval 28903 | Define a function to augme... |
| ttgvalOLD 28904 | Obsolete proof of ~ ttgval... |
| ttglem 28905 | Lemma for ~ ttgbas , ~ ttg... |
| ttglemOLD 28906 | Obsolete version of ~ ttgl... |
| ttgbas 28907 | The base set of a subcompl... |
| ttgbasOLD 28908 | Obsolete proof of ~ ttgbas... |
| ttgplusg 28909 | The addition operation of ... |
| ttgplusgOLD 28910 | Obsolete proof of ~ ttgplu... |
| ttgsub 28911 | The subtraction operation ... |
| ttgvsca 28912 | The scalar product of a su... |
| ttgvscaOLD 28913 | Obsolete proof of ~ ttgvsc... |
| ttgds 28914 | The metric of a subcomplex... |
| ttgdsOLD 28915 | Obsolete proof of ~ ttgds ... |
| ttgitvval 28916 | Betweenness for a subcompl... |
| ttgelitv 28917 | Betweenness for a subcompl... |
| ttgbtwnid 28918 | Any subcomplex module equi... |
| ttgcontlem1 28919 | Lemma for % ttgcont . (Co... |
| xmstrkgc 28920 | Any metric space fulfills ... |
| cchhllem 28921 | Lemma for chlbas and chlvs... |
| cchhllemOLD 28922 | Obsolete version of ~ cchh... |
| elee 28929 | Membership in a Euclidean ... |
| mptelee 28930 | A condition for a mapping ... |
| eleenn 28931 | If ` A ` is in ` ( EE `` N... |
| eleei 28932 | The forward direction of ~... |
| eedimeq 28933 | A point belongs to at most... |
| brbtwn 28934 | The binary relation form o... |
| brcgr 28935 | The binary relation form o... |
| fveere 28936 | The function value of a po... |
| fveecn 28937 | The function value of a po... |
| eqeefv 28938 | Two points are equal iff t... |
| eqeelen 28939 | Two points are equal iff t... |
| brbtwn2 28940 | Alternate characterization... |
| colinearalglem1 28941 | Lemma for ~ colinearalg . ... |
| colinearalglem2 28942 | Lemma for ~ colinearalg . ... |
| colinearalglem3 28943 | Lemma for ~ colinearalg . ... |
| colinearalglem4 28944 | Lemma for ~ colinearalg . ... |
| colinearalg 28945 | An algebraic characterizat... |
| eleesub 28946 | Membership of a subtractio... |
| eleesubd 28947 | Membership of a subtractio... |
| axdimuniq 28948 | The unique dimension axiom... |
| axcgrrflx 28949 | ` A ` is as far from ` B `... |
| axcgrtr 28950 | Congruence is transitive. ... |
| axcgrid 28951 | If there is no distance be... |
| axsegconlem1 28952 | Lemma for ~ axsegcon . Ha... |
| axsegconlem2 28953 | Lemma for ~ axsegcon . Sh... |
| axsegconlem3 28954 | Lemma for ~ axsegcon . Sh... |
| axsegconlem4 28955 | Lemma for ~ axsegcon . Sh... |
| axsegconlem5 28956 | Lemma for ~ axsegcon . Sh... |
| axsegconlem6 28957 | Lemma for ~ axsegcon . Sh... |
| axsegconlem7 28958 | Lemma for ~ axsegcon . Sh... |
| axsegconlem8 28959 | Lemma for ~ axsegcon . Sh... |
| axsegconlem9 28960 | Lemma for ~ axsegcon . Sh... |
| axsegconlem10 28961 | Lemma for ~ axsegcon . Sh... |
| axsegcon 28962 | Any segment ` A B ` can be... |
| ax5seglem1 28963 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem2 28964 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem3a 28965 | Lemma for ~ ax5seg . (Con... |
| ax5seglem3 28966 | Lemma for ~ ax5seg . Comb... |
| ax5seglem4 28967 | Lemma for ~ ax5seg . Give... |
| ax5seglem5 28968 | Lemma for ~ ax5seg . If `... |
| ax5seglem6 28969 | Lemma for ~ ax5seg . Give... |
| ax5seglem7 28970 | Lemma for ~ ax5seg . An a... |
| ax5seglem8 28971 | Lemma for ~ ax5seg . Use ... |
| ax5seglem9 28972 | Lemma for ~ ax5seg . Take... |
| ax5seg 28973 | The five segment axiom. T... |
| axbtwnid 28974 | Points are indivisible. T... |
| axpaschlem 28975 | Lemma for ~ axpasch . Set... |
| axpasch 28976 | The inner Pasch axiom. Ta... |
| axlowdimlem1 28977 | Lemma for ~ axlowdim . Es... |
| axlowdimlem2 28978 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem3 28979 | Lemma for ~ axlowdim . Se... |
| axlowdimlem4 28980 | Lemma for ~ axlowdim . Se... |
| axlowdimlem5 28981 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem6 28982 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem7 28983 | Lemma for ~ axlowdim . Se... |
| axlowdimlem8 28984 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem9 28985 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem10 28986 | Lemma for ~ axlowdim . Se... |
| axlowdimlem11 28987 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem12 28988 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem13 28989 | Lemma for ~ axlowdim . Es... |
| axlowdimlem14 28990 | Lemma for ~ axlowdim . Ta... |
| axlowdimlem15 28991 | Lemma for ~ axlowdim . Se... |
| axlowdimlem16 28992 | Lemma for ~ axlowdim . Se... |
| axlowdimlem17 28993 | Lemma for ~ axlowdim . Es... |
| axlowdim1 28994 | The lower dimension axiom ... |
| axlowdim2 28995 | The lower two-dimensional ... |
| axlowdim 28996 | The general lower dimensio... |
| axeuclidlem 28997 | Lemma for ~ axeuclid . Ha... |
| axeuclid 28998 | Euclid's axiom. Take an a... |
| axcontlem1 28999 | Lemma for ~ axcont . Chan... |
| axcontlem2 29000 | Lemma for ~ axcont . The ... |
| axcontlem3 29001 | Lemma for ~ axcont . Give... |
| axcontlem4 29002 | Lemma for ~ axcont . Give... |
| axcontlem5 29003 | Lemma for ~ axcont . Comp... |
| axcontlem6 29004 | Lemma for ~ axcont . Stat... |
| axcontlem7 29005 | Lemma for ~ axcont . Give... |
| axcontlem8 29006 | Lemma for ~ axcont . A po... |
| axcontlem9 29007 | Lemma for ~ axcont . Give... |
| axcontlem10 29008 | Lemma for ~ axcont . Give... |
| axcontlem11 29009 | Lemma for ~ axcont . Elim... |
| axcontlem12 29010 | Lemma for ~ axcont . Elim... |
| axcont 29011 | The axiom of continuity. ... |
| eengv 29014 | The value of the Euclidean... |
| eengstr 29015 | The Euclidean geometry as ... |
| eengbas 29016 | The Base of the Euclidean ... |
| ebtwntg 29017 | The betweenness relation u... |
| ecgrtg 29018 | The congruence relation us... |
| elntg 29019 | The line definition in the... |
| elntg2 29020 | The line definition in the... |
| eengtrkg 29021 | The geometry structure for... |
| eengtrkge 29022 | The geometry structure for... |
| edgfid 29025 | Utility theorem: index-ind... |
| edgfndx 29026 | Index value of the ~ df-ed... |
| edgfndxnn 29027 | The index value of the edg... |
| edgfndxid 29028 | The value of the edge func... |
| edgfndxidOLD 29029 | Obsolete version of ~ edgf... |
| basendxltedgfndx 29030 | The index value of the ` B... |
| baseltedgfOLD 29031 | Obsolete proof of ~ basend... |
| basendxnedgfndx 29032 | The slots ` Base ` and ` .... |
| vtxval 29037 | The set of vertices of a g... |
| iedgval 29038 | The set of indexed edges o... |
| 1vgrex 29039 | A graph with at least one ... |
| opvtxval 29040 | The set of vertices of a g... |
| opvtxfv 29041 | The set of vertices of a g... |
| opvtxov 29042 | The set of vertices of a g... |
| opiedgval 29043 | The set of indexed edges o... |
| opiedgfv 29044 | The set of indexed edges o... |
| opiedgov 29045 | The set of indexed edges o... |
| opvtxfvi 29046 | The set of vertices of a g... |
| opiedgfvi 29047 | The set of indexed edges o... |
| funvtxdmge2val 29048 | The set of vertices of an ... |
| funiedgdmge2val 29049 | The set of indexed edges o... |
| funvtxdm2val 29050 | The set of vertices of an ... |
| funiedgdm2val 29051 | The set of indexed edges o... |
| funvtxval0 29052 | The set of vertices of an ... |
| basvtxval 29053 | The set of vertices of a g... |
| edgfiedgval 29054 | The set of indexed edges o... |
| funvtxval 29055 | The set of vertices of a g... |
| funiedgval 29056 | The set of indexed edges o... |
| structvtxvallem 29057 | Lemma for ~ structvtxval a... |
| structvtxval 29058 | The set of vertices of an ... |
| structiedg0val 29059 | The set of indexed edges o... |
| structgrssvtxlem 29060 | Lemma for ~ structgrssvtx ... |
| structgrssvtx 29061 | The set of vertices of a g... |
| structgrssiedg 29062 | The set of indexed edges o... |
| struct2grstr 29063 | A graph represented as an ... |
| struct2grvtx 29064 | The set of vertices of a g... |
| struct2griedg 29065 | The set of indexed edges o... |
| graop 29066 | Any representation of a gr... |
| grastruct 29067 | Any representation of a gr... |
| gropd 29068 | If any representation of a... |
| grstructd 29069 | If any representation of a... |
| gropeld 29070 | If any representation of a... |
| grstructeld 29071 | If any representation of a... |
| setsvtx 29072 | The vertices of a structur... |
| setsiedg 29073 | The (indexed) edges of a s... |
| snstrvtxval 29074 | The set of vertices of a g... |
| snstriedgval 29075 | The set of indexed edges o... |
| vtxval0 29076 | Degenerated case 1 for ver... |
| iedgval0 29077 | Degenerated case 1 for edg... |
| vtxvalsnop 29078 | Degenerated case 2 for ver... |
| iedgvalsnop 29079 | Degenerated case 2 for edg... |
| vtxval3sn 29080 | Degenerated case 3 for ver... |
| iedgval3sn 29081 | Degenerated case 3 for edg... |
| vtxvalprc 29082 | Degenerated case 4 for ver... |
| iedgvalprc 29083 | Degenerated case 4 for edg... |
| edgval 29086 | The edges of a graph. (Co... |
| iedgedg 29087 | An indexed edge is an edge... |
| edgopval 29088 | The edges of a graph repre... |
| edgov 29089 | The edges of a graph repre... |
| edgstruct 29090 | The edges of a graph repre... |
| edgiedgb 29091 | A set is an edge iff it is... |
| edg0iedg0 29092 | There is no edge in a grap... |
| isuhgr 29097 | The predicate "is an undir... |
| isushgr 29098 | The predicate "is an undir... |
| uhgrf 29099 | The edge function of an un... |
| ushgrf 29100 | The edge function of an un... |
| uhgrss 29101 | An edge is a subset of ver... |
| uhgreq12g 29102 | If two sets have the same ... |
| uhgrfun 29103 | The edge function of an un... |
| uhgrn0 29104 | An edge is a nonempty subs... |
| lpvtx 29105 | The endpoints of a loop (w... |
| ushgruhgr 29106 | An undirected simple hyper... |
| isuhgrop 29107 | The property of being an u... |
| uhgr0e 29108 | The empty graph, with vert... |
| uhgr0vb 29109 | The null graph, with no ve... |
| uhgr0 29110 | The null graph represented... |
| uhgrun 29111 | The union ` U ` of two (un... |
| uhgrunop 29112 | The union of two (undirect... |
| ushgrun 29113 | The union ` U ` of two (un... |
| ushgrunop 29114 | The union of two (undirect... |
| uhgrstrrepe 29115 | Replacing (or adding) the ... |
| incistruhgr 29116 | An _incidence structure_ `... |
| isupgr 29121 | The property of being an u... |
| wrdupgr 29122 | The property of being an u... |
| upgrf 29123 | The edge function of an un... |
| upgrfn 29124 | The edge function of an un... |
| upgrss 29125 | An edge is a subset of ver... |
| upgrn0 29126 | An edge is a nonempty subs... |
| upgrle 29127 | An edge of an undirected p... |
| upgrfi 29128 | An edge is a finite subset... |
| upgrex 29129 | An edge is an unordered pa... |
| upgrbi 29130 | Show that an unordered pai... |
| upgrop 29131 | A pseudograph represented ... |
| isumgr 29132 | The property of being an u... |
| isumgrs 29133 | The simplified property of... |
| wrdumgr 29134 | The property of being an u... |
| umgrf 29135 | The edge function of an un... |
| umgrfn 29136 | The edge function of an un... |
| umgredg2 29137 | An edge of a multigraph ha... |
| umgrbi 29138 | Show that an unordered pai... |
| upgruhgr 29139 | An undirected pseudograph ... |
| umgrupgr 29140 | An undirected multigraph i... |
| umgruhgr 29141 | An undirected multigraph i... |
| upgrle2 29142 | An edge of an undirected p... |
| umgrnloopv 29143 | In a multigraph, there is ... |
| umgredgprv 29144 | In a multigraph, an edge i... |
| umgrnloop 29145 | In a multigraph, there is ... |
| umgrnloop0 29146 | A multigraph has no loops.... |
| umgr0e 29147 | The empty graph, with vert... |
| upgr0e 29148 | The empty graph, with vert... |
| upgr1elem 29149 | Lemma for ~ upgr1e and ~ u... |
| upgr1e 29150 | A pseudograph with one edg... |
| upgr0eop 29151 | The empty graph, with vert... |
| upgr1eop 29152 | A pseudograph with one edg... |
| upgr0eopALT 29153 | Alternate proof of ~ upgr0... |
| upgr1eopALT 29154 | Alternate proof of ~ upgr1... |
| upgrun 29155 | The union ` U ` of two pse... |
| upgrunop 29156 | The union of two pseudogra... |
| umgrun 29157 | The union ` U ` of two mul... |
| umgrunop 29158 | The union of two multigrap... |
| umgrislfupgrlem 29159 | Lemma for ~ umgrislfupgr a... |
| umgrislfupgr 29160 | A multigraph is a loop-fre... |
| lfgredgge2 29161 | An edge of a loop-free gra... |
| lfgrnloop 29162 | A loop-free graph has no l... |
| uhgredgiedgb 29163 | In a hypergraph, a set is ... |
| uhgriedg0edg0 29164 | A hypergraph has no edges ... |
| uhgredgn0 29165 | An edge of a hypergraph is... |
| edguhgr 29166 | An edge of a hypergraph is... |
| uhgredgrnv 29167 | An edge of a hypergraph co... |
| uhgredgss 29168 | The set of edges of a hype... |
| upgredgss 29169 | The set of edges of a pseu... |
| umgredgss 29170 | The set of edges of a mult... |
| edgupgr 29171 | Properties of an edge of a... |
| edgumgr 29172 | Properties of an edge of a... |
| uhgrvtxedgiedgb 29173 | In a hypergraph, a vertex ... |
| upgredg 29174 | For each edge in a pseudog... |
| umgredg 29175 | For each edge in a multigr... |
| upgrpredgv 29176 | An edge of a pseudograph a... |
| umgrpredgv 29177 | An edge of a multigraph al... |
| upgredg2vtx 29178 | For a vertex incident to a... |
| upgredgpr 29179 | If a proper pair (of verti... |
| edglnl 29180 | The edges incident with a ... |
| numedglnl 29181 | The number of edges incide... |
| umgredgne 29182 | An edge of a multigraph al... |
| umgrnloop2 29183 | A multigraph has no loops.... |
| umgredgnlp 29184 | An edge of a multigraph is... |
| isuspgr 29189 | The property of being a si... |
| isusgr 29190 | The property of being a si... |
| uspgrf 29191 | The edge function of a sim... |
| usgrf 29192 | The edge function of a sim... |
| isusgrs 29193 | The property of being a si... |
| usgrfs 29194 | The edge function of a sim... |
| usgrfun 29195 | The edge function of a sim... |
| usgredgss 29196 | The set of edges of a simp... |
| edgusgr 29197 | An edge of a simple graph ... |
| isuspgrop 29198 | The property of being an u... |
| isusgrop 29199 | The property of being an u... |
| usgrop 29200 | A simple graph represented... |
| isausgr 29201 | The property of an unorder... |
| ausgrusgrb 29202 | The equivalence of the def... |
| usgrausgri 29203 | A simple graph represented... |
| ausgrumgri 29204 | If an alternatively define... |
| ausgrusgri 29205 | The equivalence of the def... |
| usgrausgrb 29206 | The equivalence of the def... |
| usgredgop 29207 | An edge of a simple graph ... |
| usgrf1o 29208 | The edge function of a sim... |
| usgrf1 29209 | The edge function of a sim... |
| uspgrf1oedg 29210 | The edge function of a sim... |
| usgrss 29211 | An edge is a subset of ver... |
| uspgredgiedg 29212 | In a simple pseudograph, f... |
| uspgriedgedg 29213 | In a simple pseudograph, f... |
| uspgrushgr 29214 | A simple pseudograph is an... |
| uspgrupgr 29215 | A simple pseudograph is an... |
| uspgrupgrushgr 29216 | A graph is a simple pseudo... |
| usgruspgr 29217 | A simple graph is a simple... |
| usgrumgr 29218 | A simple graph is an undir... |
| usgrumgruspgr 29219 | A graph is a simple graph ... |
| usgruspgrb 29220 | A class is a simple graph ... |
| uspgruhgr 29221 | An undirected simple pseud... |
| usgrupgr 29222 | A simple graph is an undir... |
| usgruhgr 29223 | A simple graph is an undir... |
| usgrislfuspgr 29224 | A simple graph is a loop-f... |
| uspgrun 29225 | The union ` U ` of two sim... |
| uspgrunop 29226 | The union of two simple ps... |
| usgrun 29227 | The union ` U ` of two sim... |
| usgrunop 29228 | The union of two simple gr... |
| usgredg2 29229 | The value of the "edge fun... |
| usgredg2ALT 29230 | Alternate proof of ~ usgre... |
| usgredgprv 29231 | In a simple graph, an edge... |
| usgredgprvALT 29232 | Alternate proof of ~ usgre... |
| usgredgppr 29233 | An edge of a simple graph ... |
| usgrpredgv 29234 | An edge of a simple graph ... |
| edgssv2 29235 | An edge of a simple graph ... |
| usgredg 29236 | For each edge in a simple ... |
| usgrnloopv 29237 | In a simple graph, there i... |
| usgrnloopvALT 29238 | Alternate proof of ~ usgrn... |
| usgrnloop 29239 | In a simple graph, there i... |
| usgrnloopALT 29240 | Alternate proof of ~ usgrn... |
| usgrnloop0 29241 | A simple graph has no loop... |
| usgrnloop0ALT 29242 | Alternate proof of ~ usgrn... |
| usgredgne 29243 | An edge of a simple graph ... |
| usgrf1oedg 29244 | The edge function of a sim... |
| uhgr2edg 29245 | If a vertex is adjacent to... |
| umgr2edg 29246 | If a vertex is adjacent to... |
| usgr2edg 29247 | If a vertex is adjacent to... |
| umgr2edg1 29248 | If a vertex is adjacent to... |
| usgr2edg1 29249 | If a vertex is adjacent to... |
| umgrvad2edg 29250 | If a vertex is adjacent to... |
| umgr2edgneu 29251 | If a vertex is adjacent to... |
| usgrsizedg 29252 | In a simple graph, the siz... |
| usgredg3 29253 | The value of the "edge fun... |
| usgredg4 29254 | For a vertex incident to a... |
| usgredgreu 29255 | For a vertex incident to a... |
| usgredg2vtx 29256 | For a vertex incident to a... |
| uspgredg2vtxeu 29257 | For a vertex incident to a... |
| usgredg2vtxeu 29258 | For a vertex incident to a... |
| usgredg2vtxeuALT 29259 | Alternate proof of ~ usgre... |
| uspgredg2vlem 29260 | Lemma for ~ uspgredg2v . ... |
| uspgredg2v 29261 | In a simple pseudograph, t... |
| usgredg2vlem1 29262 | Lemma 1 for ~ usgredg2v . ... |
| usgredg2vlem2 29263 | Lemma 2 for ~ usgredg2v . ... |
| usgredg2v 29264 | In a simple graph, the map... |
| usgriedgleord 29265 | Alternate version of ~ usg... |
| ushgredgedg 29266 | In a simple hypergraph the... |
| usgredgedg 29267 | In a simple graph there is... |
| ushgredgedgloop 29268 | In a simple hypergraph the... |
| uspgredgleord 29269 | In a simple pseudograph th... |
| usgredgleord 29270 | In a simple graph the numb... |
| usgredgleordALT 29271 | Alternate proof for ~ usgr... |
| usgrstrrepe 29272 | Replacing (or adding) the ... |
| usgr0e 29273 | The empty graph, with vert... |
| usgr0vb 29274 | The null graph, with no ve... |
| uhgr0v0e 29275 | The null graph, with no ve... |
| uhgr0vsize0 29276 | The size of a hypergraph w... |
| uhgr0edgfi 29277 | A graph of order 0 (i.e. w... |
| usgr0v 29278 | The null graph, with no ve... |
| uhgr0vusgr 29279 | The null graph, with no ve... |
| usgr0 29280 | The null graph represented... |
| uspgr1e 29281 | A simple pseudograph with ... |
| usgr1e 29282 | A simple graph with one ed... |
| usgr0eop 29283 | The empty graph, with vert... |
| uspgr1eop 29284 | A simple pseudograph with ... |
| uspgr1ewop 29285 | A simple pseudograph with ... |
| uspgr1v1eop 29286 | A simple pseudograph with ... |
| usgr1eop 29287 | A simple graph with (at le... |
| uspgr2v1e2w 29288 | A simple pseudograph with ... |
| usgr2v1e2w 29289 | A simple graph with two ve... |
| edg0usgr 29290 | A class without edges is a... |
| lfuhgr1v0e 29291 | A loop-free hypergraph wit... |
| usgr1vr 29292 | A simple graph with one ve... |
| usgr1v 29293 | A class with one (or no) v... |
| usgr1v0edg 29294 | A class with one (or no) v... |
| usgrexmpldifpr 29295 | Lemma for ~ usgrexmpledg :... |
| usgrexmplef 29296 | Lemma for ~ usgrexmpl . (... |
| usgrexmpllem 29297 | Lemma for ~ usgrexmpl . (... |
| usgrexmplvtx 29298 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpledg 29299 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl 29300 | ` G ` is a simple graph of... |
| griedg0prc 29301 | The class of empty graphs ... |
| griedg0ssusgr 29302 | The class of all simple gr... |
| usgrprc 29303 | The class of simple graphs... |
| relsubgr 29306 | The class of the subgraph ... |
| subgrv 29307 | If a class is a subgraph o... |
| issubgr 29308 | The property of a set to b... |
| issubgr2 29309 | The property of a set to b... |
| subgrprop 29310 | The properties of a subgra... |
| subgrprop2 29311 | The properties of a subgra... |
| uhgrissubgr 29312 | The property of a hypergra... |
| subgrprop3 29313 | The properties of a subgra... |
| egrsubgr 29314 | An empty graph consisting ... |
| 0grsubgr 29315 | The null graph (represente... |
| 0uhgrsubgr 29316 | The null graph (as hypergr... |
| uhgrsubgrself 29317 | A hypergraph is a subgraph... |
| subgrfun 29318 | The edge function of a sub... |
| subgruhgrfun 29319 | The edge function of a sub... |
| subgreldmiedg 29320 | An element of the domain o... |
| subgruhgredgd 29321 | An edge of a subgraph of a... |
| subumgredg2 29322 | An edge of a subgraph of a... |
| subuhgr 29323 | A subgraph of a hypergraph... |
| subupgr 29324 | A subgraph of a pseudograp... |
| subumgr 29325 | A subgraph of a multigraph... |
| subusgr 29326 | A subgraph of a simple gra... |
| uhgrspansubgrlem 29327 | Lemma for ~ uhgrspansubgr ... |
| uhgrspansubgr 29328 | A spanning subgraph ` S ` ... |
| uhgrspan 29329 | A spanning subgraph ` S ` ... |
| upgrspan 29330 | A spanning subgraph ` S ` ... |
| umgrspan 29331 | A spanning subgraph ` S ` ... |
| usgrspan 29332 | A spanning subgraph ` S ` ... |
| uhgrspanop 29333 | A spanning subgraph of a h... |
| upgrspanop 29334 | A spanning subgraph of a p... |
| umgrspanop 29335 | A spanning subgraph of a m... |
| usgrspanop 29336 | A spanning subgraph of a s... |
| uhgrspan1lem1 29337 | Lemma 1 for ~ uhgrspan1 . ... |
| uhgrspan1lem2 29338 | Lemma 2 for ~ uhgrspan1 . ... |
| uhgrspan1lem3 29339 | Lemma 3 for ~ uhgrspan1 . ... |
| uhgrspan1 29340 | The induced subgraph ` S `... |
| upgrreslem 29341 | Lemma for ~ upgrres . (Co... |
| umgrreslem 29342 | Lemma for ~ umgrres and ~ ... |
| upgrres 29343 | A subgraph obtained by rem... |
| umgrres 29344 | A subgraph obtained by rem... |
| usgrres 29345 | A subgraph obtained by rem... |
| upgrres1lem1 29346 | Lemma 1 for ~ upgrres1 . ... |
| umgrres1lem 29347 | Lemma for ~ umgrres1 . (C... |
| upgrres1lem2 29348 | Lemma 2 for ~ upgrres1 . ... |
| upgrres1lem3 29349 | Lemma 3 for ~ upgrres1 . ... |
| upgrres1 29350 | A pseudograph obtained by ... |
| umgrres1 29351 | A multigraph obtained by r... |
| usgrres1 29352 | Restricting a simple graph... |
| isfusgr 29355 | The property of being a fi... |
| fusgrvtxfi 29356 | A finite simple graph has ... |
| isfusgrf1 29357 | The property of being a fi... |
| isfusgrcl 29358 | The property of being a fi... |
| fusgrusgr 29359 | A finite simple graph is a... |
| opfusgr 29360 | A finite simple graph repr... |
| usgredgffibi 29361 | The number of edges in a s... |
| fusgredgfi 29362 | In a finite simple graph t... |
| usgr1v0e 29363 | The size of a (finite) sim... |
| usgrfilem 29364 | In a finite simple graph, ... |
| fusgrfisbase 29365 | Induction base for ~ fusgr... |
| fusgrfisstep 29366 | Induction step in ~ fusgrf... |
| fusgrfis 29367 | A finite simple graph is o... |
| fusgrfupgrfs 29368 | A finite simple graph is a... |
| nbgrprc0 29371 | The set of neighbors is em... |
| nbgrcl 29372 | If a class ` X ` has at le... |
| nbgrval 29373 | The set of neighbors of a ... |
| dfnbgr2 29374 | Alternate definition of th... |
| dfnbgr3 29375 | Alternate definition of th... |
| nbgrnvtx0 29376 | If a class ` X ` is not a ... |
| nbgrel 29377 | Characterization of a neig... |
| nbgrisvtx 29378 | Every neighbor ` N ` of a ... |
| nbgrssvtx 29379 | The neighbors of a vertex ... |
| nbuhgr 29380 | The set of neighbors of a ... |
| nbupgr 29381 | The set of neighbors of a ... |
| nbupgrel 29382 | A neighbor of a vertex in ... |
| nbumgrvtx 29383 | The set of neighbors of a ... |
| nbumgr 29384 | The set of neighbors of an... |
| nbusgrvtx 29385 | The set of neighbors of a ... |
| nbusgr 29386 | The set of neighbors of an... |
| nbgr2vtx1edg 29387 | If a graph has two vertice... |
| nbuhgr2vtx1edgblem 29388 | Lemma for ~ nbuhgr2vtx1edg... |
| nbuhgr2vtx1edgb 29389 | If a hypergraph has two ve... |
| nbusgreledg 29390 | A class/vertex is a neighb... |
| uhgrnbgr0nb 29391 | A vertex which is not endp... |
| nbgr0vtx 29392 | In a null graph (with no v... |
| nbgr0edglem 29393 | Lemma for ~ nbgr0edg and ~... |
| nbgr0edg 29394 | In an empty graph (with no... |
| nbgr1vtx 29395 | In a graph with one vertex... |
| nbgrnself 29396 | A vertex in a graph is not... |
| nbgrnself2 29397 | A class ` X ` is not a nei... |
| nbgrssovtx 29398 | The neighbors of a vertex ... |
| nbgrssvwo2 29399 | The neighbors of a vertex ... |
| nbgrsym 29400 | In a graph, the neighborho... |
| nbupgrres 29401 | The neighborhood of a vert... |
| usgrnbcnvfv 29402 | Applying the edge function... |
| nbusgredgeu 29403 | For each neighbor of a ver... |
| edgnbusgreu 29404 | For each edge incident to ... |
| nbusgredgeu0 29405 | For each neighbor of a ver... |
| nbusgrf1o0 29406 | The mapping of neighbors o... |
| nbusgrf1o1 29407 | The set of neighbors of a ... |
| nbusgrf1o 29408 | The set of neighbors of a ... |
| nbedgusgr 29409 | The number of neighbors of... |
| edgusgrnbfin 29410 | The number of neighbors of... |
| nbusgrfi 29411 | The class of neighbors of ... |
| nbfiusgrfi 29412 | The class of neighbors of ... |
| hashnbusgrnn0 29413 | The number of neighbors of... |
| nbfusgrlevtxm1 29414 | The number of neighbors of... |
| nbfusgrlevtxm2 29415 | If there is a vertex which... |
| nbusgrvtxm1 29416 | If the number of neighbors... |
| nb3grprlem1 29417 | Lemma 1 for ~ nb3grpr . (... |
| nb3grprlem2 29418 | Lemma 2 for ~ nb3grpr . (... |
| nb3grpr 29419 | The neighbors of a vertex ... |
| nb3grpr2 29420 | The neighbors of a vertex ... |
| nb3gr2nb 29421 | If the neighbors of two ve... |
| uvtxval 29424 | The set of all universal v... |
| uvtxel 29425 | A universal vertex, i.e. a... |
| uvtxisvtx 29426 | A universal vertex is a ve... |
| uvtxssvtx 29427 | The set of the universal v... |
| vtxnbuvtx 29428 | A universal vertex has all... |
| uvtxnbgrss 29429 | A universal vertex has all... |
| uvtxnbgrvtx 29430 | A universal vertex is neig... |
| uvtx0 29431 | There is no universal vert... |
| isuvtx 29432 | The set of all universal v... |
| uvtxel1 29433 | Characterization of a univ... |
| uvtx01vtx 29434 | If a graph/class has no ed... |
| uvtx2vtx1edg 29435 | If a graph has two vertice... |
| uvtx2vtx1edgb 29436 | If a hypergraph has two ve... |
| uvtxnbgr 29437 | A universal vertex has all... |
| uvtxnbgrb 29438 | A vertex is universal iff ... |
| uvtxusgr 29439 | The set of all universal v... |
| uvtxusgrel 29440 | A universal vertex, i.e. a... |
| uvtxnm1nbgr 29441 | A universal vertex has ` n... |
| nbusgrvtxm1uvtx 29442 | If the number of neighbors... |
| uvtxnbvtxm1 29443 | A universal vertex has ` n... |
| nbupgruvtxres 29444 | The neighborhood of a univ... |
| uvtxupgrres 29445 | A universal vertex is univ... |
| cplgruvtxb 29450 | A graph ` G ` is complete ... |
| prcliscplgr 29451 | A proper class (representi... |
| iscplgr 29452 | The property of being a co... |
| iscplgrnb 29453 | A graph is complete iff al... |
| iscplgredg 29454 | A graph ` G ` is complete ... |
| iscusgr 29455 | The property of being a co... |
| cusgrusgr 29456 | A complete simple graph is... |
| cusgrcplgr 29457 | A complete simple graph is... |
| iscusgrvtx 29458 | A simple graph is complete... |
| cusgruvtxb 29459 | A simple graph is complete... |
| iscusgredg 29460 | A simple graph is complete... |
| cusgredg 29461 | In a complete simple graph... |
| cplgr0 29462 | The null graph (with no ve... |
| cusgr0 29463 | The null graph (with no ve... |
| cplgr0v 29464 | A null graph (with no vert... |
| cusgr0v 29465 | A graph with no vertices a... |
| cplgr1vlem 29466 | Lemma for ~ cplgr1v and ~ ... |
| cplgr1v 29467 | A graph with one vertex is... |
| cusgr1v 29468 | A graph with one vertex an... |
| cplgr2v 29469 | An undirected hypergraph w... |
| cplgr2vpr 29470 | An undirected hypergraph w... |
| nbcplgr 29471 | In a complete graph, each ... |
| cplgr3v 29472 | A pseudograph with three (... |
| cusgr3vnbpr 29473 | The neighbors of a vertex ... |
| cplgrop 29474 | A complete graph represent... |
| cusgrop 29475 | A complete simple graph re... |
| cusgrexilem1 29476 | Lemma 1 for ~ cusgrexi . ... |
| usgrexilem 29477 | Lemma for ~ usgrexi . (Co... |
| usgrexi 29478 | An arbitrary set regarded ... |
| cusgrexilem2 29479 | Lemma 2 for ~ cusgrexi . ... |
| cusgrexi 29480 | An arbitrary set ` V ` reg... |
| cusgrexg 29481 | For each set there is a se... |
| structtousgr 29482 | Any (extensible) structure... |
| structtocusgr 29483 | Any (extensible) structure... |
| cffldtocusgr 29484 | The field of complex numbe... |
| cffldtocusgrOLD 29485 | Obsolete version of ~ cffl... |
| cusgrres 29486 | Restricting a complete sim... |
| cusgrsizeindb0 29487 | Base case of the induction... |
| cusgrsizeindb1 29488 | Base case of the induction... |
| cusgrsizeindslem 29489 | Lemma for ~ cusgrsizeinds ... |
| cusgrsizeinds 29490 | Part 1 of induction step i... |
| cusgrsize2inds 29491 | Induction step in ~ cusgrs... |
| cusgrsize 29492 | The size of a finite compl... |
| cusgrfilem1 29493 | Lemma 1 for ~ cusgrfi . (... |
| cusgrfilem2 29494 | Lemma 2 for ~ cusgrfi . (... |
| cusgrfilem3 29495 | Lemma 3 for ~ cusgrfi . (... |
| cusgrfi 29496 | If the size of a complete ... |
| usgredgsscusgredg 29497 | A simple graph is a subgra... |
| usgrsscusgr 29498 | A simple graph is a subgra... |
| sizusglecusglem1 29499 | Lemma 1 for ~ sizusglecusg... |
| sizusglecusglem2 29500 | Lemma 2 for ~ sizusglecusg... |
| sizusglecusg 29501 | The size of a simple graph... |
| fusgrmaxsize 29502 | The maximum size of a fini... |
| vtxdgfval 29505 | The value of the vertex de... |
| vtxdgval 29506 | The degree of a vertex. (... |
| vtxdgfival 29507 | The degree of a vertex for... |
| vtxdgop 29508 | The vertex degree expresse... |
| vtxdgf 29509 | The vertex degree function... |
| vtxdgelxnn0 29510 | The degree of a vertex is ... |
| vtxdg0v 29511 | The degree of a vertex in ... |
| vtxdg0e 29512 | The degree of a vertex in ... |
| vtxdgfisnn0 29513 | The degree of a vertex in ... |
| vtxdgfisf 29514 | The vertex degree function... |
| vtxdeqd 29515 | Equality theorem for the v... |
| vtxduhgr0e 29516 | The degree of a vertex in ... |
| vtxdlfuhgr1v 29517 | The degree of the vertex i... |
| vdumgr0 29518 | A vertex in a multigraph h... |
| vtxdun 29519 | The degree of a vertex in ... |
| vtxdfiun 29520 | The degree of a vertex in ... |
| vtxduhgrun 29521 | The degree of a vertex in ... |
| vtxduhgrfiun 29522 | The degree of a vertex in ... |
| vtxdlfgrval 29523 | The value of the vertex de... |
| vtxdumgrval 29524 | The value of the vertex de... |
| vtxdusgrval 29525 | The value of the vertex de... |
| vtxd0nedgb 29526 | A vertex has degree 0 iff ... |
| vtxdushgrfvedglem 29527 | Lemma for ~ vtxdushgrfvedg... |
| vtxdushgrfvedg 29528 | The value of the vertex de... |
| vtxdusgrfvedg 29529 | The value of the vertex de... |
| vtxduhgr0nedg 29530 | If a vertex in a hypergrap... |
| vtxdumgr0nedg 29531 | If a vertex in a multigrap... |
| vtxduhgr0edgnel 29532 | A vertex in a hypergraph h... |
| vtxdusgr0edgnel 29533 | A vertex in a simple graph... |
| vtxdusgr0edgnelALT 29534 | Alternate proof of ~ vtxdu... |
| vtxdgfusgrf 29535 | The vertex degree function... |
| vtxdgfusgr 29536 | In a finite simple graph, ... |
| fusgrn0degnn0 29537 | In a nonempty, finite grap... |
| 1loopgruspgr 29538 | A graph with one edge whic... |
| 1loopgredg 29539 | The set of edges in a grap... |
| 1loopgrnb0 29540 | In a graph (simple pseudog... |
| 1loopgrvd2 29541 | The vertex degree of a one... |
| 1loopgrvd0 29542 | The vertex degree of a one... |
| 1hevtxdg0 29543 | The vertex degree of verte... |
| 1hevtxdg1 29544 | The vertex degree of verte... |
| 1hegrvtxdg1 29545 | The vertex degree of a gra... |
| 1hegrvtxdg1r 29546 | The vertex degree of a gra... |
| 1egrvtxdg1 29547 | The vertex degree of a one... |
| 1egrvtxdg1r 29548 | The vertex degree of a one... |
| 1egrvtxdg0 29549 | The vertex degree of a one... |
| p1evtxdeqlem 29550 | Lemma for ~ p1evtxdeq and ... |
| p1evtxdeq 29551 | If an edge ` E ` which doe... |
| p1evtxdp1 29552 | If an edge ` E ` (not bein... |
| uspgrloopvtx 29553 | The set of vertices in a g... |
| uspgrloopvtxel 29554 | A vertex in a graph (simpl... |
| uspgrloopiedg 29555 | The set of edges in a grap... |
| uspgrloopedg 29556 | The set of edges in a grap... |
| uspgrloopnb0 29557 | In a graph (simple pseudog... |
| uspgrloopvd2 29558 | The vertex degree of a one... |
| umgr2v2evtx 29559 | The set of vertices in a m... |
| umgr2v2evtxel 29560 | A vertex in a multigraph w... |
| umgr2v2eiedg 29561 | The edge function in a mul... |
| umgr2v2eedg 29562 | The set of edges in a mult... |
| umgr2v2e 29563 | A multigraph with two edge... |
| umgr2v2enb1 29564 | In a multigraph with two e... |
| umgr2v2evd2 29565 | In a multigraph with two e... |
| hashnbusgrvd 29566 | In a simple graph, the num... |
| usgruvtxvdb 29567 | In a finite simple graph w... |
| vdiscusgrb 29568 | A finite simple graph with... |
| vdiscusgr 29569 | In a finite complete simpl... |
| vtxdusgradjvtx 29570 | The degree of a vertex in ... |
| usgrvd0nedg 29571 | If a vertex in a simple gr... |
| uhgrvd00 29572 | If every vertex in a hyper... |
| usgrvd00 29573 | If every vertex in a simpl... |
| vdegp1ai 29574 | The induction step for a v... |
| vdegp1bi 29575 | The induction step for a v... |
| vdegp1ci 29576 | The induction step for a v... |
| vtxdginducedm1lem1 29577 | Lemma 1 for ~ vtxdginduced... |
| vtxdginducedm1lem2 29578 | Lemma 2 for ~ vtxdginduced... |
| vtxdginducedm1lem3 29579 | Lemma 3 for ~ vtxdginduced... |
| vtxdginducedm1lem4 29580 | Lemma 4 for ~ vtxdginduced... |
| vtxdginducedm1 29581 | The degree of a vertex ` v... |
| vtxdginducedm1fi 29582 | The degree of a vertex ` v... |
| finsumvtxdg2ssteplem1 29583 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem2 29584 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem3 29585 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem4 29586 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2sstep 29587 | Induction step of ~ finsum... |
| finsumvtxdg2size 29588 | The sum of the degrees of ... |
| fusgr1th 29589 | The sum of the degrees of ... |
| finsumvtxdgeven 29590 | The sum of the degrees of ... |
| vtxdgoddnumeven 29591 | The number of vertices of ... |
| fusgrvtxdgonume 29592 | The number of vertices of ... |
| isrgr 29597 | The property of a class be... |
| rgrprop 29598 | The properties of a k-regu... |
| isrusgr 29599 | The property of being a k-... |
| rusgrprop 29600 | The properties of a k-regu... |
| rusgrrgr 29601 | A k-regular simple graph i... |
| rusgrusgr 29602 | A k-regular simple graph i... |
| finrusgrfusgr 29603 | A finite regular simple gr... |
| isrusgr0 29604 | The property of being a k-... |
| rusgrprop0 29605 | The properties of a k-regu... |
| usgreqdrusgr 29606 | If all vertices in a simpl... |
| fusgrregdegfi 29607 | In a nonempty finite simpl... |
| fusgrn0eqdrusgr 29608 | If all vertices in a nonem... |
| frusgrnn0 29609 | In a nonempty finite k-reg... |
| 0edg0rgr 29610 | A graph is 0-regular if it... |
| uhgr0edg0rgr 29611 | A hypergraph is 0-regular ... |
| uhgr0edg0rgrb 29612 | A hypergraph is 0-regular ... |
| usgr0edg0rusgr 29613 | A simple graph is 0-regula... |
| 0vtxrgr 29614 | A null graph (with no vert... |
| 0vtxrusgr 29615 | A graph with no vertices a... |
| 0uhgrrusgr 29616 | The null graph as hypergra... |
| 0grrusgr 29617 | The null graph represented... |
| 0grrgr 29618 | The null graph represented... |
| cusgrrusgr 29619 | A complete simple graph wi... |
| cusgrm1rusgr 29620 | A finite simple graph with... |
| rusgrpropnb 29621 | The properties of a k-regu... |
| rusgrpropedg 29622 | The properties of a k-regu... |
| rusgrpropadjvtx 29623 | The properties of a k-regu... |
| rusgrnumwrdl2 29624 | In a k-regular simple grap... |
| rusgr1vtxlem 29625 | Lemma for ~ rusgr1vtx . (... |
| rusgr1vtx 29626 | If a k-regular simple grap... |
| rgrusgrprc 29627 | The class of 0-regular sim... |
| rusgrprc 29628 | The class of 0-regular sim... |
| rgrprc 29629 | The class of 0-regular gra... |
| rgrprcx 29630 | The class of 0-regular gra... |
| rgrx0ndm 29631 | 0 is not in the domain of ... |
| rgrx0nd 29632 | The potentially alternativ... |
| ewlksfval 29639 | The set of s-walks of edge... |
| isewlk 29640 | Conditions for a function ... |
| ewlkprop 29641 | Properties of an s-walk of... |
| ewlkinedg 29642 | The intersection (common v... |
| ewlkle 29643 | An s-walk of edges is also... |
| upgrewlkle2 29644 | In a pseudograph, there is... |
| wkslem1 29645 | Lemma 1 for walks to subst... |
| wkslem2 29646 | Lemma 2 for walks to subst... |
| wksfval 29647 | The set of walks (in an un... |
| iswlk 29648 | Properties of a pair of fu... |
| wlkprop 29649 | Properties of a walk. (Co... |
| wlkv 29650 | The classes involved in a ... |
| iswlkg 29651 | Generalization of ~ iswlk ... |
| wlkf 29652 | The mapping enumerating th... |
| wlkcl 29653 | A walk has length ` # ( F ... |
| wlkp 29654 | The mapping enumerating th... |
| wlkpwrd 29655 | The sequence of vertices o... |
| wlklenvp1 29656 | The number of vertices of ... |
| wksv 29657 | The class of walks is a se... |
| wksvOLD 29658 | Obsolete version of ~ wksv... |
| wlkn0 29659 | The sequence of vertices o... |
| wlklenvm1 29660 | The number of edges of a w... |
| ifpsnprss 29661 | Lemma for ~ wlkvtxeledg : ... |
| wlkvtxeledg 29662 | Each pair of adjacent vert... |
| wlkvtxiedg 29663 | The vertices of a walk are... |
| relwlk 29664 | The set ` ( Walks `` G ) `... |
| wlkvv 29665 | If there is at least one w... |
| wlkop 29666 | A walk is an ordered pair.... |
| wlkcpr 29667 | A walk as class with two c... |
| wlk2f 29668 | If there is a walk ` W ` t... |
| wlkcomp 29669 | A walk expressed by proper... |
| wlkcompim 29670 | Implications for the prope... |
| wlkelwrd 29671 | The components of a walk a... |
| wlkeq 29672 | Conditions for two walks (... |
| edginwlk 29673 | The value of the edge func... |
| upgredginwlk 29674 | The value of the edge func... |
| iedginwlk 29675 | The value of the edge func... |
| wlkl1loop 29676 | A walk of length 1 from a ... |
| wlk1walk 29677 | A walk is a 1-walk "on the... |
| wlk1ewlk 29678 | A walk is an s-walk "on th... |
| upgriswlk 29679 | Properties of a pair of fu... |
| upgrwlkedg 29680 | The edges of a walk in a p... |
| upgrwlkcompim 29681 | Implications for the prope... |
| wlkvtxedg 29682 | The vertices of a walk are... |
| upgrwlkvtxedg 29683 | The pairs of connected ver... |
| uspgr2wlkeq 29684 | Conditions for two walks w... |
| uspgr2wlkeq2 29685 | Conditions for two walks w... |
| uspgr2wlkeqi 29686 | Conditions for two walks w... |
| umgrwlknloop 29687 | In a multigraph, each walk... |
| wlkResOLD 29688 | Obsolete version of ~ opab... |
| wlkv0 29689 | If there is a walk in the ... |
| g0wlk0 29690 | There is no walk in a null... |
| 0wlk0 29691 | There is no walk for the e... |
| wlk0prc 29692 | There is no walk in a null... |
| wlklenvclwlk 29693 | The number of vertices in ... |
| wlkson 29694 | The set of walks between t... |
| iswlkon 29695 | Properties of a pair of fu... |
| wlkonprop 29696 | Properties of a walk betwe... |
| wlkpvtx 29697 | A walk connects vertices. ... |
| wlkepvtx 29698 | The endpoints of a walk ar... |
| wlkoniswlk 29699 | A walk between two vertice... |
| wlkonwlk 29700 | A walk is a walk between i... |
| wlkonwlk1l 29701 | A walk is a walk from its ... |
| wlksoneq1eq2 29702 | Two walks with identical s... |
| wlkonl1iedg 29703 | If there is a walk between... |
| wlkon2n0 29704 | The length of a walk betwe... |
| 2wlklem 29705 | Lemma for theorems for wal... |
| upgr2wlk 29706 | Properties of a pair of fu... |
| wlkreslem 29707 | Lemma for ~ wlkres . (Con... |
| wlkres 29708 | The restriction ` <. H , Q... |
| redwlklem 29709 | Lemma for ~ redwlk . (Con... |
| redwlk 29710 | A walk ending at the last ... |
| wlkp1lem1 29711 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem2 29712 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem3 29713 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem4 29714 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem5 29715 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem6 29716 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem7 29717 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem8 29718 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1 29719 | Append one path segment (e... |
| wlkdlem1 29720 | Lemma 1 for ~ wlkd . (Con... |
| wlkdlem2 29721 | Lemma 2 for ~ wlkd . (Con... |
| wlkdlem3 29722 | Lemma 3 for ~ wlkd . (Con... |
| wlkdlem4 29723 | Lemma 4 for ~ wlkd . (Con... |
| wlkd 29724 | Two words representing a w... |
| lfgrwlkprop 29725 | Two adjacent vertices in a... |
| lfgriswlk 29726 | Conditions for a pair of f... |
| lfgrwlknloop 29727 | In a loop-free graph, each... |
| reltrls 29732 | The set ` ( Trails `` G ) ... |
| trlsfval 29733 | The set of trails (in an u... |
| istrl 29734 | Conditions for a pair of c... |
| trliswlk 29735 | A trail is a walk. (Contr... |
| trlf1 29736 | The enumeration ` F ` of a... |
| trlreslem 29737 | Lemma for ~ trlres . Form... |
| trlres 29738 | The restriction ` <. H , Q... |
| upgrtrls 29739 | The set of trails in a pse... |
| upgristrl 29740 | Properties of a pair of fu... |
| upgrf1istrl 29741 | Properties of a pair of a ... |
| wksonproplem 29742 | Lemma for theorems for pro... |
| wksonproplemOLD 29743 | Obsolete version of ~ wkso... |
| trlsonfval 29744 | The set of trails between ... |
| istrlson 29745 | Properties of a pair of fu... |
| trlsonprop 29746 | Properties of a trail betw... |
| trlsonistrl 29747 | A trail between two vertic... |
| trlsonwlkon 29748 | A trail between two vertic... |
| trlontrl 29749 | A trail is a trail between... |
| relpths 29758 | The set ` ( Paths `` G ) `... |
| pthsfval 29759 | The set of paths (in an un... |
| spthsfval 29760 | The set of simple paths (i... |
| ispth 29761 | Conditions for a pair of c... |
| isspth 29762 | Conditions for a pair of c... |
| pthistrl 29763 | A path is a trail (in an u... |
| spthispth 29764 | A simple path is a path (i... |
| pthiswlk 29765 | A path is a walk (in an un... |
| spthiswlk 29766 | A simple path is a walk (i... |
| pthdivtx 29767 | The inner vertices of a pa... |
| pthdadjvtx 29768 | The adjacent vertices of a... |
| 2pthnloop 29769 | A path of length at least ... |
| upgr2pthnlp 29770 | A path of length at least ... |
| spthdifv 29771 | The vertices of a simple p... |
| spthdep 29772 | A simple path (at least of... |
| pthdepisspth 29773 | A path with different star... |
| upgrwlkdvdelem 29774 | Lemma for ~ upgrwlkdvde . ... |
| upgrwlkdvde 29775 | In a pseudograph, all edge... |
| upgrspthswlk 29776 | The set of simple paths in... |
| upgrwlkdvspth 29777 | A walk consisting of diffe... |
| pthsonfval 29778 | The set of paths between t... |
| spthson 29779 | The set of simple paths be... |
| ispthson 29780 | Properties of a pair of fu... |
| isspthson 29781 | Properties of a pair of fu... |
| pthsonprop 29782 | Properties of a path betwe... |
| spthonprop 29783 | Properties of a simple pat... |
| pthonispth 29784 | A path between two vertice... |
| pthontrlon 29785 | A path between two vertice... |
| pthonpth 29786 | A path is a path between i... |
| isspthonpth 29787 | A pair of functions is a s... |
| spthonisspth 29788 | A simple path between to v... |
| spthonpthon 29789 | A simple path between two ... |
| spthonepeq 29790 | The endpoints of a simple ... |
| uhgrwkspthlem1 29791 | Lemma 1 for ~ uhgrwkspth .... |
| uhgrwkspthlem2 29792 | Lemma 2 for ~ uhgrwkspth .... |
| uhgrwkspth 29793 | Any walk of length 1 betwe... |
| usgr2wlkneq 29794 | The vertices and edges are... |
| usgr2wlkspthlem1 29795 | Lemma 1 for ~ usgr2wlkspth... |
| usgr2wlkspthlem2 29796 | Lemma 2 for ~ usgr2wlkspth... |
| usgr2wlkspth 29797 | In a simple graph, any wal... |
| usgr2trlncl 29798 | In a simple graph, any tra... |
| usgr2trlspth 29799 | In a simple graph, any tra... |
| usgr2pthspth 29800 | In a simple graph, any pat... |
| usgr2pthlem 29801 | Lemma for ~ usgr2pth . (C... |
| usgr2pth 29802 | In a simple graph, there i... |
| usgr2pth0 29803 | In a simply graph, there i... |
| pthdlem1 29804 | Lemma 1 for ~ pthd . (Con... |
| pthdlem2lem 29805 | Lemma for ~ pthdlem2 . (C... |
| pthdlem2 29806 | Lemma 2 for ~ pthd . (Con... |
| pthd 29807 | Two words representing a t... |
| clwlks 29810 | The set of closed walks (i... |
| isclwlk 29811 | A pair of functions repres... |
| clwlkiswlk 29812 | A closed walk is a walk (i... |
| clwlkwlk 29813 | Closed walks are walks (in... |
| clwlkswks 29814 | Closed walks are walks (in... |
| isclwlke 29815 | Properties of a pair of fu... |
| isclwlkupgr 29816 | Properties of a pair of fu... |
| clwlkcomp 29817 | A closed walk expressed by... |
| clwlkcompim 29818 | Implications for the prope... |
| upgrclwlkcompim 29819 | Implications for the prope... |
| clwlkcompbp 29820 | Basic properties of the co... |
| clwlkl1loop 29821 | A closed walk of length 1 ... |
| crcts 29826 | The set of circuits (in an... |
| cycls 29827 | The set of cycles (in an u... |
| iscrct 29828 | Sufficient and necessary c... |
| iscycl 29829 | Sufficient and necessary c... |
| crctprop 29830 | The properties of a circui... |
| cyclprop 29831 | The properties of a cycle:... |
| crctisclwlk 29832 | A circuit is a closed walk... |
| crctistrl 29833 | A circuit is a trail. (Co... |
| crctiswlk 29834 | A circuit is a walk. (Con... |
| cyclispth 29835 | A cycle is a path. (Contr... |
| cycliswlk 29836 | A cycle is a walk. (Contr... |
| cycliscrct 29837 | A cycle is a circuit. (Co... |
| cyclnspth 29838 | A (non-trivial) cycle is n... |
| cyclispthon 29839 | A cycle is a path starting... |
| lfgrn1cycl 29840 | In a loop-free graph there... |
| usgr2trlncrct 29841 | In a simple graph, any tra... |
| umgrn1cycl 29842 | In a multigraph graph (wit... |
| uspgrn2crct 29843 | In a simple pseudograph th... |
| usgrn2cycl 29844 | In a simple graph there ar... |
| crctcshwlkn0lem1 29845 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem2 29846 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem3 29847 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem4 29848 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem5 29849 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem6 29850 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem7 29851 | Lemma for ~ crctcshwlkn0 .... |
| crctcshlem1 29852 | Lemma for ~ crctcsh . (Co... |
| crctcshlem2 29853 | Lemma for ~ crctcsh . (Co... |
| crctcshlem3 29854 | Lemma for ~ crctcsh . (Co... |
| crctcshlem4 29855 | Lemma for ~ crctcsh . (Co... |
| crctcshwlkn0 29856 | Cyclically shifting the in... |
| crctcshwlk 29857 | Cyclically shifting the in... |
| crctcshtrl 29858 | Cyclically shifting the in... |
| crctcsh 29859 | Cyclically shifting the in... |
| wwlks 29870 | The set of walks (in an un... |
| iswwlks 29871 | A word over the set of ver... |
| wwlksn 29872 | The set of walks (in an un... |
| iswwlksn 29873 | A word over the set of ver... |
| wwlksnprcl 29874 | Derivation of the length o... |
| iswwlksnx 29875 | Properties of a word to re... |
| wwlkbp 29876 | Basic properties of a walk... |
| wwlknbp 29877 | Basic properties of a walk... |
| wwlknp 29878 | Properties of a set being ... |
| wwlknbp1 29879 | Other basic properties of ... |
| wwlknvtx 29880 | The symbols of a word ` W ... |
| wwlknllvtx 29881 | If a word ` W ` represents... |
| wwlknlsw 29882 | If a word represents a wal... |
| wspthsn 29883 | The set of simple paths of... |
| iswspthn 29884 | An element of the set of s... |
| wspthnp 29885 | Properties of a set being ... |
| wwlksnon 29886 | The set of walks of a fixe... |
| wspthsnon 29887 | The set of simple paths of... |
| iswwlksnon 29888 | The set of walks of a fixe... |
| wwlksnon0 29889 | Sufficient conditions for ... |
| wwlksonvtx 29890 | If a word ` W ` represents... |
| iswspthsnon 29891 | The set of simple paths of... |
| wwlknon 29892 | An element of the set of w... |
| wspthnon 29893 | An element of the set of s... |
| wspthnonp 29894 | Properties of a set being ... |
| wspthneq1eq2 29895 | Two simple paths with iden... |
| wwlksn0s 29896 | The set of all walks as wo... |
| wwlkssswrd 29897 | Walks (represented by word... |
| wwlksn0 29898 | A walk of length 0 is repr... |
| 0enwwlksnge1 29899 | In graphs without edges, t... |
| wwlkswwlksn 29900 | A walk of a fixed length a... |
| wwlkssswwlksn 29901 | The walks of a fixed lengt... |
| wlkiswwlks1 29902 | The sequence of vertices i... |
| wlklnwwlkln1 29903 | The sequence of vertices i... |
| wlkiswwlks2lem1 29904 | Lemma 1 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem2 29905 | Lemma 2 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem3 29906 | Lemma 3 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem4 29907 | Lemma 4 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem5 29908 | Lemma 5 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem6 29909 | Lemma 6 for ~ wlkiswwlks2 ... |
| wlkiswwlks2 29910 | A walk as word corresponds... |
| wlkiswwlks 29911 | A walk as word corresponds... |
| wlkiswwlksupgr2 29912 | A walk as word corresponds... |
| wlkiswwlkupgr 29913 | A walk as word corresponds... |
| wlkswwlksf1o 29914 | The mapping of (ordinary) ... |
| wlkswwlksen 29915 | The set of walks as words ... |
| wwlksm1edg 29916 | Removing the trailing edge... |
| wlklnwwlkln2lem 29917 | Lemma for ~ wlklnwwlkln2 a... |
| wlklnwwlkln2 29918 | A walk of length ` N ` as ... |
| wlklnwwlkn 29919 | A walk of length ` N ` as ... |
| wlklnwwlklnupgr2 29920 | A walk of length ` N ` as ... |
| wlklnwwlknupgr 29921 | A walk of length ` N ` as ... |
| wlknewwlksn 29922 | If a walk in a pseudograph... |
| wlknwwlksnbij 29923 | The mapping ` ( t e. T |->... |
| wlknwwlksnen 29924 | In a simple pseudograph, t... |
| wlknwwlksneqs 29925 | The set of walks of a fixe... |
| wwlkseq 29926 | Equality of two walks (as ... |
| wwlksnred 29927 | Reduction of a walk (as wo... |
| wwlksnext 29928 | Extension of a walk (as wo... |
| wwlksnextbi 29929 | Extension of a walk (as wo... |
| wwlksnredwwlkn 29930 | For each walk (as word) of... |
| wwlksnredwwlkn0 29931 | For each walk (as word) of... |
| wwlksnextwrd 29932 | Lemma for ~ wwlksnextbij .... |
| wwlksnextfun 29933 | Lemma for ~ wwlksnextbij .... |
| wwlksnextinj 29934 | Lemma for ~ wwlksnextbij .... |
| wwlksnextsurj 29935 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij0 29936 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij 29937 | There is a bijection betwe... |
| wwlksnexthasheq 29938 | The number of the extensio... |
| disjxwwlksn 29939 | Sets of walks (as words) e... |
| wwlksnndef 29940 | Conditions for ` WWalksN `... |
| wwlksnfi 29941 | The number of walks repres... |
| wlksnfi 29942 | The number of walks of fix... |
| wlksnwwlknvbij 29943 | There is a bijection betwe... |
| wwlksnextproplem1 29944 | Lemma 1 for ~ wwlksnextpro... |
| wwlksnextproplem2 29945 | Lemma 2 for ~ wwlksnextpro... |
| wwlksnextproplem3 29946 | Lemma 3 for ~ wwlksnextpro... |
| wwlksnextprop 29947 | Adding additional properti... |
| disjxwwlkn 29948 | Sets of walks (as words) e... |
| hashwwlksnext 29949 | Number of walks (as words)... |
| wwlksnwwlksnon 29950 | A walk of fixed length is ... |
| wspthsnwspthsnon 29951 | A simple path of fixed len... |
| wspthsnonn0vne 29952 | If the set of simple paths... |
| wspthsswwlkn 29953 | The set of simple paths of... |
| wspthnfi 29954 | In a finite graph, the set... |
| wwlksnonfi 29955 | In a finite graph, the set... |
| wspthsswwlknon 29956 | The set of simple paths of... |
| wspthnonfi 29957 | In a finite graph, the set... |
| wspniunwspnon 29958 | The set of nonempty simple... |
| wspn0 29959 | If there are no vertices, ... |
| 2wlkdlem1 29960 | Lemma 1 for ~ 2wlkd . (Co... |
| 2wlkdlem2 29961 | Lemma 2 for ~ 2wlkd . (Co... |
| 2wlkdlem3 29962 | Lemma 3 for ~ 2wlkd . (Co... |
| 2wlkdlem4 29963 | Lemma 4 for ~ 2wlkd . (Co... |
| 2wlkdlem5 29964 | Lemma 5 for ~ 2wlkd . (Co... |
| 2pthdlem1 29965 | Lemma 1 for ~ 2pthd . (Co... |
| 2wlkdlem6 29966 | Lemma 6 for ~ 2wlkd . (Co... |
| 2wlkdlem7 29967 | Lemma 7 for ~ 2wlkd . (Co... |
| 2wlkdlem8 29968 | Lemma 8 for ~ 2wlkd . (Co... |
| 2wlkdlem9 29969 | Lemma 9 for ~ 2wlkd . (Co... |
| 2wlkdlem10 29970 | Lemma 10 for ~ 3wlkd . (C... |
| 2wlkd 29971 | Construction of a walk fro... |
| 2wlkond 29972 | A walk of length 2 from on... |
| 2trld 29973 | Construction of a trail fr... |
| 2trlond 29974 | A trail of length 2 from o... |
| 2pthd 29975 | A path of length 2 from on... |
| 2spthd 29976 | A simple path of length 2 ... |
| 2pthond 29977 | A simple path of length 2 ... |
| 2pthon3v 29978 | For a vertex adjacent to t... |
| umgr2adedgwlklem 29979 | Lemma for ~ umgr2adedgwlk ... |
| umgr2adedgwlk 29980 | In a multigraph, two adjac... |
| umgr2adedgwlkon 29981 | In a multigraph, two adjac... |
| umgr2adedgwlkonALT 29982 | Alternate proof for ~ umgr... |
| umgr2adedgspth 29983 | In a multigraph, two adjac... |
| umgr2wlk 29984 | In a multigraph, there is ... |
| umgr2wlkon 29985 | For each pair of adjacent ... |
| elwwlks2s3 29986 | A walk of length 2 as word... |
| midwwlks2s3 29987 | There is a vertex between ... |
| wwlks2onv 29988 | If a length 3 string repre... |
| elwwlks2ons3im 29989 | A walk as word of length 2... |
| elwwlks2ons3 29990 | For each walk of length 2 ... |
| s3wwlks2on 29991 | A length 3 string which re... |
| umgrwwlks2on 29992 | A walk of length 2 between... |
| wwlks2onsym 29993 | There is a walk of length ... |
| elwwlks2on 29994 | A walk of length 2 between... |
| elwspths2on 29995 | A simple path of length 2 ... |
| wpthswwlks2on 29996 | For two different vertices... |
| 2wspdisj 29997 | All simple paths of length... |
| 2wspiundisj 29998 | All simple paths of length... |
| usgr2wspthons3 29999 | A simple path of length 2 ... |
| usgr2wspthon 30000 | A simple path of length 2 ... |
| elwwlks2 30001 | A walk of length 2 between... |
| elwspths2spth 30002 | A simple path of length 2 ... |
| rusgrnumwwlkl1 30003 | In a k-regular graph, ther... |
| rusgrnumwwlkslem 30004 | Lemma for ~ rusgrnumwwlks ... |
| rusgrnumwwlklem 30005 | Lemma for ~ rusgrnumwwlk e... |
| rusgrnumwwlkb0 30006 | Induction base 0 for ~ rus... |
| rusgrnumwwlkb1 30007 | Induction base 1 for ~ rus... |
| rusgr0edg 30008 | Special case for graphs wi... |
| rusgrnumwwlks 30009 | Induction step for ~ rusgr... |
| rusgrnumwwlk 30010 | In a ` K `-regular graph, ... |
| rusgrnumwwlkg 30011 | In a ` K `-regular graph, ... |
| rusgrnumwlkg 30012 | In a k-regular graph, the ... |
| clwwlknclwwlkdif 30013 | The set ` A ` of walks of ... |
| clwwlknclwwlkdifnum 30014 | In a ` K `-regular graph, ... |
| clwwlk 30017 | The set of closed walks (i... |
| isclwwlk 30018 | Properties of a word to re... |
| clwwlkbp 30019 | Basic properties of a clos... |
| clwwlkgt0 30020 | There is no empty closed w... |
| clwwlksswrd 30021 | Closed walks (represented ... |
| clwwlk1loop 30022 | A closed walk of length 1 ... |
| clwwlkccatlem 30023 | Lemma for ~ clwwlkccat : i... |
| clwwlkccat 30024 | The concatenation of two w... |
| umgrclwwlkge2 30025 | A closed walk in a multigr... |
| clwlkclwwlklem2a1 30026 | Lemma 1 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a2 30027 | Lemma 2 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a3 30028 | Lemma 3 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2fv1 30029 | Lemma 4a for ~ clwlkclwwlk... |
| clwlkclwwlklem2fv2 30030 | Lemma 4b for ~ clwlkclwwlk... |
| clwlkclwwlklem2a4 30031 | Lemma 4 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a 30032 | Lemma for ~ clwlkclwwlklem... |
| clwlkclwwlklem1 30033 | Lemma 1 for ~ clwlkclwwlk ... |
| clwlkclwwlklem2 30034 | Lemma 2 for ~ clwlkclwwlk ... |
| clwlkclwwlklem3 30035 | Lemma 3 for ~ clwlkclwwlk ... |
| clwlkclwwlk 30036 | A closed walk as word of l... |
| clwlkclwwlk2 30037 | A closed walk corresponds ... |
| clwlkclwwlkflem 30038 | Lemma for ~ clwlkclwwlkf .... |
| clwlkclwwlkf1lem2 30039 | Lemma 2 for ~ clwlkclwwlkf... |
| clwlkclwwlkf1lem3 30040 | Lemma 3 for ~ clwlkclwwlkf... |
| clwlkclwwlkfolem 30041 | Lemma for ~ clwlkclwwlkfo ... |
| clwlkclwwlkf 30042 | ` F ` is a function from t... |
| clwlkclwwlkfo 30043 | ` F ` is a function from t... |
| clwlkclwwlkf1 30044 | ` F ` is a one-to-one func... |
| clwlkclwwlkf1o 30045 | ` F ` is a bijection betwe... |
| clwlkclwwlken 30046 | The set of the nonempty cl... |
| clwwisshclwwslemlem 30047 | Lemma for ~ clwwisshclwwsl... |
| clwwisshclwwslem 30048 | Lemma for ~ clwwisshclwws ... |
| clwwisshclwws 30049 | Cyclically shifting a clos... |
| clwwisshclwwsn 30050 | Cyclically shifting a clos... |
| erclwwlkrel 30051 | ` .~ ` is a relation. (Co... |
| erclwwlkeq 30052 | Two classes are equivalent... |
| erclwwlkeqlen 30053 | If two classes are equival... |
| erclwwlkref 30054 | ` .~ ` is a reflexive rela... |
| erclwwlksym 30055 | ` .~ ` is a symmetric rela... |
| erclwwlktr 30056 | ` .~ ` is a transitive rel... |
| erclwwlk 30057 | ` .~ ` is an equivalence r... |
| clwwlkn 30060 | The set of closed walks of... |
| isclwwlkn 30061 | A word over the set of ver... |
| clwwlkn0 30062 | There is no closed walk of... |
| clwwlkneq0 30063 | Sufficient conditions for ... |
| clwwlkclwwlkn 30064 | A closed walk of a fixed l... |
| clwwlksclwwlkn 30065 | The closed walks of a fixe... |
| clwwlknlen 30066 | The length of a word repre... |
| clwwlknnn 30067 | The length of a closed wal... |
| clwwlknwrd 30068 | A closed walk of a fixed l... |
| clwwlknbp 30069 | Basic properties of a clos... |
| isclwwlknx 30070 | Characterization of a word... |
| clwwlknp 30071 | Properties of a set being ... |
| clwwlknwwlksn 30072 | A word representing a clos... |
| clwwlknlbonbgr1 30073 | The last but one vertex in... |
| clwwlkinwwlk 30074 | If the initial vertex of a... |
| clwwlkn1 30075 | A closed walk of length 1 ... |
| loopclwwlkn1b 30076 | The singleton word consist... |
| clwwlkn1loopb 30077 | A word represents a closed... |
| clwwlkn2 30078 | A closed walk of length 2 ... |
| clwwlknfi 30079 | If there is only a finite ... |
| clwwlkel 30080 | Obtaining a closed walk (a... |
| clwwlkf 30081 | Lemma 1 for ~ clwwlkf1o : ... |
| clwwlkfv 30082 | Lemma 2 for ~ clwwlkf1o : ... |
| clwwlkf1 30083 | Lemma 3 for ~ clwwlkf1o : ... |
| clwwlkfo 30084 | Lemma 4 for ~ clwwlkf1o : ... |
| clwwlkf1o 30085 | F is a 1-1 onto function, ... |
| clwwlken 30086 | The set of closed walks of... |
| clwwlknwwlkncl 30087 | Obtaining a closed walk (a... |
| clwwlkwwlksb 30088 | A nonempty word over verti... |
| clwwlknwwlksnb 30089 | A word over vertices repre... |
| clwwlkext2edg 30090 | If a word concatenated wit... |
| wwlksext2clwwlk 30091 | If a word represents a wal... |
| wwlksubclwwlk 30092 | Any prefix of a word repre... |
| clwwnisshclwwsn 30093 | Cyclically shifting a clos... |
| eleclclwwlknlem1 30094 | Lemma 1 for ~ eleclclwwlkn... |
| eleclclwwlknlem2 30095 | Lemma 2 for ~ eleclclwwlkn... |
| clwwlknscsh 30096 | The set of cyclical shifts... |
| clwwlknccat 30097 | The concatenation of two w... |
| umgr2cwwk2dif 30098 | If a word represents a clo... |
| umgr2cwwkdifex 30099 | If a word represents a clo... |
| erclwwlknrel 30100 | ` .~ ` is a relation. (Co... |
| erclwwlkneq 30101 | Two classes are equivalent... |
| erclwwlkneqlen 30102 | If two classes are equival... |
| erclwwlknref 30103 | ` .~ ` is a reflexive rela... |
| erclwwlknsym 30104 | ` .~ ` is a symmetric rela... |
| erclwwlkntr 30105 | ` .~ ` is a transitive rel... |
| erclwwlkn 30106 | ` .~ ` is an equivalence r... |
| qerclwwlknfi 30107 | The quotient set of the se... |
| hashclwwlkn0 30108 | The number of closed walks... |
| eclclwwlkn1 30109 | An equivalence class accor... |
| eleclclwwlkn 30110 | A member of an equivalence... |
| hashecclwwlkn1 30111 | The size of every equivale... |
| umgrhashecclwwlk 30112 | The size of every equivale... |
| fusgrhashclwwlkn 30113 | The size of the set of clo... |
| clwwlkndivn 30114 | The size of the set of clo... |
| clwlknf1oclwwlknlem1 30115 | Lemma 1 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem2 30116 | Lemma 2 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem3 30117 | Lemma 3 for ~ clwlknf1oclw... |
| clwlknf1oclwwlkn 30118 | There is a one-to-one onto... |
| clwlkssizeeq 30119 | The size of the set of clo... |
| clwlksndivn 30120 | The size of the set of clo... |
| clwwlknonmpo 30123 | ` ( ClWWalksNOn `` G ) ` i... |
| clwwlknon 30124 | The set of closed walks on... |
| isclwwlknon 30125 | A word over the set of ver... |
| clwwlk0on0 30126 | There is no word over the ... |
| clwwlknon0 30127 | Sufficient conditions for ... |
| clwwlknonfin 30128 | In a finite graph ` G ` , ... |
| clwwlknonel 30129 | Characterization of a word... |
| clwwlknonccat 30130 | The concatenation of two w... |
| clwwlknon1 30131 | The set of closed walks on... |
| clwwlknon1loop 30132 | If there is a loop at vert... |
| clwwlknon1nloop 30133 | If there is no loop at ver... |
| clwwlknon1sn 30134 | The set of (closed) walks ... |
| clwwlknon1le1 30135 | There is at most one (clos... |
| clwwlknon2 30136 | The set of closed walks on... |
| clwwlknon2x 30137 | The set of closed walks on... |
| s2elclwwlknon2 30138 | Sufficient conditions of a... |
| clwwlknon2num 30139 | In a ` K `-regular graph `... |
| clwwlknonwwlknonb 30140 | A word over vertices repre... |
| clwwlknonex2lem1 30141 | Lemma 1 for ~ clwwlknonex2... |
| clwwlknonex2lem2 30142 | Lemma 2 for ~ clwwlknonex2... |
| clwwlknonex2 30143 | Extending a closed walk ` ... |
| clwwlknonex2e 30144 | Extending a closed walk ` ... |
| clwwlknondisj 30145 | The sets of closed walks o... |
| clwwlknun 30146 | The set of closed walks of... |
| clwwlkvbij 30147 | There is a bijection betwe... |
| 0ewlk 30148 | The empty set (empty seque... |
| 1ewlk 30149 | A sequence of 1 edge is an... |
| 0wlk 30150 | A pair of an empty set (of... |
| is0wlk 30151 | A pair of an empty set (of... |
| 0wlkonlem1 30152 | Lemma 1 for ~ 0wlkon and ~... |
| 0wlkonlem2 30153 | Lemma 2 for ~ 0wlkon and ~... |
| 0wlkon 30154 | A walk of length 0 from a ... |
| 0wlkons1 30155 | A walk of length 0 from a ... |
| 0trl 30156 | A pair of an empty set (of... |
| is0trl 30157 | A pair of an empty set (of... |
| 0trlon 30158 | A trail of length 0 from a... |
| 0pth 30159 | A pair of an empty set (of... |
| 0spth 30160 | A pair of an empty set (of... |
| 0pthon 30161 | A path of length 0 from a ... |
| 0pthon1 30162 | A path of length 0 from a ... |
| 0pthonv 30163 | For each vertex there is a... |
| 0clwlk 30164 | A pair of an empty set (of... |
| 0clwlkv 30165 | Any vertex (more precisely... |
| 0clwlk0 30166 | There is no closed walk in... |
| 0crct 30167 | A pair of an empty set (of... |
| 0cycl 30168 | A pair of an empty set (of... |
| 1pthdlem1 30169 | Lemma 1 for ~ 1pthd . (Co... |
| 1pthdlem2 30170 | Lemma 2 for ~ 1pthd . (Co... |
| 1wlkdlem1 30171 | Lemma 1 for ~ 1wlkd . (Co... |
| 1wlkdlem2 30172 | Lemma 2 for ~ 1wlkd . (Co... |
| 1wlkdlem3 30173 | Lemma 3 for ~ 1wlkd . (Co... |
| 1wlkdlem4 30174 | Lemma 4 for ~ 1wlkd . (Co... |
| 1wlkd 30175 | In a graph with two vertic... |
| 1trld 30176 | In a graph with two vertic... |
| 1pthd 30177 | In a graph with two vertic... |
| 1pthond 30178 | In a graph with two vertic... |
| upgr1wlkdlem1 30179 | Lemma 1 for ~ upgr1wlkd . ... |
| upgr1wlkdlem2 30180 | Lemma 2 for ~ upgr1wlkd . ... |
| upgr1wlkd 30181 | In a pseudograph with two ... |
| upgr1trld 30182 | In a pseudograph with two ... |
| upgr1pthd 30183 | In a pseudograph with two ... |
| upgr1pthond 30184 | In a pseudograph with two ... |
| lppthon 30185 | A loop (which is an edge a... |
| lp1cycl 30186 | A loop (which is an edge a... |
| 1pthon2v 30187 | For each pair of adjacent ... |
| 1pthon2ve 30188 | For each pair of adjacent ... |
| wlk2v2elem1 30189 | Lemma 1 for ~ wlk2v2e : ` ... |
| wlk2v2elem2 30190 | Lemma 2 for ~ wlk2v2e : T... |
| wlk2v2e 30191 | In a graph with two vertic... |
| ntrl2v2e 30192 | A walk which is not a trai... |
| 3wlkdlem1 30193 | Lemma 1 for ~ 3wlkd . (Co... |
| 3wlkdlem2 30194 | Lemma 2 for ~ 3wlkd . (Co... |
| 3wlkdlem3 30195 | Lemma 3 for ~ 3wlkd . (Co... |
| 3wlkdlem4 30196 | Lemma 4 for ~ 3wlkd . (Co... |
| 3wlkdlem5 30197 | Lemma 5 for ~ 3wlkd . (Co... |
| 3pthdlem1 30198 | Lemma 1 for ~ 3pthd . (Co... |
| 3wlkdlem6 30199 | Lemma 6 for ~ 3wlkd . (Co... |
| 3wlkdlem7 30200 | Lemma 7 for ~ 3wlkd . (Co... |
| 3wlkdlem8 30201 | Lemma 8 for ~ 3wlkd . (Co... |
| 3wlkdlem9 30202 | Lemma 9 for ~ 3wlkd . (Co... |
| 3wlkdlem10 30203 | Lemma 10 for ~ 3wlkd . (C... |
| 3wlkd 30204 | Construction of a walk fro... |
| 3wlkond 30205 | A walk of length 3 from on... |
| 3trld 30206 | Construction of a trail fr... |
| 3trlond 30207 | A trail of length 3 from o... |
| 3pthd 30208 | A path of length 3 from on... |
| 3pthond 30209 | A path of length 3 from on... |
| 3spthd 30210 | A simple path of length 3 ... |
| 3spthond 30211 | A simple path of length 3 ... |
| 3cycld 30212 | Construction of a 3-cycle ... |
| 3cyclpd 30213 | Construction of a 3-cycle ... |
| upgr3v3e3cycl 30214 | If there is a cycle of len... |
| uhgr3cyclexlem 30215 | Lemma for ~ uhgr3cyclex . ... |
| uhgr3cyclex 30216 | If there are three differe... |
| umgr3cyclex 30217 | If there are three (differ... |
| umgr3v3e3cycl 30218 | If and only if there is a ... |
| upgr4cycl4dv4e 30219 | If there is a cycle of len... |
| dfconngr1 30222 | Alternative definition of ... |
| isconngr 30223 | The property of being a co... |
| isconngr1 30224 | The property of being a co... |
| cusconngr 30225 | A complete hypergraph is c... |
| 0conngr 30226 | A graph without vertices i... |
| 0vconngr 30227 | A graph without vertices i... |
| 1conngr 30228 | A graph with (at most) one... |
| conngrv2edg 30229 | A vertex in a connected gr... |
| vdn0conngrumgrv2 30230 | A vertex in a connected mu... |
| releupth 30233 | The set ` ( EulerPaths `` ... |
| eupths 30234 | The Eulerian paths on the ... |
| iseupth 30235 | The property " ` <. F , P ... |
| iseupthf1o 30236 | The property " ` <. F , P ... |
| eupthi 30237 | Properties of an Eulerian ... |
| eupthf1o 30238 | The ` F ` function in an E... |
| eupthfi 30239 | Any graph with an Eulerian... |
| eupthseg 30240 | The ` N ` -th edge in an e... |
| upgriseupth 30241 | The property " ` <. F , P ... |
| upgreupthi 30242 | Properties of an Eulerian ... |
| upgreupthseg 30243 | The ` N ` -th edge in an e... |
| eupthcl 30244 | An Eulerian path has lengt... |
| eupthistrl 30245 | An Eulerian path is a trai... |
| eupthiswlk 30246 | An Eulerian path is a walk... |
| eupthpf 30247 | The ` P ` function in an E... |
| eupth0 30248 | There is an Eulerian path ... |
| eupthres 30249 | The restriction ` <. H , Q... |
| eupthp1 30250 | Append one path segment to... |
| eupth2eucrct 30251 | Append one path segment to... |
| eupth2lem1 30252 | Lemma for ~ eupth2 . (Con... |
| eupth2lem2 30253 | Lemma for ~ eupth2 . (Con... |
| trlsegvdeglem1 30254 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem2 30255 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem3 30256 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem4 30257 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem5 30258 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem6 30259 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem7 30260 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeg 30261 | Formerly part of proof of ... |
| eupth2lem3lem1 30262 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem2 30263 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem3 30264 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem4 30265 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem5 30266 | Lemma for ~ eupth2 . (Con... |
| eupth2lem3lem6 30267 | Formerly part of proof of ... |
| eupth2lem3lem7 30268 | Lemma for ~ eupth2lem3 : ... |
| eupthvdres 30269 | Formerly part of proof of ... |
| eupth2lem3 30270 | Lemma for ~ eupth2 . (Con... |
| eupth2lemb 30271 | Lemma for ~ eupth2 (induct... |
| eupth2lems 30272 | Lemma for ~ eupth2 (induct... |
| eupth2 30273 | The only vertices of odd d... |
| eulerpathpr 30274 | A graph with an Eulerian p... |
| eulerpath 30275 | A pseudograph with an Eule... |
| eulercrct 30276 | A pseudograph with an Eule... |
| eucrctshift 30277 | Cyclically shifting the in... |
| eucrct2eupth1 30278 | Removing one edge ` ( I ``... |
| eucrct2eupth 30279 | Removing one edge ` ( I ``... |
| konigsbergvtx 30280 | The set of vertices of the... |
| konigsbergiedg 30281 | The indexed edges of the K... |
| konigsbergiedgw 30282 | The indexed edges of the K... |
| konigsbergssiedgwpr 30283 | Each subset of the indexed... |
| konigsbergssiedgw 30284 | Each subset of the indexed... |
| konigsbergumgr 30285 | The Königsberg graph ... |
| konigsberglem1 30286 | Lemma 1 for ~ konigsberg :... |
| konigsberglem2 30287 | Lemma 2 for ~ konigsberg :... |
| konigsberglem3 30288 | Lemma 3 for ~ konigsberg :... |
| konigsberglem4 30289 | Lemma 4 for ~ konigsberg :... |
| konigsberglem5 30290 | Lemma 5 for ~ konigsberg :... |
| konigsberg 30291 | The Königsberg Bridge... |
| isfrgr 30294 | The property of being a fr... |
| frgrusgr 30295 | A friendship graph is a si... |
| frgr0v 30296 | Any null graph (set with n... |
| frgr0vb 30297 | Any null graph (without ve... |
| frgruhgr0v 30298 | Any null graph (without ve... |
| frgr0 30299 | The null graph (graph with... |
| frcond1 30300 | The friendship condition: ... |
| frcond2 30301 | The friendship condition: ... |
| frgreu 30302 | Variant of ~ frcond2 : An... |
| frcond3 30303 | The friendship condition, ... |
| frcond4 30304 | The friendship condition, ... |
| frgr1v 30305 | Any graph with (at most) o... |
| nfrgr2v 30306 | Any graph with two (differ... |
| frgr3vlem1 30307 | Lemma 1 for ~ frgr3v . (C... |
| frgr3vlem2 30308 | Lemma 2 for ~ frgr3v . (C... |
| frgr3v 30309 | Any graph with three verti... |
| 1vwmgr 30310 | Every graph with one verte... |
| 3vfriswmgrlem 30311 | Lemma for ~ 3vfriswmgr . ... |
| 3vfriswmgr 30312 | Every friendship graph wit... |
| 1to2vfriswmgr 30313 | Every friendship graph wit... |
| 1to3vfriswmgr 30314 | Every friendship graph wit... |
| 1to3vfriendship 30315 | The friendship theorem for... |
| 2pthfrgrrn 30316 | Between any two (different... |
| 2pthfrgrrn2 30317 | Between any two (different... |
| 2pthfrgr 30318 | Between any two (different... |
| 3cyclfrgrrn1 30319 | Every vertex in a friendsh... |
| 3cyclfrgrrn 30320 | Every vertex in a friendsh... |
| 3cyclfrgrrn2 30321 | Every vertex in a friendsh... |
| 3cyclfrgr 30322 | Every vertex in a friendsh... |
| 4cycl2v2nb 30323 | In a (maybe degenerate) 4-... |
| 4cycl2vnunb 30324 | In a 4-cycle, two distinct... |
| n4cyclfrgr 30325 | There is no 4-cycle in a f... |
| 4cyclusnfrgr 30326 | A graph with a 4-cycle is ... |
| frgrnbnb 30327 | If two neighbors ` U ` and... |
| frgrconngr 30328 | A friendship graph is conn... |
| vdgn0frgrv2 30329 | A vertex in a friendship g... |
| vdgn1frgrv2 30330 | Any vertex in a friendship... |
| vdgn1frgrv3 30331 | Any vertex in a friendship... |
| vdgfrgrgt2 30332 | Any vertex in a friendship... |
| frgrncvvdeqlem1 30333 | Lemma 1 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem2 30334 | Lemma 2 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem3 30335 | Lemma 3 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem4 30336 | Lemma 4 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem5 30337 | Lemma 5 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem6 30338 | Lemma 6 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem7 30339 | Lemma 7 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem8 30340 | Lemma 8 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem9 30341 | Lemma 9 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem10 30342 | Lemma 10 for ~ frgrncvvdeq... |
| frgrncvvdeq 30343 | In a friendship graph, two... |
| frgrwopreglem4a 30344 | In a friendship graph any ... |
| frgrwopreglem5a 30345 | If a friendship graph has ... |
| frgrwopreglem1 30346 | Lemma 1 for ~ frgrwopreg :... |
| frgrwopreglem2 30347 | Lemma 2 for ~ frgrwopreg .... |
| frgrwopreglem3 30348 | Lemma 3 for ~ frgrwopreg .... |
| frgrwopreglem4 30349 | Lemma 4 for ~ frgrwopreg .... |
| frgrwopregasn 30350 | According to statement 5 i... |
| frgrwopregbsn 30351 | According to statement 5 i... |
| frgrwopreg1 30352 | According to statement 5 i... |
| frgrwopreg2 30353 | According to statement 5 i... |
| frgrwopreglem5lem 30354 | Lemma for ~ frgrwopreglem5... |
| frgrwopreglem5 30355 | Lemma 5 for ~ frgrwopreg .... |
| frgrwopreglem5ALT 30356 | Alternate direct proof of ... |
| frgrwopreg 30357 | In a friendship graph ther... |
| frgrregorufr0 30358 | In a friendship graph ther... |
| frgrregorufr 30359 | If there is a vertex havin... |
| frgrregorufrg 30360 | If there is a vertex havin... |
| frgr2wwlkeu 30361 | For two different vertices... |
| frgr2wwlkn0 30362 | In a friendship graph, the... |
| frgr2wwlk1 30363 | In a friendship graph, the... |
| frgr2wsp1 30364 | In a friendship graph, the... |
| frgr2wwlkeqm 30365 | If there is a (simple) pat... |
| frgrhash2wsp 30366 | The number of simple paths... |
| fusgreg2wsplem 30367 | Lemma for ~ fusgreg2wsp an... |
| fusgr2wsp2nb 30368 | The set of paths of length... |
| fusgreghash2wspv 30369 | According to statement 7 i... |
| fusgreg2wsp 30370 | In a finite simple graph, ... |
| 2wspmdisj 30371 | The sets of paths of lengt... |
| fusgreghash2wsp 30372 | In a finite k-regular grap... |
| frrusgrord0lem 30373 | Lemma for ~ frrusgrord0 . ... |
| frrusgrord0 30374 | If a nonempty finite frien... |
| frrusgrord 30375 | If a nonempty finite frien... |
| numclwwlk2lem1lem 30376 | Lemma for ~ numclwwlk2lem1... |
| 2clwwlklem 30377 | Lemma for ~ clwwnonrepclww... |
| clwwnrepclwwn 30378 | If the initial vertex of a... |
| clwwnonrepclwwnon 30379 | If the initial vertex of a... |
| 2clwwlk2clwwlklem 30380 | Lemma for ~ 2clwwlk2clwwlk... |
| 2clwwlk 30381 | Value of operation ` C ` ,... |
| 2clwwlk2 30382 | The set ` ( X C 2 ) ` of d... |
| 2clwwlkel 30383 | Characterization of an ele... |
| 2clwwlk2clwwlk 30384 | An element of the value of... |
| numclwwlk1lem2foalem 30385 | Lemma for ~ numclwwlk1lem2... |
| extwwlkfab 30386 | The set ` ( X C N ) ` of d... |
| extwwlkfabel 30387 | Characterization of an ele... |
| numclwwlk1lem2foa 30388 | Going forth and back from ... |
| numclwwlk1lem2f 30389 | ` T ` is a function, mappi... |
| numclwwlk1lem2fv 30390 | Value of the function ` T ... |
| numclwwlk1lem2f1 30391 | ` T ` is a 1-1 function. ... |
| numclwwlk1lem2fo 30392 | ` T ` is an onto function.... |
| numclwwlk1lem2f1o 30393 | ` T ` is a 1-1 onto functi... |
| numclwwlk1lem2 30394 | The set of double loops of... |
| numclwwlk1 30395 | Statement 9 in [Huneke] p.... |
| clwwlknonclwlknonf1o 30396 | ` F ` is a bijection betwe... |
| clwwlknonclwlknonen 30397 | The sets of the two repres... |
| dlwwlknondlwlknonf1olem1 30398 | Lemma 1 for ~ dlwwlknondlw... |
| dlwwlknondlwlknonf1o 30399 | ` F ` is a bijection betwe... |
| dlwwlknondlwlknonen 30400 | The sets of the two repres... |
| wlkl0 30401 | There is exactly one walk ... |
| clwlknon2num 30402 | There are k walks of lengt... |
| numclwlk1lem1 30403 | Lemma 1 for ~ numclwlk1 (S... |
| numclwlk1lem2 30404 | Lemma 2 for ~ numclwlk1 (S... |
| numclwlk1 30405 | Statement 9 in [Huneke] p.... |
| numclwwlkovh0 30406 | Value of operation ` H ` ,... |
| numclwwlkovh 30407 | Value of operation ` H ` ,... |
| numclwwlkovq 30408 | Value of operation ` Q ` ,... |
| numclwwlkqhash 30409 | In a ` K `-regular graph, ... |
| numclwwlk2lem1 30410 | In a friendship graph, for... |
| numclwlk2lem2f 30411 | ` R ` is a function mappin... |
| numclwlk2lem2fv 30412 | Value of the function ` R ... |
| numclwlk2lem2f1o 30413 | ` R ` is a 1-1 onto functi... |
| numclwwlk2lem3 30414 | In a friendship graph, the... |
| numclwwlk2 30415 | Statement 10 in [Huneke] p... |
| numclwwlk3lem1 30416 | Lemma 2 for ~ numclwwlk3 .... |
| numclwwlk3lem2lem 30417 | Lemma for ~ numclwwlk3lem2... |
| numclwwlk3lem2 30418 | Lemma 1 for ~ numclwwlk3 :... |
| numclwwlk3 30419 | Statement 12 in [Huneke] p... |
| numclwwlk4 30420 | The total number of closed... |
| numclwwlk5lem 30421 | Lemma for ~ numclwwlk5 . ... |
| numclwwlk5 30422 | Statement 13 in [Huneke] p... |
| numclwwlk7lem 30423 | Lemma for ~ numclwwlk7 , ~... |
| numclwwlk6 30424 | For a prime divisor ` P ` ... |
| numclwwlk7 30425 | Statement 14 in [Huneke] p... |
| numclwwlk8 30426 | The size of the set of clo... |
| frgrreggt1 30427 | If a finite nonempty frien... |
| frgrreg 30428 | If a finite nonempty frien... |
| frgrregord013 30429 | If a finite friendship gra... |
| frgrregord13 30430 | If a nonempty finite frien... |
| frgrogt3nreg 30431 | If a finite friendship gra... |
| friendshipgt3 30432 | The friendship theorem for... |
| friendship 30433 | The friendship theorem: I... |
| conventions 30434 |
H... |
| conventions-labels 30435 |
... |
| conventions-comments 30436 |
... |
| natded 30437 | Here are typical n... |
| ex-natded5.2 30438 | Theorem 5.2 of [Clemente] ... |
| ex-natded5.2-2 30439 | A more efficient proof of ... |
| ex-natded5.2i 30440 | The same as ~ ex-natded5.2... |
| ex-natded5.3 30441 | Theorem 5.3 of [Clemente] ... |
| ex-natded5.3-2 30442 | A more efficient proof of ... |
| ex-natded5.3i 30443 | The same as ~ ex-natded5.3... |
| ex-natded5.5 30444 | Theorem 5.5 of [Clemente] ... |
| ex-natded5.7 30445 | Theorem 5.7 of [Clemente] ... |
| ex-natded5.7-2 30446 | A more efficient proof of ... |
| ex-natded5.8 30447 | Theorem 5.8 of [Clemente] ... |
| ex-natded5.8-2 30448 | A more efficient proof of ... |
| ex-natded5.13 30449 | Theorem 5.13 of [Clemente]... |
| ex-natded5.13-2 30450 | A more efficient proof of ... |
| ex-natded9.20 30451 | Theorem 9.20 of [Clemente]... |
| ex-natded9.20-2 30452 | A more efficient proof of ... |
| ex-natded9.26 30453 | Theorem 9.26 of [Clemente]... |
| ex-natded9.26-2 30454 | A more efficient proof of ... |
| ex-or 30455 | Example for ~ df-or . Exa... |
| ex-an 30456 | Example for ~ df-an . Exa... |
| ex-dif 30457 | Example for ~ df-dif . Ex... |
| ex-un 30458 | Example for ~ df-un . Exa... |
| ex-in 30459 | Example for ~ df-in . Exa... |
| ex-uni 30460 | Example for ~ df-uni . Ex... |
| ex-ss 30461 | Example for ~ df-ss . Exa... |
| ex-pss 30462 | Example for ~ df-pss . Ex... |
| ex-pw 30463 | Example for ~ df-pw . Exa... |
| ex-pr 30464 | Example for ~ df-pr . (Co... |
| ex-br 30465 | Example for ~ df-br . Exa... |
| ex-opab 30466 | Example for ~ df-opab . E... |
| ex-eprel 30467 | Example for ~ df-eprel . ... |
| ex-id 30468 | Example for ~ df-id . Exa... |
| ex-po 30469 | Example for ~ df-po . Exa... |
| ex-xp 30470 | Example for ~ df-xp . Exa... |
| ex-cnv 30471 | Example for ~ df-cnv . Ex... |
| ex-co 30472 | Example for ~ df-co . Exa... |
| ex-dm 30473 | Example for ~ df-dm . Exa... |
| ex-rn 30474 | Example for ~ df-rn . Exa... |
| ex-res 30475 | Example for ~ df-res . Ex... |
| ex-ima 30476 | Example for ~ df-ima . Ex... |
| ex-fv 30477 | Example for ~ df-fv . Exa... |
| ex-1st 30478 | Example for ~ df-1st . Ex... |
| ex-2nd 30479 | Example for ~ df-2nd . Ex... |
| 1kp2ke3k 30480 | Example for ~ df-dec , 100... |
| ex-fl 30481 | Example for ~ df-fl . Exa... |
| ex-ceil 30482 | Example for ~ df-ceil . (... |
| ex-mod 30483 | Example for ~ df-mod . (C... |
| ex-exp 30484 | Example for ~ df-exp . (C... |
| ex-fac 30485 | Example for ~ df-fac . (C... |
| ex-bc 30486 | Example for ~ df-bc . (Co... |
| ex-hash 30487 | Example for ~ df-hash . (... |
| ex-sqrt 30488 | Example for ~ df-sqrt . (... |
| ex-abs 30489 | Example for ~ df-abs . (C... |
| ex-dvds 30490 | Example for ~ df-dvds : 3 ... |
| ex-gcd 30491 | Example for ~ df-gcd . (C... |
| ex-lcm 30492 | Example for ~ df-lcm . (C... |
| ex-prmo 30493 | Example for ~ df-prmo : ` ... |
| aevdemo 30494 | Proof illustrating the com... |
| ex-ind-dvds 30495 | Example of a proof by indu... |
| ex-fpar 30496 | Formalized example provide... |
| avril1 30497 | Poisson d'Avril's Theorem.... |
| 2bornot2b 30498 | The law of excluded middle... |
| helloworld 30499 | The classic "Hello world" ... |
| 1p1e2apr1 30500 | One plus one equals two. ... |
| eqid1 30501 | Law of identity (reflexivi... |
| 1div0apr 30502 | Division by zero is forbid... |
| topnfbey 30503 | Nothing seems to be imposs... |
| 9p10ne21 30504 | 9 + 10 is not equal to 21.... |
| 9p10ne21fool 30505 | 9 + 10 equals 21. This as... |
| nrt2irr 30507 | The ` N ` -th root of 2 is... |
| isplig 30510 | The predicate "is a planar... |
| ispligb 30511 | The predicate "is a planar... |
| tncp 30512 | In any planar incidence ge... |
| l2p 30513 | For any line in a planar i... |
| lpni 30514 | For any line in a planar i... |
| nsnlplig 30515 | There is no "one-point lin... |
| nsnlpligALT 30516 | Alternate version of ~ nsn... |
| n0lplig 30517 | There is no "empty line" i... |
| n0lpligALT 30518 | Alternate version of ~ n0l... |
| eulplig 30519 | Through two distinct point... |
| pliguhgr 30520 | Any planar incidence geome... |
| dummylink 30521 | Alias for ~ a1ii that may ... |
| id1 30522 | Alias for ~ idALT that may... |
| isgrpo 30531 | The predicate "is a group ... |
| isgrpoi 30532 | Properties that determine ... |
| grpofo 30533 | A group operation maps ont... |
| grpocl 30534 | Closure law for a group op... |
| grpolidinv 30535 | A group has a left identit... |
| grpon0 30536 | The base set of a group is... |
| grpoass 30537 | A group operation is assoc... |
| grpoidinvlem1 30538 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem2 30539 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem3 30540 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem4 30541 | Lemma for ~ grpoidinv . (... |
| grpoidinv 30542 | A group has a left and rig... |
| grpoideu 30543 | The left identity element ... |
| grporndm 30544 | A group's range in terms o... |
| 0ngrp 30545 | The empty set is not a gro... |
| gidval 30546 | The value of the identity ... |
| grpoidval 30547 | Lemma for ~ grpoidcl and o... |
| grpoidcl 30548 | The identity element of a ... |
| grpoidinv2 30549 | A group's properties using... |
| grpolid 30550 | The identity element of a ... |
| grporid 30551 | The identity element of a ... |
| grporcan 30552 | Right cancellation law for... |
| grpoinveu 30553 | The left inverse element o... |
| grpoid 30554 | Two ways of saying that an... |
| grporn 30555 | The range of a group opera... |
| grpoinvfval 30556 | The inverse function of a ... |
| grpoinvval 30557 | The inverse of a group ele... |
| grpoinvcl 30558 | A group element's inverse ... |
| grpoinv 30559 | The properties of a group ... |
| grpolinv 30560 | The left inverse of a grou... |
| grporinv 30561 | The right inverse of a gro... |
| grpoinvid1 30562 | The inverse of a group ele... |
| grpoinvid2 30563 | The inverse of a group ele... |
| grpolcan 30564 | Left cancellation law for ... |
| grpo2inv 30565 | Double inverse law for gro... |
| grpoinvf 30566 | Mapping of the inverse fun... |
| grpoinvop 30567 | The inverse of the group o... |
| grpodivfval 30568 | Group division (or subtrac... |
| grpodivval 30569 | Group division (or subtrac... |
| grpodivinv 30570 | Group division by an inver... |
| grpoinvdiv 30571 | Inverse of a group divisio... |
| grpodivf 30572 | Mapping for group division... |
| grpodivcl 30573 | Closure of group division ... |
| grpodivdiv 30574 | Double group division. (C... |
| grpomuldivass 30575 | Associative-type law for m... |
| grpodivid 30576 | Division of a group member... |
| grponpcan 30577 | Cancellation law for group... |
| isablo 30580 | The predicate "is an Abeli... |
| ablogrpo 30581 | An Abelian group operation... |
| ablocom 30582 | An Abelian group operation... |
| ablo32 30583 | Commutative/associative la... |
| ablo4 30584 | Commutative/associative la... |
| isabloi 30585 | Properties that determine ... |
| ablomuldiv 30586 | Law for group multiplicati... |
| ablodivdiv 30587 | Law for double group divis... |
| ablodivdiv4 30588 | Law for double group divis... |
| ablodiv32 30589 | Swap the second and third ... |
| ablonncan 30590 | Cancellation law for group... |
| ablonnncan1 30591 | Cancellation law for group... |
| vcrel 30594 | The class of all complex v... |
| vciOLD 30595 | Obsolete version of ~ cvsi... |
| vcsm 30596 | Functionality of th scalar... |
| vccl 30597 | Closure of the scalar prod... |
| vcidOLD 30598 | Identity element for the s... |
| vcdi 30599 | Distributive law for the s... |
| vcdir 30600 | Distributive law for the s... |
| vcass 30601 | Associative law for the sc... |
| vc2OLD 30602 | A vector plus itself is tw... |
| vcablo 30603 | Vector addition is an Abel... |
| vcgrp 30604 | Vector addition is a group... |
| vclcan 30605 | Left cancellation law for ... |
| vczcl 30606 | The zero vector is a vecto... |
| vc0rid 30607 | The zero vector is a right... |
| vc0 30608 | Zero times a vector is the... |
| vcz 30609 | Anything times the zero ve... |
| vcm 30610 | Minus 1 times a vector is ... |
| isvclem 30611 | Lemma for ~ isvcOLD . (Co... |
| vcex 30612 | The components of a comple... |
| isvcOLD 30613 | The predicate "is a comple... |
| isvciOLD 30614 | Properties that determine ... |
| cnaddabloOLD 30615 | Obsolete version of ~ cnad... |
| cnidOLD 30616 | Obsolete version of ~ cnad... |
| cncvcOLD 30617 | Obsolete version of ~ cncv... |
| nvss 30627 | Structure of the class of ... |
| nvvcop 30628 | A normed complex vector sp... |
| nvrel 30636 | The class of all normed co... |
| vafval 30637 | Value of the function for ... |
| bafval 30638 | Value of the function for ... |
| smfval 30639 | Value of the function for ... |
| 0vfval 30640 | Value of the function for ... |
| nmcvfval 30641 | Value of the norm function... |
| nvop2 30642 | A normed complex vector sp... |
| nvvop 30643 | The vector space component... |
| isnvlem 30644 | Lemma for ~ isnv . (Contr... |
| nvex 30645 | The components of a normed... |
| isnv 30646 | The predicate "is a normed... |
| isnvi 30647 | Properties that determine ... |
| nvi 30648 | The properties of a normed... |
| nvvc 30649 | The vector space component... |
| nvablo 30650 | The vector addition operat... |
| nvgrp 30651 | The vector addition operat... |
| nvgf 30652 | Mapping for the vector add... |
| nvsf 30653 | Mapping for the scalar mul... |
| nvgcl 30654 | Closure law for the vector... |
| nvcom 30655 | The vector addition (group... |
| nvass 30656 | The vector addition (group... |
| nvadd32 30657 | Commutative/associative la... |
| nvrcan 30658 | Right cancellation law for... |
| nvadd4 30659 | Rearrangement of 4 terms i... |
| nvscl 30660 | Closure law for the scalar... |
| nvsid 30661 | Identity element for the s... |
| nvsass 30662 | Associative law for the sc... |
| nvscom 30663 | Commutative law for the sc... |
| nvdi 30664 | Distributive law for the s... |
| nvdir 30665 | Distributive law for the s... |
| nv2 30666 | A vector plus itself is tw... |
| vsfval 30667 | Value of the function for ... |
| nvzcl 30668 | Closure law for the zero v... |
| nv0rid 30669 | The zero vector is a right... |
| nv0lid 30670 | The zero vector is a left ... |
| nv0 30671 | Zero times a vector is the... |
| nvsz 30672 | Anything times the zero ve... |
| nvinv 30673 | Minus 1 times a vector is ... |
| nvinvfval 30674 | Function for the negative ... |
| nvm 30675 | Vector subtraction in term... |
| nvmval 30676 | Value of vector subtractio... |
| nvmval2 30677 | Value of vector subtractio... |
| nvmfval 30678 | Value of the function for ... |
| nvmf 30679 | Mapping for the vector sub... |
| nvmcl 30680 | Closure law for the vector... |
| nvnnncan1 30681 | Cancellation law for vecto... |
| nvmdi 30682 | Distributive law for scala... |
| nvnegneg 30683 | Double negative of a vecto... |
| nvmul0or 30684 | If a scalar product is zer... |
| nvrinv 30685 | A vector minus itself. (C... |
| nvlinv 30686 | Minus a vector plus itself... |
| nvpncan2 30687 | Cancellation law for vecto... |
| nvpncan 30688 | Cancellation law for vecto... |
| nvaddsub 30689 | Commutative/associative la... |
| nvnpcan 30690 | Cancellation law for a nor... |
| nvaddsub4 30691 | Rearrangement of 4 terms i... |
| nvmeq0 30692 | The difference between two... |
| nvmid 30693 | A vector minus itself is t... |
| nvf 30694 | Mapping for the norm funct... |
| nvcl 30695 | The norm of a normed compl... |
| nvcli 30696 | The norm of a normed compl... |
| nvs 30697 | Proportionality property o... |
| nvsge0 30698 | The norm of a scalar produ... |
| nvm1 30699 | The norm of the negative o... |
| nvdif 30700 | The norm of the difference... |
| nvpi 30701 | The norm of a vector plus ... |
| nvz0 30702 | The norm of a zero vector ... |
| nvz 30703 | The norm of a vector is ze... |
| nvtri 30704 | Triangle inequality for th... |
| nvmtri 30705 | Triangle inequality for th... |
| nvabs 30706 | Norm difference property o... |
| nvge0 30707 | The norm of a normed compl... |
| nvgt0 30708 | A nonzero norm is positive... |
| nv1 30709 | From any nonzero vector, c... |
| nvop 30710 | A complex inner product sp... |
| cnnv 30711 | The set of complex numbers... |
| cnnvg 30712 | The vector addition (group... |
| cnnvba 30713 | The base set of the normed... |
| cnnvs 30714 | The scalar product operati... |
| cnnvnm 30715 | The norm operation of the ... |
| cnnvm 30716 | The vector subtraction ope... |
| elimnv 30717 | Hypothesis elimination lem... |
| elimnvu 30718 | Hypothesis elimination lem... |
| imsval 30719 | Value of the induced metri... |
| imsdval 30720 | Value of the induced metri... |
| imsdval2 30721 | Value of the distance func... |
| nvnd 30722 | The norm of a normed compl... |
| imsdf 30723 | Mapping for the induced me... |
| imsmetlem 30724 | Lemma for ~ imsmet . (Con... |
| imsmet 30725 | The induced metric of a no... |
| imsxmet 30726 | The induced metric of a no... |
| cnims 30727 | The metric induced on the ... |
| vacn 30728 | Vector addition is jointly... |
| nmcvcn 30729 | The norm of a normed compl... |
| nmcnc 30730 | The norm of a normed compl... |
| smcnlem 30731 | Lemma for ~ smcn . (Contr... |
| smcn 30732 | Scalar multiplication is j... |
| vmcn 30733 | Vector subtraction is join... |
| dipfval 30736 | The inner product function... |
| ipval 30737 | Value of the inner product... |
| ipval2lem2 30738 | Lemma for ~ ipval3 . (Con... |
| ipval2lem3 30739 | Lemma for ~ ipval3 . (Con... |
| ipval2lem4 30740 | Lemma for ~ ipval3 . (Con... |
| ipval2 30741 | Expansion of the inner pro... |
| 4ipval2 30742 | Four times the inner produ... |
| ipval3 30743 | Expansion of the inner pro... |
| ipidsq 30744 | The inner product of a vec... |
| ipnm 30745 | Norm expressed in terms of... |
| dipcl 30746 | An inner product is a comp... |
| ipf 30747 | Mapping for the inner prod... |
| dipcj 30748 | The complex conjugate of a... |
| ipipcj 30749 | An inner product times its... |
| diporthcom 30750 | Orthogonality (meaning inn... |
| dip0r 30751 | Inner product with a zero ... |
| dip0l 30752 | Inner product with a zero ... |
| ipz 30753 | The inner product of a vec... |
| dipcn 30754 | Inner product is jointly c... |
| sspval 30757 | The set of all subspaces o... |
| isssp 30758 | The predicate "is a subspa... |
| sspid 30759 | A normed complex vector sp... |
| sspnv 30760 | A subspace is a normed com... |
| sspba 30761 | The base set of a subspace... |
| sspg 30762 | Vector addition on a subsp... |
| sspgval 30763 | Vector addition on a subsp... |
| ssps 30764 | Scalar multiplication on a... |
| sspsval 30765 | Scalar multiplication on a... |
| sspmlem 30766 | Lemma for ~ sspm and other... |
| sspmval 30767 | Vector addition on a subsp... |
| sspm 30768 | Vector subtraction on a su... |
| sspz 30769 | The zero vector of a subsp... |
| sspn 30770 | The norm on a subspace is ... |
| sspnval 30771 | The norm on a subspace in ... |
| sspimsval 30772 | The induced metric on a su... |
| sspims 30773 | The induced metric on a su... |
| lnoval 30786 | The set of linear operator... |
| islno 30787 | The predicate "is a linear... |
| lnolin 30788 | Basic linearity property o... |
| lnof 30789 | A linear operator is a map... |
| lno0 30790 | The value of a linear oper... |
| lnocoi 30791 | The composition of two lin... |
| lnoadd 30792 | Addition property of a lin... |
| lnosub 30793 | Subtraction property of a ... |
| lnomul 30794 | Scalar multiplication prop... |
| nvo00 30795 | Two ways to express a zero... |
| nmoofval 30796 | The operator norm function... |
| nmooval 30797 | The operator norm function... |
| nmosetre 30798 | The set in the supremum of... |
| nmosetn0 30799 | The set in the supremum of... |
| nmoxr 30800 | The norm of an operator is... |
| nmooge0 30801 | The norm of an operator is... |
| nmorepnf 30802 | The norm of an operator is... |
| nmoreltpnf 30803 | The norm of any operator i... |
| nmogtmnf 30804 | The norm of an operator is... |
| nmoolb 30805 | A lower bound for an opera... |
| nmoubi 30806 | An upper bound for an oper... |
| nmoub3i 30807 | An upper bound for an oper... |
| nmoub2i 30808 | An upper bound for an oper... |
| nmobndi 30809 | Two ways to express that a... |
| nmounbi 30810 | Two ways two express that ... |
| nmounbseqi 30811 | An unbounded operator dete... |
| nmounbseqiALT 30812 | Alternate shorter proof of... |
| nmobndseqi 30813 | A bounded sequence determi... |
| nmobndseqiALT 30814 | Alternate shorter proof of... |
| bloval 30815 | The class of bounded linea... |
| isblo 30816 | The predicate "is a bounde... |
| isblo2 30817 | The predicate "is a bounde... |
| bloln 30818 | A bounded operator is a li... |
| blof 30819 | A bounded operator is an o... |
| nmblore 30820 | The norm of a bounded oper... |
| 0ofval 30821 | The zero operator between ... |
| 0oval 30822 | Value of the zero operator... |
| 0oo 30823 | The zero operator is an op... |
| 0lno 30824 | The zero operator is linea... |
| nmoo0 30825 | The operator norm of the z... |
| 0blo 30826 | The zero operator is a bou... |
| nmlno0lem 30827 | Lemma for ~ nmlno0i . (Co... |
| nmlno0i 30828 | The norm of a linear opera... |
| nmlno0 30829 | The norm of a linear opera... |
| nmlnoubi 30830 | An upper bound for the ope... |
| nmlnogt0 30831 | The norm of a nonzero line... |
| lnon0 30832 | The domain of a nonzero li... |
| nmblolbii 30833 | A lower bound for the norm... |
| nmblolbi 30834 | A lower bound for the norm... |
| isblo3i 30835 | The predicate "is a bounde... |
| blo3i 30836 | Properties that determine ... |
| blometi 30837 | Upper bound for the distan... |
| blocnilem 30838 | Lemma for ~ blocni and ~ l... |
| blocni 30839 | A linear operator is conti... |
| lnocni 30840 | If a linear operator is co... |
| blocn 30841 | A linear operator is conti... |
| blocn2 30842 | A bounded linear operator ... |
| ajfval 30843 | The adjoint function. (Co... |
| hmoval 30844 | The set of Hermitian (self... |
| ishmo 30845 | The predicate "is a hermit... |
| phnv 30848 | Every complex inner produc... |
| phrel 30849 | The class of all complex i... |
| phnvi 30850 | Every complex inner produc... |
| isphg 30851 | The predicate "is a comple... |
| phop 30852 | A complex inner product sp... |
| cncph 30853 | The set of complex numbers... |
| elimph 30854 | Hypothesis elimination lem... |
| elimphu 30855 | Hypothesis elimination lem... |
| isph 30856 | The predicate "is an inner... |
| phpar2 30857 | The parallelogram law for ... |
| phpar 30858 | The parallelogram law for ... |
| ip0i 30859 | A slight variant of Equati... |
| ip1ilem 30860 | Lemma for ~ ip1i . (Contr... |
| ip1i 30861 | Equation 6.47 of [Ponnusam... |
| ip2i 30862 | Equation 6.48 of [Ponnusam... |
| ipdirilem 30863 | Lemma for ~ ipdiri . (Con... |
| ipdiri 30864 | Distributive law for inner... |
| ipasslem1 30865 | Lemma for ~ ipassi . Show... |
| ipasslem2 30866 | Lemma for ~ ipassi . Show... |
| ipasslem3 30867 | Lemma for ~ ipassi . Show... |
| ipasslem4 30868 | Lemma for ~ ipassi . Show... |
| ipasslem5 30869 | Lemma for ~ ipassi . Show... |
| ipasslem7 30870 | Lemma for ~ ipassi . Show... |
| ipasslem8 30871 | Lemma for ~ ipassi . By ~... |
| ipasslem9 30872 | Lemma for ~ ipassi . Conc... |
| ipasslem10 30873 | Lemma for ~ ipassi . Show... |
| ipasslem11 30874 | Lemma for ~ ipassi . Show... |
| ipassi 30875 | Associative law for inner ... |
| dipdir 30876 | Distributive law for inner... |
| dipdi 30877 | Distributive law for inner... |
| ip2dii 30878 | Inner product of two sums.... |
| dipass 30879 | Associative law for inner ... |
| dipassr 30880 | "Associative" law for seco... |
| dipassr2 30881 | "Associative" law for inne... |
| dipsubdir 30882 | Distributive law for inner... |
| dipsubdi 30883 | Distributive law for inner... |
| pythi 30884 | The Pythagorean theorem fo... |
| siilem1 30885 | Lemma for ~ sii . (Contri... |
| siilem2 30886 | Lemma for ~ sii . (Contri... |
| siii 30887 | Inference from ~ sii . (C... |
| sii 30888 | Obsolete version of ~ ipca... |
| ipblnfi 30889 | A function ` F ` generated... |
| ip2eqi 30890 | Two vectors are equal iff ... |
| phoeqi 30891 | A condition implying that ... |
| ajmoi 30892 | Every operator has at most... |
| ajfuni 30893 | The adjoint function is a ... |
| ajfun 30894 | The adjoint function is a ... |
| ajval 30895 | Value of the adjoint funct... |
| iscbn 30898 | A complex Banach space is ... |
| cbncms 30899 | The induced metric on comp... |
| bnnv 30900 | Every complex Banach space... |
| bnrel 30901 | The class of all complex B... |
| bnsscmcl 30902 | A subspace of a Banach spa... |
| cnbn 30903 | The set of complex numbers... |
| ubthlem1 30904 | Lemma for ~ ubth . The fu... |
| ubthlem2 30905 | Lemma for ~ ubth . Given ... |
| ubthlem3 30906 | Lemma for ~ ubth . Prove ... |
| ubth 30907 | Uniform Boundedness Theore... |
| minvecolem1 30908 | Lemma for ~ minveco . The... |
| minvecolem2 30909 | Lemma for ~ minveco . Any... |
| minvecolem3 30910 | Lemma for ~ minveco . The... |
| minvecolem4a 30911 | Lemma for ~ minveco . ` F ... |
| minvecolem4b 30912 | Lemma for ~ minveco . The... |
| minvecolem4c 30913 | Lemma for ~ minveco . The... |
| minvecolem4 30914 | Lemma for ~ minveco . The... |
| minvecolem5 30915 | Lemma for ~ minveco . Dis... |
| minvecolem6 30916 | Lemma for ~ minveco . Any... |
| minvecolem7 30917 | Lemma for ~ minveco . Sin... |
| minveco 30918 | Minimizing vector theorem,... |
| ishlo 30921 | The predicate "is a comple... |
| hlobn 30922 | Every complex Hilbert spac... |
| hlph 30923 | Every complex Hilbert spac... |
| hlrel 30924 | The class of all complex H... |
| hlnv 30925 | Every complex Hilbert spac... |
| hlnvi 30926 | Every complex Hilbert spac... |
| hlvc 30927 | Every complex Hilbert spac... |
| hlcmet 30928 | The induced metric on a co... |
| hlmet 30929 | The induced metric on a co... |
| hlpar2 30930 | The parallelogram law sati... |
| hlpar 30931 | The parallelogram law sati... |
| hlex 30932 | The base set of a Hilbert ... |
| hladdf 30933 | Mapping for Hilbert space ... |
| hlcom 30934 | Hilbert space vector addit... |
| hlass 30935 | Hilbert space vector addit... |
| hl0cl 30936 | The Hilbert space zero vec... |
| hladdid 30937 | Hilbert space addition wit... |
| hlmulf 30938 | Mapping for Hilbert space ... |
| hlmulid 30939 | Hilbert space scalar multi... |
| hlmulass 30940 | Hilbert space scalar multi... |
| hldi 30941 | Hilbert space scalar multi... |
| hldir 30942 | Hilbert space scalar multi... |
| hlmul0 30943 | Hilbert space scalar multi... |
| hlipf 30944 | Mapping for Hilbert space ... |
| hlipcj 30945 | Conjugate law for Hilbert ... |
| hlipdir 30946 | Distributive law for Hilbe... |
| hlipass 30947 | Associative law for Hilber... |
| hlipgt0 30948 | The inner product of a Hil... |
| hlcompl 30949 | Completeness of a Hilbert ... |
| cnchl 30950 | The set of complex numbers... |
| htthlem 30951 | Lemma for ~ htth . The co... |
| htth 30952 | Hellinger-Toeplitz Theorem... |
| The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here | |
| h2hva 31008 | The group (addition) opera... |
| h2hsm 31009 | The scalar product operati... |
| h2hnm 31010 | The norm function of Hilbe... |
| h2hvs 31011 | The vector subtraction ope... |
| h2hmetdval 31012 | Value of the distance func... |
| h2hcau 31013 | The Cauchy sequences of Hi... |
| h2hlm 31014 | The limit sequences of Hil... |
| axhilex-zf 31015 | Derive Axiom ~ ax-hilex fr... |
| axhfvadd-zf 31016 | Derive Axiom ~ ax-hfvadd f... |
| axhvcom-zf 31017 | Derive Axiom ~ ax-hvcom fr... |
| axhvass-zf 31018 | Derive Axiom ~ ax-hvass fr... |
| axhv0cl-zf 31019 | Derive Axiom ~ ax-hv0cl fr... |
| axhvaddid-zf 31020 | Derive Axiom ~ ax-hvaddid ... |
| axhfvmul-zf 31021 | Derive Axiom ~ ax-hfvmul f... |
| axhvmulid-zf 31022 | Derive Axiom ~ ax-hvmulid ... |
| axhvmulass-zf 31023 | Derive Axiom ~ ax-hvmulass... |
| axhvdistr1-zf 31024 | Derive Axiom ~ ax-hvdistr1... |
| axhvdistr2-zf 31025 | Derive Axiom ~ ax-hvdistr2... |
| axhvmul0-zf 31026 | Derive Axiom ~ ax-hvmul0 f... |
| axhfi-zf 31027 | Derive Axiom ~ ax-hfi from... |
| axhis1-zf 31028 | Derive Axiom ~ ax-his1 fro... |
| axhis2-zf 31029 | Derive Axiom ~ ax-his2 fro... |
| axhis3-zf 31030 | Derive Axiom ~ ax-his3 fro... |
| axhis4-zf 31031 | Derive Axiom ~ ax-his4 fro... |
| axhcompl-zf 31032 | Derive Axiom ~ ax-hcompl f... |
| hvmulex 31045 | The Hilbert space scalar p... |
| hvaddcl 31046 | Closure of vector addition... |
| hvmulcl 31047 | Closure of scalar multipli... |
| hvmulcli 31048 | Closure inference for scal... |
| hvsubf 31049 | Mapping domain and codomai... |
| hvsubval 31050 | Value of vector subtractio... |
| hvsubcl 31051 | Closure of vector subtract... |
| hvaddcli 31052 | Closure of vector addition... |
| hvcomi 31053 | Commutation of vector addi... |
| hvsubvali 31054 | Value of vector subtractio... |
| hvsubcli 31055 | Closure of vector subtract... |
| ifhvhv0 31056 | Prove ` if ( A e. ~H , A ,... |
| hvaddlid 31057 | Addition with the zero vec... |
| hvmul0 31058 | Scalar multiplication with... |
| hvmul0or 31059 | If a scalar product is zer... |
| hvsubid 31060 | Subtraction of a vector fr... |
| hvnegid 31061 | Addition of negative of a ... |
| hv2neg 31062 | Two ways to express the ne... |
| hvaddlidi 31063 | Addition with the zero vec... |
| hvnegidi 31064 | Addition of negative of a ... |
| hv2negi 31065 | Two ways to express the ne... |
| hvm1neg 31066 | Convert minus one times a ... |
| hvaddsubval 31067 | Value of vector addition i... |
| hvadd32 31068 | Commutative/associative la... |
| hvadd12 31069 | Commutative/associative la... |
| hvadd4 31070 | Hilbert vector space addit... |
| hvsub4 31071 | Hilbert vector space addit... |
| hvaddsub12 31072 | Commutative/associative la... |
| hvpncan 31073 | Addition/subtraction cance... |
| hvpncan2 31074 | Addition/subtraction cance... |
| hvaddsubass 31075 | Associativity of sum and d... |
| hvpncan3 31076 | Subtraction and addition o... |
| hvmulcom 31077 | Scalar multiplication comm... |
| hvsubass 31078 | Hilbert vector space assoc... |
| hvsub32 31079 | Hilbert vector space commu... |
| hvmulassi 31080 | Scalar multiplication asso... |
| hvmulcomi 31081 | Scalar multiplication comm... |
| hvmul2negi 31082 | Double negative in scalar ... |
| hvsubdistr1 31083 | Scalar multiplication dist... |
| hvsubdistr2 31084 | Scalar multiplication dist... |
| hvdistr1i 31085 | Scalar multiplication dist... |
| hvsubdistr1i 31086 | Scalar multiplication dist... |
| hvassi 31087 | Hilbert vector space assoc... |
| hvadd32i 31088 | Hilbert vector space commu... |
| hvsubassi 31089 | Hilbert vector space assoc... |
| hvsub32i 31090 | Hilbert vector space commu... |
| hvadd12i 31091 | Hilbert vector space commu... |
| hvadd4i 31092 | Hilbert vector space addit... |
| hvsubsub4i 31093 | Hilbert vector space addit... |
| hvsubsub4 31094 | Hilbert vector space addit... |
| hv2times 31095 | Two times a vector. (Cont... |
| hvnegdii 31096 | Distribution of negative o... |
| hvsubeq0i 31097 | If the difference between ... |
| hvsubcan2i 31098 | Vector cancellation law. ... |
| hvaddcani 31099 | Cancellation law for vecto... |
| hvsubaddi 31100 | Relationship between vecto... |
| hvnegdi 31101 | Distribution of negative o... |
| hvsubeq0 31102 | If the difference between ... |
| hvaddeq0 31103 | If the sum of two vectors ... |
| hvaddcan 31104 | Cancellation law for vecto... |
| hvaddcan2 31105 | Cancellation law for vecto... |
| hvmulcan 31106 | Cancellation law for scala... |
| hvmulcan2 31107 | Cancellation law for scala... |
| hvsubcan 31108 | Cancellation law for vecto... |
| hvsubcan2 31109 | Cancellation law for vecto... |
| hvsub0 31110 | Subtraction of a zero vect... |
| hvsubadd 31111 | Relationship between vecto... |
| hvaddsub4 31112 | Hilbert vector space addit... |
| hicl 31114 | Closure of inner product. ... |
| hicli 31115 | Closure inference for inne... |
| his5 31120 | Associative law for inner ... |
| his52 31121 | Associative law for inner ... |
| his35 31122 | Move scalar multiplication... |
| his35i 31123 | Move scalar multiplication... |
| his7 31124 | Distributive law for inner... |
| hiassdi 31125 | Distributive/associative l... |
| his2sub 31126 | Distributive law for inner... |
| his2sub2 31127 | Distributive law for inner... |
| hire 31128 | A necessary and sufficient... |
| hiidrcl 31129 | Real closure of inner prod... |
| hi01 31130 | Inner product with the 0 v... |
| hi02 31131 | Inner product with the 0 v... |
| hiidge0 31132 | Inner product with self is... |
| his6 31133 | Zero inner product with se... |
| his1i 31134 | Conjugate law for inner pr... |
| abshicom 31135 | Commuted inner products ha... |
| hial0 31136 | A vector whose inner produ... |
| hial02 31137 | A vector whose inner produ... |
| hisubcomi 31138 | Two vector subtractions si... |
| hi2eq 31139 | Lemma used to prove equali... |
| hial2eq 31140 | Two vectors whose inner pr... |
| hial2eq2 31141 | Two vectors whose inner pr... |
| orthcom 31142 | Orthogonality commutes. (... |
| normlem0 31143 | Lemma used to derive prope... |
| normlem1 31144 | Lemma used to derive prope... |
| normlem2 31145 | Lemma used to derive prope... |
| normlem3 31146 | Lemma used to derive prope... |
| normlem4 31147 | Lemma used to derive prope... |
| normlem5 31148 | Lemma used to derive prope... |
| normlem6 31149 | Lemma used to derive prope... |
| normlem7 31150 | Lemma used to derive prope... |
| normlem8 31151 | Lemma used to derive prope... |
| normlem9 31152 | Lemma used to derive prope... |
| normlem7tALT 31153 | Lemma used to derive prope... |
| bcseqi 31154 | Equality case of Bunjakova... |
| normlem9at 31155 | Lemma used to derive prope... |
| dfhnorm2 31156 | Alternate definition of th... |
| normf 31157 | The norm function maps fro... |
| normval 31158 | The value of the norm of a... |
| normcl 31159 | Real closure of the norm o... |
| normge0 31160 | The norm of a vector is no... |
| normgt0 31161 | The norm of nonzero vector... |
| norm0 31162 | The norm of a zero vector.... |
| norm-i 31163 | Theorem 3.3(i) of [Beran] ... |
| normne0 31164 | A norm is nonzero iff its ... |
| normcli 31165 | Real closure of the norm o... |
| normsqi 31166 | The square of a norm. (Co... |
| norm-i-i 31167 | Theorem 3.3(i) of [Beran] ... |
| normsq 31168 | The square of a norm. (Co... |
| normsub0i 31169 | Two vectors are equal iff ... |
| normsub0 31170 | Two vectors are equal iff ... |
| norm-ii-i 31171 | Triangle inequality for no... |
| norm-ii 31172 | Triangle inequality for no... |
| norm-iii-i 31173 | Theorem 3.3(iii) of [Beran... |
| norm-iii 31174 | Theorem 3.3(iii) of [Beran... |
| normsubi 31175 | Negative doesn't change th... |
| normpythi 31176 | Analogy to Pythagorean the... |
| normsub 31177 | Swapping order of subtract... |
| normneg 31178 | The norm of a vector equal... |
| normpyth 31179 | Analogy to Pythagorean the... |
| normpyc 31180 | Corollary to Pythagorean t... |
| norm3difi 31181 | Norm of differences around... |
| norm3adifii 31182 | Norm of differences around... |
| norm3lem 31183 | Lemma involving norm of di... |
| norm3dif 31184 | Norm of differences around... |
| norm3dif2 31185 | Norm of differences around... |
| norm3lemt 31186 | Lemma involving norm of di... |
| norm3adifi 31187 | Norm of differences around... |
| normpari 31188 | Parallelogram law for norm... |
| normpar 31189 | Parallelogram law for norm... |
| normpar2i 31190 | Corollary of parallelogram... |
| polid2i 31191 | Generalized polarization i... |
| polidi 31192 | Polarization identity. Re... |
| polid 31193 | Polarization identity. Re... |
| hilablo 31194 | Hilbert space vector addit... |
| hilid 31195 | The group identity element... |
| hilvc 31196 | Hilbert space is a complex... |
| hilnormi 31197 | Hilbert space norm in term... |
| hilhhi 31198 | Deduce the structure of Hi... |
| hhnv 31199 | Hilbert space is a normed ... |
| hhva 31200 | The group (addition) opera... |
| hhba 31201 | The base set of Hilbert sp... |
| hh0v 31202 | The zero vector of Hilbert... |
| hhsm 31203 | The scalar product operati... |
| hhvs 31204 | The vector subtraction ope... |
| hhnm 31205 | The norm function of Hilbe... |
| hhims 31206 | The induced metric of Hilb... |
| hhims2 31207 | Hilbert space distance met... |
| hhmet 31208 | The induced metric of Hilb... |
| hhxmet 31209 | The induced metric of Hilb... |
| hhmetdval 31210 | Value of the distance func... |
| hhip 31211 | The inner product operatio... |
| hhph 31212 | The Hilbert space of the H... |
| bcsiALT 31213 | Bunjakovaskij-Cauchy-Schwa... |
| bcsiHIL 31214 | Bunjakovaskij-Cauchy-Schwa... |
| bcs 31215 | Bunjakovaskij-Cauchy-Schwa... |
| bcs2 31216 | Corollary of the Bunjakova... |
| bcs3 31217 | Corollary of the Bunjakova... |
| hcau 31218 | Member of the set of Cauch... |
| hcauseq 31219 | A Cauchy sequences on a Hi... |
| hcaucvg 31220 | A Cauchy sequence on a Hil... |
| seq1hcau 31221 | A sequence on a Hilbert sp... |
| hlimi 31222 | Express the predicate: Th... |
| hlimseqi 31223 | A sequence with a limit on... |
| hlimveci 31224 | Closure of the limit of a ... |
| hlimconvi 31225 | Convergence of a sequence ... |
| hlim2 31226 | The limit of a sequence on... |
| hlimadd 31227 | Limit of the sum of two se... |
| hilmet 31228 | The Hilbert space norm det... |
| hilxmet 31229 | The Hilbert space norm det... |
| hilmetdval 31230 | Value of the distance func... |
| hilims 31231 | Hilbert space distance met... |
| hhcau 31232 | The Cauchy sequences of Hi... |
| hhlm 31233 | The limit sequences of Hil... |
| hhcmpl 31234 | Lemma used for derivation ... |
| hilcompl 31235 | Lemma used for derivation ... |
| hhcms 31237 | The Hilbert space induced ... |
| hhhl 31238 | The Hilbert space structur... |
| hilcms 31239 | The Hilbert space norm det... |
| hilhl 31240 | The Hilbert space of the H... |
| issh 31242 | Subspace ` H ` of a Hilber... |
| issh2 31243 | Subspace ` H ` of a Hilber... |
| shss 31244 | A subspace is a subset of ... |
| shel 31245 | A member of a subspace of ... |
| shex 31246 | The set of subspaces of a ... |
| shssii 31247 | A closed subspace of a Hil... |
| sheli 31248 | A member of a subspace of ... |
| shelii 31249 | A member of a subspace of ... |
| sh0 31250 | The zero vector belongs to... |
| shaddcl 31251 | Closure of vector addition... |
| shmulcl 31252 | Closure of vector scalar m... |
| issh3 31253 | Subspace ` H ` of a Hilber... |
| shsubcl 31254 | Closure of vector subtract... |
| isch 31256 | Closed subspace ` H ` of a... |
| isch2 31257 | Closed subspace ` H ` of a... |
| chsh 31258 | A closed subspace is a sub... |
| chsssh 31259 | Closed subspaces are subsp... |
| chex 31260 | The set of closed subspace... |
| chshii 31261 | A closed subspace is a sub... |
| ch0 31262 | The zero vector belongs to... |
| chss 31263 | A closed subspace of a Hil... |
| chel 31264 | A member of a closed subsp... |
| chssii 31265 | A closed subspace of a Hil... |
| cheli 31266 | A member of a closed subsp... |
| chelii 31267 | A member of a closed subsp... |
| chlimi 31268 | The limit property of a cl... |
| hlim0 31269 | The zero sequence in Hilbe... |
| hlimcaui 31270 | If a sequence in Hilbert s... |
| hlimf 31271 | Function-like behavior of ... |
| hlimuni 31272 | A Hilbert space sequence c... |
| hlimreui 31273 | The limit of a Hilbert spa... |
| hlimeui 31274 | The limit of a Hilbert spa... |
| isch3 31275 | A Hilbert subspace is clos... |
| chcompl 31276 | Completeness of a closed s... |
| helch 31277 | The Hilbert lattice one (w... |
| ifchhv 31278 | Prove ` if ( A e. CH , A ,... |
| helsh 31279 | Hilbert space is a subspac... |
| shsspwh 31280 | Subspaces are subsets of H... |
| chsspwh 31281 | Closed subspaces are subse... |
| hsn0elch 31282 | The zero subspace belongs ... |
| norm1 31283 | From any nonzero Hilbert s... |
| norm1exi 31284 | A normalized vector exists... |
| norm1hex 31285 | A normalized vector can ex... |
| elch0 31288 | Membership in zero for clo... |
| h0elch 31289 | The zero subspace is a clo... |
| h0elsh 31290 | The zero subspace is a sub... |
| hhssva 31291 | The vector addition operat... |
| hhsssm 31292 | The scalar multiplication ... |
| hhssnm 31293 | The norm operation on a su... |
| issubgoilem 31294 | Lemma for ~ hhssabloilem .... |
| hhssabloilem 31295 | Lemma for ~ hhssabloi . F... |
| hhssabloi 31296 | Abelian group property of ... |
| hhssablo 31297 | Abelian group property of ... |
| hhssnv 31298 | Normed complex vector spac... |
| hhssnvt 31299 | Normed complex vector spac... |
| hhsst 31300 | A member of ` SH ` is a su... |
| hhshsslem1 31301 | Lemma for ~ hhsssh . (Con... |
| hhshsslem2 31302 | Lemma for ~ hhsssh . (Con... |
| hhsssh 31303 | The predicate " ` H ` is a... |
| hhsssh2 31304 | The predicate " ` H ` is a... |
| hhssba 31305 | The base set of a subspace... |
| hhssvs 31306 | The vector subtraction ope... |
| hhssvsf 31307 | Mapping of the vector subt... |
| hhssims 31308 | Induced metric of a subspa... |
| hhssims2 31309 | Induced metric of a subspa... |
| hhssmet 31310 | Induced metric of a subspa... |
| hhssmetdval 31311 | Value of the distance func... |
| hhsscms 31312 | The induced metric of a cl... |
| hhssbnOLD 31313 | Obsolete version of ~ cssb... |
| ocval 31314 | Value of orthogonal comple... |
| ocel 31315 | Membership in orthogonal c... |
| shocel 31316 | Membership in orthogonal c... |
| ocsh 31317 | The orthogonal complement ... |
| shocsh 31318 | The orthogonal complement ... |
| ocss 31319 | An orthogonal complement i... |
| shocss 31320 | An orthogonal complement i... |
| occon 31321 | Contraposition law for ort... |
| occon2 31322 | Double contraposition for ... |
| occon2i 31323 | Double contraposition for ... |
| oc0 31324 | The zero vector belongs to... |
| ocorth 31325 | Members of a subset and it... |
| shocorth 31326 | Members of a subspace and ... |
| ococss 31327 | Inclusion in complement of... |
| shococss 31328 | Inclusion in complement of... |
| shorth 31329 | Members of orthogonal subs... |
| ocin 31330 | Intersection of a Hilbert ... |
| occon3 31331 | Hilbert lattice contraposi... |
| ocnel 31332 | A nonzero vector in the co... |
| chocvali 31333 | Value of the orthogonal co... |
| shuni 31334 | Two subspaces with trivial... |
| chocunii 31335 | Lemma for uniqueness part ... |
| pjhthmo 31336 | Projection Theorem, unique... |
| occllem 31337 | Lemma for ~ occl . (Contr... |
| occl 31338 | Closure of complement of H... |
| shoccl 31339 | Closure of complement of H... |
| choccl 31340 | Closure of complement of H... |
| choccli 31341 | Closure of ` CH ` orthocom... |
| shsval 31346 | Value of subspace sum of t... |
| shsss 31347 | The subspace sum is a subs... |
| shsel 31348 | Membership in the subspace... |
| shsel3 31349 | Membership in the subspace... |
| shseli 31350 | Membership in subspace sum... |
| shscli 31351 | Closure of subspace sum. ... |
| shscl 31352 | Closure of subspace sum. ... |
| shscom 31353 | Commutative law for subspa... |
| shsva 31354 | Vector sum belongs to subs... |
| shsel1 31355 | A subspace sum contains a ... |
| shsel2 31356 | A subspace sum contains a ... |
| shsvs 31357 | Vector subtraction belongs... |
| shsub1 31358 | Subspace sum is an upper b... |
| shsub2 31359 | Subspace sum is an upper b... |
| choc0 31360 | The orthocomplement of the... |
| choc1 31361 | The orthocomplement of the... |
| chocnul 31362 | Orthogonal complement of t... |
| shintcli 31363 | Closure of intersection of... |
| shintcl 31364 | The intersection of a none... |
| chintcli 31365 | The intersection of a none... |
| chintcl 31366 | The intersection (infimum)... |
| spanval 31367 | Value of the linear span o... |
| hsupval 31368 | Value of supremum of set o... |
| chsupval 31369 | The value of the supremum ... |
| spancl 31370 | The span of a subset of Hi... |
| elspancl 31371 | A member of a span is a ve... |
| shsupcl 31372 | Closure of the subspace su... |
| hsupcl 31373 | Closure of supremum of set... |
| chsupcl 31374 | Closure of supremum of sub... |
| hsupss 31375 | Subset relation for suprem... |
| chsupss 31376 | Subset relation for suprem... |
| hsupunss 31377 | The union of a set of Hilb... |
| chsupunss 31378 | The union of a set of clos... |
| spanss2 31379 | A subset of Hilbert space ... |
| shsupunss 31380 | The union of a set of subs... |
| spanid 31381 | A subspace of Hilbert spac... |
| spanss 31382 | Ordering relationship for ... |
| spanssoc 31383 | The span of a subset of Hi... |
| sshjval 31384 | Value of join for subsets ... |
| shjval 31385 | Value of join in ` SH ` . ... |
| chjval 31386 | Value of join in ` CH ` . ... |
| chjvali 31387 | Value of join in ` CH ` . ... |
| sshjval3 31388 | Value of join for subsets ... |
| sshjcl 31389 | Closure of join for subset... |
| shjcl 31390 | Closure of join in ` SH ` ... |
| chjcl 31391 | Closure of join in ` CH ` ... |
| shjcom 31392 | Commutative law for Hilber... |
| shless 31393 | Subset implies subset of s... |
| shlej1 31394 | Add disjunct to both sides... |
| shlej2 31395 | Add disjunct to both sides... |
| shincli 31396 | Closure of intersection of... |
| shscomi 31397 | Commutative law for subspa... |
| shsvai 31398 | Vector sum belongs to subs... |
| shsel1i 31399 | A subspace sum contains a ... |
| shsel2i 31400 | A subspace sum contains a ... |
| shsvsi 31401 | Vector subtraction belongs... |
| shunssi 31402 | Union is smaller than subs... |
| shunssji 31403 | Union is smaller than Hilb... |
| shsleji 31404 | Subspace sum is smaller th... |
| shjcomi 31405 | Commutative law for join i... |
| shsub1i 31406 | Subspace sum is an upper b... |
| shsub2i 31407 | Subspace sum is an upper b... |
| shub1i 31408 | Hilbert lattice join is an... |
| shjcli 31409 | Closure of ` CH ` join. (... |
| shjshcli 31410 | ` SH ` closure of join. (... |
| shlessi 31411 | Subset implies subset of s... |
| shlej1i 31412 | Add disjunct to both sides... |
| shlej2i 31413 | Add disjunct to both sides... |
| shslej 31414 | Subspace sum is smaller th... |
| shincl 31415 | Closure of intersection of... |
| shub1 31416 | Hilbert lattice join is an... |
| shub2 31417 | A subspace is a subset of ... |
| shsidmi 31418 | Idempotent law for Hilbert... |
| shslubi 31419 | The least upper bound law ... |
| shlesb1i 31420 | Hilbert lattice ordering i... |
| shsval2i 31421 | An alternate way to expres... |
| shsval3i 31422 | An alternate way to expres... |
| shmodsi 31423 | The modular law holds for ... |
| shmodi 31424 | The modular law is implied... |
| pjhthlem1 31425 | Lemma for ~ pjhth . (Cont... |
| pjhthlem2 31426 | Lemma for ~ pjhth . (Cont... |
| pjhth 31427 | Projection Theorem: Any H... |
| pjhtheu 31428 | Projection Theorem: Any H... |
| pjhfval 31430 | The value of the projectio... |
| pjhval 31431 | Value of a projection. (C... |
| pjpreeq 31432 | Equality with a projection... |
| pjeq 31433 | Equality with a projection... |
| axpjcl 31434 | Closure of a projection in... |
| pjhcl 31435 | Closure of a projection in... |
| omlsilem 31436 | Lemma for orthomodular law... |
| omlsii 31437 | Subspace inference form of... |
| omlsi 31438 | Subspace form of orthomodu... |
| ococi 31439 | Complement of complement o... |
| ococ 31440 | Complement of complement o... |
| dfch2 31441 | Alternate definition of th... |
| ococin 31442 | The double complement is t... |
| hsupval2 31443 | Alternate definition of su... |
| chsupval2 31444 | The value of the supremum ... |
| sshjval2 31445 | Value of join in the set o... |
| chsupid 31446 | A subspace is the supremum... |
| chsupsn 31447 | Value of supremum of subse... |
| shlub 31448 | Hilbert lattice join is th... |
| shlubi 31449 | Hilbert lattice join is th... |
| pjhtheu2 31450 | Uniqueness of ` y ` for th... |
| pjcli 31451 | Closure of a projection in... |
| pjhcli 31452 | Closure of a projection in... |
| pjpjpre 31453 | Decomposition of a vector ... |
| axpjpj 31454 | Decomposition of a vector ... |
| pjclii 31455 | Closure of a projection in... |
| pjhclii 31456 | Closure of a projection in... |
| pjpj0i 31457 | Decomposition of a vector ... |
| pjpji 31458 | Decomposition of a vector ... |
| pjpjhth 31459 | Projection Theorem: Any H... |
| pjpjhthi 31460 | Projection Theorem: Any H... |
| pjop 31461 | Orthocomplement projection... |
| pjpo 31462 | Projection in terms of ort... |
| pjopi 31463 | Orthocomplement projection... |
| pjpoi 31464 | Projection in terms of ort... |
| pjoc1i 31465 | Projection of a vector in ... |
| pjchi 31466 | Projection of a vector in ... |
| pjoccl 31467 | The part of a vector that ... |
| pjoc1 31468 | Projection of a vector in ... |
| pjomli 31469 | Subspace form of orthomodu... |
| pjoml 31470 | Subspace form of orthomodu... |
| pjococi 31471 | Proof of orthocomplement t... |
| pjoc2i 31472 | Projection of a vector in ... |
| pjoc2 31473 | Projection of a vector in ... |
| sh0le 31474 | The zero subspace is the s... |
| ch0le 31475 | The zero subspace is the s... |
| shle0 31476 | No subspace is smaller tha... |
| chle0 31477 | No Hilbert lattice element... |
| chnlen0 31478 | A Hilbert lattice element ... |
| ch0pss 31479 | The zero subspace is a pro... |
| orthin 31480 | The intersection of orthog... |
| ssjo 31481 | The lattice join of a subs... |
| shne0i 31482 | A nonzero subspace has a n... |
| shs0i 31483 | Hilbert subspace sum with ... |
| shs00i 31484 | Two subspaces are zero iff... |
| ch0lei 31485 | The closed subspace zero i... |
| chle0i 31486 | No Hilbert closed subspace... |
| chne0i 31487 | A nonzero closed subspace ... |
| chocini 31488 | Intersection of a closed s... |
| chj0i 31489 | Join with lattice zero in ... |
| chm1i 31490 | Meet with lattice one in `... |
| chjcli 31491 | Closure of ` CH ` join. (... |
| chsleji 31492 | Subspace sum is smaller th... |
| chseli 31493 | Membership in subspace sum... |
| chincli 31494 | Closure of Hilbert lattice... |
| chsscon3i 31495 | Hilbert lattice contraposi... |
| chsscon1i 31496 | Hilbert lattice contraposi... |
| chsscon2i 31497 | Hilbert lattice contraposi... |
| chcon2i 31498 | Hilbert lattice contraposi... |
| chcon1i 31499 | Hilbert lattice contraposi... |
| chcon3i 31500 | Hilbert lattice contraposi... |
| chunssji 31501 | Union is smaller than ` CH... |
| chjcomi 31502 | Commutative law for join i... |
| chub1i 31503 | ` CH ` join is an upper bo... |
| chub2i 31504 | ` CH ` join is an upper bo... |
| chlubi 31505 | Hilbert lattice join is th... |
| chlubii 31506 | Hilbert lattice join is th... |
| chlej1i 31507 | Add join to both sides of ... |
| chlej2i 31508 | Add join to both sides of ... |
| chlej12i 31509 | Add join to both sides of ... |
| chlejb1i 31510 | Hilbert lattice ordering i... |
| chdmm1i 31511 | De Morgan's law for meet i... |
| chdmm2i 31512 | De Morgan's law for meet i... |
| chdmm3i 31513 | De Morgan's law for meet i... |
| chdmm4i 31514 | De Morgan's law for meet i... |
| chdmj1i 31515 | De Morgan's law for join i... |
| chdmj2i 31516 | De Morgan's law for join i... |
| chdmj3i 31517 | De Morgan's law for join i... |
| chdmj4i 31518 | De Morgan's law for join i... |
| chnlei 31519 | Equivalent expressions for... |
| chjassi 31520 | Associative law for Hilber... |
| chj00i 31521 | Two Hilbert lattice elemen... |
| chjoi 31522 | The join of a closed subsp... |
| chj1i 31523 | Join with Hilbert lattice ... |
| chm0i 31524 | Meet with Hilbert lattice ... |
| chm0 31525 | Meet with Hilbert lattice ... |
| shjshsi 31526 | Hilbert lattice join equal... |
| shjshseli 31527 | A closed subspace sum equa... |
| chne0 31528 | A nonzero closed subspace ... |
| chocin 31529 | Intersection of a closed s... |
| chssoc 31530 | A closed subspace less tha... |
| chj0 31531 | Join with Hilbert lattice ... |
| chslej 31532 | Subspace sum is smaller th... |
| chincl 31533 | Closure of Hilbert lattice... |
| chsscon3 31534 | Hilbert lattice contraposi... |
| chsscon1 31535 | Hilbert lattice contraposi... |
| chsscon2 31536 | Hilbert lattice contraposi... |
| chpsscon3 31537 | Hilbert lattice contraposi... |
| chpsscon1 31538 | Hilbert lattice contraposi... |
| chpsscon2 31539 | Hilbert lattice contraposi... |
| chjcom 31540 | Commutative law for Hilber... |
| chub1 31541 | Hilbert lattice join is gr... |
| chub2 31542 | Hilbert lattice join is gr... |
| chlub 31543 | Hilbert lattice join is th... |
| chlej1 31544 | Add join to both sides of ... |
| chlej2 31545 | Add join to both sides of ... |
| chlejb1 31546 | Hilbert lattice ordering i... |
| chlejb2 31547 | Hilbert lattice ordering i... |
| chnle 31548 | Equivalent expressions for... |
| chjo 31549 | The join of a closed subsp... |
| chabs1 31550 | Hilbert lattice absorption... |
| chabs2 31551 | Hilbert lattice absorption... |
| chabs1i 31552 | Hilbert lattice absorption... |
| chabs2i 31553 | Hilbert lattice absorption... |
| chjidm 31554 | Idempotent law for Hilbert... |
| chjidmi 31555 | Idempotent law for Hilbert... |
| chj12i 31556 | A rearrangement of Hilbert... |
| chj4i 31557 | Rearrangement of the join ... |
| chjjdiri 31558 | Hilbert lattice join distr... |
| chdmm1 31559 | De Morgan's law for meet i... |
| chdmm2 31560 | De Morgan's law for meet i... |
| chdmm3 31561 | De Morgan's law for meet i... |
| chdmm4 31562 | De Morgan's law for meet i... |
| chdmj1 31563 | De Morgan's law for join i... |
| chdmj2 31564 | De Morgan's law for join i... |
| chdmj3 31565 | De Morgan's law for join i... |
| chdmj4 31566 | De Morgan's law for join i... |
| chjass 31567 | Associative law for Hilber... |
| chj12 31568 | A rearrangement of Hilbert... |
| chj4 31569 | Rearrangement of the join ... |
| ledii 31570 | An ortholattice is distrib... |
| lediri 31571 | An ortholattice is distrib... |
| lejdii 31572 | An ortholattice is distrib... |
| lejdiri 31573 | An ortholattice is distrib... |
| ledi 31574 | An ortholattice is distrib... |
| spansn0 31575 | The span of the singleton ... |
| span0 31576 | The span of the empty set ... |
| elspani 31577 | Membership in the span of ... |
| spanuni 31578 | The span of a union is the... |
| spanun 31579 | The span of a union is the... |
| sshhococi 31580 | The join of two Hilbert sp... |
| hne0 31581 | Hilbert space has a nonzer... |
| chsup0 31582 | The supremum of the empty ... |
| h1deoi 31583 | Membership in orthocomplem... |
| h1dei 31584 | Membership in 1-dimensiona... |
| h1did 31585 | A generating vector belong... |
| h1dn0 31586 | A nonzero vector generates... |
| h1de2i 31587 | Membership in 1-dimensiona... |
| h1de2bi 31588 | Membership in 1-dimensiona... |
| h1de2ctlem 31589 | Lemma for ~ h1de2ci . (Co... |
| h1de2ci 31590 | Membership in 1-dimensiona... |
| spansni 31591 | The span of a singleton in... |
| elspansni 31592 | Membership in the span of ... |
| spansn 31593 | The span of a singleton in... |
| spansnch 31594 | The span of a Hilbert spac... |
| spansnsh 31595 | The span of a Hilbert spac... |
| spansnchi 31596 | The span of a singleton in... |
| spansnid 31597 | A vector belongs to the sp... |
| spansnmul 31598 | A scalar product with a ve... |
| elspansncl 31599 | A member of a span of a si... |
| elspansn 31600 | Membership in the span of ... |
| elspansn2 31601 | Membership in the span of ... |
| spansncol 31602 | The singletons of collinea... |
| spansneleqi 31603 | Membership relation implie... |
| spansneleq 31604 | Membership relation that i... |
| spansnss 31605 | The span of the singleton ... |
| elspansn3 31606 | A member of the span of th... |
| elspansn4 31607 | A span membership conditio... |
| elspansn5 31608 | A vector belonging to both... |
| spansnss2 31609 | The span of the singleton ... |
| normcan 31610 | Cancellation-type law that... |
| pjspansn 31611 | A projection on the span o... |
| spansnpji 31612 | A subset of Hilbert space ... |
| spanunsni 31613 | The span of the union of a... |
| spanpr 31614 | The span of a pair of vect... |
| h1datomi 31615 | A 1-dimensional subspace i... |
| h1datom 31616 | A 1-dimensional subspace i... |
| cmbr 31618 | Binary relation expressing... |
| pjoml2i 31619 | Variation of orthomodular ... |
| pjoml3i 31620 | Variation of orthomodular ... |
| pjoml4i 31621 | Variation of orthomodular ... |
| pjoml5i 31622 | The orthomodular law. Rem... |
| pjoml6i 31623 | An equivalent of the ortho... |
| cmbri 31624 | Binary relation expressing... |
| cmcmlem 31625 | Commutation is symmetric. ... |
| cmcmi 31626 | Commutation is symmetric. ... |
| cmcm2i 31627 | Commutation with orthocomp... |
| cmcm3i 31628 | Commutation with orthocomp... |
| cmcm4i 31629 | Commutation with orthocomp... |
| cmbr2i 31630 | Alternate definition of th... |
| cmcmii 31631 | Commutation is symmetric. ... |
| cmcm2ii 31632 | Commutation with orthocomp... |
| cmcm3ii 31633 | Commutation with orthocomp... |
| cmbr3i 31634 | Alternate definition for t... |
| cmbr4i 31635 | Alternate definition for t... |
| lecmi 31636 | Comparable Hilbert lattice... |
| lecmii 31637 | Comparable Hilbert lattice... |
| cmj1i 31638 | A Hilbert lattice element ... |
| cmj2i 31639 | A Hilbert lattice element ... |
| cmm1i 31640 | A Hilbert lattice element ... |
| cmm2i 31641 | A Hilbert lattice element ... |
| cmbr3 31642 | Alternate definition for t... |
| cm0 31643 | The zero Hilbert lattice e... |
| cmidi 31644 | The commutes relation is r... |
| pjoml2 31645 | Variation of orthomodular ... |
| pjoml3 31646 | Variation of orthomodular ... |
| pjoml5 31647 | The orthomodular law. Rem... |
| cmcm 31648 | Commutation is symmetric. ... |
| cmcm3 31649 | Commutation with orthocomp... |
| cmcm2 31650 | Commutation with orthocomp... |
| lecm 31651 | Comparable Hilbert lattice... |
| fh1 31652 | Foulis-Holland Theorem. I... |
| fh2 31653 | Foulis-Holland Theorem. I... |
| cm2j 31654 | A lattice element that com... |
| fh1i 31655 | Foulis-Holland Theorem. I... |
| fh2i 31656 | Foulis-Holland Theorem. I... |
| fh3i 31657 | Variation of the Foulis-Ho... |
| fh4i 31658 | Variation of the Foulis-Ho... |
| cm2ji 31659 | A lattice element that com... |
| cm2mi 31660 | A lattice element that com... |
| qlax1i 31661 | One of the equations showi... |
| qlax2i 31662 | One of the equations showi... |
| qlax3i 31663 | One of the equations showi... |
| qlax4i 31664 | One of the equations showi... |
| qlax5i 31665 | One of the equations showi... |
| qlaxr1i 31666 | One of the conditions show... |
| qlaxr2i 31667 | One of the conditions show... |
| qlaxr4i 31668 | One of the conditions show... |
| qlaxr5i 31669 | One of the conditions show... |
| qlaxr3i 31670 | A variation of the orthomo... |
| chscllem1 31671 | Lemma for ~ chscl . (Cont... |
| chscllem2 31672 | Lemma for ~ chscl . (Cont... |
| chscllem3 31673 | Lemma for ~ chscl . (Cont... |
| chscllem4 31674 | Lemma for ~ chscl . (Cont... |
| chscl 31675 | The subspace sum of two cl... |
| osumi 31676 | If two closed subspaces of... |
| osumcori 31677 | Corollary of ~ osumi . (C... |
| osumcor2i 31678 | Corollary of ~ osumi , sho... |
| osum 31679 | If two closed subspaces of... |
| spansnji 31680 | The subspace sum of a clos... |
| spansnj 31681 | The subspace sum of a clos... |
| spansnscl 31682 | The subspace sum of a clos... |
| sumspansn 31683 | The sum of two vectors bel... |
| spansnm0i 31684 | The meet of different one-... |
| nonbooli 31685 | A Hilbert lattice with two... |
| spansncvi 31686 | Hilbert space has the cove... |
| spansncv 31687 | Hilbert space has the cove... |
| 5oalem1 31688 | Lemma for orthoarguesian l... |
| 5oalem2 31689 | Lemma for orthoarguesian l... |
| 5oalem3 31690 | Lemma for orthoarguesian l... |
| 5oalem4 31691 | Lemma for orthoarguesian l... |
| 5oalem5 31692 | Lemma for orthoarguesian l... |
| 5oalem6 31693 | Lemma for orthoarguesian l... |
| 5oalem7 31694 | Lemma for orthoarguesian l... |
| 5oai 31695 | Orthoarguesian law 5OA. Th... |
| 3oalem1 31696 | Lemma for 3OA (weak) ortho... |
| 3oalem2 31697 | Lemma for 3OA (weak) ortho... |
| 3oalem3 31698 | Lemma for 3OA (weak) ortho... |
| 3oalem4 31699 | Lemma for 3OA (weak) ortho... |
| 3oalem5 31700 | Lemma for 3OA (weak) ortho... |
| 3oalem6 31701 | Lemma for 3OA (weak) ortho... |
| 3oai 31702 | 3OA (weak) orthoarguesian ... |
| pjorthi 31703 | Projection components on o... |
| pjch1 31704 | Property of identity proje... |
| pjo 31705 | The orthogonal projection.... |
| pjcompi 31706 | Component of a projection.... |
| pjidmi 31707 | A projection is idempotent... |
| pjadjii 31708 | A projection is self-adjoi... |
| pjaddii 31709 | Projection of vector sum i... |
| pjinormii 31710 | The inner product of a pro... |
| pjmulii 31711 | Projection of (scalar) pro... |
| pjsubii 31712 | Projection of vector diffe... |
| pjsslem 31713 | Lemma for subset relations... |
| pjss2i 31714 | Subset relationship for pr... |
| pjssmii 31715 | Projection meet property. ... |
| pjssge0ii 31716 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormii 31717 | Theorem 4.5(v)<->(vi) of [... |
| pjcji 31718 | The projection on a subspa... |
| pjadji 31719 | A projection is self-adjoi... |
| pjaddi 31720 | Projection of vector sum i... |
| pjinormi 31721 | The inner product of a pro... |
| pjsubi 31722 | Projection of vector diffe... |
| pjmuli 31723 | Projection of scalar produ... |
| pjige0i 31724 | The inner product of a pro... |
| pjige0 31725 | The inner product of a pro... |
| pjcjt2 31726 | The projection on a subspa... |
| pj0i 31727 | The projection of the zero... |
| pjch 31728 | Projection of a vector in ... |
| pjid 31729 | The projection of a vector... |
| pjvec 31730 | The set of vectors belongi... |
| pjocvec 31731 | The set of vectors belongi... |
| pjocini 31732 | Membership of projection i... |
| pjini 31733 | Membership of projection i... |
| pjjsi 31734 | A sufficient condition for... |
| pjfni 31735 | Functionality of a project... |
| pjrni 31736 | The range of a projection.... |
| pjfoi 31737 | A projection maps onto its... |
| pjfi 31738 | The mapping of a projectio... |
| pjvi 31739 | The value of a projection ... |
| pjhfo 31740 | A projection maps onto its... |
| pjrn 31741 | The range of a projection.... |
| pjhf 31742 | The mapping of a projectio... |
| pjfn 31743 | Functionality of a project... |
| pjsumi 31744 | The projection on a subspa... |
| pj11i 31745 | One-to-one correspondence ... |
| pjdsi 31746 | Vector decomposition into ... |
| pjds3i 31747 | Vector decomposition into ... |
| pj11 31748 | One-to-one correspondence ... |
| pjmfn 31749 | Functionality of the proje... |
| pjmf1 31750 | The projector function map... |
| pjoi0 31751 | The inner product of proje... |
| pjoi0i 31752 | The inner product of proje... |
| pjopythi 31753 | Pythagorean theorem for pr... |
| pjopyth 31754 | Pythagorean theorem for pr... |
| pjnormi 31755 | The norm of the projection... |
| pjpythi 31756 | Pythagorean theorem for pr... |
| pjneli 31757 | If a vector does not belon... |
| pjnorm 31758 | The norm of the projection... |
| pjpyth 31759 | Pythagorean theorem for pr... |
| pjnel 31760 | If a vector does not belon... |
| pjnorm2 31761 | A vector belongs to the su... |
| mayete3i 31762 | Mayet's equation E_3. Par... |
| mayetes3i 31763 | Mayet's equation E^*_3, de... |
| hosmval 31769 | Value of the sum of two Hi... |
| hommval 31770 | Value of the scalar produc... |
| hodmval 31771 | Value of the difference of... |
| hfsmval 31772 | Value of the sum of two Hi... |
| hfmmval 31773 | Value of the scalar produc... |
| hosval 31774 | Value of the sum of two Hi... |
| homval 31775 | Value of the scalar produc... |
| hodval 31776 | Value of the difference of... |
| hfsval 31777 | Value of the sum of two Hi... |
| hfmval 31778 | Value of the scalar produc... |
| hoscl 31779 | Closure of the sum of two ... |
| homcl 31780 | Closure of the scalar prod... |
| hodcl 31781 | Closure of the difference ... |
| ho0val 31784 | Value of the zero Hilbert ... |
| ho0f 31785 | Functionality of the zero ... |
| df0op2 31786 | Alternate definition of Hi... |
| dfiop2 31787 | Alternate definition of Hi... |
| hoif 31788 | Functionality of the Hilbe... |
| hoival 31789 | The value of the Hilbert s... |
| hoico1 31790 | Composition with the Hilbe... |
| hoico2 31791 | Composition with the Hilbe... |
| hoaddcl 31792 | The sum of Hilbert space o... |
| homulcl 31793 | The scalar product of a Hi... |
| hoeq 31794 | Equality of Hilbert space ... |
| hoeqi 31795 | Equality of Hilbert space ... |
| hoscli 31796 | Closure of Hilbert space o... |
| hodcli 31797 | Closure of Hilbert space o... |
| hocoi 31798 | Composition of Hilbert spa... |
| hococli 31799 | Closure of composition of ... |
| hocofi 31800 | Mapping of composition of ... |
| hocofni 31801 | Functionality of compositi... |
| hoaddcli 31802 | Mapping of sum of Hilbert ... |
| hosubcli 31803 | Mapping of difference of H... |
| hoaddfni 31804 | Functionality of sum of Hi... |
| hosubfni 31805 | Functionality of differenc... |
| hoaddcomi 31806 | Commutativity of sum of Hi... |
| hosubcl 31807 | Mapping of difference of H... |
| hoaddcom 31808 | Commutativity of sum of Hi... |
| hodsi 31809 | Relationship between Hilbe... |
| hoaddassi 31810 | Associativity of sum of Hi... |
| hoadd12i 31811 | Commutative/associative la... |
| hoadd32i 31812 | Commutative/associative la... |
| hocadddiri 31813 | Distributive law for Hilbe... |
| hocsubdiri 31814 | Distributive law for Hilbe... |
| ho2coi 31815 | Double composition of Hilb... |
| hoaddass 31816 | Associativity of sum of Hi... |
| hoadd32 31817 | Commutative/associative la... |
| hoadd4 31818 | Rearrangement of 4 terms i... |
| hocsubdir 31819 | Distributive law for Hilbe... |
| hoaddridi 31820 | Sum of a Hilbert space ope... |
| hodidi 31821 | Difference of a Hilbert sp... |
| ho0coi 31822 | Composition of the zero op... |
| hoid1i 31823 | Composition of Hilbert spa... |
| hoid1ri 31824 | Composition of Hilbert spa... |
| hoaddrid 31825 | Sum of a Hilbert space ope... |
| hodid 31826 | Difference of a Hilbert sp... |
| hon0 31827 | A Hilbert space operator i... |
| hodseqi 31828 | Subtraction and addition o... |
| ho0subi 31829 | Subtraction of Hilbert spa... |
| honegsubi 31830 | Relationship between Hilbe... |
| ho0sub 31831 | Subtraction of Hilbert spa... |
| hosubid1 31832 | The zero operator subtract... |
| honegsub 31833 | Relationship between Hilbe... |
| homullid 31834 | An operator equals its sca... |
| homco1 31835 | Associative law for scalar... |
| homulass 31836 | Scalar product associative... |
| hoadddi 31837 | Scalar product distributiv... |
| hoadddir 31838 | Scalar product reverse dis... |
| homul12 31839 | Swap first and second fact... |
| honegneg 31840 | Double negative of a Hilbe... |
| hosubneg 31841 | Relationship between opera... |
| hosubdi 31842 | Scalar product distributiv... |
| honegdi 31843 | Distribution of negative o... |
| honegsubdi 31844 | Distribution of negative o... |
| honegsubdi2 31845 | Distribution of negative o... |
| hosubsub2 31846 | Law for double subtraction... |
| hosub4 31847 | Rearrangement of 4 terms i... |
| hosubadd4 31848 | Rearrangement of 4 terms i... |
| hoaddsubass 31849 | Associative-type law for a... |
| hoaddsub 31850 | Law for operator addition ... |
| hosubsub 31851 | Law for double subtraction... |
| hosubsub4 31852 | Law for double subtraction... |
| ho2times 31853 | Two times a Hilbert space ... |
| hoaddsubassi 31854 | Associativity of sum and d... |
| hoaddsubi 31855 | Law for sum and difference... |
| hosd1i 31856 | Hilbert space operator sum... |
| hosd2i 31857 | Hilbert space operator sum... |
| hopncani 31858 | Hilbert space operator can... |
| honpcani 31859 | Hilbert space operator can... |
| hosubeq0i 31860 | If the difference between ... |
| honpncani 31861 | Hilbert space operator can... |
| ho01i 31862 | A condition implying that ... |
| ho02i 31863 | A condition implying that ... |
| hoeq1 31864 | A condition implying that ... |
| hoeq2 31865 | A condition implying that ... |
| adjmo 31866 | Every Hilbert space operat... |
| adjsym 31867 | Symmetry property of an ad... |
| eigrei 31868 | A necessary and sufficient... |
| eigre 31869 | A necessary and sufficient... |
| eigposi 31870 | A sufficient condition (fi... |
| eigorthi 31871 | A necessary and sufficient... |
| eigorth 31872 | A necessary and sufficient... |
| nmopval 31890 | Value of the norm of a Hil... |
| elcnop 31891 | Property defining a contin... |
| ellnop 31892 | Property defining a linear... |
| lnopf 31893 | A linear Hilbert space ope... |
| elbdop 31894 | Property defining a bounde... |
| bdopln 31895 | A bounded linear Hilbert s... |
| bdopf 31896 | A bounded linear Hilbert s... |
| nmopsetretALT 31897 | The set in the supremum of... |
| nmopsetretHIL 31898 | The set in the supremum of... |
| nmopsetn0 31899 | The set in the supremum of... |
| nmopxr 31900 | The norm of a Hilbert spac... |
| nmoprepnf 31901 | The norm of a Hilbert spac... |
| nmopgtmnf 31902 | The norm of a Hilbert spac... |
| nmopreltpnf 31903 | The norm of a Hilbert spac... |
| nmopre 31904 | The norm of a bounded oper... |
| elbdop2 31905 | Property defining a bounde... |
| elunop 31906 | Property defining a unitar... |
| elhmop 31907 | Property defining a Hermit... |
| hmopf 31908 | A Hermitian operator is a ... |
| hmopex 31909 | The class of Hermitian ope... |
| nmfnval 31910 | Value of the norm of a Hil... |
| nmfnsetre 31911 | The set in the supremum of... |
| nmfnsetn0 31912 | The set in the supremum of... |
| nmfnxr 31913 | The norm of any Hilbert sp... |
| nmfnrepnf 31914 | The norm of a Hilbert spac... |
| nlfnval 31915 | Value of the null space of... |
| elcnfn 31916 | Property defining a contin... |
| ellnfn 31917 | Property defining a linear... |
| lnfnf 31918 | A linear Hilbert space fun... |
| dfadj2 31919 | Alternate definition of th... |
| funadj 31920 | Functionality of the adjoi... |
| dmadjss 31921 | The domain of the adjoint ... |
| dmadjop 31922 | A member of the domain of ... |
| adjeu 31923 | Elementhood in the domain ... |
| adjval 31924 | Value of the adjoint funct... |
| adjval2 31925 | Value of the adjoint funct... |
| cnvadj 31926 | The adjoint function equal... |
| funcnvadj 31927 | The converse of the adjoin... |
| adj1o 31928 | The adjoint function maps ... |
| dmadjrn 31929 | The adjoint of an operator... |
| eigvecval 31930 | The set of eigenvectors of... |
| eigvalfval 31931 | The eigenvalues of eigenve... |
| specval 31932 | The value of the spectrum ... |
| speccl 31933 | The spectrum of an operato... |
| hhlnoi 31934 | The linear operators of Hi... |
| hhnmoi 31935 | The norm of an operator in... |
| hhbloi 31936 | A bounded linear operator ... |
| hh0oi 31937 | The zero operator in Hilbe... |
| hhcno 31938 | The continuous operators o... |
| hhcnf 31939 | The continuous functionals... |
| dmadjrnb 31940 | The adjoint of an operator... |
| nmoplb 31941 | A lower bound for an opera... |
| nmopub 31942 | An upper bound for an oper... |
| nmopub2tALT 31943 | An upper bound for an oper... |
| nmopub2tHIL 31944 | An upper bound for an oper... |
| nmopge0 31945 | The norm of any Hilbert sp... |
| nmopgt0 31946 | A linear Hilbert space ope... |
| cnopc 31947 | Basic continuity property ... |
| lnopl 31948 | Basic linearity property o... |
| unop 31949 | Basic inner product proper... |
| unopf1o 31950 | A unitary operator in Hilb... |
| unopnorm 31951 | A unitary operator is idem... |
| cnvunop 31952 | The inverse (converse) of ... |
| unopadj 31953 | The inverse (converse) of ... |
| unoplin 31954 | A unitary operator is line... |
| counop 31955 | The composition of two uni... |
| hmop 31956 | Basic inner product proper... |
| hmopre 31957 | The inner product of the v... |
| nmfnlb 31958 | A lower bound for a functi... |
| nmfnleub 31959 | An upper bound for the nor... |
| nmfnleub2 31960 | An upper bound for the nor... |
| nmfnge0 31961 | The norm of any Hilbert sp... |
| elnlfn 31962 | Membership in the null spa... |
| elnlfn2 31963 | Membership in the null spa... |
| cnfnc 31964 | Basic continuity property ... |
| lnfnl 31965 | Basic linearity property o... |
| adjcl 31966 | Closure of the adjoint of ... |
| adj1 31967 | Property of an adjoint Hil... |
| adj2 31968 | Property of an adjoint Hil... |
| adjeq 31969 | A property that determines... |
| adjadj 31970 | Double adjoint. Theorem 3... |
| adjvalval 31971 | Value of the value of the ... |
| unopadj2 31972 | The adjoint of a unitary o... |
| hmopadj 31973 | A Hermitian operator is se... |
| hmdmadj 31974 | Every Hermitian operator h... |
| hmopadj2 31975 | An operator is Hermitian i... |
| hmoplin 31976 | A Hermitian operator is li... |
| brafval 31977 | The bra of a vector, expre... |
| braval 31978 | A bra-ket juxtaposition, e... |
| braadd 31979 | Linearity property of bra ... |
| bramul 31980 | Linearity property of bra ... |
| brafn 31981 | The bra function is a func... |
| bralnfn 31982 | The Dirac bra function is ... |
| bracl 31983 | Closure of the bra functio... |
| bra0 31984 | The Dirac bra of the zero ... |
| brafnmul 31985 | Anti-linearity property of... |
| kbfval 31986 | The outer product of two v... |
| kbop 31987 | The outer product of two v... |
| kbval 31988 | The value of the operator ... |
| kbmul 31989 | Multiplication property of... |
| kbpj 31990 | If a vector ` A ` has norm... |
| eleigvec 31991 | Membership in the set of e... |
| eleigvec2 31992 | Membership in the set of e... |
| eleigveccl 31993 | Closure of an eigenvector ... |
| eigvalval 31994 | The eigenvalue of an eigen... |
| eigvalcl 31995 | An eigenvalue is a complex... |
| eigvec1 31996 | Property of an eigenvector... |
| eighmre 31997 | The eigenvalues of a Hermi... |
| eighmorth 31998 | Eigenvectors of a Hermitia... |
| nmopnegi 31999 | Value of the norm of the n... |
| lnop0 32000 | The value of a linear Hilb... |
| lnopmul 32001 | Multiplicative property of... |
| lnopli 32002 | Basic scalar product prope... |
| lnopfi 32003 | A linear Hilbert space ope... |
| lnop0i 32004 | The value of a linear Hilb... |
| lnopaddi 32005 | Additive property of a lin... |
| lnopmuli 32006 | Multiplicative property of... |
| lnopaddmuli 32007 | Sum/product property of a ... |
| lnopsubi 32008 | Subtraction property for a... |
| lnopsubmuli 32009 | Subtraction/product proper... |
| lnopmulsubi 32010 | Product/subtraction proper... |
| homco2 32011 | Move a scalar product out ... |
| idunop 32012 | The identity function (res... |
| 0cnop 32013 | The identically zero funct... |
| 0cnfn 32014 | The identically zero funct... |
| idcnop 32015 | The identity function (res... |
| idhmop 32016 | The Hilbert space identity... |
| 0hmop 32017 | The identically zero funct... |
| 0lnop 32018 | The identically zero funct... |
| 0lnfn 32019 | The identically zero funct... |
| nmop0 32020 | The norm of the zero opera... |
| nmfn0 32021 | The norm of the identicall... |
| hmopbdoptHIL 32022 | A Hermitian operator is a ... |
| hoddii 32023 | Distributive law for Hilbe... |
| hoddi 32024 | Distributive law for Hilbe... |
| nmop0h 32025 | The norm of any operator o... |
| idlnop 32026 | The identity function (res... |
| 0bdop 32027 | The identically zero opera... |
| adj0 32028 | Adjoint of the zero operat... |
| nmlnop0iALT 32029 | A linear operator with a z... |
| nmlnop0iHIL 32030 | A linear operator with a z... |
| nmlnopgt0i 32031 | A linear Hilbert space ope... |
| nmlnop0 32032 | A linear operator with a z... |
| nmlnopne0 32033 | A linear operator with a n... |
| lnopmi 32034 | The scalar product of a li... |
| lnophsi 32035 | The sum of two linear oper... |
| lnophdi 32036 | The difference of two line... |
| lnopcoi 32037 | The composition of two lin... |
| lnopco0i 32038 | The composition of a linea... |
| lnopeq0lem1 32039 | Lemma for ~ lnopeq0i . Ap... |
| lnopeq0lem2 32040 | Lemma for ~ lnopeq0i . (C... |
| lnopeq0i 32041 | A condition implying that ... |
| lnopeqi 32042 | Two linear Hilbert space o... |
| lnopeq 32043 | Two linear Hilbert space o... |
| lnopunilem1 32044 | Lemma for ~ lnopunii . (C... |
| lnopunilem2 32045 | Lemma for ~ lnopunii . (C... |
| lnopunii 32046 | If a linear operator (whos... |
| elunop2 32047 | An operator is unitary iff... |
| nmopun 32048 | Norm of a unitary Hilbert ... |
| unopbd 32049 | A unitary operator is a bo... |
| lnophmlem1 32050 | Lemma for ~ lnophmi . (Co... |
| lnophmlem2 32051 | Lemma for ~ lnophmi . (Co... |
| lnophmi 32052 | A linear operator is Hermi... |
| lnophm 32053 | A linear operator is Hermi... |
| hmops 32054 | The sum of two Hermitian o... |
| hmopm 32055 | The scalar product of a He... |
| hmopd 32056 | The difference of two Herm... |
| hmopco 32057 | The composition of two com... |
| nmbdoplbi 32058 | A lower bound for the norm... |
| nmbdoplb 32059 | A lower bound for the norm... |
| nmcexi 32060 | Lemma for ~ nmcopexi and ~... |
| nmcopexi 32061 | The norm of a continuous l... |
| nmcoplbi 32062 | A lower bound for the norm... |
| nmcopex 32063 | The norm of a continuous l... |
| nmcoplb 32064 | A lower bound for the norm... |
| nmophmi 32065 | The norm of the scalar pro... |
| bdophmi 32066 | The scalar product of a bo... |
| lnconi 32067 | Lemma for ~ lnopconi and ~... |
| lnopconi 32068 | A condition equivalent to ... |
| lnopcon 32069 | A condition equivalent to ... |
| lnopcnbd 32070 | A linear operator is conti... |
| lncnopbd 32071 | A continuous linear operat... |
| lncnbd 32072 | A continuous linear operat... |
| lnopcnre 32073 | A linear operator is conti... |
| lnfnli 32074 | Basic property of a linear... |
| lnfnfi 32075 | A linear Hilbert space fun... |
| lnfn0i 32076 | The value of a linear Hilb... |
| lnfnaddi 32077 | Additive property of a lin... |
| lnfnmuli 32078 | Multiplicative property of... |
| lnfnaddmuli 32079 | Sum/product property of a ... |
| lnfnsubi 32080 | Subtraction property for a... |
| lnfn0 32081 | The value of a linear Hilb... |
| lnfnmul 32082 | Multiplicative property of... |
| nmbdfnlbi 32083 | A lower bound for the norm... |
| nmbdfnlb 32084 | A lower bound for the norm... |
| nmcfnexi 32085 | The norm of a continuous l... |
| nmcfnlbi 32086 | A lower bound for the norm... |
| nmcfnex 32087 | The norm of a continuous l... |
| nmcfnlb 32088 | A lower bound of the norm ... |
| lnfnconi 32089 | A condition equivalent to ... |
| lnfncon 32090 | A condition equivalent to ... |
| lnfncnbd 32091 | A linear functional is con... |
| imaelshi 32092 | The image of a subspace un... |
| rnelshi 32093 | The range of a linear oper... |
| nlelshi 32094 | The null space of a linear... |
| nlelchi 32095 | The null space of a contin... |
| riesz3i 32096 | A continuous linear functi... |
| riesz4i 32097 | A continuous linear functi... |
| riesz4 32098 | A continuous linear functi... |
| riesz1 32099 | Part 1 of the Riesz repres... |
| riesz2 32100 | Part 2 of the Riesz repres... |
| cnlnadjlem1 32101 | Lemma for ~ cnlnadji (Theo... |
| cnlnadjlem2 32102 | Lemma for ~ cnlnadji . ` G... |
| cnlnadjlem3 32103 | Lemma for ~ cnlnadji . By... |
| cnlnadjlem4 32104 | Lemma for ~ cnlnadji . Th... |
| cnlnadjlem5 32105 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem6 32106 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem7 32107 | Lemma for ~ cnlnadji . He... |
| cnlnadjlem8 32108 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem9 32109 | Lemma for ~ cnlnadji . ` F... |
| cnlnadji 32110 | Every continuous linear op... |
| cnlnadjeui 32111 | Every continuous linear op... |
| cnlnadjeu 32112 | Every continuous linear op... |
| cnlnadj 32113 | Every continuous linear op... |
| cnlnssadj 32114 | Every continuous linear Hi... |
| bdopssadj 32115 | Every bounded linear Hilbe... |
| bdopadj 32116 | Every bounded linear Hilbe... |
| adjbdln 32117 | The adjoint of a bounded l... |
| adjbdlnb 32118 | An operator is bounded and... |
| adjbd1o 32119 | The mapping of adjoints of... |
| adjlnop 32120 | The adjoint of an operator... |
| adjsslnop 32121 | Every operator with an adj... |
| nmopadjlei 32122 | Property of the norm of an... |
| nmopadjlem 32123 | Lemma for ~ nmopadji . (C... |
| nmopadji 32124 | Property of the norm of an... |
| adjeq0 32125 | An operator is zero iff it... |
| adjmul 32126 | The adjoint of the scalar ... |
| adjadd 32127 | The adjoint of the sum of ... |
| nmoptrii 32128 | Triangle inequality for th... |
| nmopcoi 32129 | Upper bound for the norm o... |
| bdophsi 32130 | The sum of two bounded lin... |
| bdophdi 32131 | The difference between two... |
| bdopcoi 32132 | The composition of two bou... |
| nmoptri2i 32133 | Triangle-type inequality f... |
| adjcoi 32134 | The adjoint of a compositi... |
| nmopcoadji 32135 | The norm of an operator co... |
| nmopcoadj2i 32136 | The norm of an operator co... |
| nmopcoadj0i 32137 | An operator composed with ... |
| unierri 32138 | If we approximate a chain ... |
| branmfn 32139 | The norm of the bra functi... |
| brabn 32140 | The bra of a vector is a b... |
| rnbra 32141 | The set of bras equals the... |
| bra11 32142 | The bra function maps vect... |
| bracnln 32143 | A bra is a continuous line... |
| cnvbraval 32144 | Value of the converse of t... |
| cnvbracl 32145 | Closure of the converse of... |
| cnvbrabra 32146 | The converse bra of the br... |
| bracnvbra 32147 | The bra of the converse br... |
| bracnlnval 32148 | The vector that a continuo... |
| cnvbramul 32149 | Multiplication property of... |
| kbass1 32150 | Dirac bra-ket associative ... |
| kbass2 32151 | Dirac bra-ket associative ... |
| kbass3 32152 | Dirac bra-ket associative ... |
| kbass4 32153 | Dirac bra-ket associative ... |
| kbass5 32154 | Dirac bra-ket associative ... |
| kbass6 32155 | Dirac bra-ket associative ... |
| leopg 32156 | Ordering relation for posi... |
| leop 32157 | Ordering relation for oper... |
| leop2 32158 | Ordering relation for oper... |
| leop3 32159 | Operator ordering in terms... |
| leoppos 32160 | Binary relation defining a... |
| leoprf2 32161 | The ordering relation for ... |
| leoprf 32162 | The ordering relation for ... |
| leopsq 32163 | The square of a Hermitian ... |
| 0leop 32164 | The zero operator is a pos... |
| idleop 32165 | The identity operator is a... |
| leopadd 32166 | The sum of two positive op... |
| leopmuli 32167 | The scalar product of a no... |
| leopmul 32168 | The scalar product of a po... |
| leopmul2i 32169 | Scalar product applied to ... |
| leoptri 32170 | The positive operator orde... |
| leoptr 32171 | The positive operator orde... |
| leopnmid 32172 | A bounded Hermitian operat... |
| nmopleid 32173 | A nonzero, bounded Hermiti... |
| opsqrlem1 32174 | Lemma for opsqri . (Contr... |
| opsqrlem2 32175 | Lemma for opsqri . ` F `` ... |
| opsqrlem3 32176 | Lemma for opsqri . (Contr... |
| opsqrlem4 32177 | Lemma for opsqri . (Contr... |
| opsqrlem5 32178 | Lemma for opsqri . (Contr... |
| opsqrlem6 32179 | Lemma for opsqri . (Contr... |
| pjhmopi 32180 | A projector is a Hermitian... |
| pjlnopi 32181 | A projector is a linear op... |
| pjnmopi 32182 | The operator norm of a pro... |
| pjbdlni 32183 | A projector is a bounded l... |
| pjhmop 32184 | A projection is a Hermitia... |
| hmopidmchi 32185 | An idempotent Hermitian op... |
| hmopidmpji 32186 | An idempotent Hermitian op... |
| hmopidmch 32187 | An idempotent Hermitian op... |
| hmopidmpj 32188 | An idempotent Hermitian op... |
| pjsdii 32189 | Distributive law for Hilbe... |
| pjddii 32190 | Distributive law for Hilbe... |
| pjsdi2i 32191 | Chained distributive law f... |
| pjcoi 32192 | Composition of projections... |
| pjcocli 32193 | Closure of composition of ... |
| pjcohcli 32194 | Closure of composition of ... |
| pjadjcoi 32195 | Adjoint of composition of ... |
| pjcofni 32196 | Functionality of compositi... |
| pjss1coi 32197 | Subset relationship for pr... |
| pjss2coi 32198 | Subset relationship for pr... |
| pjssmi 32199 | Projection meet property. ... |
| pjssge0i 32200 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormi 32201 | Theorem 4.5(v)<->(vi) of [... |
| pjnormssi 32202 | Theorem 4.5(i)<->(vi) of [... |
| pjorthcoi 32203 | Composition of projections... |
| pjscji 32204 | The projection of orthogon... |
| pjssumi 32205 | The projection on a subspa... |
| pjssposi 32206 | Projector ordering can be ... |
| pjordi 32207 | The definition of projecto... |
| pjssdif2i 32208 | The projection subspace of... |
| pjssdif1i 32209 | A necessary and sufficient... |
| pjimai 32210 | The image of a projection.... |
| pjidmcoi 32211 | A projection is idempotent... |
| pjoccoi 32212 | Composition of projections... |
| pjtoi 32213 | Subspace sum of projection... |
| pjoci 32214 | Projection of orthocomplem... |
| pjidmco 32215 | A projection operator is i... |
| dfpjop 32216 | Definition of projection o... |
| pjhmopidm 32217 | Two ways to express the se... |
| elpjidm 32218 | A projection operator is i... |
| elpjhmop 32219 | A projection operator is H... |
| 0leopj 32220 | A projector is a positive ... |
| pjadj2 32221 | A projector is self-adjoin... |
| pjadj3 32222 | A projector is self-adjoin... |
| elpjch 32223 | Reconstruction of the subs... |
| elpjrn 32224 | Reconstruction of the subs... |
| pjinvari 32225 | A closed subspace ` H ` wi... |
| pjin1i 32226 | Lemma for Theorem 1.22 of ... |
| pjin2i 32227 | Lemma for Theorem 1.22 of ... |
| pjin3i 32228 | Lemma for Theorem 1.22 of ... |
| pjclem1 32229 | Lemma for projection commu... |
| pjclem2 32230 | Lemma for projection commu... |
| pjclem3 32231 | Lemma for projection commu... |
| pjclem4a 32232 | Lemma for projection commu... |
| pjclem4 32233 | Lemma for projection commu... |
| pjci 32234 | Two subspaces commute iff ... |
| pjcmul1i 32235 | A necessary and sufficient... |
| pjcmul2i 32236 | The projection subspace of... |
| pjcohocli 32237 | Closure of composition of ... |
| pjadj2coi 32238 | Adjoint of double composit... |
| pj2cocli 32239 | Closure of double composit... |
| pj3lem1 32240 | Lemma for projection tripl... |
| pj3si 32241 | Stronger projection triple... |
| pj3i 32242 | Projection triplet theorem... |
| pj3cor1i 32243 | Projection triplet corolla... |
| pjs14i 32244 | Theorem S-14 of Watanabe, ... |
| isst 32247 | Property of a state. (Con... |
| ishst 32248 | Property of a complex Hilb... |
| sticl 32249 | ` [ 0 , 1 ] ` closure of t... |
| stcl 32250 | Real closure of the value ... |
| hstcl 32251 | Closure of the value of a ... |
| hst1a 32252 | Unit value of a Hilbert-sp... |
| hstel2 32253 | Properties of a Hilbert-sp... |
| hstorth 32254 | Orthogonality property of ... |
| hstosum 32255 | Orthogonal sum property of... |
| hstoc 32256 | Sum of a Hilbert-space-val... |
| hstnmoc 32257 | Sum of norms of a Hilbert-... |
| stge0 32258 | The value of a state is no... |
| stle1 32259 | The value of a state is le... |
| hstle1 32260 | The norm of the value of a... |
| hst1h 32261 | The norm of a Hilbert-spac... |
| hst0h 32262 | The norm of a Hilbert-spac... |
| hstpyth 32263 | Pythagorean property of a ... |
| hstle 32264 | Ordering property of a Hil... |
| hstles 32265 | Ordering property of a Hil... |
| hstoh 32266 | A Hilbert-space-valued sta... |
| hst0 32267 | A Hilbert-space-valued sta... |
| sthil 32268 | The value of a state at th... |
| stj 32269 | The value of a state on a ... |
| sto1i 32270 | The state of a subspace pl... |
| sto2i 32271 | The state of the orthocomp... |
| stge1i 32272 | If a state is greater than... |
| stle0i 32273 | If a state is less than or... |
| stlei 32274 | Ordering law for states. ... |
| stlesi 32275 | Ordering law for states. ... |
| stji1i 32276 | Join of components of Sasa... |
| stm1i 32277 | State of component of unit... |
| stm1ri 32278 | State of component of unit... |
| stm1addi 32279 | Sum of states whose meet i... |
| staddi 32280 | If the sum of 2 states is ... |
| stm1add3i 32281 | Sum of states whose meet i... |
| stadd3i 32282 | If the sum of 3 states is ... |
| st0 32283 | The state of the zero subs... |
| strlem1 32284 | Lemma for strong state the... |
| strlem2 32285 | Lemma for strong state the... |
| strlem3a 32286 | Lemma for strong state the... |
| strlem3 32287 | Lemma for strong state the... |
| strlem4 32288 | Lemma for strong state the... |
| strlem5 32289 | Lemma for strong state the... |
| strlem6 32290 | Lemma for strong state the... |
| stri 32291 | Strong state theorem. The... |
| strb 32292 | Strong state theorem (bidi... |
| hstrlem2 32293 | Lemma for strong set of CH... |
| hstrlem3a 32294 | Lemma for strong set of CH... |
| hstrlem3 32295 | Lemma for strong set of CH... |
| hstrlem4 32296 | Lemma for strong set of CH... |
| hstrlem5 32297 | Lemma for strong set of CH... |
| hstrlem6 32298 | Lemma for strong set of CH... |
| hstri 32299 | Hilbert space admits a str... |
| hstrbi 32300 | Strong CH-state theorem (b... |
| largei 32301 | A Hilbert lattice admits a... |
| jplem1 32302 | Lemma for Jauch-Piron theo... |
| jplem2 32303 | Lemma for Jauch-Piron theo... |
| jpi 32304 | The function ` S ` , that ... |
| golem1 32305 | Lemma for Godowski's equat... |
| golem2 32306 | Lemma for Godowski's equat... |
| goeqi 32307 | Godowski's equation, shown... |
| stcltr1i 32308 | Property of a strong class... |
| stcltr2i 32309 | Property of a strong class... |
| stcltrlem1 32310 | Lemma for strong classical... |
| stcltrlem2 32311 | Lemma for strong classical... |
| stcltrthi 32312 | Theorem for classically st... |
| cvbr 32316 | Binary relation expressing... |
| cvbr2 32317 | Binary relation expressing... |
| cvcon3 32318 | Contraposition law for the... |
| cvpss 32319 | The covers relation implie... |
| cvnbtwn 32320 | The covers relation implie... |
| cvnbtwn2 32321 | The covers relation implie... |
| cvnbtwn3 32322 | The covers relation implie... |
| cvnbtwn4 32323 | The covers relation implie... |
| cvnsym 32324 | The covers relation is not... |
| cvnref 32325 | The covers relation is not... |
| cvntr 32326 | The covers relation is not... |
| spansncv2 32327 | Hilbert space has the cove... |
| mdbr 32328 | Binary relation expressing... |
| mdi 32329 | Consequence of the modular... |
| mdbr2 32330 | Binary relation expressing... |
| mdbr3 32331 | Binary relation expressing... |
| mdbr4 32332 | Binary relation expressing... |
| dmdbr 32333 | Binary relation expressing... |
| dmdmd 32334 | The dual modular pair prop... |
| mddmd 32335 | The modular pair property ... |
| dmdi 32336 | Consequence of the dual mo... |
| dmdbr2 32337 | Binary relation expressing... |
| dmdi2 32338 | Consequence of the dual mo... |
| dmdbr3 32339 | Binary relation expressing... |
| dmdbr4 32340 | Binary relation expressing... |
| dmdi4 32341 | Consequence of the dual mo... |
| dmdbr5 32342 | Binary relation expressing... |
| mddmd2 32343 | Relationship between modul... |
| mdsl0 32344 | A sublattice condition tha... |
| ssmd1 32345 | Ordering implies the modul... |
| ssmd2 32346 | Ordering implies the modul... |
| ssdmd1 32347 | Ordering implies the dual ... |
| ssdmd2 32348 | Ordering implies the dual ... |
| dmdsl3 32349 | Sublattice mapping for a d... |
| mdsl3 32350 | Sublattice mapping for a m... |
| mdslle1i 32351 | Order preservation of the ... |
| mdslle2i 32352 | Order preservation of the ... |
| mdslj1i 32353 | Join preservation of the o... |
| mdslj2i 32354 | Meet preservation of the r... |
| mdsl1i 32355 | If the modular pair proper... |
| mdsl2i 32356 | If the modular pair proper... |
| mdsl2bi 32357 | If the modular pair proper... |
| cvmdi 32358 | The covering property impl... |
| mdslmd1lem1 32359 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem2 32360 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem3 32361 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem4 32362 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1i 32363 | Preservation of the modula... |
| mdslmd2i 32364 | Preservation of the modula... |
| mdsldmd1i 32365 | Preservation of the dual m... |
| mdslmd3i 32366 | Modular pair conditions th... |
| mdslmd4i 32367 | Modular pair condition tha... |
| csmdsymi 32368 | Cross-symmetry implies M-s... |
| mdexchi 32369 | An exchange lemma for modu... |
| cvmd 32370 | The covering property impl... |
| cvdmd 32371 | The covering property impl... |
| ela 32373 | Atoms in a Hilbert lattice... |
| elat2 32374 | Expanded membership relati... |
| elatcv0 32375 | A Hilbert lattice element ... |
| atcv0 32376 | An atom covers the zero su... |
| atssch 32377 | Atoms are a subset of the ... |
| atelch 32378 | An atom is a Hilbert latti... |
| atne0 32379 | An atom is not the Hilbert... |
| atss 32380 | A lattice element smaller ... |
| atsseq 32381 | Two atoms in a subset rela... |
| atcveq0 32382 | A Hilbert lattice element ... |
| h1da 32383 | A 1-dimensional subspace i... |
| spansna 32384 | The span of the singleton ... |
| sh1dle 32385 | A 1-dimensional subspace i... |
| ch1dle 32386 | A 1-dimensional subspace i... |
| atom1d 32387 | The 1-dimensional subspace... |
| superpos 32388 | Superposition Principle. ... |
| chcv1 32389 | The Hilbert lattice has th... |
| chcv2 32390 | The Hilbert lattice has th... |
| chjatom 32391 | The join of a closed subsp... |
| shatomici 32392 | The lattice of Hilbert sub... |
| hatomici 32393 | The Hilbert lattice is ato... |
| hatomic 32394 | A Hilbert lattice is atomi... |
| shatomistici 32395 | The lattice of Hilbert sub... |
| hatomistici 32396 | ` CH ` is atomistic, i.e. ... |
| chpssati 32397 | Two Hilbert lattice elemen... |
| chrelati 32398 | The Hilbert lattice is rel... |
| chrelat2i 32399 | A consequence of relative ... |
| cvati 32400 | If a Hilbert lattice eleme... |
| cvbr4i 32401 | An alternate way to expres... |
| cvexchlem 32402 | Lemma for ~ cvexchi . (Co... |
| cvexchi 32403 | The Hilbert lattice satisf... |
| chrelat2 32404 | A consequence of relative ... |
| chrelat3 32405 | A consequence of relative ... |
| chrelat3i 32406 | A consequence of the relat... |
| chrelat4i 32407 | A consequence of relative ... |
| cvexch 32408 | The Hilbert lattice satisf... |
| cvp 32409 | The Hilbert lattice satisf... |
| atnssm0 32410 | The meet of a Hilbert latt... |
| atnemeq0 32411 | The meet of distinct atoms... |
| atssma 32412 | The meet with an atom's su... |
| atcv0eq 32413 | Two atoms covering the zer... |
| atcv1 32414 | Two atoms covering the zer... |
| atexch 32415 | The Hilbert lattice satisf... |
| atomli 32416 | An assertion holding in at... |
| atoml2i 32417 | An assertion holding in at... |
| atordi 32418 | An ordering law for a Hilb... |
| atcvatlem 32419 | Lemma for ~ atcvati . (Co... |
| atcvati 32420 | A nonzero Hilbert lattice ... |
| atcvat2i 32421 | A Hilbert lattice element ... |
| atord 32422 | An ordering law for a Hilb... |
| atcvat2 32423 | A Hilbert lattice element ... |
| chirredlem1 32424 | Lemma for ~ chirredi . (C... |
| chirredlem2 32425 | Lemma for ~ chirredi . (C... |
| chirredlem3 32426 | Lemma for ~ chirredi . (C... |
| chirredlem4 32427 | Lemma for ~ chirredi . (C... |
| chirredi 32428 | The Hilbert lattice is irr... |
| chirred 32429 | The Hilbert lattice is irr... |
| atcvat3i 32430 | A condition implying that ... |
| atcvat4i 32431 | A condition implying exist... |
| atdmd 32432 | Two Hilbert lattice elemen... |
| atmd 32433 | Two Hilbert lattice elemen... |
| atmd2 32434 | Two Hilbert lattice elemen... |
| atabsi 32435 | Absorption of an incompara... |
| atabs2i 32436 | Absorption of an incompara... |
| mdsymlem1 32437 | Lemma for ~ mdsymi . (Con... |
| mdsymlem2 32438 | Lemma for ~ mdsymi . (Con... |
| mdsymlem3 32439 | Lemma for ~ mdsymi . (Con... |
| mdsymlem4 32440 | Lemma for ~ mdsymi . This... |
| mdsymlem5 32441 | Lemma for ~ mdsymi . (Con... |
| mdsymlem6 32442 | Lemma for ~ mdsymi . This... |
| mdsymlem7 32443 | Lemma for ~ mdsymi . Lemm... |
| mdsymlem8 32444 | Lemma for ~ mdsymi . Lemm... |
| mdsymi 32445 | M-symmetry of the Hilbert ... |
| mdsym 32446 | M-symmetry of the Hilbert ... |
| dmdsym 32447 | Dual M-symmetry of the Hil... |
| atdmd2 32448 | Two Hilbert lattice elemen... |
| sumdmdii 32449 | If the subspace sum of two... |
| cmmdi 32450 | Commuting subspaces form a... |
| cmdmdi 32451 | Commuting subspaces form a... |
| sumdmdlem 32452 | Lemma for ~ sumdmdi . The... |
| sumdmdlem2 32453 | Lemma for ~ sumdmdi . (Co... |
| sumdmdi 32454 | The subspace sum of two Hi... |
| dmdbr4ati 32455 | Dual modular pair property... |
| dmdbr5ati 32456 | Dual modular pair property... |
| dmdbr6ati 32457 | Dual modular pair property... |
| dmdbr7ati 32458 | Dual modular pair property... |
| mdoc1i 32459 | Orthocomplements form a mo... |
| mdoc2i 32460 | Orthocomplements form a mo... |
| dmdoc1i 32461 | Orthocomplements form a du... |
| dmdoc2i 32462 | Orthocomplements form a du... |
| mdcompli 32463 | A condition equivalent to ... |
| dmdcompli 32464 | A condition equivalent to ... |
| mddmdin0i 32465 | If dual modular implies mo... |
| cdjreui 32466 | A member of the sum of dis... |
| cdj1i 32467 | Two ways to express " ` A ... |
| cdj3lem1 32468 | A property of " ` A ` and ... |
| cdj3lem2 32469 | Lemma for ~ cdj3i . Value... |
| cdj3lem2a 32470 | Lemma for ~ cdj3i . Closu... |
| cdj3lem2b 32471 | Lemma for ~ cdj3i . The f... |
| cdj3lem3 32472 | Lemma for ~ cdj3i . Value... |
| cdj3lem3a 32473 | Lemma for ~ cdj3i . Closu... |
| cdj3lem3b 32474 | Lemma for ~ cdj3i . The s... |
| cdj3i 32475 | Two ways to express " ` A ... |
| The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here | |
| mathbox 32476 | (_This theorem is a dummy ... |
| sa-abvi 32477 | A theorem about the univer... |
| xfree 32478 | A partial converse to ~ 19... |
| xfree2 32479 | A partial converse to ~ 19... |
| addltmulALT 32480 | A proof readability experi... |
| an42ds 32481 | Inference exchanging the l... |
| an52ds 32482 | Inference exchanging the l... |
| an62ds 32483 | Inference exchanging the l... |
| an72ds 32484 | Inference exchanging the l... |
| an82ds 32485 | Inference exchanging the l... |
| bian1d 32486 | Adding a superfluous conju... |
| bian1dOLD 32487 | Obsolete version of ~ bian... |
| bibiad 32488 | Eliminate an hypothesis ` ... |
| orim12da 32489 | Deduce a disjunction from ... |
| or3di 32490 | Distributive law for disju... |
| or3dir 32491 | Distributive law for disju... |
| 3o1cs 32492 | Deduction eliminating disj... |
| 3o2cs 32493 | Deduction eliminating disj... |
| 3o3cs 32494 | Deduction eliminating disj... |
| 13an22anass 32495 | Associative law for four c... |
| sbc2iedf 32496 | Conversion of implicit sub... |
| rspc2daf 32497 | Double restricted speciali... |
| ralcom4f 32498 | Commutation of restricted ... |
| rexcom4f 32499 | Commutation of restricted ... |
| 19.9d2rf 32500 | A deduction version of one... |
| 19.9d2r 32501 | A deduction version of one... |
| r19.29ffa 32502 | A commonly used pattern ba... |
| n0limd 32503 | Deduction rule for nonempt... |
| eqtrb 32504 | A transposition of equalit... |
| eqelbid 32505 | A variable elimination law... |
| opsbc2ie 32506 | Conversion of implicit sub... |
| opreu2reuALT 32507 | Correspondence between uni... |
| 2reucom 32510 | Double restricted existent... |
| 2reu2rex1 32511 | Double restricted existent... |
| 2reureurex 32512 | Double restricted existent... |
| 2reu2reu2 32513 | Double restricted existent... |
| opreu2reu1 32514 | Equivalent definition of t... |
| sq2reunnltb 32515 | There exists a unique deco... |
| addsqnot2reu 32516 | For each complex number ` ... |
| sbceqbidf 32517 | Equality theorem for class... |
| sbcies 32518 | A special version of class... |
| mo5f 32519 | Alternate definition of "a... |
| nmo 32520 | Negation of "at most one".... |
| reuxfrdf 32521 | Transfer existential uniqu... |
| rexunirn 32522 | Restricted existential qua... |
| rmoxfrd 32523 | Transfer "at most one" res... |
| rmoun 32524 | "At most one" restricted e... |
| rmounid 32525 | A case where an "at most o... |
| riotaeqbidva 32526 | Equivalent wff's yield equ... |
| dmrab 32527 | Domain of a restricted cla... |
| difrab2 32528 | Difference of two restrict... |
| rabexgfGS 32529 | Separation Scheme in terms... |
| rabsnel 32530 | Truth implied by equality ... |
| rabsspr 32531 | Conditions for a restricte... |
| rabsstp 32532 | Conditions for a restricte... |
| 3unrab 32533 | Union of three restricted ... |
| foresf1o 32534 | From a surjective function... |
| rabfodom 32535 | Domination relation for re... |
| rabrexfi 32536 | Conditions for a class abs... |
| abrexdomjm 32537 | An indexed set is dominate... |
| abrexdom2jm 32538 | An indexed set is dominate... |
| abrexexd 32539 | Existence of a class abstr... |
| elabreximd 32540 | Class substitution in an i... |
| elabreximdv 32541 | Class substitution in an i... |
| abrexss 32542 | A necessary condition for ... |
| elunsn 32543 | Elementhood to a union wit... |
| nelun 32544 | Negated membership for a u... |
| snsssng 32545 | If a singleton is a subset... |
| n0nsnel 32546 | If a class with one elemen... |
| inin 32547 | Intersection with an inter... |
| inindif 32548 | See ~ inundif . (Contribu... |
| difininv 32549 | Condition for the intersec... |
| difeq 32550 | Rewriting an equation with... |
| eqdif 32551 | If both set differences of... |
| indifbi 32552 | Two ways to express equali... |
| diffib 32553 | Case where ~ diffi is a bi... |
| difxp1ss 32554 | Difference law for Cartesi... |
| difxp2ss 32555 | Difference law for Cartesi... |
| indifundif 32556 | A remarkable equation with... |
| elpwincl1 32557 | Closure of intersection wi... |
| elpwdifcl 32558 | Closure of class differenc... |
| elpwiuncl 32559 | Closure of indexed union w... |
| elpreq 32560 | Equality wihin a pair. (C... |
| nelpr 32561 | A set ` A ` not in a pair ... |
| inpr0 32562 | Rewrite an empty intersect... |
| neldifpr1 32563 | The first element of a pai... |
| neldifpr2 32564 | The second element of a pa... |
| unidifsnel 32565 | The other element of a pai... |
| unidifsnne 32566 | The other element of a pai... |
| ifeqeqx 32567 | An equality theorem tailor... |
| elimifd 32568 | Elimination of a condition... |
| elim2if 32569 | Elimination of two conditi... |
| elim2ifim 32570 | Elimination of two conditi... |
| ifeq3da 32571 | Given an expression ` C ` ... |
| ifnetrue 32572 | Deduce truth from a condit... |
| ifnefals 32573 | Deduce falsehood from a co... |
| ifnebib 32574 | The converse of ~ ifbi hol... |
| uniinn0 32575 | Sufficient and necessary c... |
| uniin1 32576 | Union of intersection. Ge... |
| uniin2 32577 | Union of intersection. Ge... |
| difuncomp 32578 | Express a class difference... |
| elpwunicl 32579 | Closure of a set union wit... |
| cbviunf 32580 | Rule used to change the bo... |
| iuneq12daf 32581 | Equality deduction for ind... |
| iunin1f 32582 | Indexed union of intersect... |
| ssiun3 32583 | Subset equivalence for an ... |
| ssiun2sf 32584 | Subset relationship for an... |
| iuninc 32585 | The union of an increasing... |
| iundifdifd 32586 | The intersection of a set ... |
| iundifdif 32587 | The intersection of a set ... |
| iunrdx 32588 | Re-index an indexed union.... |
| iunpreima 32589 | Preimage of an indexed uni... |
| iunrnmptss 32590 | A subset relation for an i... |
| iunxunsn 32591 | Appending a set to an inde... |
| iunxunpr 32592 | Appending two sets to an i... |
| iinabrex 32593 | Rewriting an indexed inter... |
| disjnf 32594 | In case ` x ` is not free ... |
| cbvdisjf 32595 | Change bound variables in ... |
| disjss1f 32596 | A subset of a disjoint col... |
| disjeq1f 32597 | Equality theorem for disjo... |
| disjxun0 32598 | Simplify a disjoint union.... |
| disjdifprg 32599 | A trivial partition into a... |
| disjdifprg2 32600 | A trivial partition of a s... |
| disji2f 32601 | Property of a disjoint col... |
| disjif 32602 | Property of a disjoint col... |
| disjorf 32603 | Two ways to say that a col... |
| disjorsf 32604 | Two ways to say that a col... |
| disjif2 32605 | Property of a disjoint col... |
| disjabrex 32606 | Rewriting a disjoint colle... |
| disjabrexf 32607 | Rewriting a disjoint colle... |
| disjpreima 32608 | A preimage of a disjoint s... |
| disjrnmpt 32609 | Rewriting a disjoint colle... |
| disjin 32610 | If a collection is disjoin... |
| disjin2 32611 | If a collection is disjoin... |
| disjxpin 32612 | Derive a disjunction over ... |
| iundisjf 32613 | Rewrite a countable union ... |
| iundisj2f 32614 | A disjoint union is disjoi... |
| disjrdx 32615 | Re-index a disjunct collec... |
| disjex 32616 | Two ways to say that two c... |
| disjexc 32617 | A variant of ~ disjex , ap... |
| disjunsn 32618 | Append an element to a dis... |
| disjun0 32619 | Adding the empty element p... |
| disjiunel 32620 | A set of elements B of a d... |
| disjuniel 32621 | A set of elements B of a d... |
| xpdisjres 32622 | Restriction of a constant ... |
| opeldifid 32623 | Ordered pair elementhood o... |
| difres 32624 | Case when class difference... |
| imadifxp 32625 | Image of the difference wi... |
| relfi 32626 | A relation (set) is finite... |
| 0res 32627 | Restriction of the empty f... |
| fcoinver 32628 | Build an equivalence relat... |
| fcoinvbr 32629 | Binary relation for the eq... |
| copsex2dv 32630 | Implicit substitution dedu... |
| brab2d 32631 | Expressing that two sets a... |
| brabgaf 32632 | The law of concretion for ... |
| brelg 32633 | Two things in a binary rel... |
| br8d 32634 | Substitution for an eight-... |
| opabdm 32635 | Domain of an ordered-pair ... |
| opabrn 32636 | Range of an ordered-pair c... |
| opabssi 32637 | Sufficient condition for a... |
| opabid2ss 32638 | One direction of ~ opabid2... |
| ssrelf 32639 | A subclass relationship de... |
| eqrelrd2 32640 | A version of ~ eqrelrdv2 w... |
| erbr3b 32641 | Biconditional for equivale... |
| iunsnima 32642 | Image of a singleton by an... |
| iunsnima2 32643 | Version of ~ iunsnima with... |
| feq2dd 32644 | Equality deduction for fun... |
| feq3dd 32645 | Equality deduction for fun... |
| ac6sf2 32646 | Alternate version of ~ ac6... |
| fnresin 32647 | Restriction of a function ... |
| f1o3d 32648 | Describe an implicit one-t... |
| eldmne0 32649 | A function of nonempty dom... |
| f1rnen 32650 | Equinumerosity of the rang... |
| rinvf1o 32651 | Sufficient conditions for ... |
| fresf1o 32652 | Conditions for a restricti... |
| nfpconfp 32653 | The set of fixed points of... |
| fmptco1f1o 32654 | The action of composing (t... |
| cofmpt2 32655 | Express composition of a m... |
| f1mptrn 32656 | Express injection for a ma... |
| dfimafnf 32657 | Alternate definition of th... |
| funimass4f 32658 | Membership relation for th... |
| suppss2f 32659 | Show that the support of a... |
| ofrn 32660 | The range of the function ... |
| ofrn2 32661 | The range of the function ... |
| off2 32662 | The function operation pro... |
| ofresid 32663 | Applying an operation rest... |
| unipreima 32664 | Preimage of a class union.... |
| opfv 32665 | Value of a function produc... |
| xppreima 32666 | The preimage of a Cartesia... |
| 2ndimaxp 32667 | Image of a cartesian produ... |
| djussxp2 32668 | Stronger version of ~ djus... |
| 2ndresdju 32669 | The ` 2nd ` function restr... |
| 2ndresdjuf1o 32670 | The ` 2nd ` function restr... |
| xppreima2 32671 | The preimage of a Cartesia... |
| abfmpunirn 32672 | Membership in a union of a... |
| rabfmpunirn 32673 | Membership in a union of a... |
| abfmpeld 32674 | Membership in an element o... |
| abfmpel 32675 | Membership in an element o... |
| fmptdF 32676 | Domain and codomain of the... |
| fmptcof2 32677 | Composition of two functio... |
| fcomptf 32678 | Express composition of two... |
| acunirnmpt 32679 | Axiom of choice for the un... |
| acunirnmpt2 32680 | Axiom of choice for the un... |
| acunirnmpt2f 32681 | Axiom of choice for the un... |
| aciunf1lem 32682 | Choice in an index union. ... |
| aciunf1 32683 | Choice in an index union. ... |
| ofoprabco 32684 | Function operation as a co... |
| ofpreima 32685 | Express the preimage of a ... |
| ofpreima2 32686 | Express the preimage of a ... |
| funcnvmpt 32687 | Condition for a function i... |
| funcnv5mpt 32688 | Two ways to say that a fun... |
| funcnv4mpt 32689 | Two ways to say that a fun... |
| preimane 32690 | Different elements have di... |
| fnpreimac 32691 | Choose a set ` x ` contain... |
| fgreu 32692 | Exactly one point of a fun... |
| fcnvgreu 32693 | If the converse of a relat... |
| rnmposs 32694 | The range of an operation ... |
| mptssALT 32695 | Deduce subset relation of ... |
| dfcnv2 32696 | Alternative definition of ... |
| mpomptxf 32697 | Express a two-argument fun... |
| of0r 32698 | Function operation with th... |
| suppovss 32699 | A bound for the support of... |
| suppiniseg 32700 | Relation between the suppo... |
| fsuppinisegfi 32701 | The initial segment ` ( ``... |
| fressupp 32702 | The restriction of a funct... |
| fdifsuppconst 32703 | A function is a zero const... |
| ressupprn 32704 | The range of a function re... |
| supppreima 32705 | Express the support of a f... |
| fsupprnfi 32706 | Finite support implies fin... |
| mptiffisupp 32707 | Conditions for a mapping f... |
| cosnopne 32708 | Composition of two ordered... |
| cosnop 32709 | Composition of two ordered... |
| cnvprop 32710 | Converse of a pair of orde... |
| brprop 32711 | Binary relation for a pair... |
| mptprop 32712 | Rewrite pairs of ordered p... |
| coprprop 32713 | Composition of two pairs o... |
| gtiso 32714 | Two ways to write a strict... |
| isoun 32715 | Infer an isomorphism from ... |
| disjdsct 32716 | A disjoint collection is d... |
| df1stres 32717 | Definition for a restricti... |
| df2ndres 32718 | Definition for a restricti... |
| 1stpreimas 32719 | The preimage of a singleto... |
| 1stpreima 32720 | The preimage by ` 1st ` is... |
| 2ndpreima 32721 | The preimage by ` 2nd ` is... |
| curry2ima 32722 | The image of a curried fun... |
| preiman0 32723 | The preimage of a nonempty... |
| intimafv 32724 | The intersection of an ima... |
| supssd 32725 | Inequality deduction for s... |
| infssd 32726 | Inequality deduction for i... |
| imafi2 32727 | The image by a finite set ... |
| unifi3 32728 | If a union is finite, then... |
| snct 32729 | A singleton is countable. ... |
| prct 32730 | An unordered pair is count... |
| mpocti 32731 | An operation is countable ... |
| abrexct 32732 | An image set of a countabl... |
| mptctf 32733 | A countable mapping set is... |
| abrexctf 32734 | An image set of a countabl... |
| padct 32735 | Index a countable set with... |
| cnvoprabOLD 32736 | The converse of a class ab... |
| f1od2 32737 | Sufficient condition for a... |
| fcobij 32738 | Composing functions with a... |
| fcobijfs 32739 | Composing finitely support... |
| suppss3 32740 | Deduce a function's suppor... |
| fsuppcurry1 32741 | Finite support of a currie... |
| fsuppcurry2 32742 | Finite support of a currie... |
| offinsupp1 32743 | Finite support for a funct... |
| ffs2 32744 | Rewrite a function's suppo... |
| ffsrn 32745 | The range of a finitely su... |
| resf1o 32746 | Restriction of functions t... |
| maprnin 32747 | Restricting the range of t... |
| fpwrelmapffslem 32748 | Lemma for ~ fpwrelmapffs .... |
| fpwrelmap 32749 | Define a canonical mapping... |
| fpwrelmapffs 32750 | Define a canonical mapping... |
| creq0 32751 | The real representation of... |
| 1nei 32752 | The imaginary unit ` _i ` ... |
| 1neg1t1neg1 32753 | An integer unit times itse... |
| nnmulge 32754 | Multiplying by a positive ... |
| submuladdd 32755 | The product of a differenc... |
| muldivdid 32756 | Distribution of division o... |
| cjsubd 32757 | Complex conjugate distribu... |
| re0cj 32758 | The conjugate of a pure im... |
| quad3d 32759 | Variant of quadratic equat... |
| lt2addrd 32760 | If the right-hand side of ... |
| xrlelttric 32761 | Trichotomy law for extende... |
| xaddeq0 32762 | Two extended reals which a... |
| xrinfm 32763 | The extended real numbers ... |
| le2halvesd 32764 | A sum is less than the who... |
| xraddge02 32765 | A number is less than or e... |
| xrge0addge 32766 | A number is less than or e... |
| xlt2addrd 32767 | If the right-hand side of ... |
| xrsupssd 32768 | Inequality deduction for s... |
| xrge0infss 32769 | Any subset of nonnegative ... |
| xrge0infssd 32770 | Inequality deduction for i... |
| xrge0addcld 32771 | Nonnegative extended reals... |
| xrge0subcld 32772 | Condition for closure of n... |
| infxrge0lb 32773 | A member of a set of nonne... |
| infxrge0glb 32774 | The infimum of a set of no... |
| infxrge0gelb 32775 | The infimum of a set of no... |
| xrofsup 32776 | The supremum is preserved ... |
| supxrnemnf 32777 | The supremum of a nonempty... |
| xnn0gt0 32778 | Nonzero extended nonnegati... |
| xnn01gt 32779 | An extended nonnegative in... |
| nn0xmulclb 32780 | Finite multiplication in t... |
| joiniooico 32781 | Disjoint joining an open i... |
| ubico 32782 | A right-open interval does... |
| xeqlelt 32783 | Equality in terms of 'less... |
| eliccelico 32784 | Relate elementhood to a cl... |
| elicoelioo 32785 | Relate elementhood to a cl... |
| iocinioc2 32786 | Intersection between two o... |
| xrdifh 32787 | Class difference of a half... |
| iocinif 32788 | Relate intersection of two... |
| difioo 32789 | The difference between two... |
| difico 32790 | The difference between two... |
| uzssico 32791 | Upper integer sets are a s... |
| fz2ssnn0 32792 | A finite set of sequential... |
| nndiffz1 32793 | Upper set of the positive ... |
| ssnnssfz 32794 | For any finite subset of `... |
| fzne1 32795 | Elementhood in a finite se... |
| fzm1ne1 32796 | Elementhood of an integer ... |
| fzspl 32797 | Split the last element of ... |
| fzdif2 32798 | Split the last element of ... |
| fzodif2 32799 | Split the last element of ... |
| fzodif1 32800 | Set difference of two half... |
| fzsplit3 32801 | Split a finite interval of... |
| bcm1n 32802 | The proportion of one bino... |
| iundisjfi 32803 | Rewrite a countable union ... |
| iundisj2fi 32804 | A disjoint union is disjoi... |
| iundisjcnt 32805 | Rewrite a countable union ... |
| iundisj2cnt 32806 | A countable disjoint union... |
| fzone1 32807 | Elementhood in a half-open... |
| fzom1ne1 32808 | Elementhood in a half-open... |
| f1ocnt 32809 | Given a countable set ` A ... |
| fz1nnct 32810 | NN and integer ranges star... |
| fz1nntr 32811 | NN and integer ranges star... |
| fzo0opth 32812 | Equality for a half open i... |
| nn0difffzod 32813 | A nonnegative integer that... |
| suppssnn0 32814 | Show that the support of a... |
| hashunif 32815 | The cardinality of a disjo... |
| hashxpe 32816 | The size of the Cartesian ... |
| hashgt1 32817 | Restate "set contains at l... |
| znumd 32818 | Numerator of an integer. ... |
| zdend 32819 | Denominator of an integer.... |
| numdenneg 32820 | Numerator and denominator ... |
| divnumden2 32821 | Calculate the reduced form... |
| expgt0b 32822 | A real number ` A ` raised... |
| nn0split01 32823 | Split 0 and 1 from the non... |
| nn0disj01 32824 | The pair ` { 0 , 1 } ` doe... |
| nnindf 32825 | Principle of Mathematical ... |
| nn0min 32826 | Extracting the minimum pos... |
| subne0nn 32827 | A nonnegative difference i... |
| ltesubnnd 32828 | Subtracting an integer num... |
| fprodeq02 32829 | If one of the factors is z... |
| pr01ssre 32830 | The range of the indicator... |
| fprodex01 32831 | A product of factors equal... |
| prodpr 32832 | A product over a pair is t... |
| prodtp 32833 | A product over a triple is... |
| fsumub 32834 | An upper bound for a term ... |
| fsumiunle 32835 | Upper bound for a sum of n... |
| dfdec100 32836 | Split the hundreds from a ... |
| dp2eq1 32839 | Equality theorem for the d... |
| dp2eq2 32840 | Equality theorem for the d... |
| dp2eq1i 32841 | Equality theorem for the d... |
| dp2eq2i 32842 | Equality theorem for the d... |
| dp2eq12i 32843 | Equality theorem for the d... |
| dp20u 32844 | Add a zero in the tenths (... |
| dp20h 32845 | Add a zero in the unit pla... |
| dp2cl 32846 | Closure for the decimal fr... |
| dp2clq 32847 | Closure for a decimal frac... |
| rpdp2cl 32848 | Closure for a decimal frac... |
| rpdp2cl2 32849 | Closure for a decimal frac... |
| dp2lt10 32850 | Decimal fraction builds re... |
| dp2lt 32851 | Comparing two decimal frac... |
| dp2ltsuc 32852 | Comparing a decimal fracti... |
| dp2ltc 32853 | Comparing two decimal expa... |
| dpval 32856 | Define the value of the de... |
| dpcl 32857 | Prove that the closure of ... |
| dpfrac1 32858 | Prove a simple equivalence... |
| dpval2 32859 | Value of the decimal point... |
| dpval3 32860 | Value of the decimal point... |
| dpmul10 32861 | Multiply by 10 a decimal e... |
| decdiv10 32862 | Divide a decimal number by... |
| dpmul100 32863 | Multiply by 100 a decimal ... |
| dp3mul10 32864 | Multiply by 10 a decimal e... |
| dpmul1000 32865 | Multiply by 1000 a decimal... |
| dpval3rp 32866 | Value of the decimal point... |
| dp0u 32867 | Add a zero in the tenths p... |
| dp0h 32868 | Remove a zero in the units... |
| rpdpcl 32869 | Closure of the decimal poi... |
| dplt 32870 | Comparing two decimal expa... |
| dplti 32871 | Comparing a decimal expans... |
| dpgti 32872 | Comparing a decimal expans... |
| dpltc 32873 | Comparing two decimal inte... |
| dpexpp1 32874 | Add one zero to the mantis... |
| 0dp2dp 32875 | Multiply by 10 a decimal e... |
| dpadd2 32876 | Addition with one decimal,... |
| dpadd 32877 | Addition with one decimal.... |
| dpadd3 32878 | Addition with two decimals... |
| dpmul 32879 | Multiplication with one de... |
| dpmul4 32880 | An upper bound to multipli... |
| threehalves 32881 | Example theorem demonstrat... |
| 1mhdrd 32882 | Example theorem demonstrat... |
| xdivval 32885 | Value of division: the (un... |
| xrecex 32886 | Existence of reciprocal of... |
| xmulcand 32887 | Cancellation law for exten... |
| xreceu 32888 | Existential uniqueness of ... |
| xdivcld 32889 | Closure law for the extend... |
| xdivcl 32890 | Closure law for the extend... |
| xdivmul 32891 | Relationship between divis... |
| rexdiv 32892 | The extended real division... |
| xdivrec 32893 | Relationship between divis... |
| xdivid 32894 | A number divided by itself... |
| xdiv0 32895 | Division into zero is zero... |
| xdiv0rp 32896 | Division into zero is zero... |
| eliccioo 32897 | Membership in a closed int... |
| elxrge02 32898 | Elementhood in the set of ... |
| xdivpnfrp 32899 | Plus infinity divided by a... |
| rpxdivcld 32900 | Closure law for extended d... |
| xrpxdivcld 32901 | Closure law for extended d... |
| wrdfd 32902 | A word is a zero-based seq... |
| wrdres 32903 | Condition for the restrict... |
| wrdsplex 32904 | Existence of a split of a ... |
| wrdfsupp 32905 | A word has finite support.... |
| wrdpmcl 32906 | Closure of a word with per... |
| pfx1s2 32907 | The prefix of length 1 of ... |
| pfxrn2 32908 | The range of a prefix of a... |
| pfxrn3 32909 | Express the range of a pre... |
| pfxf1 32910 | Condition for a prefix to ... |
| s1f1 32911 | Conditions for a length 1 ... |
| s2rnOLD 32912 | Obsolete version of ~ s2rn... |
| s2f1 32913 | Conditions for a length 2 ... |
| s3rnOLD 32914 | Obsolete version of ~ s2rn... |
| s3f1 32915 | Conditions for a length 3 ... |
| s3clhash 32916 | Closure of the words of le... |
| ccatf1 32917 | Conditions for a concatena... |
| ccatdmss 32918 | The domain of a concatenat... |
| pfxlsw2ccat 32919 | Reconstruct a word from it... |
| ccatws1f1o 32920 | Conditions for the concate... |
| ccatws1f1olast 32921 | Two ways to reorder symbol... |
| wrdt2ind 32922 | Perform an induction over ... |
| swrdrn2 32923 | The range of a subword is ... |
| swrdrn3 32924 | Express the range of a sub... |
| swrdf1 32925 | Condition for a subword to... |
| swrdrndisj 32926 | Condition for the range of... |
| splfv3 32927 | Symbols to the right of a ... |
| 1cshid 32928 | Cyclically shifting a sing... |
| cshw1s2 32929 | Cyclically shifting a leng... |
| cshwrnid 32930 | Cyclically shifting a word... |
| cshf1o 32931 | Condition for the cyclic s... |
| ressplusf 32932 | The group operation functi... |
| ressnm 32933 | The norm in a restricted s... |
| abvpropd2 32934 | Weaker version of ~ abvpro... |
| oppgle 32935 | less-than relation of an o... |
| oppgleOLD 32936 | Obsolete version of ~ oppg... |
| oppglt 32937 | less-than relation of an o... |
| ressprs 32938 | The restriction of a prose... |
| oduprs 32939 | Being a proset is a self-d... |
| posrasymb 32940 | A poset ordering is asymet... |
| resspos 32941 | The restriction of a Poset... |
| resstos 32942 | The restriction of a Toset... |
| odutos 32943 | Being a toset is a self-du... |
| tlt2 32944 | In a Toset, two elements m... |
| tlt3 32945 | In a Toset, two elements m... |
| trleile 32946 | In a Toset, two elements m... |
| toslublem 32947 | Lemma for ~ toslub and ~ x... |
| toslub 32948 | In a toset, the lowest upp... |
| tosglblem 32949 | Lemma for ~ tosglb and ~ x... |
| tosglb 32950 | Same theorem as ~ toslub ,... |
| clatp0cl 32951 | The poset zero of a comple... |
| clatp1cl 32952 | The poset one of a complet... |
| mntoval 32957 | Operation value of the mon... |
| ismnt 32958 | Express the statement " ` ... |
| ismntd 32959 | Property of being a monoto... |
| mntf 32960 | A monotone function is a f... |
| mgcoval 32961 | Operation value of the mon... |
| mgcval 32962 | Monotone Galois connection... |
| mgcf1 32963 | The lower adjoint ` F ` of... |
| mgcf2 32964 | The upper adjoint ` G ` of... |
| mgccole1 32965 | An inequality for the kern... |
| mgccole2 32966 | Inequality for the closure... |
| mgcmnt1 32967 | The lower adjoint ` F ` of... |
| mgcmnt2 32968 | The upper adjoint ` G ` of... |
| mgcmntco 32969 | A Galois connection like s... |
| dfmgc2lem 32970 | Lemma for dfmgc2, backward... |
| dfmgc2 32971 | Alternate definition of th... |
| mgcmnt1d 32972 | Galois connection implies ... |
| mgcmnt2d 32973 | Galois connection implies ... |
| mgccnv 32974 | The inverse Galois connect... |
| pwrssmgc 32975 | Given a function ` F ` , e... |
| mgcf1olem1 32976 | Property of a Galois conne... |
| mgcf1olem2 32977 | Property of a Galois conne... |
| mgcf1o 32978 | Given a Galois connection,... |
| ischn 32981 | Property of being a chain.... |
| chnwrd 32982 | A chain is an ordered sequ... |
| chnltm1 32983 | Basic property of a chain.... |
| pfxchn 32984 | A prefix of a chain is sti... |
| chnind 32985 | Induction over a chain. S... |
| chnub 32986 | In a chain, the last eleme... |
| chnlt 32987 | Compare any two elements i... |
| chnso 32988 | A chain induces a total or... |
| xrs0 32991 | The zero of the extended r... |
| xrslt 32992 | The "strictly less than" r... |
| xrsinvgval 32993 | The inversion operation in... |
| xrsmulgzz 32994 | The "multiple" function in... |
| xrstos 32995 | The extended real numbers ... |
| xrsclat 32996 | The extended real numbers ... |
| xrsp0 32997 | The poset 0 of the extende... |
| xrsp1 32998 | The poset 1 of the extende... |
| xrge0base 32999 | The base of the extended n... |
| xrge00 33000 | The zero of the extended n... |
| xrge0plusg 33001 | The additive law of the ex... |
| xrge0le 33002 | The "less than or equal to... |
| xrge0mulgnn0 33003 | The group multiple functio... |
| xrge0addass 33004 | Associativity of extended ... |
| xrge0addgt0 33005 | The sum of nonnegative and... |
| xrge0adddir 33006 | Right-distributivity of ex... |
| xrge0adddi 33007 | Left-distributivity of ext... |
| xrge0npcan 33008 | Extended nonnegative real ... |
| fsumrp0cl 33009 | Closure of a finite sum of... |
| mndcld 33010 | Closure of the operation o... |
| mndassd 33011 | A monoid operation is asso... |
| mndlrinv 33012 | In a monoid, if an element... |
| mndlrinvb 33013 | In a monoid, if an element... |
| mndlactf1 33014 | If an element ` X ` of a m... |
| mndlactfo 33015 | An element ` X ` of a mono... |
| mndractf1 33016 | If an element ` X ` of a m... |
| mndractfo 33017 | An element ` X ` of a mono... |
| mndlactf1o 33018 | An element ` X ` of a mono... |
| mndractf1o 33019 | An element ` X ` of a mono... |
| cmn4d 33020 | Commutative/associative la... |
| cmn246135 33021 | Rearrange terms in a commu... |
| cmn145236 33022 | Rearrange terms in a commu... |
| submcld 33023 | Submonoids are closed unde... |
| abliso 33024 | The image of an Abelian gr... |
| lmhmghmd 33025 | A module homomorphism is a... |
| mhmimasplusg 33026 | Value of the operation of ... |
| lmhmimasvsca 33027 | Value of the scalar produc... |
| grpsubcld 33028 | Closure of group subtracti... |
| subgcld 33029 | A subgroup is closed under... |
| subgsubcld 33030 | A subgroup is closed under... |
| gsumsubg 33031 | The group sum in a subgrou... |
| gsumsra 33032 | The group sum in a subring... |
| gsummpt2co 33033 | Split a finite sum into a ... |
| gsummpt2d 33034 | Express a finite sum over ... |
| lmodvslmhm 33035 | Scalar multiplication in a... |
| gsumvsmul1 33036 | Pull a scalar multiplicati... |
| gsummptres 33037 | Extend a finite group sum ... |
| gsummptres2 33038 | Extend a finite group sum ... |
| gsumzresunsn 33039 | Append an element to a fin... |
| gsumpart 33040 | Express a group sum as a d... |
| gsumtp 33041 | Group sum of an unordered ... |
| gsumhashmul 33042 | Express a group sum by gro... |
| xrge0tsmsd 33043 | Any finite or infinite sum... |
| xrge0tsmsbi 33044 | Any limit of a finite or i... |
| xrge0tsmseq 33045 | Any limit of a finite or i... |
| cntzun 33046 | The centralizer of a union... |
| cntzsnid 33047 | The centralizer of the ide... |
| cntrcrng 33048 | The center of a ring is a ... |
| isomnd 33053 | A (left) ordered monoid is... |
| isogrp 33054 | A (left-)ordered group is ... |
| ogrpgrp 33055 | A left-ordered group is a ... |
| omndmnd 33056 | A left-ordered monoid is a... |
| omndtos 33057 | A left-ordered monoid is a... |
| omndadd 33058 | In an ordered monoid, the ... |
| omndaddr 33059 | In a right ordered monoid,... |
| omndadd2d 33060 | In a commutative left orde... |
| omndadd2rd 33061 | In a left- and right- orde... |
| submomnd 33062 | A submonoid of an ordered ... |
| xrge0omnd 33063 | The nonnegative extended r... |
| omndmul2 33064 | In an ordered monoid, the ... |
| omndmul3 33065 | In an ordered monoid, the ... |
| omndmul 33066 | In a commutative ordered m... |
| ogrpinv0le 33067 | In an ordered group, the o... |
| ogrpsub 33068 | In an ordered group, the o... |
| ogrpaddlt 33069 | In an ordered group, stric... |
| ogrpaddltbi 33070 | In a right ordered group, ... |
| ogrpaddltrd 33071 | In a right ordered group, ... |
| ogrpaddltrbid 33072 | In a right ordered group, ... |
| ogrpsublt 33073 | In an ordered group, stric... |
| ogrpinv0lt 33074 | In an ordered group, the o... |
| ogrpinvlt 33075 | In an ordered group, the o... |
| gsumle 33076 | A finite sum in an ordered... |
| symgfcoeu 33077 | Uniqueness property of per... |
| symgcom 33078 | Two permutations ` X ` and... |
| symgcom2 33079 | Two permutations ` X ` and... |
| symgcntz 33080 | All elements of a (finite)... |
| odpmco 33081 | The composition of two odd... |
| symgsubg 33082 | The value of the group sub... |
| pmtrprfv2 33083 | In a transposition of two ... |
| pmtrcnel 33084 | Composing a permutation ` ... |
| pmtrcnel2 33085 | Variation on ~ pmtrcnel . ... |
| pmtrcnelor 33086 | Composing a permutation ` ... |
| fzo0pmtrlast 33087 | Reorder a half-open intege... |
| wrdpmtrlast 33088 | Reorder a word, so that th... |
| pmtridf1o 33089 | Transpositions of ` X ` an... |
| pmtridfv1 33090 | Value at X of the transpos... |
| pmtridfv2 33091 | Value at Y of the transpos... |
| psgnid 33092 | Permutation sign of the id... |
| psgndmfi 33093 | For a finite base set, the... |
| pmtrto1cl 33094 | Useful lemma for the follo... |
| psgnfzto1stlem 33095 | Lemma for ~ psgnfzto1st . ... |
| fzto1stfv1 33096 | Value of our permutation `... |
| fzto1st1 33097 | Special case where the per... |
| fzto1st 33098 | The function moving one el... |
| fzto1stinvn 33099 | Value of the inverse of ou... |
| psgnfzto1st 33100 | The permutation sign for m... |
| tocycval 33103 | Value of the cycle builder... |
| tocycfv 33104 | Function value of a permut... |
| tocycfvres1 33105 | A cyclic permutation is a ... |
| tocycfvres2 33106 | A cyclic permutation is th... |
| cycpmfvlem 33107 | Lemma for ~ cycpmfv1 and ~... |
| cycpmfv1 33108 | Value of a cycle function ... |
| cycpmfv2 33109 | Value of a cycle function ... |
| cycpmfv3 33110 | Values outside of the orbi... |
| cycpmcl 33111 | Cyclic permutations are pe... |
| tocycf 33112 | The permutation cycle buil... |
| tocyc01 33113 | Permutation cycles built f... |
| cycpm2tr 33114 | A cyclic permutation of 2 ... |
| cycpm2cl 33115 | Closure for the 2-cycles. ... |
| cyc2fv1 33116 | Function value of a 2-cycl... |
| cyc2fv2 33117 | Function value of a 2-cycl... |
| trsp2cyc 33118 | Exhibit the word a transpo... |
| cycpmco2f1 33119 | The word U used in ~ cycpm... |
| cycpmco2rn 33120 | The orbit of the compositi... |
| cycpmco2lem1 33121 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem2 33122 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem3 33123 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem4 33124 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem5 33125 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem6 33126 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem7 33127 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2 33128 | The composition of a cycli... |
| cyc2fvx 33129 | Function value of a 2-cycl... |
| cycpm3cl 33130 | Closure of the 3-cycles in... |
| cycpm3cl2 33131 | Closure of the 3-cycles in... |
| cyc3fv1 33132 | Function value of a 3-cycl... |
| cyc3fv2 33133 | Function value of a 3-cycl... |
| cyc3fv3 33134 | Function value of a 3-cycl... |
| cyc3co2 33135 | Represent a 3-cycle as a c... |
| cycpmconjvlem 33136 | Lemma for ~ cycpmconjv . ... |
| cycpmconjv 33137 | A formula for computing co... |
| cycpmrn 33138 | The range of the word used... |
| tocyccntz 33139 | All elements of a (finite)... |
| evpmval 33140 | Value of the set of even p... |
| cnmsgn0g 33141 | The neutral element of the... |
| evpmsubg 33142 | The alternating group is a... |
| evpmid 33143 | The identity is an even pe... |
| altgnsg 33144 | The alternating group ` ( ... |
| cyc3evpm 33145 | 3-Cycles are even permutat... |
| cyc3genpmlem 33146 | Lemma for ~ cyc3genpm . (... |
| cyc3genpm 33147 | The alternating group ` A ... |
| cycpmgcl 33148 | Cyclic permutations are pe... |
| cycpmconjslem1 33149 | Lemma for ~ cycpmconjs . ... |
| cycpmconjslem2 33150 | Lemma for ~ cycpmconjs . ... |
| cycpmconjs 33151 | All cycles of the same len... |
| cyc3conja 33152 | All 3-cycles are conjugate... |
| sgnsv 33155 | The sign mapping. (Contri... |
| sgnsval 33156 | The sign value. (Contribu... |
| sgnsf 33157 | The sign function. (Contr... |
| inftmrel 33162 | The infinitesimal relation... |
| isinftm 33163 | Express ` x ` is infinites... |
| isarchi 33164 | Express the predicate " ` ... |
| pnfinf 33165 | Plus infinity is an infini... |
| xrnarchi 33166 | The completed real line is... |
| isarchi2 33167 | Alternative way to express... |
| submarchi 33168 | A submonoid is archimedean... |
| isarchi3 33169 | This is the usual definiti... |
| archirng 33170 | Property of Archimedean or... |
| archirngz 33171 | Property of Archimedean le... |
| archiexdiv 33172 | In an Archimedean group, g... |
| archiabllem1a 33173 | Lemma for ~ archiabl : In... |
| archiabllem1b 33174 | Lemma for ~ archiabl . (C... |
| archiabllem1 33175 | Archimedean ordered groups... |
| archiabllem2a 33176 | Lemma for ~ archiabl , whi... |
| archiabllem2c 33177 | Lemma for ~ archiabl . (C... |
| archiabllem2b 33178 | Lemma for ~ archiabl . (C... |
| archiabllem2 33179 | Archimedean ordered groups... |
| archiabl 33180 | Archimedean left- and righ... |
| isslmd 33183 | The predicate "is a semimo... |
| slmdlema 33184 | Lemma for properties of a ... |
| lmodslmd 33185 | Left semimodules generaliz... |
| slmdcmn 33186 | A semimodule is a commutat... |
| slmdmnd 33187 | A semimodule is a monoid. ... |
| slmdsrg 33188 | The scalar component of a ... |
| slmdbn0 33189 | The base set of a semimodu... |
| slmdacl 33190 | Closure of ring addition f... |
| slmdmcl 33191 | Closure of ring multiplica... |
| slmdsn0 33192 | The set of scalars in a se... |
| slmdvacl 33193 | Closure of vector addition... |
| slmdass 33194 | Semiring left module vecto... |
| slmdvscl 33195 | Closure of scalar product ... |
| slmdvsdi 33196 | Distributive law for scala... |
| slmdvsdir 33197 | Distributive law for scala... |
| slmdvsass 33198 | Associative law for scalar... |
| slmd0cl 33199 | The ring zero in a semimod... |
| slmd1cl 33200 | The ring unity in a semiri... |
| slmdvs1 33201 | Scalar product with ring u... |
| slmd0vcl 33202 | The zero vector is a vecto... |
| slmd0vlid 33203 | Left identity law for the ... |
| slmd0vrid 33204 | Right identity law for the... |
| slmd0vs 33205 | Zero times a vector is the... |
| slmdvs0 33206 | Anything times the zero ve... |
| gsumvsca1 33207 | Scalar product of a finite... |
| gsumvsca2 33208 | Scalar product of a finite... |
| prmsimpcyc 33209 | A group of prime order is ... |
| cringmul32d 33210 | Commutative/associative la... |
| ringdid 33211 | Distributive law for the m... |
| ringdird 33212 | Distributive law for the m... |
| ringdi22 33213 | Expand the product of two ... |
| urpropd 33214 | Sufficient condition for r... |
| subrgmcld 33215 | A subring is closed under ... |
| ress1r 33216 | ` 1r ` is unaffected by re... |
| ringinvval 33217 | The ring inverse expressed... |
| dvrcan5 33218 | Cancellation law for commo... |
| subrgchr 33219 | If ` A ` is a subring of `... |
| rmfsupp2 33220 | A mapping of a multiplicat... |
| unitnz 33221 | In a nonzero ring, a unit ... |
| isunit2 33222 | Alternate definition of be... |
| isunit3 33223 | Alternate definition of be... |
| irrednzr 33224 | A ring with an irreducible... |
| 0ringsubrg 33225 | A subring of a zero ring i... |
| 0ringcring 33226 | The zero ring is commutati... |
| reldmrloc 33231 | Ring localization is a pro... |
| erlval 33232 | Value of the ring localiza... |
| rlocval 33233 | Expand the value of the ri... |
| erlcl1 33234 | Closure for the ring local... |
| erlcl2 33235 | Closure for the ring local... |
| erldi 33236 | Main property of the ring ... |
| erlbrd 33237 | Deduce the ring localizati... |
| erlbr2d 33238 | Deduce the ring localizati... |
| erler 33239 | The relation used to build... |
| elrlocbasi 33240 | Membership in the basis of... |
| rlocbas 33241 | The base set of a ring loc... |
| rlocaddval 33242 | Value of the addition in t... |
| rlocmulval 33243 | Value of the addition in t... |
| rloccring 33244 | The ring localization ` L ... |
| rloc0g 33245 | The zero of a ring localiz... |
| rloc1r 33246 | The multiplicative identit... |
| rlocf1 33247 | The embedding ` F ` of a r... |
| domnmuln0rd 33248 | In a domain, factors of a ... |
| domnprodn0 33249 | In a domain, a finite prod... |
| idomrcan 33250 | Right-cancellation law for... |
| domnlcanOLD 33251 | Obsolete version of ~ domn... |
| domnlcanbOLD 33252 | Obsolete version of ~ domn... |
| idomrcanOLD 33253 | Obsolete version of ~ idom... |
| 1rrg 33254 | The multiplicative identit... |
| rrgsubm 33255 | The left regular elements ... |
| subrdom 33256 | A subring of a domain is a... |
| subridom 33257 | A subring of an integral d... |
| subrfld 33258 | A subring of a field is an... |
| eufndx 33261 | Index value of the Euclide... |
| eufid 33262 | Utility theorem: index-ind... |
| ringinveu 33265 | If a ring unit element ` X... |
| isdrng4 33266 | A division ring is a ring ... |
| rndrhmcl 33267 | The image of a division ri... |
| sdrgdvcl 33268 | A sub-division-ring is clo... |
| sdrginvcl 33269 | A sub-division-ring is clo... |
| primefldchr 33270 | The characteristic of a pr... |
| fracval 33273 | Value of the field of frac... |
| fracbas 33274 | The base of the field of f... |
| fracerl 33275 | Rewrite the ring localizat... |
| fracf1 33276 | The embedding of a commuta... |
| fracfld 33277 | The field of fractions of ... |
| idomsubr 33278 | Every integral domain is i... |
| fldgenval 33281 | Value of the field generat... |
| fldgenssid 33282 | The field generated by a s... |
| fldgensdrg 33283 | A generated subfield is a ... |
| fldgenssv 33284 | A generated subfield is a ... |
| fldgenss 33285 | Generated subfields preser... |
| fldgenidfld 33286 | The subfield generated by ... |
| fldgenssp 33287 | The field generated by a s... |
| fldgenid 33288 | The subfield of a field ` ... |
| fldgenfld 33289 | A generated subfield is a ... |
| primefldgen1 33290 | The prime field of a divis... |
| 1fldgenq 33291 | The field of rational numb... |
| isorng 33296 | An ordered ring is a ring ... |
| orngring 33297 | An ordered ring is a ring.... |
| orngogrp 33298 | An ordered ring is an orde... |
| isofld 33299 | An ordered field is a fiel... |
| orngmul 33300 | In an ordered ring, the or... |
| orngsqr 33301 | In an ordered ring, all sq... |
| ornglmulle 33302 | In an ordered ring, multip... |
| orngrmulle 33303 | In an ordered ring, multip... |
| ornglmullt 33304 | In an ordered ring, multip... |
| orngrmullt 33305 | In an ordered ring, multip... |
| orngmullt 33306 | In an ordered ring, the st... |
| ofldfld 33307 | An ordered field is a fiel... |
| ofldtos 33308 | An ordered field is a tota... |
| orng0le1 33309 | In an ordered ring, the ri... |
| ofldlt1 33310 | In an ordered field, the r... |
| ofldchr 33311 | The characteristic of an o... |
| suborng 33312 | Every subring of an ordere... |
| subofld 33313 | Every subfield of an order... |
| isarchiofld 33314 | Axiom of Archimedes : a ch... |
| rhmdvd 33315 | A ring homomorphism preser... |
| kerunit 33316 | If a unit element lies in ... |
| reldmresv 33319 | The scalar restriction is ... |
| resvval 33320 | Value of structure restric... |
| resvid2 33321 | General behavior of trivia... |
| resvval2 33322 | Value of nontrivial struct... |
| resvsca 33323 | Base set of a structure re... |
| resvlem 33324 | Other elements of a scalar... |
| resvlemOLD 33325 | Obsolete version of ~ resv... |
| resvbas 33326 | ` Base ` is unaffected by ... |
| resvbasOLD 33327 | Obsolete proof of ~ resvba... |
| resvplusg 33328 | ` +g ` is unaffected by sc... |
| resvplusgOLD 33329 | Obsolete proof of ~ resvpl... |
| resvvsca 33330 | ` .s ` is unaffected by sc... |
| resvvscaOLD 33331 | Obsolete proof of ~ resvvs... |
| resvmulr 33332 | ` .r ` is unaffected by sc... |
| resvmulrOLD 33333 | Obsolete proof of ~ resvmu... |
| resv0g 33334 | ` 0g ` is unaffected by sc... |
| resv1r 33335 | ` 1r ` is unaffected by sc... |
| resvcmn 33336 | Scalar restriction preserv... |
| gzcrng 33337 | The gaussian integers form... |
| cnfldfld 33338 | The complex numbers form a... |
| reofld 33339 | The real numbers form an o... |
| nn0omnd 33340 | The nonnegative integers f... |
| rearchi 33341 | The field of the real numb... |
| nn0archi 33342 | The monoid of the nonnegat... |
| xrge0slmod 33343 | The extended nonnegative r... |
| qusker 33344 | The kernel of a quotient m... |
| eqgvscpbl 33345 | The left coset equivalence... |
| qusvscpbl 33346 | The quotient map distribut... |
| qusvsval 33347 | Value of the scalar multip... |
| imaslmod 33348 | The image structure of a l... |
| imasmhm 33349 | Given a function ` F ` wit... |
| imasghm 33350 | Given a function ` F ` wit... |
| imasrhm 33351 | Given a function ` F ` wit... |
| imaslmhm 33352 | Given a function ` F ` wit... |
| quslmod 33353 | If ` G ` is a submodule in... |
| quslmhm 33354 | If ` G ` is a submodule of... |
| quslvec 33355 | If ` S ` is a vector subsp... |
| ecxpid 33356 | The equivalence class of a... |
| qsxpid 33357 | The quotient set of a cart... |
| qusxpid 33358 | The Group quotient equival... |
| qustriv 33359 | The quotient of a group ` ... |
| qustrivr 33360 | Converse of ~ qustriv . (... |
| znfermltl 33361 | Fermat's little theorem in... |
| islinds5 33362 | A set is linearly independ... |
| ellspds 33363 | Variation on ~ ellspd . (... |
| 0ellsp 33364 | Zero is in all spans. (Co... |
| 0nellinds 33365 | The group identity cannot ... |
| rspsnid 33366 | A principal ideal contains... |
| elrsp 33367 | Write the elements of a ri... |
| ellpi 33368 | Elementhood in a left prin... |
| lpirlidllpi 33369 | In a principal ideal ring,... |
| rspidlid 33370 | The ideal span of an ideal... |
| pidlnz 33371 | A principal ideal generate... |
| lbslsp 33372 | Any element of a left modu... |
| lindssn 33373 | Any singleton of a nonzero... |
| lindflbs 33374 | Conditions for an independ... |
| islbs5 33375 | An equivalent formulation ... |
| linds2eq 33376 | Deduce equality of element... |
| lindfpropd 33377 | Property deduction for lin... |
| lindspropd 33378 | Property deduction for lin... |
| dvdsruassoi 33379 | If two elements ` X ` and ... |
| dvdsruasso 33380 | Two elements ` X ` and ` Y... |
| dvdsruasso2 33381 | A reformulation of ~ dvdsr... |
| dvdsrspss 33382 | In a ring, an element ` X ... |
| rspsnasso 33383 | Two elements ` X ` and ` Y... |
| unitprodclb 33384 | A finite product is a unit... |
| elgrplsmsn 33385 | Membership in a sumset wit... |
| lsmsnorb 33386 | The sumset of a group with... |
| lsmsnorb2 33387 | The sumset of a single ele... |
| elringlsm 33388 | Membership in a product of... |
| elringlsmd 33389 | Membership in a product of... |
| ringlsmss 33390 | Closure of the product of ... |
| ringlsmss1 33391 | The product of an ideal ` ... |
| ringlsmss2 33392 | The product with an ideal ... |
| lsmsnpridl 33393 | The product of the ring wi... |
| lsmsnidl 33394 | The product of the ring wi... |
| lsmidllsp 33395 | The sum of two ideals is t... |
| lsmidl 33396 | The sum of two ideals is a... |
| lsmssass 33397 | Group sum is associative, ... |
| grplsm0l 33398 | Sumset with the identity s... |
| grplsmid 33399 | The direct sum of an eleme... |
| quslsm 33400 | Express the image by the q... |
| qusbas2 33401 | Alternate definition of th... |
| qus0g 33402 | The identity element of a ... |
| qusima 33403 | The image of a subgroup by... |
| qusrn 33404 | The natural map from eleme... |
| nsgqus0 33405 | A normal subgroup ` N ` is... |
| nsgmgclem 33406 | Lemma for ~ nsgmgc . (Con... |
| nsgmgc 33407 | There is a monotone Galois... |
| nsgqusf1olem1 33408 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem2 33409 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem3 33410 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1o 33411 | The canonical projection h... |
| lmhmqusker 33412 | A surjective module homomo... |
| lmicqusker 33413 | The image ` H ` of a modul... |
| lidlmcld 33414 | An ideal is closed under l... |
| intlidl 33415 | The intersection of a none... |
| 0ringidl 33416 | The zero ideal is the only... |
| pidlnzb 33417 | A principal ideal is nonze... |
| lidlunitel 33418 | If an ideal ` I ` contains... |
| unitpidl1 33419 | The ideal ` I ` generated ... |
| rhmquskerlem 33420 | The mapping ` J ` induced ... |
| rhmqusker 33421 | A surjective ring homomorp... |
| ricqusker 33422 | The image ` H ` of a ring ... |
| elrspunidl 33423 | Elementhood in the span of... |
| elrspunsn 33424 | Membership to the span of ... |
| lidlincl 33425 | Ideals are closed under in... |
| idlinsubrg 33426 | The intersection between a... |
| rhmimaidl 33427 | The image of an ideal ` I ... |
| drngidl 33428 | A nonzero ring is a divisi... |
| drngidlhash 33429 | A ring is a division ring ... |
| prmidlval 33432 | The class of prime ideals ... |
| isprmidl 33433 | The predicate "is a prime ... |
| prmidlnr 33434 | A prime ideal is a proper ... |
| prmidl 33435 | The main property of a pri... |
| prmidl2 33436 | A condition that shows an ... |
| idlmulssprm 33437 | Let ` P ` be a prime ideal... |
| pridln1 33438 | A proper ideal cannot cont... |
| prmidlidl 33439 | A prime ideal is an ideal.... |
| prmidlssidl 33440 | Prime ideals as a subset o... |
| cringm4 33441 | Commutative/associative la... |
| isprmidlc 33442 | The predicate "is prime id... |
| prmidlc 33443 | Property of a prime ideal ... |
| 0ringprmidl 33444 | The trivial ring does not ... |
| prmidl0 33445 | The zero ideal of a commut... |
| rhmpreimaprmidl 33446 | The preimage of a prime id... |
| qsidomlem1 33447 | If the quotient ring of a ... |
| qsidomlem2 33448 | A quotient by a prime idea... |
| qsidom 33449 | An ideal ` I ` in the comm... |
| qsnzr 33450 | A quotient of a non-zero r... |
| ssdifidllem 33451 | Lemma for ~ ssdifidl : Th... |
| ssdifidl 33452 | Let ` R ` be a ring, and l... |
| ssdifidlprm 33453 | If the set ` S ` of ~ ssdi... |
| mxidlval 33456 | The set of maximal ideals ... |
| ismxidl 33457 | The predicate "is a maxima... |
| mxidlidl 33458 | A maximal ideal is an idea... |
| mxidlnr 33459 | A maximal ideal is proper.... |
| mxidlmax 33460 | A maximal ideal is a maxim... |
| mxidln1 33461 | One is not contained in an... |
| mxidlnzr 33462 | A ring with a maximal idea... |
| mxidlmaxv 33463 | An ideal ` I ` strictly co... |
| crngmxidl 33464 | In a commutative ring, max... |
| mxidlprm 33465 | Every maximal ideal is pri... |
| mxidlirredi 33466 | In an integral domain, the... |
| mxidlirred 33467 | In a principal ideal domai... |
| ssmxidllem 33468 | The set ` P ` used in the ... |
| ssmxidl 33469 | Let ` R ` be a ring, and l... |
| drnglidl1ne0 33470 | In a nonzero ring, the zer... |
| drng0mxidl 33471 | In a division ring, the ze... |
| drngmxidl 33472 | The zero ideal is the only... |
| drngmxidlr 33473 | If a ring's only maximal i... |
| krull 33474 | Krull's theorem: Any nonz... |
| mxidlnzrb 33475 | A ring is nonzero if and o... |
| krullndrng 33476 | Krull's theorem for non-di... |
| opprabs 33477 | The opposite ring of the o... |
| oppreqg 33478 | Group coset equivalence re... |
| opprnsg 33479 | Normal subgroups of the op... |
| opprlidlabs 33480 | The ideals of the opposite... |
| oppr2idl 33481 | Two sided ideal of the opp... |
| opprmxidlabs 33482 | The maximal ideal of the o... |
| opprqusbas 33483 | The base of the quotient o... |
| opprqusplusg 33484 | The group operation of the... |
| opprqus0g 33485 | The group identity element... |
| opprqusmulr 33486 | The multiplication operati... |
| opprqus1r 33487 | The ring unity of the quot... |
| opprqusdrng 33488 | The quotient of the opposi... |
| qsdrngilem 33489 | Lemma for ~ qsdrngi . (Co... |
| qsdrngi 33490 | A quotient by a maximal le... |
| qsdrnglem2 33491 | Lemma for ~ qsdrng . (Con... |
| qsdrng 33492 | An ideal ` M ` is both lef... |
| qsfld 33493 | An ideal ` M ` in the comm... |
| mxidlprmALT 33494 | Every maximal ideal is pri... |
| idlsrgstr 33497 | A constructed semiring of ... |
| idlsrgval 33498 | Lemma for ~ idlsrgbas thro... |
| idlsrgbas 33499 | Base of the ideals of a ri... |
| idlsrgplusg 33500 | Additive operation of the ... |
| idlsrg0g 33501 | The zero ideal is the addi... |
| idlsrgmulr 33502 | Multiplicative operation o... |
| idlsrgtset 33503 | Topology component of the ... |
| idlsrgmulrval 33504 | Value of the ring multipli... |
| idlsrgmulrcl 33505 | Ideals of a ring ` R ` are... |
| idlsrgmulrss1 33506 | In a commutative ring, the... |
| idlsrgmulrss2 33507 | The product of two ideals ... |
| idlsrgmulrssin 33508 | In a commutative ring, the... |
| idlsrgmnd 33509 | The ideals of a ring form ... |
| idlsrgcmnd 33510 | The ideals of a ring form ... |
| rprmval 33511 | The prime elements of a ri... |
| isrprm 33512 | Property for ` P ` to be a... |
| rprmcl 33513 | A ring prime is an element... |
| rprmdvds 33514 | If a ring prime ` Q ` divi... |
| rprmnz 33515 | A ring prime is nonzero. ... |
| rprmnunit 33516 | A ring prime is not a unit... |
| rsprprmprmidl 33517 | In a commutative ring, ide... |
| rsprprmprmidlb 33518 | In an integral domain, an ... |
| rprmndvdsr1 33519 | A ring prime element does ... |
| rprmasso 33520 | In an integral domain, the... |
| rprmasso2 33521 | In an integral domain, if ... |
| rprmasso3 33522 | In an integral domain, if ... |
| unitmulrprm 33523 | A ring unit multiplied by ... |
| rprmndvdsru 33524 | A ring prime element does ... |
| rprmirredlem 33525 | Lemma for ~ rprmirred . (... |
| rprmirred 33526 | In an integral domain, rin... |
| rprmirredb 33527 | In a principal ideal domai... |
| rprmdvdspow 33528 | If a prime element divides... |
| rprmdvdsprod 33529 | If a prime element ` Q ` d... |
| 1arithidomlem1 33530 | Lemma for ~ 1arithidom . ... |
| 1arithidomlem2 33531 | Lemma for ~ 1arithidom : i... |
| 1arithidom 33532 | Uniqueness of prime factor... |
| isufd 33535 | The property of being a Un... |
| ufdprmidl 33536 | In a unique factorization ... |
| ufdidom 33537 | A nonzero unique factoriza... |
| pidufd 33538 | Every principal ideal doma... |
| 1arithufdlem1 33539 | Lemma for ~ 1arithufd . T... |
| 1arithufdlem2 33540 | Lemma for ~ 1arithufd . T... |
| 1arithufdlem3 33541 | Lemma for ~ 1arithufd . I... |
| 1arithufdlem4 33542 | Lemma for ~ 1arithufd . N... |
| 1arithufd 33543 | Existence of a factorizati... |
| dfufd2lem 33544 | Lemma for ~ dfufd2 . (Con... |
| dfufd2 33545 | Alternative definition of ... |
| zringidom 33546 | The ring of integers is an... |
| zringpid 33547 | The ring of integers is a ... |
| dfprm3 33548 | The (positive) prime eleme... |
| zringfrac 33549 | The field of fractions of ... |
| 0ringmon1p 33550 | There are no monic polynom... |
| fply1 33551 | Conditions for a function ... |
| ply1lvec 33552 | In a division ring, the un... |
| evls1fn 33553 | Functionality of the subri... |
| evls1dm 33554 | The domain of the subring ... |
| evls1fvf 33555 | The subring evaluation fun... |
| evl1fvf 33556 | The univariate polynomial ... |
| evl1fpws 33557 | Evaluation of a univariate... |
| ressdeg1 33558 | The degree of a univariate... |
| ressply10g 33559 | A restricted polynomial al... |
| ressply1mon1p 33560 | The monic polynomials of a... |
| ressply1invg 33561 | An element of a restricted... |
| ressply1sub 33562 | A restricted polynomial al... |
| ressasclcl 33563 | Closure of the univariate ... |
| evls1subd 33564 | Univariate polynomial eval... |
| deg1le0eq0 33565 | A polynomial with nonposit... |
| ply1asclunit 33566 | A non-zero scalar polynomi... |
| ply1unit 33567 | In a field ` F ` , a polyn... |
| evl1deg1 33568 | Evaluation of a univariate... |
| evl1deg2 33569 | Evaluation of a univariate... |
| evl1deg3 33570 | Evaluation of a univariate... |
| ply1dg1rt 33571 | Express the root ` - B / A... |
| ply1dg1rtn0 33572 | Polynomials of degree 1 ov... |
| ply1mulrtss 33573 | The roots of a factor ` F ... |
| ply1dg3rt0irred 33574 | If a cubic polynomial over... |
| m1pmeq 33575 | If two monic polynomials `... |
| ply1fermltl 33576 | Fermat's little theorem fo... |
| coe1mon 33577 | Coefficient vector of a mo... |
| ply1moneq 33578 | Two monomials are equal if... |
| coe1zfv 33579 | The coefficients of the ze... |
| coe1vr1 33580 | Polynomial coefficient of ... |
| deg1vr 33581 | The degree of the variable... |
| ply1degltel 33582 | Characterize elementhood i... |
| ply1degleel 33583 | Characterize elementhood i... |
| ply1degltlss 33584 | The space ` S ` of the uni... |
| gsummoncoe1fzo 33585 | A coefficient of the polyn... |
| ply1gsumz 33586 | If a polynomial given as a... |
| deg1addlt 33587 | If both factors have degre... |
| ig1pnunit 33588 | The polynomial ideal gener... |
| ig1pmindeg 33589 | The polynomial ideal gener... |
| q1pdir 33590 | Distribution of univariate... |
| q1pvsca 33591 | Scalar multiplication prop... |
| r1pvsca 33592 | Scalar multiplication prop... |
| r1p0 33593 | Polynomial remainder opera... |
| r1pcyc 33594 | The polynomial remainder o... |
| r1padd1 33595 | Addition property of the p... |
| r1pid2OLD 33596 | Obsolete version of ~ r1pi... |
| r1plmhm 33597 | The univariate polynomial ... |
| r1pquslmic 33598 | The univariate polynomial ... |
| sra1r 33599 | The unity element of a sub... |
| sradrng 33600 | Condition for a subring al... |
| srasubrg 33601 | A subring of the original ... |
| sralvec 33602 | Given a sub division ring ... |
| srafldlvec 33603 | Given a subfield ` F ` of ... |
| resssra 33604 | The subring algebra of a r... |
| lsssra 33605 | A subring is a subspace of... |
| drgext0g 33606 | The additive neutral eleme... |
| drgextvsca 33607 | The scalar multiplication ... |
| drgext0gsca 33608 | The additive neutral eleme... |
| drgextsubrg 33609 | The scalar field is a subr... |
| drgextlsp 33610 | The scalar field is a subs... |
| drgextgsum 33611 | Group sum in a division ri... |
| lvecdimfi 33612 | Finite version of ~ lvecdi... |
| dimval 33615 | The dimension of a vector ... |
| dimvalfi 33616 | The dimension of a vector ... |
| dimcl 33617 | Closure of the vector spac... |
| lmimdim 33618 | Module isomorphisms preser... |
| lmicdim 33619 | Module isomorphisms preser... |
| lvecdim0i 33620 | A vector space of dimensio... |
| lvecdim0 33621 | A vector space of dimensio... |
| lssdimle 33622 | The dimension of a linear ... |
| dimpropd 33623 | If two structures have the... |
| rlmdim 33624 | The left vector space indu... |
| rgmoddimOLD 33625 | Obsolete version of ~ rlmd... |
| frlmdim 33626 | Dimension of a free left m... |
| tnglvec 33627 | Augmenting a structure wit... |
| tngdim 33628 | Dimension of a left vector... |
| rrxdim 33629 | Dimension of the generaliz... |
| matdim 33630 | Dimension of the space of ... |
| lbslsat 33631 | A nonzero vector ` X ` is ... |
| lsatdim 33632 | A line, spanned by a nonze... |
| drngdimgt0 33633 | The dimension of a vector ... |
| lmhmlvec2 33634 | A homomorphism of left vec... |
| kerlmhm 33635 | The kernel of a vector spa... |
| imlmhm 33636 | The image of a vector spac... |
| ply1degltdimlem 33637 | Lemma for ~ ply1degltdim .... |
| ply1degltdim 33638 | The space ` S ` of the uni... |
| lindsunlem 33639 | Lemma for ~ lindsun . (Co... |
| lindsun 33640 | Condition for the union of... |
| lbsdiflsp0 33641 | The linear spans of two di... |
| dimkerim 33642 | Given a linear map ` F ` b... |
| qusdimsum 33643 | Let ` W ` be a vector spac... |
| fedgmullem1 33644 | Lemma for ~ fedgmul . (Co... |
| fedgmullem2 33645 | Lemma for ~ fedgmul . (Co... |
| fedgmul 33646 | The multiplicativity formu... |
| dimlssid 33647 | If the dimension of a line... |
| lvecendof1f1o 33648 | If an endomorphism ` U ` o... |
| lactlmhm 33649 | In an associative algebra ... |
| assalactf1o 33650 | In an associative algebra ... |
| assarrginv 33651 | If an element ` X ` of an ... |
| assafld 33652 | If an algebra ` A ` of fin... |
| relfldext 33661 | The field extension is a r... |
| brfldext 33662 | The field extension relati... |
| ccfldextrr 33663 | The field of the complex n... |
| fldextfld1 33664 | A field extension is only ... |
| fldextfld2 33665 | A field extension is only ... |
| fldextsubrg 33666 | Field extension implies a ... |
| fldextress 33667 | Field extension implies a ... |
| brfinext 33668 | The finite field extension... |
| extdgval 33669 | Value of the field extensi... |
| fldextsralvec 33670 | The subring algebra associ... |
| extdgcl 33671 | Closure of the field exten... |
| extdggt0 33672 | Degrees of field extension... |
| fldexttr 33673 | Field extension is a trans... |
| fldextid 33674 | The field extension relati... |
| extdgid 33675 | A trivial field extension ... |
| extdgmul 33676 | The multiplicativity formu... |
| finexttrb 33677 | The extension ` E ` of ` K... |
| extdg1id 33678 | If the degree of the exten... |
| extdg1b 33679 | The degree of the extensio... |
| fldgenfldext 33680 | A subfield ` F ` extended ... |
| fldextchr 33681 | The characteristic of a su... |
| evls1fldgencl 33682 | Closure of the subring pol... |
| ccfldsrarelvec 33683 | The subring algebra of the... |
| ccfldextdgrr 33684 | The degree of the field ex... |
| irngval 33687 | The elements of a field ` ... |
| elirng 33688 | Property for an element ` ... |
| irngss 33689 | All elements of a subring ... |
| irngssv 33690 | An integral element is an ... |
| 0ringirng 33691 | A zero ring ` R ` has no i... |
| irngnzply1lem 33692 | In the case of a field ` E... |
| irngnzply1 33693 | In the case of a field ` E... |
| ply1annidllem 33696 | Write the set ` Q ` of pol... |
| ply1annidl 33697 | The set ` Q ` of polynomia... |
| ply1annnr 33698 | The set ` Q ` of polynomia... |
| ply1annig1p 33699 | The ideal ` Q ` of polynom... |
| minplyval 33700 | Expand the value of the mi... |
| minplycl 33701 | The minimal polynomial is ... |
| ply1annprmidl 33702 | The set ` Q ` of polynomia... |
| minplymindeg 33703 | The minimal polynomial of ... |
| minplyann 33704 | The minimal polynomial for... |
| minplyirredlem 33705 | Lemma for ~ minplyirred . ... |
| minplyirred 33706 | A nonzero minimal polynomi... |
| irngnminplynz 33707 | Integral elements have non... |
| minplym1p 33708 | A minimal polynomial is mo... |
| irredminply 33709 | An irreducible, monic, ann... |
| algextdeglem1 33710 | Lemma for ~ algextdeg . (... |
| algextdeglem2 33711 | Lemma for ~ algextdeg . B... |
| algextdeglem3 33712 | Lemma for ~ algextdeg . T... |
| algextdeglem4 33713 | Lemma for ~ algextdeg . B... |
| algextdeglem5 33714 | Lemma for ~ algextdeg . T... |
| algextdeglem6 33715 | Lemma for ~ algextdeg . B... |
| algextdeglem7 33716 | Lemma for ~ algextdeg . T... |
| algextdeglem8 33717 | Lemma for ~ algextdeg . T... |
| algextdeg 33718 | The degree of an algebraic... |
| rtelextdg2lem 33719 | Lemma for ~ rtelextdg2 : ... |
| rtelextdg2 33720 | If an element ` X ` is a s... |
| fldext2chn 33721 | In a non-empty tower ` T `... |
| constrrtll 33724 | In the construction of con... |
| constrrtlc1 33725 | In the construction of con... |
| constrrtlc2 33726 | In the construction of con... |
| constrrtcclem 33727 | In the construction of con... |
| constrrtcc 33728 | In the construction of con... |
| constr0 33729 | The first step of the cons... |
| constrsuc 33730 | Membership in the successo... |
| constrlim 33731 | Limit step of the construc... |
| constrsscn 33732 | Closure of the constructib... |
| constrsslem 33733 | Lemma for ~ constrss . Th... |
| constr01 33734 | ` 0 ` and ` 1 ` are in all... |
| constrss 33735 | Constructed points are in ... |
| constrmon 33736 | The construction of constr... |
| constrconj 33737 | If a point ` X ` of the co... |
| constrfin 33738 | Each step of the construct... |
| constrelextdg2 33739 | If the ` N ` -th step ` ( ... |
| 2sqr3minply 33740 | The polynomial ` ( ( X ^ 3... |
| smatfval 33743 | Value of the submatrix. (... |
| smatrcl 33744 | Closure of the rectangular... |
| smatlem 33745 | Lemma for the next theorem... |
| smattl 33746 | Entries of a submatrix, to... |
| smattr 33747 | Entries of a submatrix, to... |
| smatbl 33748 | Entries of a submatrix, bo... |
| smatbr 33749 | Entries of a submatrix, bo... |
| smatcl 33750 | Closure of the square subm... |
| matmpo 33751 | Write a square matrix as a... |
| 1smat1 33752 | The submatrix of the ident... |
| submat1n 33753 | One case where the submatr... |
| submatres 33754 | Special case where the sub... |
| submateqlem1 33755 | Lemma for ~ submateq . (C... |
| submateqlem2 33756 | Lemma for ~ submateq . (C... |
| submateq 33757 | Sufficient condition for t... |
| submatminr1 33758 | If we take a submatrix by ... |
| lmatval 33761 | Value of the literal matri... |
| lmatfval 33762 | Entries of a literal matri... |
| lmatfvlem 33763 | Useful lemma to extract li... |
| lmatcl 33764 | Closure of the literal mat... |
| lmat22lem 33765 | Lemma for ~ lmat22e11 and ... |
| lmat22e11 33766 | Entry of a 2x2 literal mat... |
| lmat22e12 33767 | Entry of a 2x2 literal mat... |
| lmat22e21 33768 | Entry of a 2x2 literal mat... |
| lmat22e22 33769 | Entry of a 2x2 literal mat... |
| lmat22det 33770 | The determinant of a liter... |
| mdetpmtr1 33771 | The determinant of a matri... |
| mdetpmtr2 33772 | The determinant of a matri... |
| mdetpmtr12 33773 | The determinant of a matri... |
| mdetlap1 33774 | A Laplace expansion of the... |
| madjusmdetlem1 33775 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem2 33776 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem3 33777 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem4 33778 | Lemma for ~ madjusmdet . ... |
| madjusmdet 33779 | Express the cofactor of th... |
| mdetlap 33780 | Laplace expansion of the d... |
| ist0cld 33781 | The predicate "is a T_0 sp... |
| txomap 33782 | Given two open maps ` F ` ... |
| qtopt1 33783 | If every equivalence class... |
| qtophaus 33784 | If an open map's graph in ... |
| circtopn 33785 | The topology of the unit c... |
| circcn 33786 | The function gluing the re... |
| reff 33787 | For any cover refinement, ... |
| locfinreflem 33788 | A locally finite refinemen... |
| locfinref 33789 | A locally finite refinemen... |
| iscref 33792 | The property that every op... |
| crefeq 33793 | Equality theorem for the "... |
| creftop 33794 | A space where every open c... |
| crefi 33795 | The property that every op... |
| crefdf 33796 | A formulation of ~ crefi e... |
| crefss 33797 | The "every open cover has ... |
| cmpcref 33798 | Equivalent definition of c... |
| cmpfiref 33799 | Every open cover of a Comp... |
| ldlfcntref 33802 | Every open cover of a Lind... |
| ispcmp 33805 | The predicate "is a paraco... |
| cmppcmp 33806 | Every compact space is par... |
| dispcmp 33807 | Every discrete space is pa... |
| pcmplfin 33808 | Given a paracompact topolo... |
| pcmplfinf 33809 | Given a paracompact topolo... |
| rspecval 33812 | Value of the spectrum of t... |
| rspecbas 33813 | The prime ideals form the ... |
| rspectset 33814 | Topology component of the ... |
| rspectopn 33815 | The topology component of ... |
| zarcls0 33816 | The closure of the identit... |
| zarcls1 33817 | The unit ideal ` B ` is th... |
| zarclsun 33818 | The union of two closed se... |
| zarclsiin 33819 | In a Zariski topology, the... |
| zarclsint 33820 | The intersection of a fami... |
| zarclssn 33821 | The closed points of Zaris... |
| zarcls 33822 | The open sets of the Zaris... |
| zartopn 33823 | The Zariski topology is a ... |
| zartop 33824 | The Zariski topology is a ... |
| zartopon 33825 | The points of the Zariski ... |
| zar0ring 33826 | The Zariski Topology of th... |
| zart0 33827 | The Zariski topology is T_... |
| zarmxt1 33828 | The Zariski topology restr... |
| zarcmplem 33829 | Lemma for ~ zarcmp . (Con... |
| zarcmp 33830 | The Zariski topology is co... |
| rspectps 33831 | The spectrum of a ring ` R... |
| rhmpreimacnlem 33832 | Lemma for ~ rhmpreimacn . ... |
| rhmpreimacn 33833 | The function mapping a pri... |
| metidval 33838 | Value of the metric identi... |
| metidss 33839 | As a relation, the metric ... |
| metidv 33840 | ` A ` and ` B ` identify b... |
| metideq 33841 | Basic property of the metr... |
| metider 33842 | The metric identification ... |
| pstmval 33843 | Value of the metric induce... |
| pstmfval 33844 | Function value of the metr... |
| pstmxmet 33845 | The metric induced by a ps... |
| hauseqcn 33846 | In a Hausdorff topology, t... |
| elunitge0 33847 | An element of the closed u... |
| unitssxrge0 33848 | The closed unit interval i... |
| unitdivcld 33849 | Necessary conditions for a... |
| iistmd 33850 | The closed unit interval f... |
| unicls 33851 | The union of the closed se... |
| tpr2tp 33852 | The usual topology on ` ( ... |
| tpr2uni 33853 | The usual topology on ` ( ... |
| xpinpreima 33854 | Rewrite the cartesian prod... |
| xpinpreima2 33855 | Rewrite the cartesian prod... |
| sqsscirc1 33856 | The complex square of side... |
| sqsscirc2 33857 | The complex square of side... |
| cnre2csqlem 33858 | Lemma for ~ cnre2csqima . ... |
| cnre2csqima 33859 | Image of a centered square... |
| tpr2rico 33860 | For any point of an open s... |
| cnvordtrestixx 33861 | The restriction of the 'gr... |
| prsdm 33862 | Domain of the relation of ... |
| prsrn 33863 | Range of the relation of a... |
| prsss 33864 | Relation of a subproset. ... |
| prsssdm 33865 | Domain of a subproset rela... |
| ordtprsval 33866 | Value of the order topolog... |
| ordtprsuni 33867 | Value of the order topolog... |
| ordtcnvNEW 33868 | The order dual generates t... |
| ordtrestNEW 33869 | The subspace topology of a... |
| ordtrest2NEWlem 33870 | Lemma for ~ ordtrest2NEW .... |
| ordtrest2NEW 33871 | An interval-closed set ` A... |
| ordtconnlem1 33872 | Connectedness in the order... |
| ordtconn 33873 | Connectedness in the order... |
| mndpluscn 33874 | A mapping that is both a h... |
| mhmhmeotmd 33875 | Deduce a Topological Monoi... |
| rmulccn 33876 | Multiplication by a real c... |
| raddcn 33877 | Addition in the real numbe... |
| xrmulc1cn 33878 | The operation multiplying ... |
| fmcncfil 33879 | The image of a Cauchy filt... |
| xrge0hmph 33880 | The extended nonnegative r... |
| xrge0iifcnv 33881 | Define a bijection from ` ... |
| xrge0iifcv 33882 | The defined function's val... |
| xrge0iifiso 33883 | The defined bijection from... |
| xrge0iifhmeo 33884 | Expose a homeomorphism fro... |
| xrge0iifhom 33885 | The defined function from ... |
| xrge0iif1 33886 | Condition for the defined ... |
| xrge0iifmhm 33887 | The defined function from ... |
| xrge0pluscn 33888 | The addition operation of ... |
| xrge0mulc1cn 33889 | The operation multiplying ... |
| xrge0tps 33890 | The extended nonnegative r... |
| xrge0topn 33891 | The topology of the extend... |
| xrge0haus 33892 | The topology of the extend... |
| xrge0tmd 33893 | The extended nonnegative r... |
| xrge0tmdALT 33894 | Alternate proof of ~ xrge0... |
| lmlim 33895 | Relate a limit in a given ... |
| lmlimxrge0 33896 | Relate a limit in the nonn... |
| rge0scvg 33897 | Implication of convergence... |
| fsumcvg4 33898 | A serie with finite suppor... |
| pnfneige0 33899 | A neighborhood of ` +oo ` ... |
| lmxrge0 33900 | Express "sequence ` F ` co... |
| lmdvg 33901 | If a monotonic sequence of... |
| lmdvglim 33902 | If a monotonic real number... |
| pl1cn 33903 | A univariate polynomial is... |
| zringnm 33906 | The norm (function) for a ... |
| zzsnm 33907 | The norm of the ring of th... |
| zlm0 33908 | Zero of a ` ZZ ` -module. ... |
| zlm1 33909 | Unity element of a ` ZZ ` ... |
| zlmds 33910 | Distance in a ` ZZ ` -modu... |
| zlmdsOLD 33911 | Obsolete proof of ~ zlmds ... |
| zlmtset 33912 | Topology in a ` ZZ ` -modu... |
| zlmtsetOLD 33913 | Obsolete proof of ~ zlmtse... |
| zlmnm 33914 | Norm of a ` ZZ ` -module (... |
| zhmnrg 33915 | The ` ZZ ` -module built f... |
| nmmulg 33916 | The norm of a group produc... |
| zrhnm 33917 | The norm of the image by `... |
| cnzh 33918 | The ` ZZ ` -module of ` CC... |
| rezh 33919 | The ` ZZ ` -module of ` RR... |
| qqhval 33922 | Value of the canonical hom... |
| zrhf1ker 33923 | The kernel of the homomorp... |
| zrhchr 33924 | The kernel of the homomorp... |
| zrhker 33925 | The kernel of the homomorp... |
| zrhunitpreima 33926 | The preimage by ` ZRHom ` ... |
| elzrhunit 33927 | Condition for the image by... |
| elzdif0 33928 | Lemma for ~ qqhval2 . (Co... |
| qqhval2lem 33929 | Lemma for ~ qqhval2 . (Co... |
| qqhval2 33930 | Value of the canonical hom... |
| qqhvval 33931 | Value of the canonical hom... |
| qqh0 33932 | The image of ` 0 ` by the ... |
| qqh1 33933 | The image of ` 1 ` by the ... |
| qqhf 33934 | ` QQHom ` as a function. ... |
| qqhvq 33935 | The image of a quotient by... |
| qqhghm 33936 | The ` QQHom ` homomorphism... |
| qqhrhm 33937 | The ` QQHom ` homomorphism... |
| qqhnm 33938 | The norm of the image by `... |
| qqhcn 33939 | The ` QQHom ` homomorphism... |
| qqhucn 33940 | The ` QQHom ` homomorphism... |
| rrhval 33944 | Value of the canonical hom... |
| rrhcn 33945 | If the topology of ` R ` i... |
| rrhf 33946 | If the topology of ` R ` i... |
| isrrext 33948 | Express the property " ` R... |
| rrextnrg 33949 | An extension of ` RR ` is ... |
| rrextdrg 33950 | An extension of ` RR ` is ... |
| rrextnlm 33951 | The norm of an extension o... |
| rrextchr 33952 | The ring characteristic of... |
| rrextcusp 33953 | An extension of ` RR ` is ... |
| rrexttps 33954 | An extension of ` RR ` is ... |
| rrexthaus 33955 | The topology of an extensi... |
| rrextust 33956 | The uniformity of an exten... |
| rerrext 33957 | The field of the real numb... |
| cnrrext 33958 | The field of the complex n... |
| qqtopn 33959 | The topology of the field ... |
| rrhfe 33960 | If ` R ` is an extension o... |
| rrhcne 33961 | If ` R ` is an extension o... |
| rrhqima 33962 | The ` RRHom ` homomorphism... |
| rrh0 33963 | The image of ` 0 ` by the ... |
| xrhval 33966 | The value of the embedding... |
| zrhre 33967 | The ` ZRHom ` homomorphism... |
| qqhre 33968 | The ` QQHom ` homomorphism... |
| rrhre 33969 | The ` RRHom ` homomorphism... |
| relmntop 33972 | Manifold is a relation. (... |
| ismntoplly 33973 | Property of being a manifo... |
| ismntop 33974 | Property of being a manifo... |
| nexple 33975 | A lower bound for an expon... |
| indv 33978 | Value of the indicator fun... |
| indval 33979 | Value of the indicator fun... |
| indval2 33980 | Alternate value of the ind... |
| indf 33981 | An indicator function as a... |
| indfval 33982 | Value of the indicator fun... |
| ind1 33983 | Value of the indicator fun... |
| ind0 33984 | Value of the indicator fun... |
| ind1a 33985 | Value of the indicator fun... |
| indpi1 33986 | Preimage of the singleton ... |
| indsum 33987 | Finite sum of a product wi... |
| indsumin 33988 | Finite sum of a product wi... |
| prodindf 33989 | The product of indicators ... |
| indf1o 33990 | The bijection between a po... |
| indpreima 33991 | A function with range ` { ... |
| indf1ofs 33992 | The bijection between fini... |
| esumex 33995 | An extended sum is a set b... |
| esumcl 33996 | Closure for extended sum i... |
| esumeq12dvaf 33997 | Equality deduction for ext... |
| esumeq12dva 33998 | Equality deduction for ext... |
| esumeq12d 33999 | Equality deduction for ext... |
| esumeq1 34000 | Equality theorem for an ex... |
| esumeq1d 34001 | Equality theorem for an ex... |
| esumeq2 34002 | Equality theorem for exten... |
| esumeq2d 34003 | Equality deduction for ext... |
| esumeq2dv 34004 | Equality deduction for ext... |
| esumeq2sdv 34005 | Equality deduction for ext... |
| nfesum1 34006 | Bound-variable hypothesis ... |
| nfesum2 34007 | Bound-variable hypothesis ... |
| cbvesum 34008 | Change bound variable in a... |
| cbvesumv 34009 | Change bound variable in a... |
| esumid 34010 | Identify the extended sum ... |
| esumgsum 34011 | A finite extended sum is t... |
| esumval 34012 | Develop the value of the e... |
| esumel 34013 | The extended sum is a limi... |
| esumnul 34014 | Extended sum over the empt... |
| esum0 34015 | Extended sum of zero. (Co... |
| esumf1o 34016 | Re-index an extended sum u... |
| esumc 34017 | Convert from the collectio... |
| esumrnmpt 34018 | Rewrite an extended sum in... |
| esumsplit 34019 | Split an extended sum into... |
| esummono 34020 | Extended sum is monotonic.... |
| esumpad 34021 | Extend an extended sum by ... |
| esumpad2 34022 | Remove zeroes from an exte... |
| esumadd 34023 | Addition of infinite sums.... |
| esumle 34024 | If all of the terms of an ... |
| gsumesum 34025 | Relate a group sum on ` ( ... |
| esumlub 34026 | The extended sum is the lo... |
| esumaddf 34027 | Addition of infinite sums.... |
| esumlef 34028 | If all of the terms of an ... |
| esumcst 34029 | The extended sum of a cons... |
| esumsnf 34030 | The extended sum of a sing... |
| esumsn 34031 | The extended sum of a sing... |
| esumpr 34032 | Extended sum over a pair. ... |
| esumpr2 34033 | Extended sum over a pair, ... |
| esumrnmpt2 34034 | Rewrite an extended sum in... |
| esumfzf 34035 | Formulating a partial exte... |
| esumfsup 34036 | Formulating an extended su... |
| esumfsupre 34037 | Formulating an extended su... |
| esumss 34038 | Change the index set to a ... |
| esumpinfval 34039 | The value of the extended ... |
| esumpfinvallem 34040 | Lemma for ~ esumpfinval . ... |
| esumpfinval 34041 | The value of the extended ... |
| esumpfinvalf 34042 | Same as ~ esumpfinval , mi... |
| esumpinfsum 34043 | The value of the extended ... |
| esumpcvgval 34044 | The value of the extended ... |
| esumpmono 34045 | The partial sums in an ext... |
| esumcocn 34046 | Lemma for ~ esummulc2 and ... |
| esummulc1 34047 | An extended sum multiplied... |
| esummulc2 34048 | An extended sum multiplied... |
| esumdivc 34049 | An extended sum divided by... |
| hashf2 34050 | Lemma for ~ hasheuni . (C... |
| hasheuni 34051 | The cardinality of a disjo... |
| esumcvg 34052 | The sequence of partial su... |
| esumcvg2 34053 | Simpler version of ~ esumc... |
| esumcvgsum 34054 | The value of the extended ... |
| esumsup 34055 | Express an extended sum as... |
| esumgect 34056 | "Send ` n ` to ` +oo ` " i... |
| esumcvgre 34057 | All terms of a converging ... |
| esum2dlem 34058 | Lemma for ~ esum2d (finite... |
| esum2d 34059 | Write a double extended su... |
| esumiun 34060 | Sum over a nonnecessarily ... |
| ofceq 34063 | Equality theorem for funct... |
| ofcfval 34064 | Value of an operation appl... |
| ofcval 34065 | Evaluate a function/consta... |
| ofcfn 34066 | The function operation pro... |
| ofcfeqd2 34067 | Equality theorem for funct... |
| ofcfval3 34068 | General value of ` ( F oFC... |
| ofcf 34069 | The function/constant oper... |
| ofcfval2 34070 | The function operation exp... |
| ofcfval4 34071 | The function/constant oper... |
| ofcc 34072 | Left operation by a consta... |
| ofcof 34073 | Relate function operation ... |
| sigaex 34076 | Lemma for ~ issiga and ~ i... |
| sigaval 34077 | The set of sigma-algebra w... |
| issiga 34078 | An alternative definition ... |
| isrnsiga 34079 | The property of being a si... |
| 0elsiga 34080 | A sigma-algebra contains t... |
| baselsiga 34081 | A sigma-algebra contains i... |
| sigasspw 34082 | A sigma-algebra is a set o... |
| sigaclcu 34083 | A sigma-algebra is closed ... |
| sigaclcuni 34084 | A sigma-algebra is closed ... |
| sigaclfu 34085 | A sigma-algebra is closed ... |
| sigaclcu2 34086 | A sigma-algebra is closed ... |
| sigaclfu2 34087 | A sigma-algebra is closed ... |
| sigaclcu3 34088 | A sigma-algebra is closed ... |
| issgon 34089 | Property of being a sigma-... |
| sgon 34090 | A sigma-algebra is a sigma... |
| elsigass 34091 | An element of a sigma-alge... |
| elrnsiga 34092 | Dropping the base informat... |
| isrnsigau 34093 | The property of being a si... |
| unielsiga 34094 | A sigma-algebra contains i... |
| dmvlsiga 34095 | Lebesgue-measurable subset... |
| pwsiga 34096 | Any power set forms a sigm... |
| prsiga 34097 | The smallest possible sigm... |
| sigaclci 34098 | A sigma-algebra is closed ... |
| difelsiga 34099 | A sigma-algebra is closed ... |
| unelsiga 34100 | A sigma-algebra is closed ... |
| inelsiga 34101 | A sigma-algebra is closed ... |
| sigainb 34102 | Building a sigma-algebra f... |
| insiga 34103 | The intersection of a coll... |
| sigagenval 34106 | Value of the generated sig... |
| sigagensiga 34107 | A generated sigma-algebra ... |
| sgsiga 34108 | A generated sigma-algebra ... |
| unisg 34109 | The sigma-algebra generate... |
| dmsigagen 34110 | A sigma-algebra can be gen... |
| sssigagen 34111 | A set is a subset of the s... |
| sssigagen2 34112 | A subset of the generating... |
| elsigagen 34113 | Any element of a set is al... |
| elsigagen2 34114 | Any countable union of ele... |
| sigagenss 34115 | The generated sigma-algebr... |
| sigagenss2 34116 | Sufficient condition for i... |
| sigagenid 34117 | The sigma-algebra generate... |
| ispisys 34118 | The property of being a pi... |
| ispisys2 34119 | The property of being a pi... |
| inelpisys 34120 | Pi-systems are closed unde... |
| sigapisys 34121 | All sigma-algebras are pi-... |
| isldsys 34122 | The property of being a la... |
| pwldsys 34123 | The power set of the unive... |
| unelldsys 34124 | Lambda-systems are closed ... |
| sigaldsys 34125 | All sigma-algebras are lam... |
| ldsysgenld 34126 | The intersection of all la... |
| sigapildsyslem 34127 | Lemma for ~ sigapildsys . ... |
| sigapildsys 34128 | Sigma-algebra are exactly ... |
| ldgenpisyslem1 34129 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem2 34130 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem3 34131 | Lemma for ~ ldgenpisys . ... |
| ldgenpisys 34132 | The lambda system ` E ` ge... |
| dynkin 34133 | Dynkin's lambda-pi theorem... |
| isros 34134 | The property of being a ri... |
| rossspw 34135 | A ring of sets is a collec... |
| 0elros 34136 | A ring of sets contains th... |
| unelros 34137 | A ring of sets is closed u... |
| difelros 34138 | A ring of sets is closed u... |
| inelros 34139 | A ring of sets is closed u... |
| fiunelros 34140 | A ring of sets is closed u... |
| issros 34141 | The property of being a se... |
| srossspw 34142 | A semiring of sets is a co... |
| 0elsros 34143 | A semiring of sets contain... |
| inelsros 34144 | A semiring of sets is clos... |
| diffiunisros 34145 | In semiring of sets, compl... |
| rossros 34146 | Rings of sets are semiring... |
| brsiga 34149 | The Borel Algebra on real ... |
| brsigarn 34150 | The Borel Algebra is a sig... |
| brsigasspwrn 34151 | The Borel Algebra is a set... |
| unibrsiga 34152 | The union of the Borel Alg... |
| cldssbrsiga 34153 | A Borel Algebra contains a... |
| sxval 34156 | Value of the product sigma... |
| sxsiga 34157 | A product sigma-algebra is... |
| sxsigon 34158 | A product sigma-algebra is... |
| sxuni 34159 | The base set of a product ... |
| elsx 34160 | The cartesian product of t... |
| measbase 34163 | The base set of a measure ... |
| measval 34164 | The value of the ` measure... |
| ismeas 34165 | The property of being a me... |
| isrnmeas 34166 | The property of being a me... |
| dmmeas 34167 | The domain of a measure is... |
| measbasedom 34168 | The base set of a measure ... |
| measfrge0 34169 | A measure is a function ov... |
| measfn 34170 | A measure is a function on... |
| measvxrge0 34171 | The values of a measure ar... |
| measvnul 34172 | The measure of the empty s... |
| measge0 34173 | A measure is nonnegative. ... |
| measle0 34174 | If the measure of a given ... |
| measvun 34175 | The measure of a countable... |
| measxun2 34176 | The measure the union of t... |
| measun 34177 | The measure the union of t... |
| measvunilem 34178 | Lemma for ~ measvuni . (C... |
| measvunilem0 34179 | Lemma for ~ measvuni . (C... |
| measvuni 34180 | The measure of a countable... |
| measssd 34181 | A measure is monotone with... |
| measunl 34182 | A measure is sub-additive ... |
| measiuns 34183 | The measure of the union o... |
| measiun 34184 | A measure is sub-additive.... |
| meascnbl 34185 | A measure is continuous fr... |
| measinblem 34186 | Lemma for ~ measinb . (Co... |
| measinb 34187 | Building a measure restric... |
| measres 34188 | Building a measure restric... |
| measinb2 34189 | Building a measure restric... |
| measdivcst 34190 | Division of a measure by a... |
| measdivcstALTV 34191 | Alternate version of ~ mea... |
| cntmeas 34192 | The Counting measure is a ... |
| pwcntmeas 34193 | The counting measure is a ... |
| cntnevol 34194 | Counting and Lebesgue meas... |
| voliune 34195 | The Lebesgue measure funct... |
| volfiniune 34196 | The Lebesgue measure funct... |
| volmeas 34197 | The Lebesgue measure is a ... |
| ddeval1 34200 | Value of the delta measure... |
| ddeval0 34201 | Value of the delta measure... |
| ddemeas 34202 | The Dirac delta measure is... |
| relae 34206 | 'almost everywhere' is a r... |
| brae 34207 | 'almost everywhere' relati... |
| braew 34208 | 'almost everywhere' relati... |
| truae 34209 | A truth holds almost every... |
| aean 34210 | A conjunction holds almost... |
| faeval 34212 | Value of the 'almost every... |
| relfae 34213 | The 'almost everywhere' bu... |
| brfae 34214 | 'almost everywhere' relati... |
| ismbfm 34217 | The predicate " ` F ` is a... |
| elunirnmbfm 34218 | The property of being a me... |
| mbfmfun 34219 | A measurable function is a... |
| mbfmf 34220 | A measurable function as a... |
| isanmbfmOLD 34221 | Obsolete version of ~ isan... |
| mbfmcnvima 34222 | The preimage by a measurab... |
| isanmbfm 34223 | The predicate to be a meas... |
| mbfmbfmOLD 34224 | A measurable function to a... |
| mbfmbfm 34225 | A measurable function to a... |
| mbfmcst 34226 | A constant function is mea... |
| 1stmbfm 34227 | The first projection map i... |
| 2ndmbfm 34228 | The second projection map ... |
| imambfm 34229 | If the sigma-algebra in th... |
| cnmbfm 34230 | A continuous function is m... |
| mbfmco 34231 | The composition of two mea... |
| mbfmco2 34232 | The pair building of two m... |
| mbfmvolf 34233 | Measurable functions with ... |
| elmbfmvol2 34234 | Measurable functions with ... |
| mbfmcnt 34235 | All functions are measurab... |
| br2base 34236 | The base set for the gener... |
| dya2ub 34237 | An upper bound for a dyadi... |
| sxbrsigalem0 34238 | The closed half-spaces of ... |
| sxbrsigalem3 34239 | The sigma-algebra generate... |
| dya2iocival 34240 | The function ` I ` returns... |
| dya2iocress 34241 | Dyadic intervals are subse... |
| dya2iocbrsiga 34242 | Dyadic intervals are Borel... |
| dya2icobrsiga 34243 | Dyadic intervals are Borel... |
| dya2icoseg 34244 | For any point and any clos... |
| dya2icoseg2 34245 | For any point and any open... |
| dya2iocrfn 34246 | The function returning dya... |
| dya2iocct 34247 | The dyadic rectangle set i... |
| dya2iocnrect 34248 | For any point of an open r... |
| dya2iocnei 34249 | For any point of an open s... |
| dya2iocuni 34250 | Every open set of ` ( RR X... |
| dya2iocucvr 34251 | The dyadic rectangular set... |
| sxbrsigalem1 34252 | The Borel algebra on ` ( R... |
| sxbrsigalem2 34253 | The sigma-algebra generate... |
| sxbrsigalem4 34254 | The Borel algebra on ` ( R... |
| sxbrsigalem5 34255 | First direction for ~ sxbr... |
| sxbrsigalem6 34256 | First direction for ~ sxbr... |
| sxbrsiga 34257 | The product sigma-algebra ... |
| omsval 34260 | Value of the function mapp... |
| omsfval 34261 | Value of the outer measure... |
| omscl 34262 | A closure lemma for the co... |
| omsf 34263 | A constructed outer measur... |
| oms0 34264 | A constructed outer measur... |
| omsmon 34265 | A constructed outer measur... |
| omssubaddlem 34266 | For any small margin ` E `... |
| omssubadd 34267 | A constructed outer measur... |
| carsgval 34270 | Value of the Caratheodory ... |
| carsgcl 34271 | Closure of the Caratheodor... |
| elcarsg 34272 | Property of being a Carath... |
| baselcarsg 34273 | The universe set, ` O ` , ... |
| 0elcarsg 34274 | The empty set is Caratheod... |
| carsguni 34275 | The union of all Caratheod... |
| elcarsgss 34276 | Caratheodory measurable se... |
| difelcarsg 34277 | The Caratheodory measurabl... |
| inelcarsg 34278 | The Caratheodory measurabl... |
| unelcarsg 34279 | The Caratheodory-measurabl... |
| difelcarsg2 34280 | The Caratheodory-measurabl... |
| carsgmon 34281 | Utility lemma: Apply mono... |
| carsgsigalem 34282 | Lemma for the following th... |
| fiunelcarsg 34283 | The Caratheodory measurabl... |
| carsgclctunlem1 34284 | Lemma for ~ carsgclctun . ... |
| carsggect 34285 | The outer measure is count... |
| carsgclctunlem2 34286 | Lemma for ~ carsgclctun . ... |
| carsgclctunlem3 34287 | Lemma for ~ carsgclctun . ... |
| carsgclctun 34288 | The Caratheodory measurabl... |
| carsgsiga 34289 | The Caratheodory measurabl... |
| omsmeas 34290 | The restriction of a const... |
| pmeasmono 34291 | This theorem's hypotheses ... |
| pmeasadd 34292 | A premeasure on a ring of ... |
| itgeq12dv 34293 | Equality theorem for an in... |
| sitgval 34299 | Value of the simple functi... |
| issibf 34300 | The predicate " ` F ` is a... |
| sibf0 34301 | The constant zero function... |
| sibfmbl 34302 | A simple function is measu... |
| sibff 34303 | A simple function is a fun... |
| sibfrn 34304 | A simple function has fini... |
| sibfima 34305 | Any preimage of a singleto... |
| sibfinima 34306 | The measure of the interse... |
| sibfof 34307 | Applying function operatio... |
| sitgfval 34308 | Value of the Bochner integ... |
| sitgclg 34309 | Closure of the Bochner int... |
| sitgclbn 34310 | Closure of the Bochner int... |
| sitgclcn 34311 | Closure of the Bochner int... |
| sitgclre 34312 | Closure of the Bochner int... |
| sitg0 34313 | The integral of the consta... |
| sitgf 34314 | The integral for simple fu... |
| sitgaddlemb 34315 | Lemma for * sitgadd . (Co... |
| sitmval 34316 | Value of the simple functi... |
| sitmfval 34317 | Value of the integral dist... |
| sitmcl 34318 | Closure of the integral di... |
| sitmf 34319 | The integral metric as a f... |
| oddpwdc 34321 | Lemma for ~ eulerpart . T... |
| oddpwdcv 34322 | Lemma for ~ eulerpart : va... |
| eulerpartlemsv1 34323 | Lemma for ~ eulerpart . V... |
| eulerpartlemelr 34324 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv2 34325 | Lemma for ~ eulerpart . V... |
| eulerpartlemsf 34326 | Lemma for ~ eulerpart . (... |
| eulerpartlems 34327 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv3 34328 | Lemma for ~ eulerpart . V... |
| eulerpartlemgc 34329 | Lemma for ~ eulerpart . (... |
| eulerpartleme 34330 | Lemma for ~ eulerpart . (... |
| eulerpartlemv 34331 | Lemma for ~ eulerpart . (... |
| eulerpartlemo 34332 | Lemma for ~ eulerpart : ` ... |
| eulerpartlemd 34333 | Lemma for ~ eulerpart : ` ... |
| eulerpartlem1 34334 | Lemma for ~ eulerpart . (... |
| eulerpartlemb 34335 | Lemma for ~ eulerpart . T... |
| eulerpartlemt0 34336 | Lemma for ~ eulerpart . (... |
| eulerpartlemf 34337 | Lemma for ~ eulerpart : O... |
| eulerpartlemt 34338 | Lemma for ~ eulerpart . (... |
| eulerpartgbij 34339 | Lemma for ~ eulerpart : T... |
| eulerpartlemgv 34340 | Lemma for ~ eulerpart : va... |
| eulerpartlemr 34341 | Lemma for ~ eulerpart . (... |
| eulerpartlemmf 34342 | Lemma for ~ eulerpart . (... |
| eulerpartlemgvv 34343 | Lemma for ~ eulerpart : va... |
| eulerpartlemgu 34344 | Lemma for ~ eulerpart : R... |
| eulerpartlemgh 34345 | Lemma for ~ eulerpart : T... |
| eulerpartlemgf 34346 | Lemma for ~ eulerpart : I... |
| eulerpartlemgs2 34347 | Lemma for ~ eulerpart : T... |
| eulerpartlemn 34348 | Lemma for ~ eulerpart . (... |
| eulerpart 34349 | Euler's theorem on partiti... |
| subiwrd 34352 | Lemma for ~ sseqp1 . (Con... |
| subiwrdlen 34353 | Length of a subword of an ... |
| iwrdsplit 34354 | Lemma for ~ sseqp1 . (Con... |
| sseqval 34355 | Value of the strong sequen... |
| sseqfv1 34356 | Value of the strong sequen... |
| sseqfn 34357 | A strong recursive sequenc... |
| sseqmw 34358 | Lemma for ~ sseqf amd ~ ss... |
| sseqf 34359 | A strong recursive sequenc... |
| sseqfres 34360 | The first elements in the ... |
| sseqfv2 34361 | Value of the strong sequen... |
| sseqp1 34362 | Value of the strong sequen... |
| fiblem 34365 | Lemma for ~ fib0 , ~ fib1 ... |
| fib0 34366 | Value of the Fibonacci seq... |
| fib1 34367 | Value of the Fibonacci seq... |
| fibp1 34368 | Value of the Fibonacci seq... |
| fib2 34369 | Value of the Fibonacci seq... |
| fib3 34370 | Value of the Fibonacci seq... |
| fib4 34371 | Value of the Fibonacci seq... |
| fib5 34372 | Value of the Fibonacci seq... |
| fib6 34373 | Value of the Fibonacci seq... |
| elprob 34376 | The property of being a pr... |
| domprobmeas 34377 | A probability measure is a... |
| domprobsiga 34378 | The domain of a probabilit... |
| probtot 34379 | The probability of the uni... |
| prob01 34380 | A probability is an elemen... |
| probnul 34381 | The probability of the emp... |
| unveldomd 34382 | The universe is an element... |
| unveldom 34383 | The universe is an element... |
| nuleldmp 34384 | The empty set is an elemen... |
| probcun 34385 | The probability of the uni... |
| probun 34386 | The probability of the uni... |
| probdif 34387 | The probability of the dif... |
| probinc 34388 | A probability law is incre... |
| probdsb 34389 | The probability of the com... |
| probmeasd 34390 | A probability measure is a... |
| probvalrnd 34391 | The value of a probability... |
| probtotrnd 34392 | The probability of the uni... |
| totprobd 34393 | Law of total probability, ... |
| totprob 34394 | Law of total probability. ... |
| probfinmeasb 34395 | Build a probability measur... |
| probfinmeasbALTV 34396 | Alternate version of ~ pro... |
| probmeasb 34397 | Build a probability from a... |
| cndprobval 34400 | The value of the condition... |
| cndprobin 34401 | An identity linking condit... |
| cndprob01 34402 | The conditional probabilit... |
| cndprobtot 34403 | The conditional probabilit... |
| cndprobnul 34404 | The conditional probabilit... |
| cndprobprob 34405 | The conditional probabilit... |
| bayesth 34406 | Bayes Theorem. (Contribut... |
| rrvmbfm 34409 | A real-valued random varia... |
| isrrvv 34410 | Elementhood to the set of ... |
| rrvvf 34411 | A real-valued random varia... |
| rrvfn 34412 | A real-valued random varia... |
| rrvdm 34413 | The domain of a random var... |
| rrvrnss 34414 | The range of a random vari... |
| rrvf2 34415 | A real-valued random varia... |
| rrvdmss 34416 | The domain of a random var... |
| rrvfinvima 34417 | For a real-value random va... |
| 0rrv 34418 | The constant function equa... |
| rrvadd 34419 | The sum of two random vari... |
| rrvmulc 34420 | A random variable multipli... |
| rrvsum 34421 | An indexed sum of random v... |
| orvcval 34424 | Value of the preimage mapp... |
| orvcval2 34425 | Another way to express the... |
| elorvc 34426 | Elementhood of a preimage.... |
| orvcval4 34427 | The value of the preimage ... |
| orvcoel 34428 | If the relation produces o... |
| orvccel 34429 | If the relation produces c... |
| elorrvc 34430 | Elementhood of a preimage ... |
| orrvcval4 34431 | The value of the preimage ... |
| orrvcoel 34432 | If the relation produces o... |
| orrvccel 34433 | If the relation produces c... |
| orvcgteel 34434 | Preimage maps produced by ... |
| orvcelval 34435 | Preimage maps produced by ... |
| orvcelel 34436 | Preimage maps produced by ... |
| dstrvval 34437 | The value of the distribut... |
| dstrvprob 34438 | The distribution of a rand... |
| orvclteel 34439 | Preimage maps produced by ... |
| dstfrvel 34440 | Elementhood of preimage ma... |
| dstfrvunirn 34441 | The limit of all preimage ... |
| orvclteinc 34442 | Preimage maps produced by ... |
| dstfrvinc 34443 | A cumulative distribution ... |
| dstfrvclim1 34444 | The limit of the cumulativ... |
| coinfliplem 34445 | Division in the extended r... |
| coinflipprob 34446 | The ` P ` we defined for c... |
| coinflipspace 34447 | The space of our coin-flip... |
| coinflipuniv 34448 | The universe of our coin-f... |
| coinfliprv 34449 | The ` X ` we defined for c... |
| coinflippv 34450 | The probability of heads i... |
| coinflippvt 34451 | The probability of tails i... |
| ballotlemoex 34452 | ` O ` is a set. (Contribu... |
| ballotlem1 34453 | The size of the universe i... |
| ballotlemelo 34454 | Elementhood in ` O ` . (C... |
| ballotlem2 34455 | The probability that the f... |
| ballotlemfval 34456 | The value of ` F ` . (Con... |
| ballotlemfelz 34457 | ` ( F `` C ) ` has values ... |
| ballotlemfp1 34458 | If the ` J ` th ballot is ... |
| ballotlemfc0 34459 | ` F ` takes value 0 betwee... |
| ballotlemfcc 34460 | ` F ` takes value 0 betwee... |
| ballotlemfmpn 34461 | ` ( F `` C ) ` finishes co... |
| ballotlemfval0 34462 | ` ( F `` C ) ` always star... |
| ballotleme 34463 | Elements of ` E ` . (Cont... |
| ballotlemodife 34464 | Elements of ` ( O \ E ) ` ... |
| ballotlem4 34465 | If the first pick is a vot... |
| ballotlem5 34466 | If A is not ahead througho... |
| ballotlemi 34467 | Value of ` I ` for a given... |
| ballotlemiex 34468 | Properties of ` ( I `` C )... |
| ballotlemi1 34469 | The first tie cannot be re... |
| ballotlemii 34470 | The first tie cannot be re... |
| ballotlemsup 34471 | The set of zeroes of ` F `... |
| ballotlemimin 34472 | ` ( I `` C ) ` is the firs... |
| ballotlemic 34473 | If the first vote is for B... |
| ballotlem1c 34474 | If the first vote is for A... |
| ballotlemsval 34475 | Value of ` S ` . (Contrib... |
| ballotlemsv 34476 | Value of ` S ` evaluated a... |
| ballotlemsgt1 34477 | ` S ` maps values less tha... |
| ballotlemsdom 34478 | Domain of ` S ` for a give... |
| ballotlemsel1i 34479 | The range ` ( 1 ... ( I ``... |
| ballotlemsf1o 34480 | The defined ` S ` is a bij... |
| ballotlemsi 34481 | The image by ` S ` of the ... |
| ballotlemsima 34482 | The image by ` S ` of an i... |
| ballotlemieq 34483 | If two countings share the... |
| ballotlemrval 34484 | Value of ` R ` . (Contrib... |
| ballotlemscr 34485 | The image of ` ( R `` C ) ... |
| ballotlemrv 34486 | Value of ` R ` evaluated a... |
| ballotlemrv1 34487 | Value of ` R ` before the ... |
| ballotlemrv2 34488 | Value of ` R ` after the t... |
| ballotlemro 34489 | Range of ` R ` is included... |
| ballotlemgval 34490 | Expand the value of ` .^ `... |
| ballotlemgun 34491 | A property of the defined ... |
| ballotlemfg 34492 | Express the value of ` ( F... |
| ballotlemfrc 34493 | Express the value of ` ( F... |
| ballotlemfrci 34494 | Reverse counting preserves... |
| ballotlemfrceq 34495 | Value of ` F ` for a rever... |
| ballotlemfrcn0 34496 | Value of ` F ` for a rever... |
| ballotlemrc 34497 | Range of ` R ` . (Contrib... |
| ballotlemirc 34498 | Applying ` R ` does not ch... |
| ballotlemrinv0 34499 | Lemma for ~ ballotlemrinv ... |
| ballotlemrinv 34500 | ` R ` is its own inverse :... |
| ballotlem1ri 34501 | When the vote on the first... |
| ballotlem7 34502 | ` R ` is a bijection betwe... |
| ballotlem8 34503 | There are as many counting... |
| ballotth 34504 | Bertrand's ballot problem ... |
| sgncl 34505 | Closure of the signum. (C... |
| sgnclre 34506 | Closure of the signum. (C... |
| sgnneg 34507 | Negation of the signum. (... |
| sgn3da 34508 | A conditional containing a... |
| sgnmul 34509 | Signum of a product. (Con... |
| sgnmulrp2 34510 | Multiplication by a positi... |
| sgnsub 34511 | Subtraction of a number of... |
| sgnnbi 34512 | Negative signum. (Contrib... |
| sgnpbi 34513 | Positive signum. (Contrib... |
| sgn0bi 34514 | Zero signum. (Contributed... |
| sgnsgn 34515 | Signum is idempotent. (Co... |
| sgnmulsgn 34516 | If two real numbers are of... |
| sgnmulsgp 34517 | If two real numbers are of... |
| fzssfzo 34518 | Condition for an integer i... |
| gsumncl 34519 | Closure of a group sum in ... |
| gsumnunsn 34520 | Closure of a group sum in ... |
| ccatmulgnn0dir 34521 | Concatenation of words fol... |
| ofcccat 34522 | Letterwise operations on w... |
| ofcs1 34523 | Letterwise operations on a... |
| ofcs2 34524 | Letterwise operations on a... |
| plymul02 34525 | Product of a polynomial wi... |
| plymulx0 34526 | Coefficients of a polynomi... |
| plymulx 34527 | Coefficients of a polynomi... |
| plyrecld 34528 | Closure of a polynomial wi... |
| signsplypnf 34529 | The quotient of a polynomi... |
| signsply0 34530 | Lemma for the rule of sign... |
| signspval 34531 | The value of the skipping ... |
| signsw0glem 34532 | Neutral element property o... |
| signswbase 34533 | The base of ` W ` is the u... |
| signswplusg 34534 | The operation of ` W ` . ... |
| signsw0g 34535 | The neutral element of ` W... |
| signswmnd 34536 | ` W ` is a monoid structur... |
| signswrid 34537 | The zero-skipping operatio... |
| signswlid 34538 | The zero-skipping operatio... |
| signswn0 34539 | The zero-skipping operatio... |
| signswch 34540 | The zero-skipping operatio... |
| signslema 34541 | Computational part of ~~? ... |
| signstfv 34542 | Value of the zero-skipping... |
| signstfval 34543 | Value of the zero-skipping... |
| signstcl 34544 | Closure of the zero skippi... |
| signstf 34545 | The zero skipping sign wor... |
| signstlen 34546 | Length of the zero skippin... |
| signstf0 34547 | Sign of a single letter wo... |
| signstfvn 34548 | Zero-skipping sign in a wo... |
| signsvtn0 34549 | If the last letter is nonz... |
| signstfvp 34550 | Zero-skipping sign in a wo... |
| signstfvneq0 34551 | In case the first letter i... |
| signstfvcl 34552 | Closure of the zero skippi... |
| signstfvc 34553 | Zero-skipping sign in a wo... |
| signstres 34554 | Restriction of a zero skip... |
| signstfveq0a 34555 | Lemma for ~ signstfveq0 . ... |
| signstfveq0 34556 | In case the last letter is... |
| signsvvfval 34557 | The value of ` V ` , which... |
| signsvvf 34558 | ` V ` is a function. (Con... |
| signsvf0 34559 | There is no change of sign... |
| signsvf1 34560 | In a single-letter word, w... |
| signsvfn 34561 | Number of changes in a wor... |
| signsvtp 34562 | Adding a letter of the sam... |
| signsvtn 34563 | Adding a letter of a diffe... |
| signsvfpn 34564 | Adding a letter of the sam... |
| signsvfnn 34565 | Adding a letter of a diffe... |
| signlem0 34566 | Adding a zero as the highe... |
| signshf 34567 | ` H ` , corresponding to t... |
| signshwrd 34568 | ` H ` , corresponding to t... |
| signshlen 34569 | Length of ` H ` , correspo... |
| signshnz 34570 | ` H ` is not the empty wor... |
| iblidicc 34571 | The identity function is i... |
| rpsqrtcn 34572 | Continuity of the real pos... |
| divsqrtid 34573 | A real number divided by i... |
| cxpcncf1 34574 | The power function on comp... |
| efmul2picn 34575 | Multiplying by ` ( _i x. (... |
| fct2relem 34576 | Lemma for ~ ftc2re . (Con... |
| ftc2re 34577 | The Fundamental Theorem of... |
| fdvposlt 34578 | Functions with a positive ... |
| fdvneggt 34579 | Functions with a negative ... |
| fdvposle 34580 | Functions with a nonnegati... |
| fdvnegge 34581 | Functions with a nonpositi... |
| prodfzo03 34582 | A product of three factors... |
| actfunsnf1o 34583 | The action ` F ` of extend... |
| actfunsnrndisj 34584 | The action ` F ` of extend... |
| itgexpif 34585 | The basis for the circle m... |
| fsum2dsub 34586 | Lemma for ~ breprexp - Re-... |
| reprval 34589 | Value of the representatio... |
| repr0 34590 | There is exactly one repre... |
| reprf 34591 | Members of the representat... |
| reprsum 34592 | Sums of values of the memb... |
| reprle 34593 | Upper bound to the terms i... |
| reprsuc 34594 | Express the representation... |
| reprfi 34595 | Bounded representations ar... |
| reprss 34596 | Representations with terms... |
| reprinrn 34597 | Representations with term ... |
| reprlt 34598 | There are no representatio... |
| hashreprin 34599 | Express a sum of represent... |
| reprgt 34600 | There are no representatio... |
| reprinfz1 34601 | For the representation of ... |
| reprfi2 34602 | Corollary of ~ reprinfz1 .... |
| reprfz1 34603 | Corollary of ~ reprinfz1 .... |
| hashrepr 34604 | Develop the number of repr... |
| reprpmtf1o 34605 | Transposing ` 0 ` and ` X ... |
| reprdifc 34606 | Express the representation... |
| chpvalz 34607 | Value of the second Chebys... |
| chtvalz 34608 | Value of the Chebyshev fun... |
| breprexplema 34609 | Lemma for ~ breprexp (indu... |
| breprexplemb 34610 | Lemma for ~ breprexp (clos... |
| breprexplemc 34611 | Lemma for ~ breprexp (indu... |
| breprexp 34612 | Express the ` S ` th power... |
| breprexpnat 34613 | Express the ` S ` th power... |
| vtsval 34616 | Value of the Vinogradov tr... |
| vtscl 34617 | Closure of the Vinogradov ... |
| vtsprod 34618 | Express the Vinogradov tri... |
| circlemeth 34619 | The Hardy, Littlewood and ... |
| circlemethnat 34620 | The Hardy, Littlewood and ... |
| circlevma 34621 | The Circle Method, where t... |
| circlemethhgt 34622 | The circle method, where t... |
| hgt750lemc 34626 | An upper bound to the summ... |
| hgt750lemd 34627 | An upper bound to the summ... |
| hgt749d 34628 | A deduction version of ~ a... |
| logdivsqrle 34629 | Conditions for ` ( ( log `... |
| hgt750lem 34630 | Lemma for ~ tgoldbachgtd .... |
| hgt750lem2 34631 | Decimal multiplication gal... |
| hgt750lemf 34632 | Lemma for the statement 7.... |
| hgt750lemg 34633 | Lemma for the statement 7.... |
| oddprm2 34634 | Two ways to write the set ... |
| hgt750lemb 34635 | An upper bound on the cont... |
| hgt750lema 34636 | An upper bound on the cont... |
| hgt750leme 34637 | An upper bound on the cont... |
| tgoldbachgnn 34638 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtde 34639 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtda 34640 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtd 34641 | Odd integers greater than ... |
| tgoldbachgt 34642 | Odd integers greater than ... |
| istrkg2d 34645 | Property of fulfilling dim... |
| axtglowdim2ALTV 34646 | Alternate version of ~ axt... |
| axtgupdim2ALTV 34647 | Alternate version of ~ axt... |
| afsval 34650 | Value of the AFS relation ... |
| brafs 34651 | Binary relation form of th... |
| tg5segofs 34652 | Rephrase ~ axtg5seg using ... |
| lpadval 34655 | Value of the ` leftpad ` f... |
| lpadlem1 34656 | Lemma for the ` leftpad ` ... |
| lpadlem3 34657 | Lemma for ~ lpadlen1 . (C... |
| lpadlen1 34658 | Length of a left-padded wo... |
| lpadlem2 34659 | Lemma for the ` leftpad ` ... |
| lpadlen2 34660 | Length of a left-padded wo... |
| lpadmax 34661 | Length of a left-padded wo... |
| lpadleft 34662 | The contents of prefix of ... |
| lpadright 34663 | The suffix of a left-padde... |
| bnj170 34676 | ` /\ ` -manipulation. (Co... |
| bnj240 34677 | ` /\ ` -manipulation. (Co... |
| bnj248 34678 | ` /\ ` -manipulation. (Co... |
| bnj250 34679 | ` /\ ` -manipulation. (Co... |
| bnj251 34680 | ` /\ ` -manipulation. (Co... |
| bnj252 34681 | ` /\ ` -manipulation. (Co... |
| bnj253 34682 | ` /\ ` -manipulation. (Co... |
| bnj255 34683 | ` /\ ` -manipulation. (Co... |
| bnj256 34684 | ` /\ ` -manipulation. (Co... |
| bnj257 34685 | ` /\ ` -manipulation. (Co... |
| bnj258 34686 | ` /\ ` -manipulation. (Co... |
| bnj268 34687 | ` /\ ` -manipulation. (Co... |
| bnj290 34688 | ` /\ ` -manipulation. (Co... |
| bnj291 34689 | ` /\ ` -manipulation. (Co... |
| bnj312 34690 | ` /\ ` -manipulation. (Co... |
| bnj334 34691 | ` /\ ` -manipulation. (Co... |
| bnj345 34692 | ` /\ ` -manipulation. (Co... |
| bnj422 34693 | ` /\ ` -manipulation. (Co... |
| bnj432 34694 | ` /\ ` -manipulation. (Co... |
| bnj446 34695 | ` /\ ` -manipulation. (Co... |
| bnj23 34696 | First-order logic and set ... |
| bnj31 34697 | First-order logic and set ... |
| bnj62 34698 | First-order logic and set ... |
| bnj89 34699 | First-order logic and set ... |
| bnj90 34700 | First-order logic and set ... |
| bnj101 34701 | First-order logic and set ... |
| bnj105 34702 | First-order logic and set ... |
| bnj115 34703 | First-order logic and set ... |
| bnj132 34704 | First-order logic and set ... |
| bnj133 34705 | First-order logic and set ... |
| bnj156 34706 | First-order logic and set ... |
| bnj158 34707 | First-order logic and set ... |
| bnj168 34708 | First-order logic and set ... |
| bnj206 34709 | First-order logic and set ... |
| bnj216 34710 | First-order logic and set ... |
| bnj219 34711 | First-order logic and set ... |
| bnj226 34712 | First-order logic and set ... |
| bnj228 34713 | First-order logic and set ... |
| bnj519 34714 | First-order logic and set ... |
| bnj524 34715 | First-order logic and set ... |
| bnj525 34716 | First-order logic and set ... |
| bnj534 34717 | First-order logic and set ... |
| bnj538 34718 | First-order logic and set ... |
| bnj529 34719 | First-order logic and set ... |
| bnj551 34720 | First-order logic and set ... |
| bnj563 34721 | First-order logic and set ... |
| bnj564 34722 | First-order logic and set ... |
| bnj593 34723 | First-order logic and set ... |
| bnj596 34724 | First-order logic and set ... |
| bnj610 34725 | Pass from equality ( ` x =... |
| bnj642 34726 | ` /\ ` -manipulation. (Co... |
| bnj643 34727 | ` /\ ` -manipulation. (Co... |
| bnj645 34728 | ` /\ ` -manipulation. (Co... |
| bnj658 34729 | ` /\ ` -manipulation. (Co... |
| bnj667 34730 | ` /\ ` -manipulation. (Co... |
| bnj705 34731 | ` /\ ` -manipulation. (Co... |
| bnj706 34732 | ` /\ ` -manipulation. (Co... |
| bnj707 34733 | ` /\ ` -manipulation. (Co... |
| bnj708 34734 | ` /\ ` -manipulation. (Co... |
| bnj721 34735 | ` /\ ` -manipulation. (Co... |
| bnj832 34736 | ` /\ ` -manipulation. (Co... |
| bnj835 34737 | ` /\ ` -manipulation. (Co... |
| bnj836 34738 | ` /\ ` -manipulation. (Co... |
| bnj837 34739 | ` /\ ` -manipulation. (Co... |
| bnj769 34740 | ` /\ ` -manipulation. (Co... |
| bnj770 34741 | ` /\ ` -manipulation. (Co... |
| bnj771 34742 | ` /\ ` -manipulation. (Co... |
| bnj887 34743 | ` /\ ` -manipulation. (Co... |
| bnj918 34744 | First-order logic and set ... |
| bnj919 34745 | First-order logic and set ... |
| bnj923 34746 | First-order logic and set ... |
| bnj927 34747 | First-order logic and set ... |
| bnj931 34748 | First-order logic and set ... |
| bnj937 34749 | First-order logic and set ... |
| bnj941 34750 | First-order logic and set ... |
| bnj945 34751 | Technical lemma for ~ bnj6... |
| bnj946 34752 | First-order logic and set ... |
| bnj951 34753 | ` /\ ` -manipulation. (Co... |
| bnj956 34754 | First-order logic and set ... |
| bnj976 34755 | First-order logic and set ... |
| bnj982 34756 | First-order logic and set ... |
| bnj1019 34757 | First-order logic and set ... |
| bnj1023 34758 | First-order logic and set ... |
| bnj1095 34759 | First-order logic and set ... |
| bnj1096 34760 | First-order logic and set ... |
| bnj1098 34761 | First-order logic and set ... |
| bnj1101 34762 | First-order logic and set ... |
| bnj1113 34763 | First-order logic and set ... |
| bnj1109 34764 | First-order logic and set ... |
| bnj1131 34765 | First-order logic and set ... |
| bnj1138 34766 | First-order logic and set ... |
| bnj1142 34767 | First-order logic and set ... |
| bnj1143 34768 | First-order logic and set ... |
| bnj1146 34769 | First-order logic and set ... |
| bnj1149 34770 | First-order logic and set ... |
| bnj1185 34771 | First-order logic and set ... |
| bnj1196 34772 | First-order logic and set ... |
| bnj1198 34773 | First-order logic and set ... |
| bnj1209 34774 | First-order logic and set ... |
| bnj1211 34775 | First-order logic and set ... |
| bnj1213 34776 | First-order logic and set ... |
| bnj1212 34777 | First-order logic and set ... |
| bnj1219 34778 | First-order logic and set ... |
| bnj1224 34779 | First-order logic and set ... |
| bnj1230 34780 | First-order logic and set ... |
| bnj1232 34781 | First-order logic and set ... |
| bnj1235 34782 | First-order logic and set ... |
| bnj1239 34783 | First-order logic and set ... |
| bnj1238 34784 | First-order logic and set ... |
| bnj1241 34785 | First-order logic and set ... |
| bnj1247 34786 | First-order logic and set ... |
| bnj1254 34787 | First-order logic and set ... |
| bnj1262 34788 | First-order logic and set ... |
| bnj1266 34789 | First-order logic and set ... |
| bnj1265 34790 | First-order logic and set ... |
| bnj1275 34791 | First-order logic and set ... |
| bnj1276 34792 | First-order logic and set ... |
| bnj1292 34793 | First-order logic and set ... |
| bnj1293 34794 | First-order logic and set ... |
| bnj1294 34795 | First-order logic and set ... |
| bnj1299 34796 | First-order logic and set ... |
| bnj1304 34797 | First-order logic and set ... |
| bnj1316 34798 | First-order logic and set ... |
| bnj1317 34799 | First-order logic and set ... |
| bnj1322 34800 | First-order logic and set ... |
| bnj1340 34801 | First-order logic and set ... |
| bnj1345 34802 | First-order logic and set ... |
| bnj1350 34803 | First-order logic and set ... |
| bnj1351 34804 | First-order logic and set ... |
| bnj1352 34805 | First-order logic and set ... |
| bnj1361 34806 | First-order logic and set ... |
| bnj1366 34807 | First-order logic and set ... |
| bnj1379 34808 | First-order logic and set ... |
| bnj1383 34809 | First-order logic and set ... |
| bnj1385 34810 | First-order logic and set ... |
| bnj1386 34811 | First-order logic and set ... |
| bnj1397 34812 | First-order logic and set ... |
| bnj1400 34813 | First-order logic and set ... |
| bnj1405 34814 | First-order logic and set ... |
| bnj1422 34815 | First-order logic and set ... |
| bnj1424 34816 | First-order logic and set ... |
| bnj1436 34817 | First-order logic and set ... |
| bnj1441 34818 | First-order logic and set ... |
| bnj1441g 34819 | First-order logic and set ... |
| bnj1454 34820 | First-order logic and set ... |
| bnj1459 34821 | First-order logic and set ... |
| bnj1464 34822 | Conversion of implicit sub... |
| bnj1465 34823 | First-order logic and set ... |
| bnj1468 34824 | Conversion of implicit sub... |
| bnj1476 34825 | First-order logic and set ... |
| bnj1502 34826 | First-order logic and set ... |
| bnj1503 34827 | First-order logic and set ... |
| bnj1517 34828 | First-order logic and set ... |
| bnj1521 34829 | First-order logic and set ... |
| bnj1533 34830 | First-order logic and set ... |
| bnj1534 34831 | First-order logic and set ... |
| bnj1536 34832 | First-order logic and set ... |
| bnj1538 34833 | First-order logic and set ... |
| bnj1541 34834 | First-order logic and set ... |
| bnj1542 34835 | First-order logic and set ... |
| bnj110 34836 | Well-founded induction res... |
| bnj157 34837 | Well-founded induction res... |
| bnj66 34838 | Technical lemma for ~ bnj6... |
| bnj91 34839 | First-order logic and set ... |
| bnj92 34840 | First-order logic and set ... |
| bnj93 34841 | Technical lemma for ~ bnj9... |
| bnj95 34842 | Technical lemma for ~ bnj1... |
| bnj96 34843 | Technical lemma for ~ bnj1... |
| bnj97 34844 | Technical lemma for ~ bnj1... |
| bnj98 34845 | Technical lemma for ~ bnj1... |
| bnj106 34846 | First-order logic and set ... |
| bnj118 34847 | First-order logic and set ... |
| bnj121 34848 | First-order logic and set ... |
| bnj124 34849 | Technical lemma for ~ bnj1... |
| bnj125 34850 | Technical lemma for ~ bnj1... |
| bnj126 34851 | Technical lemma for ~ bnj1... |
| bnj130 34852 | Technical lemma for ~ bnj1... |
| bnj149 34853 | Technical lemma for ~ bnj1... |
| bnj150 34854 | Technical lemma for ~ bnj1... |
| bnj151 34855 | Technical lemma for ~ bnj1... |
| bnj154 34856 | Technical lemma for ~ bnj1... |
| bnj155 34857 | Technical lemma for ~ bnj1... |
| bnj153 34858 | Technical lemma for ~ bnj8... |
| bnj207 34859 | Technical lemma for ~ bnj8... |
| bnj213 34860 | First-order logic and set ... |
| bnj222 34861 | Technical lemma for ~ bnj2... |
| bnj229 34862 | Technical lemma for ~ bnj5... |
| bnj517 34863 | Technical lemma for ~ bnj5... |
| bnj518 34864 | Technical lemma for ~ bnj8... |
| bnj523 34865 | Technical lemma for ~ bnj8... |
| bnj526 34866 | Technical lemma for ~ bnj8... |
| bnj528 34867 | Technical lemma for ~ bnj8... |
| bnj535 34868 | Technical lemma for ~ bnj8... |
| bnj539 34869 | Technical lemma for ~ bnj8... |
| bnj540 34870 | Technical lemma for ~ bnj8... |
| bnj543 34871 | Technical lemma for ~ bnj8... |
| bnj544 34872 | Technical lemma for ~ bnj8... |
| bnj545 34873 | Technical lemma for ~ bnj8... |
| bnj546 34874 | Technical lemma for ~ bnj8... |
| bnj548 34875 | Technical lemma for ~ bnj8... |
| bnj553 34876 | Technical lemma for ~ bnj8... |
| bnj554 34877 | Technical lemma for ~ bnj8... |
| bnj556 34878 | Technical lemma for ~ bnj8... |
| bnj557 34879 | Technical lemma for ~ bnj8... |
| bnj558 34880 | Technical lemma for ~ bnj8... |
| bnj561 34881 | Technical lemma for ~ bnj8... |
| bnj562 34882 | Technical lemma for ~ bnj8... |
| bnj570 34883 | Technical lemma for ~ bnj8... |
| bnj571 34884 | Technical lemma for ~ bnj8... |
| bnj605 34885 | Technical lemma. This lem... |
| bnj581 34886 | Technical lemma for ~ bnj5... |
| bnj589 34887 | Technical lemma for ~ bnj8... |
| bnj590 34888 | Technical lemma for ~ bnj8... |
| bnj591 34889 | Technical lemma for ~ bnj8... |
| bnj594 34890 | Technical lemma for ~ bnj8... |
| bnj580 34891 | Technical lemma for ~ bnj5... |
| bnj579 34892 | Technical lemma for ~ bnj8... |
| bnj602 34893 | Equality theorem for the `... |
| bnj607 34894 | Technical lemma for ~ bnj8... |
| bnj609 34895 | Technical lemma for ~ bnj8... |
| bnj611 34896 | Technical lemma for ~ bnj8... |
| bnj600 34897 | Technical lemma for ~ bnj8... |
| bnj601 34898 | Technical lemma for ~ bnj8... |
| bnj852 34899 | Technical lemma for ~ bnj6... |
| bnj864 34900 | Technical lemma for ~ bnj6... |
| bnj865 34901 | Technical lemma for ~ bnj6... |
| bnj873 34902 | Technical lemma for ~ bnj6... |
| bnj849 34903 | Technical lemma for ~ bnj6... |
| bnj882 34904 | Definition (using hypothes... |
| bnj18eq1 34905 | Equality theorem for trans... |
| bnj893 34906 | Property of ` _trCl ` . U... |
| bnj900 34907 | Technical lemma for ~ bnj6... |
| bnj906 34908 | Property of ` _trCl ` . (... |
| bnj908 34909 | Technical lemma for ~ bnj6... |
| bnj911 34910 | Technical lemma for ~ bnj6... |
| bnj916 34911 | Technical lemma for ~ bnj6... |
| bnj917 34912 | Technical lemma for ~ bnj6... |
| bnj934 34913 | Technical lemma for ~ bnj6... |
| bnj929 34914 | Technical lemma for ~ bnj6... |
| bnj938 34915 | Technical lemma for ~ bnj6... |
| bnj944 34916 | Technical lemma for ~ bnj6... |
| bnj953 34917 | Technical lemma for ~ bnj6... |
| bnj958 34918 | Technical lemma for ~ bnj6... |
| bnj1000 34919 | Technical lemma for ~ bnj8... |
| bnj965 34920 | Technical lemma for ~ bnj8... |
| bnj964 34921 | Technical lemma for ~ bnj6... |
| bnj966 34922 | Technical lemma for ~ bnj6... |
| bnj967 34923 | Technical lemma for ~ bnj6... |
| bnj969 34924 | Technical lemma for ~ bnj6... |
| bnj970 34925 | Technical lemma for ~ bnj6... |
| bnj910 34926 | Technical lemma for ~ bnj6... |
| bnj978 34927 | Technical lemma for ~ bnj6... |
| bnj981 34928 | Technical lemma for ~ bnj6... |
| bnj983 34929 | Technical lemma for ~ bnj6... |
| bnj984 34930 | Technical lemma for ~ bnj6... |
| bnj985v 34931 | Version of ~ bnj985 with a... |
| bnj985 34932 | Technical lemma for ~ bnj6... |
| bnj986 34933 | Technical lemma for ~ bnj6... |
| bnj996 34934 | Technical lemma for ~ bnj6... |
| bnj998 34935 | Technical lemma for ~ bnj6... |
| bnj999 34936 | Technical lemma for ~ bnj6... |
| bnj1001 34937 | Technical lemma for ~ bnj6... |
| bnj1006 34938 | Technical lemma for ~ bnj6... |
| bnj1014 34939 | Technical lemma for ~ bnj6... |
| bnj1015 34940 | Technical lemma for ~ bnj6... |
| bnj1018g 34941 | Version of ~ bnj1018 with ... |
| bnj1018 34942 | Technical lemma for ~ bnj6... |
| bnj1020 34943 | Technical lemma for ~ bnj6... |
| bnj1021 34944 | Technical lemma for ~ bnj6... |
| bnj907 34945 | Technical lemma for ~ bnj6... |
| bnj1029 34946 | Property of ` _trCl ` . (... |
| bnj1033 34947 | Technical lemma for ~ bnj6... |
| bnj1034 34948 | Technical lemma for ~ bnj6... |
| bnj1039 34949 | Technical lemma for ~ bnj6... |
| bnj1040 34950 | Technical lemma for ~ bnj6... |
| bnj1047 34951 | Technical lemma for ~ bnj6... |
| bnj1049 34952 | Technical lemma for ~ bnj6... |
| bnj1052 34953 | Technical lemma for ~ bnj6... |
| bnj1053 34954 | Technical lemma for ~ bnj6... |
| bnj1071 34955 | Technical lemma for ~ bnj6... |
| bnj1083 34956 | Technical lemma for ~ bnj6... |
| bnj1090 34957 | Technical lemma for ~ bnj6... |
| bnj1093 34958 | Technical lemma for ~ bnj6... |
| bnj1097 34959 | Technical lemma for ~ bnj6... |
| bnj1110 34960 | Technical lemma for ~ bnj6... |
| bnj1112 34961 | Technical lemma for ~ bnj6... |
| bnj1118 34962 | Technical lemma for ~ bnj6... |
| bnj1121 34963 | Technical lemma for ~ bnj6... |
| bnj1123 34964 | Technical lemma for ~ bnj6... |
| bnj1030 34965 | Technical lemma for ~ bnj6... |
| bnj1124 34966 | Property of ` _trCl ` . (... |
| bnj1133 34967 | Technical lemma for ~ bnj6... |
| bnj1128 34968 | Technical lemma for ~ bnj6... |
| bnj1127 34969 | Property of ` _trCl ` . (... |
| bnj1125 34970 | Property of ` _trCl ` . (... |
| bnj1145 34971 | Technical lemma for ~ bnj6... |
| bnj1147 34972 | Property of ` _trCl ` . (... |
| bnj1137 34973 | Property of ` _trCl ` . (... |
| bnj1148 34974 | Property of ` _pred ` . (... |
| bnj1136 34975 | Technical lemma for ~ bnj6... |
| bnj1152 34976 | Technical lemma for ~ bnj6... |
| bnj1154 34977 | Property of ` Fr ` . (Con... |
| bnj1171 34978 | Technical lemma for ~ bnj6... |
| bnj1172 34979 | Technical lemma for ~ bnj6... |
| bnj1173 34980 | Technical lemma for ~ bnj6... |
| bnj1174 34981 | Technical lemma for ~ bnj6... |
| bnj1175 34982 | Technical lemma for ~ bnj6... |
| bnj1176 34983 | Technical lemma for ~ bnj6... |
| bnj1177 34984 | Technical lemma for ~ bnj6... |
| bnj1186 34985 | Technical lemma for ~ bnj6... |
| bnj1190 34986 | Technical lemma for ~ bnj6... |
| bnj1189 34987 | Technical lemma for ~ bnj6... |
| bnj69 34988 | Existence of a minimal ele... |
| bnj1228 34989 | Existence of a minimal ele... |
| bnj1204 34990 | Well-founded induction. T... |
| bnj1234 34991 | Technical lemma for ~ bnj6... |
| bnj1245 34992 | Technical lemma for ~ bnj6... |
| bnj1256 34993 | Technical lemma for ~ bnj6... |
| bnj1259 34994 | Technical lemma for ~ bnj6... |
| bnj1253 34995 | Technical lemma for ~ bnj6... |
| bnj1279 34996 | Technical lemma for ~ bnj6... |
| bnj1286 34997 | Technical lemma for ~ bnj6... |
| bnj1280 34998 | Technical lemma for ~ bnj6... |
| bnj1296 34999 | Technical lemma for ~ bnj6... |
| bnj1309 35000 | Technical lemma for ~ bnj6... |
| bnj1307 35001 | Technical lemma for ~ bnj6... |
| bnj1311 35002 | Technical lemma for ~ bnj6... |
| bnj1318 35003 | Technical lemma for ~ bnj6... |
| bnj1326 35004 | Technical lemma for ~ bnj6... |
| bnj1321 35005 | Technical lemma for ~ bnj6... |
| bnj1364 35006 | Property of ` _FrSe ` . (... |
| bnj1371 35007 | Technical lemma for ~ bnj6... |
| bnj1373 35008 | Technical lemma for ~ bnj6... |
| bnj1374 35009 | Technical lemma for ~ bnj6... |
| bnj1384 35010 | Technical lemma for ~ bnj6... |
| bnj1388 35011 | Technical lemma for ~ bnj6... |
| bnj1398 35012 | Technical lemma for ~ bnj6... |
| bnj1413 35013 | Property of ` _trCl ` . (... |
| bnj1408 35014 | Technical lemma for ~ bnj1... |
| bnj1414 35015 | Property of ` _trCl ` . (... |
| bnj1415 35016 | Technical lemma for ~ bnj6... |
| bnj1416 35017 | Technical lemma for ~ bnj6... |
| bnj1418 35018 | Property of ` _pred ` . (... |
| bnj1417 35019 | Technical lemma for ~ bnj6... |
| bnj1421 35020 | Technical lemma for ~ bnj6... |
| bnj1444 35021 | Technical lemma for ~ bnj6... |
| bnj1445 35022 | Technical lemma for ~ bnj6... |
| bnj1446 35023 | Technical lemma for ~ bnj6... |
| bnj1447 35024 | Technical lemma for ~ bnj6... |
| bnj1448 35025 | Technical lemma for ~ bnj6... |
| bnj1449 35026 | Technical lemma for ~ bnj6... |
| bnj1442 35027 | Technical lemma for ~ bnj6... |
| bnj1450 35028 | Technical lemma for ~ bnj6... |
| bnj1423 35029 | Technical lemma for ~ bnj6... |
| bnj1452 35030 | Technical lemma for ~ bnj6... |
| bnj1466 35031 | Technical lemma for ~ bnj6... |
| bnj1467 35032 | Technical lemma for ~ bnj6... |
| bnj1463 35033 | Technical lemma for ~ bnj6... |
| bnj1489 35034 | Technical lemma for ~ bnj6... |
| bnj1491 35035 | Technical lemma for ~ bnj6... |
| bnj1312 35036 | Technical lemma for ~ bnj6... |
| bnj1493 35037 | Technical lemma for ~ bnj6... |
| bnj1497 35038 | Technical lemma for ~ bnj6... |
| bnj1498 35039 | Technical lemma for ~ bnj6... |
| bnj60 35040 | Well-founded recursion, pa... |
| bnj1514 35041 | Technical lemma for ~ bnj1... |
| bnj1518 35042 | Technical lemma for ~ bnj1... |
| bnj1519 35043 | Technical lemma for ~ bnj1... |
| bnj1520 35044 | Technical lemma for ~ bnj1... |
| bnj1501 35045 | Technical lemma for ~ bnj1... |
| bnj1500 35046 | Well-founded recursion, pa... |
| bnj1525 35047 | Technical lemma for ~ bnj1... |
| bnj1529 35048 | Technical lemma for ~ bnj1... |
| bnj1523 35049 | Technical lemma for ~ bnj1... |
| bnj1522 35050 | Well-founded recursion, pa... |
| nfan1c 35051 | Variant of ~ nfan and comm... |
| cbvex1v 35052 | Rule used to change bound ... |
| dvelimalcased 35053 | Eliminate a disjoint varia... |
| dvelimalcasei 35054 | Eliminate a disjoint varia... |
| dvelimexcased 35055 | Eliminate a disjoint varia... |
| dvelimexcasei 35056 | Eliminate a disjoint varia... |
| exdifsn 35057 | There exists an element in... |
| srcmpltd 35058 | If a statement is true for... |
| prsrcmpltd 35059 | If a statement is true for... |
| axsepg2 35060 | A generalization of ~ ax-s... |
| axsepg2ALT 35061 | Alternate proof of ~ axsep... |
| dff15 35062 | A one-to-one function in t... |
| f1resveqaeq 35063 | If a function restricted t... |
| f1resrcmplf1dlem 35064 | Lemma for ~ f1resrcmplf1d ... |
| f1resrcmplf1d 35065 | If a function's restrictio... |
| funen1cnv 35066 | If a function is equinumer... |
| fnrelpredd 35067 | A function that preserves ... |
| cardpred 35068 | The cardinality function p... |
| nummin 35069 | Every nonempty class of nu... |
| axnulg 35070 | A generalization of ~ ax-n... |
| axnulALT2 35071 | Alternate proof of ~ axnul... |
| prcinf 35072 | Any proper class is litera... |
| fineqvrep 35073 | If the Axiom of Infinity i... |
| fineqvpow 35074 | If the Axiom of Infinity i... |
| fineqvac 35075 | If the Axiom of Infinity i... |
| fineqvacALT 35076 | Shorter proof of ~ fineqva... |
| gblacfnacd 35077 | If ` F ` is a global choic... |
| wevgblacfn 35078 | If ` R ` is a well-orderin... |
| zltp1ne 35079 | Integer ordering relation.... |
| nnltp1ne 35080 | Positive integer ordering ... |
| nn0ltp1ne 35081 | Nonnegative integer orderi... |
| 0nn0m1nnn0 35082 | A number is zero if and on... |
| f1resfz0f1d 35083 | If a function with a seque... |
| fisshasheq 35084 | A finite set is equal to i... |
| revpfxsfxrev 35085 | The reverse of a prefix of... |
| swrdrevpfx 35086 | A subword expressed in ter... |
| lfuhgr 35087 | A hypergraph is loop-free ... |
| lfuhgr2 35088 | A hypergraph is loop-free ... |
| lfuhgr3 35089 | A hypergraph is loop-free ... |
| cplgredgex 35090 | Any two (distinct) vertice... |
| cusgredgex 35091 | Any two (distinct) vertice... |
| cusgredgex2 35092 | Any two distinct vertices ... |
| pfxwlk 35093 | A prefix of a walk is a wa... |
| revwlk 35094 | The reverse of a walk is a... |
| revwlkb 35095 | Two words represent a walk... |
| swrdwlk 35096 | Two matching subwords of a... |
| pthhashvtx 35097 | A graph containing a path ... |
| pthisspthorcycl 35098 | A path is either a simple ... |
| spthcycl 35099 | A walk is a trivial path i... |
| usgrgt2cycl 35100 | A non-trivial cycle in a s... |
| usgrcyclgt2v 35101 | A simple graph with a non-... |
| subgrwlk 35102 | If a walk exists in a subg... |
| subgrtrl 35103 | If a trail exists in a sub... |
| subgrpth 35104 | If a path exists in a subg... |
| subgrcycl 35105 | If a cycle exists in a sub... |
| cusgr3cyclex 35106 | Every complete simple grap... |
| loop1cycl 35107 | A hypergraph has a cycle o... |
| 2cycld 35108 | Construction of a 2-cycle ... |
| 2cycl2d 35109 | Construction of a 2-cycle ... |
| umgr2cycllem 35110 | Lemma for ~ umgr2cycl . (... |
| umgr2cycl 35111 | A multigraph with two dist... |
| dfacycgr1 35114 | An alternate definition of... |
| isacycgr 35115 | The property of being an a... |
| isacycgr1 35116 | The property of being an a... |
| acycgrcycl 35117 | Any cycle in an acyclic gr... |
| acycgr0v 35118 | A null graph (with no vert... |
| acycgr1v 35119 | A multigraph with one vert... |
| acycgr2v 35120 | A simple graph with two ve... |
| prclisacycgr 35121 | A proper class (representi... |
| acycgrislfgr 35122 | An acyclic hypergraph is a... |
| upgracycumgr 35123 | An acyclic pseudograph is ... |
| umgracycusgr 35124 | An acyclic multigraph is a... |
| upgracycusgr 35125 | An acyclic pseudograph is ... |
| cusgracyclt3v 35126 | A complete simple graph is... |
| pthacycspth 35127 | A path in an acyclic graph... |
| acycgrsubgr 35128 | The subgraph of an acyclic... |
| quartfull 35135 | The quartic equation, writ... |
| deranglem 35136 | Lemma for derangements. (... |
| derangval 35137 | Define the derangement fun... |
| derangf 35138 | The derangement number is ... |
| derang0 35139 | The derangement number of ... |
| derangsn 35140 | The derangement number of ... |
| derangenlem 35141 | One half of ~ derangen . ... |
| derangen 35142 | The derangement number is ... |
| subfacval 35143 | The subfactorial is define... |
| derangen2 35144 | Write the derangement numb... |
| subfacf 35145 | The subfactorial is a func... |
| subfaclefac 35146 | The subfactorial is less t... |
| subfac0 35147 | The subfactorial at zero. ... |
| subfac1 35148 | The subfactorial at one. ... |
| subfacp1lem1 35149 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem2a 35150 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem2b 35151 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem3 35152 | Lemma for ~ subfacp1 . In... |
| subfacp1lem4 35153 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem5 35154 | Lemma for ~ subfacp1 . In... |
| subfacp1lem6 35155 | Lemma for ~ subfacp1 . By... |
| subfacp1 35156 | A two-term recurrence for ... |
| subfacval2 35157 | A closed-form expression f... |
| subfaclim 35158 | The subfactorial converges... |
| subfacval3 35159 | Another closed form expres... |
| derangfmla 35160 | The derangements formula, ... |
| erdszelem1 35161 | Lemma for ~ erdsze . (Con... |
| erdszelem2 35162 | Lemma for ~ erdsze . (Con... |
| erdszelem3 35163 | Lemma for ~ erdsze . (Con... |
| erdszelem4 35164 | Lemma for ~ erdsze . (Con... |
| erdszelem5 35165 | Lemma for ~ erdsze . (Con... |
| erdszelem6 35166 | Lemma for ~ erdsze . (Con... |
| erdszelem7 35167 | Lemma for ~ erdsze . (Con... |
| erdszelem8 35168 | Lemma for ~ erdsze . (Con... |
| erdszelem9 35169 | Lemma for ~ erdsze . (Con... |
| erdszelem10 35170 | Lemma for ~ erdsze . (Con... |
| erdszelem11 35171 | Lemma for ~ erdsze . (Con... |
| erdsze 35172 | The Erdős-Szekeres th... |
| erdsze2lem1 35173 | Lemma for ~ erdsze2 . (Co... |
| erdsze2lem2 35174 | Lemma for ~ erdsze2 . (Co... |
| erdsze2 35175 | Generalize the statement o... |
| kur14lem1 35176 | Lemma for ~ kur14 . (Cont... |
| kur14lem2 35177 | Lemma for ~ kur14 . Write... |
| kur14lem3 35178 | Lemma for ~ kur14 . A clo... |
| kur14lem4 35179 | Lemma for ~ kur14 . Compl... |
| kur14lem5 35180 | Lemma for ~ kur14 . Closu... |
| kur14lem6 35181 | Lemma for ~ kur14 . If ` ... |
| kur14lem7 35182 | Lemma for ~ kur14 : main p... |
| kur14lem8 35183 | Lemma for ~ kur14 . Show ... |
| kur14lem9 35184 | Lemma for ~ kur14 . Since... |
| kur14lem10 35185 | Lemma for ~ kur14 . Disch... |
| kur14 35186 | Kuratowski's closure-compl... |
| ispconn 35193 | The property of being a pa... |
| pconncn 35194 | The property of being a pa... |
| pconntop 35195 | A simply connected space i... |
| issconn 35196 | The property of being a si... |
| sconnpconn 35197 | A simply connected space i... |
| sconntop 35198 | A simply connected space i... |
| sconnpht 35199 | A closed path in a simply ... |
| cnpconn 35200 | An image of a path-connect... |
| pconnconn 35201 | A path-connected space is ... |
| txpconn 35202 | The topological product of... |
| ptpconn 35203 | The topological product of... |
| indispconn 35204 | The indiscrete topology (o... |
| connpconn 35205 | A connected and locally pa... |
| qtoppconn 35206 | A quotient of a path-conne... |
| pconnpi1 35207 | All fundamental groups in ... |
| sconnpht2 35208 | Any two paths in a simply ... |
| sconnpi1 35209 | A path-connected topologic... |
| txsconnlem 35210 | Lemma for ~ txsconn . (Co... |
| txsconn 35211 | The topological product of... |
| cvxpconn 35212 | A convex subset of the com... |
| cvxsconn 35213 | A convex subset of the com... |
| blsconn 35214 | An open ball in the comple... |
| cnllysconn 35215 | The topology of the comple... |
| resconn 35216 | A subset of ` RR ` is simp... |
| ioosconn 35217 | An open interval is simply... |
| iccsconn 35218 | A closed interval is simpl... |
| retopsconn 35219 | The real numbers are simpl... |
| iccllysconn 35220 | A closed interval is local... |
| rellysconn 35221 | The real numbers are local... |
| iisconn 35222 | The unit interval is simpl... |
| iillysconn 35223 | The unit interval is local... |
| iinllyconn 35224 | The unit interval is local... |
| fncvm 35227 | Lemma for covering maps. ... |
| cvmscbv 35228 | Change bound variables in ... |
| iscvm 35229 | The property of being a co... |
| cvmtop1 35230 | Reverse closure for a cove... |
| cvmtop2 35231 | Reverse closure for a cove... |
| cvmcn 35232 | A covering map is a contin... |
| cvmcov 35233 | Property of a covering map... |
| cvmsrcl 35234 | Reverse closure for an eve... |
| cvmsi 35235 | One direction of ~ cvmsval... |
| cvmsval 35236 | Elementhood in the set ` S... |
| cvmsss 35237 | An even covering is a subs... |
| cvmsn0 35238 | An even covering is nonemp... |
| cvmsuni 35239 | An even covering of ` U ` ... |
| cvmsdisj 35240 | An even covering of ` U ` ... |
| cvmshmeo 35241 | Every element of an even c... |
| cvmsf1o 35242 | ` F ` , localized to an el... |
| cvmscld 35243 | The sets of an even coveri... |
| cvmsss2 35244 | An open subset of an evenl... |
| cvmcov2 35245 | The covering map property ... |
| cvmseu 35246 | Every element in ` U. T ` ... |
| cvmsiota 35247 | Identify the unique elemen... |
| cvmopnlem 35248 | Lemma for ~ cvmopn . (Con... |
| cvmfolem 35249 | Lemma for ~ cvmfo . (Cont... |
| cvmopn 35250 | A covering map is an open ... |
| cvmliftmolem1 35251 | Lemma for ~ cvmliftmo . (... |
| cvmliftmolem2 35252 | Lemma for ~ cvmliftmo . (... |
| cvmliftmoi 35253 | A lift of a continuous fun... |
| cvmliftmo 35254 | A lift of a continuous fun... |
| cvmliftlem1 35255 | Lemma for ~ cvmlift . In ... |
| cvmliftlem2 35256 | Lemma for ~ cvmlift . ` W ... |
| cvmliftlem3 35257 | Lemma for ~ cvmlift . Sin... |
| cvmliftlem4 35258 | Lemma for ~ cvmlift . The... |
| cvmliftlem5 35259 | Lemma for ~ cvmlift . Def... |
| cvmliftlem6 35260 | Lemma for ~ cvmlift . Ind... |
| cvmliftlem7 35261 | Lemma for ~ cvmlift . Pro... |
| cvmliftlem8 35262 | Lemma for ~ cvmlift . The... |
| cvmliftlem9 35263 | Lemma for ~ cvmlift . The... |
| cvmliftlem10 35264 | Lemma for ~ cvmlift . The... |
| cvmliftlem11 35265 | Lemma for ~ cvmlift . (Co... |
| cvmliftlem13 35266 | Lemma for ~ cvmlift . The... |
| cvmliftlem14 35267 | Lemma for ~ cvmlift . Put... |
| cvmliftlem15 35268 | Lemma for ~ cvmlift . Dis... |
| cvmlift 35269 | One of the important prope... |
| cvmfo 35270 | A covering map is an onto ... |
| cvmliftiota 35271 | Write out a function ` H `... |
| cvmlift2lem1 35272 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9a 35273 | Lemma for ~ cvmlift2 and ~... |
| cvmlift2lem2 35274 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem3 35275 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem4 35276 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem5 35277 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem6 35278 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem7 35279 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem8 35280 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9 35281 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem10 35282 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem11 35283 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem12 35284 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem13 35285 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2 35286 | A two-dimensional version ... |
| cvmliftphtlem 35287 | Lemma for ~ cvmliftpht . ... |
| cvmliftpht 35288 | If ` G ` and ` H ` are pat... |
| cvmlift3lem1 35289 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem2 35290 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem3 35291 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem4 35292 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem5 35293 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem6 35294 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem7 35295 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem8 35296 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem9 35297 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3 35298 | A general version of ~ cvm... |
| snmlff 35299 | The function ` F ` from ~ ... |
| snmlfval 35300 | The function ` F ` from ~ ... |
| snmlval 35301 | The property " ` A ` is si... |
| snmlflim 35302 | If ` A ` is simply normal,... |
| goel 35317 | A "Godel-set of membership... |
| goelel3xp 35318 | A "Godel-set of membership... |
| goeleq12bg 35319 | Two "Godel-set of membersh... |
| gonafv 35320 | The "Godel-set for the She... |
| goaleq12d 35321 | Equality of the "Godel-set... |
| gonanegoal 35322 | The Godel-set for the Shef... |
| satf 35323 | The satisfaction predicate... |
| satfsucom 35324 | The satisfaction predicate... |
| satfn 35325 | The satisfaction predicate... |
| satom 35326 | The satisfaction predicate... |
| satfvsucom 35327 | The satisfaction predicate... |
| satfv0 35328 | The value of the satisfact... |
| satfvsuclem1 35329 | Lemma 1 for ~ satfvsuc . ... |
| satfvsuclem2 35330 | Lemma 2 for ~ satfvsuc . ... |
| satfvsuc 35331 | The value of the satisfact... |
| satfv1lem 35332 | Lemma for ~ satfv1 . (Con... |
| satfv1 35333 | The value of the satisfact... |
| satfsschain 35334 | The binary relation of a s... |
| satfvsucsuc 35335 | The satisfaction predicate... |
| satfbrsuc 35336 | The binary relation of a s... |
| satfrel 35337 | The value of the satisfact... |
| satfdmlem 35338 | Lemma for ~ satfdm . (Con... |
| satfdm 35339 | The domain of the satisfac... |
| satfrnmapom 35340 | The range of the satisfact... |
| satfv0fun 35341 | The value of the satisfact... |
| satf0 35342 | The satisfaction predicate... |
| satf0sucom 35343 | The satisfaction predicate... |
| satf00 35344 | The value of the satisfact... |
| satf0suclem 35345 | Lemma for ~ satf0suc , ~ s... |
| satf0suc 35346 | The value of the satisfact... |
| satf0op 35347 | An element of a value of t... |
| satf0n0 35348 | The value of the satisfact... |
| sat1el2xp 35349 | The first component of an ... |
| fmlafv 35350 | The valid Godel formulas o... |
| fmla 35351 | The set of all valid Godel... |
| fmla0 35352 | The valid Godel formulas o... |
| fmla0xp 35353 | The valid Godel formulas o... |
| fmlasuc0 35354 | The valid Godel formulas o... |
| fmlafvel 35355 | A class is a valid Godel f... |
| fmlasuc 35356 | The valid Godel formulas o... |
| fmla1 35357 | The valid Godel formulas o... |
| isfmlasuc 35358 | The characterization of a ... |
| fmlasssuc 35359 | The Godel formulas of heig... |
| fmlaomn0 35360 | The empty set is not a God... |
| fmlan0 35361 | The empty set is not a God... |
| gonan0 35362 | The "Godel-set of NAND" is... |
| goaln0 35363 | The "Godel-set of universa... |
| gonarlem 35364 | Lemma for ~ gonar (inducti... |
| gonar 35365 | If the "Godel-set of NAND"... |
| goalrlem 35366 | Lemma for ~ goalr (inducti... |
| goalr 35367 | If the "Godel-set of unive... |
| fmla0disjsuc 35368 | The set of valid Godel for... |
| fmlasucdisj 35369 | The valid Godel formulas o... |
| satfdmfmla 35370 | The domain of the satisfac... |
| satffunlem 35371 | Lemma for ~ satffunlem1lem... |
| satffunlem1lem1 35372 | Lemma for ~ satffunlem1 . ... |
| satffunlem1lem2 35373 | Lemma 2 for ~ satffunlem1 ... |
| satffunlem2lem1 35374 | Lemma 1 for ~ satffunlem2 ... |
| dmopab3rexdif 35375 | The domain of an ordered p... |
| satffunlem2lem2 35376 | Lemma 2 for ~ satffunlem2 ... |
| satffunlem1 35377 | Lemma 1 for ~ satffun : in... |
| satffunlem2 35378 | Lemma 2 for ~ satffun : in... |
| satffun 35379 | The value of the satisfact... |
| satff 35380 | The satisfaction predicate... |
| satfun 35381 | The satisfaction predicate... |
| satfvel 35382 | An element of the value of... |
| satfv0fvfmla0 35383 | The value of the satisfact... |
| satefv 35384 | The simplified satisfactio... |
| sate0 35385 | The simplified satisfactio... |
| satef 35386 | The simplified satisfactio... |
| sate0fv0 35387 | A simplified satisfaction ... |
| satefvfmla0 35388 | The simplified satisfactio... |
| sategoelfvb 35389 | Characterization of a valu... |
| sategoelfv 35390 | Condition of a valuation `... |
| ex-sategoelel 35391 | Example of a valuation of ... |
| ex-sategoel 35392 | Instance of ~ sategoelfv f... |
| satfv1fvfmla1 35393 | The value of the satisfact... |
| 2goelgoanfmla1 35394 | Two Godel-sets of membersh... |
| satefvfmla1 35395 | The simplified satisfactio... |
| ex-sategoelelomsuc 35396 | Example of a valuation of ... |
| ex-sategoelel12 35397 | Example of a valuation of ... |
| prv 35398 | The "proves" relation on a... |
| elnanelprv 35399 | The wff ` ( A e. B -/\ B e... |
| prv0 35400 | Every wff encoded as ` U `... |
| prv1n 35401 | No wff encoded as a Godel-... |
| mvtval 35470 | The set of variable typeco... |
| mrexval 35471 | The set of "raw expression... |
| mexval 35472 | The set of expressions, wh... |
| mexval2 35473 | The set of expressions, wh... |
| mdvval 35474 | The set of disjoint variab... |
| mvrsval 35475 | The set of variables in an... |
| mvrsfpw 35476 | The set of variables in an... |
| mrsubffval 35477 | The substitution of some v... |
| mrsubfval 35478 | The substitution of some v... |
| mrsubval 35479 | The substitution of some v... |
| mrsubcv 35480 | The value of a substituted... |
| mrsubvr 35481 | The value of a substituted... |
| mrsubff 35482 | A substitution is a functi... |
| mrsubrn 35483 | Although it is defined for... |
| mrsubff1 35484 | When restricted to complet... |
| mrsubff1o 35485 | When restricted to complet... |
| mrsub0 35486 | The value of the substitut... |
| mrsubf 35487 | A substitution is a functi... |
| mrsubccat 35488 | Substitution distributes o... |
| mrsubcn 35489 | A substitution does not ch... |
| elmrsubrn 35490 | Characterization of the su... |
| mrsubco 35491 | The composition of two sub... |
| mrsubvrs 35492 | The set of variables in a ... |
| msubffval 35493 | A substitution applied to ... |
| msubfval 35494 | A substitution applied to ... |
| msubval 35495 | A substitution applied to ... |
| msubrsub 35496 | A substitution applied to ... |
| msubty 35497 | The type of a substituted ... |
| elmsubrn 35498 | Characterization of substi... |
| msubrn 35499 | Although it is defined for... |
| msubff 35500 | A substitution is a functi... |
| msubco 35501 | The composition of two sub... |
| msubf 35502 | A substitution is a functi... |
| mvhfval 35503 | Value of the function mapp... |
| mvhval 35504 | Value of the function mapp... |
| mpstval 35505 | A pre-statement is an orde... |
| elmpst 35506 | Property of being a pre-st... |
| msrfval 35507 | Value of the reduct of a p... |
| msrval 35508 | Value of the reduct of a p... |
| mpstssv 35509 | A pre-statement is an orde... |
| mpst123 35510 | Decompose a pre-statement ... |
| mpstrcl 35511 | The elements of a pre-stat... |
| msrf 35512 | The reduct of a pre-statem... |
| msrrcl 35513 | If ` X ` and ` Y ` have th... |
| mstaval 35514 | Value of the set of statem... |
| msrid 35515 | The reduct of a statement ... |
| msrfo 35516 | The reduct of a pre-statem... |
| mstapst 35517 | A statement is a pre-state... |
| elmsta 35518 | Property of being a statem... |
| ismfs 35519 | A formal system is a tuple... |
| mfsdisj 35520 | The constants and variable... |
| mtyf2 35521 | The type function maps var... |
| mtyf 35522 | The type function maps var... |
| mvtss 35523 | The set of variable typeco... |
| maxsta 35524 | An axiom is a statement. ... |
| mvtinf 35525 | Each variable typecode has... |
| msubff1 35526 | When restricted to complet... |
| msubff1o 35527 | When restricted to complet... |
| mvhf 35528 | The function mapping varia... |
| mvhf1 35529 | The function mapping varia... |
| msubvrs 35530 | The set of variables in a ... |
| mclsrcl 35531 | Reverse closure for the cl... |
| mclsssvlem 35532 | Lemma for ~ mclsssv . (Co... |
| mclsval 35533 | The function mapping varia... |
| mclsssv 35534 | The closure of a set of ex... |
| ssmclslem 35535 | Lemma for ~ ssmcls . (Con... |
| vhmcls 35536 | All variable hypotheses ar... |
| ssmcls 35537 | The original expressions a... |
| ss2mcls 35538 | The closure is monotonic u... |
| mclsax 35539 | The closure is closed unde... |
| mclsind 35540 | Induction theorem for clos... |
| mppspstlem 35541 | Lemma for ~ mppspst . (Co... |
| mppsval 35542 | Definition of a provable p... |
| elmpps 35543 | Definition of a provable p... |
| mppspst 35544 | A provable pre-statement i... |
| mthmval 35545 | A theorem is a pre-stateme... |
| elmthm 35546 | A theorem is a pre-stateme... |
| mthmi 35547 | A statement whose reduct i... |
| mthmsta 35548 | A theorem is a pre-stateme... |
| mppsthm 35549 | A provable pre-statement i... |
| mthmblem 35550 | Lemma for ~ mthmb . (Cont... |
| mthmb 35551 | If two statements have the... |
| mthmpps 35552 | Given a theorem, there is ... |
| mclsppslem 35553 | The closure is closed unde... |
| mclspps 35554 | The closure is closed unde... |
| rexxfr3d 35608 | Transfer existential quant... |
| rexxfr3dALT 35609 | Longer proof of ~ rexxfr3d... |
| rspssbasd 35610 | The span of a set of ring ... |
| ellcsrspsn 35611 | Membership in a left coset... |
| ply1divalg3 35612 | Uniqueness of polynomial r... |
| r1peuqusdeg1 35613 | Uniqueness of polynomial r... |
| problem1 35635 | Practice problem 1. Clues... |
| problem2 35636 | Practice problem 2. Clues... |
| problem3 35637 | Practice problem 3. Clues... |
| problem4 35638 | Practice problem 4. Clues... |
| problem5 35639 | Practice problem 5. Clues... |
| quad3 35640 | Variant of quadratic equat... |
| climuzcnv 35641 | Utility lemma to convert b... |
| sinccvglem 35642 | ` ( ( sin `` x ) / x ) ~~>... |
| sinccvg 35643 | ` ( ( sin `` x ) / x ) ~~>... |
| circum 35644 | The circumference of a cir... |
| elfzm12 35645 | Membership in a curtailed ... |
| nn0seqcvg 35646 | A strictly-decreasing nonn... |
| lediv2aALT 35647 | Division of both sides of ... |
| abs2sqlei 35648 | The absolute values of two... |
| abs2sqlti 35649 | The absolute values of two... |
| abs2sqle 35650 | The absolute values of two... |
| abs2sqlt 35651 | The absolute values of two... |
| abs2difi 35652 | Difference of absolute val... |
| abs2difabsi 35653 | Absolute value of differen... |
| 2thALT 35654 | Alternate proof of ~ 2th .... |
| orbi2iALT 35655 | Alternate proof of ~ orbi2... |
| pm3.48ALT 35656 | Alternate proof of ~ pm3.4... |
| 3jcadALT 35657 | Alternate proof of ~ 3jcad... |
| currybi 35658 | Biconditional version of C... |
| axextprim 35665 | ~ ax-ext without distinct ... |
| axrepprim 35666 | ~ ax-rep without distinct ... |
| axunprim 35667 | ~ ax-un without distinct v... |
| axpowprim 35668 | ~ ax-pow without distinct ... |
| axregprim 35669 | ~ ax-reg without distinct ... |
| axinfprim 35670 | ~ ax-inf without distinct ... |
| axacprim 35671 | ~ ax-ac without distinct v... |
| untelirr 35672 | We call a class "untanged"... |
| untuni 35673 | The union of a class is un... |
| untsucf 35674 | If a class is untangled, t... |
| unt0 35675 | The null set is untangled.... |
| untint 35676 | If there is an untangled e... |
| efrunt 35677 | If ` A ` is well-founded b... |
| untangtr 35678 | A transitive class is unta... |
| 3jaodd 35679 | Double deduction form of ~... |
| 3orit 35680 | Closed form of ~ 3ori . (... |
| biimpexp 35681 | A biconditional in the ant... |
| nepss 35682 | Two classes are unequal if... |
| 3ccased 35683 | Triple disjunction form of... |
| dfso3 35684 | Expansion of the definitio... |
| brtpid1 35685 | A binary relation involvin... |
| brtpid2 35686 | A binary relation involvin... |
| brtpid3 35687 | A binary relation involvin... |
| iota5f 35688 | A method for computing iot... |
| jath 35689 | Closed form of ~ ja . Pro... |
| xpab 35690 | Cartesian product of two c... |
| nnuni 35691 | The union of a finite ordi... |
| sqdivzi 35692 | Distribution of square ove... |
| supfz 35693 | The supremum of a finite s... |
| inffz 35694 | The infimum of a finite se... |
| fz0n 35695 | The sequence ` ( 0 ... ( N... |
| shftvalg 35696 | Value of a sequence shifte... |
| divcnvlin 35697 | Limit of the ratio of two ... |
| climlec3 35698 | Comparison of a constant t... |
| iexpire 35699 | ` _i ` raised to itself is... |
| bcneg1 35700 | The binomial coefficient o... |
| bcm1nt 35701 | The proportion of one bino... |
| bcprod 35702 | A product identity for bin... |
| bccolsum 35703 | A column-sum rule for bino... |
| iprodefisumlem 35704 | Lemma for ~ iprodefisum . ... |
| iprodefisum 35705 | Applying the exponential f... |
| iprodgam 35706 | An infinite product versio... |
| faclimlem1 35707 | Lemma for ~ faclim . Clos... |
| faclimlem2 35708 | Lemma for ~ faclim . Show... |
| faclimlem3 35709 | Lemma for ~ faclim . Alge... |
| faclim 35710 | An infinite product expres... |
| iprodfac 35711 | An infinite product expres... |
| faclim2 35712 | Another factorial limit du... |
| gcd32 35713 | Swap the second and third ... |
| gcdabsorb 35714 | Absorption law for gcd. (... |
| dftr6 35715 | A potential definition of ... |
| coep 35716 | Composition with the membe... |
| coepr 35717 | Composition with the conve... |
| dffr5 35718 | A quantifier-free definiti... |
| dfso2 35719 | Quantifier-free definition... |
| br8 35720 | Substitution for an eight-... |
| br6 35721 | Substitution for a six-pla... |
| br4 35722 | Substitution for a four-pl... |
| cnvco1 35723 | Another distributive law o... |
| cnvco2 35724 | Another distributive law o... |
| eldm3 35725 | Quantifier-free definition... |
| elrn3 35726 | Quantifier-free definition... |
| pocnv 35727 | The converse of a partial ... |
| socnv 35728 | The converse of a strict o... |
| sotrd 35729 | Transitivity law for stric... |
| elintfv 35730 | Membership in an intersect... |
| funpsstri 35731 | A condition for subset tri... |
| fundmpss 35732 | If a class ` F ` is a prop... |
| funsseq 35733 | Given two functions with e... |
| fununiq 35734 | The uniqueness condition o... |
| funbreq 35735 | An equality condition for ... |
| br1steq 35736 | Uniqueness condition for t... |
| br2ndeq 35737 | Uniqueness condition for t... |
| dfdm5 35738 | Definition of domain in te... |
| dfrn5 35739 | Definition of range in ter... |
| opelco3 35740 | Alternate way of saying th... |
| elima4 35741 | Quantifier-free expression... |
| fv1stcnv 35742 | The value of the converse ... |
| fv2ndcnv 35743 | The value of the converse ... |
| setinds 35744 | Principle of set induction... |
| setinds2f 35745 | ` _E ` induction schema, u... |
| setinds2 35746 | ` _E ` induction schema, u... |
| elpotr 35747 | A class of transitive sets... |
| dford5reg 35748 | Given ~ ax-reg , an ordina... |
| dfon2lem1 35749 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem2 35750 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem3 35751 | Lemma for ~ dfon2 . All s... |
| dfon2lem4 35752 | Lemma for ~ dfon2 . If tw... |
| dfon2lem5 35753 | Lemma for ~ dfon2 . Two s... |
| dfon2lem6 35754 | Lemma for ~ dfon2 . A tra... |
| dfon2lem7 35755 | Lemma for ~ dfon2 . All e... |
| dfon2lem8 35756 | Lemma for ~ dfon2 . The i... |
| dfon2lem9 35757 | Lemma for ~ dfon2 . A cla... |
| dfon2 35758 | ` On ` consists of all set... |
| rdgprc0 35759 | The value of the recursive... |
| rdgprc 35760 | The value of the recursive... |
| dfrdg2 35761 | Alternate definition of th... |
| dfrdg3 35762 | Generalization of ~ dfrdg2... |
| axextdfeq 35763 | A version of ~ ax-ext for ... |
| ax8dfeq 35764 | A version of ~ ax-8 for us... |
| axextdist 35765 | ~ ax-ext with distinctors ... |
| axextbdist 35766 | ~ axextb with distinctors ... |
| 19.12b 35767 | Version of ~ 19.12vv with ... |
| exnel 35768 | There is always a set not ... |
| distel 35769 | Distinctors in terms of me... |
| axextndbi 35770 | ~ axextnd as a bicondition... |
| hbntg 35771 | A more general form of ~ h... |
| hbimtg 35772 | A more general and closed ... |
| hbaltg 35773 | A more general and closed ... |
| hbng 35774 | A more general form of ~ h... |
| hbimg 35775 | A more general form of ~ h... |
| wsuceq123 35780 | Equality theorem for well-... |
| wsuceq1 35781 | Equality theorem for well-... |
| wsuceq2 35782 | Equality theorem for well-... |
| wsuceq3 35783 | Equality theorem for well-... |
| nfwsuc 35784 | Bound-variable hypothesis ... |
| wlimeq12 35785 | Equality theorem for the l... |
| wlimeq1 35786 | Equality theorem for the l... |
| wlimeq2 35787 | Equality theorem for the l... |
| nfwlim 35788 | Bound-variable hypothesis ... |
| elwlim 35789 | Membership in the limit cl... |
| wzel 35790 | The zero of a well-founded... |
| wsuclem 35791 | Lemma for the supremum pro... |
| wsucex 35792 | Existence theorem for well... |
| wsuccl 35793 | If ` X ` is a set with an ... |
| wsuclb 35794 | A well-founded successor i... |
| wlimss 35795 | The class of limit points ... |
| txpss3v 35844 | A tail Cartesian product i... |
| txprel 35845 | A tail Cartesian product i... |
| brtxp 35846 | Characterize a ternary rel... |
| brtxp2 35847 | The binary relation over a... |
| dfpprod2 35848 | Expanded definition of par... |
| pprodcnveq 35849 | A converse law for paralle... |
| pprodss4v 35850 | The parallel product is a ... |
| brpprod 35851 | Characterize a quaternary ... |
| brpprod3a 35852 | Condition for parallel pro... |
| brpprod3b 35853 | Condition for parallel pro... |
| relsset 35854 | The subset class is a bina... |
| brsset 35855 | For sets, the ` SSet ` bin... |
| idsset 35856 | ` _I ` is equal to the int... |
| eltrans 35857 | Membership in the class of... |
| dfon3 35858 | A quantifier-free definiti... |
| dfon4 35859 | Another quantifier-free de... |
| brtxpsd 35860 | Expansion of a common form... |
| brtxpsd2 35861 | Another common abbreviatio... |
| brtxpsd3 35862 | A third common abbreviatio... |
| relbigcup 35863 | The ` Bigcup ` relationshi... |
| brbigcup 35864 | Binary relation over ` Big... |
| dfbigcup2 35865 | ` Bigcup ` using maps-to n... |
| fobigcup 35866 | ` Bigcup ` maps the univer... |
| fnbigcup 35867 | ` Bigcup ` is a function o... |
| fvbigcup 35868 | For sets, ` Bigcup ` yield... |
| elfix 35869 | Membership in the fixpoint... |
| elfix2 35870 | Alternative membership in ... |
| dffix2 35871 | The fixpoints of a class i... |
| fixssdm 35872 | The fixpoints of a class a... |
| fixssrn 35873 | The fixpoints of a class a... |
| fixcnv 35874 | The fixpoints of a class a... |
| fixun 35875 | The fixpoint operator dist... |
| ellimits 35876 | Membership in the class of... |
| limitssson 35877 | The class of all limit ord... |
| dfom5b 35878 | A quantifier-free definiti... |
| sscoid 35879 | A condition for subset and... |
| dffun10 35880 | Another potential definiti... |
| elfuns 35881 | Membership in the class of... |
| elfunsg 35882 | Closed form of ~ elfuns . ... |
| brsingle 35883 | The binary relation form o... |
| elsingles 35884 | Membership in the class of... |
| fnsingle 35885 | The singleton relationship... |
| fvsingle 35886 | The value of the singleton... |
| dfsingles2 35887 | Alternate definition of th... |
| snelsingles 35888 | A singleton is a member of... |
| dfiota3 35889 | A definition of iota using... |
| dffv5 35890 | Another quantifier-free de... |
| unisnif 35891 | Express union of singleton... |
| brimage 35892 | Binary relation form of th... |
| brimageg 35893 | Closed form of ~ brimage .... |
| funimage 35894 | ` Image A ` is a function.... |
| fnimage 35895 | ` Image R ` is a function ... |
| imageval 35896 | The image functor in maps-... |
| fvimage 35897 | Value of the image functor... |
| brcart 35898 | Binary relation form of th... |
| brdomain 35899 | Binary relation form of th... |
| brrange 35900 | Binary relation form of th... |
| brdomaing 35901 | Closed form of ~ brdomain ... |
| brrangeg 35902 | Closed form of ~ brrange .... |
| brimg 35903 | Binary relation form of th... |
| brapply 35904 | Binary relation form of th... |
| brcup 35905 | Binary relation form of th... |
| brcap 35906 | Binary relation form of th... |
| brsuccf 35907 | Binary relation form of th... |
| funpartlem 35908 | Lemma for ~ funpartfun . ... |
| funpartfun 35909 | The functional part of ` F... |
| funpartss 35910 | The functional part of ` F... |
| funpartfv 35911 | The function value of the ... |
| fullfunfnv 35912 | The full functional part o... |
| fullfunfv 35913 | The function value of the ... |
| brfullfun 35914 | A binary relation form con... |
| brrestrict 35915 | Binary relation form of th... |
| dfrecs2 35916 | A quantifier-free definiti... |
| dfrdg4 35917 | A quantifier-free definiti... |
| dfint3 35918 | Quantifier-free definition... |
| imagesset 35919 | The Image functor applied ... |
| brub 35920 | Binary relation form of th... |
| brlb 35921 | Binary relation form of th... |
| altopex 35926 | Alternative ordered pairs ... |
| altopthsn 35927 | Two alternate ordered pair... |
| altopeq12 35928 | Equality for alternate ord... |
| altopeq1 35929 | Equality for alternate ord... |
| altopeq2 35930 | Equality for alternate ord... |
| altopth1 35931 | Equality of the first memb... |
| altopth2 35932 | Equality of the second mem... |
| altopthg 35933 | Alternate ordered pair the... |
| altopthbg 35934 | Alternate ordered pair the... |
| altopth 35935 | The alternate ordered pair... |
| altopthb 35936 | Alternate ordered pair the... |
| altopthc 35937 | Alternate ordered pair the... |
| altopthd 35938 | Alternate ordered pair the... |
| altxpeq1 35939 | Equality for alternate Car... |
| altxpeq2 35940 | Equality for alternate Car... |
| elaltxp 35941 | Membership in alternate Ca... |
| altopelaltxp 35942 | Alternate ordered pair mem... |
| altxpsspw 35943 | An inclusion rule for alte... |
| altxpexg 35944 | The alternate Cartesian pr... |
| rankaltopb 35945 | Compute the rank of an alt... |
| nfaltop 35946 | Bound-variable hypothesis ... |
| sbcaltop 35947 | Distribution of class subs... |
| cgrrflx2d 35950 | Deduction form of ~ axcgrr... |
| cgrtr4d 35951 | Deduction form of ~ axcgrt... |
| cgrtr4and 35952 | Deduction form of ~ axcgrt... |
| cgrrflx 35953 | Reflexivity law for congru... |
| cgrrflxd 35954 | Deduction form of ~ cgrrfl... |
| cgrcomim 35955 | Congruence commutes on the... |
| cgrcom 35956 | Congruence commutes betwee... |
| cgrcomand 35957 | Deduction form of ~ cgrcom... |
| cgrtr 35958 | Transitivity law for congr... |
| cgrtrand 35959 | Deduction form of ~ cgrtr ... |
| cgrtr3 35960 | Transitivity law for congr... |
| cgrtr3and 35961 | Deduction form of ~ cgrtr3... |
| cgrcoml 35962 | Congruence commutes on the... |
| cgrcomr 35963 | Congruence commutes on the... |
| cgrcomlr 35964 | Congruence commutes on bot... |
| cgrcomland 35965 | Deduction form of ~ cgrcom... |
| cgrcomrand 35966 | Deduction form of ~ cgrcom... |
| cgrcomlrand 35967 | Deduction form of ~ cgrcom... |
| cgrtriv 35968 | Degenerate segments are co... |
| cgrid2 35969 | Identity law for congruenc... |
| cgrdegen 35970 | Two congruent segments are... |
| brofs 35971 | Binary relation form of th... |
| 5segofs 35972 | Rephrase ~ ax5seg using th... |
| ofscom 35973 | The outer five segment pre... |
| cgrextend 35974 | Link congruence over a pai... |
| cgrextendand 35975 | Deduction form of ~ cgrext... |
| segconeq 35976 | Two points that satisfy th... |
| segconeu 35977 | Existential uniqueness ver... |
| btwntriv2 35978 | Betweenness always holds f... |
| btwncomim 35979 | Betweenness commutes. Imp... |
| btwncom 35980 | Betweenness commutes. (Co... |
| btwncomand 35981 | Deduction form of ~ btwnco... |
| btwntriv1 35982 | Betweenness always holds f... |
| btwnswapid 35983 | If you can swap the first ... |
| btwnswapid2 35984 | If you can swap arguments ... |
| btwnintr 35985 | Inner transitivity law for... |
| btwnexch3 35986 | Exchange the first endpoin... |
| btwnexch3and 35987 | Deduction form of ~ btwnex... |
| btwnouttr2 35988 | Outer transitivity law for... |
| btwnexch2 35989 | Exchange the outer point o... |
| btwnouttr 35990 | Outer transitivity law for... |
| btwnexch 35991 | Outer transitivity law for... |
| btwnexchand 35992 | Deduction form of ~ btwnex... |
| btwndiff 35993 | There is always a ` c ` di... |
| trisegint 35994 | A line segment between two... |
| funtransport 35997 | The ` TransportTo ` relati... |
| fvtransport 35998 | Calculate the value of the... |
| transportcl 35999 | Closure law for segment tr... |
| transportprops 36000 | Calculate the defining pro... |
| brifs 36009 | Binary relation form of th... |
| ifscgr 36010 | Inner five segment congrue... |
| cgrsub 36011 | Removing identical parts f... |
| brcgr3 36012 | Binary relation form of th... |
| cgr3permute3 36013 | Permutation law for three-... |
| cgr3permute1 36014 | Permutation law for three-... |
| cgr3permute2 36015 | Permutation law for three-... |
| cgr3permute4 36016 | Permutation law for three-... |
| cgr3permute5 36017 | Permutation law for three-... |
| cgr3tr4 36018 | Transitivity law for three... |
| cgr3com 36019 | Commutativity law for thre... |
| cgr3rflx 36020 | Identity law for three-pla... |
| cgrxfr 36021 | A line segment can be divi... |
| btwnxfr 36022 | A condition for extending ... |
| colinrel 36023 | Colinearity is a relations... |
| brcolinear2 36024 | Alternate colinearity bina... |
| brcolinear 36025 | The binary relation form o... |
| colinearex 36026 | The colinear predicate exi... |
| colineardim1 36027 | If ` A ` is colinear with ... |
| colinearperm1 36028 | Permutation law for coline... |
| colinearperm3 36029 | Permutation law for coline... |
| colinearperm2 36030 | Permutation law for coline... |
| colinearperm4 36031 | Permutation law for coline... |
| colinearperm5 36032 | Permutation law for coline... |
| colineartriv1 36033 | Trivial case of colinearit... |
| colineartriv2 36034 | Trivial case of colinearit... |
| btwncolinear1 36035 | Betweenness implies coline... |
| btwncolinear2 36036 | Betweenness implies coline... |
| btwncolinear3 36037 | Betweenness implies coline... |
| btwncolinear4 36038 | Betweenness implies coline... |
| btwncolinear5 36039 | Betweenness implies coline... |
| btwncolinear6 36040 | Betweenness implies coline... |
| colinearxfr 36041 | Transfer law for colineari... |
| lineext 36042 | Extend a line with a missi... |
| brofs2 36043 | Change some conditions for... |
| brifs2 36044 | Change some conditions for... |
| brfs 36045 | Binary relation form of th... |
| fscgr 36046 | Congruence law for the gen... |
| linecgr 36047 | Congruence rule for lines.... |
| linecgrand 36048 | Deduction form of ~ linecg... |
| lineid 36049 | Identity law for points on... |
| idinside 36050 | Law for finding a point in... |
| endofsegid 36051 | If ` A ` , ` B ` , and ` C... |
| endofsegidand 36052 | Deduction form of ~ endofs... |
| btwnconn1lem1 36053 | Lemma for ~ btwnconn1 . T... |
| btwnconn1lem2 36054 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem3 36055 | Lemma for ~ btwnconn1 . E... |
| btwnconn1lem4 36056 | Lemma for ~ btwnconn1 . A... |
| btwnconn1lem5 36057 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem6 36058 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem7 36059 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem8 36060 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem9 36061 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem10 36062 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem11 36063 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem12 36064 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem13 36065 | Lemma for ~ btwnconn1 . B... |
| btwnconn1lem14 36066 | Lemma for ~ btwnconn1 . F... |
| btwnconn1 36067 | Connectitivy law for betwe... |
| btwnconn2 36068 | Another connectivity law f... |
| btwnconn3 36069 | Inner connectivity law for... |
| midofsegid 36070 | If two points fall in the ... |
| segcon2 36071 | Generalization of ~ axsegc... |
| brsegle 36074 | Binary relation form of th... |
| brsegle2 36075 | Alternate characterization... |
| seglecgr12im 36076 | Substitution law for segme... |
| seglecgr12 36077 | Substitution law for segme... |
| seglerflx 36078 | Segment comparison is refl... |
| seglemin 36079 | Any segment is at least as... |
| segletr 36080 | Segment less than is trans... |
| segleantisym 36081 | Antisymmetry law for segme... |
| seglelin 36082 | Linearity law for segment ... |
| btwnsegle 36083 | If ` B ` falls between ` A... |
| colinbtwnle 36084 | Given three colinear point... |
| broutsideof 36087 | Binary relation form of ` ... |
| broutsideof2 36088 | Alternate form of ` Outsid... |
| outsidene1 36089 | Outsideness implies inequa... |
| outsidene2 36090 | Outsideness implies inequa... |
| btwnoutside 36091 | A principle linking outsid... |
| broutsideof3 36092 | Characterization of outsid... |
| outsideofrflx 36093 | Reflexivity of outsideness... |
| outsideofcom 36094 | Commutativity law for outs... |
| outsideoftr 36095 | Transitivity law for outsi... |
| outsideofeq 36096 | Uniqueness law for ` Outsi... |
| outsideofeu 36097 | Given a nondegenerate ray,... |
| outsidele 36098 | Relate ` OutsideOf ` to ` ... |
| outsideofcol 36099 | Outside of implies colinea... |
| funray 36106 | Show that the ` Ray ` rela... |
| fvray 36107 | Calculate the value of the... |
| funline 36108 | Show that the ` Line ` rel... |
| linedegen 36109 | When ` Line ` is applied w... |
| fvline 36110 | Calculate the value of the... |
| liness 36111 | A line is a subset of the ... |
| fvline2 36112 | Alternate definition of a ... |
| lineunray 36113 | A line is composed of a po... |
| lineelsb2 36114 | If ` S ` lies on ` P Q ` ,... |
| linerflx1 36115 | Reflexivity law for line m... |
| linecom 36116 | Commutativity law for line... |
| linerflx2 36117 | Reflexivity law for line m... |
| ellines 36118 | Membership in the set of a... |
| linethru 36119 | If ` A ` is a line contain... |
| hilbert1.1 36120 | There is a line through an... |
| hilbert1.2 36121 | There is at most one line ... |
| linethrueu 36122 | There is a unique line goi... |
| lineintmo 36123 | Two distinct lines interse... |
| fwddifval 36128 | Calculate the value of the... |
| fwddifnval 36129 | The value of the forward d... |
| fwddifn0 36130 | The value of the n-iterate... |
| fwddifnp1 36131 | The value of the n-iterate... |
| rankung 36132 | The rank of the union of t... |
| ranksng 36133 | The rank of a singleton. ... |
| rankelg 36134 | The membership relation is... |
| rankpwg 36135 | The rank of a power set. ... |
| rank0 36136 | The rank of the empty set ... |
| rankeq1o 36137 | The only set with rank ` 1... |
| elhf 36140 | Membership in the heredita... |
| elhf2 36141 | Alternate form of membersh... |
| elhf2g 36142 | Hereditarily finiteness vi... |
| 0hf 36143 | The empty set is a heredit... |
| hfun 36144 | The union of two HF sets i... |
| hfsn 36145 | The singleton of an HF set... |
| hfadj 36146 | Adjoining one HF element t... |
| hfelhf 36147 | Any member of an HF set is... |
| hftr 36148 | The class of all hereditar... |
| hfext 36149 | Extensionality for HF sets... |
| hfuni 36150 | The union of an HF set is ... |
| hfpw 36151 | The power class of an HF s... |
| hfninf 36152 | ` _om ` is not hereditaril... |
| rmoeqi 36153 | Equality inference for res... |
| rmoeqbii 36154 | Equality inference for res... |
| reueqi 36155 | Equality inference for res... |
| reueqbii 36156 | Equality inference for res... |
| sbceqbii 36157 | Formula-building inference... |
| disjeq1i 36158 | Equality theorem for disjo... |
| disjeq12i 36159 | Equality theorem for disjo... |
| rabeqbii 36160 | Equality theorem for restr... |
| iuneq12i 36161 | Equality theorem for index... |
| iineq1i 36162 | Equality theorem for index... |
| iineq12i 36163 | Equality theorem for index... |
| riotaeqbii 36164 | Equivalent wff's and equal... |
| riotaeqi 36165 | Equal domains yield equal ... |
| ixpeq1i 36166 | Equality inference for inf... |
| ixpeq12i 36167 | Equality inference for inf... |
| sumeq2si 36168 | Equality inference for sum... |
| sumeq12si 36169 | Equality inference for sum... |
| prodeq2si 36170 | Equality inference for pro... |
| prodeq12si 36171 | Equality inference for pro... |
| itgeq12i 36172 | Equality inference for an ... |
| itgeq1i 36173 | Equality inference for an ... |
| itgeq2i 36174 | Equality inference for an ... |
| ditgeq123i 36175 | Equality inference for the... |
| ditgeq12i 36176 | Equality inference for the... |
| ditgeq3i 36177 | Equality inference for the... |
| rmoeqdv 36178 | Formula-building rule for ... |
| rmoeqbidv 36179 | Formula-building rule for ... |
| reueqdv 36180 | Formula-building rule for ... |
| reueqbidv 36181 | Formula-building rule for ... |
| sbequbidv 36182 | Deduction substituting bot... |
| disjeq12dv 36183 | Equality theorem for disjo... |
| ixpeq12dv 36184 | Equality theorem for infin... |
| sumeq12sdv 36185 | Equality deduction for sum... |
| prodeq12sdv 36186 | Equality deduction for pro... |
| itgeq12sdv 36187 | Equality theorem for an in... |
| itgeq2sdv 36188 | Equality theorem for an in... |
| ditgeq123dv 36189 | Equality theorem for the d... |
| ditgeq12d 36190 | Equality theorem for the d... |
| ditgeq3sdv 36191 | Equality theorem for the d... |
| in-ax8 36192 | A proof of ~ ax-8 that doe... |
| ss-ax8 36193 | A proof of ~ ax-8 that doe... |
| cbvralvw2 36194 | Change bound variable and ... |
| cbvrexvw2 36195 | Change bound variable and ... |
| cbvrmovw2 36196 | Change bound variable and ... |
| cbvreuvw2 36197 | Change bound variable and ... |
| cbvsbcvw2 36198 | Change bound variable of a... |
| cbvcsbvw2 36199 | Change bound variable of a... |
| cbviunvw2 36200 | Change bound variable and ... |
| cbviinvw2 36201 | Change bound variable and ... |
| cbvmptvw2 36202 | Change bound variable and ... |
| cbvdisjvw2 36203 | Change bound variable and ... |
| cbvriotavw2 36204 | Change bound variable and ... |
| cbvoprab1vw 36205 | Change the first bound var... |
| cbvoprab2vw 36206 | Change the second bound va... |
| cbvoprab123vw 36207 | Change all bound variables... |
| cbvoprab23vw 36208 | Change the second and thir... |
| cbvoprab13vw 36209 | Change the first and third... |
| cbvmpovw2 36210 | Change bound variables and... |
| cbvmpo1vw2 36211 | Change domains and the fir... |
| cbvmpo2vw2 36212 | Change domains and the sec... |
| cbvixpvw2 36213 | Change bound variable and ... |
| cbvsumvw2 36214 | Change bound variable and ... |
| cbvprodvw2 36215 | Change bound variable and ... |
| cbvitgvw2 36216 | Change bound variable and ... |
| cbvditgvw2 36217 | Change bound variable and ... |
| cbvmodavw 36218 | Change bound variable in t... |
| cbveudavw 36219 | Change bound variable in t... |
| cbvrmodavw 36220 | Change bound variable in t... |
| cbvreudavw 36221 | Change bound variable in t... |
| cbvsbdavw 36222 | Change bound variable in p... |
| cbvsbdavw2 36223 | Change bound variable in p... |
| cbvabdavw 36224 | Change bound variable in c... |
| cbvsbcdavw 36225 | Change bound variable of a... |
| cbvsbcdavw2 36226 | Change bound variable of a... |
| cbvcsbdavw 36227 | Change bound variable of a... |
| cbvcsbdavw2 36228 | Change bound variable of a... |
| cbvrabdavw 36229 | Change bound variable in r... |
| cbviundavw 36230 | Change bound variable in i... |
| cbviindavw 36231 | Change bound variable in i... |
| cbvopab1davw 36232 | Change the first bound var... |
| cbvopab2davw 36233 | Change the second bound va... |
| cbvopabdavw 36234 | Change bound variables in ... |
| cbvmptdavw 36235 | Change bound variable in a... |
| cbvdisjdavw 36236 | Change bound variable in a... |
| cbviotadavw 36237 | Change bound variable in a... |
| cbvriotadavw 36238 | Change bound variable in a... |
| cbvoprab1davw 36239 | Change the first bound var... |
| cbvoprab2davw 36240 | Change the second bound va... |
| cbvoprab3davw 36241 | Change the third bound var... |
| cbvoprab123davw 36242 | Change all bound variables... |
| cbvoprab12davw 36243 | Change the first and secon... |
| cbvoprab23davw 36244 | Change the second and thir... |
| cbvoprab13davw 36245 | Change the first and third... |
| cbvixpdavw 36246 | Change bound variable in a... |
| cbvsumdavw 36247 | Change bound variable in a... |
| cbvproddavw 36248 | Change bound variable in a... |
| cbvitgdavw 36249 | Change bound variable in a... |
| cbvditgdavw 36250 | Change bound variable in a... |
| cbvrmodavw2 36251 | Change bound variable and ... |
| cbvreudavw2 36252 | Change bound variable and ... |
| cbvrabdavw2 36253 | Change bound variable and ... |
| cbviundavw2 36254 | Change bound variable and ... |
| cbviindavw2 36255 | Change bound variable and ... |
| cbvmptdavw2 36256 | Change bound variable and ... |
| cbvdisjdavw2 36257 | Change bound variable and ... |
| cbvriotadavw2 36258 | Change bound variable and ... |
| cbvmpodavw2 36259 | Change bound variable and ... |
| cbvmpo1davw2 36260 | Change first bound variabl... |
| cbvmpo2davw2 36261 | Change second bound variab... |
| cbvixpdavw2 36262 | Change bound variable and ... |
| cbvsumdavw2 36263 | Change bound variable and ... |
| cbvproddavw2 36264 | Change bound variable and ... |
| cbvitgdavw2 36265 | Change bound variable and ... |
| cbvditgdavw2 36266 | Change bound variable and ... |
| mpomulnzcnf 36267 | Multiplication maps nonzer... |
| a1i14 36268 | Add two antecedents to a w... |
| a1i24 36269 | Add two antecedents to a w... |
| exp5d 36270 | An exportation inference. ... |
| exp5g 36271 | An exportation inference. ... |
| exp5k 36272 | An exportation inference. ... |
| exp56 36273 | An exportation inference. ... |
| exp58 36274 | An exportation inference. ... |
| exp510 36275 | An exportation inference. ... |
| exp511 36276 | An exportation inference. ... |
| exp512 36277 | An exportation inference. ... |
| 3com12d 36278 | Commutation in consequent.... |
| imp5p 36279 | A triple importation infer... |
| imp5q 36280 | A triple importation infer... |
| ecase13d 36281 | Deduction for elimination ... |
| subtr 36282 | Transitivity of implicit s... |
| subtr2 36283 | Transitivity of implicit s... |
| trer 36284 | A relation intersected wit... |
| elicc3 36285 | An equivalent membership c... |
| finminlem 36286 | A useful lemma about finit... |
| gtinf 36287 | Any number greater than an... |
| opnrebl 36288 | A set is open in the stand... |
| opnrebl2 36289 | A set is open in the stand... |
| nn0prpwlem 36290 | Lemma for ~ nn0prpw . Use... |
| nn0prpw 36291 | Two nonnegative integers a... |
| topbnd 36292 | Two equivalent expressions... |
| opnbnd 36293 | A set is open iff it is di... |
| cldbnd 36294 | A set is closed iff it con... |
| ntruni 36295 | A union of interiors is a ... |
| clsun 36296 | A pairwise union of closur... |
| clsint2 36297 | The closure of an intersec... |
| opnregcld 36298 | A set is regularly closed ... |
| cldregopn 36299 | A set if regularly open if... |
| neiin 36300 | Two neighborhoods intersec... |
| hmeoclda 36301 | Homeomorphisms preserve cl... |
| hmeocldb 36302 | Homeomorphisms preserve cl... |
| ivthALT 36303 | An alternate proof of the ... |
| fnerel 36306 | Fineness is a relation. (... |
| isfne 36307 | The predicate " ` B ` is f... |
| isfne4 36308 | The predicate " ` B ` is f... |
| isfne4b 36309 | A condition for a topology... |
| isfne2 36310 | The predicate " ` B ` is f... |
| isfne3 36311 | The predicate " ` B ` is f... |
| fnebas 36312 | A finer cover covers the s... |
| fnetg 36313 | A finer cover generates a ... |
| fnessex 36314 | If ` B ` is finer than ` A... |
| fneuni 36315 | If ` B ` is finer than ` A... |
| fneint 36316 | If a cover is finer than a... |
| fness 36317 | A cover is finer than its ... |
| fneref 36318 | Reflexivity of the finenes... |
| fnetr 36319 | Transitivity of the finene... |
| fneval 36320 | Two covers are finer than ... |
| fneer 36321 | Fineness intersected with ... |
| topfne 36322 | Fineness for covers corres... |
| topfneec 36323 | A cover is equivalent to a... |
| topfneec2 36324 | A topology is precisely id... |
| fnessref 36325 | A cover is finer iff it ha... |
| refssfne 36326 | A cover is a refinement if... |
| neibastop1 36327 | A collection of neighborho... |
| neibastop2lem 36328 | Lemma for ~ neibastop2 . ... |
| neibastop2 36329 | In the topology generated ... |
| neibastop3 36330 | The topology generated by ... |
| topmtcl 36331 | The meet of a collection o... |
| topmeet 36332 | Two equivalent formulation... |
| topjoin 36333 | Two equivalent formulation... |
| fnemeet1 36334 | The meet of a collection o... |
| fnemeet2 36335 | The meet of equivalence cl... |
| fnejoin1 36336 | Join of equivalence classe... |
| fnejoin2 36337 | Join of equivalence classe... |
| fgmin 36338 | Minimality property of a g... |
| neifg 36339 | The neighborhood filter of... |
| tailfval 36340 | The tail function for a di... |
| tailval 36341 | The tail of an element in ... |
| eltail 36342 | An element of a tail. (Co... |
| tailf 36343 | The tail function of a dir... |
| tailini 36344 | A tail contains its initia... |
| tailfb 36345 | The collection of tails of... |
| filnetlem1 36346 | Lemma for ~ filnet . Chan... |
| filnetlem2 36347 | Lemma for ~ filnet . The ... |
| filnetlem3 36348 | Lemma for ~ filnet . (Con... |
| filnetlem4 36349 | Lemma for ~ filnet . (Con... |
| filnet 36350 | A filter has the same conv... |
| tb-ax1 36351 | The first of three axioms ... |
| tb-ax2 36352 | The second of three axioms... |
| tb-ax3 36353 | The third of three axioms ... |
| tbsyl 36354 | The weak syllogism from Ta... |
| re1ax2lem 36355 | Lemma for ~ re1ax2 . (Con... |
| re1ax2 36356 | ~ ax-2 rederived from the ... |
| naim1 36357 | Constructor theorem for ` ... |
| naim2 36358 | Constructor theorem for ` ... |
| naim1i 36359 | Constructor rule for ` -/\... |
| naim2i 36360 | Constructor rule for ` -/\... |
| naim12i 36361 | Constructor rule for ` -/\... |
| nabi1i 36362 | Constructor rule for ` -/\... |
| nabi2i 36363 | Constructor rule for ` -/\... |
| nabi12i 36364 | Constructor rule for ` -/\... |
| df3nandALT1 36367 | The double nand expressed ... |
| df3nandALT2 36368 | The double nand expressed ... |
| andnand1 36369 | Double and in terms of dou... |
| imnand2 36370 | An ` -> ` nand relation. ... |
| nalfal 36371 | Not all sets hold ` F. ` a... |
| nexntru 36372 | There does not exist a set... |
| nexfal 36373 | There does not exist a set... |
| neufal 36374 | There does not exist exact... |
| neutru 36375 | There does not exist exact... |
| nmotru 36376 | There does not exist at mo... |
| mofal 36377 | There exist at most one se... |
| nrmo 36378 | "At most one" restricted e... |
| meran1 36379 | A single axiom for proposi... |
| meran2 36380 | A single axiom for proposi... |
| meran3 36381 | A single axiom for proposi... |
| waj-ax 36382 | A single axiom for proposi... |
| lukshef-ax2 36383 | A single axiom for proposi... |
| arg-ax 36384 | A single axiom for proposi... |
| negsym1 36385 | In the paper "On Variable ... |
| imsym1 36386 | A symmetry with ` -> ` . ... |
| bisym1 36387 | A symmetry with ` <-> ` . ... |
| consym1 36388 | A symmetry with ` /\ ` . ... |
| dissym1 36389 | A symmetry with ` \/ ` . ... |
| nandsym1 36390 | A symmetry with ` -/\ ` . ... |
| unisym1 36391 | A symmetry with ` A. ` . ... |
| exisym1 36392 | A symmetry with ` E. ` . ... |
| unqsym1 36393 | A symmetry with ` E! ` . ... |
| amosym1 36394 | A symmetry with ` E* ` . ... |
| subsym1 36395 | A symmetry with ` [ x / y ... |
| ontopbas 36396 | An ordinal number is a top... |
| onsstopbas 36397 | The class of ordinal numbe... |
| onpsstopbas 36398 | The class of ordinal numbe... |
| ontgval 36399 | The topology generated fro... |
| ontgsucval 36400 | The topology generated fro... |
| onsuctop 36401 | A successor ordinal number... |
| onsuctopon 36402 | One of the topologies on a... |
| ordtoplem 36403 | Membership of the class of... |
| ordtop 36404 | An ordinal is a topology i... |
| onsucconni 36405 | A successor ordinal number... |
| onsucconn 36406 | A successor ordinal number... |
| ordtopconn 36407 | An ordinal topology is con... |
| onintopssconn 36408 | An ordinal topology is con... |
| onsuct0 36409 | A successor ordinal number... |
| ordtopt0 36410 | An ordinal topology is T_0... |
| onsucsuccmpi 36411 | The successor of a success... |
| onsucsuccmp 36412 | The successor of a success... |
| limsucncmpi 36413 | The successor of a limit o... |
| limsucncmp 36414 | The successor of a limit o... |
| ordcmp 36415 | An ordinal topology is com... |
| ssoninhaus 36416 | The ordinal topologies ` 1... |
| onint1 36417 | The ordinal T_1 spaces are... |
| oninhaus 36418 | The ordinal Hausdorff spac... |
| fveleq 36419 | Please add description her... |
| findfvcl 36420 | Please add description her... |
| findreccl 36421 | Please add description her... |
| findabrcl 36422 | Please add description her... |
| nnssi2 36423 | Convert a theorem for real... |
| nnssi3 36424 | Convert a theorem for real... |
| nndivsub 36425 | Please add description her... |
| nndivlub 36426 | A factor of a positive int... |
| ee7.2aOLD 36429 | Lemma for Euclid's Element... |
| weiunlem1 36430 | Lemma for ~ weiunpo , ~ we... |
| weiunlem2 36431 | Lemma for ~ weiunpo , ~ we... |
| weiunfrlem 36432 | Lemma for ~ weiunfr . (Co... |
| weiunpo 36433 | A partial ordering on an i... |
| weiunso 36434 | A strict ordering on an in... |
| weiunfr 36435 | A well-founded relation on... |
| weiunse 36436 | The relation constructed i... |
| weiunwe 36437 | A well-ordering on an inde... |
| numiunnum 36438 | An indexed union of sets i... |
| dnival 36439 | Value of the "distance to ... |
| dnicld1 36440 | Closure theorem for the "d... |
| dnicld2 36441 | Closure theorem for the "d... |
| dnif 36442 | The "distance to nearest i... |
| dnizeq0 36443 | The distance to nearest in... |
| dnizphlfeqhlf 36444 | The distance to nearest in... |
| rddif2 36445 | Variant of ~ rddif . (Con... |
| dnibndlem1 36446 | Lemma for ~ dnibnd . (Con... |
| dnibndlem2 36447 | Lemma for ~ dnibnd . (Con... |
| dnibndlem3 36448 | Lemma for ~ dnibnd . (Con... |
| dnibndlem4 36449 | Lemma for ~ dnibnd . (Con... |
| dnibndlem5 36450 | Lemma for ~ dnibnd . (Con... |
| dnibndlem6 36451 | Lemma for ~ dnibnd . (Con... |
| dnibndlem7 36452 | Lemma for ~ dnibnd . (Con... |
| dnibndlem8 36453 | Lemma for ~ dnibnd . (Con... |
| dnibndlem9 36454 | Lemma for ~ dnibnd . (Con... |
| dnibndlem10 36455 | Lemma for ~ dnibnd . (Con... |
| dnibndlem11 36456 | Lemma for ~ dnibnd . (Con... |
| dnibndlem12 36457 | Lemma for ~ dnibnd . (Con... |
| dnibndlem13 36458 | Lemma for ~ dnibnd . (Con... |
| dnibnd 36459 | The "distance to nearest i... |
| dnicn 36460 | The "distance to nearest i... |
| knoppcnlem1 36461 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem2 36462 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem3 36463 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem4 36464 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem5 36465 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem6 36466 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem7 36467 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem8 36468 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem9 36469 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem10 36470 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem11 36471 | Lemma for ~ knoppcn . (Co... |
| knoppcn 36472 | The continuous nowhere dif... |
| knoppcld 36473 | Closure theorem for Knopp'... |
| unblimceq0lem 36474 | Lemma for ~ unblimceq0 . ... |
| unblimceq0 36475 | If ` F ` is unbounded near... |
| unbdqndv1 36476 | If the difference quotient... |
| unbdqndv2lem1 36477 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2lem2 36478 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2 36479 | Variant of ~ unbdqndv1 wit... |
| knoppndvlem1 36480 | Lemma for ~ knoppndv . (C... |
| knoppndvlem2 36481 | Lemma for ~ knoppndv . (C... |
| knoppndvlem3 36482 | Lemma for ~ knoppndv . (C... |
| knoppndvlem4 36483 | Lemma for ~ knoppndv . (C... |
| knoppndvlem5 36484 | Lemma for ~ knoppndv . (C... |
| knoppndvlem6 36485 | Lemma for ~ knoppndv . (C... |
| knoppndvlem7 36486 | Lemma for ~ knoppndv . (C... |
| knoppndvlem8 36487 | Lemma for ~ knoppndv . (C... |
| knoppndvlem9 36488 | Lemma for ~ knoppndv . (C... |
| knoppndvlem10 36489 | Lemma for ~ knoppndv . (C... |
| knoppndvlem11 36490 | Lemma for ~ knoppndv . (C... |
| knoppndvlem12 36491 | Lemma for ~ knoppndv . (C... |
| knoppndvlem13 36492 | Lemma for ~ knoppndv . (C... |
| knoppndvlem14 36493 | Lemma for ~ knoppndv . (C... |
| knoppndvlem15 36494 | Lemma for ~ knoppndv . (C... |
| knoppndvlem16 36495 | Lemma for ~ knoppndv . (C... |
| knoppndvlem17 36496 | Lemma for ~ knoppndv . (C... |
| knoppndvlem18 36497 | Lemma for ~ knoppndv . (C... |
| knoppndvlem19 36498 | Lemma for ~ knoppndv . (C... |
| knoppndvlem20 36499 | Lemma for ~ knoppndv . (C... |
| knoppndvlem21 36500 | Lemma for ~ knoppndv . (C... |
| knoppndvlem22 36501 | Lemma for ~ knoppndv . (C... |
| knoppndv 36502 | The continuous nowhere dif... |
| knoppf 36503 | Knopp's function is a func... |
| knoppcn2 36504 | Variant of ~ knoppcn with ... |
| cnndvlem1 36505 | Lemma for ~ cnndv . (Cont... |
| cnndvlem2 36506 | Lemma for ~ cnndv . (Cont... |
| cnndv 36507 | There exists a continuous ... |
| bj-mp2c 36508 | A double _modus ponens_ in... |
| bj-mp2d 36509 | A double _modus ponens_ in... |
| bj-0 36510 | A syntactic theorem. See ... |
| bj-1 36511 | In this proof, the use of ... |
| bj-a1k 36512 | Weakening of ~ ax-1 . As ... |
| bj-poni 36513 | Inference associated with ... |
| bj-nnclav 36514 | When ` F. ` is substituted... |
| bj-nnclavi 36515 | Inference associated with ... |
| bj-nnclavc 36516 | Commuted form of ~ bj-nncl... |
| bj-nnclavci 36517 | Inference associated with ... |
| bj-jarrii 36518 | Inference associated with ... |
| bj-imim21 36519 | The propositional function... |
| bj-imim21i 36520 | Inference associated with ... |
| bj-peircestab 36521 | Over minimal implicational... |
| bj-stabpeirce 36522 | This minimal implicational... |
| bj-syl66ib 36523 | A mixed syllogism inferenc... |
| bj-orim2 36524 | Proof of ~ orim2 from the ... |
| bj-currypeirce 36525 | Curry's axiom ~ curryax (a... |
| bj-peircecurry 36526 | Peirce's axiom ~ peirce im... |
| bj-animbi 36527 | Conjunction in terms of im... |
| bj-currypara 36528 | Curry's paradox. Note tha... |
| bj-con2com 36529 | A commuted form of the con... |
| bj-con2comi 36530 | Inference associated with ... |
| bj-nimn 36531 | If a formula is true, then... |
| bj-nimni 36532 | Inference associated with ... |
| bj-peircei 36533 | Inference associated with ... |
| bj-looinvi 36534 | Inference associated with ... |
| bj-looinvii 36535 | Inference associated with ... |
| bj-mt2bi 36536 | Version of ~ mt2 where the... |
| bj-ntrufal 36537 | The negation of a theorem ... |
| bj-fal 36538 | Shortening of ~ fal using ... |
| bj-jaoi1 36539 | Shortens ~ orfa2 (58>53), ... |
| bj-jaoi2 36540 | Shortens ~ consensus (110>... |
| bj-dfbi4 36541 | Alternate definition of th... |
| bj-dfbi5 36542 | Alternate definition of th... |
| bj-dfbi6 36543 | Alternate definition of th... |
| bj-bijust0ALT 36544 | Alternate proof of ~ bijus... |
| bj-bijust00 36545 | A self-implication does no... |
| bj-consensus 36546 | Version of ~ consensus exp... |
| bj-consensusALT 36547 | Alternate proof of ~ bj-co... |
| bj-df-ifc 36548 | Candidate definition for t... |
| bj-dfif 36549 | Alternate definition of th... |
| bj-ififc 36550 | A biconditional connecting... |
| bj-imbi12 36551 | Uncurried (imported) form ... |
| bj-falor 36552 | Dual of ~ truan (which has... |
| bj-falor2 36553 | Dual of ~ truan . (Contri... |
| bj-bibibi 36554 | A property of the bicondit... |
| bj-imn3ani 36555 | Duplication of ~ bnj1224 .... |
| bj-andnotim 36556 | Two ways of expressing a c... |
| bj-bi3ant 36557 | This used to be in the mai... |
| bj-bisym 36558 | This used to be in the mai... |
| bj-bixor 36559 | Equivalence of two ternary... |
| bj-axdd2 36560 | This implication, proved u... |
| bj-axd2d 36561 | This implication, proved u... |
| bj-axtd 36562 | This implication, proved f... |
| bj-gl4 36563 | In a normal modal logic, t... |
| bj-axc4 36564 | Over minimal calculus, the... |
| prvlem1 36569 | An elementary property of ... |
| prvlem2 36570 | An elementary property of ... |
| bj-babygodel 36571 | See the section header com... |
| bj-babylob 36572 | See the section header com... |
| bj-godellob 36573 | Proof of Gödel's theo... |
| bj-genr 36574 | Generalization rule on the... |
| bj-genl 36575 | Generalization rule on the... |
| bj-genan 36576 | Generalization rule on a c... |
| bj-mpgs 36577 | From a closed form theorem... |
| bj-2alim 36578 | Closed form of ~ 2alimi . ... |
| bj-2exim 36579 | Closed form of ~ 2eximi . ... |
| bj-alanim 36580 | Closed form of ~ alanimi .... |
| bj-2albi 36581 | Closed form of ~ 2albii . ... |
| bj-notalbii 36582 | Equivalence of universal q... |
| bj-2exbi 36583 | Closed form of ~ 2exbii . ... |
| bj-3exbi 36584 | Closed form of ~ 3exbii . ... |
| bj-sylggt 36585 | Stronger form of ~ sylgt ,... |
| bj-sylgt2 36586 | Uncurried (imported) form ... |
| bj-alrimg 36587 | The general form of the *a... |
| bj-alrimd 36588 | A slightly more general ~ ... |
| bj-sylget 36589 | Dual statement of ~ sylgt ... |
| bj-sylget2 36590 | Uncurried (imported) form ... |
| bj-exlimg 36591 | The general form of the *e... |
| bj-sylge 36592 | Dual statement of ~ sylg (... |
| bj-exlimd 36593 | A slightly more general ~ ... |
| bj-nfimexal 36594 | A weak from of nonfreeness... |
| bj-alexim 36595 | Closed form of ~ aleximi .... |
| bj-nexdh 36596 | Closed form of ~ nexdh (ac... |
| bj-nexdh2 36597 | Uncurried (imported) form ... |
| bj-hbxfrbi 36598 | Closed form of ~ hbxfrbi .... |
| bj-hbyfrbi 36599 | Version of ~ bj-hbxfrbi wi... |
| bj-exalim 36600 | Distribute quantifiers ove... |
| bj-exalimi 36601 | An inference for distribut... |
| bj-exalims 36602 | Distributing quantifiers o... |
| bj-exalimsi 36603 | An inference for distribut... |
| bj-ax12ig 36604 | A lemma used to prove a we... |
| bj-ax12i 36605 | A weakening of ~ bj-ax12ig... |
| bj-nfimt 36606 | Closed form of ~ nfim and ... |
| bj-cbvalimt 36607 | A lemma in closed form use... |
| bj-cbveximt 36608 | A lemma in closed form use... |
| bj-eximALT 36609 | Alternate proof of ~ exim ... |
| bj-aleximiALT 36610 | Alternate proof of ~ alexi... |
| bj-eximcom 36611 | A commuted form of ~ exim ... |
| bj-ax12wlem 36612 | A lemma used to prove a we... |
| bj-cbvalim 36613 | A lemma used to prove ~ bj... |
| bj-cbvexim 36614 | A lemma used to prove ~ bj... |
| bj-cbvalimi 36615 | An equality-free general i... |
| bj-cbveximi 36616 | An equality-free general i... |
| bj-cbval 36617 | Changing a bound variable ... |
| bj-cbvex 36618 | Changing a bound variable ... |
| bj-ssbeq 36621 | Substitution in an equalit... |
| bj-ssblem1 36622 | A lemma for the definiens ... |
| bj-ssblem2 36623 | An instance of ~ ax-11 pro... |
| bj-ax12v 36624 | A weaker form of ~ ax-12 a... |
| bj-ax12 36625 | Remove a DV condition from... |
| bj-ax12ssb 36626 | Axiom ~ bj-ax12 expressed ... |
| bj-19.41al 36627 | Special case of ~ 19.41 pr... |
| bj-equsexval 36628 | Special case of ~ equsexv ... |
| bj-subst 36629 | Proof of ~ sbalex from cor... |
| bj-ssbid2 36630 | A special case of ~ sbequ2... |
| bj-ssbid2ALT 36631 | Alternate proof of ~ bj-ss... |
| bj-ssbid1 36632 | A special case of ~ sbequ1... |
| bj-ssbid1ALT 36633 | Alternate proof of ~ bj-ss... |
| bj-ax6elem1 36634 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6elem2 36635 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6e 36636 | Proof of ~ ax6e (hence ~ a... |
| bj-spimvwt 36637 | Closed form of ~ spimvw . ... |
| bj-spnfw 36638 | Theorem close to a closed ... |
| bj-cbvexiw 36639 | Change bound variable. Th... |
| bj-cbvexivw 36640 | Change bound variable. Th... |
| bj-modald 36641 | A short form of the axiom ... |
| bj-denot 36642 | A weakening of ~ ax-6 and ... |
| bj-eqs 36643 | A lemma for substitutions,... |
| bj-cbvexw 36644 | Change bound variable. Th... |
| bj-ax12w 36645 | The general statement that... |
| bj-ax89 36646 | A theorem which could be u... |
| bj-cleljusti 36647 | One direction of ~ cleljus... |
| bj-alcomexcom 36648 | Commutation of two existen... |
| bj-hbalt 36649 | Closed form of ~ hbal . W... |
| axc11n11 36650 | Proof of ~ axc11n from { ~... |
| axc11n11r 36651 | Proof of ~ axc11n from { ~... |
| bj-axc16g16 36652 | Proof of ~ axc16g from { ~... |
| bj-ax12v3 36653 | A weak version of ~ ax-12 ... |
| bj-ax12v3ALT 36654 | Alternate proof of ~ bj-ax... |
| bj-sb 36655 | A weak variant of ~ sbid2 ... |
| bj-modalbe 36656 | The predicate-calculus ver... |
| bj-spst 36657 | Closed form of ~ sps . On... |
| bj-19.21bit 36658 | Closed form of ~ 19.21bi .... |
| bj-19.23bit 36659 | Closed form of ~ 19.23bi .... |
| bj-nexrt 36660 | Closed form of ~ nexr . C... |
| bj-alrim 36661 | Closed form of ~ alrimi . ... |
| bj-alrim2 36662 | Uncurried (imported) form ... |
| bj-nfdt0 36663 | A theorem close to a close... |
| bj-nfdt 36664 | Closed form of ~ nf5d and ... |
| bj-nexdt 36665 | Closed form of ~ nexd . (... |
| bj-nexdvt 36666 | Closed form of ~ nexdv . ... |
| bj-alexbiex 36667 | Adding a second quantifier... |
| bj-exexbiex 36668 | Adding a second quantifier... |
| bj-alalbial 36669 | Adding a second quantifier... |
| bj-exalbial 36670 | Adding a second quantifier... |
| bj-19.9htbi 36671 | Strengthening ~ 19.9ht by ... |
| bj-hbntbi 36672 | Strengthening ~ hbnt by re... |
| bj-biexal1 36673 | A general FOL biconditiona... |
| bj-biexal2 36674 | When ` ph ` is substituted... |
| bj-biexal3 36675 | When ` ph ` is substituted... |
| bj-bialal 36676 | When ` ph ` is substituted... |
| bj-biexex 36677 | When ` ph ` is substituted... |
| bj-hbext 36678 | Closed form of ~ hbex . (... |
| bj-nfalt 36679 | Closed form of ~ nfal . (... |
| bj-nfext 36680 | Closed form of ~ nfex . (... |
| bj-eeanvw 36681 | Version of ~ exdistrv with... |
| bj-modal4 36682 | First-order logic form of ... |
| bj-modal4e 36683 | First-order logic form of ... |
| bj-modalb 36684 | A short form of the axiom ... |
| bj-wnf1 36685 | When ` ph ` is substituted... |
| bj-wnf2 36686 | When ` ph ` is substituted... |
| bj-wnfanf 36687 | When ` ph ` is substituted... |
| bj-wnfenf 36688 | When ` ph ` is substituted... |
| bj-substax12 36689 | Equivalent form of the axi... |
| bj-substw 36690 | Weak form of the LHS of ~ ... |
| bj-nnfbi 36693 | If two formulas are equiva... |
| bj-nnfbd 36694 | If two formulas are equiva... |
| bj-nnfbii 36695 | If two formulas are equiva... |
| bj-nnfa 36696 | Nonfreeness implies the eq... |
| bj-nnfad 36697 | Nonfreeness implies the eq... |
| bj-nnfai 36698 | Nonfreeness implies the eq... |
| bj-nnfe 36699 | Nonfreeness implies the eq... |
| bj-nnfed 36700 | Nonfreeness implies the eq... |
| bj-nnfei 36701 | Nonfreeness implies the eq... |
| bj-nnfea 36702 | Nonfreeness implies the eq... |
| bj-nnfead 36703 | Nonfreeness implies the eq... |
| bj-nnfeai 36704 | Nonfreeness implies the eq... |
| bj-dfnnf2 36705 | Alternate definition of ~ ... |
| bj-nnfnfTEMP 36706 | New nonfreeness implies ol... |
| bj-wnfnf 36707 | When ` ph ` is substituted... |
| bj-nnfnt 36708 | A variable is nonfree in a... |
| bj-nnftht 36709 | A variable is nonfree in a... |
| bj-nnfth 36710 | A variable is nonfree in a... |
| bj-nnfnth 36711 | A variable is nonfree in t... |
| bj-nnfim1 36712 | A consequence of nonfreene... |
| bj-nnfim2 36713 | A consequence of nonfreene... |
| bj-nnfim 36714 | Nonfreeness in the anteced... |
| bj-nnfimd 36715 | Nonfreeness in the anteced... |
| bj-nnfan 36716 | Nonfreeness in both conjun... |
| bj-nnfand 36717 | Nonfreeness in both conjun... |
| bj-nnfor 36718 | Nonfreeness in both disjun... |
| bj-nnford 36719 | Nonfreeness in both disjun... |
| bj-nnfbit 36720 | Nonfreeness in both sides ... |
| bj-nnfbid 36721 | Nonfreeness in both sides ... |
| bj-nnfv 36722 | A non-occurring variable i... |
| bj-nnf-alrim 36723 | Proof of the closed form o... |
| bj-nnf-exlim 36724 | Proof of the closed form o... |
| bj-dfnnf3 36725 | Alternate definition of no... |
| bj-nfnnfTEMP 36726 | New nonfreeness is equival... |
| bj-nnfa1 36727 | See ~ nfa1 . (Contributed... |
| bj-nnfe1 36728 | See ~ nfe1 . (Contributed... |
| bj-19.12 36729 | See ~ 19.12 . Could be la... |
| bj-nnflemaa 36730 | One of four lemmas for non... |
| bj-nnflemee 36731 | One of four lemmas for non... |
| bj-nnflemae 36732 | One of four lemmas for non... |
| bj-nnflemea 36733 | One of four lemmas for non... |
| bj-nnfalt 36734 | See ~ nfal and ~ bj-nfalt ... |
| bj-nnfext 36735 | See ~ nfex and ~ bj-nfext ... |
| bj-stdpc5t 36736 | Alias of ~ bj-nnf-alrim fo... |
| bj-19.21t 36737 | Statement ~ 19.21t proved ... |
| bj-19.23t 36738 | Statement ~ 19.23t proved ... |
| bj-19.36im 36739 | One direction of ~ 19.36 f... |
| bj-19.37im 36740 | One direction of ~ 19.37 f... |
| bj-19.42t 36741 | Closed form of ~ 19.42 fro... |
| bj-19.41t 36742 | Closed form of ~ 19.41 fro... |
| bj-sbft 36743 | Version of ~ sbft using ` ... |
| bj-pm11.53vw 36744 | Version of ~ pm11.53v with... |
| bj-pm11.53v 36745 | Version of ~ pm11.53v with... |
| bj-pm11.53a 36746 | A variant of ~ pm11.53v . ... |
| bj-equsvt 36747 | A variant of ~ equsv . (C... |
| bj-equsalvwd 36748 | Variant of ~ equsalvw . (... |
| bj-equsexvwd 36749 | Variant of ~ equsexvw . (... |
| bj-sbievwd 36750 | Variant of ~ sbievw . (Co... |
| bj-axc10 36751 | Alternate proof of ~ axc10... |
| bj-alequex 36752 | A fol lemma. See ~ aleque... |
| bj-spimt2 36753 | A step in the proof of ~ s... |
| bj-cbv3ta 36754 | Closed form of ~ cbv3 . (... |
| bj-cbv3tb 36755 | Closed form of ~ cbv3 . (... |
| bj-hbsb3t 36756 | A theorem close to a close... |
| bj-hbsb3 36757 | Shorter proof of ~ hbsb3 .... |
| bj-nfs1t 36758 | A theorem close to a close... |
| bj-nfs1t2 36759 | A theorem close to a close... |
| bj-nfs1 36760 | Shorter proof of ~ nfs1 (t... |
| bj-axc10v 36761 | Version of ~ axc10 with a ... |
| bj-spimtv 36762 | Version of ~ spimt with a ... |
| bj-cbv3hv2 36763 | Version of ~ cbv3h with tw... |
| bj-cbv1hv 36764 | Version of ~ cbv1h with a ... |
| bj-cbv2hv 36765 | Version of ~ cbv2h with a ... |
| bj-cbv2v 36766 | Version of ~ cbv2 with a d... |
| bj-cbvaldv 36767 | Version of ~ cbvald with a... |
| bj-cbvexdv 36768 | Version of ~ cbvexd with a... |
| bj-cbval2vv 36769 | Version of ~ cbval2vv with... |
| bj-cbvex2vv 36770 | Version of ~ cbvex2vv with... |
| bj-cbvaldvav 36771 | Version of ~ cbvaldva with... |
| bj-cbvexdvav 36772 | Version of ~ cbvexdva with... |
| bj-cbvex4vv 36773 | Version of ~ cbvex4v with ... |
| bj-equsalhv 36774 | Version of ~ equsalh with ... |
| bj-axc11nv 36775 | Version of ~ axc11n with a... |
| bj-aecomsv 36776 | Version of ~ aecoms with a... |
| bj-axc11v 36777 | Version of ~ axc11 with a ... |
| bj-drnf2v 36778 | Version of ~ drnf2 with a ... |
| bj-equs45fv 36779 | Version of ~ equs45f with ... |
| bj-hbs1 36780 | Version of ~ hbsb2 with a ... |
| bj-nfs1v 36781 | Version of ~ nfsb2 with a ... |
| bj-hbsb2av 36782 | Version of ~ hbsb2a with a... |
| bj-hbsb3v 36783 | Version of ~ hbsb3 with a ... |
| bj-nfsab1 36784 | Remove dependency on ~ ax-... |
| bj-dtrucor2v 36785 | Version of ~ dtrucor2 with... |
| bj-hbaeb2 36786 | Biconditional version of a... |
| bj-hbaeb 36787 | Biconditional version of ~... |
| bj-hbnaeb 36788 | Biconditional version of ~... |
| bj-dvv 36789 | A special instance of ~ bj... |
| bj-equsal1t 36790 | Duplication of ~ wl-equsal... |
| bj-equsal1ti 36791 | Inference associated with ... |
| bj-equsal1 36792 | One direction of ~ equsal ... |
| bj-equsal2 36793 | One direction of ~ equsal ... |
| bj-equsal 36794 | Shorter proof of ~ equsal ... |
| stdpc5t 36795 | Closed form of ~ stdpc5 . ... |
| bj-stdpc5 36796 | More direct proof of ~ std... |
| 2stdpc5 36797 | A double ~ stdpc5 (one dir... |
| bj-19.21t0 36798 | Proof of ~ 19.21t from ~ s... |
| exlimii 36799 | Inference associated with ... |
| ax11-pm 36800 | Proof of ~ ax-11 similar t... |
| ax6er 36801 | Commuted form of ~ ax6e . ... |
| exlimiieq1 36802 | Inferring a theorem when i... |
| exlimiieq2 36803 | Inferring a theorem when i... |
| ax11-pm2 36804 | Proof of ~ ax-11 from the ... |
| bj-sbsb 36805 | Biconditional showing two ... |
| bj-dfsb2 36806 | Alternate (dual) definitio... |
| bj-sbf3 36807 | Substitution has no effect... |
| bj-sbf4 36808 | Substitution has no effect... |
| bj-eu3f 36809 | Version of ~ eu3v where th... |
| bj-sblem1 36810 | Lemma for substitution. (... |
| bj-sblem2 36811 | Lemma for substitution. (... |
| bj-sblem 36812 | Lemma for substitution. (... |
| bj-sbievw1 36813 | Lemma for substitution. (... |
| bj-sbievw2 36814 | Lemma for substitution. (... |
| bj-sbievw 36815 | Lemma for substitution. C... |
| bj-sbievv 36816 | Version of ~ sbie with a s... |
| bj-moeub 36817 | Uniqueness is equivalent t... |
| bj-sbidmOLD 36818 | Obsolete proof of ~ sbidm ... |
| bj-dvelimdv 36819 | Deduction form of ~ dvelim... |
| bj-dvelimdv1 36820 | Curried (exported) form of... |
| bj-dvelimv 36821 | A version of ~ dvelim usin... |
| bj-nfeel2 36822 | Nonfreeness in a membershi... |
| bj-axc14nf 36823 | Proof of a version of ~ ax... |
| bj-axc14 36824 | Alternate proof of ~ axc14... |
| mobidvALT 36825 | Alternate proof of ~ mobid... |
| sbn1ALT 36826 | Alternate proof of ~ sbn1 ... |
| eliminable1 36827 | A theorem used to prove th... |
| eliminable2a 36828 | A theorem used to prove th... |
| eliminable2b 36829 | A theorem used to prove th... |
| eliminable2c 36830 | A theorem used to prove th... |
| eliminable3a 36831 | A theorem used to prove th... |
| eliminable3b 36832 | A theorem used to prove th... |
| eliminable-velab 36833 | A theorem used to prove th... |
| eliminable-veqab 36834 | A theorem used to prove th... |
| eliminable-abeqv 36835 | A theorem used to prove th... |
| eliminable-abeqab 36836 | A theorem used to prove th... |
| eliminable-abelv 36837 | A theorem used to prove th... |
| eliminable-abelab 36838 | A theorem used to prove th... |
| bj-denoteslem 36839 | Duplicate of ~ issettru an... |
| bj-denotesALTV 36840 | Moved to main as ~ iseqset... |
| bj-issettruALTV 36841 | Moved to main as ~ issettr... |
| bj-elabtru 36842 | This is as close as we can... |
| bj-issetwt 36843 | Closed form of ~ bj-issetw... |
| bj-issetw 36844 | The closest one can get to... |
| bj-issetiv 36845 | Version of ~ bj-isseti wit... |
| bj-isseti 36846 | Version of ~ isseti with a... |
| bj-ralvw 36847 | A weak version of ~ ralv n... |
| bj-rexvw 36848 | A weak version of ~ rexv n... |
| bj-rababw 36849 | A weak version of ~ rabab ... |
| bj-rexcom4bv 36850 | Version of ~ rexcom4b and ... |
| bj-rexcom4b 36851 | Remove from ~ rexcom4b dep... |
| bj-ceqsalt0 36852 | The FOL content of ~ ceqsa... |
| bj-ceqsalt1 36853 | The FOL content of ~ ceqsa... |
| bj-ceqsalt 36854 | Remove from ~ ceqsalt depe... |
| bj-ceqsaltv 36855 | Version of ~ bj-ceqsalt wi... |
| bj-ceqsalg0 36856 | The FOL content of ~ ceqsa... |
| bj-ceqsalg 36857 | Remove from ~ ceqsalg depe... |
| bj-ceqsalgALT 36858 | Alternate proof of ~ bj-ce... |
| bj-ceqsalgv 36859 | Version of ~ bj-ceqsalg wi... |
| bj-ceqsalgvALT 36860 | Alternate proof of ~ bj-ce... |
| bj-ceqsal 36861 | Remove from ~ ceqsal depen... |
| bj-ceqsalv 36862 | Remove from ~ ceqsalv depe... |
| bj-spcimdv 36863 | Remove from ~ spcimdv depe... |
| bj-spcimdvv 36864 | Remove from ~ spcimdv depe... |
| elelb 36865 | Equivalence between two co... |
| bj-pwvrelb 36866 | Characterization of the el... |
| bj-nfcsym 36867 | The nonfreeness quantifier... |
| bj-sbeqALT 36868 | Substitution in an equalit... |
| bj-sbeq 36869 | Distribute proper substitu... |
| bj-sbceqgALT 36870 | Distribute proper substitu... |
| bj-csbsnlem 36871 | Lemma for ~ bj-csbsn (in t... |
| bj-csbsn 36872 | Substitution in a singleto... |
| bj-sbel1 36873 | Version of ~ sbcel1g when ... |
| bj-abv 36874 | The class of sets verifyin... |
| bj-abvALT 36875 | Alternate version of ~ bj-... |
| bj-ab0 36876 | The class of sets verifyin... |
| bj-abf 36877 | Shorter proof of ~ abf (wh... |
| bj-csbprc 36878 | More direct proof of ~ csb... |
| bj-exlimvmpi 36879 | A Fol lemma ( ~ exlimiv fo... |
| bj-exlimmpi 36880 | Lemma for ~ bj-vtoclg1f1 (... |
| bj-exlimmpbi 36881 | Lemma for theorems of the ... |
| bj-exlimmpbir 36882 | Lemma for theorems of the ... |
| bj-vtoclf 36883 | Remove dependency on ~ ax-... |
| bj-vtocl 36884 | Remove dependency on ~ ax-... |
| bj-vtoclg1f1 36885 | The FOL content of ~ vtocl... |
| bj-vtoclg1f 36886 | Reprove ~ vtoclg1f from ~ ... |
| bj-vtoclg1fv 36887 | Version of ~ bj-vtoclg1f w... |
| bj-vtoclg 36888 | A version of ~ vtoclg with... |
| bj-rabeqbid 36889 | Version of ~ rabeqbidv wit... |
| bj-seex 36890 | Version of ~ seex with a d... |
| bj-nfcf 36891 | Version of ~ df-nfc with a... |
| bj-zfauscl 36892 | General version of ~ zfaus... |
| bj-elabd2ALT 36893 | Alternate proof of ~ elabd... |
| bj-unrab 36894 | Generalization of ~ unrab ... |
| bj-inrab 36895 | Generalization of ~ inrab ... |
| bj-inrab2 36896 | Shorter proof of ~ inrab .... |
| bj-inrab3 36897 | Generalization of ~ dfrab3... |
| bj-rabtr 36898 | Restricted class abstracti... |
| bj-rabtrALT 36899 | Alternate proof of ~ bj-ra... |
| bj-rabtrAUTO 36900 | Proof of ~ bj-rabtr found ... |
| bj-gabss 36903 | Inclusion of generalized c... |
| bj-gabssd 36904 | Inclusion of generalized c... |
| bj-gabeqd 36905 | Equality of generalized cl... |
| bj-gabeqis 36906 | Equality of generalized cl... |
| bj-elgab 36907 | Elements of a generalized ... |
| bj-gabima 36908 | Generalized class abstract... |
| bj-ru1 36911 | A version of Russell's par... |
| bj-ru 36912 | Remove dependency on ~ ax-... |
| currysetlem 36913 | Lemma for ~ currysetlem , ... |
| curryset 36914 | Curry's paradox in set the... |
| currysetlem1 36915 | Lemma for ~ currysetALT . ... |
| currysetlem2 36916 | Lemma for ~ currysetALT . ... |
| currysetlem3 36917 | Lemma for ~ currysetALT . ... |
| currysetALT 36918 | Alternate proof of ~ curry... |
| bj-n0i 36919 | Inference associated with ... |
| bj-disjsn01 36920 | Disjointness of the single... |
| bj-0nel1 36921 | The empty set does not bel... |
| bj-1nel0 36922 | ` 1o ` does not belong to ... |
| bj-xpimasn 36923 | The image of a singleton, ... |
| bj-xpima1sn 36924 | The image of a singleton b... |
| bj-xpima1snALT 36925 | Alternate proof of ~ bj-xp... |
| bj-xpima2sn 36926 | The image of a singleton b... |
| bj-xpnzex 36927 | If the first factor of a p... |
| bj-xpexg2 36928 | Curried (exported) form of... |
| bj-xpnzexb 36929 | If the first factor of a p... |
| bj-cleq 36930 | Substitution property for ... |
| bj-snsetex 36931 | The class of sets "whose s... |
| bj-clexab 36932 | Sethood of certain classes... |
| bj-sngleq 36935 | Substitution property for ... |
| bj-elsngl 36936 | Characterization of the el... |
| bj-snglc 36937 | Characterization of the el... |
| bj-snglss 36938 | The singletonization of a ... |
| bj-0nelsngl 36939 | The empty set is not a mem... |
| bj-snglinv 36940 | Inverse of singletonizatio... |
| bj-snglex 36941 | A class is a set if and on... |
| bj-tageq 36944 | Substitution property for ... |
| bj-eltag 36945 | Characterization of the el... |
| bj-0eltag 36946 | The empty set belongs to t... |
| bj-tagn0 36947 | The tagging of a class is ... |
| bj-tagss 36948 | The tagging of a class is ... |
| bj-snglsstag 36949 | The singletonization is in... |
| bj-sngltagi 36950 | The singletonization is in... |
| bj-sngltag 36951 | The singletonization and t... |
| bj-tagci 36952 | Characterization of the el... |
| bj-tagcg 36953 | Characterization of the el... |
| bj-taginv 36954 | Inverse of tagging. (Cont... |
| bj-tagex 36955 | A class is a set if and on... |
| bj-xtageq 36956 | The products of a given cl... |
| bj-xtagex 36957 | The product of a set and t... |
| bj-projeq 36960 | Substitution property for ... |
| bj-projeq2 36961 | Substitution property for ... |
| bj-projun 36962 | The class projection on a ... |
| bj-projex 36963 | Sethood of the class proje... |
| bj-projval 36964 | Value of the class project... |
| bj-1upleq 36967 | Substitution property for ... |
| bj-pr1eq 36970 | Substitution property for ... |
| bj-pr1un 36971 | The first projection prese... |
| bj-pr1val 36972 | Value of the first project... |
| bj-pr11val 36973 | Value of the first project... |
| bj-pr1ex 36974 | Sethood of the first proje... |
| bj-1uplth 36975 | The characteristic propert... |
| bj-1uplex 36976 | A monuple is a set if and ... |
| bj-1upln0 36977 | A monuple is nonempty. (C... |
| bj-2upleq 36980 | Substitution property for ... |
| bj-pr21val 36981 | Value of the first project... |
| bj-pr2eq 36984 | Substitution property for ... |
| bj-pr2un 36985 | The second projection pres... |
| bj-pr2val 36986 | Value of the second projec... |
| bj-pr22val 36987 | Value of the second projec... |
| bj-pr2ex 36988 | Sethood of the second proj... |
| bj-2uplth 36989 | The characteristic propert... |
| bj-2uplex 36990 | A couple is a set if and o... |
| bj-2upln0 36991 | A couple is nonempty. (Co... |
| bj-2upln1upl 36992 | A couple is never equal to... |
| bj-rcleqf 36993 | Relative version of ~ cleq... |
| bj-rcleq 36994 | Relative version of ~ dfcl... |
| bj-reabeq 36995 | Relative form of ~ eqabb .... |
| bj-disj2r 36996 | Relative version of ~ ssdi... |
| bj-sscon 36997 | Contraposition law for rel... |
| bj-abex 36998 | Two ways of stating that t... |
| bj-clex 36999 | Two ways of stating that a... |
| bj-axsn 37000 | Two ways of stating the ax... |
| bj-snexg 37002 | A singleton built on a set... |
| bj-snex 37003 | A singleton is a set. See... |
| bj-axbun 37004 | Two ways of stating the ax... |
| bj-unexg 37006 | Existence of binary unions... |
| bj-prexg 37007 | Existence of unordered pai... |
| bj-prex 37008 | Existence of unordered pai... |
| bj-axadj 37009 | Two ways of stating the ax... |
| bj-adjg1 37011 | Existence of the result of... |
| bj-snfromadj 37012 | Singleton from adjunction ... |
| bj-prfromadj 37013 | Unordered pair from adjunc... |
| bj-adjfrombun 37014 | Adjunction from singleton ... |
| eleq2w2ALT 37015 | Alternate proof of ~ eleq2... |
| bj-clel3gALT 37016 | Alternate proof of ~ clel3... |
| bj-pw0ALT 37017 | Alternate proof of ~ pw0 .... |
| bj-sselpwuni 37018 | Quantitative version of ~ ... |
| bj-unirel 37019 | Quantitative version of ~ ... |
| bj-elpwg 37020 | If the intersection of two... |
| bj-velpwALT 37021 | This theorem ~ bj-velpwALT... |
| bj-elpwgALT 37022 | Alternate proof of ~ elpwg... |
| bj-vjust 37023 | Justification theorem for ... |
| bj-nul 37024 | Two formulations of the ax... |
| bj-nuliota 37025 | Definition of the empty se... |
| bj-nuliotaALT 37026 | Alternate proof of ~ bj-nu... |
| bj-vtoclgfALT 37027 | Alternate proof of ~ vtocl... |
| bj-elsn12g 37028 | Join of ~ elsng and ~ elsn... |
| bj-elsnb 37029 | Biconditional version of ~... |
| bj-pwcfsdom 37030 | Remove hypothesis from ~ p... |
| bj-grur1 37031 | Remove hypothesis from ~ g... |
| bj-bm1.3ii 37032 | The extension of a predica... |
| bj-dfid2ALT 37033 | Alternate version of ~ dfi... |
| bj-0nelopab 37034 | The empty set is never an ... |
| bj-brrelex12ALT 37035 | Two classes related by a b... |
| bj-epelg 37036 | The membership relation an... |
| bj-epelb 37037 | Two classes are related by... |
| bj-nsnid 37038 | A set does not contain the... |
| bj-rdg0gALT 37039 | Alternate proof of ~ rdg0g... |
| bj-evaleq 37040 | Equality theorem for the `... |
| bj-evalfun 37041 | The evaluation at a class ... |
| bj-evalfn 37042 | The evaluation at a class ... |
| bj-evalval 37043 | Value of the evaluation at... |
| bj-evalid 37044 | The evaluation at a set of... |
| bj-ndxarg 37045 | Proof of ~ ndxarg from ~ b... |
| bj-evalidval 37046 | Closed general form of ~ s... |
| bj-rest00 37049 | An elementwise intersectio... |
| bj-restsn 37050 | An elementwise intersectio... |
| bj-restsnss 37051 | Special case of ~ bj-rests... |
| bj-restsnss2 37052 | Special case of ~ bj-rests... |
| bj-restsn0 37053 | An elementwise intersectio... |
| bj-restsn10 37054 | Special case of ~ bj-rests... |
| bj-restsnid 37055 | The elementwise intersecti... |
| bj-rest10 37056 | An elementwise intersectio... |
| bj-rest10b 37057 | Alternate version of ~ bj-... |
| bj-restn0 37058 | An elementwise intersectio... |
| bj-restn0b 37059 | Alternate version of ~ bj-... |
| bj-restpw 37060 | The elementwise intersecti... |
| bj-rest0 37061 | An elementwise intersectio... |
| bj-restb 37062 | An elementwise intersectio... |
| bj-restv 37063 | An elementwise intersectio... |
| bj-resta 37064 | An elementwise intersectio... |
| bj-restuni 37065 | The union of an elementwis... |
| bj-restuni2 37066 | The union of an elementwis... |
| bj-restreg 37067 | A reformulation of the axi... |
| bj-raldifsn 37068 | All elements in a set sati... |
| bj-0int 37069 | If ` A ` is a collection o... |
| bj-mooreset 37070 | A Moore collection is a se... |
| bj-ismoore 37073 | Characterization of Moore ... |
| bj-ismoored0 37074 | Necessary condition to be ... |
| bj-ismoored 37075 | Necessary condition to be ... |
| bj-ismoored2 37076 | Necessary condition to be ... |
| bj-ismooredr 37077 | Sufficient condition to be... |
| bj-ismooredr2 37078 | Sufficient condition to be... |
| bj-discrmoore 37079 | The powerclass ` ~P A ` is... |
| bj-0nmoore 37080 | The empty set is not a Moo... |
| bj-snmoore 37081 | A singleton is a Moore col... |
| bj-snmooreb 37082 | A singleton is a Moore col... |
| bj-prmoore 37083 | A pair formed of two neste... |
| bj-0nelmpt 37084 | The empty set is not an el... |
| bj-mptval 37085 | Value of a function given ... |
| bj-dfmpoa 37086 | An equivalent definition o... |
| bj-mpomptALT 37087 | Alternate proof of ~ mpomp... |
| setsstrset 37104 | Relation between ~ df-sets... |
| bj-nfald 37105 | Variant of ~ nfald . (Con... |
| bj-nfexd 37106 | Variant of ~ nfexd . (Con... |
| copsex2d 37107 | Implicit substitution dedu... |
| copsex2b 37108 | Biconditional form of ~ co... |
| opelopabd 37109 | Membership of an ordere pa... |
| opelopabb 37110 | Membership of an ordered p... |
| opelopabbv 37111 | Membership of an ordered p... |
| bj-opelrelex 37112 | The coordinates of an orde... |
| bj-opelresdm 37113 | If an ordered pair is in a... |
| bj-brresdm 37114 | If two classes are related... |
| brabd0 37115 | Expressing that two sets a... |
| brabd 37116 | Expressing that two sets a... |
| bj-brab2a1 37117 | "Unbounded" version of ~ b... |
| bj-opabssvv 37118 | A variant of ~ relopabiv (... |
| bj-funidres 37119 | The restricted identity re... |
| bj-opelidb 37120 | Characterization of the or... |
| bj-opelidb1 37121 | Characterization of the or... |
| bj-inexeqex 37122 | Lemma for ~ bj-opelid (but... |
| bj-elsn0 37123 | If the intersection of two... |
| bj-opelid 37124 | Characterization of the or... |
| bj-ideqg 37125 | Characterization of the cl... |
| bj-ideqgALT 37126 | Alternate proof of ~ bj-id... |
| bj-ideqb 37127 | Characterization of classe... |
| bj-idres 37128 | Alternate expression for t... |
| bj-opelidres 37129 | Characterization of the or... |
| bj-idreseq 37130 | Sufficient condition for t... |
| bj-idreseqb 37131 | Characterization for two c... |
| bj-ideqg1 37132 | For sets, the identity rel... |
| bj-ideqg1ALT 37133 | Alternate proof of bj-ideq... |
| bj-opelidb1ALT 37134 | Characterization of the co... |
| bj-elid3 37135 | Characterization of the co... |
| bj-elid4 37136 | Characterization of the el... |
| bj-elid5 37137 | Characterization of the el... |
| bj-elid6 37138 | Characterization of the el... |
| bj-elid7 37139 | Characterization of the el... |
| bj-diagval 37142 | Value of the functionalize... |
| bj-diagval2 37143 | Value of the functionalize... |
| bj-eldiag 37144 | Characterization of the el... |
| bj-eldiag2 37145 | Characterization of the el... |
| bj-imdirvallem 37148 | Lemma for ~ bj-imdirval an... |
| bj-imdirval 37149 | Value of the functionalize... |
| bj-imdirval2lem 37150 | Lemma for ~ bj-imdirval2 a... |
| bj-imdirval2 37151 | Value of the functionalize... |
| bj-imdirval3 37152 | Value of the functionalize... |
| bj-imdiridlem 37153 | Lemma for ~ bj-imdirid and... |
| bj-imdirid 37154 | Functorial property of the... |
| bj-opelopabid 37155 | Membership in an ordered-p... |
| bj-opabco 37156 | Composition of ordered-pai... |
| bj-xpcossxp 37157 | The composition of two Car... |
| bj-imdirco 37158 | Functorial property of the... |
| bj-iminvval 37161 | Value of the functionalize... |
| bj-iminvval2 37162 | Value of the functionalize... |
| bj-iminvid 37163 | Functorial property of the... |
| bj-inftyexpitaufo 37170 | The function ` inftyexpita... |
| bj-inftyexpitaudisj 37173 | An element of the circle a... |
| bj-inftyexpiinv 37176 | Utility theorem for the in... |
| bj-inftyexpiinj 37177 | Injectivity of the paramet... |
| bj-inftyexpidisj 37178 | An element of the circle a... |
| bj-ccinftydisj 37181 | The circle at infinity is ... |
| bj-elccinfty 37182 | A lemma for infinite exten... |
| bj-ccssccbar 37185 | Complex numbers are extend... |
| bj-ccinftyssccbar 37186 | Infinite extended complex ... |
| bj-pinftyccb 37189 | The class ` pinfty ` is an... |
| bj-pinftynrr 37190 | The extended complex numbe... |
| bj-minftyccb 37193 | The class ` minfty ` is an... |
| bj-minftynrr 37194 | The extended complex numbe... |
| bj-pinftynminfty 37195 | The extended complex numbe... |
| bj-rrhatsscchat 37204 | The real projective line i... |
| bj-imafv 37219 | If the direct image of a s... |
| bj-funun 37220 | Value of a function expres... |
| bj-fununsn1 37221 | Value of a function expres... |
| bj-fununsn2 37222 | Value of a function expres... |
| bj-fvsnun1 37223 | The value of a function wi... |
| bj-fvsnun2 37224 | The value of a function wi... |
| bj-fvmptunsn1 37225 | Value of a function expres... |
| bj-fvmptunsn2 37226 | Value of a function expres... |
| bj-iomnnom 37227 | The canonical bijection fr... |
| bj-smgrpssmgm 37236 | Semigroups are magmas. (C... |
| bj-smgrpssmgmel 37237 | Semigroups are magmas (ele... |
| bj-mndsssmgrp 37238 | Monoids are semigroups. (... |
| bj-mndsssmgrpel 37239 | Monoids are semigroups (el... |
| bj-cmnssmnd 37240 | Commutative monoids are mo... |
| bj-cmnssmndel 37241 | Commutative monoids are mo... |
| bj-grpssmnd 37242 | Groups are monoids. (Cont... |
| bj-grpssmndel 37243 | Groups are monoids (elemen... |
| bj-ablssgrp 37244 | Abelian groups are groups.... |
| bj-ablssgrpel 37245 | Abelian groups are groups ... |
| bj-ablsscmn 37246 | Abelian groups are commuta... |
| bj-ablsscmnel 37247 | Abelian groups are commuta... |
| bj-modssabl 37248 | (The additive groups of) m... |
| bj-vecssmod 37249 | Vector spaces are modules.... |
| bj-vecssmodel 37250 | Vector spaces are modules ... |
| bj-finsumval0 37253 | Value of a finite sum. (C... |
| bj-fvimacnv0 37254 | Variant of ~ fvimacnv wher... |
| bj-isvec 37255 | The predicate "is a vector... |
| bj-fldssdrng 37256 | Fields are division rings.... |
| bj-flddrng 37257 | Fields are division rings ... |
| bj-rrdrg 37258 | The field of real numbers ... |
| bj-isclm 37259 | The predicate "is a subcom... |
| bj-isrvec 37262 | The predicate "is a real v... |
| bj-rvecmod 37263 | Real vector spaces are mod... |
| bj-rvecssmod 37264 | Real vector spaces are mod... |
| bj-rvecrr 37265 | The field of scalars of a ... |
| bj-isrvecd 37266 | The predicate "is a real v... |
| bj-rvecvec 37267 | Real vector spaces are vec... |
| bj-isrvec2 37268 | The predicate "is a real v... |
| bj-rvecssvec 37269 | Real vector spaces are vec... |
| bj-rveccmod 37270 | Real vector spaces are sub... |
| bj-rvecsscmod 37271 | Real vector spaces are sub... |
| bj-rvecsscvec 37272 | Real vector spaces are sub... |
| bj-rveccvec 37273 | Real vector spaces are sub... |
| bj-rvecssabl 37274 | (The additive groups of) r... |
| bj-rvecabl 37275 | (The additive groups of) r... |
| bj-subcom 37276 | A consequence of commutati... |
| bj-lineqi 37277 | Solution of a (scalar) lin... |
| bj-bary1lem 37278 | Lemma for ~ bj-bary1 : exp... |
| bj-bary1lem1 37279 | Lemma for ~ bj-bary1 : com... |
| bj-bary1 37280 | Barycentric coordinates in... |
| bj-endval 37283 | Value of the monoid of end... |
| bj-endbase 37284 | Base set of the monoid of ... |
| bj-endcomp 37285 | Composition law of the mon... |
| bj-endmnd 37286 | The monoid of endomorphism... |
| taupilem3 37287 | Lemma for tau-related theo... |
| taupilemrplb 37288 | A set of positive reals ha... |
| taupilem1 37289 | Lemma for ~ taupi . A pos... |
| taupilem2 37290 | Lemma for ~ taupi . The s... |
| taupi 37291 | Relationship between ` _ta... |
| dfgcd3 37292 | Alternate definition of th... |
| irrdifflemf 37293 | Lemma for ~ irrdiff . The... |
| irrdiff 37294 | The irrationals are exactl... |
| iccioo01 37295 | The closed unit interval i... |
| csbrecsg 37296 | Move class substitution in... |
| csbrdgg 37297 | Move class substitution in... |
| csboprabg 37298 | Move class substitution in... |
| csbmpo123 37299 | Move class substitution in... |
| con1bii2 37300 | A contraposition inference... |
| con2bii2 37301 | A contraposition inference... |
| vtoclefex 37302 | Implicit substitution of a... |
| rnmptsn 37303 | The range of a function ma... |
| f1omptsnlem 37304 | This is the core of the pr... |
| f1omptsn 37305 | A function mapping to sing... |
| mptsnunlem 37306 | This is the core of the pr... |
| mptsnun 37307 | A class ` B ` is equal to ... |
| dissneqlem 37308 | This is the core of the pr... |
| dissneq 37309 | Any topology that contains... |
| exlimim 37310 | Closed form of ~ exlimimd ... |
| exlimimd 37311 | Existential elimination ru... |
| exellim 37312 | Closed form of ~ exellimdd... |
| exellimddv 37313 | Eliminate an antecedent wh... |
| topdifinfindis 37314 | Part of Exercise 3 of [Mun... |
| topdifinffinlem 37315 | This is the core of the pr... |
| topdifinffin 37316 | Part of Exercise 3 of [Mun... |
| topdifinf 37317 | Part of Exercise 3 of [Mun... |
| topdifinfeq 37318 | Two different ways of defi... |
| icorempo 37319 | Closed-below, open-above i... |
| icoreresf 37320 | Closed-below, open-above i... |
| icoreval 37321 | Value of the closed-below,... |
| icoreelrnab 37322 | Elementhood in the set of ... |
| isbasisrelowllem1 37323 | Lemma for ~ isbasisrelowl ... |
| isbasisrelowllem2 37324 | Lemma for ~ isbasisrelowl ... |
| icoreclin 37325 | The set of closed-below, o... |
| isbasisrelowl 37326 | The set of all closed-belo... |
| icoreunrn 37327 | The union of all closed-be... |
| istoprelowl 37328 | The set of all closed-belo... |
| icoreelrn 37329 | A class abstraction which ... |
| iooelexlt 37330 | An element of an open inte... |
| relowlssretop 37331 | The lower limit topology o... |
| relowlpssretop 37332 | The lower limit topology o... |
| sucneqond 37333 | Inequality of an ordinal s... |
| sucneqoni 37334 | Inequality of an ordinal s... |
| onsucuni3 37335 | If an ordinal number has a... |
| 1oequni2o 37336 | The ordinal number ` 1o ` ... |
| rdgsucuni 37337 | If an ordinal number has a... |
| rdgeqoa 37338 | If a recursive function wi... |
| elxp8 37339 | Membership in a Cartesian ... |
| cbveud 37340 | Deduction used to change b... |
| cbvreud 37341 | Deduction used to change b... |
| difunieq 37342 | The difference of unions i... |
| inunissunidif 37343 | Theorem about subsets of t... |
| rdgellim 37344 | Elementhood in a recursive... |
| rdglimss 37345 | A recursive definition at ... |
| rdgssun 37346 | In a recursive definition ... |
| exrecfnlem 37347 | Lemma for ~ exrecfn . (Co... |
| exrecfn 37348 | Theorem about the existenc... |
| exrecfnpw 37349 | For any base set, a set wh... |
| finorwe 37350 | If the Axiom of Infinity i... |
| dffinxpf 37353 | This theorem is the same a... |
| finxpeq1 37354 | Equality theorem for Carte... |
| finxpeq2 37355 | Equality theorem for Carte... |
| csbfinxpg 37356 | Distribute proper substitu... |
| finxpreclem1 37357 | Lemma for ` ^^ ` recursion... |
| finxpreclem2 37358 | Lemma for ` ^^ ` recursion... |
| finxp0 37359 | The value of Cartesian exp... |
| finxp1o 37360 | The value of Cartesian exp... |
| finxpreclem3 37361 | Lemma for ` ^^ ` recursion... |
| finxpreclem4 37362 | Lemma for ` ^^ ` recursion... |
| finxpreclem5 37363 | Lemma for ` ^^ ` recursion... |
| finxpreclem6 37364 | Lemma for ` ^^ ` recursion... |
| finxpsuclem 37365 | Lemma for ~ finxpsuc . (C... |
| finxpsuc 37366 | The value of Cartesian exp... |
| finxp2o 37367 | The value of Cartesian exp... |
| finxp3o 37368 | The value of Cartesian exp... |
| finxpnom 37369 | Cartesian exponentiation w... |
| finxp00 37370 | Cartesian exponentiation o... |
| iunctb2 37371 | Using the axiom of countab... |
| domalom 37372 | A class which dominates ev... |
| isinf2 37373 | The converse of ~ isinf . ... |
| ctbssinf 37374 | Using the axiom of choice,... |
| ralssiun 37375 | The index set of an indexe... |
| nlpineqsn 37376 | For every point ` p ` of a... |
| nlpfvineqsn 37377 | Given a subset ` A ` of ` ... |
| fvineqsnf1 37378 | A theorem about functions ... |
| fvineqsneu 37379 | A theorem about functions ... |
| fvineqsneq 37380 | A theorem about functions ... |
| pibp16 37381 | Property P000016 of pi-bas... |
| pibp19 37382 | Property P000019 of pi-bas... |
| pibp21 37383 | Property P000021 of pi-bas... |
| pibt1 37384 | Theorem T000001 of pi-base... |
| pibt2 37385 | Theorem T000002 of pi-base... |
| wl-section-prop 37386 | Intuitionistic logic is no... |
| wl-section-boot 37390 | In this section, I provide... |
| wl-luk-imim1i 37391 | Inference adding common co... |
| wl-luk-syl 37392 | An inference version of th... |
| wl-luk-imtrid 37393 | A syllogism rule of infere... |
| wl-luk-pm2.18d 37394 | Deduction based on reducti... |
| wl-luk-con4i 37395 | Inference rule. Copy of ~... |
| wl-luk-pm2.24i 37396 | Inference rule. Copy of ~... |
| wl-luk-a1i 37397 | Inference rule. Copy of ~... |
| wl-luk-mpi 37398 | A nested _modus ponens_ in... |
| wl-luk-imim2i 37399 | Inference adding common an... |
| wl-luk-imtrdi 37400 | A syllogism rule of infere... |
| wl-luk-ax3 37401 | ~ ax-3 proved from Lukasie... |
| wl-luk-ax1 37402 | ~ ax-1 proved from Lukasie... |
| wl-luk-pm2.27 37403 | This theorem, called "Asse... |
| wl-luk-com12 37404 | Inference that swaps (comm... |
| wl-luk-pm2.21 37405 | From a wff and its negatio... |
| wl-luk-con1i 37406 | A contraposition inference... |
| wl-luk-ja 37407 | Inference joining the ante... |
| wl-luk-imim2 37408 | A closed form of syllogism... |
| wl-luk-a1d 37409 | Deduction introducing an e... |
| wl-luk-ax2 37410 | ~ ax-2 proved from Lukasie... |
| wl-luk-id 37411 | Principle of identity. Th... |
| wl-luk-notnotr 37412 | Converse of double negatio... |
| wl-luk-pm2.04 37413 | Swap antecedents. Theorem... |
| wl-section-impchain 37414 | An implication like ` ( ps... |
| wl-impchain-mp-x 37415 | This series of theorems pr... |
| wl-impchain-mp-0 37416 | This theorem is the start ... |
| wl-impchain-mp-1 37417 | This theorem is in fact a ... |
| wl-impchain-mp-2 37418 | This theorem is in fact a ... |
| wl-impchain-com-1.x 37419 | It is often convenient to ... |
| wl-impchain-com-1.1 37420 | A degenerate form of antec... |
| wl-impchain-com-1.2 37421 | This theorem is in fact a ... |
| wl-impchain-com-1.3 37422 | This theorem is in fact a ... |
| wl-impchain-com-1.4 37423 | This theorem is in fact a ... |
| wl-impchain-com-n.m 37424 | This series of theorems al... |
| wl-impchain-com-2.3 37425 | This theorem is in fact a ... |
| wl-impchain-com-2.4 37426 | This theorem is in fact a ... |
| wl-impchain-com-3.2.1 37427 | This theorem is in fact a ... |
| wl-impchain-a1-x 37428 | If an implication chain is... |
| wl-impchain-a1-1 37429 | Inference rule, a copy of ... |
| wl-impchain-a1-2 37430 | Inference rule, a copy of ... |
| wl-impchain-a1-3 37431 | Inference rule, a copy of ... |
| wl-ifp-ncond1 37432 | If one case of an ` if- ` ... |
| wl-ifp-ncond2 37433 | If one case of an ` if- ` ... |
| wl-ifpimpr 37434 | If one case of an ` if- ` ... |
| wl-ifp4impr 37435 | If one case of an ` if- ` ... |
| wl-df-3xor 37436 | Alternative definition of ... |
| wl-df3xor2 37437 | Alternative definition of ... |
| wl-df3xor3 37438 | Alternative form of ~ wl-d... |
| wl-3xortru 37439 | If the first input is true... |
| wl-3xorfal 37440 | If the first input is fals... |
| wl-3xorbi 37441 | Triple xor can be replaced... |
| wl-3xorbi2 37442 | Alternative form of ~ wl-3... |
| wl-3xorbi123d 37443 | Equivalence theorem for tr... |
| wl-3xorbi123i 37444 | Equivalence theorem for tr... |
| wl-3xorrot 37445 | Rotation law for triple xo... |
| wl-3xorcoma 37446 | Commutative law for triple... |
| wl-3xorcomb 37447 | Commutative law for triple... |
| wl-3xornot1 37448 | Flipping the first input f... |
| wl-3xornot 37449 | Triple xor distributes ove... |
| wl-1xor 37450 | In the recursive scheme ... |
| wl-2xor 37451 | In the recursive scheme ... |
| wl-df-3mintru2 37452 | Alternative definition of ... |
| wl-df2-3mintru2 37453 | The adder carry in disjunc... |
| wl-df3-3mintru2 37454 | The adder carry in conjunc... |
| wl-df4-3mintru2 37455 | An alternative definition ... |
| wl-1mintru1 37456 | Using the recursion formul... |
| wl-1mintru2 37457 | Using the recursion formul... |
| wl-2mintru1 37458 | Using the recursion formul... |
| wl-2mintru2 37459 | Using the recursion formul... |
| wl-df3maxtru1 37460 | Assuming "(n+1)-maxtru1" `... |
| wl-ax13lem1 37462 | A version of ~ ax-wl-13v w... |
| wl-mps 37463 | Replacing a nested consequ... |
| wl-syls1 37464 | Replacing a nested consequ... |
| wl-syls2 37465 | Replacing a nested anteced... |
| wl-embant 37466 | A true wff can always be a... |
| wl-orel12 37467 | In a conjunctive normal fo... |
| wl-cases2-dnf 37468 | A particular instance of ~... |
| wl-cbvmotv 37469 | Change bound variable. Us... |
| wl-moteq 37470 | Change bound variable. Us... |
| wl-motae 37471 | Change bound variable. Us... |
| wl-moae 37472 | Two ways to express "at mo... |
| wl-euae 37473 | Two ways to express "exact... |
| wl-nax6im 37474 | The following series of th... |
| wl-hbae1 37475 | This specialization of ~ h... |
| wl-naevhba1v 37476 | An instance of ~ hbn1w app... |
| wl-spae 37477 | Prove an instance of ~ sp ... |
| wl-speqv 37478 | Under the assumption ` -. ... |
| wl-19.8eqv 37479 | Under the assumption ` -. ... |
| wl-19.2reqv 37480 | Under the assumption ` -. ... |
| wl-nfalv 37481 | If ` x ` is not present in... |
| wl-nfimf1 37482 | An antecedent is irrelevan... |
| wl-nfae1 37483 | Unlike ~ nfae , this speci... |
| wl-nfnae1 37484 | Unlike ~ nfnae , this spec... |
| wl-aetr 37485 | A transitive law for varia... |
| wl-axc11r 37486 | Same as ~ axc11r , but usi... |
| wl-dral1d 37487 | A version of ~ dral1 with ... |
| wl-cbvalnaed 37488 | ~ wl-cbvalnae with a conte... |
| wl-cbvalnae 37489 | A more general version of ... |
| wl-exeq 37490 | The semantics of ` E. x y ... |
| wl-aleq 37491 | The semantics of ` A. x y ... |
| wl-nfeqfb 37492 | Extend ~ nfeqf to an equiv... |
| wl-nfs1t 37493 | If ` y ` is not free in ` ... |
| wl-equsalvw 37494 | Version of ~ equsalv with ... |
| wl-equsald 37495 | Deduction version of ~ equ... |
| wl-equsaldv 37496 | Deduction version of ~ equ... |
| wl-equsal 37497 | A useful equivalence relat... |
| wl-equsal1t 37498 | The expression ` x = y ` i... |
| wl-equsalcom 37499 | This simple equivalence ea... |
| wl-equsal1i 37500 | The antecedent ` x = y ` i... |
| wl-sbid2ft 37501 | A more general version of ... |
| wl-cbvalsbi 37502 | Change bounded variables i... |
| wl-sbrimt 37503 | Substitution with a variab... |
| wl-sblimt 37504 | Substitution with a variab... |
| wl-sb9v 37505 | Commutation of quantificat... |
| wl-sb8ft 37506 | Substitution of variable i... |
| wl-sb8eft 37507 | Substitution of variable i... |
| wl-sb8t 37508 | Substitution of variable i... |
| wl-sb8et 37509 | Substitution of variable i... |
| wl-sbhbt 37510 | Closed form of ~ sbhb . C... |
| wl-sbnf1 37511 | Two ways expressing that `... |
| wl-equsb3 37512 | ~ equsb3 with a distinctor... |
| wl-equsb4 37513 | Substitution applied to an... |
| wl-2sb6d 37514 | Version of ~ 2sb6 with a c... |
| wl-sbcom2d-lem1 37515 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d-lem2 37516 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d 37517 | Version of ~ sbcom2 with a... |
| wl-sbalnae 37518 | A theorem used in eliminat... |
| wl-sbal1 37519 | A theorem used in eliminat... |
| wl-sbal2 37520 | Move quantifier in and out... |
| wl-2spsbbi 37521 | ~ spsbbi applied twice. (... |
| wl-lem-exsb 37522 | This theorem provides a ba... |
| wl-lem-nexmo 37523 | This theorem provides a ba... |
| wl-lem-moexsb 37524 | The antecedent ` A. x ( ph... |
| wl-alanbii 37525 | This theorem extends ~ ala... |
| wl-mo2df 37526 | Version of ~ mof with a co... |
| wl-mo2tf 37527 | Closed form of ~ mof with ... |
| wl-eudf 37528 | Version of ~ eu6 with a co... |
| wl-eutf 37529 | Closed form of ~ eu6 with ... |
| wl-euequf 37530 | ~ euequ proved with a dist... |
| wl-mo2t 37531 | Closed form of ~ mof . (C... |
| wl-mo3t 37532 | Closed form of ~ mo3 . (C... |
| wl-nfsbtv 37533 | Closed form of ~ nfsbv . ... |
| wl-sb8eut 37534 | Substitution of variable i... |
| wl-sb8eutv 37535 | Substitution of variable i... |
| wl-sb8mot 37536 | Substitution of variable i... |
| wl-sb8motv 37537 | Substitution of variable i... |
| wl-issetft 37538 | A closed form of ~ issetf ... |
| wl-axc11rc11 37539 | Proving ~ axc11r from ~ ax... |
| wl-ax11-lem1 37541 | A transitive law for varia... |
| wl-ax11-lem2 37542 | Lemma. (Contributed by Wo... |
| wl-ax11-lem3 37543 | Lemma. (Contributed by Wo... |
| wl-ax11-lem4 37544 | Lemma. (Contributed by Wo... |
| wl-ax11-lem5 37545 | Lemma. (Contributed by Wo... |
| wl-ax11-lem6 37546 | Lemma. (Contributed by Wo... |
| wl-ax11-lem7 37547 | Lemma. (Contributed by Wo... |
| wl-ax11-lem8 37548 | Lemma. (Contributed by Wo... |
| wl-ax11-lem9 37549 | The easy part when ` x ` c... |
| wl-ax11-lem10 37550 | We now have prepared every... |
| wl-clabv 37551 | Variant of ~ df-clab , whe... |
| wl-dfclab 37552 | Rederive ~ df-clab from ~ ... |
| wl-clabtv 37553 | Using class abstraction in... |
| wl-clabt 37554 | Using class abstraction in... |
| rabiun 37555 | Abstraction restricted to ... |
| iundif1 37556 | Indexed union of class dif... |
| imadifss 37557 | The difference of images i... |
| cureq 37558 | Equality theorem for curry... |
| unceq 37559 | Equality theorem for uncur... |
| curf 37560 | Functional property of cur... |
| uncf 37561 | Functional property of unc... |
| curfv 37562 | Value of currying. (Contr... |
| uncov 37563 | Value of uncurrying. (Con... |
| curunc 37564 | Currying of uncurrying. (... |
| unccur 37565 | Uncurrying of currying. (... |
| phpreu 37566 | Theorem related to pigeonh... |
| finixpnum 37567 | A finite Cartesian product... |
| fin2solem 37568 | Lemma for ~ fin2so . (Con... |
| fin2so 37569 | Any totally ordered Tarski... |
| ltflcei 37570 | Theorem to move the floor ... |
| leceifl 37571 | Theorem to move the floor ... |
| sin2h 37572 | Half-angle rule for sine. ... |
| cos2h 37573 | Half-angle rule for cosine... |
| tan2h 37574 | Half-angle rule for tangen... |
| lindsadd 37575 | In a vector space, the uni... |
| lindsdom 37576 | A linearly independent set... |
| lindsenlbs 37577 | A maximal linearly indepen... |
| matunitlindflem1 37578 | One direction of ~ matunit... |
| matunitlindflem2 37579 | One direction of ~ matunit... |
| matunitlindf 37580 | A matrix over a field is i... |
| ptrest 37581 | Expressing a restriction o... |
| ptrecube 37582 | Any point in an open set o... |
| poimirlem1 37583 | Lemma for ~ poimir - the v... |
| poimirlem2 37584 | Lemma for ~ poimir - conse... |
| poimirlem3 37585 | Lemma for ~ poimir to add ... |
| poimirlem4 37586 | Lemma for ~ poimir connect... |
| poimirlem5 37587 | Lemma for ~ poimir to esta... |
| poimirlem6 37588 | Lemma for ~ poimir establi... |
| poimirlem7 37589 | Lemma for ~ poimir , simil... |
| poimirlem8 37590 | Lemma for ~ poimir , estab... |
| poimirlem9 37591 | Lemma for ~ poimir , estab... |
| poimirlem10 37592 | Lemma for ~ poimir establi... |
| poimirlem11 37593 | Lemma for ~ poimir connect... |
| poimirlem12 37594 | Lemma for ~ poimir connect... |
| poimirlem13 37595 | Lemma for ~ poimir - for a... |
| poimirlem14 37596 | Lemma for ~ poimir - for a... |
| poimirlem15 37597 | Lemma for ~ poimir , that ... |
| poimirlem16 37598 | Lemma for ~ poimir establi... |
| poimirlem17 37599 | Lemma for ~ poimir establi... |
| poimirlem18 37600 | Lemma for ~ poimir stating... |
| poimirlem19 37601 | Lemma for ~ poimir establi... |
| poimirlem20 37602 | Lemma for ~ poimir establi... |
| poimirlem21 37603 | Lemma for ~ poimir stating... |
| poimirlem22 37604 | Lemma for ~ poimir , that ... |
| poimirlem23 37605 | Lemma for ~ poimir , two w... |
| poimirlem24 37606 | Lemma for ~ poimir , two w... |
| poimirlem25 37607 | Lemma for ~ poimir stating... |
| poimirlem26 37608 | Lemma for ~ poimir showing... |
| poimirlem27 37609 | Lemma for ~ poimir showing... |
| poimirlem28 37610 | Lemma for ~ poimir , a var... |
| poimirlem29 37611 | Lemma for ~ poimir connect... |
| poimirlem30 37612 | Lemma for ~ poimir combini... |
| poimirlem31 37613 | Lemma for ~ poimir , assig... |
| poimirlem32 37614 | Lemma for ~ poimir , combi... |
| poimir 37615 | Poincare-Miranda theorem. ... |
| broucube 37616 | Brouwer - or as Kulpa call... |
| heicant 37617 | Heine-Cantor theorem: a co... |
| opnmbllem0 37618 | Lemma for ~ ismblfin ; cou... |
| mblfinlem1 37619 | Lemma for ~ ismblfin , ord... |
| mblfinlem2 37620 | Lemma for ~ ismblfin , eff... |
| mblfinlem3 37621 | The difference between two... |
| mblfinlem4 37622 | Backward direction of ~ is... |
| ismblfin 37623 | Measurability in terms of ... |
| ovoliunnfl 37624 | ~ ovoliun is incompatible ... |
| ex-ovoliunnfl 37625 | Demonstration of ~ ovoliun... |
| voliunnfl 37626 | ~ voliun is incompatible w... |
| volsupnfl 37627 | ~ volsup is incompatible w... |
| mbfresfi 37628 | Measurability of a piecewi... |
| mbfposadd 37629 | If the sum of two measurab... |
| cnambfre 37630 | A real-valued, a.e. contin... |
| dvtanlem 37631 | Lemma for ~ dvtan - the do... |
| dvtan 37632 | Derivative of tangent. (C... |
| itg2addnclem 37633 | An alternate expression fo... |
| itg2addnclem2 37634 | Lemma for ~ itg2addnc . T... |
| itg2addnclem3 37635 | Lemma incomprehensible in ... |
| itg2addnc 37636 | Alternate proof of ~ itg2a... |
| itg2gt0cn 37637 | ~ itg2gt0 holds on functio... |
| ibladdnclem 37638 | Lemma for ~ ibladdnc ; cf ... |
| ibladdnc 37639 | Choice-free analogue of ~ ... |
| itgaddnclem1 37640 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnclem2 37641 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnc 37642 | Choice-free analogue of ~ ... |
| iblsubnc 37643 | Choice-free analogue of ~ ... |
| itgsubnc 37644 | Choice-free analogue of ~ ... |
| iblabsnclem 37645 | Lemma for ~ iblabsnc ; cf.... |
| iblabsnc 37646 | Choice-free analogue of ~ ... |
| iblmulc2nc 37647 | Choice-free analogue of ~ ... |
| itgmulc2nclem1 37648 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nclem2 37649 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nc 37650 | Choice-free analogue of ~ ... |
| itgabsnc 37651 | Choice-free analogue of ~ ... |
| itggt0cn 37652 | ~ itggt0 holds for continu... |
| ftc1cnnclem 37653 | Lemma for ~ ftc1cnnc ; cf.... |
| ftc1cnnc 37654 | Choice-free proof of ~ ftc... |
| ftc1anclem1 37655 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem2 37656 | Lemma for ~ ftc1anc - rest... |
| ftc1anclem3 37657 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem4 37658 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem5 37659 | Lemma for ~ ftc1anc , the ... |
| ftc1anclem6 37660 | Lemma for ~ ftc1anc - cons... |
| ftc1anclem7 37661 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem8 37662 | Lemma for ~ ftc1anc . (Co... |
| ftc1anc 37663 | ~ ftc1a holds for function... |
| ftc2nc 37664 | Choice-free proof of ~ ftc... |
| asindmre 37665 | Real part of domain of dif... |
| dvasin 37666 | Derivative of arcsine. (C... |
| dvacos 37667 | Derivative of arccosine. ... |
| dvreasin 37668 | Real derivative of arcsine... |
| dvreacos 37669 | Real derivative of arccosi... |
| areacirclem1 37670 | Antiderivative of cross-se... |
| areacirclem2 37671 | Endpoint-inclusive continu... |
| areacirclem3 37672 | Integrability of cross-sec... |
| areacirclem4 37673 | Endpoint-inclusive continu... |
| areacirclem5 37674 | Finding the cross-section ... |
| areacirc 37675 | The area of a circle of ra... |
| unirep 37676 | Define a quantity whose de... |
| cover2 37677 | Two ways of expressing the... |
| cover2g 37678 | Two ways of expressing the... |
| brabg2 37679 | Relation by a binary relat... |
| opelopab3 37680 | Ordered pair membership in... |
| cocanfo 37681 | Cancellation of a surjecti... |
| brresi2 37682 | Restriction of a binary re... |
| fnopabeqd 37683 | Equality deduction for fun... |
| fvopabf4g 37684 | Function value of an opera... |
| fnopabco 37685 | Composition of a function ... |
| opropabco 37686 | Composition of an operator... |
| cocnv 37687 | Composition with a functio... |
| f1ocan1fv 37688 | Cancel a composition by a ... |
| f1ocan2fv 37689 | Cancel a composition by th... |
| inixp 37690 | Intersection of Cartesian ... |
| upixp 37691 | Universal property of the ... |
| abrexdom 37692 | An indexed set is dominate... |
| abrexdom2 37693 | An indexed set is dominate... |
| ac6gf 37694 | Axiom of Choice. (Contrib... |
| indexa 37695 | If for every element of an... |
| indexdom 37696 | If for every element of an... |
| frinfm 37697 | A subset of a well-founded... |
| welb 37698 | A nonempty subset of a wel... |
| supex2g 37699 | Existence of supremum. (C... |
| supclt 37700 | Closure of supremum. (Con... |
| supubt 37701 | Upper bound property of su... |
| filbcmb 37702 | Combine a finite set of lo... |
| fzmul 37703 | Membership of a product in... |
| sdclem2 37704 | Lemma for ~ sdc . (Contri... |
| sdclem1 37705 | Lemma for ~ sdc . (Contri... |
| sdc 37706 | Strong dependent choice. ... |
| fdc 37707 | Finite version of dependen... |
| fdc1 37708 | Variant of ~ fdc with no s... |
| seqpo 37709 | Two ways to say that a seq... |
| incsequz 37710 | An increasing sequence of ... |
| incsequz2 37711 | An increasing sequence of ... |
| nnubfi 37712 | A bounded above set of pos... |
| nninfnub 37713 | An infinite set of positiv... |
| subspopn 37714 | An open set is open in the... |
| neificl 37715 | Neighborhoods are closed u... |
| lpss2 37716 | Limit points of a subset a... |
| metf1o 37717 | Use a bijection with a met... |
| blssp 37718 | A ball in the subspace met... |
| mettrifi 37719 | Generalized triangle inequ... |
| lmclim2 37720 | A sequence in a metric spa... |
| geomcau 37721 | If the distance between co... |
| caures 37722 | The restriction of a Cauch... |
| caushft 37723 | A shifted Cauchy sequence ... |
| constcncf 37724 | A constant function is a c... |
| cnres2 37725 | The restriction of a conti... |
| cnresima 37726 | A continuous function is c... |
| cncfres 37727 | A continuous function on c... |
| istotbnd 37731 | The predicate "is a totall... |
| istotbnd2 37732 | The predicate "is a totall... |
| istotbnd3 37733 | A metric space is totally ... |
| totbndmet 37734 | The predicate "totally bou... |
| 0totbnd 37735 | The metric (there is only ... |
| sstotbnd2 37736 | Condition for a subset of ... |
| sstotbnd 37737 | Condition for a subset of ... |
| sstotbnd3 37738 | Use a net that is not nece... |
| totbndss 37739 | A subset of a totally boun... |
| equivtotbnd 37740 | If the metric ` M ` is "st... |
| isbnd 37742 | The predicate "is a bounde... |
| bndmet 37743 | A bounded metric space is ... |
| isbndx 37744 | A "bounded extended metric... |
| isbnd2 37745 | The predicate "is a bounde... |
| isbnd3 37746 | A metric space is bounded ... |
| isbnd3b 37747 | A metric space is bounded ... |
| bndss 37748 | A subset of a bounded metr... |
| blbnd 37749 | A ball is bounded. (Contr... |
| ssbnd 37750 | A subset of a metric space... |
| totbndbnd 37751 | A totally bounded metric s... |
| equivbnd 37752 | If the metric ` M ` is "st... |
| bnd2lem 37753 | Lemma for ~ equivbnd2 and ... |
| equivbnd2 37754 | If balls are totally bound... |
| prdsbnd 37755 | The product metric over fi... |
| prdstotbnd 37756 | The product metric over fi... |
| prdsbnd2 37757 | If balls are totally bound... |
| cntotbnd 37758 | A subset of the complex nu... |
| cnpwstotbnd 37759 | A subset of ` A ^ I ` , wh... |
| ismtyval 37762 | The set of isometries betw... |
| isismty 37763 | The condition "is an isome... |
| ismtycnv 37764 | The inverse of an isometry... |
| ismtyima 37765 | The image of a ball under ... |
| ismtyhmeolem 37766 | Lemma for ~ ismtyhmeo . (... |
| ismtyhmeo 37767 | An isometry is a homeomorp... |
| ismtybndlem 37768 | Lemma for ~ ismtybnd . (C... |
| ismtybnd 37769 | Isometries preserve bounde... |
| ismtyres 37770 | A restriction of an isomet... |
| heibor1lem 37771 | Lemma for ~ heibor1 . A c... |
| heibor1 37772 | One half of ~ heibor , tha... |
| heiborlem1 37773 | Lemma for ~ heibor . We w... |
| heiborlem2 37774 | Lemma for ~ heibor . Subs... |
| heiborlem3 37775 | Lemma for ~ heibor . Usin... |
| heiborlem4 37776 | Lemma for ~ heibor . Usin... |
| heiborlem5 37777 | Lemma for ~ heibor . The ... |
| heiborlem6 37778 | Lemma for ~ heibor . Sinc... |
| heiborlem7 37779 | Lemma for ~ heibor . Sinc... |
| heiborlem8 37780 | Lemma for ~ heibor . The ... |
| heiborlem9 37781 | Lemma for ~ heibor . Disc... |
| heiborlem10 37782 | Lemma for ~ heibor . The ... |
| heibor 37783 | Generalized Heine-Borel Th... |
| bfplem1 37784 | Lemma for ~ bfp . The seq... |
| bfplem2 37785 | Lemma for ~ bfp . Using t... |
| bfp 37786 | Banach fixed point theorem... |
| rrnval 37789 | The n-dimensional Euclidea... |
| rrnmval 37790 | The value of the Euclidean... |
| rrnmet 37791 | Euclidean space is a metri... |
| rrndstprj1 37792 | The distance between two p... |
| rrndstprj2 37793 | Bound on the distance betw... |
| rrncmslem 37794 | Lemma for ~ rrncms . (Con... |
| rrncms 37795 | Euclidean space is complet... |
| repwsmet 37796 | The supremum metric on ` R... |
| rrnequiv 37797 | The supremum metric on ` R... |
| rrntotbnd 37798 | A set in Euclidean space i... |
| rrnheibor 37799 | Heine-Borel theorem for Eu... |
| ismrer1 37800 | An isometry between ` RR `... |
| reheibor 37801 | Heine-Borel theorem for re... |
| iccbnd 37802 | A closed interval in ` RR ... |
| icccmpALT 37803 | A closed interval in ` RR ... |
| isass 37808 | The predicate "is an assoc... |
| isexid 37809 | The predicate ` G ` has a ... |
| ismgmOLD 37812 | Obsolete version of ~ ismg... |
| clmgmOLD 37813 | Obsolete version of ~ mgmc... |
| opidonOLD 37814 | Obsolete version of ~ mndp... |
| rngopidOLD 37815 | Obsolete version of ~ mndp... |
| opidon2OLD 37816 | Obsolete version of ~ mndp... |
| isexid2 37817 | If ` G e. ( Magma i^i ExId... |
| exidu1 37818 | Uniqueness of the left and... |
| idrval 37819 | The value of the identity ... |
| iorlid 37820 | A magma right and left ide... |
| cmpidelt 37821 | A magma right and left ide... |
| smgrpismgmOLD 37824 | Obsolete version of ~ sgrp... |
| issmgrpOLD 37825 | Obsolete version of ~ issg... |
| smgrpmgm 37826 | A semigroup is a magma. (... |
| smgrpassOLD 37827 | Obsolete version of ~ sgrp... |
| mndoissmgrpOLD 37830 | Obsolete version of ~ mnds... |
| mndoisexid 37831 | A monoid has an identity e... |
| mndoismgmOLD 37832 | Obsolete version of ~ mndm... |
| mndomgmid 37833 | A monoid is a magma with a... |
| ismndo 37834 | The predicate "is a monoid... |
| ismndo1 37835 | The predicate "is a monoid... |
| ismndo2 37836 | The predicate "is a monoid... |
| grpomndo 37837 | A group is a monoid. (Con... |
| exidcl 37838 | Closure of the binary oper... |
| exidreslem 37839 | Lemma for ~ exidres and ~ ... |
| exidres 37840 | The restriction of a binar... |
| exidresid 37841 | The restriction of a binar... |
| ablo4pnp 37842 | A commutative/associative ... |
| grpoeqdivid 37843 | Two group elements are equ... |
| grposnOLD 37844 | The group operation for th... |
| elghomlem1OLD 37847 | Obsolete as of 15-Mar-2020... |
| elghomlem2OLD 37848 | Obsolete as of 15-Mar-2020... |
| elghomOLD 37849 | Obsolete version of ~ isgh... |
| ghomlinOLD 37850 | Obsolete version of ~ ghml... |
| ghomidOLD 37851 | Obsolete version of ~ ghmi... |
| ghomf 37852 | Mapping property of a grou... |
| ghomco 37853 | The composition of two gro... |
| ghomdiv 37854 | Group homomorphisms preser... |
| grpokerinj 37855 | A group homomorphism is in... |
| relrngo 37858 | The class of all unital ri... |
| isrngo 37859 | The predicate "is a (unita... |
| isrngod 37860 | Conditions that determine ... |
| rngoi 37861 | The properties of a unital... |
| rngosm 37862 | Functionality of the multi... |
| rngocl 37863 | Closure of the multiplicat... |
| rngoid 37864 | The multiplication operati... |
| rngoideu 37865 | The unity element of a rin... |
| rngodi 37866 | Distributive law for the m... |
| rngodir 37867 | Distributive law for the m... |
| rngoass 37868 | Associative law for the mu... |
| rngo2 37869 | A ring element plus itself... |
| rngoablo 37870 | A ring's addition operatio... |
| rngoablo2 37871 | In a unital ring the addit... |
| rngogrpo 37872 | A ring's addition operatio... |
| rngone0 37873 | The base set of a ring is ... |
| rngogcl 37874 | Closure law for the additi... |
| rngocom 37875 | The addition operation of ... |
| rngoaass 37876 | The addition operation of ... |
| rngoa32 37877 | The addition operation of ... |
| rngoa4 37878 | Rearrangement of 4 terms i... |
| rngorcan 37879 | Right cancellation law for... |
| rngolcan 37880 | Left cancellation law for ... |
| rngo0cl 37881 | A ring has an additive ide... |
| rngo0rid 37882 | The additive identity of a... |
| rngo0lid 37883 | The additive identity of a... |
| rngolz 37884 | The zero of a unital ring ... |
| rngorz 37885 | The zero of a unital ring ... |
| rngosn3 37886 | Obsolete as of 25-Jan-2020... |
| rngosn4 37887 | Obsolete as of 25-Jan-2020... |
| rngosn6 37888 | Obsolete as of 25-Jan-2020... |
| rngonegcl 37889 | A ring is closed under neg... |
| rngoaddneg1 37890 | Adding the negative in a r... |
| rngoaddneg2 37891 | Adding the negative in a r... |
| rngosub 37892 | Subtraction in a ring, in ... |
| rngmgmbs4 37893 | The range of an internal o... |
| rngodm1dm2 37894 | In a unital ring the domai... |
| rngorn1 37895 | In a unital ring the range... |
| rngorn1eq 37896 | In a unital ring the range... |
| rngomndo 37897 | In a unital ring the multi... |
| rngoidmlem 37898 | The unity element of a rin... |
| rngolidm 37899 | The unity element of a rin... |
| rngoridm 37900 | The unity element of a rin... |
| rngo1cl 37901 | The unity element of a rin... |
| rngoueqz 37902 | Obsolete as of 23-Jan-2020... |
| rngonegmn1l 37903 | Negation in a ring is the ... |
| rngonegmn1r 37904 | Negation in a ring is the ... |
| rngoneglmul 37905 | Negation of a product in a... |
| rngonegrmul 37906 | Negation of a product in a... |
| rngosubdi 37907 | Ring multiplication distri... |
| rngosubdir 37908 | Ring multiplication distri... |
| zerdivemp1x 37909 | In a unital ring a left in... |
| isdivrngo 37912 | The predicate "is a divisi... |
| drngoi 37913 | The properties of a divisi... |
| gidsn 37914 | Obsolete as of 23-Jan-2020... |
| zrdivrng 37915 | The zero ring is not a div... |
| dvrunz 37916 | In a division ring the rin... |
| isgrpda 37917 | Properties that determine ... |
| isdrngo1 37918 | The predicate "is a divisi... |
| divrngcl 37919 | The product of two nonzero... |
| isdrngo2 37920 | A division ring is a ring ... |
| isdrngo3 37921 | A division ring is a ring ... |
| rngohomval 37926 | The set of ring homomorphi... |
| isrngohom 37927 | The predicate "is a ring h... |
| rngohomf 37928 | A ring homomorphism is a f... |
| rngohomcl 37929 | Closure law for a ring hom... |
| rngohom1 37930 | A ring homomorphism preser... |
| rngohomadd 37931 | Ring homomorphisms preserv... |
| rngohommul 37932 | Ring homomorphisms preserv... |
| rngogrphom 37933 | A ring homomorphism is a g... |
| rngohom0 37934 | A ring homomorphism preser... |
| rngohomsub 37935 | Ring homomorphisms preserv... |
| rngohomco 37936 | The composition of two rin... |
| rngokerinj 37937 | A ring homomorphism is inj... |
| rngoisoval 37939 | The set of ring isomorphis... |
| isrngoiso 37940 | The predicate "is a ring i... |
| rngoiso1o 37941 | A ring isomorphism is a bi... |
| rngoisohom 37942 | A ring isomorphism is a ri... |
| rngoisocnv 37943 | The inverse of a ring isom... |
| rngoisoco 37944 | The composition of two rin... |
| isriscg 37946 | The ring isomorphism relat... |
| isrisc 37947 | The ring isomorphism relat... |
| risc 37948 | The ring isomorphism relat... |
| risci 37949 | Determine that two rings a... |
| riscer 37950 | Ring isomorphism is an equ... |
| iscom2 37957 | A device to add commutativ... |
| iscrngo 37958 | The predicate "is a commut... |
| iscrngo2 37959 | The predicate "is a commut... |
| iscringd 37960 | Conditions that determine ... |
| flddivrng 37961 | A field is a division ring... |
| crngorngo 37962 | A commutative ring is a ri... |
| crngocom 37963 | The multiplication operati... |
| crngm23 37964 | Commutative/associative la... |
| crngm4 37965 | Commutative/associative la... |
| fldcrngo 37966 | A field is a commutative r... |
| isfld2 37967 | The predicate "is a field"... |
| crngohomfo 37968 | The image of a homomorphis... |
| idlval 37975 | The class of ideals of a r... |
| isidl 37976 | The predicate "is an ideal... |
| isidlc 37977 | The predicate "is an ideal... |
| idlss 37978 | An ideal of ` R ` is a sub... |
| idlcl 37979 | An element of an ideal is ... |
| idl0cl 37980 | An ideal contains ` 0 ` . ... |
| idladdcl 37981 | An ideal is closed under a... |
| idllmulcl 37982 | An ideal is closed under m... |
| idlrmulcl 37983 | An ideal is closed under m... |
| idlnegcl 37984 | An ideal is closed under n... |
| idlsubcl 37985 | An ideal is closed under s... |
| rngoidl 37986 | A ring ` R ` is an ` R ` i... |
| 0idl 37987 | The set containing only ` ... |
| 1idl 37988 | Two ways of expressing the... |
| 0rngo 37989 | In a ring, ` 0 = 1 ` iff t... |
| divrngidl 37990 | The only ideals in a divis... |
| intidl 37991 | The intersection of a none... |
| inidl 37992 | The intersection of two id... |
| unichnidl 37993 | The union of a nonempty ch... |
| keridl 37994 | The kernel of a ring homom... |
| pridlval 37995 | The class of prime ideals ... |
| ispridl 37996 | The predicate "is a prime ... |
| pridlidl 37997 | A prime ideal is an ideal.... |
| pridlnr 37998 | A prime ideal is a proper ... |
| pridl 37999 | The main property of a pri... |
| ispridl2 38000 | A condition that shows an ... |
| maxidlval 38001 | The set of maximal ideals ... |
| ismaxidl 38002 | The predicate "is a maxima... |
| maxidlidl 38003 | A maximal ideal is an idea... |
| maxidlnr 38004 | A maximal ideal is proper.... |
| maxidlmax 38005 | A maximal ideal is a maxim... |
| maxidln1 38006 | One is not contained in an... |
| maxidln0 38007 | A ring with a maximal idea... |
| isprrngo 38012 | The predicate "is a prime ... |
| prrngorngo 38013 | A prime ring is a ring. (... |
| smprngopr 38014 | A simple ring (one whose o... |
| divrngpr 38015 | A division ring is a prime... |
| isdmn 38016 | The predicate "is a domain... |
| isdmn2 38017 | The predicate "is a domain... |
| dmncrng 38018 | A domain is a commutative ... |
| dmnrngo 38019 | A domain is a ring. (Cont... |
| flddmn 38020 | A field is a domain. (Con... |
| igenval 38023 | The ideal generated by a s... |
| igenss 38024 | A set is a subset of the i... |
| igenidl 38025 | The ideal generated by a s... |
| igenmin 38026 | The ideal generated by a s... |
| igenidl2 38027 | The ideal generated by an ... |
| igenval2 38028 | The ideal generated by a s... |
| prnc 38029 | A principal ideal (an idea... |
| isfldidl 38030 | Determine if a ring is a f... |
| isfldidl2 38031 | Determine if a ring is a f... |
| ispridlc 38032 | The predicate "is a prime ... |
| pridlc 38033 | Property of a prime ideal ... |
| pridlc2 38034 | Property of a prime ideal ... |
| pridlc3 38035 | Property of a prime ideal ... |
| isdmn3 38036 | The predicate "is a domain... |
| dmnnzd 38037 | A domain has no zero-divis... |
| dmncan1 38038 | Cancellation law for domai... |
| dmncan2 38039 | Cancellation law for domai... |
| efald2 38040 | A proof by contradiction. ... |
| notbinot1 38041 | Simplification rule of neg... |
| bicontr 38042 | Biconditional of its own n... |
| impor 38043 | An equivalent formula for ... |
| orfa 38044 | The falsum ` F. ` can be r... |
| notbinot2 38045 | Commutation rule between n... |
| biimpor 38046 | A rewriting rule for bicon... |
| orfa1 38047 | Add a contradicting disjun... |
| orfa2 38048 | Remove a contradicting dis... |
| bifald 38049 | Infer the equivalence to a... |
| orsild 38050 | A lemma for not-or-not eli... |
| orsird 38051 | A lemma for not-or-not eli... |
| cnf1dd 38052 | A lemma for Conjunctive No... |
| cnf2dd 38053 | A lemma for Conjunctive No... |
| cnfn1dd 38054 | A lemma for Conjunctive No... |
| cnfn2dd 38055 | A lemma for Conjunctive No... |
| or32dd 38056 | A rearrangement of disjunc... |
| notornotel1 38057 | A lemma for not-or-not eli... |
| notornotel2 38058 | A lemma for not-or-not eli... |
| contrd 38059 | A proof by contradiction, ... |
| an12i 38060 | An inference from commutin... |
| exmid2 38061 | An excluded middle law. (... |
| selconj 38062 | An inference for selecting... |
| truconj 38063 | Add true as a conjunct. (... |
| orel 38064 | An inference for disjuncti... |
| negel 38065 | An inference for negation ... |
| botel 38066 | An inference for bottom el... |
| tradd 38067 | Add top ad a conjunct. (C... |
| gm-sbtru 38068 | Substitution does not chan... |
| sbfal 38069 | Substitution does not chan... |
| sbcani 38070 | Distribution of class subs... |
| sbcori 38071 | Distribution of class subs... |
| sbcimi 38072 | Distribution of class subs... |
| sbcni 38073 | Move class substitution in... |
| sbali 38074 | Discard class substitution... |
| sbexi 38075 | Discard class substitution... |
| sbcalf 38076 | Move universal quantifier ... |
| sbcexf 38077 | Move existential quantifie... |
| sbcalfi 38078 | Move universal quantifier ... |
| sbcexfi 38079 | Move existential quantifie... |
| spsbcdi 38080 | A lemma for eliminating a ... |
| alrimii 38081 | A lemma for introducing a ... |
| spesbcdi 38082 | A lemma for introducing an... |
| exlimddvf 38083 | A lemma for eliminating an... |
| exlimddvfi 38084 | A lemma for eliminating an... |
| sbceq1ddi 38085 | A lemma for eliminating in... |
| sbccom2lem 38086 | Lemma for ~ sbccom2 . (Co... |
| sbccom2 38087 | Commutative law for double... |
| sbccom2f 38088 | Commutative law for double... |
| sbccom2fi 38089 | Commutative law for double... |
| csbcom2fi 38090 | Commutative law for double... |
| fald 38091 | Refutation of falsity, in ... |
| tsim1 38092 | A Tseitin axiom for logica... |
| tsim2 38093 | A Tseitin axiom for logica... |
| tsim3 38094 | A Tseitin axiom for logica... |
| tsbi1 38095 | A Tseitin axiom for logica... |
| tsbi2 38096 | A Tseitin axiom for logica... |
| tsbi3 38097 | A Tseitin axiom for logica... |
| tsbi4 38098 | A Tseitin axiom for logica... |
| tsxo1 38099 | A Tseitin axiom for logica... |
| tsxo2 38100 | A Tseitin axiom for logica... |
| tsxo3 38101 | A Tseitin axiom for logica... |
| tsxo4 38102 | A Tseitin axiom for logica... |
| tsan1 38103 | A Tseitin axiom for logica... |
| tsan2 38104 | A Tseitin axiom for logica... |
| tsan3 38105 | A Tseitin axiom for logica... |
| tsna1 38106 | A Tseitin axiom for logica... |
| tsna2 38107 | A Tseitin axiom for logica... |
| tsna3 38108 | A Tseitin axiom for logica... |
| tsor1 38109 | A Tseitin axiom for logica... |
| tsor2 38110 | A Tseitin axiom for logica... |
| tsor3 38111 | A Tseitin axiom for logica... |
| ts3an1 38112 | A Tseitin axiom for triple... |
| ts3an2 38113 | A Tseitin axiom for triple... |
| ts3an3 38114 | A Tseitin axiom for triple... |
| ts3or1 38115 | A Tseitin axiom for triple... |
| ts3or2 38116 | A Tseitin axiom for triple... |
| ts3or3 38117 | A Tseitin axiom for triple... |
| iuneq2f 38118 | Equality deduction for ind... |
| rabeq12f 38119 | Equality deduction for res... |
| csbeq12 38120 | Equality deduction for sub... |
| sbeqi 38121 | Equality deduction for sub... |
| ralbi12f 38122 | Equality deduction for res... |
| oprabbi 38123 | Equality deduction for cla... |
| mpobi123f 38124 | Equality deduction for map... |
| iuneq12f 38125 | Equality deduction for ind... |
| iineq12f 38126 | Equality deduction for ind... |
| opabbi 38127 | Equality deduction for cla... |
| mptbi12f 38128 | Equality deduction for map... |
| orcomdd 38129 | Commutativity of logic dis... |
| scottexf 38130 | A version of ~ scottex wit... |
| scott0f 38131 | A version of ~ scott0 with... |
| scottn0f 38132 | A version of ~ scott0f wit... |
| ac6s3f 38133 | Generalization of the Axio... |
| ac6s6 38134 | Generalization of the Axio... |
| ac6s6f 38135 | Generalization of the Axio... |
| el2v1 38179 | New way ( ~ elv , and the ... |
| el3v1 38180 | New way ( ~ elv , and the ... |
| el3v2 38181 | New way ( ~ elv , and the ... |
| el3v12 38182 | New way ( ~ elv , and the ... |
| el3v13 38183 | New way ( ~ elv , and the ... |
| el3v23 38184 | New way ( ~ elv , and the ... |
| anan 38185 | Multiple commutations in c... |
| triantru3 38186 | A wff is equivalent to its... |
| bianim 38187 | Exchanging conjunction in ... |
| biorfd 38188 | A wff is equivalent to its... |
| eqbrtr 38189 | Substitution of equal clas... |
| eqbrb 38190 | Substitution of equal clas... |
| eqeltr 38191 | Substitution of equal clas... |
| eqelb 38192 | Substitution of equal clas... |
| eqeqan2d 38193 | Implication of introducing... |
| suceqsneq 38194 | One-to-one relationship be... |
| sucdifsn2 38195 | Absorption of union with a... |
| sucdifsn 38196 | The difference between the... |
| disjresin 38197 | The restriction to a disjo... |
| disjresdisj 38198 | The intersection of restri... |
| disjresdif 38199 | The difference between res... |
| disjresundif 38200 | Lemma for ~ ressucdifsn2 .... |
| ressucdifsn2 38201 | The difference between res... |
| ressucdifsn 38202 | The difference between res... |
| inres2 38203 | Two ways of expressing the... |
| coideq 38204 | Equality theorem for compo... |
| nexmo1 38205 | If there is no case where ... |
| ralin 38206 | Restricted universal quant... |
| r2alan 38207 | Double restricted universa... |
| ssrabi 38208 | Inference of restricted ab... |
| rabimbieq 38209 | Restricted equivalent wff'... |
| abeqin 38210 | Intersection with class ab... |
| abeqinbi 38211 | Intersection with class ab... |
| rabeqel 38212 | Class element of a restric... |
| eqrelf 38213 | The equality connective be... |
| br1cnvinxp 38214 | Binary relation on the con... |
| releleccnv 38215 | Elementhood in a converse ... |
| releccnveq 38216 | Equality of converse ` R `... |
| opelvvdif 38217 | Negated elementhood of ord... |
| vvdifopab 38218 | Ordered-pair class abstrac... |
| brvdif 38219 | Binary relation with unive... |
| brvdif2 38220 | Binary relation with unive... |
| brvvdif 38221 | Binary relation with the c... |
| brvbrvvdif 38222 | Binary relation with the c... |
| brcnvep 38223 | The converse of the binary... |
| elecALTV 38224 | Elementhood in the ` R ` -... |
| brcnvepres 38225 | Restricted converse epsilo... |
| brres2 38226 | Binary relation on a restr... |
| br1cnvres 38227 | Binary relation on the con... |
| eldmres 38228 | Elementhood in the domain ... |
| elrnres 38229 | Element of the range of a ... |
| eldmressnALTV 38230 | Element of the domain of a... |
| elrnressn 38231 | Element of the range of a ... |
| eldm4 38232 | Elementhood in a domain. ... |
| eldmres2 38233 | Elementhood in the domain ... |
| eceq1i 38234 | Equality theorem for ` C `... |
| elecres 38235 | Elementhood in the restric... |
| ecres 38236 | Restricted coset of ` B ` ... |
| ecres2 38237 | The restricted coset of ` ... |
| eccnvepres 38238 | Restricted converse epsilo... |
| eleccnvep 38239 | Elementhood in the convers... |
| eccnvep 38240 | The converse epsilon coset... |
| extep 38241 | Property of epsilon relati... |
| disjeccnvep 38242 | Property of the epsilon re... |
| eccnvepres2 38243 | The restricted converse ep... |
| eccnvepres3 38244 | Condition for a restricted... |
| eldmqsres 38245 | Elementhood in a restricte... |
| eldmqsres2 38246 | Elementhood in a restricte... |
| qsss1 38247 | Subclass theorem for quoti... |
| qseq1i 38248 | Equality theorem for quoti... |
| brinxprnres 38249 | Binary relation on a restr... |
| inxprnres 38250 | Restriction of a class as ... |
| dfres4 38251 | Alternate definition of th... |
| exan3 38252 | Equivalent expressions wit... |
| exanres 38253 | Equivalent expressions wit... |
| exanres3 38254 | Equivalent expressions wit... |
| exanres2 38255 | Equivalent expressions wit... |
| cnvepres 38256 | Restricted converse epsilo... |
| eqrel2 38257 | Equality of relations. (C... |
| rncnv 38258 | Range of converse is the d... |
| dfdm6 38259 | Alternate definition of do... |
| dfrn6 38260 | Alternate definition of ra... |
| rncnvepres 38261 | The range of the restricte... |
| dmecd 38262 | Equality of the coset of `... |
| dmec2d 38263 | Equality of the coset of `... |
| brid 38264 | Property of the identity b... |
| ideq2 38265 | For sets, the identity bin... |
| idresssidinxp 38266 | Condition for the identity... |
| idreseqidinxp 38267 | Condition for the identity... |
| extid 38268 | Property of identity relat... |
| inxpss 38269 | Two ways to say that an in... |
| idinxpss 38270 | Two ways to say that an in... |
| ref5 38271 | Two ways to say that an in... |
| inxpss3 38272 | Two ways to say that an in... |
| inxpss2 38273 | Two ways to say that inter... |
| inxpssidinxp 38274 | Two ways to say that inter... |
| idinxpssinxp 38275 | Two ways to say that inter... |
| idinxpssinxp2 38276 | Identity intersection with... |
| idinxpssinxp3 38277 | Identity intersection with... |
| idinxpssinxp4 38278 | Identity intersection with... |
| relcnveq3 38279 | Two ways of saying a relat... |
| relcnveq 38280 | Two ways of saying a relat... |
| relcnveq2 38281 | Two ways of saying a relat... |
| relcnveq4 38282 | Two ways of saying a relat... |
| qsresid 38283 | Simplification of a specia... |
| n0elqs 38284 | Two ways of expressing tha... |
| n0elqs2 38285 | Two ways of expressing tha... |
| ecex2 38286 | Condition for a coset to b... |
| uniqsALTV 38287 | The union of a quotient se... |
| imaexALTV 38288 | Existence of an image of a... |
| ecexALTV 38289 | Existence of a coset, like... |
| rnresequniqs 38290 | The range of a restriction... |
| n0el2 38291 | Two ways of expressing tha... |
| cnvepresex 38292 | Sethood condition for the ... |
| eccnvepex 38293 | The converse epsilon coset... |
| cnvepimaex 38294 | The image of converse epsi... |
| cnvepima 38295 | The image of converse epsi... |
| inex3 38296 | Sufficient condition for t... |
| inxpex 38297 | Sufficient condition for a... |
| eqres 38298 | Converting a class constan... |
| brrabga 38299 | The law of concretion for ... |
| brcnvrabga 38300 | The law of concretion for ... |
| opideq 38301 | Equality conditions for or... |
| iss2 38302 | A subclass of the identity... |
| eldmcnv 38303 | Elementhood in a domain of... |
| dfrel5 38304 | Alternate definition of th... |
| dfrel6 38305 | Alternate definition of th... |
| cnvresrn 38306 | Converse restricted to ran... |
| relssinxpdmrn 38307 | Subset of restriction, spe... |
| cnvref4 38308 | Two ways to say that a rel... |
| cnvref5 38309 | Two ways to say that a rel... |
| ecin0 38310 | Two ways of saying that th... |
| ecinn0 38311 | Two ways of saying that th... |
| ineleq 38312 | Equivalence of restricted ... |
| inecmo 38313 | Equivalence of a double re... |
| inecmo2 38314 | Equivalence of a double re... |
| ineccnvmo 38315 | Equivalence of a double re... |
| alrmomorn 38316 | Equivalence of an "at most... |
| alrmomodm 38317 | Equivalence of an "at most... |
| ineccnvmo2 38318 | Equivalence of a double un... |
| inecmo3 38319 | Equivalence of a double un... |
| moeu2 38320 | Uniqueness is equivalent t... |
| mopickr 38321 | "At most one" picks a vari... |
| moantr 38322 | Sufficient condition for t... |
| brabidgaw 38323 | The law of concretion for ... |
| brabidga 38324 | The law of concretion for ... |
| inxp2 38325 | Intersection with a Cartes... |
| opabf 38326 | A class abstraction of a c... |
| ec0 38327 | The empty-coset of a class... |
| brcnvin 38328 | Intersection with a conver... |
| xrnss3v 38330 | A range Cartesian product ... |
| xrnrel 38331 | A range Cartesian product ... |
| brxrn 38332 | Characterize a ternary rel... |
| brxrn2 38333 | A characterization of the ... |
| dfxrn2 38334 | Alternate definition of th... |
| xrneq1 38335 | Equality theorem for the r... |
| xrneq1i 38336 | Equality theorem for the r... |
| xrneq1d 38337 | Equality theorem for the r... |
| xrneq2 38338 | Equality theorem for the r... |
| xrneq2i 38339 | Equality theorem for the r... |
| xrneq2d 38340 | Equality theorem for the r... |
| xrneq12 38341 | Equality theorem for the r... |
| xrneq12i 38342 | Equality theorem for the r... |
| xrneq12d 38343 | Equality theorem for the r... |
| elecxrn 38344 | Elementhood in the ` ( R |... |
| ecxrn 38345 | The ` ( R |X. S ) ` -coset... |
| disjressuc2 38346 | Double restricted quantifi... |
| disjecxrn 38347 | Two ways of saying that ` ... |
| disjecxrncnvep 38348 | Two ways of saying that co... |
| disjsuc2 38349 | Double restricted quantifi... |
| xrninxp 38350 | Intersection of a range Ca... |
| xrninxp2 38351 | Intersection of a range Ca... |
| xrninxpex 38352 | Sufficient condition for t... |
| inxpxrn 38353 | Two ways to express the in... |
| br1cnvxrn2 38354 | The converse of a binary r... |
| elec1cnvxrn2 38355 | Elementhood in the convers... |
| rnxrn 38356 | Range of the range Cartesi... |
| rnxrnres 38357 | Range of a range Cartesian... |
| rnxrncnvepres 38358 | Range of a range Cartesian... |
| rnxrnidres 38359 | Range of a range Cartesian... |
| xrnres 38360 | Two ways to express restri... |
| xrnres2 38361 | Two ways to express restri... |
| xrnres3 38362 | Two ways to express restri... |
| xrnres4 38363 | Two ways to express restri... |
| xrnresex 38364 | Sufficient condition for a... |
| xrnidresex 38365 | Sufficient condition for a... |
| xrncnvepresex 38366 | Sufficient condition for a... |
| brin2 38367 | Binary relation on an inte... |
| brin3 38368 | Binary relation on an inte... |
| dfcoss2 38371 | Alternate definition of th... |
| dfcoss3 38372 | Alternate definition of th... |
| dfcoss4 38373 | Alternate definition of th... |
| cosscnv 38374 | Class of cosets by the con... |
| coss1cnvres 38375 | Class of cosets by the con... |
| coss2cnvepres 38376 | Special case of ~ coss1cnv... |
| cossex 38377 | If ` A ` is a set then the... |
| cosscnvex 38378 | If ` A ` is a set then the... |
| 1cosscnvepresex 38379 | Sufficient condition for a... |
| 1cossxrncnvepresex 38380 | Sufficient condition for a... |
| relcoss 38381 | Cosets by ` R ` is a relat... |
| relcoels 38382 | Coelements on ` A ` is a r... |
| cossss 38383 | Subclass theorem for the c... |
| cosseq 38384 | Equality theorem for the c... |
| cosseqi 38385 | Equality theorem for the c... |
| cosseqd 38386 | Equality theorem for the c... |
| 1cossres 38387 | The class of cosets by a r... |
| dfcoels 38388 | Alternate definition of th... |
| brcoss 38389 | ` A ` and ` B ` are cosets... |
| brcoss2 38390 | Alternate form of the ` A ... |
| brcoss3 38391 | Alternate form of the ` A ... |
| brcosscnvcoss 38392 | For sets, the ` A ` and ` ... |
| brcoels 38393 | ` B ` and ` C ` are coelem... |
| cocossss 38394 | Two ways of saying that co... |
| cnvcosseq 38395 | The converse of cosets by ... |
| br2coss 38396 | Cosets by ` ,~ R ` binary ... |
| br1cossres 38397 | ` B ` and ` C ` are cosets... |
| br1cossres2 38398 | ` B ` and ` C ` are cosets... |
| brressn 38399 | Binary relation on a restr... |
| ressn2 38400 | A class ' R ' restricted t... |
| refressn 38401 | Any class ' R ' restricted... |
| antisymressn 38402 | Every class ' R ' restrict... |
| trressn 38403 | Any class ' R ' restricted... |
| relbrcoss 38404 | ` A ` and ` B ` are cosets... |
| br1cossinres 38405 | ` B ` and ` C ` are cosets... |
| br1cossxrnres 38406 | ` <. B , C >. ` and ` <. D... |
| br1cossinidres 38407 | ` B ` and ` C ` are cosets... |
| br1cossincnvepres 38408 | ` B ` and ` C ` are cosets... |
| br1cossxrnidres 38409 | ` <. B , C >. ` and ` <. D... |
| br1cossxrncnvepres 38410 | ` <. B , C >. ` and ` <. D... |
| dmcoss3 38411 | The domain of cosets is th... |
| dmcoss2 38412 | The domain of cosets is th... |
| rncossdmcoss 38413 | The range of cosets is the... |
| dm1cosscnvepres 38414 | The domain of cosets of th... |
| dmcoels 38415 | The domain of coelements i... |
| eldmcoss 38416 | Elementhood in the domain ... |
| eldmcoss2 38417 | Elementhood in the domain ... |
| eldm1cossres 38418 | Elementhood in the domain ... |
| eldm1cossres2 38419 | Elementhood in the domain ... |
| refrelcosslem 38420 | Lemma for the left side of... |
| refrelcoss3 38421 | The class of cosets by ` R... |
| refrelcoss2 38422 | The class of cosets by ` R... |
| symrelcoss3 38423 | The class of cosets by ` R... |
| symrelcoss2 38424 | The class of cosets by ` R... |
| cossssid 38425 | Equivalent expressions for... |
| cossssid2 38426 | Equivalent expressions for... |
| cossssid3 38427 | Equivalent expressions for... |
| cossssid4 38428 | Equivalent expressions for... |
| cossssid5 38429 | Equivalent expressions for... |
| brcosscnv 38430 | ` A ` and ` B ` are cosets... |
| brcosscnv2 38431 | ` A ` and ` B ` are cosets... |
| br1cosscnvxrn 38432 | ` A ` and ` B ` are cosets... |
| 1cosscnvxrn 38433 | Cosets by the converse ran... |
| cosscnvssid3 38434 | Equivalent expressions for... |
| cosscnvssid4 38435 | Equivalent expressions for... |
| cosscnvssid5 38436 | Equivalent expressions for... |
| coss0 38437 | Cosets by the empty set ar... |
| cossid 38438 | Cosets by the identity rel... |
| cosscnvid 38439 | Cosets by the converse ide... |
| trcoss 38440 | Sufficient condition for t... |
| eleccossin 38441 | Two ways of saying that th... |
| trcoss2 38442 | Equivalent expressions for... |
| elrels2 38444 | The element of the relatio... |
| elrelsrel 38445 | The element of the relatio... |
| elrelsrelim 38446 | The element of the relatio... |
| elrels5 38447 | Equivalent expressions for... |
| elrels6 38448 | Equivalent expressions for... |
| elrelscnveq3 38449 | Two ways of saying a relat... |
| elrelscnveq 38450 | Two ways of saying a relat... |
| elrelscnveq2 38451 | Two ways of saying a relat... |
| elrelscnveq4 38452 | Two ways of saying a relat... |
| cnvelrels 38453 | The converse of a set is a... |
| cosselrels 38454 | Cosets of sets are element... |
| cosscnvelrels 38455 | Cosets of converse sets ar... |
| dfssr2 38457 | Alternate definition of th... |
| relssr 38458 | The subset relation is a r... |
| brssr 38459 | The subset relation and su... |
| brssrid 38460 | Any set is a subset of its... |
| issetssr 38461 | Two ways of expressing set... |
| brssrres 38462 | Restricted subset binary r... |
| br1cnvssrres 38463 | Restricted converse subset... |
| brcnvssr 38464 | The converse of a subset r... |
| brcnvssrid 38465 | Any set is a converse subs... |
| br1cossxrncnvssrres 38466 | ` <. B , C >. ` and ` <. D... |
| extssr 38467 | Property of subset relatio... |
| dfrefrels2 38471 | Alternate definition of th... |
| dfrefrels3 38472 | Alternate definition of th... |
| dfrefrel2 38473 | Alternate definition of th... |
| dfrefrel3 38474 | Alternate definition of th... |
| dfrefrel5 38475 | Alternate definition of th... |
| elrefrels2 38476 | Element of the class of re... |
| elrefrels3 38477 | Element of the class of re... |
| elrefrelsrel 38478 | For sets, being an element... |
| refreleq 38479 | Equality theorem for refle... |
| refrelid 38480 | Identity relation is refle... |
| refrelcoss 38481 | The class of cosets by ` R... |
| refrelressn 38482 | Any class ' R ' restricted... |
| dfcnvrefrels2 38486 | Alternate definition of th... |
| dfcnvrefrels3 38487 | Alternate definition of th... |
| dfcnvrefrel2 38488 | Alternate definition of th... |
| dfcnvrefrel3 38489 | Alternate definition of th... |
| dfcnvrefrel4 38490 | Alternate definition of th... |
| dfcnvrefrel5 38491 | Alternate definition of th... |
| elcnvrefrels2 38492 | Element of the class of co... |
| elcnvrefrels3 38493 | Element of the class of co... |
| elcnvrefrelsrel 38494 | For sets, being an element... |
| cnvrefrelcoss2 38495 | Necessary and sufficient c... |
| cosselcnvrefrels2 38496 | Necessary and sufficient c... |
| cosselcnvrefrels3 38497 | Necessary and sufficient c... |
| cosselcnvrefrels4 38498 | Necessary and sufficient c... |
| cosselcnvrefrels5 38499 | Necessary and sufficient c... |
| dfsymrels2 38503 | Alternate definition of th... |
| dfsymrels3 38504 | Alternate definition of th... |
| dfsymrels4 38505 | Alternate definition of th... |
| dfsymrels5 38506 | Alternate definition of th... |
| dfsymrel2 38507 | Alternate definition of th... |
| dfsymrel3 38508 | Alternate definition of th... |
| dfsymrel4 38509 | Alternate definition of th... |
| dfsymrel5 38510 | Alternate definition of th... |
| elsymrels2 38511 | Element of the class of sy... |
| elsymrels3 38512 | Element of the class of sy... |
| elsymrels4 38513 | Element of the class of sy... |
| elsymrels5 38514 | Element of the class of sy... |
| elsymrelsrel 38515 | For sets, being an element... |
| symreleq 38516 | Equality theorem for symme... |
| symrelim 38517 | Symmetric relation implies... |
| symrelcoss 38518 | The class of cosets by ` R... |
| idsymrel 38519 | The identity relation is s... |
| epnsymrel 38520 | The membership (epsilon) r... |
| symrefref2 38521 | Symmetry is a sufficient c... |
| symrefref3 38522 | Symmetry is a sufficient c... |
| refsymrels2 38523 | Elements of the class of r... |
| refsymrels3 38524 | Elements of the class of r... |
| refsymrel2 38525 | A relation which is reflex... |
| refsymrel3 38526 | A relation which is reflex... |
| elrefsymrels2 38527 | Elements of the class of r... |
| elrefsymrels3 38528 | Elements of the class of r... |
| elrefsymrelsrel 38529 | For sets, being an element... |
| dftrrels2 38533 | Alternate definition of th... |
| dftrrels3 38534 | Alternate definition of th... |
| dftrrel2 38535 | Alternate definition of th... |
| dftrrel3 38536 | Alternate definition of th... |
| eltrrels2 38537 | Element of the class of tr... |
| eltrrels3 38538 | Element of the class of tr... |
| eltrrelsrel 38539 | For sets, being an element... |
| trreleq 38540 | Equality theorem for the t... |
| trrelressn 38541 | Any class ' R ' restricted... |
| dfeqvrels2 38546 | Alternate definition of th... |
| dfeqvrels3 38547 | Alternate definition of th... |
| dfeqvrel2 38548 | Alternate definition of th... |
| dfeqvrel3 38549 | Alternate definition of th... |
| eleqvrels2 38550 | Element of the class of eq... |
| eleqvrels3 38551 | Element of the class of eq... |
| eleqvrelsrel 38552 | For sets, being an element... |
| elcoeleqvrels 38553 | Elementhood in the coeleme... |
| elcoeleqvrelsrel 38554 | For sets, being an element... |
| eqvrelrel 38555 | An equivalence relation is... |
| eqvrelrefrel 38556 | An equivalence relation is... |
| eqvrelsymrel 38557 | An equivalence relation is... |
| eqvreltrrel 38558 | An equivalence relation is... |
| eqvrelim 38559 | Equivalence relation impli... |
| eqvreleq 38560 | Equality theorem for equiv... |
| eqvreleqi 38561 | Equality theorem for equiv... |
| eqvreleqd 38562 | Equality theorem for equiv... |
| eqvrelsym 38563 | An equivalence relation is... |
| eqvrelsymb 38564 | An equivalence relation is... |
| eqvreltr 38565 | An equivalence relation is... |
| eqvreltrd 38566 | A transitivity relation fo... |
| eqvreltr4d 38567 | A transitivity relation fo... |
| eqvrelref 38568 | An equivalence relation is... |
| eqvrelth 38569 | Basic property of equivale... |
| eqvrelcl 38570 | Elementhood in the field o... |
| eqvrelthi 38571 | Basic property of equivale... |
| eqvreldisj 38572 | Equivalence classes do not... |
| qsdisjALTV 38573 | Elements of a quotient set... |
| eqvrelqsel 38574 | If an element of a quotien... |
| eqvrelcoss 38575 | Two ways to express equiva... |
| eqvrelcoss3 38576 | Two ways to express equiva... |
| eqvrelcoss2 38577 | Two ways to express equiva... |
| eqvrelcoss4 38578 | Two ways to express equiva... |
| dfcoeleqvrels 38579 | Alternate definition of th... |
| dfcoeleqvrel 38580 | Alternate definition of th... |
| brredunds 38584 | Binary relation on the cla... |
| brredundsredund 38585 | For sets, binary relation ... |
| redundss3 38586 | Implication of redundancy ... |
| redundeq1 38587 | Equivalence of redundancy ... |
| redundpim3 38588 | Implication of redundancy ... |
| redundpbi1 38589 | Equivalence of redundancy ... |
| refrelsredund4 38590 | The naive version of the c... |
| refrelsredund2 38591 | The naive version of the c... |
| refrelsredund3 38592 | The naive version of the c... |
| refrelredund4 38593 | The naive version of the d... |
| refrelredund2 38594 | The naive version of the d... |
| refrelredund3 38595 | The naive version of the d... |
| dmqseq 38598 | Equality theorem for domai... |
| dmqseqi 38599 | Equality theorem for domai... |
| dmqseqd 38600 | Equality theorem for domai... |
| dmqseqeq1 38601 | Equality theorem for domai... |
| dmqseqeq1i 38602 | Equality theorem for domai... |
| dmqseqeq1d 38603 | Equality theorem for domai... |
| brdmqss 38604 | The domain quotient binary... |
| brdmqssqs 38605 | If ` A ` and ` R ` are set... |
| n0eldmqs 38606 | The empty set is not an el... |
| n0eldmqseq 38607 | The empty set is not an el... |
| n0elim 38608 | Implication of that the em... |
| n0el3 38609 | Two ways of expressing tha... |
| cnvepresdmqss 38610 | The domain quotient binary... |
| cnvepresdmqs 38611 | The domain quotient predic... |
| unidmqs 38612 | The range of a relation is... |
| unidmqseq 38613 | The union of the domain qu... |
| dmqseqim 38614 | If the domain quotient of ... |
| dmqseqim2 38615 | Lemma for ~ erimeq2 . (Co... |
| releldmqs 38616 | Elementhood in the domain ... |
| eldmqs1cossres 38617 | Elementhood in the domain ... |
| releldmqscoss 38618 | Elementhood in the domain ... |
| dmqscoelseq 38619 | Two ways to express the eq... |
| dmqs1cosscnvepreseq 38620 | Two ways to express the eq... |
| brers 38625 | Binary equivalence relatio... |
| dferALTV2 38626 | Equivalence relation with ... |
| erALTVeq1 38627 | Equality theorem for equiv... |
| erALTVeq1i 38628 | Equality theorem for equiv... |
| erALTVeq1d 38629 | Equality theorem for equiv... |
| dfcomember 38630 | Alternate definition of th... |
| dfcomember2 38631 | Alternate definition of th... |
| dfcomember3 38632 | Alternate definition of th... |
| eqvreldmqs 38633 | Two ways to express comemb... |
| eqvreldmqs2 38634 | Two ways to express comemb... |
| brerser 38635 | Binary equivalence relatio... |
| erimeq2 38636 | Equivalence relation on it... |
| erimeq 38637 | Equivalence relation on it... |
| dffunsALTV 38641 | Alternate definition of th... |
| dffunsALTV2 38642 | Alternate definition of th... |
| dffunsALTV3 38643 | Alternate definition of th... |
| dffunsALTV4 38644 | Alternate definition of th... |
| dffunsALTV5 38645 | Alternate definition of th... |
| dffunALTV2 38646 | Alternate definition of th... |
| dffunALTV3 38647 | Alternate definition of th... |
| dffunALTV4 38648 | Alternate definition of th... |
| dffunALTV5 38649 | Alternate definition of th... |
| elfunsALTV 38650 | Elementhood in the class o... |
| elfunsALTV2 38651 | Elementhood in the class o... |
| elfunsALTV3 38652 | Elementhood in the class o... |
| elfunsALTV4 38653 | Elementhood in the class o... |
| elfunsALTV5 38654 | Elementhood in the class o... |
| elfunsALTVfunALTV 38655 | The element of the class o... |
| funALTVfun 38656 | Our definition of the func... |
| funALTVss 38657 | Subclass theorem for funct... |
| funALTVeq 38658 | Equality theorem for funct... |
| funALTVeqi 38659 | Equality inference for the... |
| funALTVeqd 38660 | Equality deduction for the... |
| dfdisjs 38666 | Alternate definition of th... |
| dfdisjs2 38667 | Alternate definition of th... |
| dfdisjs3 38668 | Alternate definition of th... |
| dfdisjs4 38669 | Alternate definition of th... |
| dfdisjs5 38670 | Alternate definition of th... |
| dfdisjALTV 38671 | Alternate definition of th... |
| dfdisjALTV2 38672 | Alternate definition of th... |
| dfdisjALTV3 38673 | Alternate definition of th... |
| dfdisjALTV4 38674 | Alternate definition of th... |
| dfdisjALTV5 38675 | Alternate definition of th... |
| dfeldisj2 38676 | Alternate definition of th... |
| dfeldisj3 38677 | Alternate definition of th... |
| dfeldisj4 38678 | Alternate definition of th... |
| dfeldisj5 38679 | Alternate definition of th... |
| eldisjs 38680 | Elementhood in the class o... |
| eldisjs2 38681 | Elementhood in the class o... |
| eldisjs3 38682 | Elementhood in the class o... |
| eldisjs4 38683 | Elementhood in the class o... |
| eldisjs5 38684 | Elementhood in the class o... |
| eldisjsdisj 38685 | The element of the class o... |
| eleldisjs 38686 | Elementhood in the disjoin... |
| eleldisjseldisj 38687 | The element of the disjoin... |
| disjrel 38688 | Disjoint relation is a rel... |
| disjss 38689 | Subclass theorem for disjo... |
| disjssi 38690 | Subclass theorem for disjo... |
| disjssd 38691 | Subclass theorem for disjo... |
| disjeq 38692 | Equality theorem for disjo... |
| disjeqi 38693 | Equality theorem for disjo... |
| disjeqd 38694 | Equality theorem for disjo... |
| disjdmqseqeq1 38695 | Lemma for the equality the... |
| eldisjss 38696 | Subclass theorem for disjo... |
| eldisjssi 38697 | Subclass theorem for disjo... |
| eldisjssd 38698 | Subclass theorem for disjo... |
| eldisjeq 38699 | Equality theorem for disjo... |
| eldisjeqi 38700 | Equality theorem for disjo... |
| eldisjeqd 38701 | Equality theorem for disjo... |
| disjres 38702 | Disjoint restriction. (Co... |
| eldisjn0elb 38703 | Two forms of disjoint elem... |
| disjxrn 38704 | Two ways of saying that a ... |
| disjxrnres5 38705 | Disjoint range Cartesian p... |
| disjorimxrn 38706 | Disjointness condition for... |
| disjimxrn 38707 | Disjointness condition for... |
| disjimres 38708 | Disjointness condition for... |
| disjimin 38709 | Disjointness condition for... |
| disjiminres 38710 | Disjointness condition for... |
| disjimxrnres 38711 | Disjointness condition for... |
| disjALTV0 38712 | The null class is disjoint... |
| disjALTVid 38713 | The class of identity rela... |
| disjALTVidres 38714 | The class of identity rela... |
| disjALTVinidres 38715 | The intersection with rest... |
| disjALTVxrnidres 38716 | The class of range Cartesi... |
| disjsuc 38717 | Disjoint range Cartesian p... |
| dfantisymrel4 38719 | Alternate definition of th... |
| dfantisymrel5 38720 | Alternate definition of th... |
| antisymrelres 38721 | (Contributed by Peter Mazs... |
| antisymrelressn 38722 | (Contributed by Peter Mazs... |
| dfpart2 38727 | Alternate definition of th... |
| dfmembpart2 38728 | Alternate definition of th... |
| brparts 38729 | Binary partitions relation... |
| brparts2 38730 | Binary partitions relation... |
| brpartspart 38731 | Binary partition and the p... |
| parteq1 38732 | Equality theorem for parti... |
| parteq2 38733 | Equality theorem for parti... |
| parteq12 38734 | Equality theorem for parti... |
| parteq1i 38735 | Equality theorem for parti... |
| parteq1d 38736 | Equality theorem for parti... |
| partsuc2 38737 | Property of the partition.... |
| partsuc 38738 | Property of the partition.... |
| disjim 38739 | The "Divide et Aequivalere... |
| disjimi 38740 | Every disjoint relation ge... |
| detlem 38741 | If a relation is disjoint,... |
| eldisjim 38742 | If the elements of ` A ` a... |
| eldisjim2 38743 | Alternate form of ~ eldisj... |
| eqvrel0 38744 | The null class is an equiv... |
| det0 38745 | The cosets by the null cla... |
| eqvrelcoss0 38746 | The cosets by the null cla... |
| eqvrelid 38747 | The identity relation is a... |
| eqvrel1cossidres 38748 | The cosets by a restricted... |
| eqvrel1cossinidres 38749 | The cosets by an intersect... |
| eqvrel1cossxrnidres 38750 | The cosets by a range Cart... |
| detid 38751 | The cosets by the identity... |
| eqvrelcossid 38752 | The cosets by the identity... |
| detidres 38753 | The cosets by the restrict... |
| detinidres 38754 | The cosets by the intersec... |
| detxrnidres 38755 | The cosets by the range Ca... |
| disjlem14 38756 | Lemma for ~ disjdmqseq , ~... |
| disjlem17 38757 | Lemma for ~ disjdmqseq , ~... |
| disjlem18 38758 | Lemma for ~ disjdmqseq , ~... |
| disjlem19 38759 | Lemma for ~ disjdmqseq , ~... |
| disjdmqsss 38760 | Lemma for ~ disjdmqseq via... |
| disjdmqscossss 38761 | Lemma for ~ disjdmqseq via... |
| disjdmqs 38762 | If a relation is disjoint,... |
| disjdmqseq 38763 | If a relation is disjoint,... |
| eldisjn0el 38764 | Special case of ~ disjdmqs... |
| partim2 38765 | Disjoint relation on its n... |
| partim 38766 | Partition implies equivale... |
| partimeq 38767 | Partition implies that the... |
| eldisjlem19 38768 | Special case of ~ disjlem1... |
| membpartlem19 38769 | Together with ~ disjlem19 ... |
| petlem 38770 | If you can prove that the ... |
| petlemi 38771 | If you can prove disjointn... |
| pet02 38772 | Class ` A ` is a partition... |
| pet0 38773 | Class ` A ` is a partition... |
| petid2 38774 | Class ` A ` is a partition... |
| petid 38775 | A class is a partition by ... |
| petidres2 38776 | Class ` A ` is a partition... |
| petidres 38777 | A class is a partition by ... |
| petinidres2 38778 | Class ` A ` is a partition... |
| petinidres 38779 | A class is a partition by ... |
| petxrnidres2 38780 | Class ` A ` is a partition... |
| petxrnidres 38781 | A class is a partition by ... |
| eqvreldisj1 38782 | The elements of the quotie... |
| eqvreldisj2 38783 | The elements of the quotie... |
| eqvreldisj3 38784 | The elements of the quotie... |
| eqvreldisj4 38785 | Intersection with the conv... |
| eqvreldisj5 38786 | Range Cartesian product wi... |
| eqvrelqseqdisj2 38787 | Implication of ~ eqvreldis... |
| fences3 38788 | Implication of ~ eqvrelqse... |
| eqvrelqseqdisj3 38789 | Implication of ~ eqvreldis... |
| eqvrelqseqdisj4 38790 | Lemma for ~ petincnvepres2... |
| eqvrelqseqdisj5 38791 | Lemma for the Partition-Eq... |
| mainer 38792 | The Main Theorem of Equiva... |
| partimcomember 38793 | Partition with general ` R... |
| mpet3 38794 | Member Partition-Equivalen... |
| cpet2 38795 | The conventional form of t... |
| cpet 38796 | The conventional form of M... |
| mpet 38797 | Member Partition-Equivalen... |
| mpet2 38798 | Member Partition-Equivalen... |
| mpets2 38799 | Member Partition-Equivalen... |
| mpets 38800 | Member Partition-Equivalen... |
| mainpart 38801 | Partition with general ` R... |
| fences 38802 | The Theorem of Fences by E... |
| fences2 38803 | The Theorem of Fences by E... |
| mainer2 38804 | The Main Theorem of Equiva... |
| mainerim 38805 | Every equivalence relation... |
| petincnvepres2 38806 | A partition-equivalence th... |
| petincnvepres 38807 | The shortest form of a par... |
| pet2 38808 | Partition-Equivalence Theo... |
| pet 38809 | Partition-Equivalence Theo... |
| pets 38810 | Partition-Equivalence Theo... |
| prtlem60 38811 | Lemma for ~ prter3 . (Con... |
| bicomdd 38812 | Commute two sides of a bic... |
| jca2r 38813 | Inference conjoining the c... |
| jca3 38814 | Inference conjoining the c... |
| prtlem70 38815 | Lemma for ~ prter3 : a rea... |
| ibdr 38816 | Reverse of ~ ibd . (Contr... |
| prtlem100 38817 | Lemma for ~ prter3 . (Con... |
| prtlem5 38818 | Lemma for ~ prter1 , ~ prt... |
| prtlem80 38819 | Lemma for ~ prter2 . (Con... |
| brabsb2 38820 | A closed form of ~ brabsb ... |
| eqbrrdv2 38821 | Other version of ~ eqbrrdi... |
| prtlem9 38822 | Lemma for ~ prter3 . (Con... |
| prtlem10 38823 | Lemma for ~ prter3 . (Con... |
| prtlem11 38824 | Lemma for ~ prter2 . (Con... |
| prtlem12 38825 | Lemma for ~ prtex and ~ pr... |
| prtlem13 38826 | Lemma for ~ prter1 , ~ prt... |
| prtlem16 38827 | Lemma for ~ prtex , ~ prte... |
| prtlem400 38828 | Lemma for ~ prter2 and als... |
| erprt 38831 | The quotient set of an equ... |
| prtlem14 38832 | Lemma for ~ prter1 , ~ prt... |
| prtlem15 38833 | Lemma for ~ prter1 and ~ p... |
| prtlem17 38834 | Lemma for ~ prter2 . (Con... |
| prtlem18 38835 | Lemma for ~ prter2 . (Con... |
| prtlem19 38836 | Lemma for ~ prter2 . (Con... |
| prter1 38837 | Every partition generates ... |
| prtex 38838 | The equivalence relation g... |
| prter2 38839 | The quotient set of the eq... |
| prter3 38840 | For every partition there ... |
| axc5 38851 | This theorem repeats ~ sp ... |
| ax4fromc4 38852 | Rederivation of Axiom ~ ax... |
| ax10fromc7 38853 | Rederivation of Axiom ~ ax... |
| ax6fromc10 38854 | Rederivation of Axiom ~ ax... |
| hba1-o 38855 | The setvar ` x ` is not fr... |
| axc4i-o 38856 | Inference version of ~ ax-... |
| equid1 38857 | Proof of ~ equid from our ... |
| equcomi1 38858 | Proof of ~ equcomi from ~ ... |
| aecom-o 38859 | Commutation law for identi... |
| aecoms-o 38860 | A commutation rule for ide... |
| hbae-o 38861 | All variables are effectiv... |
| dral1-o 38862 | Formula-building lemma for... |
| ax12fromc15 38863 | Rederivation of Axiom ~ ax... |
| ax13fromc9 38864 | Derive ~ ax-13 from ~ ax-c... |
| ax5ALT 38865 | Axiom to quantify a variab... |
| sps-o 38866 | Generalization of antecede... |
| hbequid 38867 | Bound-variable hypothesis ... |
| nfequid-o 38868 | Bound-variable hypothesis ... |
| axc5c7 38869 | Proof of a single axiom th... |
| axc5c7toc5 38870 | Rederivation of ~ ax-c5 fr... |
| axc5c7toc7 38871 | Rederivation of ~ ax-c7 fr... |
| axc711 38872 | Proof of a single axiom th... |
| nfa1-o 38873 | ` x ` is not free in ` A. ... |
| axc711toc7 38874 | Rederivation of ~ ax-c7 fr... |
| axc711to11 38875 | Rederivation of ~ ax-11 fr... |
| axc5c711 38876 | Proof of a single axiom th... |
| axc5c711toc5 38877 | Rederivation of ~ ax-c5 fr... |
| axc5c711toc7 38878 | Rederivation of ~ ax-c7 fr... |
| axc5c711to11 38879 | Rederivation of ~ ax-11 fr... |
| equidqe 38880 | ~ equid with existential q... |
| axc5sp1 38881 | A special case of ~ ax-c5 ... |
| equidq 38882 | ~ equid with universal qua... |
| equid1ALT 38883 | Alternate proof of ~ equid... |
| axc11nfromc11 38884 | Rederivation of ~ ax-c11n ... |
| naecoms-o 38885 | A commutation rule for dis... |
| hbnae-o 38886 | All variables are effectiv... |
| dvelimf-o 38887 | Proof of ~ dvelimh that us... |
| dral2-o 38888 | Formula-building lemma for... |
| aev-o 38889 | A "distinctor elimination"... |
| ax5eq 38890 | Theorem to add distinct qu... |
| dveeq2-o 38891 | Quantifier introduction wh... |
| axc16g-o 38892 | A generalization of Axiom ... |
| dveeq1-o 38893 | Quantifier introduction wh... |
| dveeq1-o16 38894 | Version of ~ dveeq1 using ... |
| ax5el 38895 | Theorem to add distinct qu... |
| axc11n-16 38896 | This theorem shows that, g... |
| dveel2ALT 38897 | Alternate proof of ~ dveel... |
| ax12f 38898 | Basis step for constructin... |
| ax12eq 38899 | Basis step for constructin... |
| ax12el 38900 | Basis step for constructin... |
| ax12indn 38901 | Induction step for constru... |
| ax12indi 38902 | Induction step for constru... |
| ax12indalem 38903 | Lemma for ~ ax12inda2 and ... |
| ax12inda2ALT 38904 | Alternate proof of ~ ax12i... |
| ax12inda2 38905 | Induction step for constru... |
| ax12inda 38906 | Induction step for constru... |
| ax12v2-o 38907 | Rederivation of ~ ax-c15 f... |
| ax12a2-o 38908 | Derive ~ ax-c15 from a hyp... |
| axc11-o 38909 | Show that ~ ax-c11 can be ... |
| fsumshftd 38910 | Index shift of a finite su... |
| riotaclbgBAD 38912 | Closure of restricted iota... |
| riotaclbBAD 38913 | Closure of restricted iota... |
| riotasvd 38914 | Deduction version of ~ rio... |
| riotasv2d 38915 | Value of description binde... |
| riotasv2s 38916 | The value of description b... |
| riotasv 38917 | Value of description binde... |
| riotasv3d 38918 | A property ` ch ` holding ... |
| elimhyps 38919 | A version of ~ elimhyp usi... |
| dedths 38920 | A version of weak deductio... |
| renegclALT 38921 | Closure law for negative o... |
| elimhyps2 38922 | Generalization of ~ elimhy... |
| dedths2 38923 | Generalization of ~ dedths... |
| nfcxfrdf 38924 | A utility lemma to transfe... |
| nfded 38925 | A deduction theorem that c... |
| nfded2 38926 | A deduction theorem that c... |
| nfunidALT2 38927 | Deduction version of ~ nfu... |
| nfunidALT 38928 | Deduction version of ~ nfu... |
| nfopdALT 38929 | Deduction version of bound... |
| cnaddcom 38930 | Recover the commutative la... |
| toycom 38931 | Show the commutative law f... |
| lshpset 38936 | The set of all hyperplanes... |
| islshp 38937 | The predicate "is a hyperp... |
| islshpsm 38938 | Hyperplane properties expr... |
| lshplss 38939 | A hyperplane is a subspace... |
| lshpne 38940 | A hyperplane is not equal ... |
| lshpnel 38941 | A hyperplane's generating ... |
| lshpnelb 38942 | The subspace sum of a hype... |
| lshpnel2N 38943 | Condition that determines ... |
| lshpne0 38944 | The member of the span in ... |
| lshpdisj 38945 | A hyperplane and the span ... |
| lshpcmp 38946 | If two hyperplanes are com... |
| lshpinN 38947 | The intersection of two di... |
| lsatset 38948 | The set of all 1-dim subsp... |
| islsat 38949 | The predicate "is a 1-dim ... |
| lsatlspsn2 38950 | The span of a nonzero sing... |
| lsatlspsn 38951 | The span of a nonzero sing... |
| islsati 38952 | A 1-dim subspace (atom) (o... |
| lsateln0 38953 | A 1-dim subspace (atom) (o... |
| lsatlss 38954 | The set of 1-dim subspaces... |
| lsatlssel 38955 | An atom is a subspace. (C... |
| lsatssv 38956 | An atom is a set of vector... |
| lsatn0 38957 | A 1-dim subspace (atom) of... |
| lsatspn0 38958 | The span of a vector is an... |
| lsator0sp 38959 | The span of a vector is ei... |
| lsatssn0 38960 | A subspace (or any class) ... |
| lsatcmp 38961 | If two atoms are comparabl... |
| lsatcmp2 38962 | If an atom is included in ... |
| lsatel 38963 | A nonzero vector in an ato... |
| lsatelbN 38964 | A nonzero vector in an ato... |
| lsat2el 38965 | Two atoms sharing a nonzer... |
| lsmsat 38966 | Convert comparison of atom... |
| lsatfixedN 38967 | Show equality with the spa... |
| lsmsatcv 38968 | Subspace sum has the cover... |
| lssatomic 38969 | The lattice of subspaces i... |
| lssats 38970 | The lattice of subspaces i... |
| lpssat 38971 | Two subspaces in a proper ... |
| lrelat 38972 | Subspaces are relatively a... |
| lssatle 38973 | The ordering of two subspa... |
| lssat 38974 | Two subspaces in a proper ... |
| islshpat 38975 | Hyperplane properties expr... |
| lcvfbr 38978 | The covers relation for a ... |
| lcvbr 38979 | The covers relation for a ... |
| lcvbr2 38980 | The covers relation for a ... |
| lcvbr3 38981 | The covers relation for a ... |
| lcvpss 38982 | The covers relation implie... |
| lcvnbtwn 38983 | The covers relation implie... |
| lcvntr 38984 | The covers relation is not... |
| lcvnbtwn2 38985 | The covers relation implie... |
| lcvnbtwn3 38986 | The covers relation implie... |
| lsmcv2 38987 | Subspace sum has the cover... |
| lcvat 38988 | If a subspace covers anoth... |
| lsatcv0 38989 | An atom covers the zero su... |
| lsatcveq0 38990 | A subspace covered by an a... |
| lsat0cv 38991 | A subspace is an atom iff ... |
| lcvexchlem1 38992 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem2 38993 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem3 38994 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem4 38995 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem5 38996 | Lemma for ~ lcvexch . (Co... |
| lcvexch 38997 | Subspaces satisfy the exch... |
| lcvp 38998 | Covering property of Defin... |
| lcv1 38999 | Covering property of a sub... |
| lcv2 39000 | Covering property of a sub... |
| lsatexch 39001 | The atom exchange property... |
| lsatnle 39002 | The meet of a subspace and... |
| lsatnem0 39003 | The meet of distinct atoms... |
| lsatexch1 39004 | The atom exch1ange propert... |
| lsatcv0eq 39005 | If the sum of two atoms co... |
| lsatcv1 39006 | Two atoms covering the zer... |
| lsatcvatlem 39007 | Lemma for ~ lsatcvat . (C... |
| lsatcvat 39008 | A nonzero subspace less th... |
| lsatcvat2 39009 | A subspace covered by the ... |
| lsatcvat3 39010 | A condition implying that ... |
| islshpcv 39011 | Hyperplane properties expr... |
| l1cvpat 39012 | A subspace covered by the ... |
| l1cvat 39013 | Create an atom under an el... |
| lshpat 39014 | Create an atom under a hyp... |
| lflset 39017 | The set of linear function... |
| islfl 39018 | The predicate "is a linear... |
| lfli 39019 | Property of a linear funct... |
| islfld 39020 | Properties that determine ... |
| lflf 39021 | A linear functional is a f... |
| lflcl 39022 | A linear functional value ... |
| lfl0 39023 | A linear functional is zer... |
| lfladd 39024 | Property of a linear funct... |
| lflsub 39025 | Property of a linear funct... |
| lflmul 39026 | Property of a linear funct... |
| lfl0f 39027 | The zero function is a fun... |
| lfl1 39028 | A nonzero functional has a... |
| lfladdcl 39029 | Closure of addition of two... |
| lfladdcom 39030 | Commutativity of functiona... |
| lfladdass 39031 | Associativity of functiona... |
| lfladd0l 39032 | Functional addition with t... |
| lflnegcl 39033 | Closure of the negative of... |
| lflnegl 39034 | A functional plus its nega... |
| lflvscl 39035 | Closure of a scalar produc... |
| lflvsdi1 39036 | Distributive law for (righ... |
| lflvsdi2 39037 | Reverse distributive law f... |
| lflvsdi2a 39038 | Reverse distributive law f... |
| lflvsass 39039 | Associative law for (right... |
| lfl0sc 39040 | The (right vector space) s... |
| lflsc0N 39041 | The scalar product with th... |
| lfl1sc 39042 | The (right vector space) s... |
| lkrfval 39045 | The kernel of a functional... |
| lkrval 39046 | Value of the kernel of a f... |
| ellkr 39047 | Membership in the kernel o... |
| lkrval2 39048 | Value of the kernel of a f... |
| ellkr2 39049 | Membership in the kernel o... |
| lkrcl 39050 | A member of the kernel of ... |
| lkrf0 39051 | The value of a functional ... |
| lkr0f 39052 | The kernel of the zero fun... |
| lkrlss 39053 | The kernel of a linear fun... |
| lkrssv 39054 | The kernel of a linear fun... |
| lkrsc 39055 | The kernel of a nonzero sc... |
| lkrscss 39056 | The kernel of a scalar pro... |
| eqlkr 39057 | Two functionals with the s... |
| eqlkr2 39058 | Two functionals with the s... |
| eqlkr3 39059 | Two functionals with the s... |
| lkrlsp 39060 | The subspace sum of a kern... |
| lkrlsp2 39061 | The subspace sum of a kern... |
| lkrlsp3 39062 | The subspace sum of a kern... |
| lkrshp 39063 | The kernel of a nonzero fu... |
| lkrshp3 39064 | The kernels of nonzero fun... |
| lkrshpor 39065 | The kernel of a functional... |
| lkrshp4 39066 | A kernel is a hyperplane i... |
| lshpsmreu 39067 | Lemma for ~ lshpkrex . Sh... |
| lshpkrlem1 39068 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem2 39069 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem3 39070 | Lemma for ~ lshpkrex . De... |
| lshpkrlem4 39071 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem5 39072 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem6 39073 | Lemma for ~ lshpkrex . Sh... |
| lshpkrcl 39074 | The set ` G ` defined by h... |
| lshpkr 39075 | The kernel of functional `... |
| lshpkrex 39076 | There exists a functional ... |
| lshpset2N 39077 | The set of all hyperplanes... |
| islshpkrN 39078 | The predicate "is a hyperp... |
| lfl1dim 39079 | Equivalent expressions for... |
| lfl1dim2N 39080 | Equivalent expressions for... |
| ldualset 39083 | Define the (left) dual of ... |
| ldualvbase 39084 | The vectors of a dual spac... |
| ldualelvbase 39085 | Utility theorem for conver... |
| ldualfvadd 39086 | Vector addition in the dua... |
| ldualvadd 39087 | Vector addition in the dua... |
| ldualvaddcl 39088 | The value of vector additi... |
| ldualvaddval 39089 | The value of the value of ... |
| ldualsca 39090 | The ring of scalars of the... |
| ldualsbase 39091 | Base set of scalar ring fo... |
| ldualsaddN 39092 | Scalar addition for the du... |
| ldualsmul 39093 | Scalar multiplication for ... |
| ldualfvs 39094 | Scalar product operation f... |
| ldualvs 39095 | Scalar product operation v... |
| ldualvsval 39096 | Value of scalar product op... |
| ldualvscl 39097 | The scalar product operati... |
| ldualvaddcom 39098 | Commutative law for vector... |
| ldualvsass 39099 | Associative law for scalar... |
| ldualvsass2 39100 | Associative law for scalar... |
| ldualvsdi1 39101 | Distributive law for scala... |
| ldualvsdi2 39102 | Reverse distributive law f... |
| ldualgrplem 39103 | Lemma for ~ ldualgrp . (C... |
| ldualgrp 39104 | The dual of a vector space... |
| ldual0 39105 | The zero scalar of the dua... |
| ldual1 39106 | The unit scalar of the dua... |
| ldualneg 39107 | The negative of a scalar o... |
| ldual0v 39108 | The zero vector of the dua... |
| ldual0vcl 39109 | The dual zero vector is a ... |
| lduallmodlem 39110 | Lemma for ~ lduallmod . (... |
| lduallmod 39111 | The dual of a left module ... |
| lduallvec 39112 | The dual of a left vector ... |
| ldualvsub 39113 | The value of vector subtra... |
| ldualvsubcl 39114 | Closure of vector subtract... |
| ldualvsubval 39115 | The value of the value of ... |
| ldualssvscl 39116 | Closure of scalar product ... |
| ldualssvsubcl 39117 | Closure of vector subtract... |
| ldual0vs 39118 | Scalar zero times a functi... |
| lkr0f2 39119 | The kernel of the zero fun... |
| lduallkr3 39120 | The kernels of nonzero fun... |
| lkrpssN 39121 | Proper subset relation bet... |
| lkrin 39122 | Intersection of the kernel... |
| eqlkr4 39123 | Two functionals with the s... |
| ldual1dim 39124 | Equivalent expressions for... |
| ldualkrsc 39125 | The kernel of a nonzero sc... |
| lkrss 39126 | The kernel of a scalar pro... |
| lkrss2N 39127 | Two functionals with kerne... |
| lkreqN 39128 | Proportional functionals h... |
| lkrlspeqN 39129 | Condition for colinear fun... |
| isopos 39138 | The predicate "is an ortho... |
| opposet 39139 | Every orthoposet is a pose... |
| oposlem 39140 | Lemma for orthoposet prope... |
| op01dm 39141 | Conditions necessary for z... |
| op0cl 39142 | An orthoposet has a zero e... |
| op1cl 39143 | An orthoposet has a unity ... |
| op0le 39144 | Orthoposet zero is less th... |
| ople0 39145 | An element less than or eq... |
| opnlen0 39146 | An element not less than a... |
| lub0N 39147 | The least upper bound of t... |
| opltn0 39148 | A lattice element greater ... |
| ople1 39149 | Any element is less than t... |
| op1le 39150 | If the orthoposet unity is... |
| glb0N 39151 | The greatest lower bound o... |
| opoccl 39152 | Closure of orthocomplement... |
| opococ 39153 | Double negative law for or... |
| opcon3b 39154 | Contraposition law for ort... |
| opcon2b 39155 | Orthocomplement contraposi... |
| opcon1b 39156 | Orthocomplement contraposi... |
| oplecon3 39157 | Contraposition law for ort... |
| oplecon3b 39158 | Contraposition law for ort... |
| oplecon1b 39159 | Contraposition law for str... |
| opoc1 39160 | Orthocomplement of orthopo... |
| opoc0 39161 | Orthocomplement of orthopo... |
| opltcon3b 39162 | Contraposition law for str... |
| opltcon1b 39163 | Contraposition law for str... |
| opltcon2b 39164 | Contraposition law for str... |
| opexmid 39165 | Law of excluded middle for... |
| opnoncon 39166 | Law of contradiction for o... |
| riotaocN 39167 | The orthocomplement of the... |
| cmtfvalN 39168 | Value of commutes relation... |
| cmtvalN 39169 | Equivalence for commutes r... |
| isolat 39170 | The predicate "is an ortho... |
| ollat 39171 | An ortholattice is a latti... |
| olop 39172 | An ortholattice is an orth... |
| olposN 39173 | An ortholattice is a poset... |
| isolatiN 39174 | Properties that determine ... |
| oldmm1 39175 | De Morgan's law for meet i... |
| oldmm2 39176 | De Morgan's law for meet i... |
| oldmm3N 39177 | De Morgan's law for meet i... |
| oldmm4 39178 | De Morgan's law for meet i... |
| oldmj1 39179 | De Morgan's law for join i... |
| oldmj2 39180 | De Morgan's law for join i... |
| oldmj3 39181 | De Morgan's law for join i... |
| oldmj4 39182 | De Morgan's law for join i... |
| olj01 39183 | An ortholattice element jo... |
| olj02 39184 | An ortholattice element jo... |
| olm11 39185 | The meet of an ortholattic... |
| olm12 39186 | The meet of an ortholattic... |
| latmassOLD 39187 | Ortholattice meet is assoc... |
| latm12 39188 | A rearrangement of lattice... |
| latm32 39189 | A rearrangement of lattice... |
| latmrot 39190 | Rotate lattice meet of 3 c... |
| latm4 39191 | Rearrangement of lattice m... |
| latmmdiN 39192 | Lattice meet distributes o... |
| latmmdir 39193 | Lattice meet distributes o... |
| olm01 39194 | Meet with lattice zero is ... |
| olm02 39195 | Meet with lattice zero is ... |
| isoml 39196 | The predicate "is an ortho... |
| isomliN 39197 | Properties that determine ... |
| omlol 39198 | An orthomodular lattice is... |
| omlop 39199 | An orthomodular lattice is... |
| omllat 39200 | An orthomodular lattice is... |
| omllaw 39201 | The orthomodular law. (Co... |
| omllaw2N 39202 | Variation of orthomodular ... |
| omllaw3 39203 | Orthomodular law equivalen... |
| omllaw4 39204 | Orthomodular law equivalen... |
| omllaw5N 39205 | The orthomodular law. Rem... |
| cmtcomlemN 39206 | Lemma for ~ cmtcomN . ( ~... |
| cmtcomN 39207 | Commutation is symmetric. ... |
| cmt2N 39208 | Commutation with orthocomp... |
| cmt3N 39209 | Commutation with orthocomp... |
| cmt4N 39210 | Commutation with orthocomp... |
| cmtbr2N 39211 | Alternate definition of th... |
| cmtbr3N 39212 | Alternate definition for t... |
| cmtbr4N 39213 | Alternate definition for t... |
| lecmtN 39214 | Ordered elements commute. ... |
| cmtidN 39215 | Any element commutes with ... |
| omlfh1N 39216 | Foulis-Holland Theorem, pa... |
| omlfh3N 39217 | Foulis-Holland Theorem, pa... |
| omlmod1i2N 39218 | Analogue of modular law ~ ... |
| omlspjN 39219 | Contraction of a Sasaki pr... |
| cvrfval 39226 | Value of covers relation "... |
| cvrval 39227 | Binary relation expressing... |
| cvrlt 39228 | The covers relation implie... |
| cvrnbtwn 39229 | There is no element betwee... |
| ncvr1 39230 | No element covers the latt... |
| cvrletrN 39231 | Property of an element abo... |
| cvrval2 39232 | Binary relation expressing... |
| cvrnbtwn2 39233 | The covers relation implie... |
| cvrnbtwn3 39234 | The covers relation implie... |
| cvrcon3b 39235 | Contraposition law for the... |
| cvrle 39236 | The covers relation implie... |
| cvrnbtwn4 39237 | The covers relation implie... |
| cvrnle 39238 | The covers relation implie... |
| cvrne 39239 | The covers relation implie... |
| cvrnrefN 39240 | The covers relation is not... |
| cvrcmp 39241 | If two lattice elements th... |
| cvrcmp2 39242 | If two lattice elements co... |
| pats 39243 | The set of atoms in a pose... |
| isat 39244 | The predicate "is an atom"... |
| isat2 39245 | The predicate "is an atom"... |
| atcvr0 39246 | An atom covers zero. ( ~ ... |
| atbase 39247 | An atom is a member of the... |
| atssbase 39248 | The set of atoms is a subs... |
| 0ltat 39249 | An atom is greater than ze... |
| leatb 39250 | A poset element less than ... |
| leat 39251 | A poset element less than ... |
| leat2 39252 | A nonzero poset element le... |
| leat3 39253 | A poset element less than ... |
| meetat 39254 | The meet of any element wi... |
| meetat2 39255 | The meet of any element wi... |
| isatl 39257 | The predicate "is an atomi... |
| atllat 39258 | An atomic lattice is a lat... |
| atlpos 39259 | An atomic lattice is a pos... |
| atl0dm 39260 | Condition necessary for ze... |
| atl0cl 39261 | An atomic lattice has a ze... |
| atl0le 39262 | Orthoposet zero is less th... |
| atlle0 39263 | An element less than or eq... |
| atlltn0 39264 | A lattice element greater ... |
| isat3 39265 | The predicate "is an atom"... |
| atn0 39266 | An atom is not zero. ( ~ ... |
| atnle0 39267 | An atom is not less than o... |
| atlen0 39268 | A lattice element is nonze... |
| atcmp 39269 | If two atoms are comparabl... |
| atncmp 39270 | Frequently-used variation ... |
| atnlt 39271 | Two atoms cannot satisfy t... |
| atcvreq0 39272 | An element covered by an a... |
| atncvrN 39273 | Two atoms cannot satisfy t... |
| atlex 39274 | Every nonzero element of a... |
| atnle 39275 | Two ways of expressing "an... |
| atnem0 39276 | The meet of distinct atoms... |
| atlatmstc 39277 | An atomic, complete, ortho... |
| atlatle 39278 | The ordering of two Hilber... |
| atlrelat1 39279 | An atomistic lattice with ... |
| iscvlat 39281 | The predicate "is an atomi... |
| iscvlat2N 39282 | The predicate "is an atomi... |
| cvlatl 39283 | An atomic lattice with the... |
| cvllat 39284 | An atomic lattice with the... |
| cvlposN 39285 | An atomic lattice with the... |
| cvlexch1 39286 | An atomic covering lattice... |
| cvlexch2 39287 | An atomic covering lattice... |
| cvlexchb1 39288 | An atomic covering lattice... |
| cvlexchb2 39289 | An atomic covering lattice... |
| cvlexch3 39290 | An atomic covering lattice... |
| cvlexch4N 39291 | An atomic covering lattice... |
| cvlatexchb1 39292 | A version of ~ cvlexchb1 f... |
| cvlatexchb2 39293 | A version of ~ cvlexchb2 f... |
| cvlatexch1 39294 | Atom exchange property. (... |
| cvlatexch2 39295 | Atom exchange property. (... |
| cvlatexch3 39296 | Atom exchange property. (... |
| cvlcvr1 39297 | The covering property. Pr... |
| cvlcvrp 39298 | A Hilbert lattice satisfie... |
| cvlatcvr1 39299 | An atom is covered by its ... |
| cvlatcvr2 39300 | An atom is covered by its ... |
| cvlsupr2 39301 | Two equivalent ways of exp... |
| cvlsupr3 39302 | Two equivalent ways of exp... |
| cvlsupr4 39303 | Consequence of superpositi... |
| cvlsupr5 39304 | Consequence of superpositi... |
| cvlsupr6 39305 | Consequence of superpositi... |
| cvlsupr7 39306 | Consequence of superpositi... |
| cvlsupr8 39307 | Consequence of superpositi... |
| ishlat1 39310 | The predicate "is a Hilber... |
| ishlat2 39311 | The predicate "is a Hilber... |
| ishlat3N 39312 | The predicate "is a Hilber... |
| ishlatiN 39313 | Properties that determine ... |
| hlomcmcv 39314 | A Hilbert lattice is ortho... |
| hloml 39315 | A Hilbert lattice is ortho... |
| hlclat 39316 | A Hilbert lattice is compl... |
| hlcvl 39317 | A Hilbert lattice is an at... |
| hlatl 39318 | A Hilbert lattice is atomi... |
| hlol 39319 | A Hilbert lattice is an or... |
| hlop 39320 | A Hilbert lattice is an or... |
| hllat 39321 | A Hilbert lattice is a lat... |
| hllatd 39322 | Deduction form of ~ hllat ... |
| hlomcmat 39323 | A Hilbert lattice is ortho... |
| hlpos 39324 | A Hilbert lattice is a pos... |
| hlatjcl 39325 | Closure of join operation.... |
| hlatjcom 39326 | Commutatitivity of join op... |
| hlatjidm 39327 | Idempotence of join operat... |
| hlatjass 39328 | Lattice join is associativ... |
| hlatj12 39329 | Swap 1st and 2nd members o... |
| hlatj32 39330 | Swap 2nd and 3rd members o... |
| hlatjrot 39331 | Rotate lattice join of 3 c... |
| hlatj4 39332 | Rearrangement of lattice j... |
| hlatlej1 39333 | A join's first argument is... |
| hlatlej2 39334 | A join's second argument i... |
| glbconN 39335 | De Morgan's law for GLB an... |
| glbconNOLD 39336 | Obsolete version of ~ glbc... |
| glbconxN 39337 | De Morgan's law for GLB an... |
| atnlej1 39338 | If an atom is not less tha... |
| atnlej2 39339 | If an atom is not less tha... |
| hlsuprexch 39340 | A Hilbert lattice has the ... |
| hlexch1 39341 | A Hilbert lattice has the ... |
| hlexch2 39342 | A Hilbert lattice has the ... |
| hlexchb1 39343 | A Hilbert lattice has the ... |
| hlexchb2 39344 | A Hilbert lattice has the ... |
| hlsupr 39345 | A Hilbert lattice has the ... |
| hlsupr2 39346 | A Hilbert lattice has the ... |
| hlhgt4 39347 | A Hilbert lattice has a he... |
| hlhgt2 39348 | A Hilbert lattice has a he... |
| hl0lt1N 39349 | Lattice 0 is less than lat... |
| hlexch3 39350 | A Hilbert lattice has the ... |
| hlexch4N 39351 | A Hilbert lattice has the ... |
| hlatexchb1 39352 | A version of ~ hlexchb1 fo... |
| hlatexchb2 39353 | A version of ~ hlexchb2 fo... |
| hlatexch1 39354 | Atom exchange property. (... |
| hlatexch2 39355 | Atom exchange property. (... |
| hlatmstcOLDN 39356 | An atomic, complete, ortho... |
| hlatle 39357 | The ordering of two Hilber... |
| hlateq 39358 | The equality of two Hilber... |
| hlrelat1 39359 | An atomistic lattice with ... |
| hlrelat5N 39360 | An atomistic lattice with ... |
| hlrelat 39361 | A Hilbert lattice is relat... |
| hlrelat2 39362 | A consequence of relative ... |
| exatleN 39363 | A condition for an atom to... |
| hl2at 39364 | A Hilbert lattice has at l... |
| atex 39365 | At least one atom exists. ... |
| intnatN 39366 | If the intersection with a... |
| 2llnne2N 39367 | Condition implying that tw... |
| 2llnneN 39368 | Condition implying that tw... |
| cvr1 39369 | A Hilbert lattice has the ... |
| cvr2N 39370 | Less-than and covers equiv... |
| hlrelat3 39371 | The Hilbert lattice is rel... |
| cvrval3 39372 | Binary relation expressing... |
| cvrval4N 39373 | Binary relation expressing... |
| cvrval5 39374 | Binary relation expressing... |
| cvrp 39375 | A Hilbert lattice satisfie... |
| atcvr1 39376 | An atom is covered by its ... |
| atcvr2 39377 | An atom is covered by its ... |
| cvrexchlem 39378 | Lemma for ~ cvrexch . ( ~... |
| cvrexch 39379 | A Hilbert lattice satisfie... |
| cvratlem 39380 | Lemma for ~ cvrat . ( ~ a... |
| cvrat 39381 | A nonzero Hilbert lattice ... |
| ltltncvr 39382 | A chained strong ordering ... |
| ltcvrntr 39383 | Non-transitive condition f... |
| cvrntr 39384 | The covers relation is not... |
| atcvr0eq 39385 | The covers relation is not... |
| lnnat 39386 | A line (the join of two di... |
| atcvrj0 39387 | Two atoms covering the zer... |
| cvrat2 39388 | A Hilbert lattice element ... |
| atcvrneN 39389 | Inequality derived from at... |
| atcvrj1 39390 | Condition for an atom to b... |
| atcvrj2b 39391 | Condition for an atom to b... |
| atcvrj2 39392 | Condition for an atom to b... |
| atleneN 39393 | Inequality derived from at... |
| atltcvr 39394 | An equivalence of less-tha... |
| atle 39395 | Any nonzero element has an... |
| atlt 39396 | Two atoms are unequal iff ... |
| atlelt 39397 | Transfer less-than relatio... |
| 2atlt 39398 | Given an atom less than an... |
| atexchcvrN 39399 | Atom exchange property. V... |
| atexchltN 39400 | Atom exchange property. V... |
| cvrat3 39401 | A condition implying that ... |
| cvrat4 39402 | A condition implying exist... |
| cvrat42 39403 | Commuted version of ~ cvra... |
| 2atjm 39404 | The meet of a line (expres... |
| atbtwn 39405 | Property of a 3rd atom ` R... |
| atbtwnexOLDN 39406 | There exists a 3rd atom ` ... |
| atbtwnex 39407 | Given atoms ` P ` in ` X `... |
| 3noncolr2 39408 | Two ways to express 3 non-... |
| 3noncolr1N 39409 | Two ways to express 3 non-... |
| hlatcon3 39410 | Atom exchange combined wit... |
| hlatcon2 39411 | Atom exchange combined wit... |
| 4noncolr3 39412 | A way to express 4 non-col... |
| 4noncolr2 39413 | A way to express 4 non-col... |
| 4noncolr1 39414 | A way to express 4 non-col... |
| athgt 39415 | A Hilbert lattice, whose h... |
| 3dim0 39416 | There exists a 3-dimension... |
| 3dimlem1 39417 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem2 39418 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3a 39419 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem3 39420 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3OLDN 39421 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4a 39422 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem4 39423 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4OLDN 39424 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1lem5 39425 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1 39426 | Construct a 3-dimensional ... |
| 3dim2 39427 | Construct 2 new layers on ... |
| 3dim3 39428 | Construct a new layer on t... |
| 2dim 39429 | Generate a height-3 elemen... |
| 1dimN 39430 | An atom is covered by a he... |
| 1cvrco 39431 | The orthocomplement of an ... |
| 1cvratex 39432 | There exists an atom less ... |
| 1cvratlt 39433 | An atom less than or equal... |
| 1cvrjat 39434 | An element covered by the ... |
| 1cvrat 39435 | Create an atom under an el... |
| ps-1 39436 | The join of two atoms ` R ... |
| ps-2 39437 | Lattice analogue for the p... |
| 2atjlej 39438 | Two atoms are different if... |
| hlatexch3N 39439 | Rearrange join of atoms in... |
| hlatexch4 39440 | Exchange 2 atoms. (Contri... |
| ps-2b 39441 | Variation of projective ge... |
| 3atlem1 39442 | Lemma for ~ 3at . (Contri... |
| 3atlem2 39443 | Lemma for ~ 3at . (Contri... |
| 3atlem3 39444 | Lemma for ~ 3at . (Contri... |
| 3atlem4 39445 | Lemma for ~ 3at . (Contri... |
| 3atlem5 39446 | Lemma for ~ 3at . (Contri... |
| 3atlem6 39447 | Lemma for ~ 3at . (Contri... |
| 3atlem7 39448 | Lemma for ~ 3at . (Contri... |
| 3at 39449 | Any three non-colinear ato... |
| llnset 39464 | The set of lattice lines i... |
| islln 39465 | The predicate "is a lattic... |
| islln4 39466 | The predicate "is a lattic... |
| llni 39467 | Condition implying a latti... |
| llnbase 39468 | A lattice line is a lattic... |
| islln3 39469 | The predicate "is a lattic... |
| islln2 39470 | The predicate "is a lattic... |
| llni2 39471 | The join of two different ... |
| llnnleat 39472 | An atom cannot majorize a ... |
| llnneat 39473 | A lattice line is not an a... |
| 2atneat 39474 | The join of two distinct a... |
| llnn0 39475 | A lattice line is nonzero.... |
| islln2a 39476 | The predicate "is a lattic... |
| llnle 39477 | Any element greater than 0... |
| atcvrlln2 39478 | An atom under a line is co... |
| atcvrlln 39479 | An element covering an ato... |
| llnexatN 39480 | Given an atom on a line, t... |
| llncmp 39481 | If two lattice lines are c... |
| llnnlt 39482 | Two lattice lines cannot s... |
| 2llnmat 39483 | Two intersecting lines int... |
| 2at0mat0 39484 | Special case of ~ 2atmat0 ... |
| 2atmat0 39485 | The meet of two unequal li... |
| 2atm 39486 | An atom majorized by two d... |
| ps-2c 39487 | Variation of projective ge... |
| lplnset 39488 | The set of lattice planes ... |
| islpln 39489 | The predicate "is a lattic... |
| islpln4 39490 | The predicate "is a lattic... |
| lplni 39491 | Condition implying a latti... |
| islpln3 39492 | The predicate "is a lattic... |
| lplnbase 39493 | A lattice plane is a latti... |
| islpln5 39494 | The predicate "is a lattic... |
| islpln2 39495 | The predicate "is a lattic... |
| lplni2 39496 | The join of 3 different at... |
| lvolex3N 39497 | There is an atom outside o... |
| llnmlplnN 39498 | The intersection of a line... |
| lplnle 39499 | Any element greater than 0... |
| lplnnle2at 39500 | A lattice line (or atom) c... |
| lplnnleat 39501 | A lattice plane cannot maj... |
| lplnnlelln 39502 | A lattice plane is not les... |
| 2atnelpln 39503 | The join of two atoms is n... |
| lplnneat 39504 | No lattice plane is an ato... |
| lplnnelln 39505 | No lattice plane is a latt... |
| lplnn0N 39506 | A lattice plane is nonzero... |
| islpln2a 39507 | The predicate "is a lattic... |
| islpln2ah 39508 | The predicate "is a lattic... |
| lplnriaN 39509 | Property of a lattice plan... |
| lplnribN 39510 | Property of a lattice plan... |
| lplnric 39511 | Property of a lattice plan... |
| lplnri1 39512 | Property of a lattice plan... |
| lplnri2N 39513 | Property of a lattice plan... |
| lplnri3N 39514 | Property of a lattice plan... |
| lplnllnneN 39515 | Two lattice lines defined ... |
| llncvrlpln2 39516 | A lattice line under a lat... |
| llncvrlpln 39517 | An element covering a latt... |
| 2lplnmN 39518 | If the join of two lattice... |
| 2llnmj 39519 | The meet of two lattice li... |
| 2atmat 39520 | The meet of two intersecti... |
| lplncmp 39521 | If two lattice planes are ... |
| lplnexatN 39522 | Given a lattice line on a ... |
| lplnexllnN 39523 | Given an atom on a lattice... |
| lplnnlt 39524 | Two lattice planes cannot ... |
| 2llnjaN 39525 | The join of two different ... |
| 2llnjN 39526 | The join of two different ... |
| 2llnm2N 39527 | The meet of two different ... |
| 2llnm3N 39528 | Two lattice lines in a lat... |
| 2llnm4 39529 | Two lattice lines that maj... |
| 2llnmeqat 39530 | An atom equals the interse... |
| lvolset 39531 | The set of 3-dim lattice v... |
| islvol 39532 | The predicate "is a 3-dim ... |
| islvol4 39533 | The predicate "is a 3-dim ... |
| lvoli 39534 | Condition implying a 3-dim... |
| islvol3 39535 | The predicate "is a 3-dim ... |
| lvoli3 39536 | Condition implying a 3-dim... |
| lvolbase 39537 | A 3-dim lattice volume is ... |
| islvol5 39538 | The predicate "is a 3-dim ... |
| islvol2 39539 | The predicate "is a 3-dim ... |
| lvoli2 39540 | The join of 4 different at... |
| lvolnle3at 39541 | A lattice plane (or lattic... |
| lvolnleat 39542 | An atom cannot majorize a ... |
| lvolnlelln 39543 | A lattice line cannot majo... |
| lvolnlelpln 39544 | A lattice plane cannot maj... |
| 3atnelvolN 39545 | The join of 3 atoms is not... |
| 2atnelvolN 39546 | The join of two atoms is n... |
| lvolneatN 39547 | No lattice volume is an at... |
| lvolnelln 39548 | No lattice volume is a lat... |
| lvolnelpln 39549 | No lattice volume is a lat... |
| lvoln0N 39550 | A lattice volume is nonzer... |
| islvol2aN 39551 | The predicate "is a lattic... |
| 4atlem0a 39552 | Lemma for ~ 4at . (Contri... |
| 4atlem0ae 39553 | Lemma for ~ 4at . (Contri... |
| 4atlem0be 39554 | Lemma for ~ 4at . (Contri... |
| 4atlem3 39555 | Lemma for ~ 4at . Break i... |
| 4atlem3a 39556 | Lemma for ~ 4at . Break i... |
| 4atlem3b 39557 | Lemma for ~ 4at . Break i... |
| 4atlem4a 39558 | Lemma for ~ 4at . Frequen... |
| 4atlem4b 39559 | Lemma for ~ 4at . Frequen... |
| 4atlem4c 39560 | Lemma for ~ 4at . Frequen... |
| 4atlem4d 39561 | Lemma for ~ 4at . Frequen... |
| 4atlem9 39562 | Lemma for ~ 4at . Substit... |
| 4atlem10a 39563 | Lemma for ~ 4at . Substit... |
| 4atlem10b 39564 | Lemma for ~ 4at . Substit... |
| 4atlem10 39565 | Lemma for ~ 4at . Combine... |
| 4atlem11a 39566 | Lemma for ~ 4at . Substit... |
| 4atlem11b 39567 | Lemma for ~ 4at . Substit... |
| 4atlem11 39568 | Lemma for ~ 4at . Combine... |
| 4atlem12a 39569 | Lemma for ~ 4at . Substit... |
| 4atlem12b 39570 | Lemma for ~ 4at . Substit... |
| 4atlem12 39571 | Lemma for ~ 4at . Combine... |
| 4at 39572 | Four atoms determine a lat... |
| 4at2 39573 | Four atoms determine a lat... |
| lplncvrlvol2 39574 | A lattice line under a lat... |
| lplncvrlvol 39575 | An element covering a latt... |
| lvolcmp 39576 | If two lattice planes are ... |
| lvolnltN 39577 | Two lattice volumes cannot... |
| 2lplnja 39578 | The join of two different ... |
| 2lplnj 39579 | The join of two different ... |
| 2lplnm2N 39580 | The meet of two different ... |
| 2lplnmj 39581 | The meet of two lattice pl... |
| dalemkehl 39582 | Lemma for ~ dath . Freque... |
| dalemkelat 39583 | Lemma for ~ dath . Freque... |
| dalemkeop 39584 | Lemma for ~ dath . Freque... |
| dalempea 39585 | Lemma for ~ dath . Freque... |
| dalemqea 39586 | Lemma for ~ dath . Freque... |
| dalemrea 39587 | Lemma for ~ dath . Freque... |
| dalemsea 39588 | Lemma for ~ dath . Freque... |
| dalemtea 39589 | Lemma for ~ dath . Freque... |
| dalemuea 39590 | Lemma for ~ dath . Freque... |
| dalemyeo 39591 | Lemma for ~ dath . Freque... |
| dalemzeo 39592 | Lemma for ~ dath . Freque... |
| dalemclpjs 39593 | Lemma for ~ dath . Freque... |
| dalemclqjt 39594 | Lemma for ~ dath . Freque... |
| dalemclrju 39595 | Lemma for ~ dath . Freque... |
| dalem-clpjq 39596 | Lemma for ~ dath . Freque... |
| dalemceb 39597 | Lemma for ~ dath . Freque... |
| dalempeb 39598 | Lemma for ~ dath . Freque... |
| dalemqeb 39599 | Lemma for ~ dath . Freque... |
| dalemreb 39600 | Lemma for ~ dath . Freque... |
| dalemseb 39601 | Lemma for ~ dath . Freque... |
| dalemteb 39602 | Lemma for ~ dath . Freque... |
| dalemueb 39603 | Lemma for ~ dath . Freque... |
| dalempjqeb 39604 | Lemma for ~ dath . Freque... |
| dalemsjteb 39605 | Lemma for ~ dath . Freque... |
| dalemtjueb 39606 | Lemma for ~ dath . Freque... |
| dalemqrprot 39607 | Lemma for ~ dath . Freque... |
| dalemyeb 39608 | Lemma for ~ dath . Freque... |
| dalemcnes 39609 | Lemma for ~ dath . Freque... |
| dalempnes 39610 | Lemma for ~ dath . Freque... |
| dalemqnet 39611 | Lemma for ~ dath . Freque... |
| dalempjsen 39612 | Lemma for ~ dath . Freque... |
| dalemply 39613 | Lemma for ~ dath . Freque... |
| dalemsly 39614 | Lemma for ~ dath . Freque... |
| dalemswapyz 39615 | Lemma for ~ dath . Swap t... |
| dalemrot 39616 | Lemma for ~ dath . Rotate... |
| dalemrotyz 39617 | Lemma for ~ dath . Rotate... |
| dalem1 39618 | Lemma for ~ dath . Show t... |
| dalemcea 39619 | Lemma for ~ dath . Freque... |
| dalem2 39620 | Lemma for ~ dath . Show t... |
| dalemdea 39621 | Lemma for ~ dath . Freque... |
| dalemeea 39622 | Lemma for ~ dath . Freque... |
| dalem3 39623 | Lemma for ~ dalemdnee . (... |
| dalem4 39624 | Lemma for ~ dalemdnee . (... |
| dalemdnee 39625 | Lemma for ~ dath . Axis o... |
| dalem5 39626 | Lemma for ~ dath . Atom `... |
| dalem6 39627 | Lemma for ~ dath . Analog... |
| dalem7 39628 | Lemma for ~ dath . Analog... |
| dalem8 39629 | Lemma for ~ dath . Plane ... |
| dalem-cly 39630 | Lemma for ~ dalem9 . Cent... |
| dalem9 39631 | Lemma for ~ dath . Since ... |
| dalem10 39632 | Lemma for ~ dath . Atom `... |
| dalem11 39633 | Lemma for ~ dath . Analog... |
| dalem12 39634 | Lemma for ~ dath . Analog... |
| dalem13 39635 | Lemma for ~ dalem14 . (Co... |
| dalem14 39636 | Lemma for ~ dath . Planes... |
| dalem15 39637 | Lemma for ~ dath . The ax... |
| dalem16 39638 | Lemma for ~ dath . The at... |
| dalem17 39639 | Lemma for ~ dath . When p... |
| dalem18 39640 | Lemma for ~ dath . Show t... |
| dalem19 39641 | Lemma for ~ dath . Show t... |
| dalemccea 39642 | Lemma for ~ dath . Freque... |
| dalemddea 39643 | Lemma for ~ dath . Freque... |
| dalem-ccly 39644 | Lemma for ~ dath . Freque... |
| dalem-ddly 39645 | Lemma for ~ dath . Freque... |
| dalemccnedd 39646 | Lemma for ~ dath . Freque... |
| dalemclccjdd 39647 | Lemma for ~ dath . Freque... |
| dalemcceb 39648 | Lemma for ~ dath . Freque... |
| dalemswapyzps 39649 | Lemma for ~ dath . Swap t... |
| dalemrotps 39650 | Lemma for ~ dath . Rotate... |
| dalemcjden 39651 | Lemma for ~ dath . Show t... |
| dalem20 39652 | Lemma for ~ dath . Show t... |
| dalem21 39653 | Lemma for ~ dath . Show t... |
| dalem22 39654 | Lemma for ~ dath . Show t... |
| dalem23 39655 | Lemma for ~ dath . Show t... |
| dalem24 39656 | Lemma for ~ dath . Show t... |
| dalem25 39657 | Lemma for ~ dath . Show t... |
| dalem27 39658 | Lemma for ~ dath . Show t... |
| dalem28 39659 | Lemma for ~ dath . Lemma ... |
| dalem29 39660 | Lemma for ~ dath . Analog... |
| dalem30 39661 | Lemma for ~ dath . Analog... |
| dalem31N 39662 | Lemma for ~ dath . Analog... |
| dalem32 39663 | Lemma for ~ dath . Analog... |
| dalem33 39664 | Lemma for ~ dath . Analog... |
| dalem34 39665 | Lemma for ~ dath . Analog... |
| dalem35 39666 | Lemma for ~ dath . Analog... |
| dalem36 39667 | Lemma for ~ dath . Analog... |
| dalem37 39668 | Lemma for ~ dath . Analog... |
| dalem38 39669 | Lemma for ~ dath . Plane ... |
| dalem39 39670 | Lemma for ~ dath . Auxili... |
| dalem40 39671 | Lemma for ~ dath . Analog... |
| dalem41 39672 | Lemma for ~ dath . (Contr... |
| dalem42 39673 | Lemma for ~ dath . Auxili... |
| dalem43 39674 | Lemma for ~ dath . Planes... |
| dalem44 39675 | Lemma for ~ dath . Dummy ... |
| dalem45 39676 | Lemma for ~ dath . Dummy ... |
| dalem46 39677 | Lemma for ~ dath . Analog... |
| dalem47 39678 | Lemma for ~ dath . Analog... |
| dalem48 39679 | Lemma for ~ dath . Analog... |
| dalem49 39680 | Lemma for ~ dath . Analog... |
| dalem50 39681 | Lemma for ~ dath . Analog... |
| dalem51 39682 | Lemma for ~ dath . Constr... |
| dalem52 39683 | Lemma for ~ dath . Lines ... |
| dalem53 39684 | Lemma for ~ dath . The au... |
| dalem54 39685 | Lemma for ~ dath . Line `... |
| dalem55 39686 | Lemma for ~ dath . Lines ... |
| dalem56 39687 | Lemma for ~ dath . Analog... |
| dalem57 39688 | Lemma for ~ dath . Axis o... |
| dalem58 39689 | Lemma for ~ dath . Analog... |
| dalem59 39690 | Lemma for ~ dath . Analog... |
| dalem60 39691 | Lemma for ~ dath . ` B ` i... |
| dalem61 39692 | Lemma for ~ dath . Show t... |
| dalem62 39693 | Lemma for ~ dath . Elimin... |
| dalem63 39694 | Lemma for ~ dath . Combin... |
| dath 39695 | Desargues's theorem of pro... |
| dath2 39696 | Version of Desargues's the... |
| lineset 39697 | The set of lines in a Hilb... |
| isline 39698 | The predicate "is a line".... |
| islinei 39699 | Condition implying "is a l... |
| pointsetN 39700 | The set of points in a Hil... |
| ispointN 39701 | The predicate "is a point"... |
| atpointN 39702 | The singleton of an atom i... |
| psubspset 39703 | The set of projective subs... |
| ispsubsp 39704 | The predicate "is a projec... |
| ispsubsp2 39705 | The predicate "is a projec... |
| psubspi 39706 | Property of a projective s... |
| psubspi2N 39707 | Property of a projective s... |
| 0psubN 39708 | The empty set is a project... |
| snatpsubN 39709 | The singleton of an atom i... |
| pointpsubN 39710 | A point (singleton of an a... |
| linepsubN 39711 | A line is a projective sub... |
| atpsubN 39712 | The set of all atoms is a ... |
| psubssat 39713 | A projective subspace cons... |
| psubatN 39714 | A member of a projective s... |
| pmapfval 39715 | The projective map of a Hi... |
| pmapval 39716 | Value of the projective ma... |
| elpmap 39717 | Member of a projective map... |
| pmapssat 39718 | The projective map of a Hi... |
| pmapssbaN 39719 | A weakening of ~ pmapssat ... |
| pmaple 39720 | The projective map of a Hi... |
| pmap11 39721 | The projective map of a Hi... |
| pmapat 39722 | The projective map of an a... |
| elpmapat 39723 | Member of the projective m... |
| pmap0 39724 | Value of the projective ma... |
| pmapeq0 39725 | A projective map value is ... |
| pmap1N 39726 | Value of the projective ma... |
| pmapsub 39727 | The projective map of a Hi... |
| pmapglbx 39728 | The projective map of the ... |
| pmapglb 39729 | The projective map of the ... |
| pmapglb2N 39730 | The projective map of the ... |
| pmapglb2xN 39731 | The projective map of the ... |
| pmapmeet 39732 | The projective map of a me... |
| isline2 39733 | Definition of line in term... |
| linepmap 39734 | A line described with a pr... |
| isline3 39735 | Definition of line in term... |
| isline4N 39736 | Definition of line in term... |
| lneq2at 39737 | A line equals the join of ... |
| lnatexN 39738 | There is an atom in a line... |
| lnjatN 39739 | Given an atom in a line, t... |
| lncvrelatN 39740 | A lattice element covered ... |
| lncvrat 39741 | A line covers the atoms it... |
| lncmp 39742 | If two lines are comparabl... |
| 2lnat 39743 | Two intersecting lines int... |
| 2atm2atN 39744 | Two joins with a common at... |
| 2llnma1b 39745 | Generalization of ~ 2llnma... |
| 2llnma1 39746 | Two different intersecting... |
| 2llnma3r 39747 | Two different intersecting... |
| 2llnma2 39748 | Two different intersecting... |
| 2llnma2rN 39749 | Two different intersecting... |
| cdlema1N 39750 | A condition for required f... |
| cdlema2N 39751 | A condition for required f... |
| cdlemblem 39752 | Lemma for ~ cdlemb . (Con... |
| cdlemb 39753 | Given two atoms not less t... |
| paddfval 39756 | Projective subspace sum op... |
| paddval 39757 | Projective subspace sum op... |
| elpadd 39758 | Member of a projective sub... |
| elpaddn0 39759 | Member of projective subsp... |
| paddvaln0N 39760 | Projective subspace sum op... |
| elpaddri 39761 | Condition implying members... |
| elpaddatriN 39762 | Condition implying members... |
| elpaddat 39763 | Membership in a projective... |
| elpaddatiN 39764 | Consequence of membership ... |
| elpadd2at 39765 | Membership in a projective... |
| elpadd2at2 39766 | Membership in a projective... |
| paddunssN 39767 | Projective subspace sum in... |
| elpadd0 39768 | Member of projective subsp... |
| paddval0 39769 | Projective subspace sum wi... |
| padd01 39770 | Projective subspace sum wi... |
| padd02 39771 | Projective subspace sum wi... |
| paddcom 39772 | Projective subspace sum co... |
| paddssat 39773 | A projective subspace sum ... |
| sspadd1 39774 | A projective subspace sum ... |
| sspadd2 39775 | A projective subspace sum ... |
| paddss1 39776 | Subset law for projective ... |
| paddss2 39777 | Subset law for projective ... |
| paddss12 39778 | Subset law for projective ... |
| paddasslem1 39779 | Lemma for ~ paddass . (Co... |
| paddasslem2 39780 | Lemma for ~ paddass . (Co... |
| paddasslem3 39781 | Lemma for ~ paddass . Res... |
| paddasslem4 39782 | Lemma for ~ paddass . Com... |
| paddasslem5 39783 | Lemma for ~ paddass . Sho... |
| paddasslem6 39784 | Lemma for ~ paddass . (Co... |
| paddasslem7 39785 | Lemma for ~ paddass . Com... |
| paddasslem8 39786 | Lemma for ~ paddass . (Co... |
| paddasslem9 39787 | Lemma for ~ paddass . Com... |
| paddasslem10 39788 | Lemma for ~ paddass . Use... |
| paddasslem11 39789 | Lemma for ~ paddass . The... |
| paddasslem12 39790 | Lemma for ~ paddass . The... |
| paddasslem13 39791 | Lemma for ~ paddass . The... |
| paddasslem14 39792 | Lemma for ~ paddass . Rem... |
| paddasslem15 39793 | Lemma for ~ paddass . Use... |
| paddasslem16 39794 | Lemma for ~ paddass . Use... |
| paddasslem17 39795 | Lemma for ~ paddass . The... |
| paddasslem18 39796 | Lemma for ~ paddass . Com... |
| paddass 39797 | Projective subspace sum is... |
| padd12N 39798 | Commutative/associative la... |
| padd4N 39799 | Rearrangement of 4 terms i... |
| paddidm 39800 | Projective subspace sum is... |
| paddclN 39801 | The projective sum of two ... |
| paddssw1 39802 | Subset law for projective ... |
| paddssw2 39803 | Subset law for projective ... |
| paddss 39804 | Subset law for projective ... |
| pmodlem1 39805 | Lemma for ~ pmod1i . (Con... |
| pmodlem2 39806 | Lemma for ~ pmod1i . (Con... |
| pmod1i 39807 | The modular law holds in a... |
| pmod2iN 39808 | Dual of the modular law. ... |
| pmodN 39809 | The modular law for projec... |
| pmodl42N 39810 | Lemma derived from modular... |
| pmapjoin 39811 | The projective map of the ... |
| pmapjat1 39812 | The projective map of the ... |
| pmapjat2 39813 | The projective map of the ... |
| pmapjlln1 39814 | The projective map of the ... |
| hlmod1i 39815 | A version of the modular l... |
| atmod1i1 39816 | Version of modular law ~ p... |
| atmod1i1m 39817 | Version of modular law ~ p... |
| atmod1i2 39818 | Version of modular law ~ p... |
| llnmod1i2 39819 | Version of modular law ~ p... |
| atmod2i1 39820 | Version of modular law ~ p... |
| atmod2i2 39821 | Version of modular law ~ p... |
| llnmod2i2 39822 | Version of modular law ~ p... |
| atmod3i1 39823 | Version of modular law tha... |
| atmod3i2 39824 | Version of modular law tha... |
| atmod4i1 39825 | Version of modular law tha... |
| atmod4i2 39826 | Version of modular law tha... |
| llnexchb2lem 39827 | Lemma for ~ llnexchb2 . (... |
| llnexchb2 39828 | Line exchange property (co... |
| llnexch2N 39829 | Line exchange property (co... |
| dalawlem1 39830 | Lemma for ~ dalaw . Speci... |
| dalawlem2 39831 | Lemma for ~ dalaw . Utili... |
| dalawlem3 39832 | Lemma for ~ dalaw . First... |
| dalawlem4 39833 | Lemma for ~ dalaw . Secon... |
| dalawlem5 39834 | Lemma for ~ dalaw . Speci... |
| dalawlem6 39835 | Lemma for ~ dalaw . First... |
| dalawlem7 39836 | Lemma for ~ dalaw . Secon... |
| dalawlem8 39837 | Lemma for ~ dalaw . Speci... |
| dalawlem9 39838 | Lemma for ~ dalaw . Speci... |
| dalawlem10 39839 | Lemma for ~ dalaw . Combi... |
| dalawlem11 39840 | Lemma for ~ dalaw . First... |
| dalawlem12 39841 | Lemma for ~ dalaw . Secon... |
| dalawlem13 39842 | Lemma for ~ dalaw . Speci... |
| dalawlem14 39843 | Lemma for ~ dalaw . Combi... |
| dalawlem15 39844 | Lemma for ~ dalaw . Swap ... |
| dalaw 39845 | Desargues's law, derived f... |
| pclfvalN 39848 | The projective subspace cl... |
| pclvalN 39849 | Value of the projective su... |
| pclclN 39850 | Closure of the projective ... |
| elpclN 39851 | Membership in the projecti... |
| elpcliN 39852 | Implication of membership ... |
| pclssN 39853 | Ordering is preserved by s... |
| pclssidN 39854 | A set of atoms is included... |
| pclidN 39855 | The projective subspace cl... |
| pclbtwnN 39856 | A projective subspace sand... |
| pclunN 39857 | The projective subspace cl... |
| pclun2N 39858 | The projective subspace cl... |
| pclfinN 39859 | The projective subspace cl... |
| pclcmpatN 39860 | The set of projective subs... |
| polfvalN 39863 | The projective subspace po... |
| polvalN 39864 | Value of the projective su... |
| polval2N 39865 | Alternate expression for v... |
| polsubN 39866 | The polarity of a set of a... |
| polssatN 39867 | The polarity of a set of a... |
| pol0N 39868 | The polarity of the empty ... |
| pol1N 39869 | The polarity of the whole ... |
| 2pol0N 39870 | The closed subspace closur... |
| polpmapN 39871 | The polarity of a projecti... |
| 2polpmapN 39872 | Double polarity of a proje... |
| 2polvalN 39873 | Value of double polarity. ... |
| 2polssN 39874 | A set of atoms is a subset... |
| 3polN 39875 | Triple polarity cancels to... |
| polcon3N 39876 | Contraposition law for pol... |
| 2polcon4bN 39877 | Contraposition law for pol... |
| polcon2N 39878 | Contraposition law for pol... |
| polcon2bN 39879 | Contraposition law for pol... |
| pclss2polN 39880 | The projective subspace cl... |
| pcl0N 39881 | The projective subspace cl... |
| pcl0bN 39882 | The projective subspace cl... |
| pmaplubN 39883 | The LUB of a projective ma... |
| sspmaplubN 39884 | A set of atoms is a subset... |
| 2pmaplubN 39885 | Double projective map of a... |
| paddunN 39886 | The closure of the project... |
| poldmj1N 39887 | De Morgan's law for polari... |
| pmapj2N 39888 | The projective map of the ... |
| pmapocjN 39889 | The projective map of the ... |
| polatN 39890 | The polarity of the single... |
| 2polatN 39891 | Double polarity of the sin... |
| pnonsingN 39892 | The intersection of a set ... |
| psubclsetN 39895 | The set of closed projecti... |
| ispsubclN 39896 | The predicate "is a closed... |
| psubcliN 39897 | Property of a closed proje... |
| psubcli2N 39898 | Property of a closed proje... |
| psubclsubN 39899 | A closed projective subspa... |
| psubclssatN 39900 | A closed projective subspa... |
| pmapidclN 39901 | Projective map of the LUB ... |
| 0psubclN 39902 | The empty set is a closed ... |
| 1psubclN 39903 | The set of all atoms is a ... |
| atpsubclN 39904 | A point (singleton of an a... |
| pmapsubclN 39905 | A projective map value is ... |
| ispsubcl2N 39906 | Alternate predicate for "i... |
| psubclinN 39907 | The intersection of two cl... |
| paddatclN 39908 | The projective sum of a cl... |
| pclfinclN 39909 | The projective subspace cl... |
| linepsubclN 39910 | A line is a closed project... |
| polsubclN 39911 | A polarity is a closed pro... |
| poml4N 39912 | Orthomodular law for proje... |
| poml5N 39913 | Orthomodular law for proje... |
| poml6N 39914 | Orthomodular law for proje... |
| osumcllem1N 39915 | Lemma for ~ osumclN . (Co... |
| osumcllem2N 39916 | Lemma for ~ osumclN . (Co... |
| osumcllem3N 39917 | Lemma for ~ osumclN . (Co... |
| osumcllem4N 39918 | Lemma for ~ osumclN . (Co... |
| osumcllem5N 39919 | Lemma for ~ osumclN . (Co... |
| osumcllem6N 39920 | Lemma for ~ osumclN . Use... |
| osumcllem7N 39921 | Lemma for ~ osumclN . (Co... |
| osumcllem8N 39922 | Lemma for ~ osumclN . (Co... |
| osumcllem9N 39923 | Lemma for ~ osumclN . (Co... |
| osumcllem10N 39924 | Lemma for ~ osumclN . Con... |
| osumcllem11N 39925 | Lemma for ~ osumclN . (Co... |
| osumclN 39926 | Closure of orthogonal sum.... |
| pmapojoinN 39927 | For orthogonal elements, p... |
| pexmidN 39928 | Excluded middle law for cl... |
| pexmidlem1N 39929 | Lemma for ~ pexmidN . Hol... |
| pexmidlem2N 39930 | Lemma for ~ pexmidN . (Co... |
| pexmidlem3N 39931 | Lemma for ~ pexmidN . Use... |
| pexmidlem4N 39932 | Lemma for ~ pexmidN . (Co... |
| pexmidlem5N 39933 | Lemma for ~ pexmidN . (Co... |
| pexmidlem6N 39934 | Lemma for ~ pexmidN . (Co... |
| pexmidlem7N 39935 | Lemma for ~ pexmidN . Con... |
| pexmidlem8N 39936 | Lemma for ~ pexmidN . The... |
| pexmidALTN 39937 | Excluded middle law for cl... |
| pl42lem1N 39938 | Lemma for ~ pl42N . (Cont... |
| pl42lem2N 39939 | Lemma for ~ pl42N . (Cont... |
| pl42lem3N 39940 | Lemma for ~ pl42N . (Cont... |
| pl42lem4N 39941 | Lemma for ~ pl42N . (Cont... |
| pl42N 39942 | Law holding in a Hilbert l... |
| watfvalN 39951 | The W atoms function. (Co... |
| watvalN 39952 | Value of the W atoms funct... |
| iswatN 39953 | The predicate "is a W atom... |
| lhpset 39954 | The set of co-atoms (latti... |
| islhp 39955 | The predicate "is a co-ato... |
| islhp2 39956 | The predicate "is a co-ato... |
| lhpbase 39957 | A co-atom is a member of t... |
| lhp1cvr 39958 | The lattice unity covers a... |
| lhplt 39959 | An atom under a co-atom is... |
| lhp2lt 39960 | The join of two atoms unde... |
| lhpexlt 39961 | There exists an atom less ... |
| lhp0lt 39962 | A co-atom is greater than ... |
| lhpn0 39963 | A co-atom is nonzero. TOD... |
| lhpexle 39964 | There exists an atom under... |
| lhpexnle 39965 | There exists an atom not u... |
| lhpexle1lem 39966 | Lemma for ~ lhpexle1 and o... |
| lhpexle1 39967 | There exists an atom under... |
| lhpexle2lem 39968 | Lemma for ~ lhpexle2 . (C... |
| lhpexle2 39969 | There exists atom under a ... |
| lhpexle3lem 39970 | There exists atom under a ... |
| lhpexle3 39971 | There exists atom under a ... |
| lhpex2leN 39972 | There exist at least two d... |
| lhpoc 39973 | The orthocomplement of a c... |
| lhpoc2N 39974 | The orthocomplement of an ... |
| lhpocnle 39975 | The orthocomplement of a c... |
| lhpocat 39976 | The orthocomplement of a c... |
| lhpocnel 39977 | The orthocomplement of a c... |
| lhpocnel2 39978 | The orthocomplement of a c... |
| lhpjat1 39979 | The join of a co-atom (hyp... |
| lhpjat2 39980 | The join of a co-atom (hyp... |
| lhpj1 39981 | The join of a co-atom (hyp... |
| lhpmcvr 39982 | The meet of a lattice hype... |
| lhpmcvr2 39983 | Alternate way to express t... |
| lhpmcvr3 39984 | Specialization of ~ lhpmcv... |
| lhpmcvr4N 39985 | Specialization of ~ lhpmcv... |
| lhpmcvr5N 39986 | Specialization of ~ lhpmcv... |
| lhpmcvr6N 39987 | Specialization of ~ lhpmcv... |
| lhpm0atN 39988 | If the meet of a lattice h... |
| lhpmat 39989 | An element covered by the ... |
| lhpmatb 39990 | An element covered by the ... |
| lhp2at0 39991 | Join and meet with differe... |
| lhp2atnle 39992 | Inequality for 2 different... |
| lhp2atne 39993 | Inequality for joins with ... |
| lhp2at0nle 39994 | Inequality for 2 different... |
| lhp2at0ne 39995 | Inequality for joins with ... |
| lhpelim 39996 | Eliminate an atom not unde... |
| lhpmod2i2 39997 | Modular law for hyperplane... |
| lhpmod6i1 39998 | Modular law for hyperplane... |
| lhprelat3N 39999 | The Hilbert lattice is rel... |
| cdlemb2 40000 | Given two atoms not under ... |
| lhple 40001 | Property of a lattice elem... |
| lhpat 40002 | Create an atom under a co-... |
| lhpat4N 40003 | Property of an atom under ... |
| lhpat2 40004 | Create an atom under a co-... |
| lhpat3 40005 | There is only one atom und... |
| 4atexlemk 40006 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemw 40007 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempw 40008 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemp 40009 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemq 40010 | Lemma for ~ 4atexlem7 . (... |
| 4atexlems 40011 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemt 40012 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemutvt 40013 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempnq 40014 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnslpq 40015 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkl 40016 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkc 40017 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemwb 40018 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempsb 40019 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemqtb 40020 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempns 40021 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemswapqr 40022 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemu 40023 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemv 40024 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemunv 40025 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemtlw 40026 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemntlpq 40027 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemc 40028 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnclw 40029 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex2 40030 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemcnd 40031 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex4 40032 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemex6 40033 | Lemma for ~ 4atexlem7 . (... |
| 4atexlem7 40034 | Whenever there are at leas... |
| 4atex 40035 | Whenever there are at leas... |
| 4atex2 40036 | More general version of ~ ... |
| 4atex2-0aOLDN 40037 | Same as ~ 4atex2 except th... |
| 4atex2-0bOLDN 40038 | Same as ~ 4atex2 except th... |
| 4atex2-0cOLDN 40039 | Same as ~ 4atex2 except th... |
| 4atex3 40040 | More general version of ~ ... |
| lautset 40041 | The set of lattice automor... |
| islaut 40042 | The predicate "is a lattic... |
| lautle 40043 | Less-than or equal propert... |
| laut1o 40044 | A lattice automorphism is ... |
| laut11 40045 | One-to-one property of a l... |
| lautcl 40046 | A lattice automorphism val... |
| lautcnvclN 40047 | Reverse closure of a latti... |
| lautcnvle 40048 | Less-than or equal propert... |
| lautcnv 40049 | The converse of a lattice ... |
| lautlt 40050 | Less-than property of a la... |
| lautcvr 40051 | Covering property of a lat... |
| lautj 40052 | Meet property of a lattice... |
| lautm 40053 | Meet property of a lattice... |
| lauteq 40054 | A lattice automorphism arg... |
| idlaut 40055 | The identity function is a... |
| lautco 40056 | The composition of two lat... |
| pautsetN 40057 | The set of projective auto... |
| ispautN 40058 | The predicate "is a projec... |
| ldilfset 40067 | The mapping from fiducial ... |
| ldilset 40068 | The set of lattice dilatio... |
| isldil 40069 | The predicate "is a lattic... |
| ldillaut 40070 | A lattice dilation is an a... |
| ldil1o 40071 | A lattice dilation is a on... |
| ldilval 40072 | Value of a lattice dilatio... |
| idldil 40073 | The identity function is a... |
| ldilcnv 40074 | The converse of a lattice ... |
| ldilco 40075 | The composition of two lat... |
| ltrnfset 40076 | The set of all lattice tra... |
| ltrnset 40077 | The set of lattice transla... |
| isltrn 40078 | The predicate "is a lattic... |
| isltrn2N 40079 | The predicate "is a lattic... |
| ltrnu 40080 | Uniqueness property of a l... |
| ltrnldil 40081 | A lattice translation is a... |
| ltrnlaut 40082 | A lattice translation is a... |
| ltrn1o 40083 | A lattice translation is a... |
| ltrncl 40084 | Closure of a lattice trans... |
| ltrn11 40085 | One-to-one property of a l... |
| ltrncnvnid 40086 | If a translation is differ... |
| ltrncoidN 40087 | Two translations are equal... |
| ltrnle 40088 | Less-than or equal propert... |
| ltrncnvleN 40089 | Less-than or equal propert... |
| ltrnm 40090 | Lattice translation of a m... |
| ltrnj 40091 | Lattice translation of a m... |
| ltrncvr 40092 | Covering property of a lat... |
| ltrnval1 40093 | Value of a lattice transla... |
| ltrnid 40094 | A lattice translation is t... |
| ltrnnid 40095 | If a lattice translation i... |
| ltrnatb 40096 | The lattice translation of... |
| ltrncnvatb 40097 | The converse of the lattic... |
| ltrnel 40098 | The lattice translation of... |
| ltrnat 40099 | The lattice translation of... |
| ltrncnvat 40100 | The converse of the lattic... |
| ltrncnvel 40101 | The converse of the lattic... |
| ltrncoelN 40102 | Composition of lattice tra... |
| ltrncoat 40103 | Composition of lattice tra... |
| ltrncoval 40104 | Two ways to express value ... |
| ltrncnv 40105 | The converse of a lattice ... |
| ltrn11at 40106 | Frequently used one-to-one... |
| ltrneq2 40107 | The equality of two transl... |
| ltrneq 40108 | The equality of two transl... |
| idltrn 40109 | The identity function is a... |
| ltrnmw 40110 | Property of lattice transl... |
| dilfsetN 40111 | The mapping from fiducial ... |
| dilsetN 40112 | The set of dilations for a... |
| isdilN 40113 | The predicate "is a dilati... |
| trnfsetN 40114 | The mapping from fiducial ... |
| trnsetN 40115 | The set of translations fo... |
| istrnN 40116 | The predicate "is a transl... |
| trlfset 40119 | The set of all traces of l... |
| trlset 40120 | The set of traces of latti... |
| trlval 40121 | The value of the trace of ... |
| trlval2 40122 | The value of the trace of ... |
| trlcl 40123 | Closure of the trace of a ... |
| trlcnv 40124 | The trace of the converse ... |
| trljat1 40125 | The value of a translation... |
| trljat2 40126 | The value of a translation... |
| trljat3 40127 | The value of a translation... |
| trlat 40128 | If an atom differs from it... |
| trl0 40129 | If an atom not under the f... |
| trlator0 40130 | The trace of a lattice tra... |
| trlatn0 40131 | The trace of a lattice tra... |
| trlnidat 40132 | The trace of a lattice tra... |
| ltrnnidn 40133 | If a lattice translation i... |
| ltrnideq 40134 | Property of the identity l... |
| trlid0 40135 | The trace of the identity ... |
| trlnidatb 40136 | A lattice translation is n... |
| trlid0b 40137 | A lattice translation is t... |
| trlnid 40138 | Different translations wit... |
| ltrn2ateq 40139 | Property of the equality o... |
| ltrnateq 40140 | If any atom (under ` W ` )... |
| ltrnatneq 40141 | If any atom (under ` W ` )... |
| ltrnatlw 40142 | If the value of an atom eq... |
| trlle 40143 | The trace of a lattice tra... |
| trlne 40144 | The trace of a lattice tra... |
| trlnle 40145 | The atom not under the fid... |
| trlval3 40146 | The value of the trace of ... |
| trlval4 40147 | The value of the trace of ... |
| trlval5 40148 | The value of the trace of ... |
| arglem1N 40149 | Lemma for Desargues's law.... |
| cdlemc1 40150 | Part of proof of Lemma C i... |
| cdlemc2 40151 | Part of proof of Lemma C i... |
| cdlemc3 40152 | Part of proof of Lemma C i... |
| cdlemc4 40153 | Part of proof of Lemma C i... |
| cdlemc5 40154 | Lemma for ~ cdlemc . (Con... |
| cdlemc6 40155 | Lemma for ~ cdlemc . (Con... |
| cdlemc 40156 | Lemma C in [Crawley] p. 11... |
| cdlemd1 40157 | Part of proof of Lemma D i... |
| cdlemd2 40158 | Part of proof of Lemma D i... |
| cdlemd3 40159 | Part of proof of Lemma D i... |
| cdlemd4 40160 | Part of proof of Lemma D i... |
| cdlemd5 40161 | Part of proof of Lemma D i... |
| cdlemd6 40162 | Part of proof of Lemma D i... |
| cdlemd7 40163 | Part of proof of Lemma D i... |
| cdlemd8 40164 | Part of proof of Lemma D i... |
| cdlemd9 40165 | Part of proof of Lemma D i... |
| cdlemd 40166 | If two translations agree ... |
| ltrneq3 40167 | Two translations agree at ... |
| cdleme00a 40168 | Part of proof of Lemma E i... |
| cdleme0aa 40169 | Part of proof of Lemma E i... |
| cdleme0a 40170 | Part of proof of Lemma E i... |
| cdleme0b 40171 | Part of proof of Lemma E i... |
| cdleme0c 40172 | Part of proof of Lemma E i... |
| cdleme0cp 40173 | Part of proof of Lemma E i... |
| cdleme0cq 40174 | Part of proof of Lemma E i... |
| cdleme0dN 40175 | Part of proof of Lemma E i... |
| cdleme0e 40176 | Part of proof of Lemma E i... |
| cdleme0fN 40177 | Part of proof of Lemma E i... |
| cdleme0gN 40178 | Part of proof of Lemma E i... |
| cdlemeulpq 40179 | Part of proof of Lemma E i... |
| cdleme01N 40180 | Part of proof of Lemma E i... |
| cdleme02N 40181 | Part of proof of Lemma E i... |
| cdleme0ex1N 40182 | Part of proof of Lemma E i... |
| cdleme0ex2N 40183 | Part of proof of Lemma E i... |
| cdleme0moN 40184 | Part of proof of Lemma E i... |
| cdleme1b 40185 | Part of proof of Lemma E i... |
| cdleme1 40186 | Part of proof of Lemma E i... |
| cdleme2 40187 | Part of proof of Lemma E i... |
| cdleme3b 40188 | Part of proof of Lemma E i... |
| cdleme3c 40189 | Part of proof of Lemma E i... |
| cdleme3d 40190 | Part of proof of Lemma E i... |
| cdleme3e 40191 | Part of proof of Lemma E i... |
| cdleme3fN 40192 | Part of proof of Lemma E i... |
| cdleme3g 40193 | Part of proof of Lemma E i... |
| cdleme3h 40194 | Part of proof of Lemma E i... |
| cdleme3fa 40195 | Part of proof of Lemma E i... |
| cdleme3 40196 | Part of proof of Lemma E i... |
| cdleme4 40197 | Part of proof of Lemma E i... |
| cdleme4a 40198 | Part of proof of Lemma E i... |
| cdleme5 40199 | Part of proof of Lemma E i... |
| cdleme6 40200 | Part of proof of Lemma E i... |
| cdleme7aa 40201 | Part of proof of Lemma E i... |
| cdleme7a 40202 | Part of proof of Lemma E i... |
| cdleme7b 40203 | Part of proof of Lemma E i... |
| cdleme7c 40204 | Part of proof of Lemma E i... |
| cdleme7d 40205 | Part of proof of Lemma E i... |
| cdleme7e 40206 | Part of proof of Lemma E i... |
| cdleme7ga 40207 | Part of proof of Lemma E i... |
| cdleme7 40208 | Part of proof of Lemma E i... |
| cdleme8 40209 | Part of proof of Lemma E i... |
| cdleme9a 40210 | Part of proof of Lemma E i... |
| cdleme9b 40211 | Utility lemma for Lemma E ... |
| cdleme9 40212 | Part of proof of Lemma E i... |
| cdleme10 40213 | Part of proof of Lemma E i... |
| cdleme8tN 40214 | Part of proof of Lemma E i... |
| cdleme9taN 40215 | Part of proof of Lemma E i... |
| cdleme9tN 40216 | Part of proof of Lemma E i... |
| cdleme10tN 40217 | Part of proof of Lemma E i... |
| cdleme16aN 40218 | Part of proof of Lemma E i... |
| cdleme11a 40219 | Part of proof of Lemma E i... |
| cdleme11c 40220 | Part of proof of Lemma E i... |
| cdleme11dN 40221 | Part of proof of Lemma E i... |
| cdleme11e 40222 | Part of proof of Lemma E i... |
| cdleme11fN 40223 | Part of proof of Lemma E i... |
| cdleme11g 40224 | Part of proof of Lemma E i... |
| cdleme11h 40225 | Part of proof of Lemma E i... |
| cdleme11j 40226 | Part of proof of Lemma E i... |
| cdleme11k 40227 | Part of proof of Lemma E i... |
| cdleme11l 40228 | Part of proof of Lemma E i... |
| cdleme11 40229 | Part of proof of Lemma E i... |
| cdleme12 40230 | Part of proof of Lemma E i... |
| cdleme13 40231 | Part of proof of Lemma E i... |
| cdleme14 40232 | Part of proof of Lemma E i... |
| cdleme15a 40233 | Part of proof of Lemma E i... |
| cdleme15b 40234 | Part of proof of Lemma E i... |
| cdleme15c 40235 | Part of proof of Lemma E i... |
| cdleme15d 40236 | Part of proof of Lemma E i... |
| cdleme15 40237 | Part of proof of Lemma E i... |
| cdleme16b 40238 | Part of proof of Lemma E i... |
| cdleme16c 40239 | Part of proof of Lemma E i... |
| cdleme16d 40240 | Part of proof of Lemma E i... |
| cdleme16e 40241 | Part of proof of Lemma E i... |
| cdleme16f 40242 | Part of proof of Lemma E i... |
| cdleme16g 40243 | Part of proof of Lemma E i... |
| cdleme16 40244 | Part of proof of Lemma E i... |
| cdleme17a 40245 | Part of proof of Lemma E i... |
| cdleme17b 40246 | Lemma leading to ~ cdleme1... |
| cdleme17c 40247 | Part of proof of Lemma E i... |
| cdleme17d1 40248 | Part of proof of Lemma E i... |
| cdleme0nex 40249 | Part of proof of Lemma E i... |
| cdleme18a 40250 | Part of proof of Lemma E i... |
| cdleme18b 40251 | Part of proof of Lemma E i... |
| cdleme18c 40252 | Part of proof of Lemma E i... |
| cdleme22gb 40253 | Utility lemma for Lemma E ... |
| cdleme18d 40254 | Part of proof of Lemma E i... |
| cdlemesner 40255 | Part of proof of Lemma E i... |
| cdlemedb 40256 | Part of proof of Lemma E i... |
| cdlemeda 40257 | Part of proof of Lemma E i... |
| cdlemednpq 40258 | Part of proof of Lemma E i... |
| cdlemednuN 40259 | Part of proof of Lemma E i... |
| cdleme20zN 40260 | Part of proof of Lemma E i... |
| cdleme20y 40261 | Part of proof of Lemma E i... |
| cdleme19a 40262 | Part of proof of Lemma E i... |
| cdleme19b 40263 | Part of proof of Lemma E i... |
| cdleme19c 40264 | Part of proof of Lemma E i... |
| cdleme19d 40265 | Part of proof of Lemma E i... |
| cdleme19e 40266 | Part of proof of Lemma E i... |
| cdleme19f 40267 | Part of proof of Lemma E i... |
| cdleme20aN 40268 | Part of proof of Lemma E i... |
| cdleme20bN 40269 | Part of proof of Lemma E i... |
| cdleme20c 40270 | Part of proof of Lemma E i... |
| cdleme20d 40271 | Part of proof of Lemma E i... |
| cdleme20e 40272 | Part of proof of Lemma E i... |
| cdleme20f 40273 | Part of proof of Lemma E i... |
| cdleme20g 40274 | Part of proof of Lemma E i... |
| cdleme20h 40275 | Part of proof of Lemma E i... |
| cdleme20i 40276 | Part of proof of Lemma E i... |
| cdleme20j 40277 | Part of proof of Lemma E i... |
| cdleme20k 40278 | Part of proof of Lemma E i... |
| cdleme20l1 40279 | Part of proof of Lemma E i... |
| cdleme20l2 40280 | Part of proof of Lemma E i... |
| cdleme20l 40281 | Part of proof of Lemma E i... |
| cdleme20m 40282 | Part of proof of Lemma E i... |
| cdleme20 40283 | Combine ~ cdleme19f and ~ ... |
| cdleme21a 40284 | Part of proof of Lemma E i... |
| cdleme21b 40285 | Part of proof of Lemma E i... |
| cdleme21c 40286 | Part of proof of Lemma E i... |
| cdleme21at 40287 | Part of proof of Lemma E i... |
| cdleme21ct 40288 | Part of proof of Lemma E i... |
| cdleme21d 40289 | Part of proof of Lemma E i... |
| cdleme21e 40290 | Part of proof of Lemma E i... |
| cdleme21f 40291 | Part of proof of Lemma E i... |
| cdleme21g 40292 | Part of proof of Lemma E i... |
| cdleme21h 40293 | Part of proof of Lemma E i... |
| cdleme21i 40294 | Part of proof of Lemma E i... |
| cdleme21j 40295 | Combine ~ cdleme20 and ~ c... |
| cdleme21 40296 | Part of proof of Lemma E i... |
| cdleme21k 40297 | Eliminate ` S =/= T ` cond... |
| cdleme22aa 40298 | Part of proof of Lemma E i... |
| cdleme22a 40299 | Part of proof of Lemma E i... |
| cdleme22b 40300 | Part of proof of Lemma E i... |
| cdleme22cN 40301 | Part of proof of Lemma E i... |
| cdleme22d 40302 | Part of proof of Lemma E i... |
| cdleme22e 40303 | Part of proof of Lemma E i... |
| cdleme22eALTN 40304 | Part of proof of Lemma E i... |
| cdleme22f 40305 | Part of proof of Lemma E i... |
| cdleme22f2 40306 | Part of proof of Lemma E i... |
| cdleme22g 40307 | Part of proof of Lemma E i... |
| cdleme23a 40308 | Part of proof of Lemma E i... |
| cdleme23b 40309 | Part of proof of Lemma E i... |
| cdleme23c 40310 | Part of proof of Lemma E i... |
| cdleme24 40311 | Quantified version of ~ cd... |
| cdleme25a 40312 | Lemma for ~ cdleme25b . (... |
| cdleme25b 40313 | Transform ~ cdleme24 . TO... |
| cdleme25c 40314 | Transform ~ cdleme25b . (... |
| cdleme25dN 40315 | Transform ~ cdleme25c . (... |
| cdleme25cl 40316 | Show closure of the unique... |
| cdleme25cv 40317 | Change bound variables in ... |
| cdleme26e 40318 | Part of proof of Lemma E i... |
| cdleme26ee 40319 | Part of proof of Lemma E i... |
| cdleme26eALTN 40320 | Part of proof of Lemma E i... |
| cdleme26fALTN 40321 | Part of proof of Lemma E i... |
| cdleme26f 40322 | Part of proof of Lemma E i... |
| cdleme26f2ALTN 40323 | Part of proof of Lemma E i... |
| cdleme26f2 40324 | Part of proof of Lemma E i... |
| cdleme27cl 40325 | Part of proof of Lemma E i... |
| cdleme27a 40326 | Part of proof of Lemma E i... |
| cdleme27b 40327 | Lemma for ~ cdleme27N . (... |
| cdleme27N 40328 | Part of proof of Lemma E i... |
| cdleme28a 40329 | Lemma for ~ cdleme25b . T... |
| cdleme28b 40330 | Lemma for ~ cdleme25b . T... |
| cdleme28c 40331 | Part of proof of Lemma E i... |
| cdleme28 40332 | Quantified version of ~ cd... |
| cdleme29ex 40333 | Lemma for ~ cdleme29b . (... |
| cdleme29b 40334 | Transform ~ cdleme28 . (C... |
| cdleme29c 40335 | Transform ~ cdleme28b . (... |
| cdleme29cl 40336 | Show closure of the unique... |
| cdleme30a 40337 | Part of proof of Lemma E i... |
| cdleme31so 40338 | Part of proof of Lemma E i... |
| cdleme31sn 40339 | Part of proof of Lemma E i... |
| cdleme31sn1 40340 | Part of proof of Lemma E i... |
| cdleme31se 40341 | Part of proof of Lemma D i... |
| cdleme31se2 40342 | Part of proof of Lemma D i... |
| cdleme31sc 40343 | Part of proof of Lemma E i... |
| cdleme31sde 40344 | Part of proof of Lemma D i... |
| cdleme31snd 40345 | Part of proof of Lemma D i... |
| cdleme31sdnN 40346 | Part of proof of Lemma E i... |
| cdleme31sn1c 40347 | Part of proof of Lemma E i... |
| cdleme31sn2 40348 | Part of proof of Lemma E i... |
| cdleme31fv 40349 | Part of proof of Lemma E i... |
| cdleme31fv1 40350 | Part of proof of Lemma E i... |
| cdleme31fv1s 40351 | Part of proof of Lemma E i... |
| cdleme31fv2 40352 | Part of proof of Lemma E i... |
| cdleme31id 40353 | Part of proof of Lemma E i... |
| cdlemefrs29pre00 40354 | ***START OF VALUE AT ATOM ... |
| cdlemefrs29bpre0 40355 | TODO fix comment. (Contri... |
| cdlemefrs29bpre1 40356 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29cpre1 40357 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29clN 40358 | TODO: NOT USED? Show clo... |
| cdlemefrs32fva 40359 | Part of proof of Lemma E i... |
| cdlemefrs32fva1 40360 | Part of proof of Lemma E i... |
| cdlemefr29exN 40361 | Lemma for ~ cdlemefs29bpre... |
| cdlemefr27cl 40362 | Part of proof of Lemma E i... |
| cdlemefr32sn2aw 40363 | Show that ` [_ R / s ]_ N ... |
| cdlemefr32snb 40364 | Show closure of ` [_ R / s... |
| cdlemefr29bpre0N 40365 | TODO fix comment. (Contri... |
| cdlemefr29clN 40366 | Show closure of the unique... |
| cdleme43frv1snN 40367 | Value of ` [_ R / s ]_ N `... |
| cdlemefr32fvaN 40368 | Part of proof of Lemma E i... |
| cdlemefr32fva1 40369 | Part of proof of Lemma E i... |
| cdlemefr31fv1 40370 | Value of ` ( F `` R ) ` wh... |
| cdlemefs29pre00N 40371 | FIX COMMENT. TODO: see if ... |
| cdlemefs27cl 40372 | Part of proof of Lemma E i... |
| cdlemefs32sn1aw 40373 | Show that ` [_ R / s ]_ N ... |
| cdlemefs32snb 40374 | Show closure of ` [_ R / s... |
| cdlemefs29bpre0N 40375 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29bpre1N 40376 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29cpre1N 40377 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29clN 40378 | Show closure of the unique... |
| cdleme43fsv1snlem 40379 | Value of ` [_ R / s ]_ N `... |
| cdleme43fsv1sn 40380 | Value of ` [_ R / s ]_ N `... |
| cdlemefs32fvaN 40381 | Part of proof of Lemma E i... |
| cdlemefs32fva1 40382 | Part of proof of Lemma E i... |
| cdlemefs31fv1 40383 | Value of ` ( F `` R ) ` wh... |
| cdlemefr44 40384 | Value of f(r) when r is an... |
| cdlemefs44 40385 | Value of f_s(r) when r is ... |
| cdlemefr45 40386 | Value of f(r) when r is an... |
| cdlemefr45e 40387 | Explicit expansion of ~ cd... |
| cdlemefs45 40388 | Value of f_s(r) when r is ... |
| cdlemefs45ee 40389 | Explicit expansion of ~ cd... |
| cdlemefs45eN 40390 | Explicit expansion of ~ cd... |
| cdleme32sn1awN 40391 | Show that ` [_ R / s ]_ N ... |
| cdleme41sn3a 40392 | Show that ` [_ R / s ]_ N ... |
| cdleme32sn2awN 40393 | Show that ` [_ R / s ]_ N ... |
| cdleme32snaw 40394 | Show that ` [_ R / s ]_ N ... |
| cdleme32snb 40395 | Show closure of ` [_ R / s... |
| cdleme32fva 40396 | Part of proof of Lemma D i... |
| cdleme32fva1 40397 | Part of proof of Lemma D i... |
| cdleme32fvaw 40398 | Show that ` ( F `` R ) ` i... |
| cdleme32fvcl 40399 | Part of proof of Lemma D i... |
| cdleme32a 40400 | Part of proof of Lemma D i... |
| cdleme32b 40401 | Part of proof of Lemma D i... |
| cdleme32c 40402 | Part of proof of Lemma D i... |
| cdleme32d 40403 | Part of proof of Lemma D i... |
| cdleme32e 40404 | Part of proof of Lemma D i... |
| cdleme32f 40405 | Part of proof of Lemma D i... |
| cdleme32le 40406 | Part of proof of Lemma D i... |
| cdleme35a 40407 | Part of proof of Lemma E i... |
| cdleme35fnpq 40408 | Part of proof of Lemma E i... |
| cdleme35b 40409 | Part of proof of Lemma E i... |
| cdleme35c 40410 | Part of proof of Lemma E i... |
| cdleme35d 40411 | Part of proof of Lemma E i... |
| cdleme35e 40412 | Part of proof of Lemma E i... |
| cdleme35f 40413 | Part of proof of Lemma E i... |
| cdleme35g 40414 | Part of proof of Lemma E i... |
| cdleme35h 40415 | Part of proof of Lemma E i... |
| cdleme35h2 40416 | Part of proof of Lemma E i... |
| cdleme35sn2aw 40417 | Part of proof of Lemma E i... |
| cdleme35sn3a 40418 | Part of proof of Lemma E i... |
| cdleme36a 40419 | Part of proof of Lemma E i... |
| cdleme36m 40420 | Part of proof of Lemma E i... |
| cdleme37m 40421 | Part of proof of Lemma E i... |
| cdleme38m 40422 | Part of proof of Lemma E i... |
| cdleme38n 40423 | Part of proof of Lemma E i... |
| cdleme39a 40424 | Part of proof of Lemma E i... |
| cdleme39n 40425 | Part of proof of Lemma E i... |
| cdleme40m 40426 | Part of proof of Lemma E i... |
| cdleme40n 40427 | Part of proof of Lemma E i... |
| cdleme40v 40428 | Part of proof of Lemma E i... |
| cdleme40w 40429 | Part of proof of Lemma E i... |
| cdleme42a 40430 | Part of proof of Lemma E i... |
| cdleme42c 40431 | Part of proof of Lemma E i... |
| cdleme42d 40432 | Part of proof of Lemma E i... |
| cdleme41sn3aw 40433 | Part of proof of Lemma E i... |
| cdleme41sn4aw 40434 | Part of proof of Lemma E i... |
| cdleme41snaw 40435 | Part of proof of Lemma E i... |
| cdleme41fva11 40436 | Part of proof of Lemma E i... |
| cdleme42b 40437 | Part of proof of Lemma E i... |
| cdleme42e 40438 | Part of proof of Lemma E i... |
| cdleme42f 40439 | Part of proof of Lemma E i... |
| cdleme42g 40440 | Part of proof of Lemma E i... |
| cdleme42h 40441 | Part of proof of Lemma E i... |
| cdleme42i 40442 | Part of proof of Lemma E i... |
| cdleme42k 40443 | Part of proof of Lemma E i... |
| cdleme42ke 40444 | Part of proof of Lemma E i... |
| cdleme42keg 40445 | Part of proof of Lemma E i... |
| cdleme42mN 40446 | Part of proof of Lemma E i... |
| cdleme42mgN 40447 | Part of proof of Lemma E i... |
| cdleme43aN 40448 | Part of proof of Lemma E i... |
| cdleme43bN 40449 | Lemma for Lemma E in [Craw... |
| cdleme43cN 40450 | Part of proof of Lemma E i... |
| cdleme43dN 40451 | Part of proof of Lemma E i... |
| cdleme46f2g2 40452 | Conversion for ` G ` to re... |
| cdleme46f2g1 40453 | Conversion for ` G ` to re... |
| cdleme17d2 40454 | Part of proof of Lemma E i... |
| cdleme17d3 40455 | TODO: FIX COMMENT. (Contr... |
| cdleme17d4 40456 | TODO: FIX COMMENT. (Contr... |
| cdleme17d 40457 | Part of proof of Lemma E i... |
| cdleme48fv 40458 | Part of proof of Lemma D i... |
| cdleme48fvg 40459 | Remove ` P =/= Q ` conditi... |
| cdleme46fvaw 40460 | Show that ` ( F `` R ) ` i... |
| cdleme48bw 40461 | TODO: fix comment. TODO: ... |
| cdleme48b 40462 | TODO: fix comment. (Contr... |
| cdleme46frvlpq 40463 | Show that ` ( F `` S ) ` i... |
| cdleme46fsvlpq 40464 | Show that ` ( F `` R ) ` i... |
| cdlemeg46fvcl 40465 | TODO: fix comment. (Contr... |
| cdleme4gfv 40466 | Part of proof of Lemma D i... |
| cdlemeg47b 40467 | TODO: FIX COMMENT. (Contr... |
| cdlemeg47rv 40468 | Value of g_s(r) when r is ... |
| cdlemeg47rv2 40469 | Value of g_s(r) when r is ... |
| cdlemeg49le 40470 | Part of proof of Lemma D i... |
| cdlemeg46bOLDN 40471 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46c 40472 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46rvOLDN 40473 | Value of g_s(r) when r is ... |
| cdlemeg46rv2OLDN 40474 | Value of g_s(r) when r is ... |
| cdlemeg46fvaw 40475 | Show that ` ( F `` R ) ` i... |
| cdlemeg46nlpq 40476 | Show that ` ( G `` S ) ` i... |
| cdlemeg46ngfr 40477 | TODO FIX COMMENT g(f(s))=s... |
| cdlemeg46nfgr 40478 | TODO FIX COMMENT f(g(s))=s... |
| cdlemeg46sfg 40479 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fjgN 40480 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46rjgN 40481 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46fjv 40482 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fsfv 40483 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46frv 40484 | TODO FIX COMMENT. (f(r) ` ... |
| cdlemeg46v1v2 40485 | TODO FIX COMMENT v_1 = v_2... |
| cdlemeg46vrg 40486 | TODO FIX COMMENT v_1 ` <_ ... |
| cdlemeg46rgv 40487 | TODO FIX COMMENT r ` <_ ` ... |
| cdlemeg46req 40488 | TODO FIX COMMENT r = (v_1 ... |
| cdlemeg46gfv 40489 | TODO FIX COMMENT p. 115 pe... |
| cdlemeg46gfr 40490 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gfre 40491 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gf 40492 | TODO FIX COMMENT Eliminate... |
| cdlemeg46fgN 40493 | TODO FIX COMMENT p. 116 pe... |
| cdleme48d 40494 | TODO: fix comment. (Contr... |
| cdleme48gfv1 40495 | TODO: fix comment. (Contr... |
| cdleme48gfv 40496 | TODO: fix comment. (Contr... |
| cdleme48fgv 40497 | TODO: fix comment. (Contr... |
| cdlemeg49lebilem 40498 | Part of proof of Lemma D i... |
| cdleme50lebi 40499 | Part of proof of Lemma D i... |
| cdleme50eq 40500 | Part of proof of Lemma D i... |
| cdleme50f 40501 | Part of proof of Lemma D i... |
| cdleme50f1 40502 | Part of proof of Lemma D i... |
| cdleme50rnlem 40503 | Part of proof of Lemma D i... |
| cdleme50rn 40504 | Part of proof of Lemma D i... |
| cdleme50f1o 40505 | Part of proof of Lemma D i... |
| cdleme50laut 40506 | Part of proof of Lemma D i... |
| cdleme50ldil 40507 | Part of proof of Lemma D i... |
| cdleme50trn1 40508 | Part of proof that ` F ` i... |
| cdleme50trn2a 40509 | Part of proof that ` F ` i... |
| cdleme50trn2 40510 | Part of proof that ` F ` i... |
| cdleme50trn12 40511 | Part of proof that ` F ` i... |
| cdleme50trn3 40512 | Part of proof that ` F ` i... |
| cdleme50trn123 40513 | Part of proof that ` F ` i... |
| cdleme51finvfvN 40514 | Part of proof of Lemma E i... |
| cdleme51finvN 40515 | Part of proof of Lemma E i... |
| cdleme50ltrn 40516 | Part of proof of Lemma E i... |
| cdleme51finvtrN 40517 | Part of proof of Lemma E i... |
| cdleme50ex 40518 | Part of Lemma E in [Crawle... |
| cdleme 40519 | Lemma E in [Crawley] p. 11... |
| cdlemf1 40520 | Part of Lemma F in [Crawle... |
| cdlemf2 40521 | Part of Lemma F in [Crawle... |
| cdlemf 40522 | Lemma F in [Crawley] p. 11... |
| cdlemfnid 40523 | ~ cdlemf with additional c... |
| cdlemftr3 40524 | Special case of ~ cdlemf s... |
| cdlemftr2 40525 | Special case of ~ cdlemf s... |
| cdlemftr1 40526 | Part of proof of Lemma G o... |
| cdlemftr0 40527 | Special case of ~ cdlemf s... |
| trlord 40528 | The ordering of two Hilber... |
| cdlemg1a 40529 | Shorter expression for ` G... |
| cdlemg1b2 40530 | This theorem can be used t... |
| cdlemg1idlemN 40531 | Lemma for ~ cdlemg1idN . ... |
| cdlemg1fvawlemN 40532 | Lemma for ~ ltrniotafvawN ... |
| cdlemg1ltrnlem 40533 | Lemma for ~ ltrniotacl . ... |
| cdlemg1finvtrlemN 40534 | Lemma for ~ ltrniotacnvN .... |
| cdlemg1bOLDN 40535 | This theorem can be used t... |
| cdlemg1idN 40536 | Version of ~ cdleme31id wi... |
| ltrniotafvawN 40537 | Version of ~ cdleme46fvaw ... |
| ltrniotacl 40538 | Version of ~ cdleme50ltrn ... |
| ltrniotacnvN 40539 | Version of ~ cdleme51finvt... |
| ltrniotaval 40540 | Value of the unique transl... |
| ltrniotacnvval 40541 | Converse value of the uniq... |
| ltrniotaidvalN 40542 | Value of the unique transl... |
| ltrniotavalbN 40543 | Value of the unique transl... |
| cdlemeiota 40544 | A translation is uniquely ... |
| cdlemg1ci2 40545 | Any function of the form o... |
| cdlemg1cN 40546 | Any translation belongs to... |
| cdlemg1cex 40547 | Any translation is one of ... |
| cdlemg2cN 40548 | Any translation belongs to... |
| cdlemg2dN 40549 | This theorem can be used t... |
| cdlemg2cex 40550 | Any translation is one of ... |
| cdlemg2ce 40551 | Utility theorem to elimina... |
| cdlemg2jlemOLDN 40552 | Part of proof of Lemma E i... |
| cdlemg2fvlem 40553 | Lemma for ~ cdlemg2fv . (... |
| cdlemg2klem 40554 | ~ cdleme42keg with simpler... |
| cdlemg2idN 40555 | Version of ~ cdleme31id wi... |
| cdlemg3a 40556 | Part of proof of Lemma G i... |
| cdlemg2jOLDN 40557 | TODO: Replace this with ~... |
| cdlemg2fv 40558 | Value of a translation in ... |
| cdlemg2fv2 40559 | Value of a translation in ... |
| cdlemg2k 40560 | ~ cdleme42keg with simpler... |
| cdlemg2kq 40561 | ~ cdlemg2k with ` P ` and ... |
| cdlemg2l 40562 | TODO: FIX COMMENT. (Contr... |
| cdlemg2m 40563 | TODO: FIX COMMENT. (Contr... |
| cdlemg5 40564 | TODO: Is there a simpler ... |
| cdlemb3 40565 | Given two atoms not under ... |
| cdlemg7fvbwN 40566 | Properties of a translatio... |
| cdlemg4a 40567 | TODO: FIX COMMENT If fg(p... |
| cdlemg4b1 40568 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b2 40569 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b12 40570 | TODO: FIX COMMENT. (Contr... |
| cdlemg4c 40571 | TODO: FIX COMMENT. (Contr... |
| cdlemg4d 40572 | TODO: FIX COMMENT. (Contr... |
| cdlemg4e 40573 | TODO: FIX COMMENT. (Contr... |
| cdlemg4f 40574 | TODO: FIX COMMENT. (Contr... |
| cdlemg4g 40575 | TODO: FIX COMMENT. (Contr... |
| cdlemg4 40576 | TODO: FIX COMMENT. (Contr... |
| cdlemg6a 40577 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6b 40578 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6c 40579 | TODO: FIX COMMENT. (Contr... |
| cdlemg6d 40580 | TODO: FIX COMMENT. (Contr... |
| cdlemg6e 40581 | TODO: FIX COMMENT. (Contr... |
| cdlemg6 40582 | TODO: FIX COMMENT. (Contr... |
| cdlemg7fvN 40583 | Value of a translation com... |
| cdlemg7aN 40584 | TODO: FIX COMMENT. (Contr... |
| cdlemg7N 40585 | TODO: FIX COMMENT. (Contr... |
| cdlemg8a 40586 | TODO: FIX COMMENT. (Contr... |
| cdlemg8b 40587 | TODO: FIX COMMENT. (Contr... |
| cdlemg8c 40588 | TODO: FIX COMMENT. (Contr... |
| cdlemg8d 40589 | TODO: FIX COMMENT. (Contr... |
| cdlemg8 40590 | TODO: FIX COMMENT. (Contr... |
| cdlemg9a 40591 | TODO: FIX COMMENT. (Contr... |
| cdlemg9b 40592 | The triples ` <. P , ( F `... |
| cdlemg9 40593 | The triples ` <. P , ( F `... |
| cdlemg10b 40594 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10bALTN 40595 | TODO: FIX COMMENT. TODO: ... |
| cdlemg11a 40596 | TODO: FIX COMMENT. (Contr... |
| cdlemg11aq 40597 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10c 40598 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10a 40599 | TODO: FIX COMMENT. (Contr... |
| cdlemg10 40600 | TODO: FIX COMMENT. (Contr... |
| cdlemg11b 40601 | TODO: FIX COMMENT. (Contr... |
| cdlemg12a 40602 | TODO: FIX COMMENT. (Contr... |
| cdlemg12b 40603 | The triples ` <. P , ( F `... |
| cdlemg12c 40604 | The triples ` <. P , ( F `... |
| cdlemg12d 40605 | TODO: FIX COMMENT. (Contr... |
| cdlemg12e 40606 | TODO: FIX COMMENT. (Contr... |
| cdlemg12f 40607 | TODO: FIX COMMENT. (Contr... |
| cdlemg12g 40608 | TODO: FIX COMMENT. TODO: ... |
| cdlemg12 40609 | TODO: FIX COMMENT. (Contr... |
| cdlemg13a 40610 | TODO: FIX COMMENT. (Contr... |
| cdlemg13 40611 | TODO: FIX COMMENT. (Contr... |
| cdlemg14f 40612 | TODO: FIX COMMENT. (Contr... |
| cdlemg14g 40613 | TODO: FIX COMMENT. (Contr... |
| cdlemg15a 40614 | Eliminate the ` ( F `` P )... |
| cdlemg15 40615 | Eliminate the ` ( (... |
| cdlemg16 40616 | Part of proof of Lemma G o... |
| cdlemg16ALTN 40617 | This version of ~ cdlemg16... |
| cdlemg16z 40618 | Eliminate ` ( ( F `... |
| cdlemg16zz 40619 | Eliminate ` P =/= Q ` from... |
| cdlemg17a 40620 | TODO: FIX COMMENT. (Contr... |
| cdlemg17b 40621 | Part of proof of Lemma G i... |
| cdlemg17dN 40622 | TODO: fix comment. (Contr... |
| cdlemg17dALTN 40623 | Same as ~ cdlemg17dN with ... |
| cdlemg17e 40624 | TODO: fix comment. (Contr... |
| cdlemg17f 40625 | TODO: fix comment. (Contr... |
| cdlemg17g 40626 | TODO: fix comment. (Contr... |
| cdlemg17h 40627 | TODO: fix comment. (Contr... |
| cdlemg17i 40628 | TODO: fix comment. (Contr... |
| cdlemg17ir 40629 | TODO: fix comment. (Contr... |
| cdlemg17j 40630 | TODO: fix comment. (Contr... |
| cdlemg17pq 40631 | Utility theorem for swappi... |
| cdlemg17bq 40632 | ~ cdlemg17b with ` P ` and... |
| cdlemg17iqN 40633 | ~ cdlemg17i with ` P ` and... |
| cdlemg17irq 40634 | ~ cdlemg17ir with ` P ` an... |
| cdlemg17jq 40635 | ~ cdlemg17j with ` P ` and... |
| cdlemg17 40636 | Part of Lemma G of [Crawle... |
| cdlemg18a 40637 | Show two lines are differe... |
| cdlemg18b 40638 | Lemma for ~ cdlemg18c . T... |
| cdlemg18c 40639 | Show two lines intersect a... |
| cdlemg18d 40640 | Show two lines intersect a... |
| cdlemg18 40641 | Show two lines intersect a... |
| cdlemg19a 40642 | Show two lines intersect a... |
| cdlemg19 40643 | Show two lines intersect a... |
| cdlemg20 40644 | Show two lines intersect a... |
| cdlemg21 40645 | Version of cdlemg19 with `... |
| cdlemg22 40646 | ~ cdlemg21 with ` ( F `` P... |
| cdlemg24 40647 | Combine ~ cdlemg16z and ~ ... |
| cdlemg37 40648 | Use ~ cdlemg8 to eliminate... |
| cdlemg25zz 40649 | ~ cdlemg16zz restated for ... |
| cdlemg26zz 40650 | ~ cdlemg16zz restated for ... |
| cdlemg27a 40651 | For use with case when ` (... |
| cdlemg28a 40652 | Part of proof of Lemma G o... |
| cdlemg31b0N 40653 | TODO: Fix comment. (Cont... |
| cdlemg31b0a 40654 | TODO: Fix comment. (Cont... |
| cdlemg27b 40655 | TODO: Fix comment. (Cont... |
| cdlemg31a 40656 | TODO: fix comment. (Contr... |
| cdlemg31b 40657 | TODO: fix comment. (Contr... |
| cdlemg31c 40658 | Show that when ` N ` is an... |
| cdlemg31d 40659 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33b0 40660 | TODO: Fix comment. (Cont... |
| cdlemg33c0 40661 | TODO: Fix comment. (Cont... |
| cdlemg28b 40662 | Part of proof of Lemma G o... |
| cdlemg28 40663 | Part of proof of Lemma G o... |
| cdlemg29 40664 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33a 40665 | TODO: Fix comment. (Cont... |
| cdlemg33b 40666 | TODO: Fix comment. (Cont... |
| cdlemg33c 40667 | TODO: Fix comment. (Cont... |
| cdlemg33d 40668 | TODO: Fix comment. (Cont... |
| cdlemg33e 40669 | TODO: Fix comment. (Cont... |
| cdlemg33 40670 | Combine ~ cdlemg33b , ~ cd... |
| cdlemg34 40671 | Use cdlemg33 to eliminate ... |
| cdlemg35 40672 | TODO: Fix comment. TODO:... |
| cdlemg36 40673 | Use cdlemg35 to eliminate ... |
| cdlemg38 40674 | Use ~ cdlemg37 to eliminat... |
| cdlemg39 40675 | Eliminate ` =/= ` conditio... |
| cdlemg40 40676 | Eliminate ` P =/= Q ` cond... |
| cdlemg41 40677 | Convert ~ cdlemg40 to func... |
| ltrnco 40678 | The composition of two tra... |
| trlcocnv 40679 | Swap the arguments of the ... |
| trlcoabs 40680 | Absorption into a composit... |
| trlcoabs2N 40681 | Absorption of the trace of... |
| trlcoat 40682 | The trace of a composition... |
| trlcocnvat 40683 | Commonly used special case... |
| trlconid 40684 | The composition of two dif... |
| trlcolem 40685 | Lemma for ~ trlco . (Cont... |
| trlco 40686 | The trace of a composition... |
| trlcone 40687 | If two translations have d... |
| cdlemg42 40688 | Part of proof of Lemma G o... |
| cdlemg43 40689 | Part of proof of Lemma G o... |
| cdlemg44a 40690 | Part of proof of Lemma G o... |
| cdlemg44b 40691 | Eliminate ` ( F `` P ) =/=... |
| cdlemg44 40692 | Part of proof of Lemma G o... |
| cdlemg47a 40693 | TODO: fix comment. TODO: ... |
| cdlemg46 40694 | Part of proof of Lemma G o... |
| cdlemg47 40695 | Part of proof of Lemma G o... |
| cdlemg48 40696 | Eliminate ` h ` from ~ cdl... |
| ltrncom 40697 | Composition is commutative... |
| ltrnco4 40698 | Rearrange a composition of... |
| trljco 40699 | Trace joined with trace of... |
| trljco2 40700 | Trace joined with trace of... |
| tgrpfset 40703 | The translation group maps... |
| tgrpset 40704 | The translation group for ... |
| tgrpbase 40705 | The base set of the transl... |
| tgrpopr 40706 | The group operation of the... |
| tgrpov 40707 | The group operation value ... |
| tgrpgrplem 40708 | Lemma for ~ tgrpgrp . (Co... |
| tgrpgrp 40709 | The translation group is a... |
| tgrpabl 40710 | The translation group is a... |
| tendofset 40717 | The set of all trace-prese... |
| tendoset 40718 | The set of trace-preservin... |
| istendo 40719 | The predicate "is a trace-... |
| tendotp 40720 | Trace-preserving property ... |
| istendod 40721 | Deduce the predicate "is a... |
| tendof 40722 | Functionality of a trace-p... |
| tendoeq1 40723 | Condition determining equa... |
| tendovalco 40724 | Value of composition of tr... |
| tendocoval 40725 | Value of composition of en... |
| tendocl 40726 | Closure of a trace-preserv... |
| tendoco2 40727 | Distribution of compositio... |
| tendoidcl 40728 | The identity is a trace-pr... |
| tendo1mul 40729 | Multiplicative identity mu... |
| tendo1mulr 40730 | Multiplicative identity mu... |
| tendococl 40731 | The composition of two tra... |
| tendoid 40732 | The identity value of a tr... |
| tendoeq2 40733 | Condition determining equa... |
| tendoplcbv 40734 | Define sum operation for t... |
| tendopl 40735 | Value of endomorphism sum ... |
| tendopl2 40736 | Value of result of endomor... |
| tendoplcl2 40737 | Value of result of endomor... |
| tendoplco2 40738 | Value of result of endomor... |
| tendopltp 40739 | Trace-preserving property ... |
| tendoplcl 40740 | Endomorphism sum is a trac... |
| tendoplcom 40741 | The endomorphism sum opera... |
| tendoplass 40742 | The endomorphism sum opera... |
| tendodi1 40743 | Endomorphism composition d... |
| tendodi2 40744 | Endomorphism composition d... |
| tendo0cbv 40745 | Define additive identity f... |
| tendo02 40746 | Value of additive identity... |
| tendo0co2 40747 | The additive identity trac... |
| tendo0tp 40748 | Trace-preserving property ... |
| tendo0cl 40749 | The additive identity is a... |
| tendo0pl 40750 | Property of the additive i... |
| tendo0plr 40751 | Property of the additive i... |
| tendoicbv 40752 | Define inverse function fo... |
| tendoi 40753 | Value of inverse endomorph... |
| tendoi2 40754 | Value of additive inverse ... |
| tendoicl 40755 | Closure of the additive in... |
| tendoipl 40756 | Property of the additive i... |
| tendoipl2 40757 | Property of the additive i... |
| erngfset 40758 | The division rings on trac... |
| erngset 40759 | The division ring on trace... |
| erngbase 40760 | The base set of the divisi... |
| erngfplus 40761 | Ring addition operation. ... |
| erngplus 40762 | Ring addition operation. ... |
| erngplus2 40763 | Ring addition operation. ... |
| erngfmul 40764 | Ring multiplication operat... |
| erngmul 40765 | Ring addition operation. ... |
| erngfset-rN 40766 | The division rings on trac... |
| erngset-rN 40767 | The division ring on trace... |
| erngbase-rN 40768 | The base set of the divisi... |
| erngfplus-rN 40769 | Ring addition operation. ... |
| erngplus-rN 40770 | Ring addition operation. ... |
| erngplus2-rN 40771 | Ring addition operation. ... |
| erngfmul-rN 40772 | Ring multiplication operat... |
| erngmul-rN 40773 | Ring addition operation. ... |
| cdlemh1 40774 | Part of proof of Lemma H o... |
| cdlemh2 40775 | Part of proof of Lemma H o... |
| cdlemh 40776 | Lemma H of [Crawley] p. 11... |
| cdlemi1 40777 | Part of proof of Lemma I o... |
| cdlemi2 40778 | Part of proof of Lemma I o... |
| cdlemi 40779 | Lemma I of [Crawley] p. 11... |
| cdlemj1 40780 | Part of proof of Lemma J o... |
| cdlemj2 40781 | Part of proof of Lemma J o... |
| cdlemj3 40782 | Part of proof of Lemma J o... |
| tendocan 40783 | Cancellation law: if the v... |
| tendoid0 40784 | A trace-preserving endomor... |
| tendo0mul 40785 | Additive identity multipli... |
| tendo0mulr 40786 | Additive identity multipli... |
| tendo1ne0 40787 | The identity (unity) is no... |
| tendoconid 40788 | The composition (product) ... |
| tendotr 40789 | The trace of the value of ... |
| cdlemk1 40790 | Part of proof of Lemma K o... |
| cdlemk2 40791 | Part of proof of Lemma K o... |
| cdlemk3 40792 | Part of proof of Lemma K o... |
| cdlemk4 40793 | Part of proof of Lemma K o... |
| cdlemk5a 40794 | Part of proof of Lemma K o... |
| cdlemk5 40795 | Part of proof of Lemma K o... |
| cdlemk6 40796 | Part of proof of Lemma K o... |
| cdlemk8 40797 | Part of proof of Lemma K o... |
| cdlemk9 40798 | Part of proof of Lemma K o... |
| cdlemk9bN 40799 | Part of proof of Lemma K o... |
| cdlemki 40800 | Part of proof of Lemma K o... |
| cdlemkvcl 40801 | Part of proof of Lemma K o... |
| cdlemk10 40802 | Part of proof of Lemma K o... |
| cdlemksv 40803 | Part of proof of Lemma K o... |
| cdlemksel 40804 | Part of proof of Lemma K o... |
| cdlemksat 40805 | Part of proof of Lemma K o... |
| cdlemksv2 40806 | Part of proof of Lemma K o... |
| cdlemk7 40807 | Part of proof of Lemma K o... |
| cdlemk11 40808 | Part of proof of Lemma K o... |
| cdlemk12 40809 | Part of proof of Lemma K o... |
| cdlemkoatnle 40810 | Utility lemma. (Contribut... |
| cdlemk13 40811 | Part of proof of Lemma K o... |
| cdlemkole 40812 | Utility lemma. (Contribut... |
| cdlemk14 40813 | Part of proof of Lemma K o... |
| cdlemk15 40814 | Part of proof of Lemma K o... |
| cdlemk16a 40815 | Part of proof of Lemma K o... |
| cdlemk16 40816 | Part of proof of Lemma K o... |
| cdlemk17 40817 | Part of proof of Lemma K o... |
| cdlemk1u 40818 | Part of proof of Lemma K o... |
| cdlemk5auN 40819 | Part of proof of Lemma K o... |
| cdlemk5u 40820 | Part of proof of Lemma K o... |
| cdlemk6u 40821 | Part of proof of Lemma K o... |
| cdlemkj 40822 | Part of proof of Lemma K o... |
| cdlemkuvN 40823 | Part of proof of Lemma K o... |
| cdlemkuel 40824 | Part of proof of Lemma K o... |
| cdlemkuat 40825 | Part of proof of Lemma K o... |
| cdlemkuv2 40826 | Part of proof of Lemma K o... |
| cdlemk18 40827 | Part of proof of Lemma K o... |
| cdlemk19 40828 | Part of proof of Lemma K o... |
| cdlemk7u 40829 | Part of proof of Lemma K o... |
| cdlemk11u 40830 | Part of proof of Lemma K o... |
| cdlemk12u 40831 | Part of proof of Lemma K o... |
| cdlemk21N 40832 | Part of proof of Lemma K o... |
| cdlemk20 40833 | Part of proof of Lemma K o... |
| cdlemkoatnle-2N 40834 | Utility lemma. (Contribut... |
| cdlemk13-2N 40835 | Part of proof of Lemma K o... |
| cdlemkole-2N 40836 | Utility lemma. (Contribut... |
| cdlemk14-2N 40837 | Part of proof of Lemma K o... |
| cdlemk15-2N 40838 | Part of proof of Lemma K o... |
| cdlemk16-2N 40839 | Part of proof of Lemma K o... |
| cdlemk17-2N 40840 | Part of proof of Lemma K o... |
| cdlemkj-2N 40841 | Part of proof of Lemma K o... |
| cdlemkuv-2N 40842 | Part of proof of Lemma K o... |
| cdlemkuel-2N 40843 | Part of proof of Lemma K o... |
| cdlemkuv2-2 40844 | Part of proof of Lemma K o... |
| cdlemk18-2N 40845 | Part of proof of Lemma K o... |
| cdlemk19-2N 40846 | Part of proof of Lemma K o... |
| cdlemk7u-2N 40847 | Part of proof of Lemma K o... |
| cdlemk11u-2N 40848 | Part of proof of Lemma K o... |
| cdlemk12u-2N 40849 | Part of proof of Lemma K o... |
| cdlemk21-2N 40850 | Part of proof of Lemma K o... |
| cdlemk20-2N 40851 | Part of proof of Lemma K o... |
| cdlemk22 40852 | Part of proof of Lemma K o... |
| cdlemk30 40853 | Part of proof of Lemma K o... |
| cdlemkuu 40854 | Convert between function a... |
| cdlemk31 40855 | Part of proof of Lemma K o... |
| cdlemk32 40856 | Part of proof of Lemma K o... |
| cdlemkuel-3 40857 | Part of proof of Lemma K o... |
| cdlemkuv2-3N 40858 | Part of proof of Lemma K o... |
| cdlemk18-3N 40859 | Part of proof of Lemma K o... |
| cdlemk22-3 40860 | Part of proof of Lemma K o... |
| cdlemk23-3 40861 | Part of proof of Lemma K o... |
| cdlemk24-3 40862 | Part of proof of Lemma K o... |
| cdlemk25-3 40863 | Part of proof of Lemma K o... |
| cdlemk26b-3 40864 | Part of proof of Lemma K o... |
| cdlemk26-3 40865 | Part of proof of Lemma K o... |
| cdlemk27-3 40866 | Part of proof of Lemma K o... |
| cdlemk28-3 40867 | Part of proof of Lemma K o... |
| cdlemk33N 40868 | Part of proof of Lemma K o... |
| cdlemk34 40869 | Part of proof of Lemma K o... |
| cdlemk29-3 40870 | Part of proof of Lemma K o... |
| cdlemk35 40871 | Part of proof of Lemma K o... |
| cdlemk36 40872 | Part of proof of Lemma K o... |
| cdlemk37 40873 | Part of proof of Lemma K o... |
| cdlemk38 40874 | Part of proof of Lemma K o... |
| cdlemk39 40875 | Part of proof of Lemma K o... |
| cdlemk40 40876 | TODO: fix comment. (Contr... |
| cdlemk40t 40877 | TODO: fix comment. (Contr... |
| cdlemk40f 40878 | TODO: fix comment. (Contr... |
| cdlemk41 40879 | Part of proof of Lemma K o... |
| cdlemkfid1N 40880 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid1 40881 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid2N 40882 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid2 40883 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid3N 40884 | TODO: is this useful or sh... |
| cdlemky 40885 | Part of proof of Lemma K o... |
| cdlemkyu 40886 | Convert between function a... |
| cdlemkyuu 40887 | ~ cdlemkyu with some hypot... |
| cdlemk11ta 40888 | Part of proof of Lemma K o... |
| cdlemk19ylem 40889 | Lemma for ~ cdlemk19y . (... |
| cdlemk11tb 40890 | Part of proof of Lemma K o... |
| cdlemk19y 40891 | ~ cdlemk19 with simpler hy... |
| cdlemkid3N 40892 | Lemma for ~ cdlemkid . (C... |
| cdlemkid4 40893 | Lemma for ~ cdlemkid . (C... |
| cdlemkid5 40894 | Lemma for ~ cdlemkid . (C... |
| cdlemkid 40895 | The value of the tau funct... |
| cdlemk35s 40896 | Substitution version of ~ ... |
| cdlemk35s-id 40897 | Substitution version of ~ ... |
| cdlemk39s 40898 | Substitution version of ~ ... |
| cdlemk39s-id 40899 | Substitution version of ~ ... |
| cdlemk42 40900 | Part of proof of Lemma K o... |
| cdlemk19xlem 40901 | Lemma for ~ cdlemk19x . (... |
| cdlemk19x 40902 | ~ cdlemk19 with simpler hy... |
| cdlemk42yN 40903 | Part of proof of Lemma K o... |
| cdlemk11tc 40904 | Part of proof of Lemma K o... |
| cdlemk11t 40905 | Part of proof of Lemma K o... |
| cdlemk45 40906 | Part of proof of Lemma K o... |
| cdlemk46 40907 | Part of proof of Lemma K o... |
| cdlemk47 40908 | Part of proof of Lemma K o... |
| cdlemk48 40909 | Part of proof of Lemma K o... |
| cdlemk49 40910 | Part of proof of Lemma K o... |
| cdlemk50 40911 | Part of proof of Lemma K o... |
| cdlemk51 40912 | Part of proof of Lemma K o... |
| cdlemk52 40913 | Part of proof of Lemma K o... |
| cdlemk53a 40914 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53b 40915 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53 40916 | Part of proof of Lemma K o... |
| cdlemk54 40917 | Part of proof of Lemma K o... |
| cdlemk55a 40918 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55b 40919 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55 40920 | Part of proof of Lemma K o... |
| cdlemkyyN 40921 | Part of proof of Lemma K o... |
| cdlemk43N 40922 | Part of proof of Lemma K o... |
| cdlemk35u 40923 | Substitution version of ~ ... |
| cdlemk55u1 40924 | Lemma for ~ cdlemk55u . (... |
| cdlemk55u 40925 | Part of proof of Lemma K o... |
| cdlemk39u1 40926 | Lemma for ~ cdlemk39u . (... |
| cdlemk39u 40927 | Part of proof of Lemma K o... |
| cdlemk19u1 40928 | ~ cdlemk19 with simpler hy... |
| cdlemk19u 40929 | Part of Lemma K of [Crawle... |
| cdlemk56 40930 | Part of Lemma K of [Crawle... |
| cdlemk19w 40931 | Use a fixed element to eli... |
| cdlemk56w 40932 | Use a fixed element to eli... |
| cdlemk 40933 | Lemma K of [Crawley] p. 11... |
| tendoex 40934 | Generalization of Lemma K ... |
| cdleml1N 40935 | Part of proof of Lemma L o... |
| cdleml2N 40936 | Part of proof of Lemma L o... |
| cdleml3N 40937 | Part of proof of Lemma L o... |
| cdleml4N 40938 | Part of proof of Lemma L o... |
| cdleml5N 40939 | Part of proof of Lemma L o... |
| cdleml6 40940 | Part of proof of Lemma L o... |
| cdleml7 40941 | Part of proof of Lemma L o... |
| cdleml8 40942 | Part of proof of Lemma L o... |
| cdleml9 40943 | Part of proof of Lemma L o... |
| dva1dim 40944 | Two expressions for the 1-... |
| dvhb1dimN 40945 | Two expressions for the 1-... |
| erng1lem 40946 | Value of the endomorphism ... |
| erngdvlem1 40947 | Lemma for ~ eringring . (... |
| erngdvlem2N 40948 | Lemma for ~ eringring . (... |
| erngdvlem3 40949 | Lemma for ~ eringring . (... |
| erngdvlem4 40950 | Lemma for ~ erngdv . (Con... |
| eringring 40951 | An endomorphism ring is a ... |
| erngdv 40952 | An endomorphism ring is a ... |
| erng0g 40953 | The division ring zero of ... |
| erng1r 40954 | The division ring unity of... |
| erngdvlem1-rN 40955 | Lemma for ~ eringring . (... |
| erngdvlem2-rN 40956 | Lemma for ~ eringring . (... |
| erngdvlem3-rN 40957 | Lemma for ~ eringring . (... |
| erngdvlem4-rN 40958 | Lemma for ~ erngdv . (Con... |
| erngring-rN 40959 | An endomorphism ring is a ... |
| erngdv-rN 40960 | An endomorphism ring is a ... |
| dvafset 40963 | The constructed partial ve... |
| dvaset 40964 | The constructed partial ve... |
| dvasca 40965 | The ring base set of the c... |
| dvabase 40966 | The ring base set of the c... |
| dvafplusg 40967 | Ring addition operation fo... |
| dvaplusg 40968 | Ring addition operation fo... |
| dvaplusgv 40969 | Ring addition operation fo... |
| dvafmulr 40970 | Ring multiplication operat... |
| dvamulr 40971 | Ring multiplication operat... |
| dvavbase 40972 | The vectors (vector base s... |
| dvafvadd 40973 | The vector sum operation f... |
| dvavadd 40974 | Ring addition operation fo... |
| dvafvsca 40975 | Ring addition operation fo... |
| dvavsca 40976 | Ring addition operation fo... |
| tendospcl 40977 | Closure of endomorphism sc... |
| tendospass 40978 | Associative law for endomo... |
| tendospdi1 40979 | Forward distributive law f... |
| tendocnv 40980 | Converse of a trace-preser... |
| tendospdi2 40981 | Reverse distributive law f... |
| tendospcanN 40982 | Cancellation law for trace... |
| dvaabl 40983 | The constructed partial ve... |
| dvalveclem 40984 | Lemma for ~ dvalvec . (Co... |
| dvalvec 40985 | The constructed partial ve... |
| dva0g 40986 | The zero vector of partial... |
| diaffval 40989 | The partial isomorphism A ... |
| diafval 40990 | The partial isomorphism A ... |
| diaval 40991 | The partial isomorphism A ... |
| diaelval 40992 | Member of the partial isom... |
| diafn 40993 | Functionality and domain o... |
| diadm 40994 | Domain of the partial isom... |
| diaeldm 40995 | Member of domain of the pa... |
| diadmclN 40996 | A member of domain of the ... |
| diadmleN 40997 | A member of domain of the ... |
| dian0 40998 | The value of the partial i... |
| dia0eldmN 40999 | The lattice zero belongs t... |
| dia1eldmN 41000 | The fiducial hyperplane (t... |
| diass 41001 | The value of the partial i... |
| diael 41002 | A member of the value of t... |
| diatrl 41003 | Trace of a member of the p... |
| diaelrnN 41004 | Any value of the partial i... |
| dialss 41005 | The value of partial isomo... |
| diaord 41006 | The partial isomorphism A ... |
| dia11N 41007 | The partial isomorphism A ... |
| diaf11N 41008 | The partial isomorphism A ... |
| diaclN 41009 | Closure of partial isomorp... |
| diacnvclN 41010 | Closure of partial isomorp... |
| dia0 41011 | The value of the partial i... |
| dia1N 41012 | The value of the partial i... |
| dia1elN 41013 | The largest subspace in th... |
| diaglbN 41014 | Partial isomorphism A of a... |
| diameetN 41015 | Partial isomorphism A of a... |
| diainN 41016 | Inverse partial isomorphis... |
| diaintclN 41017 | The intersection of partia... |
| diasslssN 41018 | The partial isomorphism A ... |
| diassdvaN 41019 | The partial isomorphism A ... |
| dia1dim 41020 | Two expressions for the 1-... |
| dia1dim2 41021 | Two expressions for a 1-di... |
| dia1dimid 41022 | A vector (translation) bel... |
| dia2dimlem1 41023 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem2 41024 | Lemma for ~ dia2dim . Def... |
| dia2dimlem3 41025 | Lemma for ~ dia2dim . Def... |
| dia2dimlem4 41026 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem5 41027 | Lemma for ~ dia2dim . The... |
| dia2dimlem6 41028 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem7 41029 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem8 41030 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem9 41031 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem10 41032 | Lemma for ~ dia2dim . Con... |
| dia2dimlem11 41033 | Lemma for ~ dia2dim . Con... |
| dia2dimlem12 41034 | Lemma for ~ dia2dim . Obt... |
| dia2dimlem13 41035 | Lemma for ~ dia2dim . Eli... |
| dia2dim 41036 | A two-dimensional subspace... |
| dvhfset 41039 | The constructed full vecto... |
| dvhset 41040 | The constructed full vecto... |
| dvhsca 41041 | The ring of scalars of the... |
| dvhbase 41042 | The ring base set of the c... |
| dvhfplusr 41043 | Ring addition operation fo... |
| dvhfmulr 41044 | Ring multiplication operat... |
| dvhmulr 41045 | Ring multiplication operat... |
| dvhvbase 41046 | The vectors (vector base s... |
| dvhelvbasei 41047 | Vector membership in the c... |
| dvhvaddcbv 41048 | Change bound variables to ... |
| dvhvaddval 41049 | The vector sum operation f... |
| dvhfvadd 41050 | The vector sum operation f... |
| dvhvadd 41051 | The vector sum operation f... |
| dvhopvadd 41052 | The vector sum operation f... |
| dvhopvadd2 41053 | The vector sum operation f... |
| dvhvaddcl 41054 | Closure of the vector sum ... |
| dvhvaddcomN 41055 | Commutativity of vector su... |
| dvhvaddass 41056 | Associativity of vector su... |
| dvhvscacbv 41057 | Change bound variables to ... |
| dvhvscaval 41058 | The scalar product operati... |
| dvhfvsca 41059 | Scalar product operation f... |
| dvhvsca 41060 | Scalar product operation f... |
| dvhopvsca 41061 | Scalar product operation f... |
| dvhvscacl 41062 | Closure of the scalar prod... |
| tendoinvcl 41063 | Closure of multiplicative ... |
| tendolinv 41064 | Left multiplicative invers... |
| tendorinv 41065 | Right multiplicative inver... |
| dvhgrp 41066 | The full vector space ` U ... |
| dvhlveclem 41067 | Lemma for ~ dvhlvec . TOD... |
| dvhlvec 41068 | The full vector space ` U ... |
| dvhlmod 41069 | The full vector space ` U ... |
| dvh0g 41070 | The zero vector of vector ... |
| dvheveccl 41071 | Properties of a unit vecto... |
| dvhopclN 41072 | Closure of a ` DVecH ` vec... |
| dvhopaddN 41073 | Sum of ` DVecH ` vectors e... |
| dvhopspN 41074 | Scalar product of ` DVecH ... |
| dvhopN 41075 | Decompose a ` DVecH ` vect... |
| dvhopellsm 41076 | Ordered pair membership in... |
| cdlemm10N 41077 | The image of the map ` G `... |
| docaffvalN 41080 | Subspace orthocomplement f... |
| docafvalN 41081 | Subspace orthocomplement f... |
| docavalN 41082 | Subspace orthocomplement f... |
| docaclN 41083 | Closure of subspace orthoc... |
| diaocN 41084 | Value of partial isomorphi... |
| doca2N 41085 | Double orthocomplement of ... |
| doca3N 41086 | Double orthocomplement of ... |
| dvadiaN 41087 | Any closed subspace is a m... |
| diarnN 41088 | Partial isomorphism A maps... |
| diaf1oN 41089 | The partial isomorphism A ... |
| djaffvalN 41092 | Subspace join for ` DVecA ... |
| djafvalN 41093 | Subspace join for ` DVecA ... |
| djavalN 41094 | Subspace join for ` DVecA ... |
| djaclN 41095 | Closure of subspace join f... |
| djajN 41096 | Transfer lattice join to `... |
| dibffval 41099 | The partial isomorphism B ... |
| dibfval 41100 | The partial isomorphism B ... |
| dibval 41101 | The partial isomorphism B ... |
| dibopelvalN 41102 | Member of the partial isom... |
| dibval2 41103 | Value of the partial isomo... |
| dibopelval2 41104 | Member of the partial isom... |
| dibval3N 41105 | Value of the partial isomo... |
| dibelval3 41106 | Member of the partial isom... |
| dibopelval3 41107 | Member of the partial isom... |
| dibelval1st 41108 | Membership in value of the... |
| dibelval1st1 41109 | Membership in value of the... |
| dibelval1st2N 41110 | Membership in value of the... |
| dibelval2nd 41111 | Membership in value of the... |
| dibn0 41112 | The value of the partial i... |
| dibfna 41113 | Functionality and domain o... |
| dibdiadm 41114 | Domain of the partial isom... |
| dibfnN 41115 | Functionality and domain o... |
| dibdmN 41116 | Domain of the partial isom... |
| dibeldmN 41117 | Member of domain of the pa... |
| dibord 41118 | The isomorphism B for a la... |
| dib11N 41119 | The isomorphism B for a la... |
| dibf11N 41120 | The partial isomorphism A ... |
| dibclN 41121 | Closure of partial isomorp... |
| dibvalrel 41122 | The value of partial isomo... |
| dib0 41123 | The value of partial isomo... |
| dib1dim 41124 | Two expressions for the 1-... |
| dibglbN 41125 | Partial isomorphism B of a... |
| dibintclN 41126 | The intersection of partia... |
| dib1dim2 41127 | Two expressions for a 1-di... |
| dibss 41128 | The partial isomorphism B ... |
| diblss 41129 | The value of partial isomo... |
| diblsmopel 41130 | Membership in subspace sum... |
| dicffval 41133 | The partial isomorphism C ... |
| dicfval 41134 | The partial isomorphism C ... |
| dicval 41135 | The partial isomorphism C ... |
| dicopelval 41136 | Membership in value of the... |
| dicelvalN 41137 | Membership in value of the... |
| dicval2 41138 | The partial isomorphism C ... |
| dicelval3 41139 | Member of the partial isom... |
| dicopelval2 41140 | Membership in value of the... |
| dicelval2N 41141 | Membership in value of the... |
| dicfnN 41142 | Functionality and domain o... |
| dicdmN 41143 | Domain of the partial isom... |
| dicvalrelN 41144 | The value of partial isomo... |
| dicssdvh 41145 | The partial isomorphism C ... |
| dicelval1sta 41146 | Membership in value of the... |
| dicelval1stN 41147 | Membership in value of the... |
| dicelval2nd 41148 | Membership in value of the... |
| dicvaddcl 41149 | Membership in value of the... |
| dicvscacl 41150 | Membership in value of the... |
| dicn0 41151 | The value of the partial i... |
| diclss 41152 | The value of partial isomo... |
| diclspsn 41153 | The value of isomorphism C... |
| cdlemn2 41154 | Part of proof of Lemma N o... |
| cdlemn2a 41155 | Part of proof of Lemma N o... |
| cdlemn3 41156 | Part of proof of Lemma N o... |
| cdlemn4 41157 | Part of proof of Lemma N o... |
| cdlemn4a 41158 | Part of proof of Lemma N o... |
| cdlemn5pre 41159 | Part of proof of Lemma N o... |
| cdlemn5 41160 | Part of proof of Lemma N o... |
| cdlemn6 41161 | Part of proof of Lemma N o... |
| cdlemn7 41162 | Part of proof of Lemma N o... |
| cdlemn8 41163 | Part of proof of Lemma N o... |
| cdlemn9 41164 | Part of proof of Lemma N o... |
| cdlemn10 41165 | Part of proof of Lemma N o... |
| cdlemn11a 41166 | Part of proof of Lemma N o... |
| cdlemn11b 41167 | Part of proof of Lemma N o... |
| cdlemn11c 41168 | Part of proof of Lemma N o... |
| cdlemn11pre 41169 | Part of proof of Lemma N o... |
| cdlemn11 41170 | Part of proof of Lemma N o... |
| cdlemn 41171 | Lemma N of [Crawley] p. 12... |
| dihordlem6 41172 | Part of proof of Lemma N o... |
| dihordlem7 41173 | Part of proof of Lemma N o... |
| dihordlem7b 41174 | Part of proof of Lemma N o... |
| dihjustlem 41175 | Part of proof after Lemma ... |
| dihjust 41176 | Part of proof after Lemma ... |
| dihord1 41177 | Part of proof after Lemma ... |
| dihord2a 41178 | Part of proof after Lemma ... |
| dihord2b 41179 | Part of proof after Lemma ... |
| dihord2cN 41180 | Part of proof after Lemma ... |
| dihord11b 41181 | Part of proof after Lemma ... |
| dihord10 41182 | Part of proof after Lemma ... |
| dihord11c 41183 | Part of proof after Lemma ... |
| dihord2pre 41184 | Part of proof after Lemma ... |
| dihord2pre2 41185 | Part of proof after Lemma ... |
| dihord2 41186 | Part of proof after Lemma ... |
| dihffval 41189 | The isomorphism H for a la... |
| dihfval 41190 | Isomorphism H for a lattic... |
| dihval 41191 | Value of isomorphism H for... |
| dihvalc 41192 | Value of isomorphism H for... |
| dihlsscpre 41193 | Closure of isomorphism H f... |
| dihvalcqpre 41194 | Value of isomorphism H for... |
| dihvalcq 41195 | Value of isomorphism H for... |
| dihvalb 41196 | Value of isomorphism H for... |
| dihopelvalbN 41197 | Ordered pair member of the... |
| dihvalcqat 41198 | Value of isomorphism H for... |
| dih1dimb 41199 | Two expressions for a 1-di... |
| dih1dimb2 41200 | Isomorphism H at an atom u... |
| dih1dimc 41201 | Isomorphism H at an atom n... |
| dib2dim 41202 | Extend ~ dia2dim to partia... |
| dih2dimb 41203 | Extend ~ dib2dim to isomor... |
| dih2dimbALTN 41204 | Extend ~ dia2dim to isomor... |
| dihopelvalcqat 41205 | Ordered pair member of the... |
| dihvalcq2 41206 | Value of isomorphism H for... |
| dihopelvalcpre 41207 | Member of value of isomorp... |
| dihopelvalc 41208 | Member of value of isomorp... |
| dihlss 41209 | The value of isomorphism H... |
| dihss 41210 | The value of isomorphism H... |
| dihssxp 41211 | An isomorphism H value is ... |
| dihopcl 41212 | Closure of an ordered pair... |
| xihopellsmN 41213 | Ordered pair membership in... |
| dihopellsm 41214 | Ordered pair membership in... |
| dihord6apre 41215 | Part of proof that isomorp... |
| dihord3 41216 | The isomorphism H for a la... |
| dihord4 41217 | The isomorphism H for a la... |
| dihord5b 41218 | Part of proof that isomorp... |
| dihord6b 41219 | Part of proof that isomorp... |
| dihord6a 41220 | Part of proof that isomorp... |
| dihord5apre 41221 | Part of proof that isomorp... |
| dihord5a 41222 | Part of proof that isomorp... |
| dihord 41223 | The isomorphism H is order... |
| dih11 41224 | The isomorphism H is one-t... |
| dihf11lem 41225 | Functionality of the isomo... |
| dihf11 41226 | The isomorphism H for a la... |
| dihfn 41227 | Functionality and domain o... |
| dihdm 41228 | Domain of isomorphism H. (... |
| dihcl 41229 | Closure of isomorphism H. ... |
| dihcnvcl 41230 | Closure of isomorphism H c... |
| dihcnvid1 41231 | The converse isomorphism o... |
| dihcnvid2 41232 | The isomorphism of a conve... |
| dihcnvord 41233 | Ordering property for conv... |
| dihcnv11 41234 | The converse of isomorphis... |
| dihsslss 41235 | The isomorphism H maps to ... |
| dihrnlss 41236 | The isomorphism H maps to ... |
| dihrnss 41237 | The isomorphism H maps to ... |
| dihvalrel 41238 | The value of isomorphism H... |
| dih0 41239 | The value of isomorphism H... |
| dih0bN 41240 | A lattice element is zero ... |
| dih0vbN 41241 | A vector is zero iff its s... |
| dih0cnv 41242 | The isomorphism H converse... |
| dih0rn 41243 | The zero subspace belongs ... |
| dih0sb 41244 | A subspace is zero iff the... |
| dih1 41245 | The value of isomorphism H... |
| dih1rn 41246 | The full vector space belo... |
| dih1cnv 41247 | The isomorphism H converse... |
| dihwN 41248 | Value of isomorphism H at ... |
| dihmeetlem1N 41249 | Isomorphism H of a conjunc... |
| dihglblem5apreN 41250 | A conjunction property of ... |
| dihglblem5aN 41251 | A conjunction property of ... |
| dihglblem2aN 41252 | Lemma for isomorphism H of... |
| dihglblem2N 41253 | The GLB of a set of lattic... |
| dihglblem3N 41254 | Isomorphism H of a lattice... |
| dihglblem3aN 41255 | Isomorphism H of a lattice... |
| dihglblem4 41256 | Isomorphism H of a lattice... |
| dihglblem5 41257 | Isomorphism H of a lattice... |
| dihmeetlem2N 41258 | Isomorphism H of a conjunc... |
| dihglbcpreN 41259 | Isomorphism H of a lattice... |
| dihglbcN 41260 | Isomorphism H of a lattice... |
| dihmeetcN 41261 | Isomorphism H of a lattice... |
| dihmeetbN 41262 | Isomorphism H of a lattice... |
| dihmeetbclemN 41263 | Lemma for isomorphism H of... |
| dihmeetlem3N 41264 | Lemma for isomorphism H of... |
| dihmeetlem4preN 41265 | Lemma for isomorphism H of... |
| dihmeetlem4N 41266 | Lemma for isomorphism H of... |
| dihmeetlem5 41267 | Part of proof that isomorp... |
| dihmeetlem6 41268 | Lemma for isomorphism H of... |
| dihmeetlem7N 41269 | Lemma for isomorphism H of... |
| dihjatc1 41270 | Lemma for isomorphism H of... |
| dihjatc2N 41271 | Isomorphism H of join with... |
| dihjatc3 41272 | Isomorphism H of join with... |
| dihmeetlem8N 41273 | Lemma for isomorphism H of... |
| dihmeetlem9N 41274 | Lemma for isomorphism H of... |
| dihmeetlem10N 41275 | Lemma for isomorphism H of... |
| dihmeetlem11N 41276 | Lemma for isomorphism H of... |
| dihmeetlem12N 41277 | Lemma for isomorphism H of... |
| dihmeetlem13N 41278 | Lemma for isomorphism H of... |
| dihmeetlem14N 41279 | Lemma for isomorphism H of... |
| dihmeetlem15N 41280 | Lemma for isomorphism H of... |
| dihmeetlem16N 41281 | Lemma for isomorphism H of... |
| dihmeetlem17N 41282 | Lemma for isomorphism H of... |
| dihmeetlem18N 41283 | Lemma for isomorphism H of... |
| dihmeetlem19N 41284 | Lemma for isomorphism H of... |
| dihmeetlem20N 41285 | Lemma for isomorphism H of... |
| dihmeetALTN 41286 | Isomorphism H of a lattice... |
| dih1dimatlem0 41287 | Lemma for ~ dih1dimat . (... |
| dih1dimatlem 41288 | Lemma for ~ dih1dimat . (... |
| dih1dimat 41289 | Any 1-dimensional subspace... |
| dihlsprn 41290 | The span of a vector belon... |
| dihlspsnssN 41291 | A subspace included in a 1... |
| dihlspsnat 41292 | The inverse isomorphism H ... |
| dihatlat 41293 | The isomorphism H of an at... |
| dihat 41294 | There exists at least one ... |
| dihpN 41295 | The value of isomorphism H... |
| dihlatat 41296 | The reverse isomorphism H ... |
| dihatexv 41297 | There is a nonzero vector ... |
| dihatexv2 41298 | There is a nonzero vector ... |
| dihglblem6 41299 | Isomorphism H of a lattice... |
| dihglb 41300 | Isomorphism H of a lattice... |
| dihglb2 41301 | Isomorphism H of a lattice... |
| dihmeet 41302 | Isomorphism H of a lattice... |
| dihintcl 41303 | The intersection of closed... |
| dihmeetcl 41304 | Closure of closed subspace... |
| dihmeet2 41305 | Reverse isomorphism H of a... |
| dochffval 41308 | Subspace orthocomplement f... |
| dochfval 41309 | Subspace orthocomplement f... |
| dochval 41310 | Subspace orthocomplement f... |
| dochval2 41311 | Subspace orthocomplement f... |
| dochcl 41312 | Closure of subspace orthoc... |
| dochlss 41313 | A subspace orthocomplement... |
| dochssv 41314 | A subspace orthocomplement... |
| dochfN 41315 | Domain and codomain of the... |
| dochvalr 41316 | Orthocomplement of a close... |
| doch0 41317 | Orthocomplement of the zer... |
| doch1 41318 | Orthocomplement of the uni... |
| dochoc0 41319 | The zero subspace is close... |
| dochoc1 41320 | The unit subspace (all vec... |
| dochvalr2 41321 | Orthocomplement of a close... |
| dochvalr3 41322 | Orthocomplement of a close... |
| doch2val2 41323 | Double orthocomplement for... |
| dochss 41324 | Subset law for orthocomple... |
| dochocss 41325 | Double negative law for or... |
| dochoc 41326 | Double negative law for or... |
| dochsscl 41327 | If a set of vectors is inc... |
| dochoccl 41328 | A set of vectors is closed... |
| dochord 41329 | Ordering law for orthocomp... |
| dochord2N 41330 | Ordering law for orthocomp... |
| dochord3 41331 | Ordering law for orthocomp... |
| doch11 41332 | Orthocomplement is one-to-... |
| dochsordN 41333 | Strict ordering law for or... |
| dochn0nv 41334 | An orthocomplement is nonz... |
| dihoml4c 41335 | Version of ~ dihoml4 with ... |
| dihoml4 41336 | Orthomodular law for const... |
| dochspss 41337 | The span of a set of vecto... |
| dochocsp 41338 | The span of an orthocomple... |
| dochspocN 41339 | The span of an orthocomple... |
| dochocsn 41340 | The double orthocomplement... |
| dochsncom 41341 | Swap vectors in an orthoco... |
| dochsat 41342 | The double orthocomplement... |
| dochshpncl 41343 | If a hyperplane is not clo... |
| dochlkr 41344 | Equivalent conditions for ... |
| dochkrshp 41345 | The closure of a kernel is... |
| dochkrshp2 41346 | Properties of the closure ... |
| dochkrshp3 41347 | Properties of the closure ... |
| dochkrshp4 41348 | Properties of the closure ... |
| dochdmj1 41349 | De Morgan-like law for sub... |
| dochnoncon 41350 | Law of noncontradiction. ... |
| dochnel2 41351 | A nonzero member of a subs... |
| dochnel 41352 | A nonzero vector doesn't b... |
| djhffval 41355 | Subspace join for ` DVecH ... |
| djhfval 41356 | Subspace join for ` DVecH ... |
| djhval 41357 | Subspace join for ` DVecH ... |
| djhval2 41358 | Value of subspace join for... |
| djhcl 41359 | Closure of subspace join f... |
| djhlj 41360 | Transfer lattice join to `... |
| djhljjN 41361 | Lattice join in terms of `... |
| djhjlj 41362 | ` DVecH ` vector space clo... |
| djhj 41363 | ` DVecH ` vector space clo... |
| djhcom 41364 | Subspace join commutes. (... |
| djhspss 41365 | Subspace span of union is ... |
| djhsumss 41366 | Subspace sum is a subset o... |
| dihsumssj 41367 | The subspace sum of two is... |
| djhunssN 41368 | Subspace union is a subset... |
| dochdmm1 41369 | De Morgan-like law for clo... |
| djhexmid 41370 | Excluded middle property o... |
| djh01 41371 | Closed subspace join with ... |
| djh02 41372 | Closed subspace join with ... |
| djhlsmcl 41373 | A closed subspace sum equa... |
| djhcvat42 41374 | A covering property. ( ~ ... |
| dihjatb 41375 | Isomorphism H of lattice j... |
| dihjatc 41376 | Isomorphism H of lattice j... |
| dihjatcclem1 41377 | Lemma for isomorphism H of... |
| dihjatcclem2 41378 | Lemma for isomorphism H of... |
| dihjatcclem3 41379 | Lemma for ~ dihjatcc . (C... |
| dihjatcclem4 41380 | Lemma for isomorphism H of... |
| dihjatcc 41381 | Isomorphism H of lattice j... |
| dihjat 41382 | Isomorphism H of lattice j... |
| dihprrnlem1N 41383 | Lemma for ~ dihprrn , show... |
| dihprrnlem2 41384 | Lemma for ~ dihprrn . (Co... |
| dihprrn 41385 | The span of a vector pair ... |
| djhlsmat 41386 | The sum of two subspace at... |
| dihjat1lem 41387 | Subspace sum of a closed s... |
| dihjat1 41388 | Subspace sum of a closed s... |
| dihsmsprn 41389 | Subspace sum of a closed s... |
| dihjat2 41390 | The subspace sum of a clos... |
| dihjat3 41391 | Isomorphism H of lattice j... |
| dihjat4 41392 | Transfer the subspace sum ... |
| dihjat6 41393 | Transfer the subspace sum ... |
| dihsmsnrn 41394 | The subspace sum of two si... |
| dihsmatrn 41395 | The subspace sum of a clos... |
| dihjat5N 41396 | Transfer lattice join with... |
| dvh4dimat 41397 | There is an atom that is o... |
| dvh3dimatN 41398 | There is an atom that is o... |
| dvh2dimatN 41399 | Given an atom, there exist... |
| dvh1dimat 41400 | There exists an atom. (Co... |
| dvh1dim 41401 | There exists a nonzero vec... |
| dvh4dimlem 41402 | Lemma for ~ dvh4dimN . (C... |
| dvhdimlem 41403 | Lemma for ~ dvh2dim and ~ ... |
| dvh2dim 41404 | There is a vector that is ... |
| dvh3dim 41405 | There is a vector that is ... |
| dvh4dimN 41406 | There is a vector that is ... |
| dvh3dim2 41407 | There is a vector that is ... |
| dvh3dim3N 41408 | There is a vector that is ... |
| dochsnnz 41409 | The orthocomplement of a s... |
| dochsatshp 41410 | The orthocomplement of a s... |
| dochsatshpb 41411 | The orthocomplement of a s... |
| dochsnshp 41412 | The orthocomplement of a n... |
| dochshpsat 41413 | A hyperplane is closed iff... |
| dochkrsat 41414 | The orthocomplement of a k... |
| dochkrsat2 41415 | The orthocomplement of a k... |
| dochsat0 41416 | The orthocomplement of a k... |
| dochkrsm 41417 | The subspace sum of a clos... |
| dochexmidat 41418 | Special case of excluded m... |
| dochexmidlem1 41419 | Lemma for ~ dochexmid . H... |
| dochexmidlem2 41420 | Lemma for ~ dochexmid . (... |
| dochexmidlem3 41421 | Lemma for ~ dochexmid . U... |
| dochexmidlem4 41422 | Lemma for ~ dochexmid . (... |
| dochexmidlem5 41423 | Lemma for ~ dochexmid . (... |
| dochexmidlem6 41424 | Lemma for ~ dochexmid . (... |
| dochexmidlem7 41425 | Lemma for ~ dochexmid . C... |
| dochexmidlem8 41426 | Lemma for ~ dochexmid . T... |
| dochexmid 41427 | Excluded middle law for cl... |
| dochsnkrlem1 41428 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem2 41429 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem3 41430 | Lemma for ~ dochsnkr . (C... |
| dochsnkr 41431 | A (closed) kernel expresse... |
| dochsnkr2 41432 | Kernel of the explicit fun... |
| dochsnkr2cl 41433 | The ` X ` determining func... |
| dochflcl 41434 | Closure of the explicit fu... |
| dochfl1 41435 | The value of the explicit ... |
| dochfln0 41436 | The value of a functional ... |
| dochkr1 41437 | A nonzero functional has a... |
| dochkr1OLDN 41438 | A nonzero functional has a... |
| lpolsetN 41441 | The set of polarities of a... |
| islpolN 41442 | The predicate "is a polari... |
| islpoldN 41443 | Properties that determine ... |
| lpolfN 41444 | Functionality of a polarit... |
| lpolvN 41445 | The polarity of the whole ... |
| lpolconN 41446 | Contraposition property of... |
| lpolsatN 41447 | The polarity of an atomic ... |
| lpolpolsatN 41448 | Property of a polarity. (... |
| dochpolN 41449 | The subspace orthocompleme... |
| lcfl1lem 41450 | Property of a functional w... |
| lcfl1 41451 | Property of a functional w... |
| lcfl2 41452 | Property of a functional w... |
| lcfl3 41453 | Property of a functional w... |
| lcfl4N 41454 | Property of a functional w... |
| lcfl5 41455 | Property of a functional w... |
| lcfl5a 41456 | Property of a functional w... |
| lcfl6lem 41457 | Lemma for ~ lcfl6 . A fun... |
| lcfl7lem 41458 | Lemma for ~ lcfl7N . If t... |
| lcfl6 41459 | Property of a functional w... |
| lcfl7N 41460 | Property of a functional w... |
| lcfl8 41461 | Property of a functional w... |
| lcfl8a 41462 | Property of a functional w... |
| lcfl8b 41463 | Property of a nonzero func... |
| lcfl9a 41464 | Property implying that a f... |
| lclkrlem1 41465 | The set of functionals hav... |
| lclkrlem2a 41466 | Lemma for ~ lclkr . Use ~... |
| lclkrlem2b 41467 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2c 41468 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2d 41469 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2e 41470 | Lemma for ~ lclkr . The k... |
| lclkrlem2f 41471 | Lemma for ~ lclkr . Const... |
| lclkrlem2g 41472 | Lemma for ~ lclkr . Compa... |
| lclkrlem2h 41473 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2i 41474 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2j 41475 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2k 41476 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2l 41477 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2m 41478 | Lemma for ~ lclkr . Const... |
| lclkrlem2n 41479 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2o 41480 | Lemma for ~ lclkr . When ... |
| lclkrlem2p 41481 | Lemma for ~ lclkr . When ... |
| lclkrlem2q 41482 | Lemma for ~ lclkr . The s... |
| lclkrlem2r 41483 | Lemma for ~ lclkr . When ... |
| lclkrlem2s 41484 | Lemma for ~ lclkr . Thus,... |
| lclkrlem2t 41485 | Lemma for ~ lclkr . We el... |
| lclkrlem2u 41486 | Lemma for ~ lclkr . ~ lclk... |
| lclkrlem2v 41487 | Lemma for ~ lclkr . When ... |
| lclkrlem2w 41488 | Lemma for ~ lclkr . This ... |
| lclkrlem2x 41489 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2y 41490 | Lemma for ~ lclkr . Resta... |
| lclkrlem2 41491 | The set of functionals hav... |
| lclkr 41492 | The set of functionals wit... |
| lcfls1lem 41493 | Property of a functional w... |
| lcfls1N 41494 | Property of a functional w... |
| lcfls1c 41495 | Property of a functional w... |
| lclkrslem1 41496 | The set of functionals hav... |
| lclkrslem2 41497 | The set of functionals hav... |
| lclkrs 41498 | The set of functionals hav... |
| lclkrs2 41499 | The set of functionals wit... |
| lcfrvalsnN 41500 | Reconstruction from the du... |
| lcfrlem1 41501 | Lemma for ~ lcfr . Note t... |
| lcfrlem2 41502 | Lemma for ~ lcfr . (Contr... |
| lcfrlem3 41503 | Lemma for ~ lcfr . (Contr... |
| lcfrlem4 41504 | Lemma for ~ lcfr . (Contr... |
| lcfrlem5 41505 | Lemma for ~ lcfr . The se... |
| lcfrlem6 41506 | Lemma for ~ lcfr . Closur... |
| lcfrlem7 41507 | Lemma for ~ lcfr . Closur... |
| lcfrlem8 41508 | Lemma for ~ lcf1o and ~ lc... |
| lcfrlem9 41509 | Lemma for ~ lcf1o . (This... |
| lcf1o 41510 | Define a function ` J ` th... |
| lcfrlem10 41511 | Lemma for ~ lcfr . (Contr... |
| lcfrlem11 41512 | Lemma for ~ lcfr . (Contr... |
| lcfrlem12N 41513 | Lemma for ~ lcfr . (Contr... |
| lcfrlem13 41514 | Lemma for ~ lcfr . (Contr... |
| lcfrlem14 41515 | Lemma for ~ lcfr . (Contr... |
| lcfrlem15 41516 | Lemma for ~ lcfr . (Contr... |
| lcfrlem16 41517 | Lemma for ~ lcfr . (Contr... |
| lcfrlem17 41518 | Lemma for ~ lcfr . Condit... |
| lcfrlem18 41519 | Lemma for ~ lcfr . (Contr... |
| lcfrlem19 41520 | Lemma for ~ lcfr . (Contr... |
| lcfrlem20 41521 | Lemma for ~ lcfr . (Contr... |
| lcfrlem21 41522 | Lemma for ~ lcfr . (Contr... |
| lcfrlem22 41523 | Lemma for ~ lcfr . (Contr... |
| lcfrlem23 41524 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem24 41525 | Lemma for ~ lcfr . (Contr... |
| lcfrlem25 41526 | Lemma for ~ lcfr . Specia... |
| lcfrlem26 41527 | Lemma for ~ lcfr . Specia... |
| lcfrlem27 41528 | Lemma for ~ lcfr . Specia... |
| lcfrlem28 41529 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem29 41530 | Lemma for ~ lcfr . (Contr... |
| lcfrlem30 41531 | Lemma for ~ lcfr . (Contr... |
| lcfrlem31 41532 | Lemma for ~ lcfr . (Contr... |
| lcfrlem32 41533 | Lemma for ~ lcfr . (Contr... |
| lcfrlem33 41534 | Lemma for ~ lcfr . (Contr... |
| lcfrlem34 41535 | Lemma for ~ lcfr . (Contr... |
| lcfrlem35 41536 | Lemma for ~ lcfr . (Contr... |
| lcfrlem36 41537 | Lemma for ~ lcfr . (Contr... |
| lcfrlem37 41538 | Lemma for ~ lcfr . (Contr... |
| lcfrlem38 41539 | Lemma for ~ lcfr . Combin... |
| lcfrlem39 41540 | Lemma for ~ lcfr . Elimin... |
| lcfrlem40 41541 | Lemma for ~ lcfr . Elimin... |
| lcfrlem41 41542 | Lemma for ~ lcfr . Elimin... |
| lcfrlem42 41543 | Lemma for ~ lcfr . Elimin... |
| lcfr 41544 | Reconstruction of a subspa... |
| lcdfval 41547 | Dual vector space of funct... |
| lcdval 41548 | Dual vector space of funct... |
| lcdval2 41549 | Dual vector space of funct... |
| lcdlvec 41550 | The dual vector space of f... |
| lcdlmod 41551 | The dual vector space of f... |
| lcdvbase 41552 | Vector base set of a dual ... |
| lcdvbasess 41553 | The vector base set of the... |
| lcdvbaselfl 41554 | A vector in the base set o... |
| lcdvbasecl 41555 | Closure of the value of a ... |
| lcdvadd 41556 | Vector addition for the cl... |
| lcdvaddval 41557 | The value of the value of ... |
| lcdsca 41558 | The ring of scalars of the... |
| lcdsbase 41559 | Base set of scalar ring fo... |
| lcdsadd 41560 | Scalar addition for the cl... |
| lcdsmul 41561 | Scalar multiplication for ... |
| lcdvs 41562 | Scalar product for the clo... |
| lcdvsval 41563 | Value of scalar product op... |
| lcdvscl 41564 | The scalar product operati... |
| lcdlssvscl 41565 | Closure of scalar product ... |
| lcdvsass 41566 | Associative law for scalar... |
| lcd0 41567 | The zero scalar of the clo... |
| lcd1 41568 | The unit scalar of the clo... |
| lcdneg 41569 | The unit scalar of the clo... |
| lcd0v 41570 | The zero functional in the... |
| lcd0v2 41571 | The zero functional in the... |
| lcd0vvalN 41572 | Value of the zero function... |
| lcd0vcl 41573 | Closure of the zero functi... |
| lcd0vs 41574 | A scalar zero times a func... |
| lcdvs0N 41575 | A scalar times the zero fu... |
| lcdvsub 41576 | The value of vector subtra... |
| lcdvsubval 41577 | The value of the value of ... |
| lcdlss 41578 | Subspaces of a dual vector... |
| lcdlss2N 41579 | Subspaces of a dual vector... |
| lcdlsp 41580 | Span in the set of functio... |
| lcdlkreqN 41581 | Colinear functionals have ... |
| lcdlkreq2N 41582 | Colinear functionals have ... |
| mapdffval 41585 | Projectivity from vector s... |
| mapdfval 41586 | Projectivity from vector s... |
| mapdval 41587 | Value of projectivity from... |
| mapdvalc 41588 | Value of projectivity from... |
| mapdval2N 41589 | Value of projectivity from... |
| mapdval3N 41590 | Value of projectivity from... |
| mapdval4N 41591 | Value of projectivity from... |
| mapdval5N 41592 | Value of projectivity from... |
| mapdordlem1a 41593 | Lemma for ~ mapdord . (Co... |
| mapdordlem1bN 41594 | Lemma for ~ mapdord . (Co... |
| mapdordlem1 41595 | Lemma for ~ mapdord . (Co... |
| mapdordlem2 41596 | Lemma for ~ mapdord . Ord... |
| mapdord 41597 | Ordering property of the m... |
| mapd11 41598 | The map defined by ~ df-ma... |
| mapddlssN 41599 | The mapping of a subspace ... |
| mapdsn 41600 | Value of the map defined b... |
| mapdsn2 41601 | Value of the map defined b... |
| mapdsn3 41602 | Value of the map defined b... |
| mapd1dim2lem1N 41603 | Value of the map defined b... |
| mapdrvallem2 41604 | Lemma for ~ mapdrval . TO... |
| mapdrvallem3 41605 | Lemma for ~ mapdrval . (C... |
| mapdrval 41606 | Given a dual subspace ` R ... |
| mapd1o 41607 | The map defined by ~ df-ma... |
| mapdrn 41608 | Range of the map defined b... |
| mapdunirnN 41609 | Union of the range of the ... |
| mapdrn2 41610 | Range of the map defined b... |
| mapdcnvcl 41611 | Closure of the converse of... |
| mapdcl 41612 | Closure the value of the m... |
| mapdcnvid1N 41613 | Converse of the value of t... |
| mapdsord 41614 | Strong ordering property o... |
| mapdcl2 41615 | The mapping of a subspace ... |
| mapdcnvid2 41616 | Value of the converse of t... |
| mapdcnvordN 41617 | Ordering property of the c... |
| mapdcnv11N 41618 | The converse of the map de... |
| mapdcv 41619 | Covering property of the c... |
| mapdincl 41620 | Closure of dual subspace i... |
| mapdin 41621 | Subspace intersection is p... |
| mapdlsmcl 41622 | Closure of dual subspace s... |
| mapdlsm 41623 | Subspace sum is preserved ... |
| mapd0 41624 | Projectivity map of the ze... |
| mapdcnvatN 41625 | Atoms are preserved by the... |
| mapdat 41626 | Atoms are preserved by the... |
| mapdspex 41627 | The map of a span equals t... |
| mapdn0 41628 | Transfer nonzero property ... |
| mapdncol 41629 | Transfer non-colinearity f... |
| mapdindp 41630 | Transfer (part of) vector ... |
| mapdpglem1 41631 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2 41632 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2a 41633 | Lemma for ~ mapdpg . (Con... |
| mapdpglem3 41634 | Lemma for ~ mapdpg . Baer... |
| mapdpglem4N 41635 | Lemma for ~ mapdpg . (Con... |
| mapdpglem5N 41636 | Lemma for ~ mapdpg . (Con... |
| mapdpglem6 41637 | Lemma for ~ mapdpg . Baer... |
| mapdpglem8 41638 | Lemma for ~ mapdpg . Baer... |
| mapdpglem9 41639 | Lemma for ~ mapdpg . Baer... |
| mapdpglem10 41640 | Lemma for ~ mapdpg . Baer... |
| mapdpglem11 41641 | Lemma for ~ mapdpg . (Con... |
| mapdpglem12 41642 | Lemma for ~ mapdpg . TODO... |
| mapdpglem13 41643 | Lemma for ~ mapdpg . (Con... |
| mapdpglem14 41644 | Lemma for ~ mapdpg . (Con... |
| mapdpglem15 41645 | Lemma for ~ mapdpg . (Con... |
| mapdpglem16 41646 | Lemma for ~ mapdpg . Baer... |
| mapdpglem17N 41647 | Lemma for ~ mapdpg . Baer... |
| mapdpglem18 41648 | Lemma for ~ mapdpg . Baer... |
| mapdpglem19 41649 | Lemma for ~ mapdpg . Baer... |
| mapdpglem20 41650 | Lemma for ~ mapdpg . Baer... |
| mapdpglem21 41651 | Lemma for ~ mapdpg . (Con... |
| mapdpglem22 41652 | Lemma for ~ mapdpg . Baer... |
| mapdpglem23 41653 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30a 41654 | Lemma for ~ mapdpg . (Con... |
| mapdpglem30b 41655 | Lemma for ~ mapdpg . (Con... |
| mapdpglem25 41656 | Lemma for ~ mapdpg . Baer... |
| mapdpglem26 41657 | Lemma for ~ mapdpg . Baer... |
| mapdpglem27 41658 | Lemma for ~ mapdpg . Baer... |
| mapdpglem29 41659 | Lemma for ~ mapdpg . Baer... |
| mapdpglem28 41660 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30 41661 | Lemma for ~ mapdpg . Baer... |
| mapdpglem31 41662 | Lemma for ~ mapdpg . Baer... |
| mapdpglem24 41663 | Lemma for ~ mapdpg . Exis... |
| mapdpglem32 41664 | Lemma for ~ mapdpg . Uniq... |
| mapdpg 41665 | Part 1 of proof of the fir... |
| baerlem3lem1 41666 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem1 41667 | Lemma for ~ baerlem5a . (... |
| baerlem5blem1 41668 | Lemma for ~ baerlem5b . (... |
| baerlem3lem2 41669 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem2 41670 | Lemma for ~ baerlem5a . (... |
| baerlem5blem2 41671 | Lemma for ~ baerlem5b . (... |
| baerlem3 41672 | An equality that holds whe... |
| baerlem5a 41673 | An equality that holds whe... |
| baerlem5b 41674 | An equality that holds whe... |
| baerlem5amN 41675 | An equality that holds whe... |
| baerlem5bmN 41676 | An equality that holds whe... |
| baerlem5abmN 41677 | An equality that holds whe... |
| mapdindp0 41678 | Vector independence lemma.... |
| mapdindp1 41679 | Vector independence lemma.... |
| mapdindp2 41680 | Vector independence lemma.... |
| mapdindp3 41681 | Vector independence lemma.... |
| mapdindp4 41682 | Vector independence lemma.... |
| mapdhval 41683 | Lemmma for ~~? mapdh . (C... |
| mapdhval0 41684 | Lemmma for ~~? mapdh . (C... |
| mapdhval2 41685 | Lemmma for ~~? mapdh . (C... |
| mapdhcl 41686 | Lemmma for ~~? mapdh . (C... |
| mapdheq 41687 | Lemmma for ~~? mapdh . Th... |
| mapdheq2 41688 | Lemmma for ~~? mapdh . On... |
| mapdheq2biN 41689 | Lemmma for ~~? mapdh . Pa... |
| mapdheq4lem 41690 | Lemma for ~ mapdheq4 . Pa... |
| mapdheq4 41691 | Lemma for ~~? mapdh . Par... |
| mapdh6lem1N 41692 | Lemma for ~ mapdh6N . Par... |
| mapdh6lem2N 41693 | Lemma for ~ mapdh6N . Par... |
| mapdh6aN 41694 | Lemma for ~ mapdh6N . Par... |
| mapdh6b0N 41695 | Lemmma for ~ mapdh6N . (C... |
| mapdh6bN 41696 | Lemmma for ~ mapdh6N . (C... |
| mapdh6cN 41697 | Lemmma for ~ mapdh6N . (C... |
| mapdh6dN 41698 | Lemmma for ~ mapdh6N . (C... |
| mapdh6eN 41699 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6fN 41700 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6gN 41701 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6hN 41702 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6iN 41703 | Lemmma for ~ mapdh6N . El... |
| mapdh6jN 41704 | Lemmma for ~ mapdh6N . El... |
| mapdh6kN 41705 | Lemmma for ~ mapdh6N . El... |
| mapdh6N 41706 | Part (6) of [Baer] p. 47 l... |
| mapdh7eN 41707 | Part (7) of [Baer] p. 48 l... |
| mapdh7cN 41708 | Part (7) of [Baer] p. 48 l... |
| mapdh7dN 41709 | Part (7) of [Baer] p. 48 l... |
| mapdh7fN 41710 | Part (7) of [Baer] p. 48 l... |
| mapdh75e 41711 | Part (7) of [Baer] p. 48 l... |
| mapdh75cN 41712 | Part (7) of [Baer] p. 48 l... |
| mapdh75d 41713 | Part (7) of [Baer] p. 48 l... |
| mapdh75fN 41714 | Part (7) of [Baer] p. 48 l... |
| hvmapffval 41717 | Map from nonzero vectors t... |
| hvmapfval 41718 | Map from nonzero vectors t... |
| hvmapval 41719 | Value of map from nonzero ... |
| hvmapvalvalN 41720 | Value of value of map (i.e... |
| hvmapidN 41721 | The value of the vector to... |
| hvmap1o 41722 | The vector to functional m... |
| hvmapclN 41723 | Closure of the vector to f... |
| hvmap1o2 41724 | The vector to functional m... |
| hvmapcl2 41725 | Closure of the vector to f... |
| hvmaplfl 41726 | The vector to functional m... |
| hvmaplkr 41727 | Kernel of the vector to fu... |
| mapdhvmap 41728 | Relationship between ` map... |
| lspindp5 41729 | Obtain an independent vect... |
| hdmaplem1 41730 | Lemma to convert a frequen... |
| hdmaplem2N 41731 | Lemma to convert a frequen... |
| hdmaplem3 41732 | Lemma to convert a frequen... |
| hdmaplem4 41733 | Lemma to convert a frequen... |
| mapdh8a 41734 | Part of Part (8) in [Baer]... |
| mapdh8aa 41735 | Part of Part (8) in [Baer]... |
| mapdh8ab 41736 | Part of Part (8) in [Baer]... |
| mapdh8ac 41737 | Part of Part (8) in [Baer]... |
| mapdh8ad 41738 | Part of Part (8) in [Baer]... |
| mapdh8b 41739 | Part of Part (8) in [Baer]... |
| mapdh8c 41740 | Part of Part (8) in [Baer]... |
| mapdh8d0N 41741 | Part of Part (8) in [Baer]... |
| mapdh8d 41742 | Part of Part (8) in [Baer]... |
| mapdh8e 41743 | Part of Part (8) in [Baer]... |
| mapdh8g 41744 | Part of Part (8) in [Baer]... |
| mapdh8i 41745 | Part of Part (8) in [Baer]... |
| mapdh8j 41746 | Part of Part (8) in [Baer]... |
| mapdh8 41747 | Part (8) in [Baer] p. 48. ... |
| mapdh9a 41748 | Lemma for part (9) in [Bae... |
| mapdh9aOLDN 41749 | Lemma for part (9) in [Bae... |
| hdmap1ffval 41754 | Preliminary map from vecto... |
| hdmap1fval 41755 | Preliminary map from vecto... |
| hdmap1vallem 41756 | Value of preliminary map f... |
| hdmap1val 41757 | Value of preliminary map f... |
| hdmap1val0 41758 | Value of preliminary map f... |
| hdmap1val2 41759 | Value of preliminary map f... |
| hdmap1eq 41760 | The defining equation for ... |
| hdmap1cbv 41761 | Frequently used lemma to c... |
| hdmap1valc 41762 | Connect the value of the p... |
| hdmap1cl 41763 | Convert closure theorem ~ ... |
| hdmap1eq2 41764 | Convert ~ mapdheq2 to use ... |
| hdmap1eq4N 41765 | Convert ~ mapdheq4 to use ... |
| hdmap1l6lem1 41766 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6lem2 41767 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6a 41768 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6b0N 41769 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6b 41770 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6c 41771 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6d 41772 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6e 41773 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6f 41774 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6g 41775 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6h 41776 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6i 41777 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6j 41778 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6k 41779 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6 41780 | Part (6) of [Baer] p. 47 l... |
| hdmap1eulem 41781 | Lemma for ~ hdmap1eu . TO... |
| hdmap1eulemOLDN 41782 | Lemma for ~ hdmap1euOLDN .... |
| hdmap1eu 41783 | Convert ~ mapdh9a to use t... |
| hdmap1euOLDN 41784 | Convert ~ mapdh9aOLDN to u... |
| hdmapffval 41785 | Map from vectors to functi... |
| hdmapfval 41786 | Map from vectors to functi... |
| hdmapval 41787 | Value of map from vectors ... |
| hdmapfnN 41788 | Functionality of map from ... |
| hdmapcl 41789 | Closure of map from vector... |
| hdmapval2lem 41790 | Lemma for ~ hdmapval2 . (... |
| hdmapval2 41791 | Value of map from vectors ... |
| hdmapval0 41792 | Value of map from vectors ... |
| hdmapeveclem 41793 | Lemma for ~ hdmapevec . T... |
| hdmapevec 41794 | Value of map from vectors ... |
| hdmapevec2 41795 | The inner product of the r... |
| hdmapval3lemN 41796 | Value of map from vectors ... |
| hdmapval3N 41797 | Value of map from vectors ... |
| hdmap10lem 41798 | Lemma for ~ hdmap10 . (Co... |
| hdmap10 41799 | Part 10 in [Baer] p. 48 li... |
| hdmap11lem1 41800 | Lemma for ~ hdmapadd . (C... |
| hdmap11lem2 41801 | Lemma for ~ hdmapadd . (C... |
| hdmapadd 41802 | Part 11 in [Baer] p. 48 li... |
| hdmapeq0 41803 | Part of proof of part 12 i... |
| hdmapnzcl 41804 | Nonzero vector closure of ... |
| hdmapneg 41805 | Part of proof of part 12 i... |
| hdmapsub 41806 | Part of proof of part 12 i... |
| hdmap11 41807 | Part of proof of part 12 i... |
| hdmaprnlem1N 41808 | Part of proof of part 12 i... |
| hdmaprnlem3N 41809 | Part of proof of part 12 i... |
| hdmaprnlem3uN 41810 | Part of proof of part 12 i... |
| hdmaprnlem4tN 41811 | Lemma for ~ hdmaprnN . TO... |
| hdmaprnlem4N 41812 | Part of proof of part 12 i... |
| hdmaprnlem6N 41813 | Part of proof of part 12 i... |
| hdmaprnlem7N 41814 | Part of proof of part 12 i... |
| hdmaprnlem8N 41815 | Part of proof of part 12 i... |
| hdmaprnlem9N 41816 | Part of proof of part 12 i... |
| hdmaprnlem3eN 41817 | Lemma for ~ hdmaprnN . (C... |
| hdmaprnlem10N 41818 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem11N 41819 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem15N 41820 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem16N 41821 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem17N 41822 | Lemma for ~ hdmaprnN . In... |
| hdmaprnN 41823 | Part of proof of part 12 i... |
| hdmapf1oN 41824 | Part 12 in [Baer] p. 49. ... |
| hdmap14lem1a 41825 | Prior to part 14 in [Baer]... |
| hdmap14lem2a 41826 | Prior to part 14 in [Baer]... |
| hdmap14lem1 41827 | Prior to part 14 in [Baer]... |
| hdmap14lem2N 41828 | Prior to part 14 in [Baer]... |
| hdmap14lem3 41829 | Prior to part 14 in [Baer]... |
| hdmap14lem4a 41830 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem4 41831 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem6 41832 | Case where ` F ` is zero. ... |
| hdmap14lem7 41833 | Combine cases of ` F ` . ... |
| hdmap14lem8 41834 | Part of proof of part 14 i... |
| hdmap14lem9 41835 | Part of proof of part 14 i... |
| hdmap14lem10 41836 | Part of proof of part 14 i... |
| hdmap14lem11 41837 | Part of proof of part 14 i... |
| hdmap14lem12 41838 | Lemma for proof of part 14... |
| hdmap14lem13 41839 | Lemma for proof of part 14... |
| hdmap14lem14 41840 | Part of proof of part 14 i... |
| hdmap14lem15 41841 | Part of proof of part 14 i... |
| hgmapffval 41844 | Map from the scalar divisi... |
| hgmapfval 41845 | Map from the scalar divisi... |
| hgmapval 41846 | Value of map from the scal... |
| hgmapfnN 41847 | Functionality of scalar si... |
| hgmapcl 41848 | Closure of scalar sigma ma... |
| hgmapdcl 41849 | Closure of the vector spac... |
| hgmapvs 41850 | Part 15 of [Baer] p. 50 li... |
| hgmapval0 41851 | Value of the scalar sigma ... |
| hgmapval1 41852 | Value of the scalar sigma ... |
| hgmapadd 41853 | Part 15 of [Baer] p. 50 li... |
| hgmapmul 41854 | Part 15 of [Baer] p. 50 li... |
| hgmaprnlem1N 41855 | Lemma for ~ hgmaprnN . (C... |
| hgmaprnlem2N 41856 | Lemma for ~ hgmaprnN . Pa... |
| hgmaprnlem3N 41857 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem4N 41858 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem5N 41859 | Lemma for ~ hgmaprnN . El... |
| hgmaprnN 41860 | Part of proof of part 16 i... |
| hgmap11 41861 | The scalar sigma map is on... |
| hgmapf1oN 41862 | The scalar sigma map is a ... |
| hgmapeq0 41863 | The scalar sigma map is ze... |
| hdmapipcl 41864 | The inner product (Hermiti... |
| hdmapln1 41865 | Linearity property that wi... |
| hdmaplna1 41866 | Additive property of first... |
| hdmaplns1 41867 | Subtraction property of fi... |
| hdmaplnm1 41868 | Multiplicative property of... |
| hdmaplna2 41869 | Additive property of secon... |
| hdmapglnm2 41870 | g-linear property of secon... |
| hdmapgln2 41871 | g-linear property that wil... |
| hdmaplkr 41872 | Kernel of the vector to du... |
| hdmapellkr 41873 | Membership in the kernel (... |
| hdmapip0 41874 | Zero property that will be... |
| hdmapip1 41875 | Construct a proportional v... |
| hdmapip0com 41876 | Commutation property of Ba... |
| hdmapinvlem1 41877 | Line 27 in [Baer] p. 110. ... |
| hdmapinvlem2 41878 | Line 28 in [Baer] p. 110, ... |
| hdmapinvlem3 41879 | Line 30 in [Baer] p. 110, ... |
| hdmapinvlem4 41880 | Part 1.1 of Proposition 1 ... |
| hdmapglem5 41881 | Part 1.2 in [Baer] p. 110 ... |
| hgmapvvlem1 41882 | Involution property of sca... |
| hgmapvvlem2 41883 | Lemma for ~ hgmapvv . Eli... |
| hgmapvvlem3 41884 | Lemma for ~ hgmapvv . Eli... |
| hgmapvv 41885 | Value of a double involuti... |
| hdmapglem7a 41886 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7b 41887 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7 41888 | Lemma for ~ hdmapg . Line... |
| hdmapg 41889 | Apply the scalar sigma fun... |
| hdmapoc 41890 | Express our constructed or... |
| hlhilset 41893 | The final Hilbert space co... |
| hlhilsca 41894 | The scalar of the final co... |
| hlhilbase 41895 | The base set of the final ... |
| hlhilplus 41896 | The vector addition for th... |
| hlhilslem 41897 | Lemma for ~ hlhilsbase etc... |
| hlhilslemOLD 41898 | Obsolete version of ~ hlhi... |
| hlhilsbase 41899 | The scalar base set of the... |
| hlhilsbaseOLD 41900 | Obsolete version of ~ hlhi... |
| hlhilsplus 41901 | Scalar addition for the fi... |
| hlhilsplusOLD 41902 | Obsolete version of ~ hlhi... |
| hlhilsmul 41903 | Scalar multiplication for ... |
| hlhilsmulOLD 41904 | Obsolete version of ~ hlhi... |
| hlhilsbase2 41905 | The scalar base set of the... |
| hlhilsplus2 41906 | Scalar addition for the fi... |
| hlhilsmul2 41907 | Scalar multiplication for ... |
| hlhils0 41908 | The scalar ring zero for t... |
| hlhils1N 41909 | The scalar ring unity for ... |
| hlhilvsca 41910 | The scalar product for the... |
| hlhilip 41911 | Inner product operation fo... |
| hlhilipval 41912 | Value of inner product ope... |
| hlhilnvl 41913 | The involution operation o... |
| hlhillvec 41914 | The final constructed Hilb... |
| hlhildrng 41915 | The star division ring for... |
| hlhilsrnglem 41916 | Lemma for ~ hlhilsrng . (... |
| hlhilsrng 41917 | The star division ring for... |
| hlhil0 41918 | The zero vector for the fi... |
| hlhillsm 41919 | The vector sum operation f... |
| hlhilocv 41920 | The orthocomplement for th... |
| hlhillcs 41921 | The closed subspaces of th... |
| hlhilphllem 41922 | Lemma for ~ hlhil . (Cont... |
| hlhilhillem 41923 | Lemma for ~ hlhil . (Cont... |
| hlathil 41924 | Construction of a Hilbert ... |
| iscsrg 41927 | A commutative semiring is ... |
| rhmzrhval 41928 | Evaluation of integers acr... |
| zndvdchrrhm 41929 | Construction of a ring hom... |
| leexp1ad 41930 | Weak base ordering relatio... |
| relogbcld 41931 | Closure of the general log... |
| relogbexpd 41932 | Identity law for general l... |
| relogbzexpd 41933 | Power law for the general ... |
| logblebd 41934 | The general logarithm is m... |
| uzindd 41935 | Induction on the upper int... |
| fzadd2d 41936 | Membership of a sum in a f... |
| zltlem1d 41937 | Integer ordering relation,... |
| zltp1led 41938 | Integer ordering relation,... |
| fzne2d 41939 | Elementhood in a finite se... |
| eqfnfv2d2 41940 | Equality of functions is d... |
| fzsplitnd 41941 | Split a finite interval of... |
| fzsplitnr 41942 | Split a finite interval of... |
| addassnni 41943 | Associative law for additi... |
| addcomnni 41944 | Commutative law for additi... |
| mulassnni 41945 | Associative law for multip... |
| mulcomnni 41946 | Commutative law for multip... |
| gcdcomnni 41947 | Commutative law for gcd. ... |
| gcdnegnni 41948 | Negation invariance for gc... |
| neggcdnni 41949 | Negation invariance for gc... |
| bccl2d 41950 | Closure of the binomial co... |
| recbothd 41951 | Take reciprocal on both si... |
| gcdmultiplei 41952 | The GCD of a multiple of a... |
| gcdaddmzz2nni 41953 | Adding a multiple of one o... |
| gcdaddmzz2nncomi 41954 | Adding a multiple of one o... |
| gcdnncli 41955 | Closure of the gcd operato... |
| muldvds1d 41956 | If a product divides an in... |
| muldvds2d 41957 | If a product divides an in... |
| nndivdvdsd 41958 | A positive integer divides... |
| nnproddivdvdsd 41959 | A product of natural numbe... |
| coprmdvds2d 41960 | If an integer is divisible... |
| imadomfi 41961 | An image of a function und... |
| 12gcd5e1 41962 | The gcd of 12 and 5 is 1. ... |
| 60gcd6e6 41963 | The gcd of 60 and 6 is 6. ... |
| 60gcd7e1 41964 | The gcd of 60 and 7 is 1. ... |
| 420gcd8e4 41965 | The gcd of 420 and 8 is 4.... |
| lcmeprodgcdi 41966 | Calculate the least common... |
| 12lcm5e60 41967 | The lcm of 12 and 5 is 60.... |
| 60lcm6e60 41968 | The lcm of 60 and 6 is 60.... |
| 60lcm7e420 41969 | The lcm of 60 and 7 is 420... |
| 420lcm8e840 41970 | The lcm of 420 and 8 is 84... |
| lcmfunnnd 41971 | Useful equation to calcula... |
| lcm1un 41972 | Least common multiple of n... |
| lcm2un 41973 | Least common multiple of n... |
| lcm3un 41974 | Least common multiple of n... |
| lcm4un 41975 | Least common multiple of n... |
| lcm5un 41976 | Least common multiple of n... |
| lcm6un 41977 | Least common multiple of n... |
| lcm7un 41978 | Least common multiple of n... |
| lcm8un 41979 | Least common multiple of n... |
| 3factsumint1 41980 | Move constants out of inte... |
| 3factsumint2 41981 | Move constants out of inte... |
| 3factsumint3 41982 | Move constants out of inte... |
| 3factsumint4 41983 | Move constants out of inte... |
| 3factsumint 41984 | Helpful equation for lcm i... |
| resopunitintvd 41985 | Restrict continuous functi... |
| resclunitintvd 41986 | Restrict continuous functi... |
| resdvopclptsd 41987 | Restrict derivative on uni... |
| lcmineqlem1 41988 | Part of lcm inequality lem... |
| lcmineqlem2 41989 | Part of lcm inequality lem... |
| lcmineqlem3 41990 | Part of lcm inequality lem... |
| lcmineqlem4 41991 | Part of lcm inequality lem... |
| lcmineqlem5 41992 | Technical lemma for recipr... |
| lcmineqlem6 41993 | Part of lcm inequality lem... |
| lcmineqlem7 41994 | Derivative of 1-x for chai... |
| lcmineqlem8 41995 | Derivative of (1-x)^(N-M).... |
| lcmineqlem9 41996 | (1-x)^(N-M) is continuous.... |
| lcmineqlem10 41997 | Induction step of ~ lcmine... |
| lcmineqlem11 41998 | Induction step, continuati... |
| lcmineqlem12 41999 | Base case for induction. ... |
| lcmineqlem13 42000 | Induction proof for lcm in... |
| lcmineqlem14 42001 | Technical lemma for inequa... |
| lcmineqlem15 42002 | F times the least common m... |
| lcmineqlem16 42003 | Technical divisibility lem... |
| lcmineqlem17 42004 | Inequality of 2^{2n}. (Co... |
| lcmineqlem18 42005 | Technical lemma to shift f... |
| lcmineqlem19 42006 | Dividing implies inequalit... |
| lcmineqlem20 42007 | Inequality for lcm lemma. ... |
| lcmineqlem21 42008 | The lcm inequality lemma w... |
| lcmineqlem22 42009 | The lcm inequality lemma w... |
| lcmineqlem23 42010 | Penultimate step to the lc... |
| lcmineqlem 42011 | The least common multiple ... |
| 3exp7 42012 | 3 to the power of 7 equals... |
| 3lexlogpow5ineq1 42013 | First inequality in inequa... |
| 3lexlogpow5ineq2 42014 | Second inequality in inequ... |
| 3lexlogpow5ineq4 42015 | Sharper logarithm inequali... |
| 3lexlogpow5ineq3 42016 | Combined inequality chain ... |
| 3lexlogpow2ineq1 42017 | Result for bound in AKS in... |
| 3lexlogpow2ineq2 42018 | Result for bound in AKS in... |
| 3lexlogpow5ineq5 42019 | Result for bound in AKS in... |
| intlewftc 42020 | Inequality inference by in... |
| aks4d1lem1 42021 | Technical lemma to reduce ... |
| aks4d1p1p1 42022 | Exponential law for finite... |
| dvrelog2 42023 | The derivative of the loga... |
| dvrelog3 42024 | The derivative of the loga... |
| dvrelog2b 42025 | Derivative of the binary l... |
| 0nonelalab 42026 | Technical lemma for open i... |
| dvrelogpow2b 42027 | Derivative of the power of... |
| aks4d1p1p3 42028 | Bound of a ceiling of the ... |
| aks4d1p1p2 42029 | Rewrite ` A ` in more suit... |
| aks4d1p1p4 42030 | Technical step for inequal... |
| dvle2 42031 | Collapsed ~ dvle . (Contr... |
| aks4d1p1p6 42032 | Inequality lift to differe... |
| aks4d1p1p7 42033 | Bound of intermediary of i... |
| aks4d1p1p5 42034 | Show inequality for existe... |
| aks4d1p1 42035 | Show inequality for existe... |
| aks4d1p2 42036 | Technical lemma for existe... |
| aks4d1p3 42037 | There exists a small enoug... |
| aks4d1p4 42038 | There exists a small enoug... |
| aks4d1p5 42039 | Show that ` N ` and ` R ` ... |
| aks4d1p6 42040 | The maximal prime power ex... |
| aks4d1p7d1 42041 | Technical step in AKS lemm... |
| aks4d1p7 42042 | Technical step in AKS lemm... |
| aks4d1p8d1 42043 | If a prime divides one num... |
| aks4d1p8d2 42044 | Any prime power dividing a... |
| aks4d1p8d3 42045 | The remainder of a divisio... |
| aks4d1p8 42046 | Show that ` N ` and ` R ` ... |
| aks4d1p9 42047 | Show that the order is bou... |
| aks4d1 42048 | Lemma 4.1 from ~ https://w... |
| fldhmf1 42049 | A field homomorphism is in... |
| isprimroot 42052 | The value of a primitive r... |
| isprimroot2 42053 | Alternative way of creatin... |
| mndmolinv 42054 | An element of a monoid tha... |
| linvh 42055 | If an element has a unique... |
| primrootsunit1 42056 | Primitive roots have left ... |
| primrootsunit 42057 | Primitive roots have left ... |
| primrootscoprmpow 42058 | Coprime powers of primitiv... |
| posbezout 42059 | Bezout's identity restrict... |
| primrootscoprf 42060 | Coprime powers of primitiv... |
| primrootscoprbij 42061 | A bijection between coprim... |
| primrootscoprbij2 42062 | A bijection between coprim... |
| remexz 42063 | Division with rest. (Cont... |
| primrootlekpowne0 42064 | There is no smaller power ... |
| primrootspoweq0 42065 | The power of a ` R ` -th p... |
| aks6d1c1p1 42066 | Definition of the introspe... |
| aks6d1c1p1rcl 42067 | Reverse closure of the int... |
| aks6d1c1p2 42068 | ` P ` and linear factors a... |
| aks6d1c1p3 42069 | In a field with a Frobeniu... |
| aks6d1c1p4 42070 | The product of polynomials... |
| aks6d1c1p5 42071 | The product of exponents i... |
| aks6d1c1p7 42072 | ` X ` is introspective to ... |
| aks6d1c1p6 42073 | If a polynomials ` F ` is ... |
| aks6d1c1p8 42074 | If a number ` E ` is intro... |
| aks6d1c1 42075 | Claim 1 of Theorem 6.1 ~ h... |
| evl1gprodd 42076 | Polynomial evaluation buil... |
| aks6d1c2p1 42077 | In the AKS-theorem the sub... |
| aks6d1c2p2 42078 | Injective condition for co... |
| hashscontpowcl 42079 | Closure of E for ~ https:/... |
| hashscontpow1 42080 | Helper lemma for to prove ... |
| hashscontpow 42081 | If a set contains all ` N ... |
| aks6d1c3 42082 | Claim 3 of Theorem 6.1 of ... |
| aks6d1c4 42083 | Claim 4 of Theorem 6.1 of ... |
| aks6d1c1rh 42084 | Claim 1 of AKS primality p... |
| aks6d1c2lem3 42085 | Lemma for ~ aks6d1c2 to si... |
| aks6d1c2lem4 42086 | Claim 2 of Theorem 6.1 AKS... |
| hashnexinj 42087 | If the number of elements ... |
| hashnexinjle 42088 | If the number of elements ... |
| aks6d1c2 42089 | Claim 2 of Theorem 6.1 of ... |
| rspcsbnea 42090 | Special case related to ~ ... |
| idomnnzpownz 42091 | A non-zero power in an int... |
| idomnnzgmulnz 42092 | A finite product of non-ze... |
| ringexp0nn 42093 | Zero to the power of a pos... |
| aks6d1c5lem0 42094 | Lemma for Claim 5 of Theor... |
| aks6d1c5lem1 42095 | Lemma for claim 5, evaluat... |
| aks6d1c5lem3 42096 | Lemma for Claim 5, polynom... |
| aks6d1c5lem2 42097 | Lemma for Claim 5, contrad... |
| aks6d1c5 42098 | Claim 5 of Theorem 6.1 ~ h... |
| deg1gprod 42099 | Degree multiplication is a... |
| deg1pow 42100 | Exact degree of a power of... |
| 5bc2eq10 42101 | The value of 5 choose 2. ... |
| facp2 42102 | The factorial of a success... |
| 2np3bcnp1 42103 | Part of induction step for... |
| 2ap1caineq 42104 | Inequality for Theorem 6.6... |
| sticksstones1 42105 | Different strictly monoton... |
| sticksstones2 42106 | The range function on stri... |
| sticksstones3 42107 | The range function on stri... |
| sticksstones4 42108 | Equinumerosity lemma for s... |
| sticksstones5 42109 | Count the number of strict... |
| sticksstones6 42110 | Function induces an order ... |
| sticksstones7 42111 | Closure property of sticks... |
| sticksstones8 42112 | Establish mapping between ... |
| sticksstones9 42113 | Establish mapping between ... |
| sticksstones10 42114 | Establish mapping between ... |
| sticksstones11 42115 | Establish bijective mappin... |
| sticksstones12a 42116 | Establish bijective mappin... |
| sticksstones12 42117 | Establish bijective mappin... |
| sticksstones13 42118 | Establish bijective mappin... |
| sticksstones14 42119 | Sticks and stones with def... |
| sticksstones15 42120 | Sticks and stones with alm... |
| sticksstones16 42121 | Sticks and stones with col... |
| sticksstones17 42122 | Extend sticks and stones t... |
| sticksstones18 42123 | Extend sticks and stones t... |
| sticksstones19 42124 | Extend sticks and stones t... |
| sticksstones20 42125 | Lift sticks and stones to ... |
| sticksstones21 42126 | Lift sticks and stones to ... |
| sticksstones22 42127 | Non-exhaustive sticks and ... |
| sticksstones23 42128 | Non-exhaustive sticks and ... |
| aks6d1c6lem1 42129 | Lemma for claim 6, deduce ... |
| aks6d1c6lem2 42130 | Every primitive root is ro... |
| aks6d1c6lem3 42131 | Claim 6 of Theorem 6.1 of ... |
| aks6d1c6lem4 42132 | Claim 6 of Theorem 6.1 of ... |
| aks6d1c6isolem1 42133 | Lemma to construct the map... |
| aks6d1c6isolem2 42134 | Lemma to construct the gro... |
| aks6d1c6isolem3 42135 | The preimage of a map send... |
| aks6d1c6lem5 42136 | Eliminate the size hypothe... |
| bcled 42137 | Inequality for binomial co... |
| bcle2d 42138 | Inequality for binomial co... |
| aks6d1c7lem1 42139 | The last set of inequaliti... |
| aks6d1c7lem2 42140 | Contradiction to Claim 2 a... |
| aks6d1c7lem3 42141 | Remove lots of hypotheses ... |
| aks6d1c7lem4 42142 | In the AKS algorithm there... |
| aks6d1c7 42143 | ` N ` is a prime power if ... |
| rhmqusspan 42144 | Ring homomorphism out of a... |
| aks5lem1 42145 | Section 5 of ~ https://www... |
| aks5lem2 42146 | Lemma for section 5 ~ http... |
| ply1asclzrhval 42147 | Transfer results from alge... |
| aks5lem3a 42148 | Lemma for AKS section 5. ... |
| aks5lem4a 42149 | Lemma for AKS section 5, r... |
| aks5lem5a 42150 | Lemma for AKS, section 5, ... |
| aks5lem6 42151 | Connect results of section... |
| indstrd 42152 | Strong induction, deductio... |
| grpods 42153 | Relate sums of elements of... |
| unitscyglem1 42154 | Lemma for unitscyg. (Cont... |
| unitscyglem2 42155 | Lemma for unitscyg. (Cont... |
| unitscyglem3 42156 | Lemma for unitscyg. (Cont... |
| unitscyglem4 42157 | Lemma for unitscyg (Contri... |
| unitscyglem5 42158 | Lemma for unitscyg (Contri... |
| aks5lem7 42159 | Lemma for aks5. We clean ... |
| aks5lem8 42160 | Lemma for aks5. Clean up ... |
| exfinfldd 42162 | For any prime ` P ` and an... |
| aks5 42163 | The AKS Primality test, gi... |
| metakunt1 42164 | A is an endomapping. (Con... |
| metakunt2 42165 | A is an endomapping. (Con... |
| metakunt3 42166 | Value of A. (Contributed b... |
| metakunt4 42167 | Value of A. (Contributed b... |
| metakunt5 42168 | C is the left inverse for ... |
| metakunt6 42169 | C is the left inverse for ... |
| metakunt7 42170 | C is the left inverse for ... |
| metakunt8 42171 | C is the left inverse for ... |
| metakunt9 42172 | C is the left inverse for ... |
| metakunt10 42173 | C is the right inverse for... |
| metakunt11 42174 | C is the right inverse for... |
| metakunt12 42175 | C is the right inverse for... |
| metakunt13 42176 | C is the right inverse for... |
| metakunt14 42177 | A is a primitive permutati... |
| metakunt15 42178 | Construction of another pe... |
| metakunt16 42179 | Construction of another pe... |
| metakunt17 42180 | The union of three disjoin... |
| metakunt18 42181 | Disjoint domains and codom... |
| metakunt19 42182 | Domains on restrictions of... |
| metakunt20 42183 | Show that B coincides on t... |
| metakunt21 42184 | Show that B coincides on t... |
| metakunt22 42185 | Show that B coincides on t... |
| metakunt23 42186 | B coincides on the union o... |
| metakunt24 42187 | Technical condition such t... |
| metakunt25 42188 | B is a permutation. (Cont... |
| metakunt26 42189 | Construction of one soluti... |
| metakunt27 42190 | Construction of one soluti... |
| metakunt28 42191 | Construction of one soluti... |
| metakunt29 42192 | Construction of one soluti... |
| metakunt30 42193 | Construction of one soluti... |
| metakunt31 42194 | Construction of one soluti... |
| metakunt32 42195 | Construction of one soluti... |
| metakunt33 42196 | Construction of one soluti... |
| metakunt34 42197 | ` D ` is a permutation. (... |
| fac2xp3 42198 | Factorial of 2x+3, sublemm... |
| prodsplit 42199 | Product split into two fac... |
| 2xp3dxp2ge1d 42200 | 2x+3 is greater than or eq... |
| factwoffsmonot 42201 | A factorial with offset is... |
| intnanrt 42202 | Introduction of conjunct i... |
| ioin9i8 42203 | Miscellaneous inference cr... |
| jaodd 42204 | Double deduction form of ~... |
| syl3an12 42205 | A double syllogism inferen... |
| sbtd 42206 | A true statement is true u... |
| sbor2 42207 | One direction of ~ sbor , ... |
| sbalexi 42208 | Inference form of ~ sbalex... |
| 19.9dev 42209 | ~ 19.9d in the case of an ... |
| 3rspcedvd 42210 | Triple application of ~ rs... |
| sn-axrep5v 42211 | A condensed form of ~ axre... |
| sn-axprlem3 42212 | ~ axprlem3 using only Tars... |
| sn-exelALT 42213 | Alternate proof of ~ exel ... |
| ss2ab1 42214 | Class abstractions in a su... |
| ssabdv 42215 | Deduction of abstraction s... |
| sn-iotalem 42216 | An unused lemma showing th... |
| sn-iotalemcor 42217 | Corollary of ~ sn-iotalem ... |
| abbi1sn 42218 | Originally part of ~ uniab... |
| brif2 42219 | Move a relation inside and... |
| brif12 42220 | Move a relation inside and... |
| pssexg 42221 | The proper subset of a set... |
| pssn0 42222 | A proper superset is nonem... |
| psspwb 42223 | Classes are proper subclas... |
| xppss12 42224 | Proper subset theorem for ... |
| elpwbi 42225 | Membership in a power set,... |
| imaopab 42226 | The image of a class of or... |
| fnsnbt 42227 | A function's domain is a s... |
| fnimasnd 42228 | The image of a function by... |
| eqresfnbd 42229 | Property of being the rest... |
| f1o2d2 42230 | Sufficient condition for a... |
| fmpocos 42231 | Composition of two functio... |
| ovmpogad 42232 | Value of an operation give... |
| ofun 42233 | A function operation of un... |
| dfqs2 42234 | Alternate definition of qu... |
| dfqs3 42235 | Alternate definition of qu... |
| qseq12d 42236 | Equality theorem for quoti... |
| qsalrel 42237 | The quotient set is equal ... |
| elmapssresd 42238 | A restricted mapping is a ... |
| supinf 42239 | The supremum is the infimu... |
| mapcod 42240 | Compose two mappings. (Co... |
| fisdomnn 42241 | A finite set is dominated ... |
| ltex 42242 | The less-than relation is ... |
| leex 42243 | The less-than-or-equal-to ... |
| subex 42244 | The subtraction operation ... |
| absex 42245 | The absolute value functio... |
| cjex 42246 | The conjugate function is ... |
| fzosumm1 42247 | Separate out the last term... |
| ccatcan2d 42248 | Cancellation law for conca... |
| c0exALT 42249 | Alternate proof of ~ c0ex ... |
| 0cnALT3 42250 | Alternate proof of ~ 0cn u... |
| elre0re 42251 | Specialized version of ~ 0... |
| 1t1e1ALT 42252 | Alternate proof of ~ 1t1e1... |
| lttrii 42253 | 'Less than' is transitive.... |
| remulcan2d 42254 | ~ mulcan2d for real number... |
| readdridaddlidd 42255 | Given some real number ` B... |
| sn-1ne2 42256 | A proof of ~ 1ne2 without ... |
| nnn1suc 42257 | A positive integer that is... |
| nnadd1com 42258 | Addition with 1 is commuta... |
| nnaddcom 42259 | Addition is commutative fo... |
| nnaddcomli 42260 | Version of ~ addcomli for ... |
| nnadddir 42261 | Right-distributivity for n... |
| nnmul1com 42262 | Multiplication with 1 is c... |
| nnmulcom 42263 | Multiplication is commutat... |
| readdrcl2d 42264 | Reverse closure for additi... |
| mvrrsubd 42265 | Move a subtraction in the ... |
| laddrotrd 42266 | Rotate the variables right... |
| raddswap12d 42267 | Swap the first two variabl... |
| lsubrotld 42268 | Rotate the variables left ... |
| rsubrotld 42269 | Rotate the variables left ... |
| lsubswap23d 42270 | Swap the second and third ... |
| addsubeq4com 42271 | Relation between sums and ... |
| sqsumi 42272 | A sum squared. (Contribut... |
| negn0nposznnd 42273 | Lemma for ~ dffltz . (Con... |
| sqmid3api 42274 | Value of the square of the... |
| decaddcom 42275 | Commute ones place in addi... |
| sqn5i 42276 | The square of a number end... |
| sqn5ii 42277 | The square of a number end... |
| decpmulnc 42278 | Partial products algorithm... |
| decpmul 42279 | Partial products algorithm... |
| sqdeccom12 42280 | The square of a number in ... |
| sq3deccom12 42281 | Variant of ~ sqdeccom12 wi... |
| 4t5e20 42282 | 4 times 5 equals 20. (Con... |
| sq4 42283 | The square of 4 is 16. (C... |
| sq5 42284 | The square of 5 is 25. (C... |
| sq6 42285 | The square of 6 is 36. (C... |
| sq7 42286 | The square of 7 is 49. (C... |
| sq8 42287 | The square of 8 is 64. (C... |
| sq9 42288 | The square of 9 is 81. (C... |
| 4rp 42289 | 4 is a positive real. (Co... |
| 5rp 42290 | 5 is a positive real. (Co... |
| 6rp 42291 | 6 is a positive real. (Co... |
| 7rp 42292 | 7 is a positive real. (Co... |
| 8rp 42293 | 8 is a positive real. (Co... |
| 9rp 42294 | 9 is a positive real. (Co... |
| 235t711 42295 | Calculate a product by lon... |
| ex-decpmul 42296 | Example usage of ~ decpmul... |
| eluzp1 42297 | Membership in a successor ... |
| sn-eluzp1l 42298 | Shorter proof of ~ eluzp1l... |
| fz1sumconst 42299 | The sum of ` N ` constant ... |
| fz1sump1 42300 | Add one more term to a sum... |
| oddnumth 42301 | The Odd Number Theorem. T... |
| nicomachus 42302 | Nicomachus's Theorem. The... |
| sumcubes 42303 | The sum of the first ` N `... |
| pine0 42304 | ` _pi ` is nonzero. (Cont... |
| ine1 42305 | ` _i ` is not 1. (Contrib... |
| 0tie0 42306 | 0 times ` _i ` equals 0. ... |
| it1ei 42307 | ` _i ` times 1 equals ` _i... |
| 1tiei 42308 | 1 times ` _i ` equals ` _i... |
| itrere 42309 | ` _i ` times a real is rea... |
| retire 42310 | A real times ` _i ` is rea... |
| oexpreposd 42311 | Lemma for ~ dffltz . TODO... |
| explt1d 42312 | A nonnegative real number ... |
| expeq1d 42313 | A nonnegative real number ... |
| expeqidd 42314 | A nonnegative real number ... |
| exp11d 42315 | ~ exp11nnd for nonzero int... |
| 0dvds0 42316 | 0 divides 0. (Contributed... |
| absdvdsabsb 42317 | Divisibility is invariant ... |
| gcdnn0id 42318 | The ` gcd ` of a nonnegati... |
| gcdle1d 42319 | The greatest common diviso... |
| gcdle2d 42320 | The greatest common diviso... |
| dvdsexpad 42321 | Deduction associated with ... |
| dvdsexpnn 42322 | ~ dvdssqlem generalized to... |
| dvdsexpnn0 42323 | ~ dvdsexpnn generalized to... |
| dvdsexpb 42324 | ~ dvdssq generalized to po... |
| posqsqznn 42325 | When a positive rational s... |
| zdivgd 42326 | Two ways to express " ` N ... |
| efne0d 42327 | The exponential of a compl... |
| efsubd 42328 | Difference of exponents la... |
| ef11d 42329 | General condition for the ... |
| logccne0d 42330 | The logarithm isn't 0 if i... |
| cxp112d 42331 | General condition for comp... |
| cxp111d 42332 | General condition for comp... |
| cxpi11d 42333 | ` _i ` to the powers of ` ... |
| logne0d 42334 | Deduction form of ~ logne0... |
| rxp112d 42335 | Real exponentiation is one... |
| log11d 42336 | The natural logarithm is o... |
| rplog11d 42337 | The natural logarithm is o... |
| rxp11d 42338 | Real exponentiation is one... |
| tanhalfpim 42339 | The tangent of ` _pi / 2 `... |
| tan3rdpi 42340 | The tangent of ` _pi / 3 `... |
| asin1half 42341 | The arcsine of ` 1 / 2 ` i... |
| acos1half 42342 | The arccosine of ` 1 / 2 `... |
| resubval 42345 | Value of real subtraction,... |
| renegeulemv 42346 | Lemma for ~ renegeu and si... |
| renegeulem 42347 | Lemma for ~ renegeu and si... |
| renegeu 42348 | Existential uniqueness of ... |
| rernegcl 42349 | Closure law for negative r... |
| renegadd 42350 | Relationship between real ... |
| renegid 42351 | Addition of a real number ... |
| reneg0addlid 42352 | Negative zero is a left ad... |
| resubeulem1 42353 | Lemma for ~ resubeu . A v... |
| resubeulem2 42354 | Lemma for ~ resubeu . A v... |
| resubeu 42355 | Existential uniqueness of ... |
| rersubcl 42356 | Closure for real subtracti... |
| resubadd 42357 | Relation between real subt... |
| resubaddd 42358 | Relationship between subtr... |
| resubf 42359 | Real subtraction is an ope... |
| repncan2 42360 | Addition and subtraction o... |
| repncan3 42361 | Addition and subtraction o... |
| readdsub 42362 | Law for addition and subtr... |
| reladdrsub 42363 | Move LHS of a sum into RHS... |
| reltsub1 42364 | Subtraction from both side... |
| reltsubadd2 42365 | 'Less than' relationship b... |
| resubcan2 42366 | Cancellation law for real ... |
| resubsub4 42367 | Law for double subtraction... |
| rennncan2 42368 | Cancellation law for real ... |
| renpncan3 42369 | Cancellation law for real ... |
| repnpcan 42370 | Cancellation law for addit... |
| reppncan 42371 | Cancellation law for mixed... |
| resubidaddlidlem 42372 | Lemma for ~ resubidaddlid ... |
| resubidaddlid 42373 | Any real number subtracted... |
| resubdi 42374 | Distribution of multiplica... |
| re1m1e0m0 42375 | Equality of two left-addit... |
| sn-00idlem1 42376 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem2 42377 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem3 42378 | Lemma for ~ sn-00id . (Co... |
| sn-00id 42379 | ~ 00id proven without ~ ax... |
| re0m0e0 42380 | Real number version of ~ 0... |
| readdlid 42381 | Real number version of ~ a... |
| sn-addlid 42382 | ~ addlid without ~ ax-mulc... |
| remul02 42383 | Real number version of ~ m... |
| sn-0ne2 42384 | ~ 0ne2 without ~ ax-mulcom... |
| remul01 42385 | Real number version of ~ m... |
| resubid 42386 | Subtraction of a real numb... |
| readdrid 42387 | Real number version of ~ a... |
| resubid1 42388 | Real number version of ~ s... |
| renegneg 42389 | A real number is equal to ... |
| readdcan2 42390 | Commuted version of ~ read... |
| renegid2 42391 | Commuted version of ~ rene... |
| remulneg2d 42392 | Product with negative is n... |
| sn-it0e0 42393 | Proof of ~ it0e0 without ~... |
| sn-negex12 42394 | A combination of ~ cnegex ... |
| sn-negex 42395 | Proof of ~ cnegex without ... |
| sn-negex2 42396 | Proof of ~ cnegex2 without... |
| sn-addcand 42397 | ~ addcand without ~ ax-mul... |
| sn-addrid 42398 | ~ addrid without ~ ax-mulc... |
| sn-addcan2d 42399 | ~ addcan2d without ~ ax-mu... |
| reixi 42400 | ~ ixi without ~ ax-mulcom ... |
| rei4 42401 | ~ i4 without ~ ax-mulcom .... |
| sn-addid0 42402 | A number that sums to itse... |
| sn-mul01 42403 | ~ mul01 without ~ ax-mulco... |
| sn-subeu 42404 | ~ negeu without ~ ax-mulco... |
| sn-subcl 42405 | ~ subcl without ~ ax-mulco... |
| sn-subf 42406 | ~ subf without ~ ax-mulcom... |
| resubeqsub 42407 | Equivalence between real s... |
| subresre 42408 | Subtraction restricted to ... |
| addinvcom 42409 | A number commutes with its... |
| remulinvcom 42410 | A left multiplicative inve... |
| remullid 42411 | Commuted version of ~ ax-1... |
| sn-1ticom 42412 | Lemma for ~ sn-mullid and ... |
| sn-mullid 42413 | ~ mullid without ~ ax-mulc... |
| sn-it1ei 42414 | ~ it1ei without ~ ax-mulco... |
| ipiiie0 42415 | The multiplicative inverse... |
| remulcand 42416 | Commuted version of ~ remu... |
| sn-0tie0 42417 | Lemma for ~ sn-mul02 . Co... |
| sn-mul02 42418 | ~ mul02 without ~ ax-mulco... |
| sn-ltaddpos 42419 | ~ ltaddpos without ~ ax-mu... |
| sn-ltaddneg 42420 | ~ ltaddneg without ~ ax-mu... |
| reposdif 42421 | Comparison of two numbers ... |
| relt0neg1 42422 | Comparison of a real and i... |
| relt0neg2 42423 | Comparison of a real and i... |
| sn-addlt0d 42424 | The sum of negative number... |
| sn-addgt0d 42425 | The sum of positive number... |
| sn-nnne0 42426 | ~ nnne0 without ~ ax-mulco... |
| reelznn0nn 42427 | ~ elznn0nn restated using ... |
| nn0addcom 42428 | Addition is commutative fo... |
| zaddcomlem 42429 | Lemma for ~ zaddcom . (Co... |
| zaddcom 42430 | Addition is commutative fo... |
| renegmulnnass 42431 | Move multiplication by a n... |
| nn0mulcom 42432 | Multiplication is commutat... |
| zmulcomlem 42433 | Lemma for ~ zmulcom . (Co... |
| zmulcom 42434 | Multiplication is commutat... |
| mulgt0con1dlem 42435 | Lemma for ~ mulgt0con1d . ... |
| mulgt0con1d 42436 | Counterpart to ~ mulgt0con... |
| mulgt0con2d 42437 | Lemma for ~ mulgt0b2d and ... |
| mulgt0b2d 42438 | Biconditional, deductive f... |
| sn-ltmul2d 42439 | ~ ltmul2d without ~ ax-mul... |
| sn-ltmulgt11d 42440 | ~ ltmulgt11d without ~ ax-... |
| sn-0lt1 42441 | ~ 0lt1 without ~ ax-mulcom... |
| sn-ltp1 42442 | ~ ltp1 without ~ ax-mulcom... |
| sn-mulgt1d 42443 | ~ mulgt1d without ~ ax-mul... |
| reneg1lt0 42444 | Lemma for ~ sn-inelr . (C... |
| sn-inelr 42445 | ~ inelr without ~ ax-mulco... |
| sn-itrere 42446 | ` _i ` times a real is rea... |
| sn-retire 42447 | Commuted version of ~ sn-i... |
| cnreeu 42448 | The reals in the expressio... |
| sn-sup2 42449 | ~ sup2 with exactly the sa... |
| sn-sup3d 42450 | ~ sup3 without ~ ax-mulcom... |
| sn-suprcld 42451 | ~ suprcld without ~ ax-mul... |
| sn-suprubd 42452 | ~ suprubd without ~ ax-mul... |
| nelsubginvcld 42453 | The inverse of a non-subgr... |
| nelsubgcld 42454 | A non-subgroup-member plus... |
| nelsubgsubcld 42455 | A non-subgroup-member minu... |
| rnasclg 42456 | The set of injected scalar... |
| frlmfielbas 42457 | The vectors of a finite fr... |
| frlmfzwrd 42458 | A vector of a module with ... |
| frlmfzowrd 42459 | A vector of a module with ... |
| frlmfzolen 42460 | The dimension of a vector ... |
| frlmfzowrdb 42461 | The vectors of a module wi... |
| frlmfzoccat 42462 | The concatenation of two v... |
| frlmvscadiccat 42463 | Scalar multiplication dist... |
| grpasscan2d 42464 | An associative cancellatio... |
| grpcominv1 42465 | If two elements commute, t... |
| grpcominv2 42466 | If two elements commute, t... |
| finsubmsubg 42467 | A submonoid of a finite gr... |
| opprmndb 42468 | A class is a monoid if and... |
| opprgrpb 42469 | A class is a group if and ... |
| opprablb 42470 | A class is an Abelian grou... |
| imacrhmcl 42471 | The image of a commutative... |
| rimrcl1 42472 | Reverse closure of a ring ... |
| rimrcl2 42473 | Reverse closure of a ring ... |
| rimcnv 42474 | The converse of a ring iso... |
| rimco 42475 | The composition of ring is... |
| ricsym 42476 | Ring isomorphism is symmet... |
| rictr 42477 | Ring isomorphism is transi... |
| riccrng1 42478 | Ring isomorphism preserves... |
| riccrng 42479 | A ring is commutative if a... |
| domnexpgn0cl 42480 | In a domain, a (nonnegativ... |
| drnginvrn0d 42481 | A multiplicative inverse i... |
| drngmullcan 42482 | Cancellation of a nonzero ... |
| drngmulrcan 42483 | Cancellation of a nonzero ... |
| drnginvmuld 42484 | Inverse of a nonzero produ... |
| ricdrng1 42485 | A ring isomorphism maps a ... |
| ricdrng 42486 | A ring is a division ring ... |
| ricfld 42487 | A ring is a field if and o... |
| asclf1 42488 | Two ways of saying the sca... |
| abvexp 42489 | Move exponentiation in and... |
| fimgmcyclem 42490 | Lemma for ~ fimgmcyc . (C... |
| fimgmcyc 42491 | Version of ~ odcl2 for fin... |
| fidomncyc 42492 | Version of ~ odcl2 for mul... |
| fiabv 42493 | In a finite domain (a fini... |
| lvecgrp 42494 | A vector space is a group.... |
| lvecring 42495 | The scalar component of a ... |
| frlm0vald 42496 | All coordinates of the zer... |
| frlmsnic 42497 | Given a free module with a... |
| uvccl 42498 | A unit vector is a vector.... |
| uvcn0 42499 | A unit vector is nonzero. ... |
| pwselbasr 42500 | The reverse direction of ~... |
| pwsgprod 42501 | Finite products in a power... |
| psrmnd 42502 | The ring of power series i... |
| psrbagres 42503 | Restrict a bag of variable... |
| mplcrngd 42504 | The polynomial ring is a c... |
| mplsubrgcl 42505 | An element of a polynomial... |
| mhmcopsr 42506 | The composition of a monoi... |
| mhmcoaddpsr 42507 | Show that the ring homomor... |
| rhmcomulpsr 42508 | Show that the ring homomor... |
| rhmpsr 42509 | Provide a ring homomorphis... |
| rhmpsr1 42510 | Provide a ring homomorphis... |
| mplascl0 42511 | The zero scalar as a polyn... |
| mplascl1 42512 | The one scalar as a polyno... |
| mplmapghm 42513 | The function ` H ` mapping... |
| evl0 42514 | The zero polynomial evalua... |
| evlscl 42515 | A polynomial over the ring... |
| evlsval3 42516 | Give a formula for the pol... |
| evlsvval 42517 | Give a formula for the eva... |
| evlsvvvallem 42518 | Lemma for ~ evlsvvval akin... |
| evlsvvvallem2 42519 | Lemma for theorems using ~... |
| evlsvvval 42520 | Give a formula for the eva... |
| evlsscaval 42521 | Polynomial evaluation buil... |
| evlsvarval 42522 | Polynomial evaluation buil... |
| evlsbagval 42523 | Polynomial evaluation buil... |
| evlsexpval 42524 | Polynomial evaluation buil... |
| evlsaddval 42525 | Polynomial evaluation buil... |
| evlsmulval 42526 | Polynomial evaluation buil... |
| evlsmaprhm 42527 | The function ` F ` mapping... |
| evlsevl 42528 | Evaluation in a subring is... |
| evlcl 42529 | A polynomial over the ring... |
| evlvvval 42530 | Give a formula for the eva... |
| evlvvvallem 42531 | Lemma for theorems using ~... |
| evladdval 42532 | Polynomial evaluation buil... |
| evlmulval 42533 | Polynomial evaluation buil... |
| selvcllem1 42534 | ` T ` is an associative al... |
| selvcllem2 42535 | ` D ` is a ring homomorphi... |
| selvcllem3 42536 | The third argument passed ... |
| selvcllemh 42537 | Apply the third argument (... |
| selvcllem4 42538 | The fourth argument passed... |
| selvcllem5 42539 | The fifth argument passed ... |
| selvcl 42540 | Closure of the "variable s... |
| selvval2 42541 | Value of the "variable sel... |
| selvvvval 42542 | Recover the original polyn... |
| evlselvlem 42543 | Lemma for ~ evlselv . Use... |
| evlselv 42544 | Evaluating a selection of ... |
| selvadd 42545 | The "variable selection" f... |
| selvmul 42546 | The "variable selection" f... |
| fsuppind 42547 | Induction on functions ` F... |
| fsuppssindlem1 42548 | Lemma for ~ fsuppssind . ... |
| fsuppssindlem2 42549 | Lemma for ~ fsuppssind . ... |
| fsuppssind 42550 | Induction on functions ` F... |
| mhpind 42551 | The homogeneous polynomial... |
| evlsmhpvvval 42552 | Give a formula for the eva... |
| mhphflem 42553 | Lemma for ~ mhphf . Add s... |
| mhphf 42554 | A homogeneous polynomial d... |
| mhphf2 42555 | A homogeneous polynomial d... |
| mhphf3 42556 | A homogeneous polynomial d... |
| mhphf4 42557 | A homogeneous polynomial d... |
| prjspval 42560 | Value of the projective sp... |
| prjsprel 42561 | Utility theorem regarding ... |
| prjspertr 42562 | The relation in ` PrjSp ` ... |
| prjsperref 42563 | The relation in ` PrjSp ` ... |
| prjspersym 42564 | The relation in ` PrjSp ` ... |
| prjsper 42565 | The relation used to defin... |
| prjspreln0 42566 | Two nonzero vectors are eq... |
| prjspvs 42567 | A nonzero multiple of a ve... |
| prjsprellsp 42568 | Two vectors are equivalent... |
| prjspeclsp 42569 | The vectors equivalent to ... |
| prjspval2 42570 | Alternate definition of pr... |
| prjspnval 42573 | Value of the n-dimensional... |
| prjspnerlem 42574 | A lemma showing that the e... |
| prjspnval2 42575 | Value of the n-dimensional... |
| prjspner 42576 | The relation used to defin... |
| prjspnvs 42577 | A nonzero multiple of a ve... |
| prjspnssbas 42578 | A projective point spans a... |
| prjspnn0 42579 | A projective point is none... |
| 0prjspnlem 42580 | Lemma for ~ 0prjspn . The... |
| prjspnfv01 42581 | Any vector is equivalent t... |
| prjspner01 42582 | Any vector is equivalent t... |
| prjspner1 42583 | Two vectors whose zeroth c... |
| 0prjspnrel 42584 | In the zero-dimensional pr... |
| 0prjspn 42585 | A zero-dimensional project... |
| prjcrvfval 42588 | Value of the projective cu... |
| prjcrvval 42589 | Value of the projective cu... |
| prjcrv0 42590 | The "curve" (zero set) cor... |
| dffltz 42591 | Fermat's Last Theorem (FLT... |
| fltmul 42592 | A counterexample to FLT st... |
| fltdiv 42593 | A counterexample to FLT st... |
| flt0 42594 | A counterexample for FLT d... |
| fltdvdsabdvdsc 42595 | Any factor of both ` A ` a... |
| fltabcoprmex 42596 | A counterexample to FLT im... |
| fltaccoprm 42597 | A counterexample to FLT wi... |
| fltbccoprm 42598 | A counterexample to FLT wi... |
| fltabcoprm 42599 | A counterexample to FLT wi... |
| infdesc 42600 | Infinite descent. The hyp... |
| fltne 42601 | If a counterexample to FLT... |
| flt4lem 42602 | Raising a number to the fo... |
| flt4lem1 42603 | Satisfy the antecedent use... |
| flt4lem2 42604 | If ` A ` is even, ` B ` is... |
| flt4lem3 42605 | Equivalent to ~ pythagtrip... |
| flt4lem4 42606 | If the product of two copr... |
| flt4lem5 42607 | In the context of the lemm... |
| flt4lem5elem 42608 | Version of ~ fltaccoprm an... |
| flt4lem5a 42609 | Part 1 of Equation 1 of ... |
| flt4lem5b 42610 | Part 2 of Equation 1 of ... |
| flt4lem5c 42611 | Part 2 of Equation 2 of ... |
| flt4lem5d 42612 | Part 3 of Equation 2 of ... |
| flt4lem5e 42613 | Satisfy the hypotheses of ... |
| flt4lem5f 42614 | Final equation of ~... |
| flt4lem6 42615 | Remove shared factors in a... |
| flt4lem7 42616 | Convert ~ flt4lem5f into a... |
| nna4b4nsq 42617 | Strengthening of Fermat's ... |
| fltltc 42618 | ` ( C ^ N ) ` is the large... |
| fltnltalem 42619 | Lemma for ~ fltnlta . A l... |
| fltnlta 42620 | In a Fermat counterexample... |
| iddii 42621 | Version of ~ a1ii with the... |
| bicomdALT 42622 | Alternate proof of ~ bicom... |
| alan 42623 | Alias for ~ 19.26 for easi... |
| exor 42624 | Alias for ~ 19.43 for easi... |
| rexor 42625 | Alias for ~ r19.43 for eas... |
| ruvALT 42626 | Alternate proof of ~ ruv w... |
| sn-wcdeq 42627 | Alternative to ~ wcdeq and... |
| sq45 42628 | 45 squared is 2025. (Cont... |
| sum9cubes 42629 | The sum of the first nine ... |
| sn-isghm 42630 | Longer proof of ~ isghm , ... |
| aprilfools2025 42631 | An abuse of notation. (Co... |
| nfa1w 42632 | Replace ~ ax-10 in ~ nfa1 ... |
| eu6w 42633 | Replace ~ ax-10 , ~ ax-12 ... |
| abbibw 42634 | Replace ~ ax-10 , ~ ax-11 ... |
| absnw 42635 | Replace ~ ax-10 , ~ ax-11 ... |
| euabsn2w 42636 | Replace ~ ax-10 , ~ ax-11 ... |
| sn-tz6.12-2 42637 | ~ tz6.12-2 without ~ ax-10... |
| cu3addd 42638 | Cube of sum of three numbe... |
| sqnegd 42639 | The square of the negative... |
| negexpidd 42640 | The sum of a real number t... |
| rexlimdv3d 42641 | An extended version of ~ r... |
| 3cubeslem1 42642 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem2 42643 | Lemma for ~ 3cubes . Used... |
| 3cubeslem3l 42644 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3r 42645 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3 42646 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem4 42647 | Lemma for ~ 3cubes . This... |
| 3cubes 42648 | Every rational number is a... |
| rntrclfvOAI 42649 | The range of the transitiv... |
| moxfr 42650 | Transfer at-most-one betwe... |
| imaiinfv 42651 | Indexed intersection of an... |
| elrfi 42652 | Elementhood in a set of re... |
| elrfirn 42653 | Elementhood in a set of re... |
| elrfirn2 42654 | Elementhood in a set of re... |
| cmpfiiin 42655 | In a compact topology, a s... |
| ismrcd1 42656 | Any function from the subs... |
| ismrcd2 42657 | Second half of ~ ismrcd1 .... |
| istopclsd 42658 | A closure function which s... |
| ismrc 42659 | A function is a Moore clos... |
| isnacs 42662 | Expand definition of Noeth... |
| nacsfg 42663 | In a Noetherian-type closu... |
| isnacs2 42664 | Express Noetherian-type cl... |
| mrefg2 42665 | Slight variation on finite... |
| mrefg3 42666 | Slight variation on finite... |
| nacsacs 42667 | A closure system of Noethe... |
| isnacs3 42668 | A choice-free order equiva... |
| incssnn0 42669 | Transitivity induction of ... |
| nacsfix 42670 | An increasing sequence of ... |
| constmap 42671 | A constant (represented wi... |
| mapco2g 42672 | Renaming indices in a tupl... |
| mapco2 42673 | Post-composition (renaming... |
| mapfzcons 42674 | Extending a one-based mapp... |
| mapfzcons1 42675 | Recover prefix mapping fro... |
| mapfzcons1cl 42676 | A nonempty mapping has a p... |
| mapfzcons2 42677 | Recover added element from... |
| mptfcl 42678 | Interpret range of a maps-... |
| mzpclval 42683 | Substitution lemma for ` m... |
| elmzpcl 42684 | Double substitution lemma ... |
| mzpclall 42685 | The set of all functions w... |
| mzpcln0 42686 | Corollary of ~ mzpclall : ... |
| mzpcl1 42687 | Defining property 1 of a p... |
| mzpcl2 42688 | Defining property 2 of a p... |
| mzpcl34 42689 | Defining properties 3 and ... |
| mzpval 42690 | Value of the ` mzPoly ` fu... |
| dmmzp 42691 | ` mzPoly ` is defined for ... |
| mzpincl 42692 | Polynomial closedness is a... |
| mzpconst 42693 | Constant functions are pol... |
| mzpf 42694 | A polynomial function is a... |
| mzpproj 42695 | A projection function is p... |
| mzpadd 42696 | The pointwise sum of two p... |
| mzpmul 42697 | The pointwise product of t... |
| mzpconstmpt 42698 | A constant function expres... |
| mzpaddmpt 42699 | Sum of polynomial function... |
| mzpmulmpt 42700 | Product of polynomial func... |
| mzpsubmpt 42701 | The difference of two poly... |
| mzpnegmpt 42702 | Negation of a polynomial f... |
| mzpexpmpt 42703 | Raise a polynomial functio... |
| mzpindd 42704 | "Structural" induction to ... |
| mzpmfp 42705 | Relationship between multi... |
| mzpsubst 42706 | Substituting polynomials f... |
| mzprename 42707 | Simplified version of ~ mz... |
| mzpresrename 42708 | A polynomial is a polynomi... |
| mzpcompact2lem 42709 | Lemma for ~ mzpcompact2 . ... |
| mzpcompact2 42710 | Polynomials are finitary o... |
| coeq0i 42711 | ~ coeq0 but without explic... |
| fzsplit1nn0 42712 | Split a finite 1-based set... |
| eldiophb 42715 | Initial expression of Diop... |
| eldioph 42716 | Condition for a set to be ... |
| diophrw 42717 | Renaming and adding unused... |
| eldioph2lem1 42718 | Lemma for ~ eldioph2 . Co... |
| eldioph2lem2 42719 | Lemma for ~ eldioph2 . Co... |
| eldioph2 42720 | Construct a Diophantine se... |
| eldioph2b 42721 | While Diophantine sets wer... |
| eldiophelnn0 42722 | Remove antecedent on ` B `... |
| eldioph3b 42723 | Define Diophantine sets in... |
| eldioph3 42724 | Inference version of ~ eld... |
| ellz1 42725 | Membership in a lower set ... |
| lzunuz 42726 | The union of a lower set o... |
| fz1eqin 42727 | Express a one-based finite... |
| lzenom 42728 | Lower integers are countab... |
| elmapresaunres2 42729 | ~ fresaunres2 transposed t... |
| diophin 42730 | If two sets are Diophantin... |
| diophun 42731 | If two sets are Diophantin... |
| eldiophss 42732 | Diophantine sets are sets ... |
| diophrex 42733 | Projecting a Diophantine s... |
| eq0rabdioph 42734 | This is the first of a num... |
| eqrabdioph 42735 | Diophantine set builder fo... |
| 0dioph 42736 | The null set is Diophantin... |
| vdioph 42737 | The "universal" set (as la... |
| anrabdioph 42738 | Diophantine set builder fo... |
| orrabdioph 42739 | Diophantine set builder fo... |
| 3anrabdioph 42740 | Diophantine set builder fo... |
| 3orrabdioph 42741 | Diophantine set builder fo... |
| 2sbcrex 42742 | Exchange an existential qu... |
| sbcrexgOLD 42743 | Interchange class substitu... |
| 2sbcrexOLD 42744 | Exchange an existential qu... |
| sbc2rex 42745 | Exchange a substitution wi... |
| sbc2rexgOLD 42746 | Exchange a substitution wi... |
| sbc4rex 42747 | Exchange a substitution wi... |
| sbc4rexgOLD 42748 | Exchange a substitution wi... |
| sbcrot3 42749 | Rotate a sequence of three... |
| sbcrot5 42750 | Rotate a sequence of five ... |
| sbccomieg 42751 | Commute two explicit subst... |
| rexrabdioph 42752 | Diophantine set builder fo... |
| rexfrabdioph 42753 | Diophantine set builder fo... |
| 2rexfrabdioph 42754 | Diophantine set builder fo... |
| 3rexfrabdioph 42755 | Diophantine set builder fo... |
| 4rexfrabdioph 42756 | Diophantine set builder fo... |
| 6rexfrabdioph 42757 | Diophantine set builder fo... |
| 7rexfrabdioph 42758 | Diophantine set builder fo... |
| rabdiophlem1 42759 | Lemma for arithmetic dioph... |
| rabdiophlem2 42760 | Lemma for arithmetic dioph... |
| elnn0rabdioph 42761 | Diophantine set builder fo... |
| rexzrexnn0 42762 | Rewrite an existential qua... |
| lerabdioph 42763 | Diophantine set builder fo... |
| eluzrabdioph 42764 | Diophantine set builder fo... |
| elnnrabdioph 42765 | Diophantine set builder fo... |
| ltrabdioph 42766 | Diophantine set builder fo... |
| nerabdioph 42767 | Diophantine set builder fo... |
| dvdsrabdioph 42768 | Divisibility is a Diophant... |
| eldioph4b 42769 | Membership in ` Dioph ` ex... |
| eldioph4i 42770 | Forward-only version of ~ ... |
| diophren 42771 | Change variables in a Diop... |
| rabrenfdioph 42772 | Change variable numbers in... |
| rabren3dioph 42773 | Change variable numbers in... |
| fphpd 42774 | Pigeonhole principle expre... |
| fphpdo 42775 | Pigeonhole principle for s... |
| ctbnfien 42776 | An infinite subset of a co... |
| fiphp3d 42777 | Infinite pigeonhole princi... |
| rencldnfilem 42778 | Lemma for ~ rencldnfi . (... |
| rencldnfi 42779 | A set of real numbers whic... |
| irrapxlem1 42780 | Lemma for ~ irrapx1 . Div... |
| irrapxlem2 42781 | Lemma for ~ irrapx1 . Two... |
| irrapxlem3 42782 | Lemma for ~ irrapx1 . By ... |
| irrapxlem4 42783 | Lemma for ~ irrapx1 . Eli... |
| irrapxlem5 42784 | Lemma for ~ irrapx1 . Swi... |
| irrapxlem6 42785 | Lemma for ~ irrapx1 . Exp... |
| irrapx1 42786 | Dirichlet's approximation ... |
| pellexlem1 42787 | Lemma for ~ pellex . Arit... |
| pellexlem2 42788 | Lemma for ~ pellex . Arit... |
| pellexlem3 42789 | Lemma for ~ pellex . To e... |
| pellexlem4 42790 | Lemma for ~ pellex . Invo... |
| pellexlem5 42791 | Lemma for ~ pellex . Invo... |
| pellexlem6 42792 | Lemma for ~ pellex . Doin... |
| pellex 42793 | Every Pell equation has a ... |
| pell1qrval 42804 | Value of the set of first-... |
| elpell1qr 42805 | Membership in a first-quad... |
| pell14qrval 42806 | Value of the set of positi... |
| elpell14qr 42807 | Membership in the set of p... |
| pell1234qrval 42808 | Value of the set of genera... |
| elpell1234qr 42809 | Membership in the set of g... |
| pell1234qrre 42810 | General Pell solutions are... |
| pell1234qrne0 42811 | No solution to a Pell equa... |
| pell1234qrreccl 42812 | General solutions of the P... |
| pell1234qrmulcl 42813 | General solutions of the P... |
| pell14qrss1234 42814 | A positive Pell solution i... |
| pell14qrre 42815 | A positive Pell solution i... |
| pell14qrne0 42816 | A positive Pell solution i... |
| pell14qrgt0 42817 | A positive Pell solution i... |
| pell14qrrp 42818 | A positive Pell solution i... |
| pell1234qrdich 42819 | A general Pell solution is... |
| elpell14qr2 42820 | A number is a positive Pel... |
| pell14qrmulcl 42821 | Positive Pell solutions ar... |
| pell14qrreccl 42822 | Positive Pell solutions ar... |
| pell14qrdivcl 42823 | Positive Pell solutions ar... |
| pell14qrexpclnn0 42824 | Lemma for ~ pell14qrexpcl ... |
| pell14qrexpcl 42825 | Positive Pell solutions ar... |
| pell1qrss14 42826 | First-quadrant Pell soluti... |
| pell14qrdich 42827 | A positive Pell solution i... |
| pell1qrge1 42828 | A Pell solution in the fir... |
| pell1qr1 42829 | 1 is a Pell solution and i... |
| elpell1qr2 42830 | The first quadrant solutio... |
| pell1qrgaplem 42831 | Lemma for ~ pell1qrgap . ... |
| pell1qrgap 42832 | First-quadrant Pell soluti... |
| pell14qrgap 42833 | Positive Pell solutions ar... |
| pell14qrgapw 42834 | Positive Pell solutions ar... |
| pellqrexplicit 42835 | Condition for a calculated... |
| infmrgelbi 42836 | Any lower bound of a nonem... |
| pellqrex 42837 | There is a nontrivial solu... |
| pellfundval 42838 | Value of the fundamental s... |
| pellfundre 42839 | The fundamental solution o... |
| pellfundge 42840 | Lower bound on the fundame... |
| pellfundgt1 42841 | Weak lower bound on the Pe... |
| pellfundlb 42842 | A nontrivial first quadran... |
| pellfundglb 42843 | If a real is larger than t... |
| pellfundex 42844 | The fundamental solution a... |
| pellfund14gap 42845 | There are no solutions bet... |
| pellfundrp 42846 | The fundamental Pell solut... |
| pellfundne1 42847 | The fundamental Pell solut... |
| reglogcl 42848 | General logarithm is a rea... |
| reglogltb 42849 | General logarithm preserve... |
| reglogleb 42850 | General logarithm preserve... |
| reglogmul 42851 | Multiplication law for gen... |
| reglogexp 42852 | Power law for general log.... |
| reglogbas 42853 | General log of the base is... |
| reglog1 42854 | General log of 1 is 0. (C... |
| reglogexpbas 42855 | General log of a power of ... |
| pellfund14 42856 | Every positive Pell soluti... |
| pellfund14b 42857 | The positive Pell solution... |
| rmxfval 42862 | Value of the X sequence. ... |
| rmyfval 42863 | Value of the Y sequence. ... |
| rmspecsqrtnq 42864 | The discriminant used to d... |
| rmspecnonsq 42865 | The discriminant used to d... |
| qirropth 42866 | This lemma implements the ... |
| rmspecfund 42867 | The base of exponent used ... |
| rmxyelqirr 42868 | The solutions used to cons... |
| rmxyelqirrOLD 42869 | Obsolete version of ~ rmxy... |
| rmxypairf1o 42870 | The function used to extra... |
| rmxyelxp 42871 | Lemma for ~ frmx and ~ frm... |
| frmx 42872 | The X sequence is a nonneg... |
| frmy 42873 | The Y sequence is an integ... |
| rmxyval 42874 | Main definition of the X a... |
| rmspecpos 42875 | The discriminant used to d... |
| rmxycomplete 42876 | The X and Y sequences take... |
| rmxynorm 42877 | The X and Y sequences defi... |
| rmbaserp 42878 | The base of exponentiation... |
| rmxyneg 42879 | Negation law for X and Y s... |
| rmxyadd 42880 | Addition formula for X and... |
| rmxy1 42881 | Value of the X and Y seque... |
| rmxy0 42882 | Value of the X and Y seque... |
| rmxneg 42883 | Negation law (even functio... |
| rmx0 42884 | Value of X sequence at 0. ... |
| rmx1 42885 | Value of X sequence at 1. ... |
| rmxadd 42886 | Addition formula for X seq... |
| rmyneg 42887 | Negation formula for Y seq... |
| rmy0 42888 | Value of Y sequence at 0. ... |
| rmy1 42889 | Value of Y sequence at 1. ... |
| rmyadd 42890 | Addition formula for Y seq... |
| rmxp1 42891 | Special addition-of-1 form... |
| rmyp1 42892 | Special addition of 1 form... |
| rmxm1 42893 | Subtraction of 1 formula f... |
| rmym1 42894 | Subtraction of 1 formula f... |
| rmxluc 42895 | The X sequence is a Lucas ... |
| rmyluc 42896 | The Y sequence is a Lucas ... |
| rmyluc2 42897 | Lucas sequence property of... |
| rmxdbl 42898 | "Double-angle formula" for... |
| rmydbl 42899 | "Double-angle formula" for... |
| monotuz 42900 | A function defined on an u... |
| monotoddzzfi 42901 | A function which is odd an... |
| monotoddzz 42902 | A function (given implicit... |
| oddcomabszz 42903 | An odd function which take... |
| 2nn0ind 42904 | Induction on nonnegative i... |
| zindbi 42905 | Inductively transfer a pro... |
| rmxypos 42906 | For all nonnegative indice... |
| ltrmynn0 42907 | The Y-sequence is strictly... |
| ltrmxnn0 42908 | The X-sequence is strictly... |
| lermxnn0 42909 | The X-sequence is monotoni... |
| rmxnn 42910 | The X-sequence is defined ... |
| ltrmy 42911 | The Y-sequence is strictly... |
| rmyeq0 42912 | Y is zero only at zero. (... |
| rmyeq 42913 | Y is one-to-one. (Contrib... |
| lermy 42914 | Y is monotonic (non-strict... |
| rmynn 42915 | ` rmY ` is positive for po... |
| rmynn0 42916 | ` rmY ` is nonnegative for... |
| rmyabs 42917 | ` rmY ` commutes with ` ab... |
| jm2.24nn 42918 | X(n) is strictly greater t... |
| jm2.17a 42919 | First half of lemma 2.17 o... |
| jm2.17b 42920 | Weak form of the second ha... |
| jm2.17c 42921 | Second half of lemma 2.17 ... |
| jm2.24 42922 | Lemma 2.24 of [JonesMatija... |
| rmygeid 42923 | Y(n) increases faster than... |
| congtr 42924 | A wff of the form ` A || (... |
| congadd 42925 | If two pairs of numbers ar... |
| congmul 42926 | If two pairs of numbers ar... |
| congsym 42927 | Congruence mod ` A ` is a ... |
| congneg 42928 | If two integers are congru... |
| congsub 42929 | If two pairs of numbers ar... |
| congid 42930 | Every integer is congruent... |
| mzpcong 42931 | Polynomials commute with c... |
| congrep 42932 | Every integer is congruent... |
| congabseq 42933 | If two integers are congru... |
| acongid 42934 | A wff like that in this th... |
| acongsym 42935 | Symmetry of alternating co... |
| acongneg2 42936 | Negate right side of alter... |
| acongtr 42937 | Transitivity of alternatin... |
| acongeq12d 42938 | Substitution deduction for... |
| acongrep 42939 | Every integer is alternati... |
| fzmaxdif 42940 | Bound on the difference be... |
| fzneg 42941 | Reflection of a finite ran... |
| acongeq 42942 | Two numbers in the fundame... |
| dvdsacongtr 42943 | Alternating congruence pas... |
| coprmdvdsb 42944 | Multiplication by a coprim... |
| modabsdifz 42945 | Divisibility in terms of m... |
| dvdsabsmod0 42946 | Divisibility in terms of m... |
| jm2.18 42947 | Theorem 2.18 of [JonesMati... |
| jm2.19lem1 42948 | Lemma for ~ jm2.19 . X an... |
| jm2.19lem2 42949 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem3 42950 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem4 42951 | Lemma for ~ jm2.19 . Exte... |
| jm2.19 42952 | Lemma 2.19 of [JonesMatija... |
| jm2.21 42953 | Lemma for ~ jm2.20nn . Ex... |
| jm2.22 42954 | Lemma for ~ jm2.20nn . Ap... |
| jm2.23 42955 | Lemma for ~ jm2.20nn . Tr... |
| jm2.20nn 42956 | Lemma 2.20 of [JonesMatija... |
| jm2.25lem1 42957 | Lemma for ~ jm2.26 . (Con... |
| jm2.25 42958 | Lemma for ~ jm2.26 . Rema... |
| jm2.26a 42959 | Lemma for ~ jm2.26 . Reve... |
| jm2.26lem3 42960 | Lemma for ~ jm2.26 . Use ... |
| jm2.26 42961 | Lemma 2.26 of [JonesMatija... |
| jm2.15nn0 42962 | Lemma 2.15 of [JonesMatija... |
| jm2.16nn0 42963 | Lemma 2.16 of [JonesMatija... |
| jm2.27a 42964 | Lemma for ~ jm2.27 . Reve... |
| jm2.27b 42965 | Lemma for ~ jm2.27 . Expa... |
| jm2.27c 42966 | Lemma for ~ jm2.27 . Forw... |
| jm2.27 42967 | Lemma 2.27 of [JonesMatija... |
| jm2.27dlem1 42968 | Lemma for ~ rmydioph . Su... |
| jm2.27dlem2 42969 | Lemma for ~ rmydioph . Th... |
| jm2.27dlem3 42970 | Lemma for ~ rmydioph . In... |
| jm2.27dlem4 42971 | Lemma for ~ rmydioph . In... |
| jm2.27dlem5 42972 | Lemma for ~ rmydioph . Us... |
| rmydioph 42973 | ~ jm2.27 restated in terms... |
| rmxdiophlem 42974 | X can be expressed in term... |
| rmxdioph 42975 | X is a Diophantine functio... |
| jm3.1lem1 42976 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem2 42977 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem3 42978 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1 42979 | Diophantine expression for... |
| expdiophlem1 42980 | Lemma for ~ expdioph . Fu... |
| expdiophlem2 42981 | Lemma for ~ expdioph . Ex... |
| expdioph 42982 | The exponential function i... |
| setindtr 42983 | Set induction for sets con... |
| setindtrs 42984 | Set induction scheme witho... |
| dford3lem1 42985 | Lemma for ~ dford3 . (Con... |
| dford3lem2 42986 | Lemma for ~ dford3 . (Con... |
| dford3 42987 | Ordinals are precisely the... |
| dford4 42988 | ~ dford3 expressed in prim... |
| wopprc 42989 | Unrelated: Wiener pairs t... |
| rpnnen3lem 42990 | Lemma for ~ rpnnen3 . (Co... |
| rpnnen3 42991 | Dedekind cut injection of ... |
| axac10 42992 | Characterization of choice... |
| harinf 42993 | The Hartogs number of an i... |
| wdom2d2 42994 | Deduction for weak dominan... |
| ttac 42995 | Tarski's theorem about cho... |
| pw2f1ocnv 42996 | Define a bijection between... |
| pw2f1o2 42997 | Define a bijection between... |
| pw2f1o2val 42998 | Function value of the ~ pw... |
| pw2f1o2val2 42999 | Membership in a mapped set... |
| limsuc2 43000 | Limit ordinals in the sens... |
| wepwsolem 43001 | Transfer an ordering on ch... |
| wepwso 43002 | A well-ordering induces a ... |
| dnnumch1 43003 | Define an enumeration of a... |
| dnnumch2 43004 | Define an enumeration (wea... |
| dnnumch3lem 43005 | Value of the ordinal injec... |
| dnnumch3 43006 | Define an injection from a... |
| dnwech 43007 | Define a well-ordering fro... |
| fnwe2val 43008 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem1 43009 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem2 43010 | Lemma for ~ fnwe2 . An el... |
| fnwe2lem3 43011 | Lemma for ~ fnwe2 . Trich... |
| fnwe2 43012 | A well-ordering can be con... |
| aomclem1 43013 | Lemma for ~ dfac11 . This... |
| aomclem2 43014 | Lemma for ~ dfac11 . Succ... |
| aomclem3 43015 | Lemma for ~ dfac11 . Succ... |
| aomclem4 43016 | Lemma for ~ dfac11 . Limi... |
| aomclem5 43017 | Lemma for ~ dfac11 . Comb... |
| aomclem6 43018 | Lemma for ~ dfac11 . Tran... |
| aomclem7 43019 | Lemma for ~ dfac11 . ` ( R... |
| aomclem8 43020 | Lemma for ~ dfac11 . Perf... |
| dfac11 43021 | The right-hand side of thi... |
| kelac1 43022 | Kelley's choice, basic for... |
| kelac2lem 43023 | Lemma for ~ kelac2 and ~ d... |
| kelac2 43024 | Kelley's choice, most comm... |
| dfac21 43025 | Tychonoff's theorem is a c... |
| islmodfg 43028 | Property of a finitely gen... |
| islssfg 43029 | Property of a finitely gen... |
| islssfg2 43030 | Property of a finitely gen... |
| islssfgi 43031 | Finitely spanned subspaces... |
| fglmod 43032 | Finitely generated left mo... |
| lsmfgcl 43033 | The sum of two finitely ge... |
| islnm 43036 | Property of being a Noethe... |
| islnm2 43037 | Property of being a Noethe... |
| lnmlmod 43038 | A Noetherian left module i... |
| lnmlssfg 43039 | A submodule of Noetherian ... |
| lnmlsslnm 43040 | All submodules of a Noethe... |
| lnmfg 43041 | A Noetherian left module i... |
| kercvrlsm 43042 | The domain of a linear fun... |
| lmhmfgima 43043 | A homomorphism maps finite... |
| lnmepi 43044 | Epimorphic images of Noeth... |
| lmhmfgsplit 43045 | If the kernel and range of... |
| lmhmlnmsplit 43046 | If the kernel and range of... |
| lnmlmic 43047 | Noetherian is an invariant... |
| pwssplit4 43048 | Splitting for structure po... |
| filnm 43049 | Finite left modules are No... |
| pwslnmlem0 43050 | Zeroeth powers are Noether... |
| pwslnmlem1 43051 | First powers are Noetheria... |
| pwslnmlem2 43052 | A sum of powers is Noether... |
| pwslnm 43053 | Finite powers of Noetheria... |
| unxpwdom3 43054 | Weaker version of ~ unxpwd... |
| pwfi2f1o 43055 | The ~ pw2f1o bijection rel... |
| pwfi2en 43056 | Finitely supported indicat... |
| frlmpwfi 43057 | Formal linear combinations... |
| gicabl 43058 | Being Abelian is a group i... |
| imasgim 43059 | A relabeling of the elemen... |
| isnumbasgrplem1 43060 | A set which is equipollent... |
| harn0 43061 | The Hartogs number of a se... |
| numinfctb 43062 | A numerable infinite set c... |
| isnumbasgrplem2 43063 | If the (to be thought of a... |
| isnumbasgrplem3 43064 | Every nonempty numerable s... |
| isnumbasabl 43065 | A set is numerable iff it ... |
| isnumbasgrp 43066 | A set is numerable iff it ... |
| dfacbasgrp 43067 | A choice equivalent in abs... |
| islnr 43070 | Property of a left-Noether... |
| lnrring 43071 | Left-Noetherian rings are ... |
| lnrlnm 43072 | Left-Noetherian rings have... |
| islnr2 43073 | Property of being a left-N... |
| islnr3 43074 | Relate left-Noetherian rin... |
| lnr2i 43075 | Given an ideal in a left-N... |
| lpirlnr 43076 | Left principal ideal rings... |
| lnrfrlm 43077 | Finite-dimensional free mo... |
| lnrfg 43078 | Finitely-generated modules... |
| lnrfgtr 43079 | A submodule of a finitely ... |
| hbtlem1 43082 | Value of the leading coeff... |
| hbtlem2 43083 | Leading coefficient ideals... |
| hbtlem7 43084 | Functionality of leading c... |
| hbtlem4 43085 | The leading ideal function... |
| hbtlem3 43086 | The leading ideal function... |
| hbtlem5 43087 | The leading ideal function... |
| hbtlem6 43088 | There is a finite set of p... |
| hbt 43089 | The Hilbert Basis Theorem ... |
| dgrsub2 43094 | Subtracting two polynomial... |
| elmnc 43095 | Property of a monic polyno... |
| mncply 43096 | A monic polynomial is a po... |
| mnccoe 43097 | A monic polynomial has lea... |
| mncn0 43098 | A monic polynomial is not ... |
| dgraaval 43103 | Value of the degree functi... |
| dgraalem 43104 | Properties of the degree o... |
| dgraacl 43105 | Closure of the degree func... |
| dgraaf 43106 | Degree function on algebra... |
| dgraaub 43107 | Upper bound on degree of a... |
| dgraa0p 43108 | A rational polynomial of d... |
| mpaaeu 43109 | An algebraic number has ex... |
| mpaaval 43110 | Value of the minimal polyn... |
| mpaalem 43111 | Properties of the minimal ... |
| mpaacl 43112 | Minimal polynomial is a po... |
| mpaadgr 43113 | Minimal polynomial has deg... |
| mpaaroot 43114 | The minimal polynomial of ... |
| mpaamn 43115 | Minimal polynomial is moni... |
| itgoval 43120 | Value of the integral-over... |
| aaitgo 43121 | The standard algebraic num... |
| itgoss 43122 | An integral element is int... |
| itgocn 43123 | All integral elements are ... |
| cnsrexpcl 43124 | Exponentiation is closed i... |
| fsumcnsrcl 43125 | Finite sums are closed in ... |
| cnsrplycl 43126 | Polynomials are closed in ... |
| rgspnval 43127 | Value of the ring-span of ... |
| rgspncl 43128 | The ring-span of a set is ... |
| rgspnssid 43129 | The ring-span of a set con... |
| rgspnmin 43130 | The ring-span is contained... |
| rgspnid 43131 | The span of a subring is i... |
| rngunsnply 43132 | Adjoining one element to a... |
| flcidc 43133 | Finite linear combinations... |
| algstr 43136 | Lemma to shorten proofs of... |
| algbase 43137 | The base set of a construc... |
| algaddg 43138 | The additive operation of ... |
| algmulr 43139 | The multiplicative operati... |
| algsca 43140 | The set of scalars of a co... |
| algvsca 43141 | The scalar product operati... |
| mendval 43142 | Value of the module endomo... |
| mendbas 43143 | Base set of the module end... |
| mendplusgfval 43144 | Addition in the module end... |
| mendplusg 43145 | A specific addition in the... |
| mendmulrfval 43146 | Multiplication in the modu... |
| mendmulr 43147 | A specific multiplication ... |
| mendsca 43148 | The module endomorphism al... |
| mendvscafval 43149 | Scalar multiplication in t... |
| mendvsca 43150 | A specific scalar multipli... |
| mendring 43151 | The module endomorphism al... |
| mendlmod 43152 | The module endomorphism al... |
| mendassa 43153 | The module endomorphism al... |
| idomodle 43154 | Limit on the number of ` N... |
| fiuneneq 43155 | Two finite sets of equal s... |
| idomsubgmo 43156 | The units of an integral d... |
| proot1mul 43157 | Any primitive ` N ` -th ro... |
| proot1hash 43158 | If an integral domain has ... |
| proot1ex 43159 | The complex field has prim... |
| mon1psubm 43162 | Monic polynomials are a mu... |
| deg1mhm 43163 | Homomorphic property of th... |
| cytpfn 43164 | Functionality of the cyclo... |
| cytpval 43165 | Substitutions for the Nth ... |
| fgraphopab 43166 | Express a function as a su... |
| fgraphxp 43167 | Express a function as a su... |
| hausgraph 43168 | The graph of a continuous ... |
| r1sssucd 43173 | Deductive form of ~ r1sssu... |
| iocunico 43174 | Split an open interval int... |
| iocinico 43175 | The intersection of two se... |
| iocmbl 43176 | An open-below, closed-abov... |
| cnioobibld 43177 | A bounded, continuous func... |
| arearect 43178 | The area of a rectangle wh... |
| areaquad 43179 | The area of a quadrilatera... |
| uniel 43180 | Two ways to say a union is... |
| unielss 43181 | Two ways to say the union ... |
| unielid 43182 | Two ways to say the union ... |
| ssunib 43183 | Two ways to say a class is... |
| rp-intrabeq 43184 | Equality theorem for supre... |
| rp-unirabeq 43185 | Equality theorem for infim... |
| onmaxnelsup 43186 | Two ways to say the maximu... |
| onsupneqmaxlim0 43187 | If the supremum of a class... |
| onsupcl2 43188 | The supremum of a set of o... |
| onuniintrab 43189 | The union of a set of ordi... |
| onintunirab 43190 | The intersection of a non-... |
| onsupnmax 43191 | If the union of a class of... |
| onsupuni 43192 | The supremum of a set of o... |
| onsupuni2 43193 | The supremum of a set of o... |
| onsupintrab 43194 | The supremum of a set of o... |
| onsupintrab2 43195 | The supremum of a set of o... |
| onsupcl3 43196 | The supremum of a set of o... |
| onsupex3 43197 | The supremum of a set of o... |
| onuniintrab2 43198 | The union of a set of ordi... |
| oninfint 43199 | The infimum of a non-empty... |
| oninfunirab 43200 | The infimum of a non-empty... |
| oninfcl2 43201 | The infimum of a non-empty... |
| onsupmaxb 43202 | The union of a class of or... |
| onexgt 43203 | For any ordinal, there is ... |
| onexomgt 43204 | For any ordinal, there is ... |
| omlimcl2 43205 | The product of a limit ord... |
| onexlimgt 43206 | For any ordinal, there is ... |
| onexoegt 43207 | For any ordinal, there is ... |
| oninfex2 43208 | The infimum of a non-empty... |
| onsupeqmax 43209 | Condition when the supremu... |
| onsupeqnmax 43210 | Condition when the supremu... |
| onsuplub 43211 | The supremum of a set of o... |
| onsupnub 43212 | An upper bound of a set of... |
| onfisupcl 43213 | Sufficient condition when ... |
| onelord 43214 | Every element of a ordinal... |
| onepsuc 43215 | Every ordinal is less than... |
| epsoon 43216 | The ordinals are strictly ... |
| epirron 43217 | The strict order on the or... |
| oneptr 43218 | The strict order on the or... |
| oneltr 43219 | The elementhood relation o... |
| oneptri 43220 | The strict, complete (line... |
| oneltri 43221 | The elementhood relation o... |
| ordeldif 43222 | Membership in the differen... |
| ordeldifsucon 43223 | Membership in the differen... |
| ordeldif1o 43224 | Membership in the differen... |
| ordne0gt0 43225 | Ordinal zero is less than ... |
| ondif1i 43226 | Ordinal zero is less than ... |
| onsucelab 43227 | The successor of every ord... |
| dflim6 43228 | A limit ordinal is a non-z... |
| limnsuc 43229 | A limit ordinal is not an ... |
| onsucss 43230 | If one ordinal is less tha... |
| ordnexbtwnsuc 43231 | For any distinct pair of o... |
| orddif0suc 43232 | For any distinct pair of o... |
| onsucf1lem 43233 | For ordinals, the successo... |
| onsucf1olem 43234 | The successor operation is... |
| onsucrn 43235 | The successor operation is... |
| onsucf1o 43236 | The successor operation is... |
| dflim7 43237 | A limit ordinal is a non-z... |
| onov0suclim 43238 | Compactly express rules fo... |
| oa0suclim 43239 | Closed form expression of ... |
| om0suclim 43240 | Closed form expression of ... |
| oe0suclim 43241 | Closed form expression of ... |
| oaomoecl 43242 | The operations of addition... |
| onsupsucismax 43243 | If the union of a set of o... |
| onsssupeqcond 43244 | If for every element of a ... |
| limexissup 43245 | An ordinal which is a limi... |
| limiun 43246 | A limit ordinal is the uni... |
| limexissupab 43247 | An ordinal which is a limi... |
| om1om1r 43248 | Ordinal one is both a left... |
| oe0rif 43249 | Ordinal zero raised to any... |
| oasubex 43250 | While subtraction can't be... |
| nnamecl 43251 | Natural numbers are closed... |
| onsucwordi 43252 | The successor operation pr... |
| oalim2cl 43253 | The ordinal sum of any ord... |
| oaltublim 43254 | Given ` C ` is a limit ord... |
| oaordi3 43255 | Ordinal addition of the sa... |
| oaord3 43256 | When the same ordinal is a... |
| 1oaomeqom 43257 | Ordinal one plus omega is ... |
| oaabsb 43258 | The right addend absorbs t... |
| oaordnrex 43259 | When omega is added on the... |
| oaordnr 43260 | When the same ordinal is a... |
| omge1 43261 | Any non-zero ordinal produ... |
| omge2 43262 | Any non-zero ordinal produ... |
| omlim2 43263 | The non-zero product with ... |
| omord2lim 43264 | Given a limit ordinal, the... |
| omord2i 43265 | Ordinal multiplication of ... |
| omord2com 43266 | When the same non-zero ord... |
| 2omomeqom 43267 | Ordinal two times omega is... |
| omnord1ex 43268 | When omega is multiplied o... |
| omnord1 43269 | When the same non-zero ord... |
| oege1 43270 | Any non-zero ordinal power... |
| oege2 43271 | Any power of an ordinal at... |
| rp-oelim2 43272 | The power of an ordinal at... |
| oeord2lim 43273 | Given a limit ordinal, the... |
| oeord2i 43274 | Ordinal exponentiation of ... |
| oeord2com 43275 | When the same base at leas... |
| nnoeomeqom 43276 | Any natural number at leas... |
| df3o2 43277 | Ordinal 3 is the unordered... |
| df3o3 43278 | Ordinal 3, fully expanded.... |
| oenord1ex 43279 | When ordinals two and thre... |
| oenord1 43280 | When two ordinals (both at... |
| oaomoencom 43281 | Ordinal addition, multipli... |
| oenassex 43282 | Ordinal two raised to two ... |
| oenass 43283 | Ordinal exponentiation is ... |
| cantnftermord 43284 | For terms of the form of a... |
| cantnfub 43285 | Given a finite number of t... |
| cantnfub2 43286 | Given a finite number of t... |
| bropabg 43287 | Equivalence for two classe... |
| cantnfresb 43288 | A Cantor normal form which... |
| cantnf2 43289 | For every ordinal, ` A ` ,... |
| oawordex2 43290 | If ` C ` is between ` A ` ... |
| nnawordexg 43291 | If an ordinal, ` B ` , is ... |
| succlg 43292 | Closure law for ordinal su... |
| dflim5 43293 | A limit ordinal is either ... |
| oacl2g 43294 | Closure law for ordinal ad... |
| onmcl 43295 | If an ordinal is less than... |
| omabs2 43296 | Ordinal multiplication by ... |
| omcl2 43297 | Closure law for ordinal mu... |
| omcl3g 43298 | Closure law for ordinal mu... |
| ordsssucb 43299 | An ordinal number is less ... |
| tfsconcatlem 43300 | Lemma for ~ tfsconcatun . ... |
| tfsconcatun 43301 | The concatenation of two t... |
| tfsconcatfn 43302 | The concatenation of two t... |
| tfsconcatfv1 43303 | An early value of the conc... |
| tfsconcatfv2 43304 | A latter value of the conc... |
| tfsconcatfv 43305 | The value of the concatena... |
| tfsconcatrn 43306 | The range of the concatena... |
| tfsconcatfo 43307 | The concatenation of two t... |
| tfsconcatb0 43308 | The concatentation with th... |
| tfsconcat0i 43309 | The concatentation with th... |
| tfsconcat0b 43310 | The concatentation with th... |
| tfsconcat00 43311 | The concatentation of two ... |
| tfsconcatrev 43312 | If the domain of a transfi... |
| tfsconcatrnss12 43313 | The range of the concatena... |
| tfsconcatrnss 43314 | The concatenation of trans... |
| tfsconcatrnsson 43315 | The concatenation of trans... |
| tfsnfin 43316 | A transfinite sequence is ... |
| rp-tfslim 43317 | The limit of a sequence of... |
| ofoafg 43318 | Addition operator for func... |
| ofoaf 43319 | Addition operator for func... |
| ofoafo 43320 | Addition operator for func... |
| ofoacl 43321 | Closure law for component ... |
| ofoaid1 43322 | Identity law for component... |
| ofoaid2 43323 | Identity law for component... |
| ofoaass 43324 | Component-wise addition of... |
| ofoacom 43325 | Component-wise addition of... |
| naddcnff 43326 | Addition operator for Cant... |
| naddcnffn 43327 | Addition operator for Cant... |
| naddcnffo 43328 | Addition of Cantor normal ... |
| naddcnfcl 43329 | Closure law for component-... |
| naddcnfcom 43330 | Component-wise ordinal add... |
| naddcnfid1 43331 | Identity law for component... |
| naddcnfid2 43332 | Identity law for component... |
| naddcnfass 43333 | Component-wise addition of... |
| onsucunifi 43334 | The successor to the union... |
| sucunisn 43335 | The successor to the union... |
| onsucunipr 43336 | The successor to the union... |
| onsucunitp 43337 | The successor to the union... |
| oaun3lem1 43338 | The class of all ordinal s... |
| oaun3lem2 43339 | The class of all ordinal s... |
| oaun3lem3 43340 | The class of all ordinal s... |
| oaun3lem4 43341 | The class of all ordinal s... |
| rp-abid 43342 | Two ways to express a clas... |
| oadif1lem 43343 | Express the set difference... |
| oadif1 43344 | Express the set difference... |
| oaun2 43345 | Ordinal addition as a unio... |
| oaun3 43346 | Ordinal addition as a unio... |
| naddov4 43347 | Alternate expression for n... |
| nadd2rabtr 43348 | The set of ordinals which ... |
| nadd2rabord 43349 | The set of ordinals which ... |
| nadd2rabex 43350 | The class of ordinals whic... |
| nadd2rabon 43351 | The set of ordinals which ... |
| nadd1rabtr 43352 | The set of ordinals which ... |
| nadd1rabord 43353 | The set of ordinals which ... |
| nadd1rabex 43354 | The class of ordinals whic... |
| nadd1rabon 43355 | The set of ordinals which ... |
| nadd1suc 43356 | Natural addition with 1 is... |
| naddass1 43357 | Natural addition of ordina... |
| naddgeoa 43358 | Natural addition results i... |
| naddonnn 43359 | Natural addition with a na... |
| naddwordnexlem0 43360 | When ` A ` is the sum of a... |
| naddwordnexlem1 43361 | When ` A ` is the sum of a... |
| naddwordnexlem2 43362 | When ` A ` is the sum of a... |
| naddwordnexlem3 43363 | When ` A ` is the sum of a... |
| oawordex3 43364 | When ` A ` is the sum of a... |
| naddwordnexlem4 43365 | When ` A ` is the sum of a... |
| ordsssucim 43366 | If an ordinal is less than... |
| insucid 43367 | The intersection of a clas... |
| om2 43368 | Two ways to double an ordi... |
| oaltom 43369 | Multiplication eventually ... |
| oe2 43370 | Two ways to square an ordi... |
| omltoe 43371 | Exponentiation eventually ... |
| abeqabi 43372 | Generalized condition for ... |
| abpr 43373 | Condition for a class abst... |
| abtp 43374 | Condition for a class abst... |
| ralopabb 43375 | Restricted universal quant... |
| fpwfvss 43376 | Functions into a powerset ... |
| sdomne0 43377 | A class that strictly domi... |
| sdomne0d 43378 | A class that strictly domi... |
| safesnsupfiss 43379 | If ` B ` is a finite subse... |
| safesnsupfiub 43380 | If ` B ` is a finite subse... |
| safesnsupfidom1o 43381 | If ` B ` is a finite subse... |
| safesnsupfilb 43382 | If ` B ` is a finite subse... |
| isoeq145d 43383 | Equality deduction for iso... |
| resisoeq45d 43384 | Equality deduction for equ... |
| negslem1 43385 | An equivalence between ide... |
| nvocnvb 43386 | Equivalence to saying the ... |
| rp-brsslt 43387 | Binary relation form of a ... |
| nla0002 43388 | Extending a linear order t... |
| nla0003 43389 | Extending a linear order t... |
| nla0001 43390 | Extending a linear order t... |
| faosnf0.11b 43391 | ` B ` is called a non-limi... |
| dfno2 43392 | A surreal number, in the f... |
| onnog 43393 | Every ordinal maps to a su... |
| onnobdayg 43394 | Every ordinal maps to a su... |
| bdaybndex 43395 | Bounds formed from the bir... |
| bdaybndbday 43396 | Bounds formed from the bir... |
| onno 43397 | Every ordinal maps to a su... |
| onnoi 43398 | Every ordinal maps to a su... |
| 0no 43399 | Ordinal zero maps to a sur... |
| 1no 43400 | Ordinal one maps to a surr... |
| 2no 43401 | Ordinal two maps to a surr... |
| 3no 43402 | Ordinal three maps to a su... |
| 4no 43403 | Ordinal four maps to a sur... |
| fnimafnex 43404 | The functional image of a ... |
| nlimsuc 43405 | A successor is not a limit... |
| nlim1NEW 43406 | 1 is not a limit ordinal. ... |
| nlim2NEW 43407 | 2 is not a limit ordinal. ... |
| nlim3 43408 | 3 is not a limit ordinal. ... |
| nlim4 43409 | 4 is not a limit ordinal. ... |
| oa1un 43410 | Given ` A e. On ` , let ` ... |
| oa1cl 43411 | ` A +o 1o ` is in ` On ` .... |
| 0finon 43412 | 0 is a finite ordinal. Se... |
| 1finon 43413 | 1 is a finite ordinal. Se... |
| 2finon 43414 | 2 is a finite ordinal. Se... |
| 3finon 43415 | 3 is a finite ordinal. Se... |
| 4finon 43416 | 4 is a finite ordinal. Se... |
| finona1cl 43417 | The finite ordinals are cl... |
| finonex 43418 | The finite ordinals are a ... |
| fzunt 43419 | Union of two adjacent fini... |
| fzuntd 43420 | Union of two adjacent fini... |
| fzunt1d 43421 | Union of two overlapping f... |
| fzuntgd 43422 | Union of two adjacent or o... |
| ifpan123g 43423 | Conjunction of conditional... |
| ifpan23 43424 | Conjunction of conditional... |
| ifpdfor2 43425 | Define or in terms of cond... |
| ifporcor 43426 | Corollary of commutation o... |
| ifpdfan2 43427 | Define and with conditiona... |
| ifpancor 43428 | Corollary of commutation o... |
| ifpdfor 43429 | Define or in terms of cond... |
| ifpdfan 43430 | Define and with conditiona... |
| ifpbi2 43431 | Equivalence theorem for co... |
| ifpbi3 43432 | Equivalence theorem for co... |
| ifpim1 43433 | Restate implication as con... |
| ifpnot 43434 | Restate negated wff as con... |
| ifpid2 43435 | Restate wff as conditional... |
| ifpim2 43436 | Restate implication as con... |
| ifpbi23 43437 | Equivalence theorem for co... |
| ifpbiidcor 43438 | Restatement of ~ biid . (... |
| ifpbicor 43439 | Corollary of commutation o... |
| ifpxorcor 43440 | Corollary of commutation o... |
| ifpbi1 43441 | Equivalence theorem for co... |
| ifpnot23 43442 | Negation of conditional lo... |
| ifpnotnotb 43443 | Factor conditional logic o... |
| ifpnorcor 43444 | Corollary of commutation o... |
| ifpnancor 43445 | Corollary of commutation o... |
| ifpnot23b 43446 | Negation of conditional lo... |
| ifpbiidcor2 43447 | Restatement of ~ biid . (... |
| ifpnot23c 43448 | Negation of conditional lo... |
| ifpnot23d 43449 | Negation of conditional lo... |
| ifpdfnan 43450 | Define nand as conditional... |
| ifpdfxor 43451 | Define xor as conditional ... |
| ifpbi12 43452 | Equivalence theorem for co... |
| ifpbi13 43453 | Equivalence theorem for co... |
| ifpbi123 43454 | Equivalence theorem for co... |
| ifpidg 43455 | Restate wff as conditional... |
| ifpid3g 43456 | Restate wff as conditional... |
| ifpid2g 43457 | Restate wff as conditional... |
| ifpid1g 43458 | Restate wff as conditional... |
| ifpim23g 43459 | Restate implication as con... |
| ifpim3 43460 | Restate implication as con... |
| ifpnim1 43461 | Restate negated implicatio... |
| ifpim4 43462 | Restate implication as con... |
| ifpnim2 43463 | Restate negated implicatio... |
| ifpim123g 43464 | Implication of conditional... |
| ifpim1g 43465 | Implication of conditional... |
| ifp1bi 43466 | Substitute the first eleme... |
| ifpbi1b 43467 | When the first variable is... |
| ifpimimb 43468 | Factor conditional logic o... |
| ifpororb 43469 | Factor conditional logic o... |
| ifpananb 43470 | Factor conditional logic o... |
| ifpnannanb 43471 | Factor conditional logic o... |
| ifpor123g 43472 | Disjunction of conditional... |
| ifpimim 43473 | Consequnce of implication.... |
| ifpbibib 43474 | Factor conditional logic o... |
| ifpxorxorb 43475 | Factor conditional logic o... |
| rp-fakeimass 43476 | A special case where impli... |
| rp-fakeanorass 43477 | A special case where a mix... |
| rp-fakeoranass 43478 | A special case where a mix... |
| rp-fakeinunass 43479 | A special case where a mix... |
| rp-fakeuninass 43480 | A special case where a mix... |
| rp-isfinite5 43481 | A set is said to be finite... |
| rp-isfinite6 43482 | A set is said to be finite... |
| intabssd 43483 | When for each element ` y ... |
| eu0 43484 | There is only one empty se... |
| epelon2 43485 | Over the ordinal numbers, ... |
| ontric3g 43486 | For all ` x , y e. On ` , ... |
| dfsucon 43487 | ` A ` is called a successo... |
| snen1g 43488 | A singleton is equinumerou... |
| snen1el 43489 | A singleton is equinumerou... |
| sn1dom 43490 | A singleton is dominated b... |
| pr2dom 43491 | An unordered pair is domin... |
| tr3dom 43492 | An unordered triple is dom... |
| ensucne0 43493 | A class equinumerous to a ... |
| ensucne0OLD 43494 | A class equinumerous to a ... |
| dfom6 43495 | Let ` _om ` be defined to ... |
| infordmin 43496 | ` _om ` is the smallest in... |
| iscard4 43497 | Two ways to express the pr... |
| minregex 43498 | Given any cardinal number ... |
| minregex2 43499 | Given any cardinal number ... |
| iscard5 43500 | Two ways to express the pr... |
| elrncard 43501 | Let us define a cardinal n... |
| harval3 43502 | ` ( har `` A ) ` is the le... |
| harval3on 43503 | For any ordinal number ` A... |
| omssrncard 43504 | All natural numbers are ca... |
| 0iscard 43505 | 0 is a cardinal number. (... |
| 1iscard 43506 | 1 is a cardinal number. (... |
| omiscard 43507 | ` _om ` is a cardinal numb... |
| sucomisnotcard 43508 | ` _om +o 1o ` is not a car... |
| nna1iscard 43509 | For any natural number, th... |
| har2o 43510 | The least cardinal greater... |
| en2pr 43511 | A class is equinumerous to... |
| pr2cv 43512 | If an unordered pair is eq... |
| pr2el1 43513 | If an unordered pair is eq... |
| pr2cv1 43514 | If an unordered pair is eq... |
| pr2el2 43515 | If an unordered pair is eq... |
| pr2cv2 43516 | If an unordered pair is eq... |
| pren2 43517 | An unordered pair is equin... |
| pr2eldif1 43518 | If an unordered pair is eq... |
| pr2eldif2 43519 | If an unordered pair is eq... |
| pren2d 43520 | A pair of two distinct set... |
| aleph1min 43521 | ` ( aleph `` 1o ) ` is the... |
| alephiso2 43522 | ` aleph ` is a strictly or... |
| alephiso3 43523 | ` aleph ` is a strictly or... |
| pwelg 43524 | The powerclass is an eleme... |
| pwinfig 43525 | The powerclass of an infin... |
| pwinfi2 43526 | The powerclass of an infin... |
| pwinfi3 43527 | The powerclass of an infin... |
| pwinfi 43528 | The powerclass of an infin... |
| fipjust 43529 | A definition of the finite... |
| cllem0 43530 | The class of all sets with... |
| superficl 43531 | The class of all supersets... |
| superuncl 43532 | The class of all supersets... |
| ssficl 43533 | The class of all subsets o... |
| ssuncl 43534 | The class of all subsets o... |
| ssdifcl 43535 | The class of all subsets o... |
| sssymdifcl 43536 | The class of all subsets o... |
| fiinfi 43537 | If two classes have the fi... |
| rababg 43538 | Condition when restricted ... |
| elinintab 43539 | Two ways of saying a set i... |
| elmapintrab 43540 | Two ways to say a set is a... |
| elinintrab 43541 | Two ways of saying a set i... |
| inintabss 43542 | Upper bound on intersectio... |
| inintabd 43543 | Value of the intersection ... |
| xpinintabd 43544 | Value of the intersection ... |
| relintabex 43545 | If the intersection of a c... |
| elcnvcnvintab 43546 | Two ways of saying a set i... |
| relintab 43547 | Value of the intersection ... |
| nonrel 43548 | A non-relation is equal to... |
| elnonrel 43549 | Only an ordered pair where... |
| cnvssb 43550 | Subclass theorem for conve... |
| relnonrel 43551 | The non-relation part of a... |
| cnvnonrel 43552 | The converse of the non-re... |
| brnonrel 43553 | A non-relation cannot rela... |
| dmnonrel 43554 | The domain of the non-rela... |
| rnnonrel 43555 | The range of the non-relat... |
| resnonrel 43556 | A restriction of the non-r... |
| imanonrel 43557 | An image under the non-rel... |
| cononrel1 43558 | Composition with the non-r... |
| cononrel2 43559 | Composition with the non-r... |
| elmapintab 43560 | Two ways to say a set is a... |
| fvnonrel 43561 | The function value of any ... |
| elinlem 43562 | Two ways to say a set is a... |
| elcnvcnvlem 43563 | Two ways to say a set is a... |
| cnvcnvintabd 43564 | Value of the relationship ... |
| elcnvlem 43565 | Two ways to say a set is a... |
| elcnvintab 43566 | Two ways of saying a set i... |
| cnvintabd 43567 | Value of the converse of t... |
| undmrnresiss 43568 | Two ways of saying the ide... |
| reflexg 43569 | Two ways of saying a relat... |
| cnvssco 43570 | A condition weaker than re... |
| refimssco 43571 | Reflexive relations are su... |
| cleq2lem 43572 | Equality implies bijection... |
| cbvcllem 43573 | Change of bound variable i... |
| clublem 43574 | If a superset ` Y ` of ` X... |
| clss2lem 43575 | The closure of a property ... |
| dfid7 43576 | Definition of identity rel... |
| mptrcllem 43577 | Show two versions of a clo... |
| cotrintab 43578 | The intersection of a clas... |
| rclexi 43579 | The reflexive closure of a... |
| rtrclexlem 43580 | Existence of relation impl... |
| rtrclex 43581 | The reflexive-transitive c... |
| trclubgNEW 43582 | If a relation exists then ... |
| trclubNEW 43583 | If a relation exists then ... |
| trclexi 43584 | The transitive closure of ... |
| rtrclexi 43585 | The reflexive-transitive c... |
| clrellem 43586 | When the property ` ps ` h... |
| clcnvlem 43587 | When ` A ` , an upper boun... |
| cnvtrucl0 43588 | The converse of the trivia... |
| cnvrcl0 43589 | The converse of the reflex... |
| cnvtrcl0 43590 | The converse of the transi... |
| dmtrcl 43591 | The domain of the transiti... |
| rntrcl 43592 | The range of the transitiv... |
| dfrtrcl5 43593 | Definition of reflexive-tr... |
| trcleq2lemRP 43594 | Equality implies bijection... |
| sqrtcvallem1 43595 | Two ways of saying a compl... |
| reabsifneg 43596 | Alternate expression for t... |
| reabsifnpos 43597 | Alternate expression for t... |
| reabsifpos 43598 | Alternate expression for t... |
| reabsifnneg 43599 | Alternate expression for t... |
| reabssgn 43600 | Alternate expression for t... |
| sqrtcvallem2 43601 | Equivalent to saying that ... |
| sqrtcvallem3 43602 | Equivalent to saying that ... |
| sqrtcvallem4 43603 | Equivalent to saying that ... |
| sqrtcvallem5 43604 | Equivalent to saying that ... |
| sqrtcval 43605 | Explicit formula for the c... |
| sqrtcval2 43606 | Explicit formula for the c... |
| resqrtval 43607 | Real part of the complex s... |
| imsqrtval 43608 | Imaginary part of the comp... |
| resqrtvalex 43609 | Example for ~ resqrtval . ... |
| imsqrtvalex 43610 | Example for ~ imsqrtval . ... |
| al3im 43611 | Version of ~ ax-4 for a ne... |
| intima0 43612 | Two ways of expressing the... |
| elimaint 43613 | Element of image of inters... |
| cnviun 43614 | Converse of indexed union.... |
| imaiun1 43615 | The image of an indexed un... |
| coiun1 43616 | Composition with an indexe... |
| elintima 43617 | Element of intersection of... |
| intimass 43618 | The image under the inters... |
| intimass2 43619 | The image under the inters... |
| intimag 43620 | Requirement for the image ... |
| intimasn 43621 | Two ways to express the im... |
| intimasn2 43622 | Two ways to express the im... |
| ss2iundf 43623 | Subclass theorem for index... |
| ss2iundv 43624 | Subclass theorem for index... |
| cbviuneq12df 43625 | Rule used to change the bo... |
| cbviuneq12dv 43626 | Rule used to change the bo... |
| conrel1d 43627 | Deduction about compositio... |
| conrel2d 43628 | Deduction about compositio... |
| trrelind 43629 | The intersection of transi... |
| xpintrreld 43630 | The intersection of a tran... |
| restrreld 43631 | The restriction of a trans... |
| trrelsuperreldg 43632 | Concrete construction of a... |
| trficl 43633 | The class of all transitiv... |
| cnvtrrel 43634 | The converse of a transiti... |
| trrelsuperrel2dg 43635 | Concrete construction of a... |
| dfrcl2 43638 | Reflexive closure of a rel... |
| dfrcl3 43639 | Reflexive closure of a rel... |
| dfrcl4 43640 | Reflexive closure of a rel... |
| relexp2 43641 | A set operated on by the r... |
| relexpnul 43642 | If the domain and range of... |
| eliunov2 43643 | Membership in the indexed ... |
| eltrclrec 43644 | Membership in the indexed ... |
| elrtrclrec 43645 | Membership in the indexed ... |
| briunov2 43646 | Two classes related by the... |
| brmptiunrelexpd 43647 | If two elements are connec... |
| fvmptiunrelexplb0d 43648 | If the indexed union range... |
| fvmptiunrelexplb0da 43649 | If the indexed union range... |
| fvmptiunrelexplb1d 43650 | If the indexed union range... |
| brfvid 43651 | If two elements are connec... |
| brfvidRP 43652 | If two elements are connec... |
| fvilbd 43653 | A set is a subset of its i... |
| fvilbdRP 43654 | A set is a subset of its i... |
| brfvrcld 43655 | If two elements are connec... |
| brfvrcld2 43656 | If two elements are connec... |
| fvrcllb0d 43657 | A restriction of the ident... |
| fvrcllb0da 43658 | A restriction of the ident... |
| fvrcllb1d 43659 | A set is a subset of its i... |
| brtrclrec 43660 | Two classes related by the... |
| brrtrclrec 43661 | Two classes related by the... |
| briunov2uz 43662 | Two classes related by the... |
| eliunov2uz 43663 | Membership in the indexed ... |
| ov2ssiunov2 43664 | Any particular operator va... |
| relexp0eq 43665 | The zeroth power of relati... |
| iunrelexp0 43666 | Simplification of zeroth p... |
| relexpxpnnidm 43667 | Any positive power of a Ca... |
| relexpiidm 43668 | Any power of any restricti... |
| relexpss1d 43669 | The relational power of a ... |
| comptiunov2i 43670 | The composition two indexe... |
| corclrcl 43671 | The reflexive closure is i... |
| iunrelexpmin1 43672 | The indexed union of relat... |
| relexpmulnn 43673 | With exponents limited to ... |
| relexpmulg 43674 | With ordered exponents, th... |
| trclrelexplem 43675 | The union of relational po... |
| iunrelexpmin2 43676 | The indexed union of relat... |
| relexp01min 43677 | With exponents limited to ... |
| relexp1idm 43678 | Repeated raising a relatio... |
| relexp0idm 43679 | Repeated raising a relatio... |
| relexp0a 43680 | Absorption law for zeroth ... |
| relexpxpmin 43681 | The composition of powers ... |
| relexpaddss 43682 | The composition of two pow... |
| iunrelexpuztr 43683 | The indexed union of relat... |
| dftrcl3 43684 | Transitive closure of a re... |
| brfvtrcld 43685 | If two elements are connec... |
| fvtrcllb1d 43686 | A set is a subset of its i... |
| trclfvcom 43687 | The transitive closure of ... |
| cnvtrclfv 43688 | The converse of the transi... |
| cotrcltrcl 43689 | The transitive closure is ... |
| trclimalb2 43690 | Lower bound for image unde... |
| brtrclfv2 43691 | Two ways to indicate two e... |
| trclfvdecomr 43692 | The transitive closure of ... |
| trclfvdecoml 43693 | The transitive closure of ... |
| dmtrclfvRP 43694 | The domain of the transiti... |
| rntrclfvRP 43695 | The range of the transitiv... |
| rntrclfv 43696 | The range of the transitiv... |
| dfrtrcl3 43697 | Reflexive-transitive closu... |
| brfvrtrcld 43698 | If two elements are connec... |
| fvrtrcllb0d 43699 | A restriction of the ident... |
| fvrtrcllb0da 43700 | A restriction of the ident... |
| fvrtrcllb1d 43701 | A set is a subset of its i... |
| dfrtrcl4 43702 | Reflexive-transitive closu... |
| corcltrcl 43703 | The composition of the ref... |
| cortrcltrcl 43704 | Composition with the refle... |
| corclrtrcl 43705 | Composition with the refle... |
| cotrclrcl 43706 | The composition of the ref... |
| cortrclrcl 43707 | Composition with the refle... |
| cotrclrtrcl 43708 | Composition with the refle... |
| cortrclrtrcl 43709 | The reflexive-transitive c... |
| frege77d 43710 | If the images of both ` { ... |
| frege81d 43711 | If the image of ` U ` is a... |
| frege83d 43712 | If the image of the union ... |
| frege96d 43713 | If ` C ` follows ` A ` in ... |
| frege87d 43714 | If the images of both ` { ... |
| frege91d 43715 | If ` B ` follows ` A ` in ... |
| frege97d 43716 | If ` A ` contains all elem... |
| frege98d 43717 | If ` C ` follows ` A ` and... |
| frege102d 43718 | If either ` A ` and ` C ` ... |
| frege106d 43719 | If ` B ` follows ` A ` in ... |
| frege108d 43720 | If either ` A ` and ` C ` ... |
| frege109d 43721 | If ` A ` contains all elem... |
| frege114d 43722 | If either ` R ` relates ` ... |
| frege111d 43723 | If either ` A ` and ` C ` ... |
| frege122d 43724 | If ` F ` is a function, ` ... |
| frege124d 43725 | If ` F ` is a function, ` ... |
| frege126d 43726 | If ` F ` is a function, ` ... |
| frege129d 43727 | If ` F ` is a function and... |
| frege131d 43728 | If ` F ` is a function and... |
| frege133d 43729 | If ` F ` is a function and... |
| dfxor4 43730 | Express exclusive-or in te... |
| dfxor5 43731 | Express exclusive-or in te... |
| df3or2 43732 | Express triple-or in terms... |
| df3an2 43733 | Express triple-and in term... |
| nev 43734 | Express that not every set... |
| 0pssin 43735 | Express that an intersecti... |
| dfhe2 43738 | The property of relation `... |
| dfhe3 43739 | The property of relation `... |
| heeq12 43740 | Equality law for relations... |
| heeq1 43741 | Equality law for relations... |
| heeq2 43742 | Equality law for relations... |
| sbcheg 43743 | Distribute proper substitu... |
| hess 43744 | Subclass law for relations... |
| xphe 43745 | Any Cartesian product is h... |
| 0he 43746 | The empty relation is here... |
| 0heALT 43747 | The empty relation is here... |
| he0 43748 | Any relation is hereditary... |
| unhe1 43749 | The union of two relations... |
| snhesn 43750 | Any singleton is hereditar... |
| idhe 43751 | The identity relation is h... |
| psshepw 43752 | The relation between sets ... |
| sshepw 43753 | The relation between sets ... |
| rp-simp2-frege 43756 | Simplification of triple c... |
| rp-simp2 43757 | Simplification of triple c... |
| rp-frege3g 43758 | Add antecedent to ~ ax-fre... |
| frege3 43759 | Add antecedent to ~ ax-fre... |
| rp-misc1-frege 43760 | Double-use of ~ ax-frege2 ... |
| rp-frege24 43761 | Introducing an embedded an... |
| rp-frege4g 43762 | Deduction related to distr... |
| frege4 43763 | Special case of closed for... |
| frege5 43764 | A closed form of ~ syl . ... |
| rp-7frege 43765 | Distribute antecedent and ... |
| rp-4frege 43766 | Elimination of a nested an... |
| rp-6frege 43767 | Elimination of a nested an... |
| rp-8frege 43768 | Eliminate antecedent when ... |
| rp-frege25 43769 | Closed form for ~ a1dd . ... |
| frege6 43770 | A closed form of ~ imim2d ... |
| axfrege8 43771 | Swap antecedents. Identic... |
| frege7 43772 | A closed form of ~ syl6 . ... |
| frege26 43774 | Identical to ~ idd . Prop... |
| frege27 43775 | We cannot (at the same tim... |
| frege9 43776 | Closed form of ~ syl with ... |
| frege12 43777 | A closed form of ~ com23 .... |
| frege11 43778 | Elimination of a nested an... |
| frege24 43779 | Closed form for ~ a1d . D... |
| frege16 43780 | A closed form of ~ com34 .... |
| frege25 43781 | Closed form for ~ a1dd . ... |
| frege18 43782 | Closed form of a syllogism... |
| frege22 43783 | A closed form of ~ com45 .... |
| frege10 43784 | Result commuting anteceden... |
| frege17 43785 | A closed form of ~ com3l .... |
| frege13 43786 | A closed form of ~ com3r .... |
| frege14 43787 | Closed form of a deduction... |
| frege19 43788 | A closed form of ~ syl6 . ... |
| frege23 43789 | Syllogism followed by rota... |
| frege15 43790 | A closed form of ~ com4r .... |
| frege21 43791 | Replace antecedent in ante... |
| frege20 43792 | A closed form of ~ syl8 . ... |
| axfrege28 43793 | Contraposition. Identical... |
| frege29 43795 | Closed form of ~ con3d . ... |
| frege30 43796 | Commuted, closed form of ~... |
| axfrege31 43797 | Identical to ~ notnotr . ... |
| frege32 43799 | Deduce ~ con1 from ~ con3 ... |
| frege33 43800 | If ` ph ` or ` ps ` takes ... |
| frege34 43801 | If as a consequence of the... |
| frege35 43802 | Commuted, closed form of ~... |
| frege36 43803 | The case in which ` ps ` i... |
| frege37 43804 | If ` ch ` is a necessary c... |
| frege38 43805 | Identical to ~ pm2.21 . P... |
| frege39 43806 | Syllogism between ~ pm2.18... |
| frege40 43807 | Anything implies ~ pm2.18 ... |
| axfrege41 43808 | Identical to ~ notnot . A... |
| frege42 43810 | Not not ~ id . Propositio... |
| frege43 43811 | If there is a choice only ... |
| frege44 43812 | Similar to a commuted ~ pm... |
| frege45 43813 | Deduce ~ pm2.6 from ~ con1... |
| frege46 43814 | If ` ps ` holds when ` ph ... |
| frege47 43815 | Deduce consequence follows... |
| frege48 43816 | Closed form of syllogism w... |
| frege49 43817 | Closed form of deduction w... |
| frege50 43818 | Closed form of ~ jaoi . P... |
| frege51 43819 | Compare with ~ jaod . Pro... |
| axfrege52a 43820 | Justification for ~ ax-fre... |
| frege52aid 43822 | The case when the content ... |
| frege53aid 43823 | Specialization of ~ frege5... |
| frege53a 43824 | Lemma for ~ frege55a . Pr... |
| axfrege54a 43825 | Justification for ~ ax-fre... |
| frege54cor0a 43827 | Synonym for logical equiva... |
| frege54cor1a 43828 | Reflexive equality. (Cont... |
| frege55aid 43829 | Lemma for ~ frege57aid . ... |
| frege55lem1a 43830 | Necessary deduction regard... |
| frege55lem2a 43831 | Core proof of Proposition ... |
| frege55a 43832 | Proposition 55 of [Frege18... |
| frege55cor1a 43833 | Proposition 55 of [Frege18... |
| frege56aid 43834 | Lemma for ~ frege57aid . ... |
| frege56a 43835 | Proposition 56 of [Frege18... |
| frege57aid 43836 | This is the all important ... |
| frege57a 43837 | Analogue of ~ frege57aid .... |
| axfrege58a 43838 | Identical to ~ anifp . Ju... |
| frege58acor 43840 | Lemma for ~ frege59a . (C... |
| frege59a 43841 | A kind of Aristotelian inf... |
| frege60a 43842 | Swap antecedents of ~ ax-f... |
| frege61a 43843 | Lemma for ~ frege65a . Pr... |
| frege62a 43844 | A kind of Aristotelian inf... |
| frege63a 43845 | Proposition 63 of [Frege18... |
| frege64a 43846 | Lemma for ~ frege65a . Pr... |
| frege65a 43847 | A kind of Aristotelian inf... |
| frege66a 43848 | Swap antecedents of ~ freg... |
| frege67a 43849 | Lemma for ~ frege68a . Pr... |
| frege68a 43850 | Combination of applying a ... |
| axfrege52c 43851 | Justification for ~ ax-fre... |
| frege52b 43853 | The case when the content ... |
| frege53b 43854 | Lemma for frege102 (via ~ ... |
| axfrege54c 43855 | Reflexive equality of clas... |
| frege54b 43857 | Reflexive equality of sets... |
| frege54cor1b 43858 | Reflexive equality. (Cont... |
| frege55lem1b 43859 | Necessary deduction regard... |
| frege55lem2b 43860 | Lemma for ~ frege55b . Co... |
| frege55b 43861 | Lemma for ~ frege57b . Pr... |
| frege56b 43862 | Lemma for ~ frege57b . Pr... |
| frege57b 43863 | Analogue of ~ frege57aid .... |
| axfrege58b 43864 | If ` A. x ph ` is affirmed... |
| frege58bid 43866 | If ` A. x ph ` is affirmed... |
| frege58bcor 43867 | Lemma for ~ frege59b . (C... |
| frege59b 43868 | A kind of Aristotelian inf... |
| frege60b 43869 | Swap antecedents of ~ ax-f... |
| frege61b 43870 | Lemma for ~ frege65b . Pr... |
| frege62b 43871 | A kind of Aristotelian inf... |
| frege63b 43872 | Lemma for ~ frege91 . Pro... |
| frege64b 43873 | Lemma for ~ frege65b . Pr... |
| frege65b 43874 | A kind of Aristotelian inf... |
| frege66b 43875 | Swap antecedents of ~ freg... |
| frege67b 43876 | Lemma for ~ frege68b . Pr... |
| frege68b 43877 | Combination of applying a ... |
| frege53c 43878 | Proposition 53 of [Frege18... |
| frege54cor1c 43879 | Reflexive equality. (Cont... |
| frege55lem1c 43880 | Necessary deduction regard... |
| frege55lem2c 43881 | Core proof of Proposition ... |
| frege55c 43882 | Proposition 55 of [Frege18... |
| frege56c 43883 | Lemma for ~ frege57c . Pr... |
| frege57c 43884 | Swap order of implication ... |
| frege58c 43885 | Principle related to ~ sp ... |
| frege59c 43886 | A kind of Aristotelian inf... |
| frege60c 43887 | Swap antecedents of ~ freg... |
| frege61c 43888 | Lemma for ~ frege65c . Pr... |
| frege62c 43889 | A kind of Aristotelian inf... |
| frege63c 43890 | Analogue of ~ frege63b . ... |
| frege64c 43891 | Lemma for ~ frege65c . Pr... |
| frege65c 43892 | A kind of Aristotelian inf... |
| frege66c 43893 | Swap antecedents of ~ freg... |
| frege67c 43894 | Lemma for ~ frege68c . Pr... |
| frege68c 43895 | Combination of applying a ... |
| dffrege69 43896 | If from the proposition th... |
| frege70 43897 | Lemma for ~ frege72 . Pro... |
| frege71 43898 | Lemma for ~ frege72 . Pro... |
| frege72 43899 | If property ` A ` is hered... |
| frege73 43900 | Lemma for ~ frege87 . Pro... |
| frege74 43901 | If ` X ` has a property ` ... |
| frege75 43902 | If from the proposition th... |
| dffrege76 43903 | If from the two propositio... |
| frege77 43904 | If ` Y ` follows ` X ` in ... |
| frege78 43905 | Commuted form of ~ frege77... |
| frege79 43906 | Distributed form of ~ freg... |
| frege80 43907 | Add additional condition t... |
| frege81 43908 | If ` X ` has a property ` ... |
| frege82 43909 | Closed-form deduction base... |
| frege83 43910 | Apply commuted form of ~ f... |
| frege84 43911 | Commuted form of ~ frege81... |
| frege85 43912 | Commuted form of ~ frege77... |
| frege86 43913 | Conclusion about element o... |
| frege87 43914 | If ` Z ` is a result of an... |
| frege88 43915 | Commuted form of ~ frege87... |
| frege89 43916 | One direction of ~ dffrege... |
| frege90 43917 | Add antecedent to ~ frege8... |
| frege91 43918 | Every result of an applica... |
| frege92 43919 | Inference from ~ frege91 .... |
| frege93 43920 | Necessary condition for tw... |
| frege94 43921 | Looking one past a pair re... |
| frege95 43922 | Looking one past a pair re... |
| frege96 43923 | Every result of an applica... |
| frege97 43924 | The property of following ... |
| frege98 43925 | If ` Y ` follows ` X ` and... |
| dffrege99 43926 | If ` Z ` is identical with... |
| frege100 43927 | One direction of ~ dffrege... |
| frege101 43928 | Lemma for ~ frege102 . Pr... |
| frege102 43929 | If ` Z ` belongs to the ` ... |
| frege103 43930 | Proposition 103 of [Frege1... |
| frege104 43931 | Proposition 104 of [Frege1... |
| frege105 43932 | Proposition 105 of [Frege1... |
| frege106 43933 | Whatever follows ` X ` in ... |
| frege107 43934 | Proposition 107 of [Frege1... |
| frege108 43935 | If ` Y ` belongs to the ` ... |
| frege109 43936 | The property of belonging ... |
| frege110 43937 | Proposition 110 of [Frege1... |
| frege111 43938 | If ` Y ` belongs to the ` ... |
| frege112 43939 | Identity implies belonging... |
| frege113 43940 | Proposition 113 of [Frege1... |
| frege114 43941 | If ` X ` belongs to the ` ... |
| dffrege115 43942 | If from the circumstance t... |
| frege116 43943 | One direction of ~ dffrege... |
| frege117 43944 | Lemma for ~ frege118 . Pr... |
| frege118 43945 | Simplified application of ... |
| frege119 43946 | Lemma for ~ frege120 . Pr... |
| frege120 43947 | Simplified application of ... |
| frege121 43948 | Lemma for ~ frege122 . Pr... |
| frege122 43949 | If ` X ` is a result of an... |
| frege123 43950 | Lemma for ~ frege124 . Pr... |
| frege124 43951 | If ` X ` is a result of an... |
| frege125 43952 | Lemma for ~ frege126 . Pr... |
| frege126 43953 | If ` M ` follows ` Y ` in ... |
| frege127 43954 | Communte antecedents of ~ ... |
| frege128 43955 | Lemma for ~ frege129 . Pr... |
| frege129 43956 | If the procedure ` R ` is ... |
| frege130 43957 | Lemma for ~ frege131 . Pr... |
| frege131 43958 | If the procedure ` R ` is ... |
| frege132 43959 | Lemma for ~ frege133 . Pr... |
| frege133 43960 | If the procedure ` R ` is ... |
| enrelmap 43961 | The set of all possible re... |
| enrelmapr 43962 | The set of all possible re... |
| enmappw 43963 | The set of all mappings fr... |
| enmappwid 43964 | The set of all mappings fr... |
| rfovd 43965 | Value of the operator, ` (... |
| rfovfvd 43966 | Value of the operator, ` (... |
| rfovfvfvd 43967 | Value of the operator, ` (... |
| rfovcnvf1od 43968 | Properties of the operator... |
| rfovcnvd 43969 | Value of the converse of t... |
| rfovf1od 43970 | The value of the operator,... |
| rfovcnvfvd 43971 | Value of the converse of t... |
| fsovd 43972 | Value of the operator, ` (... |
| fsovrfovd 43973 | The operator which gives a... |
| fsovfvd 43974 | Value of the operator, ` (... |
| fsovfvfvd 43975 | Value of the operator, ` (... |
| fsovfd 43976 | The operator, ` ( A O B ) ... |
| fsovcnvlem 43977 | The ` O ` operator, which ... |
| fsovcnvd 43978 | The value of the converse ... |
| fsovcnvfvd 43979 | The value of the converse ... |
| fsovf1od 43980 | The value of ` ( A O B ) `... |
| dssmapfvd 43981 | Value of the duality opera... |
| dssmapfv2d 43982 | Value of the duality opera... |
| dssmapfv3d 43983 | Value of the duality opera... |
| dssmapnvod 43984 | For any base set ` B ` the... |
| dssmapf1od 43985 | For any base set ` B ` the... |
| dssmap2d 43986 | For any base set ` B ` the... |
| or3or 43987 | Decompose disjunction into... |
| andi3or 43988 | Distribute over triple dis... |
| uneqsn 43989 | If a union of classes is e... |
| brfvimex 43990 | If a binary relation holds... |
| brovmptimex 43991 | If a binary relation holds... |
| brovmptimex1 43992 | If a binary relation holds... |
| brovmptimex2 43993 | If a binary relation holds... |
| brcoffn 43994 | Conditions allowing the de... |
| brcofffn 43995 | Conditions allowing the de... |
| brco2f1o 43996 | Conditions allowing the de... |
| brco3f1o 43997 | Conditions allowing the de... |
| ntrclsbex 43998 | If (pseudo-)interior and (... |
| ntrclsrcomplex 43999 | The relative complement of... |
| neik0imk0p 44000 | Kuratowski's K0 axiom impl... |
| ntrk2imkb 44001 | If an interior function is... |
| ntrkbimka 44002 | If the interiors of disjoi... |
| ntrk0kbimka 44003 | If the interiors of disjoi... |
| clsk3nimkb 44004 | If the base set is not emp... |
| clsk1indlem0 44005 | The ansatz closure functio... |
| clsk1indlem2 44006 | The ansatz closure functio... |
| clsk1indlem3 44007 | The ansatz closure functio... |
| clsk1indlem4 44008 | The ansatz closure functio... |
| clsk1indlem1 44009 | The ansatz closure functio... |
| clsk1independent 44010 | For generalized closure fu... |
| neik0pk1imk0 44011 | Kuratowski's K0' and K1 ax... |
| isotone1 44012 | Two different ways to say ... |
| isotone2 44013 | Two different ways to say ... |
| ntrk1k3eqk13 44014 | An interior function is bo... |
| ntrclsf1o 44015 | If (pseudo-)interior and (... |
| ntrclsnvobr 44016 | If (pseudo-)interior and (... |
| ntrclsiex 44017 | If (pseudo-)interior and (... |
| ntrclskex 44018 | If (pseudo-)interior and (... |
| ntrclsfv1 44019 | If (pseudo-)interior and (... |
| ntrclsfv2 44020 | If (pseudo-)interior and (... |
| ntrclselnel1 44021 | If (pseudo-)interior and (... |
| ntrclselnel2 44022 | If (pseudo-)interior and (... |
| ntrclsfv 44023 | The value of the interior ... |
| ntrclsfveq1 44024 | If interior and closure fu... |
| ntrclsfveq2 44025 | If interior and closure fu... |
| ntrclsfveq 44026 | If interior and closure fu... |
| ntrclsss 44027 | If interior and closure fu... |
| ntrclsneine0lem 44028 | If (pseudo-)interior and (... |
| ntrclsneine0 44029 | If (pseudo-)interior and (... |
| ntrclscls00 44030 | If (pseudo-)interior and (... |
| ntrclsiso 44031 | If (pseudo-)interior and (... |
| ntrclsk2 44032 | An interior function is co... |
| ntrclskb 44033 | The interiors of disjoint ... |
| ntrclsk3 44034 | The intersection of interi... |
| ntrclsk13 44035 | The interior of the inters... |
| ntrclsk4 44036 | Idempotence of the interio... |
| ntrneibex 44037 | If (pseudo-)interior and (... |
| ntrneircomplex 44038 | The relative complement of... |
| ntrneif1o 44039 | If (pseudo-)interior and (... |
| ntrneiiex 44040 | If (pseudo-)interior and (... |
| ntrneinex 44041 | If (pseudo-)interior and (... |
| ntrneicnv 44042 | If (pseudo-)interior and (... |
| ntrneifv1 44043 | If (pseudo-)interior and (... |
| ntrneifv2 44044 | If (pseudo-)interior and (... |
| ntrneiel 44045 | If (pseudo-)interior and (... |
| ntrneifv3 44046 | The value of the neighbors... |
| ntrneineine0lem 44047 | If (pseudo-)interior and (... |
| ntrneineine1lem 44048 | If (pseudo-)interior and (... |
| ntrneifv4 44049 | The value of the interior ... |
| ntrneiel2 44050 | Membership in iterated int... |
| ntrneineine0 44051 | If (pseudo-)interior and (... |
| ntrneineine1 44052 | If (pseudo-)interior and (... |
| ntrneicls00 44053 | If (pseudo-)interior and (... |
| ntrneicls11 44054 | If (pseudo-)interior and (... |
| ntrneiiso 44055 | If (pseudo-)interior and (... |
| ntrneik2 44056 | An interior function is co... |
| ntrneix2 44057 | An interior (closure) func... |
| ntrneikb 44058 | The interiors of disjoint ... |
| ntrneixb 44059 | The interiors (closures) o... |
| ntrneik3 44060 | The intersection of interi... |
| ntrneix3 44061 | The closure of the union o... |
| ntrneik13 44062 | The interior of the inters... |
| ntrneix13 44063 | The closure of the union o... |
| ntrneik4w 44064 | Idempotence of the interio... |
| ntrneik4 44065 | Idempotence of the interio... |
| clsneibex 44066 | If (pseudo-)closure and (p... |
| clsneircomplex 44067 | The relative complement of... |
| clsneif1o 44068 | If a (pseudo-)closure func... |
| clsneicnv 44069 | If a (pseudo-)closure func... |
| clsneikex 44070 | If closure and neighborhoo... |
| clsneinex 44071 | If closure and neighborhoo... |
| clsneiel1 44072 | If a (pseudo-)closure func... |
| clsneiel2 44073 | If a (pseudo-)closure func... |
| clsneifv3 44074 | Value of the neighborhoods... |
| clsneifv4 44075 | Value of the closure (inte... |
| neicvgbex 44076 | If (pseudo-)neighborhood a... |
| neicvgrcomplex 44077 | The relative complement of... |
| neicvgf1o 44078 | If neighborhood and conver... |
| neicvgnvo 44079 | If neighborhood and conver... |
| neicvgnvor 44080 | If neighborhood and conver... |
| neicvgmex 44081 | If the neighborhoods and c... |
| neicvgnex 44082 | If the neighborhoods and c... |
| neicvgel1 44083 | A subset being an element ... |
| neicvgel2 44084 | The complement of a subset... |
| neicvgfv 44085 | The value of the neighborh... |
| ntrrn 44086 | The range of the interior ... |
| ntrf 44087 | The interior function of a... |
| ntrf2 44088 | The interior function is a... |
| ntrelmap 44089 | The interior function is a... |
| clsf2 44090 | The closure function is a ... |
| clselmap 44091 | The closure function is a ... |
| dssmapntrcls 44092 | The interior and closure o... |
| dssmapclsntr 44093 | The closure and interior o... |
| gneispa 44094 | Each point ` p ` of the ne... |
| gneispb 44095 | Given a neighborhood ` N `... |
| gneispace2 44096 | The predicate that ` F ` i... |
| gneispace3 44097 | The predicate that ` F ` i... |
| gneispace 44098 | The predicate that ` F ` i... |
| gneispacef 44099 | A generic neighborhood spa... |
| gneispacef2 44100 | A generic neighborhood spa... |
| gneispacefun 44101 | A generic neighborhood spa... |
| gneispacern 44102 | A generic neighborhood spa... |
| gneispacern2 44103 | A generic neighborhood spa... |
| gneispace0nelrn 44104 | A generic neighborhood spa... |
| gneispace0nelrn2 44105 | A generic neighborhood spa... |
| gneispace0nelrn3 44106 | A generic neighborhood spa... |
| gneispaceel 44107 | Every neighborhood of a po... |
| gneispaceel2 44108 | Every neighborhood of a po... |
| gneispacess 44109 | All supersets of a neighbo... |
| gneispacess2 44110 | All supersets of a neighbo... |
| k0004lem1 44111 | Application of ~ ssin to r... |
| k0004lem2 44112 | A mapping with a particula... |
| k0004lem3 44113 | When the value of a mappin... |
| k0004val 44114 | The topological simplex of... |
| k0004ss1 44115 | The topological simplex of... |
| k0004ss2 44116 | The topological simplex of... |
| k0004ss3 44117 | The topological simplex of... |
| k0004val0 44118 | The topological simplex of... |
| inductionexd 44119 | Simple induction example. ... |
| wwlemuld 44120 | Natural deduction form of ... |
| leeq1d 44121 | Specialization of ~ breq1d... |
| leeq2d 44122 | Specialization of ~ breq2d... |
| absmulrposd 44123 | Specialization of absmuld ... |
| imadisjld 44124 | Natural dduction form of o... |
| wnefimgd 44125 | The image of a mapping fro... |
| fco2d 44126 | Natural deduction form of ... |
| wfximgfd 44127 | The value of a function on... |
| extoimad 44128 | If |f(x)| <= C for all x t... |
| imo72b2lem0 44129 | Lemma for ~ imo72b2 . (Co... |
| suprleubrd 44130 | Natural deduction form of ... |
| imo72b2lem2 44131 | Lemma for ~ imo72b2 . (Co... |
| suprlubrd 44132 | Natural deduction form of ... |
| imo72b2lem1 44133 | Lemma for ~ imo72b2 . (Co... |
| lemuldiv3d 44134 | 'Less than or equal to' re... |
| lemuldiv4d 44135 | 'Less than or equal to' re... |
| imo72b2 44136 | IMO 1972 B2. (14th Intern... |
| int-addcomd 44137 | AdditionCommutativity gene... |
| int-addassocd 44138 | AdditionAssociativity gene... |
| int-addsimpd 44139 | AdditionSimplification gen... |
| int-mulcomd 44140 | MultiplicationCommutativit... |
| int-mulassocd 44141 | MultiplicationAssociativit... |
| int-mulsimpd 44142 | MultiplicationSimplificati... |
| int-leftdistd 44143 | AdditionMultiplicationLeft... |
| int-rightdistd 44144 | AdditionMultiplicationRigh... |
| int-sqdefd 44145 | SquareDefinition generator... |
| int-mul11d 44146 | First MultiplicationOne ge... |
| int-mul12d 44147 | Second MultiplicationOne g... |
| int-add01d 44148 | First AdditionZero generat... |
| int-add02d 44149 | Second AdditionZero genera... |
| int-sqgeq0d 44150 | SquareGEQZero generator ru... |
| int-eqprincd 44151 | PrincipleOfEquality genera... |
| int-eqtransd 44152 | EqualityTransitivity gener... |
| int-eqmvtd 44153 | EquMoveTerm generator rule... |
| int-eqineqd 44154 | EquivalenceImpliesDoubleIn... |
| int-ineqmvtd 44155 | IneqMoveTerm generator rul... |
| int-ineq1stprincd 44156 | FirstPrincipleOfInequality... |
| int-ineq2ndprincd 44157 | SecondPrincipleOfInequalit... |
| int-ineqtransd 44158 | InequalityTransitivity gen... |
| unitadd 44159 | Theorem used in conjunctio... |
| gsumws3 44160 | Valuation of a length 3 wo... |
| gsumws4 44161 | Valuation of a length 4 wo... |
| amgm2d 44162 | Arithmetic-geometric mean ... |
| amgm3d 44163 | Arithmetic-geometric mean ... |
| amgm4d 44164 | Arithmetic-geometric mean ... |
| spALT 44165 | ~ sp can be proven from th... |
| elnelneqd 44166 | Two classes are not equal ... |
| elnelneq2d 44167 | Two classes are not equal ... |
| rr-spce 44168 | Prove an existential. (Co... |
| rexlimdvaacbv 44169 | Unpack a restricted existe... |
| rexlimddvcbvw 44170 | Unpack a restricted existe... |
| rexlimddvcbv 44171 | Unpack a restricted existe... |
| rr-elrnmpt3d 44172 | Elementhood in an image se... |
| finnzfsuppd 44173 | If a function is zero outs... |
| rr-phpd 44174 | Equivalent of ~ php withou... |
| suceqd 44175 | Deduction associated with ... |
| tfindsd 44176 | Deduction associated with ... |
| mnringvald 44179 | Value of the monoid ring f... |
| mnringnmulrd 44180 | Components of a monoid rin... |
| mnringnmulrdOLD 44181 | Obsolete version of ~ mnri... |
| mnringbased 44182 | The base set of a monoid r... |
| mnringbasedOLD 44183 | Obsolete version of ~ mnri... |
| mnringbaserd 44184 | The base set of a monoid r... |
| mnringelbased 44185 | Membership in the base set... |
| mnringbasefd 44186 | Elements of a monoid ring ... |
| mnringbasefsuppd 44187 | Elements of a monoid ring ... |
| mnringaddgd 44188 | The additive operation of ... |
| mnringaddgdOLD 44189 | Obsolete version of ~ mnri... |
| mnring0gd 44190 | The additive identity of a... |
| mnring0g2d 44191 | The additive identity of a... |
| mnringmulrd 44192 | The ring product of a mono... |
| mnringscad 44193 | The scalar ring of a monoi... |
| mnringscadOLD 44194 | Obsolete version of ~ mnri... |
| mnringvscad 44195 | The scalar product of a mo... |
| mnringvscadOLD 44196 | Obsolete version of ~ mnri... |
| mnringlmodd 44197 | Monoid rings are left modu... |
| mnringmulrvald 44198 | Value of multiplication in... |
| mnringmulrcld 44199 | Monoid rings are closed un... |
| gru0eld 44200 | A nonempty Grothendieck un... |
| grusucd 44201 | Grothendieck universes are... |
| r1rankcld 44202 | Any rank of the cumulative... |
| grur1cld 44203 | Grothendieck universes are... |
| grurankcld 44204 | Grothendieck universes are... |
| grurankrcld 44205 | If a Grothendieck universe... |
| scotteqd 44208 | Equality theorem for the S... |
| scotteq 44209 | Closed form of ~ scotteqd ... |
| nfscott 44210 | Bound-variable hypothesis ... |
| scottabf 44211 | Value of the Scott operati... |
| scottab 44212 | Value of the Scott operati... |
| scottabes 44213 | Value of the Scott operati... |
| scottss 44214 | Scott's trick produces a s... |
| elscottab 44215 | An element of the output o... |
| scottex2 44216 | ~ scottex expressed using ... |
| scotteld 44217 | The Scott operation sends ... |
| scottelrankd 44218 | Property of a Scott's tric... |
| scottrankd 44219 | Rank of a nonempty Scott's... |
| gruscottcld 44220 | If a Grothendieck universe... |
| dfcoll2 44223 | Alternate definition of th... |
| colleq12d 44224 | Equality theorem for the c... |
| colleq1 44225 | Equality theorem for the c... |
| colleq2 44226 | Equality theorem for the c... |
| nfcoll 44227 | Bound-variable hypothesis ... |
| collexd 44228 | The output of the collecti... |
| cpcolld 44229 | Property of the collection... |
| cpcoll2d 44230 | ~ cpcolld with an extra ex... |
| grucollcld 44231 | A Grothendieck universe co... |
| ismnu 44232 | The hypothesis of this the... |
| mnuop123d 44233 | Operations of a minimal un... |
| mnussd 44234 | Minimal universes are clos... |
| mnuss2d 44235 | ~ mnussd with arguments pr... |
| mnu0eld 44236 | A nonempty minimal univers... |
| mnuop23d 44237 | Second and third operation... |
| mnupwd 44238 | Minimal universes are clos... |
| mnusnd 44239 | Minimal universes are clos... |
| mnuprssd 44240 | A minimal universe contain... |
| mnuprss2d 44241 | Special case of ~ mnuprssd... |
| mnuop3d 44242 | Third operation of a minim... |
| mnuprdlem1 44243 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem2 44244 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem3 44245 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem4 44246 | Lemma for ~ mnuprd . Gene... |
| mnuprd 44247 | Minimal universes are clos... |
| mnuunid 44248 | Minimal universes are clos... |
| mnuund 44249 | Minimal universes are clos... |
| mnutrcld 44250 | Minimal universes contain ... |
| mnutrd 44251 | Minimal universes are tran... |
| mnurndlem1 44252 | Lemma for ~ mnurnd . (Con... |
| mnurndlem2 44253 | Lemma for ~ mnurnd . Dedu... |
| mnurnd 44254 | Minimal universes contain ... |
| mnugrud 44255 | Minimal universes are Grot... |
| grumnudlem 44256 | Lemma for ~ grumnud . (Co... |
| grumnud 44257 | Grothendieck universes are... |
| grumnueq 44258 | The class of Grothendieck ... |
| expandan 44259 | Expand conjunction to prim... |
| expandexn 44260 | Expand an existential quan... |
| expandral 44261 | Expand a restricted univer... |
| expandrexn 44262 | Expand a restricted existe... |
| expandrex 44263 | Expand a restricted existe... |
| expanduniss 44264 | Expand ` U. A C_ B ` to pr... |
| ismnuprim 44265 | Express the predicate on `... |
| rr-grothprimbi 44266 | Express "every set is cont... |
| inagrud 44267 | Inaccessible levels of the... |
| inaex 44268 | Assuming the Tarski-Grothe... |
| gruex 44269 | Assuming the Tarski-Grothe... |
| rr-groth 44270 | An equivalent of ~ ax-grot... |
| rr-grothprim 44271 | An equivalent of ~ ax-grot... |
| ismnushort 44272 | Express the predicate on `... |
| dfuniv2 44273 | Alternative definition of ... |
| rr-grothshortbi 44274 | Express "every set is cont... |
| rr-grothshort 44275 | A shorter equivalent of ~ ... |
| nanorxor 44276 | 'nand' is equivalent to th... |
| undisjrab 44277 | Union of two disjoint rest... |
| iso0 44278 | The empty set is an ` R , ... |
| ssrecnpr 44279 | ` RR ` is a subset of both... |
| seff 44280 | Let set ` S ` be the real ... |
| sblpnf 44281 | The infinity ball in the a... |
| prmunb2 44282 | The primes are unbounded. ... |
| dvgrat 44283 | Ratio test for divergence ... |
| cvgdvgrat 44284 | Ratio test for convergence... |
| radcnvrat 44285 | Let ` L ` be the limit, if... |
| reldvds 44286 | The divides relation is in... |
| nznngen 44287 | All positive integers in t... |
| nzss 44288 | The set of multiples of _m... |
| nzin 44289 | The intersection of the se... |
| nzprmdif 44290 | Subtract one prime's multi... |
| hashnzfz 44291 | Special case of ~ hashdvds... |
| hashnzfz2 44292 | Special case of ~ hashnzfz... |
| hashnzfzclim 44293 | As the upper bound ` K ` o... |
| caofcan 44294 | Transfer a cancellation la... |
| ofsubid 44295 | Function analogue of ~ sub... |
| ofmul12 44296 | Function analogue of ~ mul... |
| ofdivrec 44297 | Function analogue of ~ div... |
| ofdivcan4 44298 | Function analogue of ~ div... |
| ofdivdiv2 44299 | Function analogue of ~ div... |
| lhe4.4ex1a 44300 | Example of the Fundamental... |
| dvsconst 44301 | Derivative of a constant f... |
| dvsid 44302 | Derivative of the identity... |
| dvsef 44303 | Derivative of the exponent... |
| expgrowthi 44304 | Exponential growth and dec... |
| dvconstbi 44305 | The derivative of a functi... |
| expgrowth 44306 | Exponential growth and dec... |
| bccval 44309 | Value of the generalized b... |
| bcccl 44310 | Closure of the generalized... |
| bcc0 44311 | The generalized binomial c... |
| bccp1k 44312 | Generalized binomial coeff... |
| bccm1k 44313 | Generalized binomial coeff... |
| bccn0 44314 | Generalized binomial coeff... |
| bccn1 44315 | Generalized binomial coeff... |
| bccbc 44316 | The binomial coefficient a... |
| uzmptshftfval 44317 | When ` F ` is a maps-to fu... |
| dvradcnv2 44318 | The radius of convergence ... |
| binomcxplemwb 44319 | Lemma for ~ binomcxp . Th... |
| binomcxplemnn0 44320 | Lemma for ~ binomcxp . Wh... |
| binomcxplemrat 44321 | Lemma for ~ binomcxp . As... |
| binomcxplemfrat 44322 | Lemma for ~ binomcxp . ~ b... |
| binomcxplemradcnv 44323 | Lemma for ~ binomcxp . By... |
| binomcxplemdvbinom 44324 | Lemma for ~ binomcxp . By... |
| binomcxplemcvg 44325 | Lemma for ~ binomcxp . Th... |
| binomcxplemdvsum 44326 | Lemma for ~ binomcxp . Th... |
| binomcxplemnotnn0 44327 | Lemma for ~ binomcxp . Wh... |
| binomcxp 44328 | Generalize the binomial th... |
| pm10.12 44329 | Theorem *10.12 in [Whitehe... |
| pm10.14 44330 | Theorem *10.14 in [Whitehe... |
| pm10.251 44331 | Theorem *10.251 in [Whiteh... |
| pm10.252 44332 | Theorem *10.252 in [Whiteh... |
| pm10.253 44333 | Theorem *10.253 in [Whiteh... |
| albitr 44334 | Theorem *10.301 in [Whiteh... |
| pm10.42 44335 | Theorem *10.42 in [Whitehe... |
| pm10.52 44336 | Theorem *10.52 in [Whitehe... |
| pm10.53 44337 | Theorem *10.53 in [Whitehe... |
| pm10.541 44338 | Theorem *10.541 in [Whiteh... |
| pm10.542 44339 | Theorem *10.542 in [Whiteh... |
| pm10.55 44340 | Theorem *10.55 in [Whitehe... |
| pm10.56 44341 | Theorem *10.56 in [Whitehe... |
| pm10.57 44342 | Theorem *10.57 in [Whitehe... |
| 2alanimi 44343 | Removes two universal quan... |
| 2al2imi 44344 | Removes two universal quan... |
| pm11.11 44345 | Theorem *11.11 in [Whitehe... |
| pm11.12 44346 | Theorem *11.12 in [Whitehe... |
| 19.21vv 44347 | Compare Theorem *11.3 in [... |
| 2alim 44348 | Theorem *11.32 in [Whitehe... |
| 2albi 44349 | Theorem *11.33 in [Whitehe... |
| 2exim 44350 | Theorem *11.34 in [Whitehe... |
| 2exbi 44351 | Theorem *11.341 in [Whiteh... |
| spsbce-2 44352 | Theorem *11.36 in [Whitehe... |
| 19.33-2 44353 | Theorem *11.421 in [Whiteh... |
| 19.36vv 44354 | Theorem *11.43 in [Whitehe... |
| 19.31vv 44355 | Theorem *11.44 in [Whitehe... |
| 19.37vv 44356 | Theorem *11.46 in [Whitehe... |
| 19.28vv 44357 | Theorem *11.47 in [Whitehe... |
| pm11.52 44358 | Theorem *11.52 in [Whitehe... |
| aaanv 44359 | Theorem *11.56 in [Whitehe... |
| pm11.57 44360 | Theorem *11.57 in [Whitehe... |
| pm11.58 44361 | Theorem *11.58 in [Whitehe... |
| pm11.59 44362 | Theorem *11.59 in [Whitehe... |
| pm11.6 44363 | Theorem *11.6 in [Whitehea... |
| pm11.61 44364 | Theorem *11.61 in [Whitehe... |
| pm11.62 44365 | Theorem *11.62 in [Whitehe... |
| pm11.63 44366 | Theorem *11.63 in [Whitehe... |
| pm11.7 44367 | Theorem *11.7 in [Whitehea... |
| pm11.71 44368 | Theorem *11.71 in [Whitehe... |
| sbeqal1 44369 | If ` x = y ` always implie... |
| sbeqal1i 44370 | Suppose you know ` x = y `... |
| sbeqal2i 44371 | If ` x = y ` implies ` x =... |
| axc5c4c711 44372 | Proof of a theorem that ca... |
| axc5c4c711toc5 44373 | Rederivation of ~ sp from ... |
| axc5c4c711toc4 44374 | Rederivation of ~ axc4 fro... |
| axc5c4c711toc7 44375 | Rederivation of ~ axc7 fro... |
| axc5c4c711to11 44376 | Rederivation of ~ ax-11 fr... |
| axc11next 44377 | This theorem shows that, g... |
| pm13.13a 44378 | One result of theorem *13.... |
| pm13.13b 44379 | Theorem *13.13 in [Whitehe... |
| pm13.14 44380 | Theorem *13.14 in [Whitehe... |
| pm13.192 44381 | Theorem *13.192 in [Whiteh... |
| pm13.193 44382 | Theorem *13.193 in [Whiteh... |
| pm13.194 44383 | Theorem *13.194 in [Whiteh... |
| pm13.195 44384 | Theorem *13.195 in [Whiteh... |
| pm13.196a 44385 | Theorem *13.196 in [Whiteh... |
| 2sbc6g 44386 | Theorem *13.21 in [Whitehe... |
| 2sbc5g 44387 | Theorem *13.22 in [Whitehe... |
| iotain 44388 | Equivalence between two di... |
| iotaexeu 44389 | The iota class exists. Th... |
| iotasbc 44390 | Definition *14.01 in [Whit... |
| iotasbc2 44391 | Theorem *14.111 in [Whiteh... |
| pm14.12 44392 | Theorem *14.12 in [Whitehe... |
| pm14.122a 44393 | Theorem *14.122 in [Whiteh... |
| pm14.122b 44394 | Theorem *14.122 in [Whiteh... |
| pm14.122c 44395 | Theorem *14.122 in [Whiteh... |
| pm14.123a 44396 | Theorem *14.123 in [Whiteh... |
| pm14.123b 44397 | Theorem *14.123 in [Whiteh... |
| pm14.123c 44398 | Theorem *14.123 in [Whiteh... |
| pm14.18 44399 | Theorem *14.18 in [Whitehe... |
| iotaequ 44400 | Theorem *14.2 in [Whitehea... |
| iotavalb 44401 | Theorem *14.202 in [Whiteh... |
| iotasbc5 44402 | Theorem *14.205 in [Whiteh... |
| pm14.24 44403 | Theorem *14.24 in [Whitehe... |
| iotavalsb 44404 | Theorem *14.242 in [Whiteh... |
| sbiota1 44405 | Theorem *14.25 in [Whitehe... |
| sbaniota 44406 | Theorem *14.26 in [Whitehe... |
| eubiOLD 44407 | Obsolete proof of ~ eubi a... |
| iotasbcq 44408 | Theorem *14.272 in [Whiteh... |
| elnev 44409 | Any set that contains one ... |
| rusbcALT 44410 | A version of Russell's par... |
| compeq 44411 | Equality between two ways ... |
| compne 44412 | The complement of ` A ` is... |
| compab 44413 | Two ways of saying "the co... |
| conss2 44414 | Contrapositive law for sub... |
| conss1 44415 | Contrapositive law for sub... |
| ralbidar 44416 | More general form of ~ ral... |
| rexbidar 44417 | More general form of ~ rex... |
| dropab1 44418 | Theorem to aid use of the ... |
| dropab2 44419 | Theorem to aid use of the ... |
| ipo0 44420 | If the identity relation p... |
| ifr0 44421 | A class that is founded by... |
| ordpss 44422 | ~ ordelpss with an anteced... |
| fvsb 44423 | Explicit substitution of a... |
| fveqsb 44424 | Implicit substitution of a... |
| xpexb 44425 | A Cartesian product exists... |
| trelpss 44426 | An element of a transitive... |
| addcomgi 44427 | Generalization of commutat... |
| addrval 44437 | Value of the operation of ... |
| subrval 44438 | Value of the operation of ... |
| mulvval 44439 | Value of the operation of ... |
| addrfv 44440 | Vector addition at a value... |
| subrfv 44441 | Vector subtraction at a va... |
| mulvfv 44442 | Scalar multiplication at a... |
| addrfn 44443 | Vector addition produces a... |
| subrfn 44444 | Vector subtraction produce... |
| mulvfn 44445 | Scalar multiplication prod... |
| addrcom 44446 | Vector addition is commuta... |
| idiALT 44450 | Placeholder for ~ idi . T... |
| exbir 44451 | Exportation implication al... |
| 3impexpbicom 44452 | Version of ~ 3impexp where... |
| 3impexpbicomi 44453 | Inference associated with ... |
| bi1imp 44454 | Importation inference simi... |
| bi2imp 44455 | Importation inference simi... |
| bi3impb 44456 | Similar to ~ 3impb with im... |
| bi3impa 44457 | Similar to ~ 3impa with im... |
| bi23impib 44458 | ~ 3impib with the inner im... |
| bi13impib 44459 | ~ 3impib with the outer im... |
| bi123impib 44460 | ~ 3impib with the implicat... |
| bi13impia 44461 | ~ 3impia with the outer im... |
| bi123impia 44462 | ~ 3impia with the implicat... |
| bi33imp12 44463 | ~ 3imp with innermost impl... |
| bi23imp13 44464 | ~ 3imp with middle implica... |
| bi13imp23 44465 | ~ 3imp with outermost impl... |
| bi13imp2 44466 | Similar to ~ 3imp except t... |
| bi12imp3 44467 | Similar to ~ 3imp except a... |
| bi23imp1 44468 | Similar to ~ 3imp except a... |
| bi123imp0 44469 | Similar to ~ 3imp except a... |
| 4animp1 44470 | A single hypothesis unific... |
| 4an31 44471 | A rearrangement of conjunc... |
| 4an4132 44472 | A rearrangement of conjunc... |
| expcomdg 44473 | Biconditional form of ~ ex... |
| iidn3 44474 | ~ idn3 without virtual ded... |
| ee222 44475 | ~ e222 without virtual ded... |
| ee3bir 44476 | Right-biconditional form o... |
| ee13 44477 | ~ e13 without virtual dedu... |
| ee121 44478 | ~ e121 without virtual ded... |
| ee122 44479 | ~ e122 without virtual ded... |
| ee333 44480 | ~ e333 without virtual ded... |
| ee323 44481 | ~ e323 without virtual ded... |
| 3ornot23 44482 | If the second and third di... |
| orbi1r 44483 | ~ orbi1 with order of disj... |
| 3orbi123 44484 | ~ pm4.39 with a 3-conjunct... |
| syl5imp 44485 | Closed form of ~ syl5 . D... |
| impexpd 44486 | The following User's Proof... |
| com3rgbi 44487 | The following User's Proof... |
| impexpdcom 44488 | The following User's Proof... |
| ee1111 44489 | Non-virtual deduction form... |
| pm2.43bgbi 44490 | Logical equivalence of a 2... |
| pm2.43cbi 44491 | Logical equivalence of a 3... |
| ee233 44492 | Non-virtual deduction form... |
| imbi13 44493 | Join three logical equival... |
| ee33 44494 | Non-virtual deduction form... |
| con5 44495 | Biconditional contrapositi... |
| con5i 44496 | Inference form of ~ con5 .... |
| exlimexi 44497 | Inference similar to Theor... |
| sb5ALT 44498 | Equivalence for substituti... |
| eexinst01 44499 | ~ exinst01 without virtual... |
| eexinst11 44500 | ~ exinst11 without virtual... |
| vk15.4j 44501 | Excercise 4j of Unit 15 of... |
| notnotrALT 44502 | Converse of double negatio... |
| con3ALT2 44503 | Contraposition. Alternate... |
| ssralv2 44504 | Quantification restricted ... |
| sbc3or 44505 | ~ sbcor with a 3-disjuncts... |
| alrim3con13v 44506 | Closed form of ~ alrimi wi... |
| rspsbc2 44507 | ~ rspsbc with two quantify... |
| sbcoreleleq 44508 | Substitution of a setvar v... |
| tratrb 44509 | If a class is transitive a... |
| ordelordALT 44510 | An element of an ordinal c... |
| sbcim2g 44511 | Distribution of class subs... |
| sbcbi 44512 | Implication form of ~ sbcb... |
| trsbc 44513 | Formula-building inference... |
| truniALT 44514 | The union of a class of tr... |
| onfrALTlem5 44515 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem4 44516 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem3 44517 | Lemma for ~ onfrALT . (Co... |
| ggen31 44518 | ~ gen31 without virtual de... |
| onfrALTlem2 44519 | Lemma for ~ onfrALT . (Co... |
| cbvexsv 44520 | A theorem pertaining to th... |
| onfrALTlem1 44521 | Lemma for ~ onfrALT . (Co... |
| onfrALT 44522 | The membership relation is... |
| 19.41rg 44523 | Closed form of right-to-le... |
| opelopab4 44524 | Ordered pair membership in... |
| 2pm13.193 44525 | ~ pm13.193 for two variabl... |
| hbntal 44526 | A closed form of ~ hbn . ~... |
| hbimpg 44527 | A closed form of ~ hbim . ... |
| hbalg 44528 | Closed form of ~ hbal . D... |
| hbexg 44529 | Closed form of ~ nfex . D... |
| ax6e2eq 44530 | Alternate form of ~ ax6e f... |
| ax6e2nd 44531 | If at least two sets exist... |
| ax6e2ndeq 44532 | "At least two sets exist" ... |
| 2sb5nd 44533 | Equivalence for double sub... |
| 2uasbanh 44534 | Distribute the unabbreviat... |
| 2uasban 44535 | Distribute the unabbreviat... |
| e2ebind 44536 | Absorption of an existenti... |
| elpwgded 44537 | ~ elpwgdedVD in convention... |
| trelded 44538 | Deduction form of ~ trel .... |
| jaoded 44539 | Deduction form of ~ jao . ... |
| sbtT 44540 | A substitution into a theo... |
| not12an2impnot1 44541 | If a double conjunction is... |
| in1 44544 | Inference form of ~ df-vd1... |
| iin1 44545 | ~ in1 without virtual dedu... |
| dfvd1ir 44546 | Inference form of ~ df-vd1... |
| idn1 44547 | Virtual deduction identity... |
| dfvd1imp 44548 | Left-to-right part of defi... |
| dfvd1impr 44549 | Right-to-left part of defi... |
| dfvd2 44552 | Definition of a 2-hypothes... |
| dfvd2an 44555 | Definition of a 2-hypothes... |
| dfvd2ani 44556 | Inference form of ~ dfvd2a... |
| dfvd2anir 44557 | Right-to-left inference fo... |
| dfvd2i 44558 | Inference form of ~ dfvd2 ... |
| dfvd2ir 44559 | Right-to-left inference fo... |
| dfvd3 44564 | Definition of a 3-hypothes... |
| dfvd3i 44565 | Inference form of ~ dfvd3 ... |
| dfvd3ir 44566 | Right-to-left inference fo... |
| dfvd3an 44567 | Definition of a 3-hypothes... |
| dfvd3ani 44568 | Inference form of ~ dfvd3a... |
| dfvd3anir 44569 | Right-to-left inference fo... |
| vd01 44570 | A virtual hypothesis virtu... |
| vd02 44571 | Two virtual hypotheses vir... |
| vd03 44572 | A theorem is virtually inf... |
| vd12 44573 | A virtual deduction with 1... |
| vd13 44574 | A virtual deduction with 1... |
| vd23 44575 | A virtual deduction with 2... |
| dfvd2imp 44576 | The virtual deduction form... |
| dfvd2impr 44577 | A 2-antecedent nested impl... |
| in2 44578 | The virtual deduction intr... |
| int2 44579 | The virtual deduction intr... |
| iin2 44580 | ~ in2 without virtual dedu... |
| in2an 44581 | The virtual deduction intr... |
| in3 44582 | The virtual deduction intr... |
| iin3 44583 | ~ in3 without virtual dedu... |
| in3an 44584 | The virtual deduction intr... |
| int3 44585 | The virtual deduction intr... |
| idn2 44586 | Virtual deduction identity... |
| iden2 44587 | Virtual deduction identity... |
| idn3 44588 | Virtual deduction identity... |
| gen11 44589 | Virtual deduction generali... |
| gen11nv 44590 | Virtual deduction generali... |
| gen12 44591 | Virtual deduction generali... |
| gen21 44592 | Virtual deduction generali... |
| gen21nv 44593 | Virtual deduction form of ... |
| gen31 44594 | Virtual deduction generali... |
| gen22 44595 | Virtual deduction generali... |
| ggen22 44596 | ~ gen22 without virtual de... |
| exinst 44597 | Existential Instantiation.... |
| exinst01 44598 | Existential Instantiation.... |
| exinst11 44599 | Existential Instantiation.... |
| e1a 44600 | A Virtual deduction elimin... |
| el1 44601 | A Virtual deduction elimin... |
| e1bi 44602 | Biconditional form of ~ e1... |
| e1bir 44603 | Right biconditional form o... |
| e2 44604 | A virtual deduction elimin... |
| e2bi 44605 | Biconditional form of ~ e2... |
| e2bir 44606 | Right biconditional form o... |
| ee223 44607 | ~ e223 without virtual ded... |
| e223 44608 | A virtual deduction elimin... |
| e222 44609 | A virtual deduction elimin... |
| e220 44610 | A virtual deduction elimin... |
| ee220 44611 | ~ e220 without virtual ded... |
| e202 44612 | A virtual deduction elimin... |
| ee202 44613 | ~ e202 without virtual ded... |
| e022 44614 | A virtual deduction elimin... |
| ee022 44615 | ~ e022 without virtual ded... |
| e002 44616 | A virtual deduction elimin... |
| ee002 44617 | ~ e002 without virtual ded... |
| e020 44618 | A virtual deduction elimin... |
| ee020 44619 | ~ e020 without virtual ded... |
| e200 44620 | A virtual deduction elimin... |
| ee200 44621 | ~ e200 without virtual ded... |
| e221 44622 | A virtual deduction elimin... |
| ee221 44623 | ~ e221 without virtual ded... |
| e212 44624 | A virtual deduction elimin... |
| ee212 44625 | ~ e212 without virtual ded... |
| e122 44626 | A virtual deduction elimin... |
| e112 44627 | A virtual deduction elimin... |
| ee112 44628 | ~ e112 without virtual ded... |
| e121 44629 | A virtual deduction elimin... |
| e211 44630 | A virtual deduction elimin... |
| ee211 44631 | ~ e211 without virtual ded... |
| e210 44632 | A virtual deduction elimin... |
| ee210 44633 | ~ e210 without virtual ded... |
| e201 44634 | A virtual deduction elimin... |
| ee201 44635 | ~ e201 without virtual ded... |
| e120 44636 | A virtual deduction elimin... |
| ee120 44637 | Virtual deduction rule ~ e... |
| e021 44638 | A virtual deduction elimin... |
| ee021 44639 | ~ e021 without virtual ded... |
| e012 44640 | A virtual deduction elimin... |
| ee012 44641 | ~ e012 without virtual ded... |
| e102 44642 | A virtual deduction elimin... |
| ee102 44643 | ~ e102 without virtual ded... |
| e22 44644 | A virtual deduction elimin... |
| e22an 44645 | Conjunction form of ~ e22 ... |
| ee22an 44646 | ~ e22an without virtual de... |
| e111 44647 | A virtual deduction elimin... |
| e1111 44648 | A virtual deduction elimin... |
| e110 44649 | A virtual deduction elimin... |
| ee110 44650 | ~ e110 without virtual ded... |
| e101 44651 | A virtual deduction elimin... |
| ee101 44652 | ~ e101 without virtual ded... |
| e011 44653 | A virtual deduction elimin... |
| ee011 44654 | ~ e011 without virtual ded... |
| e100 44655 | A virtual deduction elimin... |
| ee100 44656 | ~ e100 without virtual ded... |
| e010 44657 | A virtual deduction elimin... |
| ee010 44658 | ~ e010 without virtual ded... |
| e001 44659 | A virtual deduction elimin... |
| ee001 44660 | ~ e001 without virtual ded... |
| e11 44661 | A virtual deduction elimin... |
| e11an 44662 | Conjunction form of ~ e11 ... |
| ee11an 44663 | ~ e11an without virtual de... |
| e01 44664 | A virtual deduction elimin... |
| e01an 44665 | Conjunction form of ~ e01 ... |
| ee01an 44666 | ~ e01an without virtual de... |
| e10 44667 | A virtual deduction elimin... |
| e10an 44668 | Conjunction form of ~ e10 ... |
| ee10an 44669 | ~ e10an without virtual de... |
| e02 44670 | A virtual deduction elimin... |
| e02an 44671 | Conjunction form of ~ e02 ... |
| ee02an 44672 | ~ e02an without virtual de... |
| eel021old 44673 | ~ el021old without virtual... |
| el021old 44674 | A virtual deduction elimin... |
| eel132 44675 | ~ syl2an with antecedents ... |
| eel000cT 44676 | An elimination deduction. ... |
| eel0TT 44677 | An elimination deduction. ... |
| eelT00 44678 | An elimination deduction. ... |
| eelTTT 44679 | An elimination deduction. ... |
| eelT11 44680 | An elimination deduction. ... |
| eelT1 44681 | Syllogism inference combin... |
| eelT12 44682 | An elimination deduction. ... |
| eelTT1 44683 | An elimination deduction. ... |
| eelT01 44684 | An elimination deduction. ... |
| eel0T1 44685 | An elimination deduction. ... |
| eel12131 44686 | An elimination deduction. ... |
| eel2131 44687 | ~ syl2an with antecedents ... |
| eel3132 44688 | ~ syl2an with antecedents ... |
| eel0321old 44689 | ~ el0321old without virtua... |
| el0321old 44690 | A virtual deduction elimin... |
| eel2122old 44691 | ~ el2122old without virtua... |
| el2122old 44692 | A virtual deduction elimin... |
| eel0000 44693 | Elimination rule similar t... |
| eel00001 44694 | An elimination deduction. ... |
| eel00000 44695 | Elimination rule similar ~... |
| eel11111 44696 | Five-hypothesis eliminatio... |
| e12 44697 | A virtual deduction elimin... |
| e12an 44698 | Conjunction form of ~ e12 ... |
| el12 44699 | Virtual deduction form of ... |
| e20 44700 | A virtual deduction elimin... |
| e20an 44701 | Conjunction form of ~ e20 ... |
| ee20an 44702 | ~ e20an without virtual de... |
| e21 44703 | A virtual deduction elimin... |
| e21an 44704 | Conjunction form of ~ e21 ... |
| ee21an 44705 | ~ e21an without virtual de... |
| e333 44706 | A virtual deduction elimin... |
| e33 44707 | A virtual deduction elimin... |
| e33an 44708 | Conjunction form of ~ e33 ... |
| ee33an 44709 | ~ e33an without virtual de... |
| e3 44710 | Meta-connective form of ~ ... |
| e3bi 44711 | Biconditional form of ~ e3... |
| e3bir 44712 | Right biconditional form o... |
| e03 44713 | A virtual deduction elimin... |
| ee03 44714 | ~ e03 without virtual dedu... |
| e03an 44715 | Conjunction form of ~ e03 ... |
| ee03an 44716 | Conjunction form of ~ ee03... |
| e30 44717 | A virtual deduction elimin... |
| ee30 44718 | ~ e30 without virtual dedu... |
| e30an 44719 | A virtual deduction elimin... |
| ee30an 44720 | Conjunction form of ~ ee30... |
| e13 44721 | A virtual deduction elimin... |
| e13an 44722 | A virtual deduction elimin... |
| ee13an 44723 | ~ e13an without virtual de... |
| e31 44724 | A virtual deduction elimin... |
| ee31 44725 | ~ e31 without virtual dedu... |
| e31an 44726 | A virtual deduction elimin... |
| ee31an 44727 | ~ e31an without virtual de... |
| e23 44728 | A virtual deduction elimin... |
| e23an 44729 | A virtual deduction elimin... |
| ee23an 44730 | ~ e23an without virtual de... |
| e32 44731 | A virtual deduction elimin... |
| ee32 44732 | ~ e32 without virtual dedu... |
| e32an 44733 | A virtual deduction elimin... |
| ee32an 44734 | ~ e33an without virtual de... |
| e123 44735 | A virtual deduction elimin... |
| ee123 44736 | ~ e123 without virtual ded... |
| el123 44737 | A virtual deduction elimin... |
| e233 44738 | A virtual deduction elimin... |
| e323 44739 | A virtual deduction elimin... |
| e000 44740 | A virtual deduction elimin... |
| e00 44741 | Elimination rule identical... |
| e00an 44742 | Elimination rule identical... |
| eel00cT 44743 | An elimination deduction. ... |
| eelTT 44744 | An elimination deduction. ... |
| e0a 44745 | Elimination rule identical... |
| eelT 44746 | An elimination deduction. ... |
| eel0cT 44747 | An elimination deduction. ... |
| eelT0 44748 | An elimination deduction. ... |
| e0bi 44749 | Elimination rule identical... |
| e0bir 44750 | Elimination rule identical... |
| uun0.1 44751 | Convention notation form o... |
| un0.1 44752 | ` T. ` is the constant tru... |
| uunT1 44753 | A deduction unionizing a n... |
| uunT1p1 44754 | A deduction unionizing a n... |
| uunT21 44755 | A deduction unionizing a n... |
| uun121 44756 | A deduction unionizing a n... |
| uun121p1 44757 | A deduction unionizing a n... |
| uun132 44758 | A deduction unionizing a n... |
| uun132p1 44759 | A deduction unionizing a n... |
| anabss7p1 44760 | A deduction unionizing a n... |
| un10 44761 | A unionizing deduction. (... |
| un01 44762 | A unionizing deduction. (... |
| un2122 44763 | A deduction unionizing a n... |
| uun2131 44764 | A deduction unionizing a n... |
| uun2131p1 44765 | A deduction unionizing a n... |
| uunTT1 44766 | A deduction unionizing a n... |
| uunTT1p1 44767 | A deduction unionizing a n... |
| uunTT1p2 44768 | A deduction unionizing a n... |
| uunT11 44769 | A deduction unionizing a n... |
| uunT11p1 44770 | A deduction unionizing a n... |
| uunT11p2 44771 | A deduction unionizing a n... |
| uunT12 44772 | A deduction unionizing a n... |
| uunT12p1 44773 | A deduction unionizing a n... |
| uunT12p2 44774 | A deduction unionizing a n... |
| uunT12p3 44775 | A deduction unionizing a n... |
| uunT12p4 44776 | A deduction unionizing a n... |
| uunT12p5 44777 | A deduction unionizing a n... |
| uun111 44778 | A deduction unionizing a n... |
| 3anidm12p1 44779 | A deduction unionizing a n... |
| 3anidm12p2 44780 | A deduction unionizing a n... |
| uun123 44781 | A deduction unionizing a n... |
| uun123p1 44782 | A deduction unionizing a n... |
| uun123p2 44783 | A deduction unionizing a n... |
| uun123p3 44784 | A deduction unionizing a n... |
| uun123p4 44785 | A deduction unionizing a n... |
| uun2221 44786 | A deduction unionizing a n... |
| uun2221p1 44787 | A deduction unionizing a n... |
| uun2221p2 44788 | A deduction unionizing a n... |
| 3impdirp1 44789 | A deduction unionizing a n... |
| 3impcombi 44790 | A 1-hypothesis proposition... |
| trsspwALT 44791 | Virtual deduction proof of... |
| trsspwALT2 44792 | Virtual deduction proof of... |
| trsspwALT3 44793 | Short predicate calculus p... |
| sspwtr 44794 | Virtual deduction proof of... |
| sspwtrALT 44795 | Virtual deduction proof of... |
| sspwtrALT2 44796 | Short predicate calculus p... |
| pwtrVD 44797 | Virtual deduction proof of... |
| pwtrrVD 44798 | Virtual deduction proof of... |
| suctrALT 44799 | The successor of a transit... |
| snssiALTVD 44800 | Virtual deduction proof of... |
| snssiALT 44801 | If a class is an element o... |
| snsslVD 44802 | Virtual deduction proof of... |
| snssl 44803 | If a singleton is a subcla... |
| snelpwrVD 44804 | Virtual deduction proof of... |
| unipwrVD 44805 | Virtual deduction proof of... |
| unipwr 44806 | A class is a subclass of t... |
| sstrALT2VD 44807 | Virtual deduction proof of... |
| sstrALT2 44808 | Virtual deduction proof of... |
| suctrALT2VD 44809 | Virtual deduction proof of... |
| suctrALT2 44810 | Virtual deduction proof of... |
| elex2VD 44811 | Virtual deduction proof of... |
| elex22VD 44812 | Virtual deduction proof of... |
| eqsbc2VD 44813 | Virtual deduction proof of... |
| zfregs2VD 44814 | Virtual deduction proof of... |
| tpid3gVD 44815 | Virtual deduction proof of... |
| en3lplem1VD 44816 | Virtual deduction proof of... |
| en3lplem2VD 44817 | Virtual deduction proof of... |
| en3lpVD 44818 | Virtual deduction proof of... |
| simplbi2VD 44819 | Virtual deduction proof of... |
| 3ornot23VD 44820 | Virtual deduction proof of... |
| orbi1rVD 44821 | Virtual deduction proof of... |
| bitr3VD 44822 | Virtual deduction proof of... |
| 3orbi123VD 44823 | Virtual deduction proof of... |
| sbc3orgVD 44824 | Virtual deduction proof of... |
| 19.21a3con13vVD 44825 | Virtual deduction proof of... |
| exbirVD 44826 | Virtual deduction proof of... |
| exbiriVD 44827 | Virtual deduction proof of... |
| rspsbc2VD 44828 | Virtual deduction proof of... |
| 3impexpVD 44829 | Virtual deduction proof of... |
| 3impexpbicomVD 44830 | Virtual deduction proof of... |
| 3impexpbicomiVD 44831 | Virtual deduction proof of... |
| sbcoreleleqVD 44832 | Virtual deduction proof of... |
| hbra2VD 44833 | Virtual deduction proof of... |
| tratrbVD 44834 | Virtual deduction proof of... |
| al2imVD 44835 | Virtual deduction proof of... |
| syl5impVD 44836 | Virtual deduction proof of... |
| idiVD 44837 | Virtual deduction proof of... |
| ancomstVD 44838 | Closed form of ~ ancoms . ... |
| ssralv2VD 44839 | Quantification restricted ... |
| ordelordALTVD 44840 | An element of an ordinal c... |
| equncomVD 44841 | If a class equals the unio... |
| equncomiVD 44842 | Inference form of ~ equnco... |
| sucidALTVD 44843 | A set belongs to its succe... |
| sucidALT 44844 | A set belongs to its succe... |
| sucidVD 44845 | A set belongs to its succe... |
| imbi12VD 44846 | Implication form of ~ imbi... |
| imbi13VD 44847 | Join three logical equival... |
| sbcim2gVD 44848 | Distribution of class subs... |
| sbcbiVD 44849 | Implication form of ~ sbcb... |
| trsbcVD 44850 | Formula-building inference... |
| truniALTVD 44851 | The union of a class of tr... |
| ee33VD 44852 | Non-virtual deduction form... |
| trintALTVD 44853 | The intersection of a clas... |
| trintALT 44854 | The intersection of a clas... |
| undif3VD 44855 | The first equality of Exer... |
| sbcssgVD 44856 | Virtual deduction proof of... |
| csbingVD 44857 | Virtual deduction proof of... |
| onfrALTlem5VD 44858 | Virtual deduction proof of... |
| onfrALTlem4VD 44859 | Virtual deduction proof of... |
| onfrALTlem3VD 44860 | Virtual deduction proof of... |
| simplbi2comtVD 44861 | Virtual deduction proof of... |
| onfrALTlem2VD 44862 | Virtual deduction proof of... |
| onfrALTlem1VD 44863 | Virtual deduction proof of... |
| onfrALTVD 44864 | Virtual deduction proof of... |
| csbeq2gVD 44865 | Virtual deduction proof of... |
| csbsngVD 44866 | Virtual deduction proof of... |
| csbxpgVD 44867 | Virtual deduction proof of... |
| csbresgVD 44868 | Virtual deduction proof of... |
| csbrngVD 44869 | Virtual deduction proof of... |
| csbima12gALTVD 44870 | Virtual deduction proof of... |
| csbunigVD 44871 | Virtual deduction proof of... |
| csbfv12gALTVD 44872 | Virtual deduction proof of... |
| con5VD 44873 | Virtual deduction proof of... |
| relopabVD 44874 | Virtual deduction proof of... |
| 19.41rgVD 44875 | Virtual deduction proof of... |
| 2pm13.193VD 44876 | Virtual deduction proof of... |
| hbimpgVD 44877 | Virtual deduction proof of... |
| hbalgVD 44878 | Virtual deduction proof of... |
| hbexgVD 44879 | Virtual deduction proof of... |
| ax6e2eqVD 44880 | The following User's Proof... |
| ax6e2ndVD 44881 | The following User's Proof... |
| ax6e2ndeqVD 44882 | The following User's Proof... |
| 2sb5ndVD 44883 | The following User's Proof... |
| 2uasbanhVD 44884 | The following User's Proof... |
| e2ebindVD 44885 | The following User's Proof... |
| sb5ALTVD 44886 | The following User's Proof... |
| vk15.4jVD 44887 | The following User's Proof... |
| notnotrALTVD 44888 | The following User's Proof... |
| con3ALTVD 44889 | The following User's Proof... |
| elpwgdedVD 44890 | Membership in a power clas... |
| sspwimp 44891 | If a class is a subclass o... |
| sspwimpVD 44892 | The following User's Proof... |
| sspwimpcf 44893 | If a class is a subclass o... |
| sspwimpcfVD 44894 | The following User's Proof... |
| suctrALTcf 44895 | The successor of a transit... |
| suctrALTcfVD 44896 | The following User's Proof... |
| suctrALT3 44897 | The successor of a transit... |
| sspwimpALT 44898 | If a class is a subclass o... |
| unisnALT 44899 | A set equals the union of ... |
| notnotrALT2 44900 | Converse of double negatio... |
| sspwimpALT2 44901 | If a class is a subclass o... |
| e2ebindALT 44902 | Absorption of an existenti... |
| ax6e2ndALT 44903 | If at least two sets exist... |
| ax6e2ndeqALT 44904 | "At least two sets exist" ... |
| 2sb5ndALT 44905 | Equivalence for double sub... |
| chordthmALT 44906 | The intersecting chords th... |
| isosctrlem1ALT 44907 | Lemma for ~ isosctr . Thi... |
| iunconnlem2 44908 | The indexed union of conne... |
| iunconnALT 44909 | The indexed union of conne... |
| sineq0ALT 44910 | A complex number whose sin... |
| trwf 44911 | The class of well-founded ... |
| traxext 44912 | A transitive class models ... |
| wfaxext 44913 | The class of well-founded ... |
| xpwf 44914 | The Cartesian product of t... |
| dmwf 44915 | The domain of a well-found... |
| rnwf 44916 | The range of a well-founde... |
| evth2f 44917 | A version of ~ evth2 using... |
| elunif 44918 | A version of ~ eluni using... |
| rzalf 44919 | A version of ~ rzal using ... |
| fvelrnbf 44920 | A version of ~ fvelrnb usi... |
| rfcnpre1 44921 | If F is a continuous funct... |
| ubelsupr 44922 | If U belongs to A and U is... |
| fsumcnf 44923 | A finite sum of functions ... |
| mulltgt0 44924 | The product of a negative ... |
| rspcegf 44925 | A version of ~ rspcev usin... |
| rabexgf 44926 | A version of ~ rabexg usin... |
| fcnre 44927 | A function continuous with... |
| sumsnd 44928 | A sum of a singleton is th... |
| evthf 44929 | A version of ~ evth using ... |
| cnfex 44930 | The class of continuous fu... |
| fnchoice 44931 | For a finite set, a choice... |
| refsumcn 44932 | A finite sum of continuous... |
| rfcnpre2 44933 | If ` F ` is a continuous f... |
| cncmpmax 44934 | When the hypothesis for th... |
| rfcnpre3 44935 | If F is a continuous funct... |
| rfcnpre4 44936 | If F is a continuous funct... |
| sumpair 44937 | Sum of two distinct comple... |
| rfcnnnub 44938 | Given a real continuous fu... |
| refsum2cnlem1 44939 | This is the core Lemma for... |
| refsum2cn 44940 | The sum of two continuus r... |
| adantlllr 44941 | Deduction adding a conjunc... |
| 3adantlr3 44942 | Deduction adding a conjunc... |
| 3adantll2 44943 | Deduction adding a conjunc... |
| 3adantll3 44944 | Deduction adding a conjunc... |
| ssnel 44945 | If not element of a set, t... |
| sncldre 44946 | A singleton is closed w.r.... |
| n0p 44947 | A polynomial with a nonzer... |
| pm2.65ni 44948 | Inference rule for proof b... |
| pwssfi 44949 | Every element of the power... |
| iuneq2df 44950 | Equality deduction for ind... |
| nnfoctb 44951 | There exists a mapping fro... |
| ssinss1d 44952 | Intersection preserves sub... |
| elpwinss 44953 | An element of the powerset... |
| unidmex 44954 | If ` F ` is a set, then ` ... |
| ndisj2 44955 | A non-disjointness conditi... |
| zenom 44956 | The set of integer numbers... |
| uzwo4 44957 | Well-ordering principle: a... |
| unisn0 44958 | The union of the singleton... |
| ssin0 44959 | If two classes are disjoin... |
| inabs3 44960 | Absorption law for interse... |
| pwpwuni 44961 | Relationship between power... |
| disjiun2 44962 | In a disjoint collection, ... |
| 0pwfi 44963 | The empty set is in any po... |
| ssinss2d 44964 | Intersection preserves sub... |
| zct 44965 | The set of integer numbers... |
| pwfin0 44966 | A finite set always belong... |
| uzct 44967 | An upper integer set is co... |
| iunxsnf 44968 | A singleton index picks ou... |
| fiiuncl 44969 | If a set is closed under t... |
| iunp1 44970 | The addition of the next s... |
| fiunicl 44971 | If a set is closed under t... |
| ixpeq2d 44972 | Equality theorem for infin... |
| disjxp1 44973 | The sets of a cartesian pr... |
| disjsnxp 44974 | The sets in the cartesian ... |
| eliind 44975 | Membership in indexed inte... |
| rspcef 44976 | Restricted existential spe... |
| ixpssmapc 44977 | An infinite Cartesian prod... |
| elintd 44978 | Membership in class inters... |
| ssdf 44979 | A sufficient condition for... |
| brneqtrd 44980 | Substitution of equal clas... |
| ssnct 44981 | A set containing an uncoun... |
| ssuniint 44982 | Sufficient condition for b... |
| elintdv 44983 | Membership in class inters... |
| ssd 44984 | A sufficient condition for... |
| ralimralim 44985 | Introducing any antecedent... |
| snelmap 44986 | Membership of the element ... |
| xrnmnfpnf 44987 | An extended real that is n... |
| nelrnmpt 44988 | Non-membership in the rang... |
| iuneq1i 44989 | Equality theorem for index... |
| nssrex 44990 | Negation of subclass relat... |
| ssinc 44991 | Inclusion relation for a m... |
| ssdec 44992 | Inclusion relation for a m... |
| elixpconstg 44993 | Membership in an infinite ... |
| iineq1d 44994 | Equality theorem for index... |
| metpsmet 44995 | A metric is a pseudometric... |
| ixpssixp 44996 | Subclass theorem for infin... |
| ballss3 44997 | A sufficient condition for... |
| iunincfi 44998 | Given a sequence of increa... |
| nsstr 44999 | If it's not a subclass, it... |
| rexanuz3 45000 | Combine two different uppe... |
| cbvmpo2 45001 | Rule to change the second ... |
| cbvmpo1 45002 | Rule to change the first b... |
| eliuniin 45003 | Indexed union of indexed i... |
| ssabf 45004 | Subclass of a class abstra... |
| pssnssi 45005 | A proper subclass does not... |
| rabidim2 45006 | Membership in a restricted... |
| eluni2f 45007 | Membership in class union.... |
| eliin2f 45008 | Membership in indexed inte... |
| nssd 45009 | Negation of subclass relat... |
| iineq12dv 45010 | Equality deduction for ind... |
| supxrcld 45011 | The supremum of an arbitra... |
| elrestd 45012 | A sufficient condition for... |
| eliuniincex 45013 | Counterexample to show tha... |
| eliincex 45014 | Counterexample to show tha... |
| eliinid 45015 | Membership in an indexed i... |
| abssf 45016 | Class abstraction in a sub... |
| supxrubd 45017 | A member of a set of exten... |
| ssrabf 45018 | Subclass of a restricted c... |
| ssrabdf 45019 | Subclass of a restricted c... |
| eliin2 45020 | Membership in indexed inte... |
| ssrab2f 45021 | Subclass relation for a re... |
| restuni3 45022 | The underlying set of a su... |
| rabssf 45023 | Restricted class abstracti... |
| eliuniin2 45024 | Indexed union of indexed i... |
| restuni4 45025 | The underlying set of a su... |
| restuni6 45026 | The underlying set of a su... |
| restuni5 45027 | The underlying set of a su... |
| unirestss 45028 | The union of an elementwis... |
| iniin1 45029 | Indexed intersection of in... |
| iniin2 45030 | Indexed intersection of in... |
| cbvrabv2 45031 | A more general version of ... |
| cbvrabv2w 45032 | A more general version of ... |
| iinssiin 45033 | Subset implication for an ... |
| eliind2 45034 | Membership in indexed inte... |
| iinssd 45035 | Subset implication for an ... |
| rabbida2 45036 | Equivalent wff's yield equ... |
| iinexd 45037 | The existence of an indexe... |
| rabexf 45038 | Separation Scheme in terms... |
| rabbida3 45039 | Equivalent wff's yield equ... |
| r19.36vf 45040 | Restricted quantifier vers... |
| raleqd 45041 | Equality deduction for res... |
| iinssf 45042 | Subset implication for an ... |
| iinssdf 45043 | Subset implication for an ... |
| resabs2i 45044 | Absorption law for restric... |
| ssdf2 45045 | A sufficient condition for... |
| rabssd 45046 | Restricted class abstracti... |
| rexnegd 45047 | Minus a real number. (Con... |
| rexlimd3 45048 | * Inference from Theorem 1... |
| resabs1i 45049 | Absorption law for restric... |
| nel1nelin 45050 | Membership in an intersect... |
| nel2nelin 45051 | Membership in an intersect... |
| nel1nelini 45052 | Membership in an intersect... |
| nel2nelini 45053 | Membership in an intersect... |
| eliunid 45054 | Membership in indexed unio... |
| reximdd 45055 | Deduction from Theorem 19.... |
| inopnd 45056 | The intersection of two op... |
| ss2rabdf 45057 | Deduction of restricted ab... |
| restopn3 45058 | If ` A ` is open, then ` A... |
| restopnssd 45059 | A topology restricted to a... |
| restsubel 45060 | A subset belongs in the sp... |
| toprestsubel 45061 | A subset is open in the to... |
| rabidd 45062 | An "identity" law of concr... |
| iunssdf 45063 | Subset theorem for an inde... |
| iinss2d 45064 | Subset implication for an ... |
| r19.3rzf 45065 | Restricted quantification ... |
| r19.28zf 45066 | Restricted quantifier vers... |
| iindif2f 45067 | Indexed intersection of cl... |
| ralfal 45068 | Two ways of expressing emp... |
| archd 45069 | Archimedean property of re... |
| eliund 45070 | Membership in indexed unio... |
| nimnbi 45071 | If an implication is false... |
| nimnbi2 45072 | If an implication is false... |
| notbicom 45073 | Commutative law for the ne... |
| rexeqif 45074 | Equality inference for res... |
| rspced 45075 | Restricted existential spe... |
| feq1dd 45076 | Equality deduction for fun... |
| fnresdmss 45077 | A function does not change... |
| fmptsnxp 45078 | Maps-to notation and Carte... |
| fvmpt2bd 45079 | Value of a function given ... |
| rnmptfi 45080 | The range of a function wi... |
| fresin2 45081 | Restriction of a function ... |
| ffi 45082 | A function with finite dom... |
| suprnmpt 45083 | An explicit bound for the ... |
| rnffi 45084 | The range of a function wi... |
| mptelpm 45085 | A function in maps-to nota... |
| rnmptpr 45086 | Range of a function define... |
| resmpti 45087 | Restriction of the mapping... |
| founiiun 45088 | Union expressed as an inde... |
| rnresun 45089 | Distribution law for range... |
| elrnmptf 45090 | The range of a function in... |
| rnmptssrn 45091 | Inclusion relation for two... |
| disjf1 45092 | A 1 to 1 mapping built fro... |
| rnsnf 45093 | The range of a function wh... |
| wessf1ornlem 45094 | Given a function ` F ` on ... |
| wessf1orn 45095 | Given a function ` F ` on ... |
| nelrnres 45096 | If ` A ` is not in the ran... |
| disjrnmpt2 45097 | Disjointness of the range ... |
| elrnmpt1sf 45098 | Elementhood in an image se... |
| founiiun0 45099 | Union expressed as an inde... |
| disjf1o 45100 | A bijection built from dis... |
| disjinfi 45101 | Only a finite number of di... |
| fvovco 45102 | Value of the composition o... |
| ssnnf1octb 45103 | There exists a bijection b... |
| nnf1oxpnn 45104 | There is a bijection betwe... |
| rnmptssd 45105 | The range of a function gi... |
| projf1o 45106 | A biijection from a set to... |
| fvmap 45107 | Function value for a membe... |
| fvixp2 45108 | Projection of a factor of ... |
| choicefi 45109 | For a finite set, a choice... |
| mpct 45110 | The exponentiation of a co... |
| cnmetcoval 45111 | Value of the distance func... |
| fcomptss 45112 | Express composition of two... |
| elmapsnd 45113 | Membership in a set expone... |
| mapss2 45114 | Subset inheritance for set... |
| fsneq 45115 | Equality condition for two... |
| difmap 45116 | Difference of two sets exp... |
| unirnmap 45117 | Given a subset of a set ex... |
| inmap 45118 | Intersection of two sets e... |
| fcoss 45119 | Composition of two mapping... |
| fsneqrn 45120 | Equality condition for two... |
| difmapsn 45121 | Difference of two sets exp... |
| mapssbi 45122 | Subset inheritance for set... |
| unirnmapsn 45123 | Equality theorem for a sub... |
| iunmapss 45124 | The indexed union of set e... |
| ssmapsn 45125 | A subset ` C ` of a set ex... |
| iunmapsn 45126 | The indexed union of set e... |
| absfico 45127 | Mapping domain and codomai... |
| icof 45128 | The set of left-closed rig... |
| elpmrn 45129 | The range of a partial fun... |
| imaexi 45130 | The image of a set is a se... |
| axccdom 45131 | Relax the constraint on ax... |
| dmmptdff 45132 | The domain of the mapping ... |
| dmmptdf 45133 | The domain of the mapping ... |
| elpmi2 45134 | The domain of a partial fu... |
| dmrelrnrel 45135 | A relation preserving func... |
| fvcod 45136 | Value of a function compos... |
| elrnmpoid 45137 | Membership in the range of... |
| axccd 45138 | An alternative version of ... |
| axccd2 45139 | An alternative version of ... |
| feqresmptf 45140 | Express a restricted funct... |
| dmmptssf 45141 | The domain of a mapping is... |
| dmmptdf2 45142 | The domain of the mapping ... |
| dmuz 45143 | Domain of the upper intege... |
| fmptd2f 45144 | Domain and codomain of the... |
| mpteq1df 45145 | An equality theorem for th... |
| mpteq1dfOLD 45146 | Obsolete version of ~ mpte... |
| mptexf 45147 | If the domain of a functio... |
| fvmpt4 45148 | Value of a function given ... |
| fmptf 45149 | Functionality of the mappi... |
| resimass 45150 | The image of a restriction... |
| mptssid 45151 | The mapping operation expr... |
| mptfnd 45152 | The maps-to notation defin... |
| mpteq12daOLD 45153 | Obsolete version of ~ mpte... |
| rnmptlb 45154 | Boundness below of the ran... |
| rnmptbddlem 45155 | Boundness of the range of ... |
| rnmptbdd 45156 | Boundness of the range of ... |
| funimaeq 45157 | Membership relation for th... |
| rnmptssf 45158 | The range of a function gi... |
| rnmptbd2lem 45159 | Boundness below of the ran... |
| rnmptbd2 45160 | Boundness below of the ran... |
| infnsuprnmpt 45161 | The indexed infimum of rea... |
| suprclrnmpt 45162 | Closure of the indexed sup... |
| suprubrnmpt2 45163 | A member of a nonempty ind... |
| suprubrnmpt 45164 | A member of a nonempty ind... |
| rnmptssdf 45165 | The range of a function gi... |
| rnmptbdlem 45166 | Boundness above of the ran... |
| rnmptbd 45167 | Boundness above of the ran... |
| rnmptss2 45168 | The range of a function gi... |
| elmptima 45169 | The image of a function in... |
| ralrnmpt3 45170 | A restricted quantifier ov... |
| fvelima2 45171 | Function value in an image... |
| rnmptssbi 45172 | The range of a function gi... |
| imass2d 45173 | Subset theorem for image. ... |
| imassmpt 45174 | Membership relation for th... |
| fpmd 45175 | A total function is a part... |
| fconst7 45176 | An alternative way to expr... |
| fnmptif 45177 | Functionality and domain o... |
| dmmptif 45178 | Domain of the mapping oper... |
| mpteq2dfa 45179 | Slightly more general equa... |
| dmmpt1 45180 | The domain of the mapping ... |
| fmptff 45181 | Functionality of the mappi... |
| fvmptelcdmf 45182 | The value of a function at... |
| fmptdff 45183 | A version of ~ fmptd using... |
| fvmpt2df 45184 | Deduction version of ~ fvm... |
| rn1st 45185 | The range of a function wi... |
| rnmptssff 45186 | The range of a function gi... |
| rnmptssdff 45187 | The range of a function gi... |
| fvmpt4d 45188 | Value of a function given ... |
| sub2times 45189 | Subtracting from a number,... |
| nnxrd 45190 | A natural number is an ext... |
| nnxr 45191 | A natural number is an ext... |
| abssubrp 45192 | The distance of two distin... |
| elfzfzo 45193 | Relationship between membe... |
| oddfl 45194 | Odd number representation ... |
| abscosbd 45195 | Bound for the absolute val... |
| mul13d 45196 | Commutative/associative la... |
| negpilt0 45197 | Negative ` _pi ` is negati... |
| dstregt0 45198 | A complex number ` A ` tha... |
| subadd4b 45199 | Rearrangement of 4 terms i... |
| xrlttri5d 45200 | Not equal and not larger i... |
| neglt 45201 | The negative of a positive... |
| zltlesub 45202 | If an integer ` N ` is les... |
| divlt0gt0d 45203 | The ratio of a negative nu... |
| subsub23d 45204 | Swap subtrahend and result... |
| 2timesgt 45205 | Double of a positive real ... |
| reopn 45206 | The reals are open with re... |
| sub31 45207 | Swap the first and third t... |
| nnne1ge2 45208 | A positive integer which i... |
| lefldiveq 45209 | A closed enough, smaller r... |
| negsubdi3d 45210 | Distribution of negative o... |
| ltdiv2dd 45211 | Division of a positive num... |
| abssinbd 45212 | Bound for the absolute val... |
| halffl 45213 | Floor of ` ( 1 / 2 ) ` . ... |
| monoords 45214 | Ordering relation for a st... |
| hashssle 45215 | The size of a subset of a ... |
| lttri5d 45216 | Not equal and not larger i... |
| fzisoeu 45217 | A finite ordered set has a... |
| lt3addmuld 45218 | If three real numbers are ... |
| absnpncan2d 45219 | Triangular inequality, com... |
| fperiodmullem 45220 | A function with period ` T... |
| fperiodmul 45221 | A function with period T i... |
| upbdrech 45222 | Choice of an upper bound f... |
| lt4addmuld 45223 | If four real numbers are l... |
| absnpncan3d 45224 | Triangular inequality, com... |
| upbdrech2 45225 | Choice of an upper bound f... |
| ssfiunibd 45226 | A finite union of bounded ... |
| fzdifsuc2 45227 | Remove a successor from th... |
| fzsscn 45228 | A finite sequence of integ... |
| divcan8d 45229 | A cancellation law for div... |
| dmmcand 45230 | Cancellation law for divis... |
| fzssre 45231 | A finite sequence of integ... |
| bccld 45232 | A binomial coefficient, in... |
| leadd12dd 45233 | Addition to both sides of ... |
| fzssnn0 45234 | A finite set of sequential... |
| xreqle 45235 | Equality implies 'less tha... |
| xaddlidd 45236 | ` 0 ` is a left identity f... |
| xadd0ge 45237 | A number is less than or e... |
| elfzolem1 45238 | A member in a half-open in... |
| xrgtned 45239 | 'Greater than' implies not... |
| xrleneltd 45240 | 'Less than or equal to' an... |
| xaddcomd 45241 | The extended real addition... |
| supxrre3 45242 | The supremum of a nonempty... |
| uzfissfz 45243 | For any finite subset of t... |
| xleadd2d 45244 | Addition of extended reals... |
| suprltrp 45245 | The supremum of a nonempty... |
| xleadd1d 45246 | Addition of extended reals... |
| xreqled 45247 | Equality implies 'less tha... |
| xrgepnfd 45248 | An extended real greater t... |
| xrge0nemnfd 45249 | A nonnegative extended rea... |
| supxrgere 45250 | If a real number can be ap... |
| iuneqfzuzlem 45251 | Lemma for ~ iuneqfzuz : he... |
| iuneqfzuz 45252 | If two unions indexed by u... |
| xle2addd 45253 | Adding both side of two in... |
| supxrgelem 45254 | If an extended real number... |
| supxrge 45255 | If an extended real number... |
| suplesup 45256 | If any element of ` A ` ca... |
| infxrglb 45257 | The infimum of a set of ex... |
| xadd0ge2 45258 | A number is less than or e... |
| nepnfltpnf 45259 | An extended real that is n... |
| ltadd12dd 45260 | Addition to both sides of ... |
| nemnftgtmnft 45261 | An extended real that is n... |
| xrgtso 45262 | 'Greater than' is a strict... |
| rpex 45263 | The positive reals form a ... |
| xrge0ge0 45264 | A nonnegative extended rea... |
| xrssre 45265 | A subset of extended reals... |
| ssuzfz 45266 | A finite subset of the upp... |
| absfun 45267 | The absolute value is a fu... |
| infrpge 45268 | The infimum of a nonempty,... |
| xrlexaddrp 45269 | If an extended real number... |
| supsubc 45270 | The supremum function dist... |
| xralrple2 45271 | Show that ` A ` is less th... |
| nnuzdisj 45272 | The first ` N ` elements o... |
| ltdivgt1 45273 | Divsion by a number greate... |
| xrltned 45274 | 'Less than' implies not eq... |
| nnsplit 45275 | Express the set of positiv... |
| divdiv3d 45276 | Division into a fraction. ... |
| abslt2sqd 45277 | Comparison of the square o... |
| qenom 45278 | The set of rational number... |
| qct 45279 | The set of rational number... |
| xrltnled 45280 | 'Less than' in terms of 'l... |
| lenlteq 45281 | 'less than or equal to' bu... |
| xrred 45282 | An extended real that is n... |
| rr2sscn2 45283 | The cartesian square of ` ... |
| infxr 45284 | The infimum of a set of ex... |
| infxrunb2 45285 | The infimum of an unbounde... |
| infxrbnd2 45286 | The infimum of a bounded-b... |
| infleinflem1 45287 | Lemma for ~ infleinf , cas... |
| infleinflem2 45288 | Lemma for ~ infleinf , whe... |
| infleinf 45289 | If any element of ` B ` ca... |
| xralrple4 45290 | Show that ` A ` is less th... |
| xralrple3 45291 | Show that ` A ` is less th... |
| eluzelzd 45292 | A member of an upper set o... |
| suplesup2 45293 | If any element of ` A ` is... |
| recnnltrp 45294 | ` N ` is a natural number ... |
| nnn0 45295 | The set of positive intege... |
| fzct 45296 | A finite set of sequential... |
| rpgtrecnn 45297 | Any positive real number i... |
| fzossuz 45298 | A half-open integer interv... |
| infxrrefi 45299 | The real and extended real... |
| xrralrecnnle 45300 | Show that ` A ` is less th... |
| fzoct 45301 | A finite set of sequential... |
| frexr 45302 | A function taking real val... |
| nnrecrp 45303 | The reciprocal of a positi... |
| reclt0d 45304 | The reciprocal of a negati... |
| lt0neg1dd 45305 | If a number is negative, i... |
| infxrcld 45306 | The infimum of an arbitrar... |
| xrralrecnnge 45307 | Show that ` A ` is less th... |
| reclt0 45308 | The reciprocal of a negati... |
| ltmulneg 45309 | Multiplying by a negative ... |
| allbutfi 45310 | For all but finitely many.... |
| ltdiv23neg 45311 | Swap denominator with othe... |
| xreqnltd 45312 | A consequence of trichotom... |
| mnfnre2 45313 | Minus infinity is not a re... |
| zssxr 45314 | The integers are a subset ... |
| fisupclrnmpt 45315 | A nonempty finite indexed ... |
| supxrunb3 45316 | The supremum of an unbound... |
| elfzod 45317 | Membership in a half-open ... |
| fimaxre4 45318 | A nonempty finite set of r... |
| ren0 45319 | The set of reals is nonemp... |
| eluzelz2 45320 | A member of an upper set o... |
| resabs2d 45321 | Absorption law for restric... |
| uzid2 45322 | Membership of the least me... |
| supxrleubrnmpt 45323 | The supremum of a nonempty... |
| uzssre2 45324 | An upper set of integers i... |
| uzssd 45325 | Subset relationship for tw... |
| eluzd 45326 | Membership in an upper set... |
| infxrlbrnmpt2 45327 | A member of a nonempty ind... |
| xrre4 45328 | An extended real is real i... |
| uz0 45329 | The upper integers functio... |
| eluzelz2d 45330 | A member of an upper set o... |
| infleinf2 45331 | If any element in ` B ` is... |
| unb2ltle 45332 | "Unbounded below" expresse... |
| uzidd2 45333 | Membership of the least me... |
| uzssd2 45334 | Subset relationship for tw... |
| rexabslelem 45335 | An indexed set of absolute... |
| rexabsle 45336 | An indexed set of absolute... |
| allbutfiinf 45337 | Given a "for all but finit... |
| supxrrernmpt 45338 | The real and extended real... |
| suprleubrnmpt 45339 | The supremum of a nonempty... |
| infrnmptle 45340 | An indexed infimum of exte... |
| infxrunb3 45341 | The infimum of an unbounde... |
| uzn0d 45342 | The upper integers are all... |
| uzssd3 45343 | Subset relationship for tw... |
| rexabsle2 45344 | An indexed set of absolute... |
| infxrunb3rnmpt 45345 | The infimum of an unbounde... |
| supxrre3rnmpt 45346 | The indexed supremum of a ... |
| uzublem 45347 | A set of reals, indexed by... |
| uzub 45348 | A set of reals, indexed by... |
| ssrexr 45349 | A subset of the reals is a... |
| supxrmnf2 45350 | Removing minus infinity fr... |
| supxrcli 45351 | The supremum of an arbitra... |
| uzid3 45352 | Membership of the least me... |
| infxrlesupxr 45353 | The supremum of a nonempty... |
| xnegeqd 45354 | Equality of two extended n... |
| xnegrecl 45355 | The extended real negative... |
| xnegnegi 45356 | Extended real version of ~... |
| xnegeqi 45357 | Equality of two extended n... |
| nfxnegd 45358 | Deduction version of ~ nfx... |
| xnegnegd 45359 | Extended real version of ~... |
| uzred 45360 | An upper integer is a real... |
| xnegcli 45361 | Closure of extended real n... |
| supminfrnmpt 45362 | The indexed supremum of a ... |
| infxrpnf 45363 | Adding plus infinity to a ... |
| infxrrnmptcl 45364 | The infimum of an arbitrar... |
| leneg2d 45365 | Negative of one side of 'l... |
| supxrltinfxr 45366 | The supremum of the empty ... |
| max1d 45367 | A number is less than or e... |
| supxrleubrnmptf 45368 | The supremum of a nonempty... |
| nleltd 45369 | 'Not less than or equal to... |
| zxrd 45370 | An integer is an extended ... |
| infxrgelbrnmpt 45371 | The infimum of an indexed ... |
| rphalfltd 45372 | Half of a positive real is... |
| uzssz2 45373 | An upper set of integers i... |
| leneg3d 45374 | Negative of one side of 'l... |
| max2d 45375 | A number is less than or e... |
| uzn0bi 45376 | The upper integers functio... |
| xnegrecl2 45377 | If the extended real negat... |
| nfxneg 45378 | Bound-variable hypothesis ... |
| uzxrd 45379 | An upper integer is an ext... |
| infxrpnf2 45380 | Removing plus infinity fro... |
| supminfxr 45381 | The extended real suprema ... |
| infrpgernmpt 45382 | The infimum of a nonempty,... |
| xnegre 45383 | An extended real is real i... |
| xnegrecl2d 45384 | If the extended real negat... |
| uzxr 45385 | An upper integer is an ext... |
| supminfxr2 45386 | The extended real suprema ... |
| xnegred 45387 | An extended real is real i... |
| supminfxrrnmpt 45388 | The indexed supremum of a ... |
| min1d 45389 | The minimum of two numbers... |
| min2d 45390 | The minimum of two numbers... |
| pnfged 45391 | Plus infinity is an upper ... |
| xrnpnfmnf 45392 | An extended real that is n... |
| uzsscn 45393 | An upper set of integers i... |
| absimnre 45394 | The absolute value of the ... |
| uzsscn2 45395 | An upper set of integers i... |
| xrtgcntopre 45396 | The standard topologies on... |
| absimlere 45397 | The absolute value of the ... |
| rpssxr 45398 | The positive reals are a s... |
| monoordxrv 45399 | Ordering relation for a mo... |
| monoordxr 45400 | Ordering relation for a mo... |
| monoord2xrv 45401 | Ordering relation for a mo... |
| monoord2xr 45402 | Ordering relation for a mo... |
| xrpnf 45403 | An extended real is plus i... |
| xlenegcon1 45404 | Extended real version of ~... |
| xlenegcon2 45405 | Extended real version of ~... |
| pimxrneun 45406 | The preimage of a set of e... |
| caucvgbf 45407 | A function is convergent i... |
| cvgcau 45408 | A convergent function is C... |
| cvgcaule 45409 | A convergent function is C... |
| rexanuz2nf 45410 | A simple counterexample re... |
| gtnelioc 45411 | A real number larger than ... |
| ioossioc 45412 | An open interval is a subs... |
| ioondisj2 45413 | A condition for two open i... |
| ioondisj1 45414 | A condition for two open i... |
| ioogtlb 45415 | An element of a closed int... |
| evthiccabs 45416 | Extreme Value Theorem on y... |
| ltnelicc 45417 | A real number smaller than... |
| eliood 45418 | Membership in an open real... |
| iooabslt 45419 | An upper bound for the dis... |
| gtnelicc 45420 | A real number greater than... |
| iooinlbub 45421 | An open interval has empty... |
| iocgtlb 45422 | An element of a left-open ... |
| iocleub 45423 | An element of a left-open ... |
| eliccd 45424 | Membership in a closed rea... |
| eliccre 45425 | A member of a closed inter... |
| eliooshift 45426 | Element of an open interva... |
| eliocd 45427 | Membership in a left-open ... |
| icoltub 45428 | An element of a left-close... |
| eliocre 45429 | A member of a left-open ri... |
| iooltub 45430 | An element of an open inte... |
| ioontr 45431 | The interior of an interva... |
| snunioo1 45432 | The closure of one end of ... |
| lbioc 45433 | A left-open right-closed i... |
| ioomidp 45434 | The midpoint is an element... |
| iccdifioo 45435 | If the open inverval is re... |
| iccdifprioo 45436 | An open interval is the cl... |
| ioossioobi 45437 | Biconditional form of ~ io... |
| iccshift 45438 | A closed interval shifted ... |
| iccsuble 45439 | An upper bound to the dist... |
| iocopn 45440 | A left-open right-closed i... |
| eliccelioc 45441 | Membership in a closed int... |
| iooshift 45442 | An open interval shifted b... |
| iccintsng 45443 | Intersection of two adiace... |
| icoiccdif 45444 | Left-closed right-open int... |
| icoopn 45445 | A left-closed right-open i... |
| icoub 45446 | A left-closed, right-open ... |
| eliccxrd 45447 | Membership in a closed rea... |
| pnfel0pnf 45448 | ` +oo ` is a nonnegative e... |
| eliccnelico 45449 | An element of a closed int... |
| eliccelicod 45450 | A member of a closed inter... |
| ge0xrre 45451 | A nonnegative extended rea... |
| ge0lere 45452 | A nonnegative extended Rea... |
| elicores 45453 | Membership in a left-close... |
| inficc 45454 | The infimum of a nonempty ... |
| qinioo 45455 | The rational numbers are d... |
| lenelioc 45456 | A real number smaller than... |
| ioonct 45457 | A nonempty open interval i... |
| xrgtnelicc 45458 | A real number greater than... |
| iccdificc 45459 | The difference of two clos... |
| iocnct 45460 | A nonempty left-open, righ... |
| iccnct 45461 | A closed interval, with mo... |
| iooiinicc 45462 | A closed interval expresse... |
| iccgelbd 45463 | An element of a closed int... |
| iooltubd 45464 | An element of an open inte... |
| icoltubd 45465 | An element of a left-close... |
| qelioo 45466 | The rational numbers are d... |
| tgqioo2 45467 | Every open set of reals is... |
| iccleubd 45468 | An element of a closed int... |
| elioored 45469 | A member of an open interv... |
| ioogtlbd 45470 | An element of a closed int... |
| ioofun 45471 | ` (,) ` is a function. (C... |
| icomnfinre 45472 | A left-closed, right-open,... |
| sqrlearg 45473 | The square compared with i... |
| ressiocsup 45474 | If the supremum belongs to... |
| ressioosup 45475 | If the supremum does not b... |
| iooiinioc 45476 | A left-open, right-closed ... |
| ressiooinf 45477 | If the infimum does not be... |
| icogelbd 45478 | An element of a left-close... |
| iocleubd 45479 | An element of a left-open ... |
| uzinico 45480 | An upper interval of integ... |
| preimaiocmnf 45481 | Preimage of a right-closed... |
| uzinico2 45482 | An upper interval of integ... |
| uzinico3 45483 | An upper interval of integ... |
| icossico2 45484 | Condition for a closed-bel... |
| dmico 45485 | The domain of the closed-b... |
| ndmico 45486 | The closed-below, open-abo... |
| uzubioo 45487 | The upper integers are unb... |
| uzubico 45488 | The upper integers are unb... |
| uzubioo2 45489 | The upper integers are unb... |
| uzubico2 45490 | The upper integers are unb... |
| iocgtlbd 45491 | An element of a left-open ... |
| xrtgioo2 45492 | The topology on the extend... |
| tgioo4 45493 | The standard topology on t... |
| fsummulc1f 45494 | Closure of a finite sum of... |
| fsumnncl 45495 | Closure of a nonempty, fin... |
| fsumge0cl 45496 | The finite sum of nonnegat... |
| fsumf1of 45497 | Re-index a finite sum usin... |
| fsumiunss 45498 | Sum over a disjoint indexe... |
| fsumreclf 45499 | Closure of a finite sum of... |
| fsumlessf 45500 | A shorter sum of nonnegati... |
| fsumsupp0 45501 | Finite sum of function val... |
| fsumsermpt 45502 | A finite sum expressed in ... |
| fmul01 45503 | Multiplying a finite numbe... |
| fmulcl 45504 | If ' Y ' is closed under t... |
| fmuldfeqlem1 45505 | induction step for the pro... |
| fmuldfeq 45506 | X and Z are two equivalent... |
| fmul01lt1lem1 45507 | Given a finite multiplicat... |
| fmul01lt1lem2 45508 | Given a finite multiplicat... |
| fmul01lt1 45509 | Given a finite multiplicat... |
| cncfmptss 45510 | A continuous complex funct... |
| rrpsscn 45511 | The positive reals are a s... |
| mulc1cncfg 45512 | A version of ~ mulc1cncf u... |
| infrglb 45513 | The infimum of a nonempty ... |
| expcnfg 45514 | If ` F ` is a complex cont... |
| prodeq2ad 45515 | Equality deduction for pro... |
| fprodsplit1 45516 | Separate out a term in a f... |
| fprodexp 45517 | Positive integer exponenti... |
| fprodabs2 45518 | The absolute value of a fi... |
| fprod0 45519 | A finite product with a ze... |
| mccllem 45520 | * Induction step for ~ mcc... |
| mccl 45521 | A multinomial coefficient,... |
| fprodcnlem 45522 | A finite product of functi... |
| fprodcn 45523 | A finite product of functi... |
| clim1fr1 45524 | A class of sequences of fr... |
| isumneg 45525 | Negation of a converging s... |
| climrec 45526 | Limit of the reciprocal of... |
| climmulf 45527 | A version of ~ climmul usi... |
| climexp 45528 | The limit of natural power... |
| climinf 45529 | A bounded monotonic noninc... |
| climsuselem1 45530 | The subsequence index ` I ... |
| climsuse 45531 | A subsequence ` G ` of a c... |
| climrecf 45532 | A version of ~ climrec usi... |
| climneg 45533 | Complex limit of the negat... |
| climinff 45534 | A version of ~ climinf usi... |
| climdivf 45535 | Limit of the ratio of two ... |
| climreeq 45536 | If ` F ` is a real functio... |
| ellimciota 45537 | An explicit value for the ... |
| climaddf 45538 | A version of ~ climadd usi... |
| mullimc 45539 | Limit of the product of tw... |
| ellimcabssub0 45540 | An equivalent condition fo... |
| limcdm0 45541 | If a function has empty do... |
| islptre 45542 | An equivalence condition f... |
| limccog 45543 | Limit of the composition o... |
| limciccioolb 45544 | The limit of a function at... |
| climf 45545 | Express the predicate: Th... |
| mullimcf 45546 | Limit of the multiplicatio... |
| constlimc 45547 | Limit of constant function... |
| rexlim2d 45548 | Inference removing two res... |
| idlimc 45549 | Limit of the identity func... |
| divcnvg 45550 | The sequence of reciprocal... |
| limcperiod 45551 | If ` F ` is a periodic fun... |
| limcrecl 45552 | If ` F ` is a real-valued ... |
| sumnnodd 45553 | A series indexed by ` NN `... |
| lptioo2 45554 | The upper bound of an open... |
| lptioo1 45555 | The lower bound of an open... |
| elprn1 45556 | A member of an unordered p... |
| elprn2 45557 | A member of an unordered p... |
| limcmptdm 45558 | The domain of a maps-to fu... |
| clim2f 45559 | Express the predicate: Th... |
| limcicciooub 45560 | The limit of a function at... |
| ltmod 45561 | A sufficient condition for... |
| islpcn 45562 | A characterization for a l... |
| lptre2pt 45563 | If a set in the real line ... |
| limsupre 45564 | If a sequence is bounded, ... |
| limcresiooub 45565 | The left limit doesn't cha... |
| limcresioolb 45566 | The right limit doesn't ch... |
| limcleqr 45567 | If the left and the right ... |
| lptioo2cn 45568 | The upper bound of an open... |
| lptioo1cn 45569 | The lower bound of an open... |
| neglimc 45570 | Limit of the negative func... |
| addlimc 45571 | Sum of two limits. (Contr... |
| 0ellimcdiv 45572 | If the numerator converges... |
| clim2cf 45573 | Express the predicate ` F ... |
| limclner 45574 | For a limit point, both fr... |
| sublimc 45575 | Subtraction of two limits.... |
| reclimc 45576 | Limit of the reciprocal of... |
| clim0cf 45577 | Express the predicate ` F ... |
| limclr 45578 | For a limit point, both fr... |
| divlimc 45579 | Limit of the quotient of t... |
| expfac 45580 | Factorial grows faster tha... |
| climconstmpt 45581 | A constant sequence conver... |
| climresmpt 45582 | A function restricted to u... |
| climsubmpt 45583 | Limit of the difference of... |
| climsubc2mpt 45584 | Limit of the difference of... |
| climsubc1mpt 45585 | Limit of the difference of... |
| fnlimfv 45586 | The value of the limit fun... |
| climreclf 45587 | The limit of a convergent ... |
| climeldmeq 45588 | Two functions that are eve... |
| climf2 45589 | Express the predicate: Th... |
| fnlimcnv 45590 | The sequence of function v... |
| climeldmeqmpt 45591 | Two functions that are eve... |
| climfveq 45592 | Two functions that are eve... |
| clim2f2 45593 | Express the predicate: Th... |
| climfveqmpt 45594 | Two functions that are eve... |
| climd 45595 | Express the predicate: Th... |
| clim2d 45596 | The limit of complex numbe... |
| fnlimfvre 45597 | The limit function of real... |
| allbutfifvre 45598 | Given a sequence of real-v... |
| climleltrp 45599 | The limit of complex numbe... |
| fnlimfvre2 45600 | The limit function of real... |
| fnlimf 45601 | The limit function of real... |
| fnlimabslt 45602 | A sequence of function val... |
| climfveqf 45603 | Two functions that are eve... |
| climmptf 45604 | Exhibit a function ` G ` w... |
| climfveqmpt3 45605 | Two functions that are eve... |
| climeldmeqf 45606 | Two functions that are eve... |
| climreclmpt 45607 | The limit of B convergent ... |
| limsupref 45608 | If a sequence is bounded, ... |
| limsupbnd1f 45609 | If a sequence is eventuall... |
| climbddf 45610 | A converging sequence of c... |
| climeqf 45611 | Two functions that are eve... |
| climeldmeqmpt3 45612 | Two functions that are eve... |
| limsupcld 45613 | Closure of the superior li... |
| climfv 45614 | The limit of a convergent ... |
| limsupval3 45615 | The superior limit of an i... |
| climfveqmpt2 45616 | Two functions that are eve... |
| limsup0 45617 | The superior limit of the ... |
| climeldmeqmpt2 45618 | Two functions that are eve... |
| limsupresre 45619 | The supremum limit of a fu... |
| climeqmpt 45620 | Two functions that are eve... |
| climfvd 45621 | The limit of a convergent ... |
| limsuplesup 45622 | An upper bound for the sup... |
| limsupresico 45623 | The superior limit doesn't... |
| limsuppnfdlem 45624 | If the restriction of a fu... |
| limsuppnfd 45625 | If the restriction of a fu... |
| limsupresuz 45626 | If the real part of the do... |
| limsupub 45627 | If the limsup is not ` +oo... |
| limsupres 45628 | The superior limit of a re... |
| climinf2lem 45629 | A convergent, nonincreasin... |
| climinf2 45630 | A convergent, nonincreasin... |
| limsupvaluz 45631 | The superior limit, when t... |
| limsupresuz2 45632 | If the domain of a functio... |
| limsuppnflem 45633 | If the restriction of a fu... |
| limsuppnf 45634 | If the restriction of a fu... |
| limsupubuzlem 45635 | If the limsup is not ` +oo... |
| limsupubuz 45636 | For a real-valued function... |
| climinf2mpt 45637 | A bounded below, monotonic... |
| climinfmpt 45638 | A bounded below, monotonic... |
| climinf3 45639 | A convergent, nonincreasin... |
| limsupvaluzmpt 45640 | The superior limit, when t... |
| limsupequzmpt2 45641 | Two functions that are eve... |
| limsupubuzmpt 45642 | If the limsup is not ` +oo... |
| limsupmnflem 45643 | The superior limit of a fu... |
| limsupmnf 45644 | The superior limit of a fu... |
| limsupequzlem 45645 | Two functions that are eve... |
| limsupequz 45646 | Two functions that are eve... |
| limsupre2lem 45647 | Given a function on the ex... |
| limsupre2 45648 | Given a function on the ex... |
| limsupmnfuzlem 45649 | The superior limit of a fu... |
| limsupmnfuz 45650 | The superior limit of a fu... |
| limsupequzmptlem 45651 | Two functions that are eve... |
| limsupequzmpt 45652 | Two functions that are eve... |
| limsupre2mpt 45653 | Given a function on the ex... |
| limsupequzmptf 45654 | Two functions that are eve... |
| limsupre3lem 45655 | Given a function on the ex... |
| limsupre3 45656 | Given a function on the ex... |
| limsupre3mpt 45657 | Given a function on the ex... |
| limsupre3uzlem 45658 | Given a function on the ex... |
| limsupre3uz 45659 | Given a function on the ex... |
| limsupreuz 45660 | Given a function on the re... |
| limsupvaluz2 45661 | The superior limit, when t... |
| limsupreuzmpt 45662 | Given a function on the re... |
| supcnvlimsup 45663 | If a function on a set of ... |
| supcnvlimsupmpt 45664 | If a function on a set of ... |
| 0cnv 45665 | If ` (/) ` is a complex nu... |
| climuzlem 45666 | Express the predicate: Th... |
| climuz 45667 | Express the predicate: Th... |
| lmbr3v 45668 | Express the binary relatio... |
| climisp 45669 | If a sequence converges to... |
| lmbr3 45670 | Express the binary relatio... |
| climrescn 45671 | A sequence converging w.r.... |
| climxrrelem 45672 | If a sequence ranging over... |
| climxrre 45673 | If a sequence ranging over... |
| limsuplt2 45676 | The defining property of t... |
| liminfgord 45677 | Ordering property of the i... |
| limsupvald 45678 | The superior limit of a se... |
| limsupresicompt 45679 | The superior limit doesn't... |
| limsupcli 45680 | Closure of the superior li... |
| liminfgf 45681 | Closure of the inferior li... |
| liminfval 45682 | The inferior limit of a se... |
| climlimsup 45683 | A sequence of real numbers... |
| limsupge 45684 | The defining property of t... |
| liminfgval 45685 | Value of the inferior limi... |
| liminfcl 45686 | Closure of the inferior li... |
| liminfvald 45687 | The inferior limit of a se... |
| liminfval5 45688 | The inferior limit of an i... |
| limsupresxr 45689 | The superior limit of a fu... |
| liminfresxr 45690 | The inferior limit of a fu... |
| liminfval2 45691 | The superior limit, relati... |
| climlimsupcex 45692 | Counterexample for ~ climl... |
| liminfcld 45693 | Closure of the inferior li... |
| liminfresico 45694 | The inferior limit doesn't... |
| limsup10exlem 45695 | The range of the given fun... |
| limsup10ex 45696 | The superior limit of a fu... |
| liminf10ex 45697 | The inferior limit of a fu... |
| liminflelimsuplem 45698 | The superior limit is grea... |
| liminflelimsup 45699 | The superior limit is grea... |
| limsupgtlem 45700 | For any positive real, the... |
| limsupgt 45701 | Given a sequence of real n... |
| liminfresre 45702 | The inferior limit of a fu... |
| liminfresicompt 45703 | The inferior limit doesn't... |
| liminfltlimsupex 45704 | An example where the ` lim... |
| liminfgelimsup 45705 | The inferior limit is grea... |
| liminfvalxr 45706 | Alternate definition of ` ... |
| liminfresuz 45707 | If the real part of the do... |
| liminflelimsupuz 45708 | The superior limit is grea... |
| liminfvalxrmpt 45709 | Alternate definition of ` ... |
| liminfresuz2 45710 | If the domain of a functio... |
| liminfgelimsupuz 45711 | The inferior limit is grea... |
| liminfval4 45712 | Alternate definition of ` ... |
| liminfval3 45713 | Alternate definition of ` ... |
| liminfequzmpt2 45714 | Two functions that are eve... |
| liminfvaluz 45715 | Alternate definition of ` ... |
| liminf0 45716 | The inferior limit of the ... |
| limsupval4 45717 | Alternate definition of ` ... |
| liminfvaluz2 45718 | Alternate definition of ` ... |
| liminfvaluz3 45719 | Alternate definition of ` ... |
| liminflelimsupcex 45720 | A counterexample for ~ lim... |
| limsupvaluz3 45721 | Alternate definition of ` ... |
| liminfvaluz4 45722 | Alternate definition of ` ... |
| limsupvaluz4 45723 | Alternate definition of ` ... |
| climliminflimsupd 45724 | If a sequence of real numb... |
| liminfreuzlem 45725 | Given a function on the re... |
| liminfreuz 45726 | Given a function on the re... |
| liminfltlem 45727 | Given a sequence of real n... |
| liminflt 45728 | Given a sequence of real n... |
| climliminf 45729 | A sequence of real numbers... |
| liminflimsupclim 45730 | A sequence of real numbers... |
| climliminflimsup 45731 | A sequence of real numbers... |
| climliminflimsup2 45732 | A sequence of real numbers... |
| climliminflimsup3 45733 | A sequence of real numbers... |
| climliminflimsup4 45734 | A sequence of real numbers... |
| limsupub2 45735 | A extended real valued fun... |
| limsupubuz2 45736 | A sequence with values in ... |
| xlimpnfxnegmnf 45737 | A sequence converges to ` ... |
| liminflbuz2 45738 | A sequence with values in ... |
| liminfpnfuz 45739 | The inferior limit of a fu... |
| liminflimsupxrre 45740 | A sequence with values in ... |
| xlimrel 45743 | The limit on extended real... |
| xlimres 45744 | A function converges iff i... |
| xlimcl 45745 | The limit of a sequence of... |
| rexlimddv2 45746 | Restricted existential eli... |
| xlimclim 45747 | Given a sequence of reals,... |
| xlimconst 45748 | A constant sequence conver... |
| climxlim 45749 | A converging sequence in t... |
| xlimbr 45750 | Express the binary relatio... |
| fuzxrpmcn 45751 | A function mapping from an... |
| cnrefiisplem 45752 | Lemma for ~ cnrefiisp (som... |
| cnrefiisp 45753 | A non-real, complex number... |
| xlimxrre 45754 | If a sequence ranging over... |
| xlimmnfvlem1 45755 | Lemma for ~ xlimmnfv : the... |
| xlimmnfvlem2 45756 | Lemma for ~ xlimmnf : the ... |
| xlimmnfv 45757 | A function converges to mi... |
| xlimconst2 45758 | A sequence that eventually... |
| xlimpnfvlem1 45759 | Lemma for ~ xlimpnfv : the... |
| xlimpnfvlem2 45760 | Lemma for ~ xlimpnfv : the... |
| xlimpnfv 45761 | A function converges to pl... |
| xlimclim2lem 45762 | Lemma for ~ xlimclim2 . H... |
| xlimclim2 45763 | Given a sequence of extend... |
| xlimmnf 45764 | A function converges to mi... |
| xlimpnf 45765 | A function converges to pl... |
| xlimmnfmpt 45766 | A function converges to pl... |
| xlimpnfmpt 45767 | A function converges to pl... |
| climxlim2lem 45768 | In this lemma for ~ climxl... |
| climxlim2 45769 | A sequence of extended rea... |
| dfxlim2v 45770 | An alternative definition ... |
| dfxlim2 45771 | An alternative definition ... |
| climresd 45772 | A function restricted to u... |
| climresdm 45773 | A real function converges ... |
| dmclimxlim 45774 | A real valued sequence tha... |
| xlimmnflimsup2 45775 | A sequence of extended rea... |
| xlimuni 45776 | An infinite sequence conve... |
| xlimclimdm 45777 | A sequence of extended rea... |
| xlimfun 45778 | The convergence relation o... |
| xlimmnflimsup 45779 | If a sequence of extended ... |
| xlimdm 45780 | Two ways to express that a... |
| xlimpnfxnegmnf2 45781 | A sequence converges to ` ... |
| xlimresdm 45782 | A function converges in th... |
| xlimpnfliminf 45783 | If a sequence of extended ... |
| xlimpnfliminf2 45784 | A sequence of extended rea... |
| xlimliminflimsup 45785 | A sequence of extended rea... |
| xlimlimsupleliminf 45786 | A sequence of extended rea... |
| coseq0 45787 | A complex number whose cos... |
| sinmulcos 45788 | Multiplication formula for... |
| coskpi2 45789 | The cosine of an integer m... |
| cosnegpi 45790 | The cosine of negative ` _... |
| sinaover2ne0 45791 | If ` A ` in ` ( 0 , 2 _pi ... |
| cosknegpi 45792 | The cosine of an integer m... |
| mulcncff 45793 | The multiplication of two ... |
| cncfmptssg 45794 | A continuous complex funct... |
| constcncfg 45795 | A constant function is a c... |
| idcncfg 45796 | The identity function is a... |
| cncfshift 45797 | A periodic continuous func... |
| resincncf 45798 | ` sin ` restricted to real... |
| addccncf2 45799 | Adding a constant is a con... |
| 0cnf 45800 | The empty set is a continu... |
| fsumcncf 45801 | The finite sum of continuo... |
| cncfperiod 45802 | A periodic continuous func... |
| subcncff 45803 | The subtraction of two con... |
| negcncfg 45804 | The opposite of a continuo... |
| cnfdmsn 45805 | A function with a singleto... |
| cncfcompt 45806 | Composition of continuous ... |
| addcncff 45807 | The sum of two continuous ... |
| ioccncflimc 45808 | Limit at the upper bound o... |
| cncfuni 45809 | A complex function on a su... |
| icccncfext 45810 | A continuous function on a... |
| cncficcgt0 45811 | A the absolute value of a ... |
| icocncflimc 45812 | Limit at the lower bound, ... |
| cncfdmsn 45813 | A complex function with a ... |
| divcncff 45814 | The quotient of two contin... |
| cncfshiftioo 45815 | A periodic continuous func... |
| cncfiooicclem1 45816 | A continuous function ` F ... |
| cncfiooicc 45817 | A continuous function ` F ... |
| cncfiooiccre 45818 | A continuous function ` F ... |
| cncfioobdlem 45819 | ` G ` actually extends ` F... |
| cncfioobd 45820 | A continuous function ` F ... |
| jumpncnp 45821 | Jump discontinuity or disc... |
| cxpcncf2 45822 | The complex power function... |
| fprodcncf 45823 | The finite product of cont... |
| add1cncf 45824 | Addition to a constant is ... |
| add2cncf 45825 | Addition to a constant is ... |
| sub1cncfd 45826 | Subtracting a constant is ... |
| sub2cncfd 45827 | Subtraction from a constan... |
| fprodsub2cncf 45828 | ` F ` is continuous. (Con... |
| fprodadd2cncf 45829 | ` F ` is continuous. (Con... |
| fprodsubrecnncnvlem 45830 | The sequence ` S ` of fini... |
| fprodsubrecnncnv 45831 | The sequence ` S ` of fini... |
| fprodaddrecnncnvlem 45832 | The sequence ` S ` of fini... |
| fprodaddrecnncnv 45833 | The sequence ` S ` of fini... |
| dvsinexp 45834 | The derivative of sin^N . ... |
| dvcosre 45835 | The real derivative of the... |
| dvsinax 45836 | Derivative exercise: the d... |
| dvsubf 45837 | The subtraction rule for e... |
| dvmptconst 45838 | Function-builder for deriv... |
| dvcnre 45839 | From complex differentiati... |
| dvmptidg 45840 | Function-builder for deriv... |
| dvresntr 45841 | Function-builder for deriv... |
| fperdvper 45842 | The derivative of a period... |
| dvasinbx 45843 | Derivative exercise: the d... |
| dvresioo 45844 | Restriction of a derivativ... |
| dvdivf 45845 | The quotient rule for ever... |
| dvdivbd 45846 | A sufficient condition for... |
| dvsubcncf 45847 | A sufficient condition for... |
| dvmulcncf 45848 | A sufficient condition for... |
| dvcosax 45849 | Derivative exercise: the d... |
| dvdivcncf 45850 | A sufficient condition for... |
| dvbdfbdioolem1 45851 | Given a function with boun... |
| dvbdfbdioolem2 45852 | A function on an open inte... |
| dvbdfbdioo 45853 | A function on an open inte... |
| ioodvbdlimc1lem1 45854 | If ` F ` has bounded deriv... |
| ioodvbdlimc1lem2 45855 | Limit at the lower bound o... |
| ioodvbdlimc1 45856 | A real function with bound... |
| ioodvbdlimc2lem 45857 | Limit at the upper bound o... |
| ioodvbdlimc2 45858 | A real function with bound... |
| dvdmsscn 45859 | ` X ` is a subset of ` CC ... |
| dvmptmulf 45860 | Function-builder for deriv... |
| dvnmptdivc 45861 | Function-builder for itera... |
| dvdsn1add 45862 | If ` K ` divides ` N ` but... |
| dvxpaek 45863 | Derivative of the polynomi... |
| dvnmptconst 45864 | The ` N ` -th derivative o... |
| dvnxpaek 45865 | The ` n ` -th derivative o... |
| dvnmul 45866 | Function-builder for the `... |
| dvmptfprodlem 45867 | Induction step for ~ dvmpt... |
| dvmptfprod 45868 | Function-builder for deriv... |
| dvnprodlem1 45869 | ` D ` is bijective. (Cont... |
| dvnprodlem2 45870 | Induction step for ~ dvnpr... |
| dvnprodlem3 45871 | The multinomial formula fo... |
| dvnprod 45872 | The multinomial formula fo... |
| itgsin0pilem1 45873 | Calculation of the integra... |
| ibliccsinexp 45874 | sin^n on a closed interval... |
| itgsin0pi 45875 | Calculation of the integra... |
| iblioosinexp 45876 | sin^n on an open integral ... |
| itgsinexplem1 45877 | Integration by parts is ap... |
| itgsinexp 45878 | A recursive formula for th... |
| iblconstmpt 45879 | A constant function is int... |
| itgeq1d 45880 | Equality theorem for an in... |
| mbfres2cn 45881 | Measurability of a piecewi... |
| vol0 45882 | The measure of the empty s... |
| ditgeqiooicc 45883 | A function ` F ` on an ope... |
| volge0 45884 | The volume of a set is alw... |
| cnbdibl 45885 | A continuous bounded funct... |
| snmbl 45886 | A singleton is measurable.... |
| ditgeq3d 45887 | Equality theorem for the d... |
| iblempty 45888 | The empty function is inte... |
| iblsplit 45889 | The union of two integrabl... |
| volsn 45890 | A singleton has 0 Lebesgue... |
| itgvol0 45891 | If the domani is negligibl... |
| itgcoscmulx 45892 | Exercise: the integral of ... |
| iblsplitf 45893 | A version of ~ iblsplit us... |
| ibliooicc 45894 | If a function is integrabl... |
| volioc 45895 | The measure of a left-open... |
| iblspltprt 45896 | If a function is integrabl... |
| itgsincmulx 45897 | Exercise: the integral of ... |
| itgsubsticclem 45898 | lemma for ~ itgsubsticc . ... |
| itgsubsticc 45899 | Integration by u-substitut... |
| itgioocnicc 45900 | The integral of a piecewis... |
| iblcncfioo 45901 | A continuous function ` F ... |
| itgspltprt 45902 | The ` S. ` integral splits... |
| itgiccshift 45903 | The integral of a function... |
| itgperiod 45904 | The integral of a periodic... |
| itgsbtaddcnst 45905 | Integral substitution, add... |
| volico 45906 | The measure of left-closed... |
| sublevolico 45907 | The Lebesgue measure of a ... |
| dmvolss 45908 | Lebesgue measurable sets a... |
| ismbl3 45909 | The predicate " ` A ` is L... |
| volioof 45910 | The function that assigns ... |
| ovolsplit 45911 | The Lebesgue outer measure... |
| fvvolioof 45912 | The function value of the ... |
| volioore 45913 | The measure of an open int... |
| fvvolicof 45914 | The function value of the ... |
| voliooico 45915 | An open interval and a lef... |
| ismbl4 45916 | The predicate " ` A ` is L... |
| volioofmpt 45917 | ` ( ( vol o. (,) ) o. F ) ... |
| volicoff 45918 | ` ( ( vol o. [,) ) o. F ) ... |
| voliooicof 45919 | The Lebesgue measure of op... |
| volicofmpt 45920 | ` ( ( vol o. [,) ) o. F ) ... |
| volicc 45921 | The Lebesgue measure of a ... |
| voliccico 45922 | A closed interval and a le... |
| mbfdmssre 45923 | The domain of a measurable... |
| stoweidlem1 45924 | Lemma for ~ stoweid . Thi... |
| stoweidlem2 45925 | lemma for ~ stoweid : here... |
| stoweidlem3 45926 | Lemma for ~ stoweid : if `... |
| stoweidlem4 45927 | Lemma for ~ stoweid : a cl... |
| stoweidlem5 45928 | There exists a δ as ... |
| stoweidlem6 45929 | Lemma for ~ stoweid : two ... |
| stoweidlem7 45930 | This lemma is used to prov... |
| stoweidlem8 45931 | Lemma for ~ stoweid : two ... |
| stoweidlem9 45932 | Lemma for ~ stoweid : here... |
| stoweidlem10 45933 | Lemma for ~ stoweid . Thi... |
| stoweidlem11 45934 | This lemma is used to prov... |
| stoweidlem12 45935 | Lemma for ~ stoweid . Thi... |
| stoweidlem13 45936 | Lemma for ~ stoweid . Thi... |
| stoweidlem14 45937 | There exists a ` k ` as in... |
| stoweidlem15 45938 | This lemma is used to prov... |
| stoweidlem16 45939 | Lemma for ~ stoweid . The... |
| stoweidlem17 45940 | This lemma proves that the... |
| stoweidlem18 45941 | This theorem proves Lemma ... |
| stoweidlem19 45942 | If a set of real functions... |
| stoweidlem20 45943 | If a set A of real functio... |
| stoweidlem21 45944 | Once the Stone Weierstrass... |
| stoweidlem22 45945 | If a set of real functions... |
| stoweidlem23 45946 | This lemma is used to prov... |
| stoweidlem24 45947 | This lemma proves that for... |
| stoweidlem25 45948 | This lemma proves that for... |
| stoweidlem26 45949 | This lemma is used to prov... |
| stoweidlem27 45950 | This lemma is used to prov... |
| stoweidlem28 45951 | There exists a δ as ... |
| stoweidlem29 45952 | When the hypothesis for th... |
| stoweidlem30 45953 | This lemma is used to prov... |
| stoweidlem31 45954 | This lemma is used to prov... |
| stoweidlem32 45955 | If a set A of real functio... |
| stoweidlem33 45956 | If a set of real functions... |
| stoweidlem34 45957 | This lemma proves that for... |
| stoweidlem35 45958 | This lemma is used to prov... |
| stoweidlem36 45959 | This lemma is used to prov... |
| stoweidlem37 45960 | This lemma is used to prov... |
| stoweidlem38 45961 | This lemma is used to prov... |
| stoweidlem39 45962 | This lemma is used to prov... |
| stoweidlem40 45963 | This lemma proves that q_n... |
| stoweidlem41 45964 | This lemma is used to prov... |
| stoweidlem42 45965 | This lemma is used to prov... |
| stoweidlem43 45966 | This lemma is used to prov... |
| stoweidlem44 45967 | This lemma is used to prov... |
| stoweidlem45 45968 | This lemma proves that, gi... |
| stoweidlem46 45969 | This lemma proves that set... |
| stoweidlem47 45970 | Subtracting a constant fro... |
| stoweidlem48 45971 | This lemma is used to prov... |
| stoweidlem49 45972 | There exists a function q_... |
| stoweidlem50 45973 | This lemma proves that set... |
| stoweidlem51 45974 | There exists a function x ... |
| stoweidlem52 45975 | There exists a neighborhoo... |
| stoweidlem53 45976 | This lemma is used to prov... |
| stoweidlem54 45977 | There exists a function ` ... |
| stoweidlem55 45978 | This lemma proves the exis... |
| stoweidlem56 45979 | This theorem proves Lemma ... |
| stoweidlem57 45980 | There exists a function x ... |
| stoweidlem58 45981 | This theorem proves Lemma ... |
| stoweidlem59 45982 | This lemma proves that the... |
| stoweidlem60 45983 | This lemma proves that the... |
| stoweidlem61 45984 | This lemma proves that the... |
| stoweidlem62 45985 | This theorem proves the St... |
| stoweid 45986 | This theorem proves the St... |
| stowei 45987 | This theorem proves the St... |
| wallispilem1 45988 | ` I ` is monotone: increas... |
| wallispilem2 45989 | A first set of properties ... |
| wallispilem3 45990 | I maps to real values. (C... |
| wallispilem4 45991 | ` F ` maps to explicit exp... |
| wallispilem5 45992 | The sequence ` H ` converg... |
| wallispi 45993 | Wallis' formula for π :... |
| wallispi2lem1 45994 | An intermediate step betwe... |
| wallispi2lem2 45995 | Two expressions are proven... |
| wallispi2 45996 | An alternative version of ... |
| stirlinglem1 45997 | A simple limit of fraction... |
| stirlinglem2 45998 | ` A ` maps to positive rea... |
| stirlinglem3 45999 | Long but simple algebraic ... |
| stirlinglem4 46000 | Algebraic manipulation of ... |
| stirlinglem5 46001 | If ` T ` is between ` 0 ` ... |
| stirlinglem6 46002 | A series that converges to... |
| stirlinglem7 46003 | Algebraic manipulation of ... |
| stirlinglem8 46004 | If ` A ` converges to ` C ... |
| stirlinglem9 46005 | ` ( ( B `` N ) - ( B `` ( ... |
| stirlinglem10 46006 | A bound for any B(N)-B(N +... |
| stirlinglem11 46007 | ` B ` is decreasing. (Con... |
| stirlinglem12 46008 | The sequence ` B ` is boun... |
| stirlinglem13 46009 | ` B ` is decreasing and ha... |
| stirlinglem14 46010 | The sequence ` A ` converg... |
| stirlinglem15 46011 | The Stirling's formula is ... |
| stirling 46012 | Stirling's approximation f... |
| stirlingr 46013 | Stirling's approximation f... |
| dirkerval 46014 | The N_th Dirichlet Kernel.... |
| dirker2re 46015 | The Dirichlet Kernel value... |
| dirkerdenne0 46016 | The Dirichlet Kernel denom... |
| dirkerval2 46017 | The N_th Dirichlet Kernel ... |
| dirkerre 46018 | The Dirichlet Kernel at an... |
| dirkerper 46019 | the Dirichlet Kernel has p... |
| dirkerf 46020 | For any natural number ` N... |
| dirkertrigeqlem1 46021 | Sum of an even number of a... |
| dirkertrigeqlem2 46022 | Trigonomic equality lemma ... |
| dirkertrigeqlem3 46023 | Trigonometric equality lem... |
| dirkertrigeq 46024 | Trigonometric equality for... |
| dirkeritg 46025 | The definite integral of t... |
| dirkercncflem1 46026 | If ` Y ` is a multiple of ... |
| dirkercncflem2 46027 | Lemma used to prove that t... |
| dirkercncflem3 46028 | The Dirichlet Kernel is co... |
| dirkercncflem4 46029 | The Dirichlet Kernel is co... |
| dirkercncf 46030 | For any natural number ` N... |
| fourierdlem1 46031 | A partition interval is a ... |
| fourierdlem2 46032 | Membership in a partition.... |
| fourierdlem3 46033 | Membership in a partition.... |
| fourierdlem4 46034 | ` E ` is a function that m... |
| fourierdlem5 46035 | ` S ` is a function. (Con... |
| fourierdlem6 46036 | ` X ` is in the periodic p... |
| fourierdlem7 46037 | The difference between the... |
| fourierdlem8 46038 | A partition interval is a ... |
| fourierdlem9 46039 | ` H ` is a complex functio... |
| fourierdlem10 46040 | Condition on the bounds of... |
| fourierdlem11 46041 | If there is a partition, t... |
| fourierdlem12 46042 | A point of a partition is ... |
| fourierdlem13 46043 | Value of ` V ` in terms of... |
| fourierdlem14 46044 | Given the partition ` V ` ... |
| fourierdlem15 46045 | The range of the partition... |
| fourierdlem16 46046 | The coefficients of the fo... |
| fourierdlem17 46047 | The defined ` L ` is actua... |
| fourierdlem18 46048 | The function ` S ` is cont... |
| fourierdlem19 46049 | If two elements of ` D ` h... |
| fourierdlem20 46050 | Every interval in the part... |
| fourierdlem21 46051 | The coefficients of the fo... |
| fourierdlem22 46052 | The coefficients of the fo... |
| fourierdlem23 46053 | If ` F ` is continuous and... |
| fourierdlem24 46054 | A sufficient condition for... |
| fourierdlem25 46055 | If ` C ` is not in the ran... |
| fourierdlem26 46056 | Periodic image of a point ... |
| fourierdlem27 46057 | A partition open interval ... |
| fourierdlem28 46058 | Derivative of ` ( F `` ( X... |
| fourierdlem29 46059 | Explicit function value fo... |
| fourierdlem30 46060 | Sum of three small pieces ... |
| fourierdlem31 46061 | If ` A ` is finite and for... |
| fourierdlem32 46062 | Limit of a continuous func... |
| fourierdlem33 46063 | Limit of a continuous func... |
| fourierdlem34 46064 | A partition is one to one.... |
| fourierdlem35 46065 | There is a single point in... |
| fourierdlem36 46066 | ` F ` is an isomorphism. ... |
| fourierdlem37 46067 | ` I ` is a function that m... |
| fourierdlem38 46068 | The function ` F ` is cont... |
| fourierdlem39 46069 | Integration by parts of ... |
| fourierdlem40 46070 | ` H ` is a continuous func... |
| fourierdlem41 46071 | Lemma used to prove that e... |
| fourierdlem42 46072 | The set of points in a mov... |
| fourierdlem43 46073 | ` K ` is a real function. ... |
| fourierdlem44 46074 | A condition for having ` (... |
| fourierdlem46 46075 | The function ` F ` has a l... |
| fourierdlem47 46076 | For ` r ` large enough, th... |
| fourierdlem48 46077 | The given periodic functio... |
| fourierdlem49 46078 | The given periodic functio... |
| fourierdlem50 46079 | Continuity of ` O ` and it... |
| fourierdlem51 46080 | ` X ` is in the periodic p... |
| fourierdlem52 46081 | d16:d17,d18:jca |- ( ph ->... |
| fourierdlem53 46082 | The limit of ` F ( s ) ` a... |
| fourierdlem54 46083 | Given a partition ` Q ` an... |
| fourierdlem55 46084 | ` U ` is a real function. ... |
| fourierdlem56 46085 | Derivative of the ` K ` fu... |
| fourierdlem57 46086 | The derivative of ` O ` . ... |
| fourierdlem58 46087 | The derivative of ` K ` is... |
| fourierdlem59 46088 | The derivative of ` H ` is... |
| fourierdlem60 46089 | Given a differentiable fun... |
| fourierdlem61 46090 | Given a differentiable fun... |
| fourierdlem62 46091 | The function ` K ` is cont... |
| fourierdlem63 46092 | The upper bound of interva... |
| fourierdlem64 46093 | The partition ` V ` is fin... |
| fourierdlem65 46094 | The distance of two adjace... |
| fourierdlem66 46095 | Value of the ` G ` functio... |
| fourierdlem67 46096 | ` G ` is a function. (Con... |
| fourierdlem68 46097 | The derivative of ` O ` is... |
| fourierdlem69 46098 | A piecewise continuous fun... |
| fourierdlem70 46099 | A piecewise continuous fun... |
| fourierdlem71 46100 | A periodic piecewise conti... |
| fourierdlem72 46101 | The derivative of ` O ` is... |
| fourierdlem73 46102 | A version of the Riemann L... |
| fourierdlem74 46103 | Given a piecewise smooth f... |
| fourierdlem75 46104 | Given a piecewise smooth f... |
| fourierdlem76 46105 | Continuity of ` O ` and it... |
| fourierdlem77 46106 | If ` H ` is bounded, then ... |
| fourierdlem78 46107 | ` G ` is continuous when r... |
| fourierdlem79 46108 | ` E ` projects every inter... |
| fourierdlem80 46109 | The derivative of ` O ` is... |
| fourierdlem81 46110 | The integral of a piecewis... |
| fourierdlem82 46111 | Integral by substitution, ... |
| fourierdlem83 46112 | The fourier partial sum fo... |
| fourierdlem84 46113 | If ` F ` is piecewise cont... |
| fourierdlem85 46114 | Limit of the function ` G ... |
| fourierdlem86 46115 | Continuity of ` O ` and it... |
| fourierdlem87 46116 | The integral of ` G ` goes... |
| fourierdlem88 46117 | Given a piecewise continuo... |
| fourierdlem89 46118 | Given a piecewise continuo... |
| fourierdlem90 46119 | Given a piecewise continuo... |
| fourierdlem91 46120 | Given a piecewise continuo... |
| fourierdlem92 46121 | The integral of a piecewis... |
| fourierdlem93 46122 | Integral by substitution (... |
| fourierdlem94 46123 | For a piecewise smooth fun... |
| fourierdlem95 46124 | Algebraic manipulation of ... |
| fourierdlem96 46125 | limit for ` F ` at the low... |
| fourierdlem97 46126 | ` F ` is continuous on the... |
| fourierdlem98 46127 | ` F ` is continuous on the... |
| fourierdlem99 46128 | limit for ` F ` at the upp... |
| fourierdlem100 46129 | A piecewise continuous fun... |
| fourierdlem101 46130 | Integral by substitution f... |
| fourierdlem102 46131 | For a piecewise smooth fun... |
| fourierdlem103 46132 | The half lower part of the... |
| fourierdlem104 46133 | The half upper part of the... |
| fourierdlem105 46134 | A piecewise continuous fun... |
| fourierdlem106 46135 | For a piecewise smooth fun... |
| fourierdlem107 46136 | The integral of a piecewis... |
| fourierdlem108 46137 | The integral of a piecewis... |
| fourierdlem109 46138 | The integral of a piecewis... |
| fourierdlem110 46139 | The integral of a piecewis... |
| fourierdlem111 46140 | The fourier partial sum fo... |
| fourierdlem112 46141 | Here abbreviations (local ... |
| fourierdlem113 46142 | Fourier series convergence... |
| fourierdlem114 46143 | Fourier series convergence... |
| fourierdlem115 46144 | Fourier serier convergence... |
| fourierd 46145 | Fourier series convergence... |
| fourierclimd 46146 | Fourier series convergence... |
| fourierclim 46147 | Fourier series convergence... |
| fourier 46148 | Fourier series convergence... |
| fouriercnp 46149 | If ` F ` is continuous at ... |
| fourier2 46150 | Fourier series convergence... |
| sqwvfoura 46151 | Fourier coefficients for t... |
| sqwvfourb 46152 | Fourier series ` B ` coeff... |
| fourierswlem 46153 | The Fourier series for the... |
| fouriersw 46154 | Fourier series convergence... |
| fouriercn 46155 | If the derivative of ` F `... |
| elaa2lem 46156 | Elementhood in the set of ... |
| elaa2 46157 | Elementhood in the set of ... |
| etransclem1 46158 | ` H ` is a function. (Con... |
| etransclem2 46159 | Derivative of ` G ` . (Co... |
| etransclem3 46160 | The given ` if ` term is a... |
| etransclem4 46161 | ` F ` expressed as a finit... |
| etransclem5 46162 | A change of bound variable... |
| etransclem6 46163 | A change of bound variable... |
| etransclem7 46164 | The given product is an in... |
| etransclem8 46165 | ` F ` is a function. (Con... |
| etransclem9 46166 | If ` K ` divides ` N ` but... |
| etransclem10 46167 | The given ` if ` term is a... |
| etransclem11 46168 | A change of bound variable... |
| etransclem12 46169 | ` C ` applied to ` N ` . ... |
| etransclem13 46170 | ` F ` applied to ` Y ` . ... |
| etransclem14 46171 | Value of the term ` T ` , ... |
| etransclem15 46172 | Value of the term ` T ` , ... |
| etransclem16 46173 | Every element in the range... |
| etransclem17 46174 | The ` N ` -th derivative o... |
| etransclem18 46175 | The given function is inte... |
| etransclem19 46176 | The ` N ` -th derivative o... |
| etransclem20 46177 | ` H ` is smooth. (Contrib... |
| etransclem21 46178 | The ` N ` -th derivative o... |
| etransclem22 46179 | The ` N ` -th derivative o... |
| etransclem23 46180 | This is the claim proof in... |
| etransclem24 46181 | ` P ` divides the I -th de... |
| etransclem25 46182 | ` P ` factorial divides th... |
| etransclem26 46183 | Every term in the sum of t... |
| etransclem27 46184 | The ` N ` -th derivative o... |
| etransclem28 46185 | ` ( P - 1 ) ` factorial di... |
| etransclem29 46186 | The ` N ` -th derivative o... |
| etransclem30 46187 | The ` N ` -th derivative o... |
| etransclem31 46188 | The ` N ` -th derivative o... |
| etransclem32 46189 | This is the proof for the ... |
| etransclem33 46190 | ` F ` is smooth. (Contrib... |
| etransclem34 46191 | The ` N ` -th derivative o... |
| etransclem35 46192 | ` P ` does not divide the ... |
| etransclem36 46193 | The ` N ` -th derivative o... |
| etransclem37 46194 | ` ( P - 1 ) ` factorial di... |
| etransclem38 46195 | ` P ` divides the I -th de... |
| etransclem39 46196 | ` G ` is a function. (Con... |
| etransclem40 46197 | The ` N ` -th derivative o... |
| etransclem41 46198 | ` P ` does not divide the ... |
| etransclem42 46199 | The ` N ` -th derivative o... |
| etransclem43 46200 | ` G ` is a continuous func... |
| etransclem44 46201 | The given finite sum is no... |
| etransclem45 46202 | ` K ` is an integer. (Con... |
| etransclem46 46203 | This is the proof for equa... |
| etransclem47 46204 | ` _e ` is transcendental. ... |
| etransclem48 46205 | ` _e ` is transcendental. ... |
| etransc 46206 | ` _e ` is transcendental. ... |
| rrxtopn 46207 | The topology of the genera... |
| rrxngp 46208 | Generalized Euclidean real... |
| rrxtps 46209 | Generalized Euclidean real... |
| rrxtopnfi 46210 | The topology of the n-dime... |
| rrxtopon 46211 | The topology on generalize... |
| rrxtop 46212 | The topology on generalize... |
| rrndistlt 46213 | Given two points in the sp... |
| rrxtoponfi 46214 | The topology on n-dimensio... |
| rrxunitopnfi 46215 | The base set of the standa... |
| rrxtopn0 46216 | The topology of the zero-d... |
| qndenserrnbllem 46217 | n-dimensional rational num... |
| qndenserrnbl 46218 | n-dimensional rational num... |
| rrxtopn0b 46219 | The topology of the zero-d... |
| qndenserrnopnlem 46220 | n-dimensional rational num... |
| qndenserrnopn 46221 | n-dimensional rational num... |
| qndenserrn 46222 | n-dimensional rational num... |
| rrxsnicc 46223 | A multidimensional singlet... |
| rrnprjdstle 46224 | The distance between two p... |
| rrndsmet 46225 | ` D ` is a metric for the ... |
| rrndsxmet 46226 | ` D ` is an extended metri... |
| ioorrnopnlem 46227 | The a point in an indexed ... |
| ioorrnopn 46228 | The indexed product of ope... |
| ioorrnopnxrlem 46229 | Given a point ` F ` that b... |
| ioorrnopnxr 46230 | The indexed product of ope... |
| issal 46237 | Express the predicate " ` ... |
| pwsal 46238 | The power set of a given s... |
| salunicl 46239 | SAlg sigma-algebra is clos... |
| saluncl 46240 | The union of two sets in a... |
| prsal 46241 | The pair of the empty set ... |
| saldifcl 46242 | The complement of an eleme... |
| 0sal 46243 | The empty set belongs to e... |
| salgenval 46244 | The sigma-algebra generate... |
| saliunclf 46245 | SAlg sigma-algebra is clos... |
| saliuncl 46246 | SAlg sigma-algebra is clos... |
| salincl 46247 | The intersection of two se... |
| saluni 46248 | A set is an element of any... |
| saliinclf 46249 | SAlg sigma-algebra is clos... |
| saliincl 46250 | SAlg sigma-algebra is clos... |
| saldifcl2 46251 | The difference of two elem... |
| intsaluni 46252 | The union of an arbitrary ... |
| intsal 46253 | The arbitrary intersection... |
| salgenn0 46254 | The set used in the defini... |
| salgencl 46255 | ` SalGen ` actually genera... |
| issald 46256 | Sufficient condition to pr... |
| salexct 46257 | An example of nontrivial s... |
| sssalgen 46258 | A set is a subset of the s... |
| salgenss 46259 | The sigma-algebra generate... |
| salgenuni 46260 | The base set of the sigma-... |
| issalgend 46261 | One side of ~ dfsalgen2 . ... |
| salexct2 46262 | An example of a subset tha... |
| unisalgen 46263 | The union of a set belongs... |
| dfsalgen2 46264 | Alternate characterization... |
| salexct3 46265 | An example of a sigma-alge... |
| salgencntex 46266 | This counterexample shows ... |
| salgensscntex 46267 | This counterexample shows ... |
| issalnnd 46268 | Sufficient condition to pr... |
| dmvolsal 46269 | Lebesgue measurable sets f... |
| saldifcld 46270 | The complement of an eleme... |
| saluncld 46271 | The union of two sets in a... |
| salgencld 46272 | ` SalGen ` actually genera... |
| 0sald 46273 | The empty set belongs to e... |
| iooborel 46274 | An open interval is a Bore... |
| salincld 46275 | The intersection of two se... |
| salunid 46276 | A set is an element of any... |
| unisalgen2 46277 | The union of a set belongs... |
| bor1sal 46278 | The Borel sigma-algebra on... |
| iocborel 46279 | A left-open, right-closed ... |
| subsaliuncllem 46280 | A subspace sigma-algebra i... |
| subsaliuncl 46281 | A subspace sigma-algebra i... |
| subsalsal 46282 | A subspace sigma-algebra i... |
| subsaluni 46283 | A set belongs to the subsp... |
| salrestss 46284 | A sigma-algebra restricted... |
| sge0rnre 46287 | When ` sum^ ` is applied t... |
| fge0icoicc 46288 | If ` F ` maps to nonnegati... |
| sge0val 46289 | The value of the sum of no... |
| fge0npnf 46290 | If ` F ` maps to nonnegati... |
| sge0rnn0 46291 | The range used in the defi... |
| sge0vald 46292 | The value of the sum of no... |
| fge0iccico 46293 | A range of nonnegative ext... |
| gsumge0cl 46294 | Closure of group sum, for ... |
| sge0reval 46295 | Value of the sum of nonneg... |
| sge0pnfval 46296 | If a term in the sum of no... |
| fge0iccre 46297 | A range of nonnegative ext... |
| sge0z 46298 | Any nonnegative extended s... |
| sge00 46299 | The sum of nonnegative ext... |
| fsumlesge0 46300 | Every finite subsum of non... |
| sge0revalmpt 46301 | Value of the sum of nonneg... |
| sge0sn 46302 | A sum of a nonnegative ext... |
| sge0tsms 46303 | ` sum^ ` applied to a nonn... |
| sge0cl 46304 | The arbitrary sum of nonne... |
| sge0f1o 46305 | Re-index a nonnegative ext... |
| sge0snmpt 46306 | A sum of a nonnegative ext... |
| sge0ge0 46307 | The sum of nonnegative ext... |
| sge0xrcl 46308 | The arbitrary sum of nonne... |
| sge0repnf 46309 | The of nonnegative extende... |
| sge0fsum 46310 | The arbitrary sum of a fin... |
| sge0rern 46311 | If the sum of nonnegative ... |
| sge0supre 46312 | If the arbitrary sum of no... |
| sge0fsummpt 46313 | The arbitrary sum of a fin... |
| sge0sup 46314 | The arbitrary sum of nonne... |
| sge0less 46315 | A shorter sum of nonnegati... |
| sge0rnbnd 46316 | The range used in the defi... |
| sge0pr 46317 | Sum of a pair of nonnegati... |
| sge0gerp 46318 | The arbitrary sum of nonne... |
| sge0pnffigt 46319 | If the sum of nonnegative ... |
| sge0ssre 46320 | If a sum of nonnegative ex... |
| sge0lefi 46321 | A sum of nonnegative exten... |
| sge0lessmpt 46322 | A shorter sum of nonnegati... |
| sge0ltfirp 46323 | If the sum of nonnegative ... |
| sge0prle 46324 | The sum of a pair of nonne... |
| sge0gerpmpt 46325 | The arbitrary sum of nonne... |
| sge0resrnlem 46326 | The sum of nonnegative ext... |
| sge0resrn 46327 | The sum of nonnegative ext... |
| sge0ssrempt 46328 | If a sum of nonnegative ex... |
| sge0resplit 46329 | ` sum^ ` splits into two p... |
| sge0le 46330 | If all of the terms of sum... |
| sge0ltfirpmpt 46331 | If the extended sum of non... |
| sge0split 46332 | Split a sum of nonnegative... |
| sge0lempt 46333 | If all of the terms of sum... |
| sge0splitmpt 46334 | Split a sum of nonnegative... |
| sge0ss 46335 | Change the index set to a ... |
| sge0iunmptlemfi 46336 | Sum of nonnegative extende... |
| sge0p1 46337 | The addition of the next t... |
| sge0iunmptlemre 46338 | Sum of nonnegative extende... |
| sge0fodjrnlem 46339 | Re-index a nonnegative ext... |
| sge0fodjrn 46340 | Re-index a nonnegative ext... |
| sge0iunmpt 46341 | Sum of nonnegative extende... |
| sge0iun 46342 | Sum of nonnegative extende... |
| sge0nemnf 46343 | The generalized sum of non... |
| sge0rpcpnf 46344 | The sum of an infinite num... |
| sge0rernmpt 46345 | If the sum of nonnegative ... |
| sge0lefimpt 46346 | A sum of nonnegative exten... |
| nn0ssge0 46347 | Nonnegative integers are n... |
| sge0clmpt 46348 | The generalized sum of non... |
| sge0ltfirpmpt2 46349 | If the extended sum of non... |
| sge0isum 46350 | If a series of nonnegative... |
| sge0xrclmpt 46351 | The generalized sum of non... |
| sge0xp 46352 | Combine two generalized su... |
| sge0isummpt 46353 | If a series of nonnegative... |
| sge0ad2en 46354 | The value of the infinite ... |
| sge0isummpt2 46355 | If a series of nonnegative... |
| sge0xaddlem1 46356 | The extended addition of t... |
| sge0xaddlem2 46357 | The extended addition of t... |
| sge0xadd 46358 | The extended addition of t... |
| sge0fsummptf 46359 | The generalized sum of a f... |
| sge0snmptf 46360 | A sum of a nonnegative ext... |
| sge0ge0mpt 46361 | The sum of nonnegative ext... |
| sge0repnfmpt 46362 | The of nonnegative extende... |
| sge0pnffigtmpt 46363 | If the generalized sum of ... |
| sge0splitsn 46364 | Separate out a term in a g... |
| sge0pnffsumgt 46365 | If the sum of nonnegative ... |
| sge0gtfsumgt 46366 | If the generalized sum of ... |
| sge0uzfsumgt 46367 | If a real number is smalle... |
| sge0pnfmpt 46368 | If a term in the sum of no... |
| sge0seq 46369 | A series of nonnegative re... |
| sge0reuz 46370 | Value of the generalized s... |
| sge0reuzb 46371 | Value of the generalized s... |
| ismea 46374 | Express the predicate " ` ... |
| dmmeasal 46375 | The domain of a measure is... |
| meaf 46376 | A measure is a function th... |
| mea0 46377 | The measure of the empty s... |
| nnfoctbdjlem 46378 | There exists a mapping fro... |
| nnfoctbdj 46379 | There exists a mapping fro... |
| meadjuni 46380 | The measure of the disjoin... |
| meacl 46381 | The measure of a set is a ... |
| iundjiunlem 46382 | The sets in the sequence `... |
| iundjiun 46383 | Given a sequence ` E ` of ... |
| meaxrcl 46384 | The measure of a set is an... |
| meadjun 46385 | The measure of the union o... |
| meassle 46386 | The measure of a set is gr... |
| meaunle 46387 | The measure of the union o... |
| meadjiunlem 46388 | The sum of nonnegative ext... |
| meadjiun 46389 | The measure of the disjoin... |
| ismeannd 46390 | Sufficient condition to pr... |
| meaiunlelem 46391 | The measure of the union o... |
| meaiunle 46392 | The measure of the union o... |
| psmeasurelem 46393 | ` M ` applied to a disjoin... |
| psmeasure 46394 | Point supported measure, R... |
| voliunsge0lem 46395 | The Lebesgue measure funct... |
| voliunsge0 46396 | The Lebesgue measure funct... |
| volmea 46397 | The Lebesgue measure on th... |
| meage0 46398 | If the measure of a measur... |
| meadjunre 46399 | The measure of the union o... |
| meassre 46400 | If the measure of a measur... |
| meale0eq0 46401 | A measure that is less tha... |
| meadif 46402 | The measure of the differe... |
| meaiuninclem 46403 | Measures are continuous fr... |
| meaiuninc 46404 | Measures are continuous fr... |
| meaiuninc2 46405 | Measures are continuous fr... |
| meaiunincf 46406 | Measures are continuous fr... |
| meaiuninc3v 46407 | Measures are continuous fr... |
| meaiuninc3 46408 | Measures are continuous fr... |
| meaiininclem 46409 | Measures are continuous fr... |
| meaiininc 46410 | Measures are continuous fr... |
| meaiininc2 46411 | Measures are continuous fr... |
| caragenval 46416 | The sigma-algebra generate... |
| isome 46417 | Express the predicate " ` ... |
| caragenel 46418 | Membership in the Caratheo... |
| omef 46419 | An outer measure is a func... |
| ome0 46420 | The outer measure of the e... |
| omessle 46421 | The outer measure of a set... |
| omedm 46422 | The domain of an outer mea... |
| caragensplit 46423 | If ` E ` is in the set gen... |
| caragenelss 46424 | An element of the Caratheo... |
| carageneld 46425 | Membership in the Caratheo... |
| omecl 46426 | The outer measure of a set... |
| caragenss 46427 | The sigma-algebra generate... |
| omeunile 46428 | The outer measure of the u... |
| caragen0 46429 | The empty set belongs to a... |
| omexrcl 46430 | The outer measure of a set... |
| caragenunidm 46431 | The base set of an outer m... |
| caragensspw 46432 | The sigma-algebra generate... |
| omessre 46433 | If the outer measure of a ... |
| caragenuni 46434 | The base set of the sigma-... |
| caragenuncllem 46435 | The Caratheodory's constru... |
| caragenuncl 46436 | The Caratheodory's constru... |
| caragendifcl 46437 | The Caratheodory's constru... |
| caragenfiiuncl 46438 | The Caratheodory's constru... |
| omeunle 46439 | The outer measure of the u... |
| omeiunle 46440 | The outer measure of the i... |
| omelesplit 46441 | The outer measure of a set... |
| omeiunltfirp 46442 | If the outer measure of a ... |
| omeiunlempt 46443 | The outer measure of the i... |
| carageniuncllem1 46444 | The outer measure of ` A i... |
| carageniuncllem2 46445 | The Caratheodory's constru... |
| carageniuncl 46446 | The Caratheodory's constru... |
| caragenunicl 46447 | The Caratheodory's constru... |
| caragensal 46448 | Caratheodory's method gene... |
| caratheodorylem1 46449 | Lemma used to prove that C... |
| caratheodorylem2 46450 | Caratheodory's constructio... |
| caratheodory 46451 | Caratheodory's constructio... |
| 0ome 46452 | The map that assigns 0 to ... |
| isomenndlem 46453 | ` O ` is sub-additive w.r.... |
| isomennd 46454 | Sufficient condition to pr... |
| caragenel2d 46455 | Membership in the Caratheo... |
| omege0 46456 | If the outer measure of a ... |
| omess0 46457 | If the outer measure of a ... |
| caragencmpl 46458 | A measure built with the C... |
| vonval 46463 | Value of the Lebesgue meas... |
| ovnval 46464 | Value of the Lebesgue oute... |
| elhoi 46465 | Membership in a multidimen... |
| icoresmbl 46466 | A closed-below, open-above... |
| hoissre 46467 | The projection of a half-o... |
| ovnval2 46468 | Value of the Lebesgue oute... |
| volicorecl 46469 | The Lebesgue measure of a ... |
| hoiprodcl 46470 | The pre-measure of half-op... |
| hoicvr 46471 | ` I ` is a countable set o... |
| hoissrrn 46472 | A half-open interval is a ... |
| ovn0val 46473 | The Lebesgue outer measure... |
| ovnn0val 46474 | The value of a (multidimen... |
| ovnval2b 46475 | Value of the Lebesgue oute... |
| volicorescl 46476 | The Lebesgue measure of a ... |
| ovnprodcl 46477 | The product used in the de... |
| hoiprodcl2 46478 | The pre-measure of half-op... |
| hoicvrrex 46479 | Any subset of the multidim... |
| ovnsupge0 46480 | The set used in the defini... |
| ovnlecvr 46481 | Given a subset of multidim... |
| ovnpnfelsup 46482 | ` +oo ` is an element of t... |
| ovnsslelem 46483 | The (multidimensional, non... |
| ovnssle 46484 | The (multidimensional) Leb... |
| ovnlerp 46485 | The Lebesgue outer measure... |
| ovnf 46486 | The Lebesgue outer measure... |
| ovncvrrp 46487 | The Lebesgue outer measure... |
| ovn0lem 46488 | For any finite dimension, ... |
| ovn0 46489 | For any finite dimension, ... |
| ovncl 46490 | The Lebesgue outer measure... |
| ovn02 46491 | For the zero-dimensional s... |
| ovnxrcl 46492 | The Lebesgue outer measure... |
| ovnsubaddlem1 46493 | The Lebesgue outer measure... |
| ovnsubaddlem2 46494 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd 46495 | ` ( voln* `` X ) ` is suba... |
| ovnome 46496 | ` ( voln* `` X ) ` is an o... |
| vonmea 46497 | ` ( voln `` X ) ` is a mea... |
| volicon0 46498 | The measure of a nonempty ... |
| hsphoif 46499 | ` H ` is a function (that ... |
| hoidmvval 46500 | The dimensional volume of ... |
| hoissrrn2 46501 | A half-open interval is a ... |
| hsphoival 46502 | ` H ` is a function (that ... |
| hoiprodcl3 46503 | The pre-measure of half-op... |
| volicore 46504 | The Lebesgue measure of a ... |
| hoidmvcl 46505 | The dimensional volume of ... |
| hoidmv0val 46506 | The dimensional volume of ... |
| hoidmvn0val 46507 | The dimensional volume of ... |
| hsphoidmvle2 46508 | The dimensional volume of ... |
| hsphoidmvle 46509 | The dimensional volume of ... |
| hoidmvval0 46510 | The dimensional volume of ... |
| hoiprodp1 46511 | The dimensional volume of ... |
| sge0hsphoire 46512 | If the generalized sum of ... |
| hoidmvval0b 46513 | The dimensional volume of ... |
| hoidmv1lelem1 46514 | The supremum of ` U ` belo... |
| hoidmv1lelem2 46515 | This is the contradiction ... |
| hoidmv1lelem3 46516 | The dimensional volume of ... |
| hoidmv1le 46517 | The dimensional volume of ... |
| hoidmvlelem1 46518 | The supremum of ` U ` belo... |
| hoidmvlelem2 46519 | This is the contradiction ... |
| hoidmvlelem3 46520 | This is the contradiction ... |
| hoidmvlelem4 46521 | The dimensional volume of ... |
| hoidmvlelem5 46522 | The dimensional volume of ... |
| hoidmvle 46523 | The dimensional volume of ... |
| ovnhoilem1 46524 | The Lebesgue outer measure... |
| ovnhoilem2 46525 | The Lebesgue outer measure... |
| ovnhoi 46526 | The Lebesgue outer measure... |
| dmovn 46527 | The domain of the Lebesgue... |
| hoicoto2 46528 | The half-open interval exp... |
| dmvon 46529 | Lebesgue measurable n-dime... |
| hoi2toco 46530 | The half-open interval exp... |
| hoidifhspval 46531 | ` D ` is a function that r... |
| hspval 46532 | The value of the half-spac... |
| ovnlecvr2 46533 | Given a subset of multidim... |
| ovncvr2 46534 | ` B ` and ` T ` are the le... |
| dmovnsal 46535 | The domain of the Lebesgue... |
| unidmovn 46536 | Base set of the n-dimensio... |
| rrnmbl 46537 | The set of n-dimensional R... |
| hoidifhspval2 46538 | ` D ` is a function that r... |
| hspdifhsp 46539 | A n-dimensional half-open ... |
| unidmvon 46540 | Base set of the n-dimensio... |
| hoidifhspf 46541 | ` D ` is a function that r... |
| hoidifhspval3 46542 | ` D ` is a function that r... |
| hoidifhspdmvle 46543 | The dimensional volume of ... |
| voncmpl 46544 | The Lebesgue measure is co... |
| hoiqssbllem1 46545 | The center of the n-dimens... |
| hoiqssbllem2 46546 | The center of the n-dimens... |
| hoiqssbllem3 46547 | A n-dimensional ball conta... |
| hoiqssbl 46548 | A n-dimensional ball conta... |
| hspmbllem1 46549 | Any half-space of the n-di... |
| hspmbllem2 46550 | Any half-space of the n-di... |
| hspmbllem3 46551 | Any half-space of the n-di... |
| hspmbl 46552 | Any half-space of the n-di... |
| hoimbllem 46553 | Any n-dimensional half-ope... |
| hoimbl 46554 | Any n-dimensional half-ope... |
| opnvonmbllem1 46555 | The half-open interval exp... |
| opnvonmbllem2 46556 | An open subset of the n-di... |
| opnvonmbl 46557 | An open subset of the n-di... |
| opnssborel 46558 | Open sets of a generalized... |
| borelmbl 46559 | All Borel subsets of the n... |
| volicorege0 46560 | The Lebesgue measure of a ... |
| isvonmbl 46561 | The predicate " ` A ` is m... |
| mblvon 46562 | The n-dimensional Lebesgue... |
| vonmblss 46563 | n-dimensional Lebesgue mea... |
| volico2 46564 | The measure of left-closed... |
| vonmblss2 46565 | n-dimensional Lebesgue mea... |
| ovolval2lem 46566 | The value of the Lebesgue ... |
| ovolval2 46567 | The value of the Lebesgue ... |
| ovnsubadd2lem 46568 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd2 46569 | ` ( voln* `` X ) ` is suba... |
| ovolval3 46570 | The value of the Lebesgue ... |
| ovnsplit 46571 | The n-dimensional Lebesgue... |
| ovolval4lem1 46572 | |- ( ( ph /\ n e. A ) -> ... |
| ovolval4lem2 46573 | The value of the Lebesgue ... |
| ovolval4 46574 | The value of the Lebesgue ... |
| ovolval5lem1 46575 | ` |- ( ph -> ( sum^ `` ( n... |
| ovolval5lem2 46576 | ` |- ( ( ph /\ n e. NN ) -... |
| ovolval5lem3 46577 | The value of the Lebesgue ... |
| ovolval5 46578 | The value of the Lebesgue ... |
| ovnovollem1 46579 | if ` F ` is a cover of ` B... |
| ovnovollem2 46580 | if ` I ` is a cover of ` (... |
| ovnovollem3 46581 | The 1-dimensional Lebesgue... |
| ovnovol 46582 | The 1-dimensional Lebesgue... |
| vonvolmbllem 46583 | If a subset ` B ` of real ... |
| vonvolmbl 46584 | A subset of Real numbers i... |
| vonvol 46585 | The 1-dimensional Lebesgue... |
| vonvolmbl2 46586 | A subset ` X ` of the spac... |
| vonvol2 46587 | The 1-dimensional Lebesgue... |
| hoimbl2 46588 | Any n-dimensional half-ope... |
| voncl 46589 | The Lebesgue measure of a ... |
| vonhoi 46590 | The Lebesgue outer measure... |
| vonxrcl 46591 | The Lebesgue measure of a ... |
| ioosshoi 46592 | A n-dimensional open inter... |
| vonn0hoi 46593 | The Lebesgue outer measure... |
| von0val 46594 | The Lebesgue measure (for ... |
| vonhoire 46595 | The Lebesgue measure of a ... |
| iinhoiicclem 46596 | A n-dimensional closed int... |
| iinhoiicc 46597 | A n-dimensional closed int... |
| iunhoiioolem 46598 | A n-dimensional open inter... |
| iunhoiioo 46599 | A n-dimensional open inter... |
| ioovonmbl 46600 | Any n-dimensional open int... |
| iccvonmbllem 46601 | Any n-dimensional closed i... |
| iccvonmbl 46602 | Any n-dimensional closed i... |
| vonioolem1 46603 | The sequence of the measur... |
| vonioolem2 46604 | The n-dimensional Lebesgue... |
| vonioo 46605 | The n-dimensional Lebesgue... |
| vonicclem1 46606 | The sequence of the measur... |
| vonicclem2 46607 | The n-dimensional Lebesgue... |
| vonicc 46608 | The n-dimensional Lebesgue... |
| snvonmbl 46609 | A n-dimensional singleton ... |
| vonn0ioo 46610 | The n-dimensional Lebesgue... |
| vonn0icc 46611 | The n-dimensional Lebesgue... |
| ctvonmbl 46612 | Any n-dimensional countabl... |
| vonn0ioo2 46613 | The n-dimensional Lebesgue... |
| vonsn 46614 | The n-dimensional Lebesgue... |
| vonn0icc2 46615 | The n-dimensional Lebesgue... |
| vonct 46616 | The n-dimensional Lebesgue... |
| vitali2 46617 | There are non-measurable s... |
| pimltmnf2f 46620 | Given a real-valued functi... |
| pimltmnf2 46621 | Given a real-valued functi... |
| preimagelt 46622 | The preimage of a right-op... |
| preimalegt 46623 | The preimage of a left-ope... |
| pimconstlt0 46624 | Given a constant function,... |
| pimconstlt1 46625 | Given a constant function,... |
| pimltpnff 46626 | Given a real-valued functi... |
| pimltpnf 46627 | Given a real-valued functi... |
| pimgtpnf2f 46628 | Given a real-valued functi... |
| pimgtpnf2 46629 | Given a real-valued functi... |
| salpreimagelt 46630 | If all the preimages of le... |
| pimrecltpos 46631 | The preimage of an unbound... |
| salpreimalegt 46632 | If all the preimages of ri... |
| pimiooltgt 46633 | The preimage of an open in... |
| preimaicomnf 46634 | Preimage of an open interv... |
| pimltpnf2f 46635 | Given a real-valued functi... |
| pimltpnf2 46636 | Given a real-valued functi... |
| pimgtmnf2 46637 | Given a real-valued functi... |
| pimdecfgtioc 46638 | Given a nonincreasing func... |
| pimincfltioc 46639 | Given a nondecreasing func... |
| pimdecfgtioo 46640 | Given a nondecreasing func... |
| pimincfltioo 46641 | Given a nondecreasing func... |
| preimaioomnf 46642 | Preimage of an open interv... |
| preimageiingt 46643 | A preimage of a left-close... |
| preimaleiinlt 46644 | A preimage of a left-open,... |
| pimgtmnff 46645 | Given a real-valued functi... |
| pimgtmnf 46646 | Given a real-valued functi... |
| pimrecltneg 46647 | The preimage of an unbound... |
| salpreimagtge 46648 | If all the preimages of le... |
| salpreimaltle 46649 | If all the preimages of ri... |
| issmflem 46650 | The predicate " ` F ` is a... |
| issmf 46651 | The predicate " ` F ` is a... |
| salpreimalelt 46652 | If all the preimages of ri... |
| salpreimagtlt 46653 | If all the preimages of le... |
| smfpreimalt 46654 | Given a function measurabl... |
| smff 46655 | A function measurable w.r.... |
| smfdmss 46656 | The domain of a function m... |
| issmff 46657 | The predicate " ` F ` is a... |
| issmfd 46658 | A sufficient condition for... |
| smfpreimaltf 46659 | Given a function measurabl... |
| issmfdf 46660 | A sufficient condition for... |
| sssmf 46661 | The restriction of a sigma... |
| mbfresmf 46662 | A real-valued measurable f... |
| cnfsmf 46663 | A continuous function is m... |
| incsmflem 46664 | A nondecreasing function i... |
| incsmf 46665 | A real-valued, nondecreasi... |
| smfsssmf 46666 | If a function is measurabl... |
| issmflelem 46667 | The predicate " ` F ` is a... |
| issmfle 46668 | The predicate " ` F ` is a... |
| smfpimltmpt 46669 | Given a function measurabl... |
| smfpimltxr 46670 | Given a function measurabl... |
| issmfdmpt 46671 | A sufficient condition for... |
| smfconst 46672 | Given a sigma-algebra over... |
| sssmfmpt 46673 | The restriction of a sigma... |
| cnfrrnsmf 46674 | A function, continuous fro... |
| smfid 46675 | The identity function is B... |
| bormflebmf 46676 | A Borel measurable functio... |
| smfpreimale 46677 | Given a function measurabl... |
| issmfgtlem 46678 | The predicate " ` F ` is a... |
| issmfgt 46679 | The predicate " ` F ` is a... |
| issmfled 46680 | A sufficient condition for... |
| smfpimltxrmptf 46681 | Given a function measurabl... |
| smfpimltxrmpt 46682 | Given a function measurabl... |
| smfmbfcex 46683 | A constant function, with ... |
| issmfgtd 46684 | A sufficient condition for... |
| smfpreimagt 46685 | Given a function measurabl... |
| smfaddlem1 46686 | Given the sum of two funct... |
| smfaddlem2 46687 | The sum of two sigma-measu... |
| smfadd 46688 | The sum of two sigma-measu... |
| decsmflem 46689 | A nonincreasing function i... |
| decsmf 46690 | A real-valued, nonincreasi... |
| smfpreimagtf 46691 | Given a function measurabl... |
| issmfgelem 46692 | The predicate " ` F ` is a... |
| issmfge 46693 | The predicate " ` F ` is a... |
| smflimlem1 46694 | Lemma for the proof that t... |
| smflimlem2 46695 | Lemma for the proof that t... |
| smflimlem3 46696 | The limit of sigma-measura... |
| smflimlem4 46697 | Lemma for the proof that t... |
| smflimlem5 46698 | Lemma for the proof that t... |
| smflimlem6 46699 | Lemma for the proof that t... |
| smflim 46700 | The limit of sigma-measura... |
| nsssmfmbflem 46701 | The sigma-measurable funct... |
| nsssmfmbf 46702 | The sigma-measurable funct... |
| smfpimgtxr 46703 | Given a function measurabl... |
| smfpimgtmpt 46704 | Given a function measurabl... |
| smfpreimage 46705 | Given a function measurabl... |
| mbfpsssmf 46706 | Real-valued measurable fun... |
| smfpimgtxrmptf 46707 | Given a function measurabl... |
| smfpimgtxrmpt 46708 | Given a function measurabl... |
| smfpimioompt 46709 | Given a function measurabl... |
| smfpimioo 46710 | Given a function measurabl... |
| smfresal 46711 | Given a sigma-measurable f... |
| smfrec 46712 | The reciprocal of a sigma-... |
| smfres 46713 | The restriction of sigma-m... |
| smfmullem1 46714 | The multiplication of two ... |
| smfmullem2 46715 | The multiplication of two ... |
| smfmullem3 46716 | The multiplication of two ... |
| smfmullem4 46717 | The multiplication of two ... |
| smfmul 46718 | The multiplication of two ... |
| smfmulc1 46719 | A sigma-measurable functio... |
| smfdiv 46720 | The fraction of two sigma-... |
| smfpimbor1lem1 46721 | Every open set belongs to ... |
| smfpimbor1lem2 46722 | Given a sigma-measurable f... |
| smfpimbor1 46723 | Given a sigma-measurable f... |
| smf2id 46724 | Twice the identity functio... |
| smfco 46725 | The composition of a Borel... |
| smfneg 46726 | The negative of a sigma-me... |
| smffmptf 46727 | A function measurable w.r.... |
| smffmpt 46728 | A function measurable w.r.... |
| smflim2 46729 | The limit of a sequence of... |
| smfpimcclem 46730 | Lemma for ~ smfpimcc given... |
| smfpimcc 46731 | Given a countable set of s... |
| issmfle2d 46732 | A sufficient condition for... |
| smflimmpt 46733 | The limit of a sequence of... |
| smfsuplem1 46734 | The supremum of a countabl... |
| smfsuplem2 46735 | The supremum of a countabl... |
| smfsuplem3 46736 | The supremum of a countabl... |
| smfsup 46737 | The supremum of a countabl... |
| smfsupmpt 46738 | The supremum of a countabl... |
| smfsupxr 46739 | The supremum of a countabl... |
| smfinflem 46740 | The infimum of a countable... |
| smfinf 46741 | The infimum of a countable... |
| smfinfmpt 46742 | The infimum of a countable... |
| smflimsuplem1 46743 | If ` H ` converges, the ` ... |
| smflimsuplem2 46744 | The superior limit of a se... |
| smflimsuplem3 46745 | The limit of the ` ( H `` ... |
| smflimsuplem4 46746 | If ` H ` converges, the ` ... |
| smflimsuplem5 46747 | ` H ` converges to the sup... |
| smflimsuplem6 46748 | The superior limit of a se... |
| smflimsuplem7 46749 | The superior limit of a se... |
| smflimsuplem8 46750 | The superior limit of a se... |
| smflimsup 46751 | The superior limit of a se... |
| smflimsupmpt 46752 | The superior limit of a se... |
| smfliminflem 46753 | The inferior limit of a co... |
| smfliminf 46754 | The inferior limit of a co... |
| smfliminfmpt 46755 | The inferior limit of a co... |
| adddmmbl 46756 | If two functions have doma... |
| adddmmbl2 46757 | If two functions have doma... |
| muldmmbl 46758 | If two functions have doma... |
| muldmmbl2 46759 | If two functions have doma... |
| smfdmmblpimne 46760 | If a measurable function w... |
| smfdivdmmbl 46761 | If a functions and a sigma... |
| smfpimne 46762 | Given a function measurabl... |
| smfpimne2 46763 | Given a function measurabl... |
| smfdivdmmbl2 46764 | If a functions and a sigma... |
| fsupdm 46765 | The domain of the sup func... |
| fsupdm2 46766 | The domain of the sup func... |
| smfsupdmmbllem 46767 | If a countable set of sigm... |
| smfsupdmmbl 46768 | If a countable set of sigm... |
| finfdm 46769 | The domain of the inf func... |
| finfdm2 46770 | The domain of the inf func... |
| smfinfdmmbllem 46771 | If a countable set of sigm... |
| smfinfdmmbl 46772 | If a countable set of sigm... |
| sigarval 46773 | Define the signed area by ... |
| sigarim 46774 | Signed area takes value in... |
| sigarac 46775 | Signed area is anticommuta... |
| sigaraf 46776 | Signed area is additive by... |
| sigarmf 46777 | Signed area is additive (w... |
| sigaras 46778 | Signed area is additive by... |
| sigarms 46779 | Signed area is additive (w... |
| sigarls 46780 | Signed area is linear by t... |
| sigarid 46781 | Signed area of a flat para... |
| sigarexp 46782 | Expand the signed area for... |
| sigarperm 46783 | Signed area ` ( A - C ) G ... |
| sigardiv 46784 | If signed area between vec... |
| sigarimcd 46785 | Signed area takes value in... |
| sigariz 46786 | If signed area is zero, th... |
| sigarcol 46787 | Given three points ` A ` ,... |
| sharhght 46788 | Let ` A B C ` be a triangl... |
| sigaradd 46789 | Subtracting (double) area ... |
| cevathlem1 46790 | Ceva's theorem first lemma... |
| cevathlem2 46791 | Ceva's theorem second lemm... |
| cevath 46792 | Ceva's theorem. Let ` A B... |
| simpcntrab 46793 | The center of a simple gro... |
| et-ltneverrefl 46794 | Less-than class is never r... |
| et-equeucl 46795 | Alternative proof that equ... |
| et-sqrtnegnre 46796 | The square root of a negat... |
| natlocalincr 46797 | Global monotonicity on hal... |
| natglobalincr 46798 | Local monotonicity on half... |
| upwordnul 46801 | Empty set is an increasing... |
| upwordisword 46802 | Any increasing sequence is... |
| singoutnword 46803 | Singleton with character o... |
| singoutnupword 46804 | Singleton with character o... |
| upwordsing 46805 | Singleton is an increasing... |
| upwordsseti 46806 | Strictly increasing sequen... |
| tworepnotupword 46807 | Concatenation of identical... |
| upwrdfi 46808 | There is a finite number o... |
| hirstL-ax3 46809 | The third axiom of a syste... |
| ax3h 46810 | Recover ~ ax-3 from ~ hirs... |
| aibandbiaiffaiffb 46811 | A closed form showing (a i... |
| aibandbiaiaiffb 46812 | A closed form showing (a i... |
| notatnand 46813 | Do not use. Use intnanr i... |
| aistia 46814 | Given a is equivalent to `... |
| aisfina 46815 | Given a is equivalent to `... |
| bothtbothsame 46816 | Given both a, b are equiva... |
| bothfbothsame 46817 | Given both a, b are equiva... |
| aiffbbtat 46818 | Given a is equivalent to b... |
| aisbbisfaisf 46819 | Given a is equivalent to b... |
| axorbtnotaiffb 46820 | Given a is exclusive to b,... |
| aiffnbandciffatnotciffb 46821 | Given a is equivalent to (... |
| axorbciffatcxorb 46822 | Given a is equivalent to (... |
| aibnbna 46823 | Given a implies b, (not b)... |
| aibnbaif 46824 | Given a implies b, not b, ... |
| aiffbtbat 46825 | Given a is equivalent to b... |
| astbstanbst 46826 | Given a is equivalent to T... |
| aistbistaandb 46827 | Given a is equivalent to T... |
| aisbnaxb 46828 | Given a is equivalent to b... |
| atbiffatnnb 46829 | If a implies b, then a imp... |
| bisaiaisb 46830 | Application of bicom1 with... |
| atbiffatnnbalt 46831 | If a implies b, then a imp... |
| abnotbtaxb 46832 | Assuming a, not b, there e... |
| abnotataxb 46833 | Assuming not a, b, there e... |
| conimpf 46834 | Assuming a, not b, and a i... |
| conimpfalt 46835 | Assuming a, not b, and a i... |
| aistbisfiaxb 46836 | Given a is equivalent to T... |
| aisfbistiaxb 46837 | Given a is equivalent to F... |
| aifftbifffaibif 46838 | Given a is equivalent to T... |
| aifftbifffaibifff 46839 | Given a is equivalent to T... |
| atnaiana 46840 | Given a, it is not the cas... |
| ainaiaandna 46841 | Given a, a implies it is n... |
| abcdta 46842 | Given (((a and b) and c) a... |
| abcdtb 46843 | Given (((a and b) and c) a... |
| abcdtc 46844 | Given (((a and b) and c) a... |
| abcdtd 46845 | Given (((a and b) and c) a... |
| abciffcbatnabciffncba 46846 | Operands in a biconditiona... |
| abciffcbatnabciffncbai 46847 | Operands in a biconditiona... |
| nabctnabc 46848 | not ( a -> ( b /\ c ) ) we... |
| jabtaib 46849 | For when pm3.4 lacks a pm3... |
| onenotinotbothi 46850 | From one negated implicati... |
| twonotinotbothi 46851 | From these two negated imp... |
| clifte 46852 | show d is the same as an i... |
| cliftet 46853 | show d is the same as an i... |
| clifteta 46854 | show d is the same as an i... |
| cliftetb 46855 | show d is the same as an i... |
| confun 46856 | Given the hypotheses there... |
| confun2 46857 | Confun simplified to two p... |
| confun3 46858 | Confun's more complex form... |
| confun4 46859 | An attempt at derivative. ... |
| confun5 46860 | An attempt at derivative. ... |
| plcofph 46861 | Given, a,b and a "definiti... |
| pldofph 46862 | Given, a,b c, d, "definiti... |
| plvcofph 46863 | Given, a,b,d, and "definit... |
| plvcofphax 46864 | Given, a,b,d, and "definit... |
| plvofpos 46865 | rh is derivable because ON... |
| mdandyv0 46866 | Given the equivalences set... |
| mdandyv1 46867 | Given the equivalences set... |
| mdandyv2 46868 | Given the equivalences set... |
| mdandyv3 46869 | Given the equivalences set... |
| mdandyv4 46870 | Given the equivalences set... |
| mdandyv5 46871 | Given the equivalences set... |
| mdandyv6 46872 | Given the equivalences set... |
| mdandyv7 46873 | Given the equivalences set... |
| mdandyv8 46874 | Given the equivalences set... |
| mdandyv9 46875 | Given the equivalences set... |
| mdandyv10 46876 | Given the equivalences set... |
| mdandyv11 46877 | Given the equivalences set... |
| mdandyv12 46878 | Given the equivalences set... |
| mdandyv13 46879 | Given the equivalences set... |
| mdandyv14 46880 | Given the equivalences set... |
| mdandyv15 46881 | Given the equivalences set... |
| mdandyvr0 46882 | Given the equivalences set... |
| mdandyvr1 46883 | Given the equivalences set... |
| mdandyvr2 46884 | Given the equivalences set... |
| mdandyvr3 46885 | Given the equivalences set... |
| mdandyvr4 46886 | Given the equivalences set... |
| mdandyvr5 46887 | Given the equivalences set... |
| mdandyvr6 46888 | Given the equivalences set... |
| mdandyvr7 46889 | Given the equivalences set... |
| mdandyvr8 46890 | Given the equivalences set... |
| mdandyvr9 46891 | Given the equivalences set... |
| mdandyvr10 46892 | Given the equivalences set... |
| mdandyvr11 46893 | Given the equivalences set... |
| mdandyvr12 46894 | Given the equivalences set... |
| mdandyvr13 46895 | Given the equivalences set... |
| mdandyvr14 46896 | Given the equivalences set... |
| mdandyvr15 46897 | Given the equivalences set... |
| mdandyvrx0 46898 | Given the exclusivities se... |
| mdandyvrx1 46899 | Given the exclusivities se... |
| mdandyvrx2 46900 | Given the exclusivities se... |
| mdandyvrx3 46901 | Given the exclusivities se... |
| mdandyvrx4 46902 | Given the exclusivities se... |
| mdandyvrx5 46903 | Given the exclusivities se... |
| mdandyvrx6 46904 | Given the exclusivities se... |
| mdandyvrx7 46905 | Given the exclusivities se... |
| mdandyvrx8 46906 | Given the exclusivities se... |
| mdandyvrx9 46907 | Given the exclusivities se... |
| mdandyvrx10 46908 | Given the exclusivities se... |
| mdandyvrx11 46909 | Given the exclusivities se... |
| mdandyvrx12 46910 | Given the exclusivities se... |
| mdandyvrx13 46911 | Given the exclusivities se... |
| mdandyvrx14 46912 | Given the exclusivities se... |
| mdandyvrx15 46913 | Given the exclusivities se... |
| H15NH16TH15IH16 46914 | Given 15 hypotheses and a ... |
| dandysum2p2e4 46915 | CONTRADICTION PROVED AT 1 ... |
| mdandysum2p2e4 46916 | CONTRADICTION PROVED AT 1 ... |
| adh-jarrsc 46917 | Replacement of a nested an... |
| adh-minim 46918 | A single axiom for minimal... |
| adh-minim-ax1-ax2-lem1 46919 | First lemma for the deriva... |
| adh-minim-ax1-ax2-lem2 46920 | Second lemma for the deriv... |
| adh-minim-ax1-ax2-lem3 46921 | Third lemma for the deriva... |
| adh-minim-ax1-ax2-lem4 46922 | Fourth lemma for the deriv... |
| adh-minim-ax1 46923 | Derivation of ~ ax-1 from ... |
| adh-minim-ax2-lem5 46924 | Fifth lemma for the deriva... |
| adh-minim-ax2-lem6 46925 | Sixth lemma for the deriva... |
| adh-minim-ax2c 46926 | Derivation of a commuted f... |
| adh-minim-ax2 46927 | Derivation of ~ ax-2 from ... |
| adh-minim-idALT 46928 | Derivation of ~ id (reflex... |
| adh-minim-pm2.43 46929 | Derivation of ~ pm2.43 Whi... |
| adh-minimp 46930 | Another single axiom for m... |
| adh-minimp-jarr-imim1-ax2c-lem1 46931 | First lemma for the deriva... |
| adh-minimp-jarr-lem2 46932 | Second lemma for the deriv... |
| adh-minimp-jarr-ax2c-lem3 46933 | Third lemma for the deriva... |
| adh-minimp-sylsimp 46934 | Derivation of ~ jarr (also... |
| adh-minimp-ax1 46935 | Derivation of ~ ax-1 from ... |
| adh-minimp-imim1 46936 | Derivation of ~ imim1 ("le... |
| adh-minimp-ax2c 46937 | Derivation of a commuted f... |
| adh-minimp-ax2-lem4 46938 | Fourth lemma for the deriv... |
| adh-minimp-ax2 46939 | Derivation of ~ ax-2 from ... |
| adh-minimp-idALT 46940 | Derivation of ~ id (reflex... |
| adh-minimp-pm2.43 46941 | Derivation of ~ pm2.43 Whi... |
| n0nsn2el 46942 | If a class with one elemen... |
| eusnsn 46943 | There is a unique element ... |
| absnsb 46944 | If the class abstraction `... |
| euabsneu 46945 | Another way to express exi... |
| elprneb 46946 | An element of a proper uno... |
| oppr 46947 | Equality for ordered pairs... |
| opprb 46948 | Equality for unordered pai... |
| or2expropbilem1 46949 | Lemma 1 for ~ or2expropbi ... |
| or2expropbilem2 46950 | Lemma 2 for ~ or2expropbi ... |
| or2expropbi 46951 | If two classes are strictl... |
| eubrv 46952 | If there is a unique set w... |
| eubrdm 46953 | If there is a unique set w... |
| eldmressn 46954 | Element of the domain of a... |
| iota0def 46955 | Example for a defined iota... |
| iota0ndef 46956 | Example for an undefined i... |
| fveqvfvv 46957 | If a function's value at a... |
| fnresfnco 46958 | Composition of two functio... |
| funcoressn 46959 | A composition restricted t... |
| funressnfv 46960 | A restriction to a singlet... |
| funressndmfvrn 46961 | The value of a function ` ... |
| funressnvmo 46962 | A function restricted to a... |
| funressnmo 46963 | A function restricted to a... |
| funressneu 46964 | There is exactly one value... |
| fresfo 46965 | Conditions for a restricti... |
| fsetsniunop 46966 | The class of all functions... |
| fsetabsnop 46967 | The class of all functions... |
| fsetsnf 46968 | The mapping of an element ... |
| fsetsnf1 46969 | The mapping of an element ... |
| fsetsnfo 46970 | The mapping of an element ... |
| fsetsnf1o 46971 | The mapping of an element ... |
| fsetsnprcnex 46972 | The class of all functions... |
| cfsetssfset 46973 | The class of constant func... |
| cfsetsnfsetfv 46974 | The function value of the ... |
| cfsetsnfsetf 46975 | The mapping of the class o... |
| cfsetsnfsetf1 46976 | The mapping of the class o... |
| cfsetsnfsetfo 46977 | The mapping of the class o... |
| cfsetsnfsetf1o 46978 | The mapping of the class o... |
| fsetprcnexALT 46979 | First version of proof for... |
| fcoreslem1 46980 | Lemma 1 for ~ fcores . (C... |
| fcoreslem2 46981 | Lemma 2 for ~ fcores . (C... |
| fcoreslem3 46982 | Lemma 3 for ~ fcores . (C... |
| fcoreslem4 46983 | Lemma 4 for ~ fcores . (C... |
| fcores 46984 | Every composite function `... |
| fcoresf1lem 46985 | Lemma for ~ fcoresf1 . (C... |
| fcoresf1 46986 | If a composition is inject... |
| fcoresf1b 46987 | A composition is injective... |
| fcoresfo 46988 | If a composition is surjec... |
| fcoresfob 46989 | A composition is surjectiv... |
| fcoresf1ob 46990 | A composition is bijective... |
| f1cof1blem 46991 | Lemma for ~ f1cof1b and ~ ... |
| 3f1oss1 46992 | The composition of three b... |
| 3f1oss2 46993 | The composition of three b... |
| f1cof1b 46994 | If the range of ` F ` equa... |
| funfocofob 46995 | If the domain of a functio... |
| fnfocofob 46996 | If the domain of a functio... |
| focofob 46997 | If the domain of a functio... |
| f1ocof1ob 46998 | If the range of ` F ` equa... |
| f1ocof1ob2 46999 | If the range of ` F ` equa... |
| aiotajust 47001 | Soundness justification th... |
| dfaiota2 47003 | Alternate definition of th... |
| reuabaiotaiota 47004 | The iota and the alternate... |
| reuaiotaiota 47005 | The iota and the alternate... |
| aiotaexb 47006 | The alternate iota over a ... |
| aiotavb 47007 | The alternate iota over a ... |
| aiotaint 47008 | This is to ~ df-aiota what... |
| dfaiota3 47009 | Alternate definition of ` ... |
| iotan0aiotaex 47010 | If the iota over a wff ` p... |
| aiotaexaiotaiota 47011 | The alternate iota over a ... |
| aiotaval 47012 | Theorem 8.19 in [Quine] p.... |
| aiota0def 47013 | Example for a defined alte... |
| aiota0ndef 47014 | Example for an undefined a... |
| r19.32 47015 | Theorem 19.32 of [Margaris... |
| rexsb 47016 | An equivalent expression f... |
| rexrsb 47017 | An equivalent expression f... |
| 2rexsb 47018 | An equivalent expression f... |
| 2rexrsb 47019 | An equivalent expression f... |
| cbvral2 47020 | Change bound variables of ... |
| cbvrex2 47021 | Change bound variables of ... |
| ralndv1 47022 | Example for a theorem abou... |
| ralndv2 47023 | Second example for a theor... |
| reuf1odnf 47024 | There is exactly one eleme... |
| reuf1od 47025 | There is exactly one eleme... |
| euoreqb 47026 | There is a set which is eq... |
| 2reu3 47027 | Double restricted existent... |
| 2reu7 47028 | Two equivalent expressions... |
| 2reu8 47029 | Two equivalent expressions... |
| 2reu8i 47030 | Implication of a double re... |
| 2reuimp0 47031 | Implication of a double re... |
| 2reuimp 47032 | Implication of a double re... |
| ralbinrald 47039 | Elemination of a restricte... |
| nvelim 47040 | If a class is the universa... |
| alneu 47041 | If a statement holds for a... |
| eu2ndop1stv 47042 | If there is a unique secon... |
| dfateq12d 47043 | Equality deduction for "de... |
| nfdfat 47044 | Bound-variable hypothesis ... |
| dfdfat2 47045 | Alternate definition of th... |
| fundmdfat 47046 | A function is defined at a... |
| dfatprc 47047 | A function is not defined ... |
| dfatelrn 47048 | The value of a function ` ... |
| dfafv2 47049 | Alternative definition of ... |
| afveq12d 47050 | Equality deduction for fun... |
| afveq1 47051 | Equality theorem for funct... |
| afveq2 47052 | Equality theorem for funct... |
| nfafv 47053 | Bound-variable hypothesis ... |
| csbafv12g 47054 | Move class substitution in... |
| afvfundmfveq 47055 | If a class is a function r... |
| afvnfundmuv 47056 | If a set is not in the dom... |
| ndmafv 47057 | The value of a class outsi... |
| afvvdm 47058 | If the function value of a... |
| nfunsnafv 47059 | If the restriction of a cl... |
| afvvfunressn 47060 | If the function value of a... |
| afvprc 47061 | A function's value at a pr... |
| afvvv 47062 | If a function's value at a... |
| afvpcfv0 47063 | If the value of the altern... |
| afvnufveq 47064 | The value of the alternati... |
| afvvfveq 47065 | The value of the alternati... |
| afv0fv0 47066 | If the value of the altern... |
| afvfvn0fveq 47067 | If the function's value at... |
| afv0nbfvbi 47068 | The function's value at an... |
| afvfv0bi 47069 | The function's value at an... |
| afveu 47070 | The value of a function at... |
| fnbrafvb 47071 | Equivalence of function va... |
| fnopafvb 47072 | Equivalence of function va... |
| funbrafvb 47073 | Equivalence of function va... |
| funopafvb 47074 | Equivalence of function va... |
| funbrafv 47075 | The second argument of a b... |
| funbrafv2b 47076 | Function value in terms of... |
| dfafn5a 47077 | Representation of a functi... |
| dfafn5b 47078 | Representation of a functi... |
| fnrnafv 47079 | The range of a function ex... |
| afvelrnb 47080 | A member of a function's r... |
| afvelrnb0 47081 | A member of a function's r... |
| dfaimafn 47082 | Alternate definition of th... |
| dfaimafn2 47083 | Alternate definition of th... |
| afvelima 47084 | Function value in an image... |
| afvelrn 47085 | A function's value belongs... |
| fnafvelrn 47086 | A function's value belongs... |
| fafvelcdm 47087 | A function's value belongs... |
| ffnafv 47088 | A function maps to a class... |
| afvres 47089 | The value of a restricted ... |
| tz6.12-afv 47090 | Function value. Theorem 6... |
| tz6.12-1-afv 47091 | Function value (Theorem 6.... |
| dmfcoafv 47092 | Domains of a function comp... |
| afvco2 47093 | Value of a function compos... |
| rlimdmafv 47094 | Two ways to express that a... |
| aoveq123d 47095 | Equality deduction for ope... |
| nfaov 47096 | Bound-variable hypothesis ... |
| csbaovg 47097 | Move class substitution in... |
| aovfundmoveq 47098 | If a class is a function r... |
| aovnfundmuv 47099 | If an ordered pair is not ... |
| ndmaov 47100 | The value of an operation ... |
| ndmaovg 47101 | The value of an operation ... |
| aovvdm 47102 | If the operation value of ... |
| nfunsnaov 47103 | If the restriction of a cl... |
| aovvfunressn 47104 | If the operation value of ... |
| aovprc 47105 | The value of an operation ... |
| aovrcl 47106 | Reverse closure for an ope... |
| aovpcov0 47107 | If the alternative value o... |
| aovnuoveq 47108 | The alternative value of t... |
| aovvoveq 47109 | The alternative value of t... |
| aov0ov0 47110 | If the alternative value o... |
| aovovn0oveq 47111 | If the operation's value a... |
| aov0nbovbi 47112 | The operation's value on a... |
| aovov0bi 47113 | The operation's value on a... |
| rspceaov 47114 | A frequently used special ... |
| fnotaovb 47115 | Equivalence of operation v... |
| ffnaov 47116 | An operation maps to a cla... |
| faovcl 47117 | Closure law for an operati... |
| aovmpt4g 47118 | Value of a function given ... |
| aoprssdm 47119 | Domain of closure of an op... |
| ndmaovcl 47120 | The "closure" of an operat... |
| ndmaovrcl 47121 | Reverse closure law, in co... |
| ndmaovcom 47122 | Any operation is commutati... |
| ndmaovass 47123 | Any operation is associati... |
| ndmaovdistr 47124 | Any operation is distribut... |
| dfatafv2iota 47127 | If a function is defined a... |
| ndfatafv2 47128 | The alternate function val... |
| ndfatafv2undef 47129 | The alternate function val... |
| dfatafv2ex 47130 | The alternate function val... |
| afv2ex 47131 | The alternate function val... |
| afv2eq12d 47132 | Equality deduction for fun... |
| afv2eq1 47133 | Equality theorem for funct... |
| afv2eq2 47134 | Equality theorem for funct... |
| nfafv2 47135 | Bound-variable hypothesis ... |
| csbafv212g 47136 | Move class substitution in... |
| fexafv2ex 47137 | The alternate function val... |
| ndfatafv2nrn 47138 | The alternate function val... |
| ndmafv2nrn 47139 | The value of a class outsi... |
| funressndmafv2rn 47140 | The alternate function val... |
| afv2ndefb 47141 | Two ways to say that an al... |
| nfunsnafv2 47142 | If the restriction of a cl... |
| afv2prc 47143 | A function's value at a pr... |
| dfatafv2rnb 47144 | The alternate function val... |
| afv2orxorb 47145 | If a set is in the range o... |
| dmafv2rnb 47146 | The alternate function val... |
| fundmafv2rnb 47147 | The alternate function val... |
| afv2elrn 47148 | An alternate function valu... |
| afv20defat 47149 | If the alternate function ... |
| fnafv2elrn 47150 | An alternate function valu... |
| fafv2elcdm 47151 | An alternate function valu... |
| fafv2elrnb 47152 | An alternate function valu... |
| fcdmvafv2v 47153 | If the codomain of a funct... |
| tz6.12-2-afv2 47154 | Function value when ` F ` ... |
| afv2eu 47155 | The value of a function at... |
| afv2res 47156 | The value of a restricted ... |
| tz6.12-afv2 47157 | Function value (Theorem 6.... |
| tz6.12-1-afv2 47158 | Function value (Theorem 6.... |
| tz6.12c-afv2 47159 | Corollary of Theorem 6.12(... |
| tz6.12i-afv2 47160 | Corollary of Theorem 6.12(... |
| funressnbrafv2 47161 | The second argument of a b... |
| dfatbrafv2b 47162 | Equivalence of function va... |
| dfatopafv2b 47163 | Equivalence of function va... |
| funbrafv2 47164 | The second argument of a b... |
| fnbrafv2b 47165 | Equivalence of function va... |
| fnopafv2b 47166 | Equivalence of function va... |
| funbrafv22b 47167 | Equivalence of function va... |
| funopafv2b 47168 | Equivalence of function va... |
| dfatsnafv2 47169 | Singleton of function valu... |
| dfafv23 47170 | A definition of function v... |
| dfatdmfcoafv2 47171 | Domain of a function compo... |
| dfatcolem 47172 | Lemma for ~ dfatco . (Con... |
| dfatco 47173 | The predicate "defined at"... |
| afv2co2 47174 | Value of a function compos... |
| rlimdmafv2 47175 | Two ways to express that a... |
| dfafv22 47176 | Alternate definition of ` ... |
| afv2ndeffv0 47177 | If the alternate function ... |
| dfatafv2eqfv 47178 | If a function is defined a... |
| afv2rnfveq 47179 | If the alternate function ... |
| afv20fv0 47180 | If the alternate function ... |
| afv2fvn0fveq 47181 | If the function's value at... |
| afv2fv0 47182 | If the function's value at... |
| afv2fv0b 47183 | The function's value at an... |
| afv2fv0xorb 47184 | If a set is in the range o... |
| an4com24 47185 | Rearrangement of 4 conjunc... |
| 3an4ancom24 47186 | Commutative law for a conj... |
| 4an21 47187 | Rearrangement of 4 conjunc... |
| dfnelbr2 47190 | Alternate definition of th... |
| nelbr 47191 | The binary relation of a s... |
| nelbrim 47192 | If a set is related to ano... |
| nelbrnel 47193 | A set is related to anothe... |
| nelbrnelim 47194 | If a set is related to ano... |
| ralralimp 47195 | Selecting one of two alter... |
| otiunsndisjX 47196 | The union of singletons co... |
| fvifeq 47197 | Equality of function value... |
| rnfdmpr 47198 | The range of a one-to-one ... |
| imarnf1pr 47199 | The image of the range of ... |
| funop1 47200 | A function is an ordered p... |
| fun2dmnopgexmpl 47201 | A function with a domain c... |
| opabresex0d 47202 | A collection of ordered pa... |
| opabbrfex0d 47203 | A collection of ordered pa... |
| opabresexd 47204 | A collection of ordered pa... |
| opabbrfexd 47205 | A collection of ordered pa... |
| f1oresf1orab 47206 | Build a bijection by restr... |
| f1oresf1o 47207 | Build a bijection by restr... |
| f1oresf1o2 47208 | Build a bijection by restr... |
| fvmptrab 47209 | Value of a function mappin... |
| fvmptrabdm 47210 | Value of a function mappin... |
| cnambpcma 47211 | ((a-b)+c)-a = c-a holds fo... |
| cnapbmcpd 47212 | ((a+b)-c)+d = ((a+d)+b)-c ... |
| addsubeq0 47213 | The sum of two complex num... |
| leaddsuble 47214 | Addition and subtraction o... |
| 2leaddle2 47215 | If two real numbers are le... |
| ltnltne 47216 | Variant of trichotomy law ... |
| p1lep2 47217 | A real number increasd by ... |
| ltsubsubaddltsub 47218 | If the result of subtracti... |
| zm1nn 47219 | An integer minus 1 is posi... |
| readdcnnred 47220 | The sum of a real number a... |
| resubcnnred 47221 | The difference of a real n... |
| recnmulnred 47222 | The product of a real numb... |
| cndivrenred 47223 | The quotient of an imagina... |
| sqrtnegnre 47224 | The square root of a negat... |
| nn0resubcl 47225 | Closure law for subtractio... |
| zgeltp1eq 47226 | If an integer is between a... |
| 1t10e1p1e11 47227 | 11 is 1 times 10 to the po... |
| deccarry 47228 | Add 1 to a 2 digit number ... |
| eluzge0nn0 47229 | If an integer is greater t... |
| nltle2tri 47230 | Negated extended trichotom... |
| ssfz12 47231 | Subset relationship for fi... |
| elfz2z 47232 | Membership of an integer i... |
| 2elfz3nn0 47233 | If there are two elements ... |
| fz0addcom 47234 | The addition of two member... |
| 2elfz2melfz 47235 | If the sum of two integers... |
| fz0addge0 47236 | The sum of two integers in... |
| elfzlble 47237 | Membership of an integer i... |
| elfzelfzlble 47238 | Membership of an element o... |
| fzopred 47239 | Join a predecessor to the ... |
| fzopredsuc 47240 | Join a predecessor and a s... |
| 1fzopredsuc 47241 | Join 0 and a successor to ... |
| el1fzopredsuc 47242 | An element of an open inte... |
| subsubelfzo0 47243 | Subtracting a difference f... |
| 2ffzoeq 47244 | Two functions over a half-... |
| m1mod0mod1 47245 | An integer decreased by 1 ... |
| elmod2 47246 | An integer modulo 2 is eit... |
| smonoord 47247 | Ordering relation for a st... |
| fsummsndifre 47248 | A finite sum with one of i... |
| fsumsplitsndif 47249 | Separate out a term in a f... |
| fsummmodsndifre 47250 | A finite sum of summands m... |
| fsummmodsnunz 47251 | A finite sum of summands m... |
| setsidel 47252 | The injected slot is an el... |
| setsnidel 47253 | The injected slot is an el... |
| setsv 47254 | The value of the structure... |
| preimafvsnel 47255 | The preimage of a function... |
| preimafvn0 47256 | The preimage of a function... |
| uniimafveqt 47257 | The union of the image of ... |
| uniimaprimaeqfv 47258 | The union of the image of ... |
| setpreimafvex 47259 | The class ` P ` of all pre... |
| elsetpreimafvb 47260 | The characterization of an... |
| elsetpreimafv 47261 | An element of the class ` ... |
| elsetpreimafvssdm 47262 | An element of the class ` ... |
| fvelsetpreimafv 47263 | There is an element in a p... |
| preimafvelsetpreimafv 47264 | The preimage of a function... |
| preimafvsspwdm 47265 | The class ` P ` of all pre... |
| 0nelsetpreimafv 47266 | The empty set is not an el... |
| elsetpreimafvbi 47267 | An element of the preimage... |
| elsetpreimafveqfv 47268 | The elements of the preima... |
| eqfvelsetpreimafv 47269 | If an element of the domai... |
| elsetpreimafvrab 47270 | An element of the preimage... |
| imaelsetpreimafv 47271 | The image of an element of... |
| uniimaelsetpreimafv 47272 | The union of the image of ... |
| elsetpreimafveq 47273 | If two preimages of functi... |
| fundcmpsurinjlem1 47274 | Lemma 1 for ~ fundcmpsurin... |
| fundcmpsurinjlem2 47275 | Lemma 2 for ~ fundcmpsurin... |
| fundcmpsurinjlem3 47276 | Lemma 3 for ~ fundcmpsurin... |
| imasetpreimafvbijlemf 47277 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv 47278 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv1 47279 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemf1 47280 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfo 47281 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbij 47282 | The mapping ` H ` is a bij... |
| fundcmpsurbijinjpreimafv 47283 | Every function ` F : A -->... |
| fundcmpsurinjpreimafv 47284 | Every function ` F : A -->... |
| fundcmpsurinj 47285 | Every function ` F : A -->... |
| fundcmpsurbijinj 47286 | Every function ` F : A -->... |
| fundcmpsurinjimaid 47287 | Every function ` F : A -->... |
| fundcmpsurinjALT 47288 | Alternate proof of ~ fundc... |
| iccpval 47291 | Partition consisting of a ... |
| iccpart 47292 | A special partition. Corr... |
| iccpartimp 47293 | Implications for a class b... |
| iccpartres 47294 | The restriction of a parti... |
| iccpartxr 47295 | If there is a partition, t... |
| iccpartgtprec 47296 | If there is a partition, t... |
| iccpartipre 47297 | If there is a partition, t... |
| iccpartiltu 47298 | If there is a partition, t... |
| iccpartigtl 47299 | If there is a partition, t... |
| iccpartlt 47300 | If there is a partition, t... |
| iccpartltu 47301 | If there is a partition, t... |
| iccpartgtl 47302 | If there is a partition, t... |
| iccpartgt 47303 | If there is a partition, t... |
| iccpartleu 47304 | If there is a partition, t... |
| iccpartgel 47305 | If there is a partition, t... |
| iccpartrn 47306 | If there is a partition, t... |
| iccpartf 47307 | The range of the partition... |
| iccpartel 47308 | If there is a partition, t... |
| iccelpart 47309 | An element of any partitio... |
| iccpartiun 47310 | A half-open interval of ex... |
| icceuelpartlem 47311 | Lemma for ~ icceuelpart . ... |
| icceuelpart 47312 | An element of a partitione... |
| iccpartdisj 47313 | The segments of a partitio... |
| iccpartnel 47314 | A point of a partition is ... |
| fargshiftfv 47315 | If a class is a function, ... |
| fargshiftf 47316 | If a class is a function, ... |
| fargshiftf1 47317 | If a function is 1-1, then... |
| fargshiftfo 47318 | If a function is onto, the... |
| fargshiftfva 47319 | The values of a shifted fu... |
| lswn0 47320 | The last symbol of a not e... |
| nfich1 47323 | The first interchangeable ... |
| nfich2 47324 | The second interchangeable... |
| ichv 47325 | Setvar variables are inter... |
| ichf 47326 | Setvar variables are inter... |
| ichid 47327 | A setvar variable is alway... |
| icht 47328 | A theorem is interchangeab... |
| ichbidv 47329 | Formula building rule for ... |
| ichcircshi 47330 | The setvar variables are i... |
| ichan 47331 | If two setvar variables ar... |
| ichn 47332 | Negation does not affect i... |
| ichim 47333 | Formula building rule for ... |
| dfich2 47334 | Alternate definition of th... |
| ichcom 47335 | The interchangeability of ... |
| ichbi12i 47336 | Equivalence for interchang... |
| icheqid 47337 | In an equality for the sam... |
| icheq 47338 | In an equality of setvar v... |
| ichnfimlem 47339 | Lemma for ~ ichnfim : A s... |
| ichnfim 47340 | If in an interchangeabilit... |
| ichnfb 47341 | If ` x ` and ` y ` are int... |
| ichal 47342 | Move a universal quantifie... |
| ich2al 47343 | Two setvar variables are a... |
| ich2ex 47344 | Two setvar variables are a... |
| ichexmpl1 47345 | Example for interchangeabl... |
| ichexmpl2 47346 | Example for interchangeabl... |
| ich2exprop 47347 | If the setvar variables ar... |
| ichnreuop 47348 | If the setvar variables ar... |
| ichreuopeq 47349 | If the setvar variables ar... |
| sprid 47350 | Two identical representati... |
| elsprel 47351 | An unordered pair is an el... |
| spr0nelg 47352 | The empty set is not an el... |
| sprval 47355 | The set of all unordered p... |
| sprvalpw 47356 | The set of all unordered p... |
| sprssspr 47357 | The set of all unordered p... |
| spr0el 47358 | The empty set is not an un... |
| sprvalpwn0 47359 | The set of all unordered p... |
| sprel 47360 | An element of the set of a... |
| prssspr 47361 | An element of a subset of ... |
| prelspr 47362 | An unordered pair of eleme... |
| prsprel 47363 | The elements of a pair fro... |
| prsssprel 47364 | The elements of a pair fro... |
| sprvalpwle2 47365 | The set of all unordered p... |
| sprsymrelfvlem 47366 | Lemma for ~ sprsymrelf and... |
| sprsymrelf1lem 47367 | Lemma for ~ sprsymrelf1 . ... |
| sprsymrelfolem1 47368 | Lemma 1 for ~ sprsymrelfo ... |
| sprsymrelfolem2 47369 | Lemma 2 for ~ sprsymrelfo ... |
| sprsymrelfv 47370 | The value of the function ... |
| sprsymrelf 47371 | The mapping ` F ` is a fun... |
| sprsymrelf1 47372 | The mapping ` F ` is a one... |
| sprsymrelfo 47373 | The mapping ` F ` is a fun... |
| sprsymrelf1o 47374 | The mapping ` F ` is a bij... |
| sprbisymrel 47375 | There is a bijection betwe... |
| sprsymrelen 47376 | The class ` P ` of subsets... |
| prpair 47377 | Characterization of a prop... |
| prproropf1olem0 47378 | Lemma 0 for ~ prproropf1o ... |
| prproropf1olem1 47379 | Lemma 1 for ~ prproropf1o ... |
| prproropf1olem2 47380 | Lemma 2 for ~ prproropf1o ... |
| prproropf1olem3 47381 | Lemma 3 for ~ prproropf1o ... |
| prproropf1olem4 47382 | Lemma 4 for ~ prproropf1o ... |
| prproropf1o 47383 | There is a bijection betwe... |
| prproropen 47384 | The set of proper pairs an... |
| prproropreud 47385 | There is exactly one order... |
| pairreueq 47386 | Two equivalent representat... |
| paireqne 47387 | Two sets are not equal iff... |
| prprval 47390 | The set of all proper unor... |
| prprvalpw 47391 | The set of all proper unor... |
| prprelb 47392 | An element of the set of a... |
| prprelprb 47393 | A set is an element of the... |
| prprspr2 47394 | The set of all proper unor... |
| prprsprreu 47395 | There is a unique proper u... |
| prprreueq 47396 | There is a unique proper u... |
| sbcpr 47397 | The proper substitution of... |
| reupr 47398 | There is a unique unordere... |
| reuprpr 47399 | There is a unique proper u... |
| poprelb 47400 | Equality for unordered pai... |
| 2exopprim 47401 | The existence of an ordere... |
| reuopreuprim 47402 | There is a unique unordere... |
| fmtno 47405 | The ` N ` th Fermat number... |
| fmtnoge3 47406 | Each Fermat number is grea... |
| fmtnonn 47407 | Each Fermat number is a po... |
| fmtnom1nn 47408 | A Fermat number minus one ... |
| fmtnoodd 47409 | Each Fermat number is odd.... |
| fmtnorn 47410 | A Fermat number is a funct... |
| fmtnof1 47411 | The enumeration of the Fer... |
| fmtnoinf 47412 | The set of Fermat numbers ... |
| fmtnorec1 47413 | The first recurrence relat... |
| sqrtpwpw2p 47414 | The floor of the square ro... |
| fmtnosqrt 47415 | The floor of the square ro... |
| fmtno0 47416 | The ` 0 ` th Fermat number... |
| fmtno1 47417 | The ` 1 ` st Fermat number... |
| fmtnorec2lem 47418 | Lemma for ~ fmtnorec2 (ind... |
| fmtnorec2 47419 | The second recurrence rela... |
| fmtnodvds 47420 | Any Fermat number divides ... |
| goldbachthlem1 47421 | Lemma 1 for ~ goldbachth .... |
| goldbachthlem2 47422 | Lemma 2 for ~ goldbachth .... |
| goldbachth 47423 | Goldbach's theorem: Two d... |
| fmtnorec3 47424 | The third recurrence relat... |
| fmtnorec4 47425 | The fourth recurrence rela... |
| fmtno2 47426 | The ` 2 ` nd Fermat number... |
| fmtno3 47427 | The ` 3 ` rd Fermat number... |
| fmtno4 47428 | The ` 4 ` th Fermat number... |
| fmtno5lem1 47429 | Lemma 1 for ~ fmtno5 . (C... |
| fmtno5lem2 47430 | Lemma 2 for ~ fmtno5 . (C... |
| fmtno5lem3 47431 | Lemma 3 for ~ fmtno5 . (C... |
| fmtno5lem4 47432 | Lemma 4 for ~ fmtno5 . (C... |
| fmtno5 47433 | The ` 5 ` th Fermat number... |
| fmtno0prm 47434 | The ` 0 ` th Fermat number... |
| fmtno1prm 47435 | The ` 1 ` st Fermat number... |
| fmtno2prm 47436 | The ` 2 ` nd Fermat number... |
| 257prm 47437 | 257 is a prime number (the... |
| fmtno3prm 47438 | The ` 3 ` rd Fermat number... |
| odz2prm2pw 47439 | Any power of two is coprim... |
| fmtnoprmfac1lem 47440 | Lemma for ~ fmtnoprmfac1 :... |
| fmtnoprmfac1 47441 | Divisor of Fermat number (... |
| fmtnoprmfac2lem1 47442 | Lemma for ~ fmtnoprmfac2 .... |
| fmtnoprmfac2 47443 | Divisor of Fermat number (... |
| fmtnofac2lem 47444 | Lemma for ~ fmtnofac2 (Ind... |
| fmtnofac2 47445 | Divisor of Fermat number (... |
| fmtnofac1 47446 | Divisor of Fermat number (... |
| fmtno4sqrt 47447 | The floor of the square ro... |
| fmtno4prmfac 47448 | If P was a (prime) factor ... |
| fmtno4prmfac193 47449 | If P was a (prime) factor ... |
| fmtno4nprmfac193 47450 | 193 is not a (prime) facto... |
| fmtno4prm 47451 | The ` 4 `-th Fermat number... |
| 65537prm 47452 | 65537 is a prime number (t... |
| fmtnofz04prm 47453 | The first five Fermat numb... |
| fmtnole4prm 47454 | The first five Fermat numb... |
| fmtno5faclem1 47455 | Lemma 1 for ~ fmtno5fac . ... |
| fmtno5faclem2 47456 | Lemma 2 for ~ fmtno5fac . ... |
| fmtno5faclem3 47457 | Lemma 3 for ~ fmtno5fac . ... |
| fmtno5fac 47458 | The factorization of the `... |
| fmtno5nprm 47459 | The ` 5 ` th Fermat number... |
| prmdvdsfmtnof1lem1 47460 | Lemma 1 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof1lem2 47461 | Lemma 2 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof 47462 | The mapping of a Fermat nu... |
| prmdvdsfmtnof1 47463 | The mapping of a Fermat nu... |
| prminf2 47464 | The set of prime numbers i... |
| 2pwp1prm 47465 | For ` ( ( 2 ^ k ) + 1 ) ` ... |
| 2pwp1prmfmtno 47466 | Every prime number of the ... |
| m2prm 47467 | The second Mersenne number... |
| m3prm 47468 | The third Mersenne number ... |
| flsqrt 47469 | A condition equivalent to ... |
| flsqrt5 47470 | The floor of the square ro... |
| 3ndvds4 47471 | 3 does not divide 4. (Con... |
| 139prmALT 47472 | 139 is a prime number. In... |
| 31prm 47473 | 31 is a prime number. In ... |
| m5prm 47474 | The fifth Mersenne number ... |
| 127prm 47475 | 127 is a prime number. (C... |
| m7prm 47476 | The seventh Mersenne numbe... |
| m11nprm 47477 | The eleventh Mersenne numb... |
| mod42tp1mod8 47478 | If a number is ` 3 ` modul... |
| sfprmdvdsmersenne 47479 | If ` Q ` is a safe prime (... |
| sgprmdvdsmersenne 47480 | If ` P ` is a Sophie Germa... |
| lighneallem1 47481 | Lemma 1 for ~ lighneal . ... |
| lighneallem2 47482 | Lemma 2 for ~ lighneal . ... |
| lighneallem3 47483 | Lemma 3 for ~ lighneal . ... |
| lighneallem4a 47484 | Lemma 1 for ~ lighneallem4... |
| lighneallem4b 47485 | Lemma 2 for ~ lighneallem4... |
| lighneallem4 47486 | Lemma 3 for ~ lighneal . ... |
| lighneal 47487 | If a power of a prime ` P ... |
| modexp2m1d 47488 | The square of an integer w... |
| proththdlem 47489 | Lemma for ~ proththd . (C... |
| proththd 47490 | Proth's theorem (1878). I... |
| 5tcu2e40 47491 | 5 times the cube of 2 is 4... |
| 3exp4mod41 47492 | 3 to the fourth power is -... |
| 41prothprmlem1 47493 | Lemma 1 for ~ 41prothprm .... |
| 41prothprmlem2 47494 | Lemma 2 for ~ 41prothprm .... |
| 41prothprm 47495 | 41 is a _Proth prime_. (C... |
| quad1 47496 | A condition for a quadrati... |
| requad01 47497 | A condition for a quadrati... |
| requad1 47498 | A condition for a quadrati... |
| requad2 47499 | A condition for a quadrati... |
| iseven 47504 | The predicate "is an even ... |
| isodd 47505 | The predicate "is an odd n... |
| evenz 47506 | An even number is an integ... |
| oddz 47507 | An odd number is an intege... |
| evendiv2z 47508 | The result of dividing an ... |
| oddp1div2z 47509 | The result of dividing an ... |
| oddm1div2z 47510 | The result of dividing an ... |
| isodd2 47511 | The predicate "is an odd n... |
| dfodd2 47512 | Alternate definition for o... |
| dfodd6 47513 | Alternate definition for o... |
| dfeven4 47514 | Alternate definition for e... |
| evenm1odd 47515 | The predecessor of an even... |
| evenp1odd 47516 | The successor of an even n... |
| oddp1eveni 47517 | The successor of an odd nu... |
| oddm1eveni 47518 | The predecessor of an odd ... |
| evennodd 47519 | An even number is not an o... |
| oddneven 47520 | An odd number is not an ev... |
| enege 47521 | The negative of an even nu... |
| onego 47522 | The negative of an odd num... |
| m1expevenALTV 47523 | Exponentiation of -1 by an... |
| m1expoddALTV 47524 | Exponentiation of -1 by an... |
| dfeven2 47525 | Alternate definition for e... |
| dfodd3 47526 | Alternate definition for o... |
| iseven2 47527 | The predicate "is an even ... |
| isodd3 47528 | The predicate "is an odd n... |
| 2dvdseven 47529 | 2 divides an even number. ... |
| m2even 47530 | A multiple of 2 is an even... |
| 2ndvdsodd 47531 | 2 does not divide an odd n... |
| 2dvdsoddp1 47532 | 2 divides an odd number in... |
| 2dvdsoddm1 47533 | 2 divides an odd number de... |
| dfeven3 47534 | Alternate definition for e... |
| dfodd4 47535 | Alternate definition for o... |
| dfodd5 47536 | Alternate definition for o... |
| zefldiv2ALTV 47537 | The floor of an even numbe... |
| zofldiv2ALTV 47538 | The floor of an odd numer ... |
| oddflALTV 47539 | Odd number representation ... |
| iseven5 47540 | The predicate "is an even ... |
| isodd7 47541 | The predicate "is an odd n... |
| dfeven5 47542 | Alternate definition for e... |
| dfodd7 47543 | Alternate definition for o... |
| gcd2odd1 47544 | The greatest common diviso... |
| zneoALTV 47545 | No even integer equals an ... |
| zeoALTV 47546 | An integer is even or odd.... |
| zeo2ALTV 47547 | An integer is even or odd ... |
| nneoALTV 47548 | A positive integer is even... |
| nneoiALTV 47549 | A positive integer is even... |
| odd2np1ALTV 47550 | An integer is odd iff it i... |
| oddm1evenALTV 47551 | An integer is odd iff its ... |
| oddp1evenALTV 47552 | An integer is odd iff its ... |
| oexpnegALTV 47553 | The exponential of the neg... |
| oexpnegnz 47554 | The exponential of the neg... |
| bits0ALTV 47555 | Value of the zeroth bit. ... |
| bits0eALTV 47556 | The zeroth bit of an even ... |
| bits0oALTV 47557 | The zeroth bit of an odd n... |
| divgcdoddALTV 47558 | Either ` A / ( A gcd B ) `... |
| opoeALTV 47559 | The sum of two odds is eve... |
| opeoALTV 47560 | The sum of an odd and an e... |
| omoeALTV 47561 | The difference of two odds... |
| omeoALTV 47562 | The difference of an odd a... |
| oddprmALTV 47563 | A prime not equal to ` 2 `... |
| 0evenALTV 47564 | 0 is an even number. (Con... |
| 0noddALTV 47565 | 0 is not an odd number. (... |
| 1oddALTV 47566 | 1 is an odd number. (Cont... |
| 1nevenALTV 47567 | 1 is not an even number. ... |
| 2evenALTV 47568 | 2 is an even number. (Con... |
| 2noddALTV 47569 | 2 is not an odd number. (... |
| nn0o1gt2ALTV 47570 | An odd nonnegative integer... |
| nnoALTV 47571 | An alternate characterizat... |
| nn0oALTV 47572 | An alternate characterizat... |
| nn0e 47573 | An alternate characterizat... |
| nneven 47574 | An alternate characterizat... |
| nn0onn0exALTV 47575 | For each odd nonnegative i... |
| nn0enn0exALTV 47576 | For each even nonnegative ... |
| nnennexALTV 47577 | For each even positive int... |
| nnpw2evenALTV 47578 | 2 to the power of a positi... |
| epoo 47579 | The sum of an even and an ... |
| emoo 47580 | The difference of an even ... |
| epee 47581 | The sum of two even number... |
| emee 47582 | The difference of two even... |
| evensumeven 47583 | If a summand is even, the ... |
| 3odd 47584 | 3 is an odd number. (Cont... |
| 4even 47585 | 4 is an even number. (Con... |
| 5odd 47586 | 5 is an odd number. (Cont... |
| 6even 47587 | 6 is an even number. (Con... |
| 7odd 47588 | 7 is an odd number. (Cont... |
| 8even 47589 | 8 is an even number. (Con... |
| evenprm2 47590 | A prime number is even iff... |
| oddprmne2 47591 | Every prime number not bei... |
| oddprmuzge3 47592 | A prime number which is od... |
| evenltle 47593 | If an even number is great... |
| odd2prm2 47594 | If an odd number is the su... |
| even3prm2 47595 | If an even number is the s... |
| mogoldbblem 47596 | Lemma for ~ mogoldbb . (C... |
| perfectALTVlem1 47597 | Lemma for ~ perfectALTV . ... |
| perfectALTVlem2 47598 | Lemma for ~ perfectALTV . ... |
| perfectALTV 47599 | The Euclid-Euler theorem, ... |
| fppr 47602 | The set of Fermat pseudopr... |
| fpprmod 47603 | The set of Fermat pseudopr... |
| fpprel 47604 | A Fermat pseudoprime to th... |
| fpprbasnn 47605 | The base of a Fermat pseud... |
| fpprnn 47606 | A Fermat pseudoprime to th... |
| fppr2odd 47607 | A Fermat pseudoprime to th... |
| 11t31e341 47608 | 341 is the product of 11 a... |
| 2exp340mod341 47609 | Eight to the eighth power ... |
| 341fppr2 47610 | 341 is the (smallest) _Pou... |
| 4fppr1 47611 | 4 is the (smallest) Fermat... |
| 8exp8mod9 47612 | Eight to the eighth power ... |
| 9fppr8 47613 | 9 is the (smallest) Fermat... |
| dfwppr 47614 | Alternate definition of a ... |
| fpprwppr 47615 | A Fermat pseudoprime to th... |
| fpprwpprb 47616 | An integer ` X ` which is ... |
| fpprel2 47617 | An alternate definition fo... |
| nfermltl8rev 47618 | Fermat's little theorem wi... |
| nfermltl2rev 47619 | Fermat's little theorem wi... |
| nfermltlrev 47620 | Fermat's little theorem re... |
| isgbe 47627 | The predicate "is an even ... |
| isgbow 47628 | The predicate "is a weak o... |
| isgbo 47629 | The predicate "is an odd G... |
| gbeeven 47630 | An even Goldbach number is... |
| gbowodd 47631 | A weak odd Goldbach number... |
| gbogbow 47632 | A (strong) odd Goldbach nu... |
| gboodd 47633 | An odd Goldbach number is ... |
| gbepos 47634 | Any even Goldbach number i... |
| gbowpos 47635 | Any weak odd Goldbach numb... |
| gbopos 47636 | Any odd Goldbach number is... |
| gbegt5 47637 | Any even Goldbach number i... |
| gbowgt5 47638 | Any weak odd Goldbach numb... |
| gbowge7 47639 | Any weak odd Goldbach numb... |
| gboge9 47640 | Any odd Goldbach number is... |
| gbege6 47641 | Any even Goldbach number i... |
| gbpart6 47642 | The Goldbach partition of ... |
| gbpart7 47643 | The (weak) Goldbach partit... |
| gbpart8 47644 | The Goldbach partition of ... |
| gbpart9 47645 | The (strong) Goldbach part... |
| gbpart11 47646 | The (strong) Goldbach part... |
| 6gbe 47647 | 6 is an even Goldbach numb... |
| 7gbow 47648 | 7 is a weak odd Goldbach n... |
| 8gbe 47649 | 8 is an even Goldbach numb... |
| 9gbo 47650 | 9 is an odd Goldbach numbe... |
| 11gbo 47651 | 11 is an odd Goldbach numb... |
| stgoldbwt 47652 | If the strong ternary Gold... |
| sbgoldbwt 47653 | If the strong binary Goldb... |
| sbgoldbst 47654 | If the strong binary Goldb... |
| sbgoldbaltlem1 47655 | Lemma 1 for ~ sbgoldbalt :... |
| sbgoldbaltlem2 47656 | Lemma 2 for ~ sbgoldbalt :... |
| sbgoldbalt 47657 | An alternate (related to t... |
| sbgoldbb 47658 | If the strong binary Goldb... |
| sgoldbeven3prm 47659 | If the binary Goldbach con... |
| sbgoldbm 47660 | If the strong binary Goldb... |
| mogoldbb 47661 | If the modern version of t... |
| sbgoldbmb 47662 | The strong binary Goldbach... |
| sbgoldbo 47663 | If the strong binary Goldb... |
| nnsum3primes4 47664 | 4 is the sum of at most 3 ... |
| nnsum4primes4 47665 | 4 is the sum of at most 4 ... |
| nnsum3primesprm 47666 | Every prime is "the sum of... |
| nnsum4primesprm 47667 | Every prime is "the sum of... |
| nnsum3primesgbe 47668 | Any even Goldbach number i... |
| nnsum4primesgbe 47669 | Any even Goldbach number i... |
| nnsum3primesle9 47670 | Every integer greater than... |
| nnsum4primesle9 47671 | Every integer greater than... |
| nnsum4primesodd 47672 | If the (weak) ternary Gold... |
| nnsum4primesoddALTV 47673 | If the (strong) ternary Go... |
| evengpop3 47674 | If the (weak) ternary Gold... |
| evengpoap3 47675 | If the (strong) ternary Go... |
| nnsum4primeseven 47676 | If the (weak) ternary Gold... |
| nnsum4primesevenALTV 47677 | If the (strong) ternary Go... |
| wtgoldbnnsum4prm 47678 | If the (weak) ternary Gold... |
| stgoldbnnsum4prm 47679 | If the (strong) ternary Go... |
| bgoldbnnsum3prm 47680 | If the binary Goldbach con... |
| bgoldbtbndlem1 47681 | Lemma 1 for ~ bgoldbtbnd :... |
| bgoldbtbndlem2 47682 | Lemma 2 for ~ bgoldbtbnd .... |
| bgoldbtbndlem3 47683 | Lemma 3 for ~ bgoldbtbnd .... |
| bgoldbtbndlem4 47684 | Lemma 4 for ~ bgoldbtbnd .... |
| bgoldbtbnd 47685 | If the binary Goldbach con... |
| tgoldbachgtALTV 47688 | Variant of Thierry Arnoux'... |
| bgoldbachlt 47689 | The binary Goldbach conjec... |
| tgblthelfgott 47691 | The ternary Goldbach conje... |
| tgoldbachlt 47692 | The ternary Goldbach conje... |
| tgoldbach 47693 | The ternary Goldbach conje... |
| clnbgrprc0 47696 | The closed neighborhood is... |
| clnbgrcl 47697 | If a class ` X ` has at le... |
| clnbgrval 47698 | The closed neighborhood of... |
| dfclnbgr2 47699 | Alternate definition of th... |
| dfclnbgr4 47700 | Alternate definition of th... |
| dfclnbgr3 47701 | Alternate definition of th... |
| clnbgrnvtx0 47702 | If a class ` X ` is not a ... |
| clnbgrel 47703 | Characterization of a memb... |
| clnbgrvtxel 47704 | Every vertex ` K ` is a me... |
| clnbgrisvtx 47705 | Every member ` N ` of the ... |
| clnbgrssvtx 47706 | The closed neighborhood of... |
| clnbgrn0 47707 | The closed neighborhood of... |
| clnbupgr 47708 | The closed neighborhood of... |
| clnbupgrel 47709 | A member of the closed nei... |
| clnbgr0vtx 47710 | In a null graph (with no v... |
| clnbgr0edg 47711 | In an empty graph (with no... |
| clnbgrsym 47712 | In a graph, the closed nei... |
| predgclnbgrel 47713 | If a (not necessarily prop... |
| clnbgredg 47714 | A vertices connected by an... |
| clnbgrssedg 47715 | The vertices connected by ... |
| edgusgrclnbfin 47716 | The size of the closed nei... |
| clnbusgrfi 47717 | The closed neighborhood of... |
| clnbfiusgrfi 47718 | The closed neighborhood of... |
| clnbgrlevtx 47719 | The size of the closed nei... |
| dfsclnbgr2 47720 | Alternate definition of th... |
| sclnbgrel 47721 | Characterization of a memb... |
| sclnbgrelself 47722 | A vertex ` N ` is a member... |
| sclnbgrisvtx 47723 | Every member ` X ` of the ... |
| dfclnbgr5 47724 | Alternate definition of th... |
| dfnbgr5 47725 | Alternate definition of th... |
| dfnbgrss 47726 | Subset chain for different... |
| dfvopnbgr2 47727 | Alternate definition of th... |
| vopnbgrel 47728 | Characterization of a memb... |
| vopnbgrelself 47729 | A vertex ` N ` is a member... |
| dfclnbgr6 47730 | Alternate definition of th... |
| dfnbgr6 47731 | Alternate definition of th... |
| dfsclnbgr6 47732 | Alternate definition of a ... |
| dfnbgrss2 47733 | Subset chain for different... |
| isisubgr 47736 | The subgraph induced by a ... |
| isubgriedg 47737 | The edges of an induced su... |
| isubgrvtxuhgr 47738 | The subgraph induced by th... |
| isubgrvtx 47739 | The vertices of an induced... |
| isubgruhgr 47740 | An induced subgraph of a h... |
| isubgrsubgr 47741 | An induced subgraph of a h... |
| isubgrupgr 47742 | An induced subgraph of a p... |
| isubgrumgr 47743 | An induced subgraph of a m... |
| isubgrusgr 47744 | An induced subgraph of a s... |
| isubgr0uhgr 47745 | The subgraph induced by an... |
| grimfn 47751 | The graph isomorphism func... |
| grimdmrel 47752 | The domain of the graph is... |
| isgrim 47754 | An isomorphism of graphs i... |
| grimprop 47755 | Properties of an isomorphi... |
| grimf1o 47756 | An isomorphism of graphs i... |
| isuspgrim0lem 47757 | An isomorphism of simple p... |
| isuspgrim0 47758 | An isomorphism of simple p... |
| uspgrimprop 47759 | An isomorphism of simple p... |
| isuspgrimlem 47760 | Lemma for ~ isuspgrim . (... |
| isuspgrim 47761 | A class is an isomorphism ... |
| grimidvtxedg 47762 | The identity relation rest... |
| grimid 47763 | The identity relation rest... |
| grimuhgr 47764 | If there is a graph isomor... |
| grimcnv 47765 | The converse of a graph is... |
| grimco 47766 | The composition of graph i... |
| brgric 47767 | The relation "is isomorphi... |
| brgrici 47768 | Prove that two graphs are ... |
| gricrcl 47769 | Reverse closure of the "is... |
| dfgric2 47770 | Alternate, explicit defini... |
| gricbri 47771 | Implications of two graphs... |
| gricushgr 47772 | The "is isomorphic to" rel... |
| gricuspgr 47773 | The "is isomorphic to" rel... |
| gricrel 47774 | The "is isomorphic to" rel... |
| gricref 47775 | Graph isomorphism is refle... |
| gricsym 47776 | Graph isomorphism is symme... |
| gricsymb 47777 | Graph isomorphism is symme... |
| grictr 47778 | Graph isomorphism is trans... |
| gricer 47779 | Isomorphism is an equivale... |
| gricen 47780 | Isomorphic graphs have equ... |
| opstrgric 47781 | A graph represented as an ... |
| ushggricedg 47782 | A simple hypergraph (with ... |
| isubgrgrim 47783 | Isomorphic subgraphs induc... |
| uhgrimisgrgriclem 47784 | Lemma for ~ uhgrimisgrgric... |
| uhgrimisgrgric 47785 | For isomorphic hypergraphs... |
| clnbgrisubgrgrim 47786 | Isomorphic subgraphs induc... |
| clnbgrgrimlem 47787 | Lemma for ~ clnbgrgrim : ... |
| clnbgrgrim 47788 | Graph isomorphisms between... |
| grimedg 47789 | Graph isomorphisms map edg... |
| grtriproplem 47792 | Lemma for ~ grtriprop . (... |
| grtri 47793 | The triangles in a graph. ... |
| grtriprop 47794 | The properties of a triang... |
| grtrif1o 47795 | Any bijection onto a trian... |
| isgrtri 47796 | A triangle in a graph. (C... |
| grtrissvtx 47797 | A triangle is a subset of ... |
| grtriclwlk3 47798 | A triangle induces a close... |
| grtrimap 47799 | Conditions for mapping tri... |
| grimgrtri 47800 | Graph isomorphisms map tri... |
| usgrgrtrirex 47801 | Conditions for a simple gr... |
| grlimfn 47805 | The graph local isomorphis... |
| grlimdmrel 47806 | The domain of the graph lo... |
| isgrlim 47808 | A local isomorphism of gra... |
| isgrlim2 47809 | A local isomorphism of gra... |
| grlimprop 47810 | Properties of a local isom... |
| grlimf1o 47811 | A local isomorphism of gra... |
| grlimprop2 47812 | Properties of a local isom... |
| uhgrimgrlim 47813 | An isomorphism of hypergra... |
| uspgrlimlem1 47814 | Lemma 1 for ~ uspgrlim . ... |
| uspgrlimlem2 47815 | Lemma 2 for ~ uspgrlim . ... |
| uspgrlimlem3 47816 | Lemma 3 for ~ uspgrlim . ... |
| uspgrlimlem4 47817 | Lemma 4 for ~ uspgrlim . ... |
| uspgrlim 47818 | A local isomorphism of sim... |
| usgrlimprop 47819 | Properties of a local isom... |
| grlimgrtrilem1 47820 | Lemma 3 for ~ grlimgrtri .... |
| grlimgrtrilem2 47821 | Lemma 3 for ~ grlimgrtri .... |
| grlimgrtri 47822 | Local isomorphisms between... |
| brgrlic 47823 | The relation "is locally i... |
| brgrilci 47824 | Prove that two graphs are ... |
| grlicrel 47825 | The "is locally isomorphic... |
| grlicrcl 47826 | Reverse closure of the "is... |
| dfgrlic2 47827 | Alternate, explicit defini... |
| grilcbri 47828 | Implications of two graphs... |
| dfgrlic3 47829 | Alternate, explicit defini... |
| grilcbri2 47830 | Implications of two graphs... |
| grlicref 47831 | Graph local isomorphism is... |
| grlicsym 47832 | Graph local isomorphism is... |
| grlicsymb 47833 | Graph local isomorphism is... |
| grlictr 47834 | Graph local isomorphism is... |
| grlicer 47835 | Local isomorphism is an eq... |
| grlicen 47836 | Locally isomorphic graphs ... |
| gricgrlic 47837 | Isomorphic hypergraphs are... |
| usgrexmpl1lem 47838 | Lemma for ~ usgrexmpl1 . ... |
| usgrexmpl1 47839 | ` G ` is a simple graph of... |
| usgrexmpl1vtx 47840 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpl1edg 47841 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl1tri 47842 | ` G ` contains a triangle ... |
| usgrexmpl2lem 47843 | Lemma for ~ usgrexmpl2 . ... |
| usgrexmpl2 47844 | ` G ` is a simple graph of... |
| usgrexmpl2vtx 47845 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpl2edg 47846 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl2nblem 47847 | Lemma for ~ usgrexmpl2nb0 ... |
| usgrexmpl2nb0 47848 | The neighborhood of the fi... |
| usgrexmpl2nb1 47849 | The neighborhood of the se... |
| usgrexmpl2nb2 47850 | The neighborhood of the th... |
| usgrexmpl2nb3 47851 | The neighborhood of the fo... |
| usgrexmpl2nb4 47852 | The neighborhood of the fi... |
| usgrexmpl2nb5 47853 | The neighborhood of the si... |
| usgrexmpl2trifr 47854 | ` G ` is triangle-free. (... |
| usgrexmpl12ngric 47855 | The graphs ` H ` and ` G `... |
| usgrexmpl12ngrlic 47856 | The graphs ` H ` and ` G `... |
| 1hegrlfgr 47857 | A graph ` G ` with one hyp... |
| upwlksfval 47860 | The set of simple walks (i... |
| isupwlk 47861 | Properties of a pair of fu... |
| isupwlkg 47862 | Generalization of ~ isupwl... |
| upwlkbprop 47863 | Basic properties of a simp... |
| upwlkwlk 47864 | A simple walk is a walk. ... |
| upgrwlkupwlk 47865 | In a pseudograph, a walk i... |
| upgrwlkupwlkb 47866 | In a pseudograph, the defi... |
| upgrisupwlkALT 47867 | Alternate proof of ~ upgri... |
| upgredgssspr 47868 | The set of edges of a pseu... |
| uspgropssxp 47869 | The set ` G ` of "simple p... |
| uspgrsprfv 47870 | The value of the function ... |
| uspgrsprf 47871 | The mapping ` F ` is a fun... |
| uspgrsprf1 47872 | The mapping ` F ` is a one... |
| uspgrsprfo 47873 | The mapping ` F ` is a fun... |
| uspgrsprf1o 47874 | The mapping ` F ` is a bij... |
| uspgrex 47875 | The class ` G ` of all "si... |
| uspgrbispr 47876 | There is a bijection betwe... |
| uspgrspren 47877 | The set ` G ` of the "simp... |
| uspgrymrelen 47878 | The set ` G ` of the "simp... |
| uspgrbisymrel 47879 | There is a bijection betwe... |
| uspgrbisymrelALT 47880 | Alternate proof of ~ uspgr... |
| ovn0dmfun 47881 | If a class operation value... |
| xpsnopab 47882 | A Cartesian product with a... |
| xpiun 47883 | A Cartesian product expres... |
| ovn0ssdmfun 47884 | If a class' operation valu... |
| fnxpdmdm 47885 | The domain of the domain o... |
| cnfldsrngbas 47886 | The base set of a subring ... |
| cnfldsrngadd 47887 | The group addition operati... |
| cnfldsrngmul 47888 | The ring multiplication op... |
| plusfreseq 47889 | If the empty set is not co... |
| mgmplusfreseq 47890 | If the empty set is not co... |
| 0mgm 47891 | A set with an empty base s... |
| opmpoismgm 47892 | A structure with a group a... |
| copissgrp 47893 | A structure with a constan... |
| copisnmnd 47894 | A structure with a constan... |
| 0nodd 47895 | 0 is not an odd integer. ... |
| 1odd 47896 | 1 is an odd integer. (Con... |
| 2nodd 47897 | 2 is not an odd integer. ... |
| oddibas 47898 | Lemma 1 for ~ oddinmgm : ... |
| oddiadd 47899 | Lemma 2 for ~ oddinmgm : ... |
| oddinmgm 47900 | The structure of all odd i... |
| nnsgrpmgm 47901 | The structure of positive ... |
| nnsgrp 47902 | The structure of positive ... |
| nnsgrpnmnd 47903 | The structure of positive ... |
| nn0mnd 47904 | The set of nonnegative int... |
| gsumsplit2f 47905 | Split a group sum into two... |
| gsumdifsndf 47906 | Extract a summand from a f... |
| gsumfsupp 47907 | A group sum of a family ca... |
| iscllaw 47914 | The predicate "is a closed... |
| iscomlaw 47915 | The predicate "is a commut... |
| clcllaw 47916 | Closure of a closed operat... |
| isasslaw 47917 | The predicate "is an assoc... |
| asslawass 47918 | Associativity of an associ... |
| mgmplusgiopALT 47919 | Slot 2 (group operation) o... |
| sgrpplusgaopALT 47920 | Slot 2 (group operation) o... |
| intopval 47927 | The internal (binary) oper... |
| intop 47928 | An internal (binary) opera... |
| clintopval 47929 | The closed (internal binar... |
| assintopval 47930 | The associative (closed in... |
| assintopmap 47931 | The associative (closed in... |
| isclintop 47932 | The predicate "is a closed... |
| clintop 47933 | A closed (internal binary)... |
| assintop 47934 | An associative (closed int... |
| isassintop 47935 | The predicate "is an assoc... |
| clintopcllaw 47936 | The closure law holds for ... |
| assintopcllaw 47937 | The closure low holds for ... |
| assintopasslaw 47938 | The associative low holds ... |
| assintopass 47939 | An associative (closed int... |
| ismgmALT 47948 | The predicate "is a magma"... |
| iscmgmALT 47949 | The predicate "is a commut... |
| issgrpALT 47950 | The predicate "is a semigr... |
| iscsgrpALT 47951 | The predicate "is a commut... |
| mgm2mgm 47952 | Equivalence of the two def... |
| sgrp2sgrp 47953 | Equivalence of the two def... |
| lmod0rng 47954 | If the scalar ring of a mo... |
| nzrneg1ne0 47955 | The additive inverse of th... |
| lidldomn1 47956 | If a (left) ideal (which i... |
| lidlabl 47957 | A (left) ideal of a ring i... |
| lidlrng 47958 | A (left) ideal of a ring i... |
| zlidlring 47959 | The zero (left) ideal of a... |
| uzlidlring 47960 | Only the zero (left) ideal... |
| lidldomnnring 47961 | A (left) ideal of a domain... |
| 0even 47962 | 0 is an even integer. (Co... |
| 1neven 47963 | 1 is not an even integer. ... |
| 2even 47964 | 2 is an even integer. (Co... |
| 2zlidl 47965 | The even integers are a (l... |
| 2zrng 47966 | The ring of integers restr... |
| 2zrngbas 47967 | The base set of R is the s... |
| 2zrngadd 47968 | The group addition operati... |
| 2zrng0 47969 | The additive identity of R... |
| 2zrngamgm 47970 | R is an (additive) magma. ... |
| 2zrngasgrp 47971 | R is an (additive) semigro... |
| 2zrngamnd 47972 | R is an (additive) monoid.... |
| 2zrngacmnd 47973 | R is a commutative (additi... |
| 2zrngagrp 47974 | R is an (additive) group. ... |
| 2zrngaabl 47975 | R is an (additive) abelian... |
| 2zrngmul 47976 | The ring multiplication op... |
| 2zrngmmgm 47977 | R is a (multiplicative) ma... |
| 2zrngmsgrp 47978 | R is a (multiplicative) se... |
| 2zrngALT 47979 | The ring of integers restr... |
| 2zrngnmlid 47980 | R has no multiplicative (l... |
| 2zrngnmrid 47981 | R has no multiplicative (r... |
| 2zrngnmlid2 47982 | R has no multiplicative (l... |
| 2zrngnring 47983 | R is not a unital ring. (... |
| cznrnglem 47984 | Lemma for ~ cznrng : The ... |
| cznabel 47985 | The ring constructed from ... |
| cznrng 47986 | The ring constructed from ... |
| cznnring 47987 | The ring constructed from ... |
| rngcvalALTV 47990 | Value of the category of n... |
| rngcbasALTV 47991 | Set of objects of the cate... |
| rngchomfvalALTV 47992 | Set of arrows of the categ... |
| rngchomALTV 47993 | Set of arrows of the categ... |
| elrngchomALTV 47994 | A morphism of non-unital r... |
| rngccofvalALTV 47995 | Composition in the categor... |
| rngccoALTV 47996 | Composition in the categor... |
| rngccatidALTV 47997 | Lemma for ~ rngccatALTV . ... |
| rngccatALTV 47998 | The category of non-unital... |
| rngcidALTV 47999 | The identity arrow in the ... |
| rngcsectALTV 48000 | A section in the category ... |
| rngcinvALTV 48001 | An inverse in the category... |
| rngcisoALTV 48002 | An isomorphism in the cate... |
| rngchomffvalALTV 48003 | The value of the functiona... |
| rngchomrnghmresALTV 48004 | The value of the functiona... |
| rngcrescrhmALTV 48005 | The category of non-unital... |
| rhmsubcALTVlem1 48006 | Lemma 1 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem2 48007 | Lemma 2 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem3 48008 | Lemma 3 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem4 48009 | Lemma 4 for ~ rhmsubcALTV ... |
| rhmsubcALTV 48010 | According to ~ df-subc , t... |
| rhmsubcALTVcat 48011 | The restriction of the cat... |
| ringcvalALTV 48014 | Value of the category of r... |
| funcringcsetcALTV2lem1 48015 | Lemma 1 for ~ funcringcset... |
| funcringcsetcALTV2lem2 48016 | Lemma 2 for ~ funcringcset... |
| funcringcsetcALTV2lem3 48017 | Lemma 3 for ~ funcringcset... |
| funcringcsetcALTV2lem4 48018 | Lemma 4 for ~ funcringcset... |
| funcringcsetcALTV2lem5 48019 | Lemma 5 for ~ funcringcset... |
| funcringcsetcALTV2lem6 48020 | Lemma 6 for ~ funcringcset... |
| funcringcsetcALTV2lem7 48021 | Lemma 7 for ~ funcringcset... |
| funcringcsetcALTV2lem8 48022 | Lemma 8 for ~ funcringcset... |
| funcringcsetcALTV2lem9 48023 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV2 48024 | The "natural forgetful fun... |
| ringcbasALTV 48025 | Set of objects of the cate... |
| ringchomfvalALTV 48026 | Set of arrows of the categ... |
| ringchomALTV 48027 | Set of arrows of the categ... |
| elringchomALTV 48028 | A morphism of rings is a f... |
| ringccofvalALTV 48029 | Composition in the categor... |
| ringccoALTV 48030 | Composition in the categor... |
| ringccatidALTV 48031 | Lemma for ~ ringccatALTV .... |
| ringccatALTV 48032 | The category of rings is a... |
| ringcidALTV 48033 | The identity arrow in the ... |
| ringcsectALTV 48034 | A section in the category ... |
| ringcinvALTV 48035 | An inverse in the category... |
| ringcisoALTV 48036 | An isomorphism in the cate... |
| ringcbasbasALTV 48037 | An element of the base set... |
| funcringcsetclem1ALTV 48038 | Lemma 1 for ~ funcringcset... |
| funcringcsetclem2ALTV 48039 | Lemma 2 for ~ funcringcset... |
| funcringcsetclem3ALTV 48040 | Lemma 3 for ~ funcringcset... |
| funcringcsetclem4ALTV 48041 | Lemma 4 for ~ funcringcset... |
| funcringcsetclem5ALTV 48042 | Lemma 5 for ~ funcringcset... |
| funcringcsetclem6ALTV 48043 | Lemma 6 for ~ funcringcset... |
| funcringcsetclem7ALTV 48044 | Lemma 7 for ~ funcringcset... |
| funcringcsetclem8ALTV 48045 | Lemma 8 for ~ funcringcset... |
| funcringcsetclem9ALTV 48046 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV 48047 | The "natural forgetful fun... |
| srhmsubcALTVlem1 48048 | Lemma 1 for ~ srhmsubcALTV... |
| srhmsubcALTVlem2 48049 | Lemma 2 for ~ srhmsubcALTV... |
| srhmsubcALTV 48050 | According to ~ df-subc , t... |
| sringcatALTV 48051 | The restriction of the cat... |
| crhmsubcALTV 48052 | According to ~ df-subc , t... |
| cringcatALTV 48053 | The restriction of the cat... |
| drhmsubcALTV 48054 | According to ~ df-subc , t... |
| drngcatALTV 48055 | The restriction of the cat... |
| fldcatALTV 48056 | The restriction of the cat... |
| fldcALTV 48057 | The restriction of the cat... |
| fldhmsubcALTV 48058 | According to ~ df-subc , t... |
| opeliun2xp 48059 | Membership of an ordered p... |
| eliunxp2 48060 | Membership in a union of C... |
| mpomptx2 48061 | Express a two-argument fun... |
| cbvmpox2 48062 | Rule to change the bound v... |
| dmmpossx2 48063 | The domain of a mapping is... |
| mpoexxg2 48064 | Existence of an operation ... |
| ovmpordxf 48065 | Value of an operation give... |
| ovmpordx 48066 | Value of an operation give... |
| ovmpox2 48067 | The value of an operation ... |
| fdmdifeqresdif 48068 | The restriction of a condi... |
| offvalfv 48069 | The function operation exp... |
| ofaddmndmap 48070 | The function operation app... |
| mapsnop 48071 | A singleton of an ordered ... |
| fprmappr 48072 | A function with a domain o... |
| mapprop 48073 | An unordered pair containi... |
| ztprmneprm 48074 | A prime is not an integer ... |
| 2t6m3t4e0 48075 | 2 times 6 minus 3 times 4 ... |
| ssnn0ssfz 48076 | For any finite subset of `... |
| nn0sumltlt 48077 | If the sum of two nonnegat... |
| bcpascm1 48078 | Pascal's rule for the bino... |
| altgsumbc 48079 | The sum of binomial coeffi... |
| altgsumbcALT 48080 | Alternate proof of ~ altgs... |
| zlmodzxzlmod 48081 | The ` ZZ `-module ` ZZ X. ... |
| zlmodzxzel 48082 | An element of the (base se... |
| zlmodzxz0 48083 | The ` 0 ` of the ` ZZ `-mo... |
| zlmodzxzscm 48084 | The scalar multiplication ... |
| zlmodzxzadd 48085 | The addition of the ` ZZ `... |
| zlmodzxzsubm 48086 | The subtraction of the ` Z... |
| zlmodzxzsub 48087 | The subtraction of the ` Z... |
| mgpsumunsn 48088 | Extract a summand/factor f... |
| mgpsumz 48089 | If the group sum for the m... |
| mgpsumn 48090 | If the group sum for the m... |
| exple2lt6 48091 | A nonnegative integer to t... |
| pgrple2abl 48092 | Every symmetric group on a... |
| pgrpgt2nabl 48093 | Every symmetric group on a... |
| invginvrid 48094 | Identity for a multiplicat... |
| rmsupp0 48095 | The support of a mapping o... |
| domnmsuppn0 48096 | The support of a mapping o... |
| rmsuppss 48097 | The support of a mapping o... |
| mndpsuppss 48098 | The support of a mapping o... |
| scmsuppss 48099 | The support of a mapping o... |
| rmsuppfi 48100 | The support of a mapping o... |
| rmfsupp 48101 | A mapping of a multiplicat... |
| mndpsuppfi 48102 | The support of a mapping o... |
| mndpfsupp 48103 | A mapping of a scalar mult... |
| scmsuppfi 48104 | The support of a mapping o... |
| scmfsupp 48105 | A mapping of a scalar mult... |
| suppmptcfin 48106 | The support of a mapping w... |
| mptcfsupp 48107 | A mapping with value 0 exc... |
| fsuppmptdmf 48108 | A mapping with a finite do... |
| lmodvsmdi 48109 | Multiple distributive law ... |
| gsumlsscl 48110 | Closure of a group sum in ... |
| assaascl0 48111 | The scalar 0 embedded into... |
| assaascl1 48112 | The scalar 1 embedded into... |
| ply1vr1smo 48113 | The variable in a polynomi... |
| ply1sclrmsm 48114 | The ring multiplication of... |
| coe1id 48115 | Coefficient vector of the ... |
| coe1sclmulval 48116 | The value of the coefficie... |
| ply1mulgsumlem1 48117 | Lemma 1 for ~ ply1mulgsum ... |
| ply1mulgsumlem2 48118 | Lemma 2 for ~ ply1mulgsum ... |
| ply1mulgsumlem3 48119 | Lemma 3 for ~ ply1mulgsum ... |
| ply1mulgsumlem4 48120 | Lemma 4 for ~ ply1mulgsum ... |
| ply1mulgsum 48121 | The product of two polynom... |
| evl1at0 48122 | Polynomial evaluation for ... |
| evl1at1 48123 | Polynomial evaluation for ... |
| linply1 48124 | A term of the form ` x - C... |
| lineval 48125 | A term of the form ` x - C... |
| linevalexample 48126 | The polynomial ` x - 3 ` o... |
| dmatALTval 48131 | The algebra of ` N ` x ` N... |
| dmatALTbas 48132 | The base set of the algebr... |
| dmatALTbasel 48133 | An element of the base set... |
| dmatbas 48134 | The set of all ` N ` x ` N... |
| lincop 48139 | A linear combination as op... |
| lincval 48140 | The value of a linear comb... |
| dflinc2 48141 | Alternative definition of ... |
| lcoop 48142 | A linear combination as op... |
| lcoval 48143 | The value of a linear comb... |
| lincfsuppcl 48144 | A linear combination of ve... |
| linccl 48145 | A linear combination of ve... |
| lincval0 48146 | The value of an empty line... |
| lincvalsng 48147 | The linear combination ove... |
| lincvalsn 48148 | The linear combination ove... |
| lincvalpr 48149 | The linear combination ove... |
| lincval1 48150 | The linear combination ove... |
| lcosn0 48151 | Properties of a linear com... |
| lincvalsc0 48152 | The linear combination whe... |
| lcoc0 48153 | Properties of a linear com... |
| linc0scn0 48154 | If a set contains the zero... |
| lincdifsn 48155 | A vector is a linear combi... |
| linc1 48156 | A vector is a linear combi... |
| lincellss 48157 | A linear combination of a ... |
| lco0 48158 | The set of empty linear co... |
| lcoel0 48159 | The zero vector is always ... |
| lincsum 48160 | The sum of two linear comb... |
| lincscm 48161 | A linear combinations mult... |
| lincsumcl 48162 | The sum of two linear comb... |
| lincscmcl 48163 | The multiplication of a li... |
| lincsumscmcl 48164 | The sum of a linear combin... |
| lincolss 48165 | According to the statement... |
| ellcoellss 48166 | Every linear combination o... |
| lcoss 48167 | A set of vectors of a modu... |
| lspsslco 48168 | Lemma for ~ lspeqlco . (C... |
| lcosslsp 48169 | Lemma for ~ lspeqlco . (C... |
| lspeqlco 48170 | Equivalence of a _span_ of... |
| rellininds 48174 | The class defining the rel... |
| linindsv 48176 | The classes of the module ... |
| islininds 48177 | The property of being a li... |
| linindsi 48178 | The implications of being ... |
| linindslinci 48179 | The implications of being ... |
| islinindfis 48180 | The property of being a li... |
| islinindfiss 48181 | The property of being a li... |
| linindscl 48182 | A linearly independent set... |
| lindepsnlininds 48183 | A linearly dependent subse... |
| islindeps 48184 | The property of being a li... |
| lincext1 48185 | Property 1 of an extension... |
| lincext2 48186 | Property 2 of an extension... |
| lincext3 48187 | Property 3 of an extension... |
| lindslinindsimp1 48188 | Implication 1 for ~ lindsl... |
| lindslinindimp2lem1 48189 | Lemma 1 for ~ lindslininds... |
| lindslinindimp2lem2 48190 | Lemma 2 for ~ lindslininds... |
| lindslinindimp2lem3 48191 | Lemma 3 for ~ lindslininds... |
| lindslinindimp2lem4 48192 | Lemma 4 for ~ lindslininds... |
| lindslinindsimp2lem5 48193 | Lemma 5 for ~ lindslininds... |
| lindslinindsimp2 48194 | Implication 2 for ~ lindsl... |
| lindslininds 48195 | Equivalence of definitions... |
| linds0 48196 | The empty set is always a ... |
| el0ldep 48197 | A set containing the zero ... |
| el0ldepsnzr 48198 | A set containing the zero ... |
| lindsrng01 48199 | Any subset of a module is ... |
| lindszr 48200 | Any subset of a module ove... |
| snlindsntorlem 48201 | Lemma for ~ snlindsntor . ... |
| snlindsntor 48202 | A singleton is linearly in... |
| ldepsprlem 48203 | Lemma for ~ ldepspr . (Co... |
| ldepspr 48204 | If a vector is a scalar mu... |
| lincresunit3lem3 48205 | Lemma 3 for ~ lincresunit3... |
| lincresunitlem1 48206 | Lemma 1 for properties of ... |
| lincresunitlem2 48207 | Lemma for properties of a ... |
| lincresunit1 48208 | Property 1 of a specially ... |
| lincresunit2 48209 | Property 2 of a specially ... |
| lincresunit3lem1 48210 | Lemma 1 for ~ lincresunit3... |
| lincresunit3lem2 48211 | Lemma 2 for ~ lincresunit3... |
| lincresunit3 48212 | Property 3 of a specially ... |
| lincreslvec3 48213 | Property 3 of a specially ... |
| islindeps2 48214 | Conditions for being a lin... |
| islininds2 48215 | Implication of being a lin... |
| isldepslvec2 48216 | Alternative definition of ... |
| lindssnlvec 48217 | A singleton not containing... |
| lmod1lem1 48218 | Lemma 1 for ~ lmod1 . (Co... |
| lmod1lem2 48219 | Lemma 2 for ~ lmod1 . (Co... |
| lmod1lem3 48220 | Lemma 3 for ~ lmod1 . (Co... |
| lmod1lem4 48221 | Lemma 4 for ~ lmod1 . (Co... |
| lmod1lem5 48222 | Lemma 5 for ~ lmod1 . (Co... |
| lmod1 48223 | The (smallest) structure r... |
| lmod1zr 48224 | The (smallest) structure r... |
| lmod1zrnlvec 48225 | There is a (left) module (... |
| lmodn0 48226 | Left modules exist. (Cont... |
| zlmodzxzequa 48227 | Example of an equation wit... |
| zlmodzxznm 48228 | Example of a linearly depe... |
| zlmodzxzldeplem 48229 | A and B are not equal. (C... |
| zlmodzxzequap 48230 | Example of an equation wit... |
| zlmodzxzldeplem1 48231 | Lemma 1 for ~ zlmodzxzldep... |
| zlmodzxzldeplem2 48232 | Lemma 2 for ~ zlmodzxzldep... |
| zlmodzxzldeplem3 48233 | Lemma 3 for ~ zlmodzxzldep... |
| zlmodzxzldeplem4 48234 | Lemma 4 for ~ zlmodzxzldep... |
| zlmodzxzldep 48235 | { A , B } is a linearly de... |
| ldepsnlinclem1 48236 | Lemma 1 for ~ ldepsnlinc .... |
| ldepsnlinclem2 48237 | Lemma 2 for ~ ldepsnlinc .... |
| lvecpsslmod 48238 | The class of all (left) ve... |
| ldepsnlinc 48239 | The reverse implication of... |
| ldepslinc 48240 | For (left) vector spaces, ... |
| suppdm 48241 | If the range of a function... |
| eluz2cnn0n1 48242 | An integer greater than 1 ... |
| divge1b 48243 | The ratio of a real number... |
| divgt1b 48244 | The ratio of a real number... |
| ltsubaddb 48245 | Equivalence for the "less ... |
| ltsubsubb 48246 | Equivalence for the "less ... |
| ltsubadd2b 48247 | Equivalence for the "less ... |
| divsub1dir 48248 | Distribution of division o... |
| expnegico01 48249 | An integer greater than 1 ... |
| elfzolborelfzop1 48250 | An element of a half-open ... |
| pw2m1lepw2m1 48251 | 2 to the power of a positi... |
| zgtp1leeq 48252 | If an integer is between a... |
| flsubz 48253 | An integer can be moved in... |
| fldivmod 48254 | Expressing the floor of a ... |
| mod0mul 48255 | If an integer is 0 modulo ... |
| modn0mul 48256 | If an integer is not 0 mod... |
| m1modmmod 48257 | An integer decreased by 1 ... |
| difmodm1lt 48258 | The difference between an ... |
| nn0onn0ex 48259 | For each odd nonnegative i... |
| nn0enn0ex 48260 | For each even nonnegative ... |
| nnennex 48261 | For each even positive int... |
| nneop 48262 | A positive integer is even... |
| nneom 48263 | A positive integer is even... |
| nn0eo 48264 | A nonnegative integer is e... |
| nnpw2even 48265 | 2 to the power of a positi... |
| zefldiv2 48266 | The floor of an even integ... |
| zofldiv2 48267 | The floor of an odd intege... |
| nn0ofldiv2 48268 | The floor of an odd nonneg... |
| flnn0div2ge 48269 | The floor of a positive in... |
| flnn0ohalf 48270 | The floor of the half of a... |
| logcxp0 48271 | Logarithm of a complex pow... |
| regt1loggt0 48272 | The natural logarithm for ... |
| fdivval 48275 | The quotient of two functi... |
| fdivmpt 48276 | The quotient of two functi... |
| fdivmptf 48277 | The quotient of two functi... |
| refdivmptf 48278 | The quotient of two functi... |
| fdivpm 48279 | The quotient of two functi... |
| refdivpm 48280 | The quotient of two functi... |
| fdivmptfv 48281 | The function value of a qu... |
| refdivmptfv 48282 | The function value of a qu... |
| bigoval 48285 | Set of functions of order ... |
| elbigofrcl 48286 | Reverse closure of the "bi... |
| elbigo 48287 | Properties of a function o... |
| elbigo2 48288 | Properties of a function o... |
| elbigo2r 48289 | Sufficient condition for a... |
| elbigof 48290 | A function of order G(x) i... |
| elbigodm 48291 | The domain of a function o... |
| elbigoimp 48292 | The defining property of a... |
| elbigolo1 48293 | A function (into the posit... |
| rege1logbrege0 48294 | The general logarithm, wit... |
| rege1logbzge0 48295 | The general logarithm, wit... |
| fllogbd 48296 | A real number is between t... |
| relogbmulbexp 48297 | The logarithm of the produ... |
| relogbdivb 48298 | The logarithm of the quoti... |
| logbge0b 48299 | The logarithm of a number ... |
| logblt1b 48300 | The logarithm of a number ... |
| fldivexpfllog2 48301 | The floor of a positive re... |
| nnlog2ge0lt1 48302 | A positive integer is 1 if... |
| logbpw2m1 48303 | The floor of the binary lo... |
| fllog2 48304 | The floor of the binary lo... |
| blenval 48307 | The binary length of an in... |
| blen0 48308 | The binary length of 0. (... |
| blenn0 48309 | The binary length of a "nu... |
| blenre 48310 | The binary length of a pos... |
| blennn 48311 | The binary length of a pos... |
| blennnelnn 48312 | The binary length of a pos... |
| blennn0elnn 48313 | The binary length of a non... |
| blenpw2 48314 | The binary length of a pow... |
| blenpw2m1 48315 | The binary length of a pow... |
| nnpw2blen 48316 | A positive integer is betw... |
| nnpw2blenfzo 48317 | A positive integer is betw... |
| nnpw2blenfzo2 48318 | A positive integer is eith... |
| nnpw2pmod 48319 | Every positive integer can... |
| blen1 48320 | The binary length of 1. (... |
| blen2 48321 | The binary length of 2. (... |
| nnpw2p 48322 | Every positive integer can... |
| nnpw2pb 48323 | A number is a positive int... |
| blen1b 48324 | The binary length of a non... |
| blennnt2 48325 | The binary length of a pos... |
| nnolog2flm1 48326 | The floor of the binary lo... |
| blennn0em1 48327 | The binary length of the h... |
| blennngt2o2 48328 | The binary length of an od... |
| blengt1fldiv2p1 48329 | The binary length of an in... |
| blennn0e2 48330 | The binary length of an ev... |
| digfval 48333 | Operation to obtain the ` ... |
| digval 48334 | The ` K ` th digit of a no... |
| digvalnn0 48335 | The ` K ` th digit of a no... |
| nn0digval 48336 | The ` K ` th digit of a no... |
| dignn0fr 48337 | The digits of the fraction... |
| dignn0ldlem 48338 | Lemma for ~ dignnld . (Co... |
| dignnld 48339 | The leading digits of a po... |
| dig2nn0ld 48340 | The leading digits of a po... |
| dig2nn1st 48341 | The first (relevant) digit... |
| dig0 48342 | All digits of 0 are 0. (C... |
| digexp 48343 | The ` K ` th digit of a po... |
| dig1 48344 | All but one digits of 1 ar... |
| 0dig1 48345 | The ` 0 ` th digit of 1 is... |
| 0dig2pr01 48346 | The integers 0 and 1 corre... |
| dig2nn0 48347 | A digit of a nonnegative i... |
| 0dig2nn0e 48348 | The last bit of an even in... |
| 0dig2nn0o 48349 | The last bit of an odd int... |
| dig2bits 48350 | The ` K ` th digit of a no... |
| dignn0flhalflem1 48351 | Lemma 1 for ~ dignn0flhalf... |
| dignn0flhalflem2 48352 | Lemma 2 for ~ dignn0flhalf... |
| dignn0ehalf 48353 | The digits of the half of ... |
| dignn0flhalf 48354 | The digits of the rounded ... |
| nn0sumshdiglemA 48355 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglemB 48356 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglem1 48357 | Lemma 1 for ~ nn0sumshdig ... |
| nn0sumshdiglem2 48358 | Lemma 2 for ~ nn0sumshdig ... |
| nn0sumshdig 48359 | A nonnegative integer can ... |
| nn0mulfsum 48360 | Trivial algorithm to calcu... |
| nn0mullong 48361 | Standard algorithm (also k... |
| naryfval 48364 | The set of the n-ary (endo... |
| naryfvalixp 48365 | The set of the n-ary (endo... |
| naryfvalel 48366 | An n-ary (endo)function on... |
| naryrcl 48367 | Reverse closure for n-ary ... |
| naryfvalelfv 48368 | The value of an n-ary (end... |
| naryfvalelwrdf 48369 | An n-ary (endo)function on... |
| 0aryfvalel 48370 | A nullary (endo)function o... |
| 0aryfvalelfv 48371 | The value of a nullary (en... |
| 1aryfvalel 48372 | A unary (endo)function on ... |
| fv1arycl 48373 | Closure of a unary (endo)f... |
| 1arympt1 48374 | A unary (endo)function in ... |
| 1arympt1fv 48375 | The value of a unary (endo... |
| 1arymaptfv 48376 | The value of the mapping o... |
| 1arymaptf 48377 | The mapping of unary (endo... |
| 1arymaptf1 48378 | The mapping of unary (endo... |
| 1arymaptfo 48379 | The mapping of unary (endo... |
| 1arymaptf1o 48380 | The mapping of unary (endo... |
| 1aryenef 48381 | The set of unary (endo)fun... |
| 1aryenefmnd 48382 | The set of unary (endo)fun... |
| 2aryfvalel 48383 | A binary (endo)function on... |
| fv2arycl 48384 | Closure of a binary (endo)... |
| 2arympt 48385 | A binary (endo)function in... |
| 2arymptfv 48386 | The value of a binary (end... |
| 2arymaptfv 48387 | The value of the mapping o... |
| 2arymaptf 48388 | The mapping of binary (end... |
| 2arymaptf1 48389 | The mapping of binary (end... |
| 2arymaptfo 48390 | The mapping of binary (end... |
| 2arymaptf1o 48391 | The mapping of binary (end... |
| 2aryenef 48392 | The set of binary (endo)fu... |
| itcoval 48397 | The value of the function ... |
| itcoval0 48398 | A function iterated zero t... |
| itcoval1 48399 | A function iterated once. ... |
| itcoval2 48400 | A function iterated twice.... |
| itcoval3 48401 | A function iterated three ... |
| itcoval0mpt 48402 | A mapping iterated zero ti... |
| itcovalsuc 48403 | The value of the function ... |
| itcovalsucov 48404 | The value of the function ... |
| itcovalendof 48405 | The n-th iterate of an end... |
| itcovalpclem1 48406 | Lemma 1 for ~ itcovalpc : ... |
| itcovalpclem2 48407 | Lemma 2 for ~ itcovalpc : ... |
| itcovalpc 48408 | The value of the function ... |
| itcovalt2lem2lem1 48409 | Lemma 1 for ~ itcovalt2lem... |
| itcovalt2lem2lem2 48410 | Lemma 2 for ~ itcovalt2lem... |
| itcovalt2lem1 48411 | Lemma 1 for ~ itcovalt2 : ... |
| itcovalt2lem2 48412 | Lemma 2 for ~ itcovalt2 : ... |
| itcovalt2 48413 | The value of the function ... |
| ackvalsuc1mpt 48414 | The Ackermann function at ... |
| ackvalsuc1 48415 | The Ackermann function at ... |
| ackval0 48416 | The Ackermann function at ... |
| ackval1 48417 | The Ackermann function at ... |
| ackval2 48418 | The Ackermann function at ... |
| ackval3 48419 | The Ackermann function at ... |
| ackendofnn0 48420 | The Ackermann function at ... |
| ackfnnn0 48421 | The Ackermann function at ... |
| ackval0val 48422 | The Ackermann function at ... |
| ackvalsuc0val 48423 | The Ackermann function at ... |
| ackvalsucsucval 48424 | The Ackermann function at ... |
| ackval0012 48425 | The Ackermann function at ... |
| ackval1012 48426 | The Ackermann function at ... |
| ackval2012 48427 | The Ackermann function at ... |
| ackval3012 48428 | The Ackermann function at ... |
| ackval40 48429 | The Ackermann function at ... |
| ackval41a 48430 | The Ackermann function at ... |
| ackval41 48431 | The Ackermann function at ... |
| ackval42 48432 | The Ackermann function at ... |
| ackval42a 48433 | The Ackermann function at ... |
| ackval50 48434 | The Ackermann function at ... |
| fv1prop 48435 | The function value of unor... |
| fv2prop 48436 | The function value of unor... |
| submuladdmuld 48437 | Transformation of a sum of... |
| affinecomb1 48438 | Combination of two real af... |
| affinecomb2 48439 | Combination of two real af... |
| affineid 48440 | Identity of an affine comb... |
| 1subrec1sub 48441 | Subtract the reciprocal of... |
| resum2sqcl 48442 | The sum of two squares of ... |
| resum2sqgt0 48443 | The sum of the square of a... |
| resum2sqrp 48444 | The sum of the square of a... |
| resum2sqorgt0 48445 | The sum of the square of t... |
| reorelicc 48446 | Membership in and outside ... |
| rrx2pxel 48447 | The x-coordinate of a poin... |
| rrx2pyel 48448 | The y-coordinate of a poin... |
| prelrrx2 48449 | An unordered pair of order... |
| prelrrx2b 48450 | An unordered pair of order... |
| rrx2pnecoorneor 48451 | If two different points ` ... |
| rrx2pnedifcoorneor 48452 | If two different points ` ... |
| rrx2pnedifcoorneorr 48453 | If two different points ` ... |
| rrx2xpref1o 48454 | There is a bijection betwe... |
| rrx2xpreen 48455 | The set of points in the t... |
| rrx2plord 48456 | The lexicographical orderi... |
| rrx2plord1 48457 | The lexicographical orderi... |
| rrx2plord2 48458 | The lexicographical orderi... |
| rrx2plordisom 48459 | The set of points in the t... |
| rrx2plordso 48460 | The lexicographical orderi... |
| ehl2eudisval0 48461 | The Euclidean distance of ... |
| ehl2eudis0lt 48462 | An upper bound of the Eucl... |
| lines 48467 | The lines passing through ... |
| line 48468 | The line passing through t... |
| rrxlines 48469 | Definition of lines passin... |
| rrxline 48470 | The line passing through t... |
| rrxlinesc 48471 | Definition of lines passin... |
| rrxlinec 48472 | The line passing through t... |
| eenglngeehlnmlem1 48473 | Lemma 1 for ~ eenglngeehln... |
| eenglngeehlnmlem2 48474 | Lemma 2 for ~ eenglngeehln... |
| eenglngeehlnm 48475 | The line definition in the... |
| rrx2line 48476 | The line passing through t... |
| rrx2vlinest 48477 | The vertical line passing ... |
| rrx2linest 48478 | The line passing through t... |
| rrx2linesl 48479 | The line passing through t... |
| rrx2linest2 48480 | The line passing through t... |
| elrrx2linest2 48481 | The line passing through t... |
| spheres 48482 | The spheres for given cent... |
| sphere 48483 | A sphere with center ` X `... |
| rrxsphere 48484 | The sphere with center ` M... |
| 2sphere 48485 | The sphere with center ` M... |
| 2sphere0 48486 | The sphere around the orig... |
| line2ylem 48487 | Lemma for ~ line2y . This... |
| line2 48488 | Example for a line ` G ` p... |
| line2xlem 48489 | Lemma for ~ line2x . This... |
| line2x 48490 | Example for a horizontal l... |
| line2y 48491 | Example for a vertical lin... |
| itsclc0lem1 48492 | Lemma for theorems about i... |
| itsclc0lem2 48493 | Lemma for theorems about i... |
| itsclc0lem3 48494 | Lemma for theorems about i... |
| itscnhlc0yqe 48495 | Lemma for ~ itsclc0 . Qua... |
| itschlc0yqe 48496 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqe 48497 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqsollem1 48498 | Lemma 1 for ~ itsclc0yqsol... |
| itsclc0yqsollem2 48499 | Lemma 2 for ~ itsclc0yqsol... |
| itsclc0yqsol 48500 | Lemma for ~ itsclc0 . Sol... |
| itscnhlc0xyqsol 48501 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol1 48502 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol 48503 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsol 48504 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolr 48505 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolb 48506 | Lemma for ~ itsclc0 . Sol... |
| itsclc0 48507 | The intersection points of... |
| itsclc0b 48508 | The intersection points of... |
| itsclinecirc0 48509 | The intersection points of... |
| itsclinecirc0b 48510 | The intersection points of... |
| itsclinecirc0in 48511 | The intersection points of... |
| itsclquadb 48512 | Quadratic equation for the... |
| itsclquadeu 48513 | Quadratic equation for the... |
| 2itscplem1 48514 | Lemma 1 for ~ 2itscp . (C... |
| 2itscplem2 48515 | Lemma 2 for ~ 2itscp . (C... |
| 2itscplem3 48516 | Lemma D for ~ 2itscp . (C... |
| 2itscp 48517 | A condition for a quadrati... |
| itscnhlinecirc02plem1 48518 | Lemma 1 for ~ itscnhlineci... |
| itscnhlinecirc02plem2 48519 | Lemma 2 for ~ itscnhlineci... |
| itscnhlinecirc02plem3 48520 | Lemma 3 for ~ itscnhlineci... |
| itscnhlinecirc02p 48521 | Intersection of a nonhoriz... |
| inlinecirc02plem 48522 | Lemma for ~ inlinecirc02p ... |
| inlinecirc02p 48523 | Intersection of a line wit... |
| inlinecirc02preu 48524 | Intersection of a line wit... |
| pm4.71da 48525 | Deduction converting a bic... |
| logic1 48526 | Distribution of implicatio... |
| logic1a 48527 | Variant of ~ logic1 . (Co... |
| logic2 48528 | Variant of ~ logic1 . (Co... |
| pm5.32dav 48529 | Distribution of implicatio... |
| pm5.32dra 48530 | Reverse distribution of im... |
| exp12bd 48531 | The import-export theorem ... |
| mpbiran3d 48532 | Equivalence with a conjunc... |
| mpbiran4d 48533 | Equivalence with a conjunc... |
| dtrucor3 48534 | An example of how ~ ax-5 w... |
| ralbidb 48535 | Formula-building rule for ... |
| ralbidc 48536 | Formula-building rule for ... |
| r19.41dv 48537 | A complex deduction form o... |
| rmotru 48538 | Two ways of expressing "at... |
| reutru 48539 | Two ways of expressing "ex... |
| reutruALT 48540 | Alternate proof for ~ reut... |
| ssdisjd 48541 | Subset preserves disjointn... |
| ssdisjdr 48542 | Subset preserves disjointn... |
| disjdifb 48543 | Relative complement is ant... |
| predisj 48544 | Preimages of disjoint sets... |
| vsn 48545 | The singleton of the unive... |
| mosn 48546 | "At most one" element in a... |
| mo0 48547 | "At most one" element in a... |
| mosssn 48548 | "At most one" element in a... |
| mo0sn 48549 | Two ways of expressing "at... |
| mosssn2 48550 | Two ways of expressing "at... |
| unilbss 48551 | Superclass of the greatest... |
| inpw 48552 | Two ways of expressing a c... |
| mof0 48553 | There is at most one funct... |
| mof02 48554 | A variant of ~ mof0 . (Co... |
| mof0ALT 48555 | Alternate proof for ~ mof0... |
| eufsnlem 48556 | There is exactly one funct... |
| eufsn 48557 | There is exactly one funct... |
| eufsn2 48558 | There is exactly one funct... |
| mofsn 48559 | There is at most one funct... |
| mofsn2 48560 | There is at most one funct... |
| mofsssn 48561 | There is at most one funct... |
| mofmo 48562 | There is at most one funct... |
| mofeu 48563 | The uniqueness of a functi... |
| elfvne0 48564 | If a function value has a ... |
| fdomne0 48565 | A function with non-empty ... |
| f1sn2g 48566 | A function that maps a sin... |
| f102g 48567 | A function that maps the e... |
| f1mo 48568 | A function that maps a set... |
| f002 48569 | A function with an empty c... |
| map0cor 48570 | A function exists iff an e... |
| fvconstr 48571 | Two ways of expressing ` A... |
| fvconstrn0 48572 | Two ways of expressing ` A... |
| fvconstr2 48573 | Two ways of expressing ` A... |
| fvconst0ci 48574 | A constant function's valu... |
| fvconstdomi 48575 | A constant function's valu... |
| f1omo 48576 | There is at most one eleme... |
| f1omoALT 48577 | There is at most one eleme... |
| iccin 48578 | Intersection of two closed... |
| iccdisj2 48579 | If the upper bound of one ... |
| iccdisj 48580 | If the upper bound of one ... |
| mreuniss 48581 | The union of a collection ... |
| clduni 48582 | The union of closed sets i... |
| opncldeqv 48583 | Conditions on open sets ar... |
| opndisj 48584 | Two ways of saying that tw... |
| clddisj 48585 | Two ways of saying that tw... |
| neircl 48586 | Reverse closure of the nei... |
| opnneilem 48587 | Lemma factoring out common... |
| opnneir 48588 | If something is true for a... |
| opnneirv 48589 | A variant of ~ opnneir wit... |
| opnneilv 48590 | The converse of ~ opnneir ... |
| opnneil 48591 | A variant of ~ opnneilv . ... |
| opnneieqv 48592 | The equivalence between ne... |
| opnneieqvv 48593 | The equivalence between ne... |
| restcls2lem 48594 | A closed set in a subspace... |
| restcls2 48595 | A closed set in a subspace... |
| restclsseplem 48596 | Lemma for ~ restclssep . ... |
| restclssep 48597 | Two disjoint closed sets i... |
| cnneiima 48598 | Given a continuous functio... |
| iooii 48599 | Open intervals are open se... |
| icccldii 48600 | Closed intervals are close... |
| i0oii 48601 | ` ( 0 [,) A ) ` is open in... |
| io1ii 48602 | ` ( A (,] 1 ) ` is open in... |
| sepnsepolem1 48603 | Lemma for ~ sepnsepo . (C... |
| sepnsepolem2 48604 | Open neighborhood and neig... |
| sepnsepo 48605 | Open neighborhood and neig... |
| sepdisj 48606 | Separated sets are disjoin... |
| seposep 48607 | If two sets are separated ... |
| sepcsepo 48608 | If two sets are separated ... |
| sepfsepc 48609 | If two sets are separated ... |
| seppsepf 48610 | If two sets are precisely ... |
| seppcld 48611 | If two sets are precisely ... |
| isnrm4 48612 | A topological space is nor... |
| dfnrm2 48613 | A topological space is nor... |
| dfnrm3 48614 | A topological space is nor... |
| iscnrm3lem1 48615 | Lemma for ~ iscnrm3 . Sub... |
| iscnrm3lem2 48616 | Lemma for ~ iscnrm3 provin... |
| iscnrm3lem3 48617 | Lemma for ~ iscnrm3lem4 . ... |
| iscnrm3lem4 48618 | Lemma for ~ iscnrm3lem5 an... |
| iscnrm3lem5 48619 | Lemma for ~ iscnrm3l . (C... |
| iscnrm3lem6 48620 | Lemma for ~ iscnrm3lem7 . ... |
| iscnrm3lem7 48621 | Lemma for ~ iscnrm3rlem8 a... |
| iscnrm3rlem1 48622 | Lemma for ~ iscnrm3rlem2 .... |
| iscnrm3rlem2 48623 | Lemma for ~ iscnrm3rlem3 .... |
| iscnrm3rlem3 48624 | Lemma for ~ iscnrm3r . Th... |
| iscnrm3rlem4 48625 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem5 48626 | Lemma for ~ iscnrm3rlem6 .... |
| iscnrm3rlem6 48627 | Lemma for ~ iscnrm3rlem7 .... |
| iscnrm3rlem7 48628 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem8 48629 | Lemma for ~ iscnrm3r . Di... |
| iscnrm3r 48630 | Lemma for ~ iscnrm3 . If ... |
| iscnrm3llem1 48631 | Lemma for ~ iscnrm3l . Cl... |
| iscnrm3llem2 48632 | Lemma for ~ iscnrm3l . If... |
| iscnrm3l 48633 | Lemma for ~ iscnrm3 . Giv... |
| iscnrm3 48634 | A completely normal topolo... |
| iscnrm3v 48635 | A topology is completely n... |
| iscnrm4 48636 | A completely normal topolo... |
| isprsd 48637 | Property of being a preord... |
| lubeldm2 48638 | Member of the domain of th... |
| glbeldm2 48639 | Member of the domain of th... |
| lubeldm2d 48640 | Member of the domain of th... |
| glbeldm2d 48641 | Member of the domain of th... |
| lubsscl 48642 | If a subset of ` S ` conta... |
| glbsscl 48643 | If a subset of ` S ` conta... |
| lubprlem 48644 | Lemma for ~ lubprdm and ~ ... |
| lubprdm 48645 | The set of two comparable ... |
| lubpr 48646 | The LUB of the set of two ... |
| glbprlem 48647 | Lemma for ~ glbprdm and ~ ... |
| glbprdm 48648 | The set of two comparable ... |
| glbpr 48649 | The GLB of the set of two ... |
| joindm2 48650 | The join of any two elemen... |
| joindm3 48651 | The join of any two elemen... |
| meetdm2 48652 | The meet of any two elemen... |
| meetdm3 48653 | The meet of any two elemen... |
| posjidm 48654 | Poset join is idempotent. ... |
| posmidm 48655 | Poset meet is idempotent. ... |
| toslat 48656 | A toset is a lattice. (Co... |
| isclatd 48657 | The predicate "is a comple... |
| intubeu 48658 | Existential uniqueness of ... |
| unilbeu 48659 | Existential uniqueness of ... |
| ipolublem 48660 | Lemma for ~ ipolubdm and ~... |
| ipolubdm 48661 | The domain of the LUB of t... |
| ipolub 48662 | The LUB of the inclusion p... |
| ipoglblem 48663 | Lemma for ~ ipoglbdm and ~... |
| ipoglbdm 48664 | The domain of the GLB of t... |
| ipoglb 48665 | The GLB of the inclusion p... |
| ipolub0 48666 | The LUB of the empty set i... |
| ipolub00 48667 | The LUB of the empty set i... |
| ipoglb0 48668 | The GLB of the empty set i... |
| mrelatlubALT 48669 | Least upper bounds in a Mo... |
| mrelatglbALT 48670 | Greatest lower bounds in a... |
| mreclat 48671 | A Moore space is a complet... |
| topclat 48672 | A topology is a complete l... |
| toplatglb0 48673 | The empty intersection in ... |
| toplatlub 48674 | Least upper bounds in a to... |
| toplatglb 48675 | Greatest lower bounds in a... |
| toplatjoin 48676 | Joins in a topology are re... |
| toplatmeet 48677 | Meets in a topology are re... |
| topdlat 48678 | A topology is a distributi... |
| elmgpcntrd 48679 | The center of a ring. (Co... |
| asclelbas 48680 | Lifted scalars are in the ... |
| asclcntr 48681 | The algebra scalars functi... |
| asclcom 48682 | Scalars are commutative af... |
| catprslem 48683 | Lemma for ~ catprs . (Con... |
| catprs 48684 | A preorder can be extracte... |
| catprs2 48685 | A category equipped with t... |
| catprsc 48686 | A construction of the preo... |
| catprsc2 48687 | An alternate construction ... |
| endmndlem 48688 | A diagonal hom-set in a ca... |
| idmon 48689 | An identity arrow, or an i... |
| idepi 48690 | An identity arrow, or an i... |
| funcf2lem 48691 | A utility theorem for prov... |
| isthinc 48694 | The predicate "is a thin c... |
| isthinc2 48695 | A thin category is a categ... |
| isthinc3 48696 | A thin category is a categ... |
| thincc 48697 | A thin category is a categ... |
| thinccd 48698 | A thin category is a categ... |
| thincssc 48699 | A thin category is a categ... |
| isthincd2lem1 48700 | Lemma for ~ isthincd2 and ... |
| thincmo2 48701 | Morphisms in the same hom-... |
| thincmo 48702 | There is at most one morph... |
| thincmoALT 48703 | Alternate proof for ~ thin... |
| thincmod 48704 | At most one morphism in ea... |
| thincn0eu 48705 | In a thin category, a hom-... |
| thincid 48706 | In a thin category, a morp... |
| thincmon 48707 | In a thin category, all mo... |
| thincepi 48708 | In a thin category, all mo... |
| isthincd2lem2 48709 | Lemma for ~ isthincd2 . (... |
| isthincd 48710 | The predicate "is a thin c... |
| isthincd2 48711 | The predicate " ` C ` is a... |
| oppcthin 48712 | The opposite category of a... |
| subthinc 48713 | A subcategory of a thin ca... |
| functhinclem1 48714 | Lemma for ~ functhinc . G... |
| functhinclem2 48715 | Lemma for ~ functhinc . (... |
| functhinclem3 48716 | Lemma for ~ functhinc . T... |
| functhinclem4 48717 | Lemma for ~ functhinc . O... |
| functhinc 48718 | A functor to a thin catego... |
| fullthinc 48719 | A functor to a thin catego... |
| fullthinc2 48720 | A full functor to a thin c... |
| thincfth 48721 | A functor from a thin cate... |
| thincciso 48722 | Two thin categories are is... |
| 0thincg 48723 | Any structure with an empt... |
| 0thinc 48724 | The empty category (see ~ ... |
| indthinc 48725 | An indiscrete category in ... |
| indthincALT 48726 | An alternate proof for ~ i... |
| prsthinc 48727 | Preordered sets as categor... |
| setcthin 48728 | A category of sets all of ... |
| setc2othin 48729 | The category ` ( SetCat ``... |
| thincsect 48730 | In a thin category, one mo... |
| thincsect2 48731 | In a thin category, ` F ` ... |
| thincinv 48732 | In a thin category, ` F ` ... |
| thinciso 48733 | In a thin category, ` F : ... |
| thinccic 48734 | In a thin category, two ob... |
| prstcval 48737 | Lemma for ~ prstcnidlem an... |
| prstcnidlem 48738 | Lemma for ~ prstcnid and ~... |
| prstcnid 48739 | Components other than ` Ho... |
| prstcbas 48740 | The base set is unchanged.... |
| prstcleval 48741 | Value of the less-than-or-... |
| prstclevalOLD 48742 | Obsolete proof of ~ prstcl... |
| prstcle 48743 | Value of the less-than-or-... |
| prstcocval 48744 | Orthocomplementation is un... |
| prstcocvalOLD 48745 | Obsolete proof of ~ prstco... |
| prstcoc 48746 | Orthocomplementation is un... |
| prstchomval 48747 | Hom-sets of the constructe... |
| prstcprs 48748 | The category is a preorder... |
| prstcthin 48749 | The preordered set is equi... |
| prstchom 48750 | Hom-sets of the constructe... |
| prstchom2 48751 | Hom-sets of the constructe... |
| prstchom2ALT 48752 | Hom-sets of the constructe... |
| postcpos 48753 | The converted category is ... |
| postcposALT 48754 | Alternate proof for ~ post... |
| postc 48755 | The converted category is ... |
| mndtcval 48758 | Value of the category buil... |
| mndtcbasval 48759 | The base set of the catego... |
| mndtcbas 48760 | The category built from a ... |
| mndtcob 48761 | Lemma for ~ mndtchom and ~... |
| mndtcbas2 48762 | Two objects in a category ... |
| mndtchom 48763 | The only hom-set of the ca... |
| mndtcco 48764 | The composition of the cat... |
| mndtcco2 48765 | The composition of the cat... |
| mndtccatid 48766 | Lemma for ~ mndtccat and ~... |
| mndtccat 48767 | The function value is a ca... |
| mndtcid 48768 | The identity morphism, or ... |
| grptcmon 48769 | All morphisms in a categor... |
| grptcepi 48770 | All morphisms in a categor... |
| nfintd 48771 | Bound-variable hypothesis ... |
| nfiund 48772 | Bound-variable hypothesis ... |
| nfiundg 48773 | Bound-variable hypothesis ... |
| iunord 48774 | The indexed union of a col... |
| iunordi 48775 | The indexed union of a col... |
| spd 48776 | Specialization deduction, ... |
| spcdvw 48777 | A version of ~ spcdv where... |
| tfis2d 48778 | Transfinite Induction Sche... |
| bnd2d 48779 | Deduction form of ~ bnd2 .... |
| dffun3f 48780 | Alternate definition of fu... |
| setrecseq 48783 | Equality theorem for set r... |
| nfsetrecs 48784 | Bound-variable hypothesis ... |
| setrec1lem1 48785 | Lemma for ~ setrec1 . Thi... |
| setrec1lem2 48786 | Lemma for ~ setrec1 . If ... |
| setrec1lem3 48787 | Lemma for ~ setrec1 . If ... |
| setrec1lem4 48788 | Lemma for ~ setrec1 . If ... |
| setrec1 48789 | This is the first of two f... |
| setrec2fun 48790 | This is the second of two ... |
| setrec2lem1 48791 | Lemma for ~ setrec2 . The... |
| setrec2lem2 48792 | Lemma for ~ setrec2 . The... |
| setrec2 48793 | This is the second of two ... |
| setrec2v 48794 | Version of ~ setrec2 with ... |
| setrec2mpt 48795 | Version of ~ setrec2 where... |
| setis 48796 | Version of ~ setrec2 expre... |
| elsetrecslem 48797 | Lemma for ~ elsetrecs . A... |
| elsetrecs 48798 | A set ` A ` is an element ... |
| setrecsss 48799 | The ` setrecs ` operator r... |
| setrecsres 48800 | A recursively generated cl... |
| vsetrec 48801 | Construct ` _V ` using set... |
| 0setrec 48802 | If a function sends the em... |
| onsetreclem1 48803 | Lemma for ~ onsetrec . (C... |
| onsetreclem2 48804 | Lemma for ~ onsetrec . (C... |
| onsetreclem3 48805 | Lemma for ~ onsetrec . (C... |
| onsetrec 48806 | Construct ` On ` using set... |
| elpglem1 48809 | Lemma for ~ elpg . (Contr... |
| elpglem2 48810 | Lemma for ~ elpg . (Contr... |
| elpglem3 48811 | Lemma for ~ elpg . (Contr... |
| elpg 48812 | Membership in the class of... |
| pgindlem 48813 | Lemma for ~ pgind . (Cont... |
| pgindnf 48814 | Version of ~ pgind with ex... |
| pgind 48815 | Induction on partizan game... |
| sbidd 48816 | An identity theorem for su... |
| sbidd-misc 48817 | An identity theorem for su... |
| gte-lte 48822 | Simple relationship betwee... |
| gt-lt 48823 | Simple relationship betwee... |
| gte-lteh 48824 | Relationship between ` <_ ... |
| gt-lth 48825 | Relationship between ` < `... |
| ex-gt 48826 | Simple example of ` > ` , ... |
| ex-gte 48827 | Simple example of ` >_ ` ,... |
| sinhval-named 48834 | Value of the named sinh fu... |
| coshval-named 48835 | Value of the named cosh fu... |
| tanhval-named 48836 | Value of the named tanh fu... |
| sinh-conventional 48837 | Conventional definition of... |
| sinhpcosh 48838 | Prove that ` ( sinh `` A )... |
| secval 48845 | Value of the secant functi... |
| cscval 48846 | Value of the cosecant func... |
| cotval 48847 | Value of the cotangent fun... |
| seccl 48848 | The closure of the secant ... |
| csccl 48849 | The closure of the cosecan... |
| cotcl 48850 | The closure of the cotange... |
| reseccl 48851 | The closure of the secant ... |
| recsccl 48852 | The closure of the cosecan... |
| recotcl 48853 | The closure of the cotange... |
| recsec 48854 | The reciprocal of secant i... |
| reccsc 48855 | The reciprocal of cosecant... |
| reccot 48856 | The reciprocal of cotangen... |
| rectan 48857 | The reciprocal of tangent ... |
| sec0 48858 | The value of the secant fu... |
| onetansqsecsq 48859 | Prove the tangent squared ... |
| cotsqcscsq 48860 | Prove the tangent squared ... |
| ifnmfalse 48861 | If A is not a member of B,... |
| logb2aval 48862 | Define the value of the ` ... |
| comraddi 48869 | Commute RHS addition. See... |
| mvlraddi 48870 | Move the right term in a s... |
| mvrladdi 48871 | Move the left term in a su... |
| assraddsubi 48872 | Associate RHS addition-sub... |
| joinlmuladdmuli 48873 | Join AB+CB into (A+C) on L... |
| joinlmulsubmuld 48874 | Join AB-CB into (A-C) on L... |
| joinlmulsubmuli 48875 | Join AB-CB into (A-C) on L... |
| mvlrmuld 48876 | Move the right term in a p... |
| mvlrmuli 48877 | Move the right term in a p... |
| i2linesi 48878 | Solve for the intersection... |
| i2linesd 48879 | Solve for the intersection... |
| alimp-surprise 48880 | Demonstrate that when usin... |
| alimp-no-surprise 48881 | There is no "surprise" in ... |
| empty-surprise 48882 | Demonstrate that when usin... |
| empty-surprise2 48883 | "Prove" that false is true... |
| eximp-surprise 48884 | Show what implication insi... |
| eximp-surprise2 48885 | Show that "there exists" w... |
| alsconv 48890 | There is an equivalence be... |
| alsi1d 48891 | Deduction rule: Given "al... |
| alsi2d 48892 | Deduction rule: Given "al... |
| alsc1d 48893 | Deduction rule: Given "al... |
| alsc2d 48894 | Deduction rule: Given "al... |
| alscn0d 48895 | Deduction rule: Given "al... |
| alsi-no-surprise 48896 | Demonstrate that there is ... |
| 5m4e1 48897 | Prove that 5 - 4 = 1. (Co... |
| 2p2ne5 48898 | Prove that ` 2 + 2 =/= 5 `... |
| resolution 48899 | Resolution rule. This is ... |
| testable 48900 | In classical logic all wff... |
| aacllem 48901 | Lemma for other theorems a... |
| amgmwlem 48902 | Weighted version of ~ amgm... |
| amgmlemALT 48903 | Alternate proof of ~ amgml... |
| amgmw2d 48904 | Weighted arithmetic-geomet... |
| young2d 48905 | Young's inequality for ` n... |
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